c^^ - f' A^' ,\'' \" ^" .0'^ 'wife' '■^■'^ :m£/k° ^^.-^ -Mil ,.S' .^■% =W •i Q .\^ '^^.■ "'/- " M ^ x'?-' ^\ t: :^^^^ -^^^ /'^o-,V'"^^*^ ^/i/^" ' '-■'' '>#^ ,0 0. ^°^- ^x-v^ -.0 0^ OO ' ""-U.v^-'.'' .N*" ,-^ -t; \V _, . . '^. * . n'^A.^" ..-^ii^.. /? "WZ /T^Si*2<^ ^^^5>r^-iiiWu <^^^C-t^^,.^^ ^^/^. '■f'i'^^^fy^Shf-^y/^^ '^%^/^. THK CONSTRUCTOR A HAND-BOOK OF MACHINE DESIGN BY f/reuleaux Professor at the Royal Technical High School at Berlin, Royal Privy Conncillor, Member of the Royal Technical Deputation, Corresponding Member of the Institute of Lombardy and of the Swedish Technical Society, Foreign Member of the Royal Academy of Sciences of Stockholm, Honorary Member of the Technical Societies of Riga and Erfurt, of the Technical Society of Frankfurt a M., of the Society of Arts of Geneva, of the Flora Society of Cologne, of the American Philosophical Society and of the American Society of Mechanical Engineers. With Portrait and over 1200 Illustrations. AUTHORIZED TRANSLATION COMPLETE AND UNABRIDGED From the Fourth Enlarged German Edition BY HENRY HARRISON SUPLEE, B. Sc. Member of the American Society of Mechanical Engineers Member of the Franklin Institute. NEW YORK \ h:. h:. sxjflee 120 Liberty St. 1899 Copyright, I890, by John M. Davis. Copyright, 1893, by Henry Harrison Suplee. Entered at Stationers Hall. A. Translator's Preface. In presenting to the engineering profession of England and America this translation of Reuleaux's Constructor, a few prefatory remarks may be permitted. Although the first edition of the German work appeared as long ago as i86i,and translations have been made into French, Swedish and Russian, no English translation has hitherto been made, notwithstandine the fact that repeated editions and enlargements of the original German work have appeared. The translation here given, therefore, is the first presentation to English speaking engi- neers of a work which during the past thirty years has acquired the highest reputation over all Europe, and is so well known to German reading engineers and students in this country that no excuse is needed for its present appearance. The freedom with which the author has drawn from English and American sources as well as from Continental practice gives the work a value not found in other treatises upon machine de- sign, while the vast improvement which has been made by the introduction of the kinematic analysis and the resulting classification of the details of the subject, cannot fail to appeal to the instructor as well as to the practising engineer. The translation has been made from the Fourth Enlarged German Edition of 1889, the last which has appeared in the original, and is complete and unabridged in every respect. The introduction to this edition is especially worthy of note, as it contains the author's summary of the principles set forth in his larger work on Theoretical Kinematics,* and the more so as it includes a brief glance at the still wider subject included in his work on Applied Kinematics, as yet unpublished in Germany, and embodying a mass of manuscript which it is trusted will at no. distant day be given to the public. The work of translation has been done with the especial sanction and exclusive authoriza- tion of Prof. Reuleaux, by whom also the portrait and special introduction to the American edition have been furnished. The transformation of the notation of the work from the metric system to the English values has involved much labor and while it is too much to expect entire freedom from errors, not- withstanding the care which has been given to this portion of the work, it is trusted that but few errors will be found. It is especially requested that any corrections which may be found neces- sary will kindly be sent to the translator for future use. HENRY HARRISON SUPLEE. Philadelphia, September, 1893. * It is to be regretted that Prof. Kennedy's translation of this valuable work is now out of print, and it is hoped that a new edition maybe issued. Author's Introduction to the American Edition, The present translation of the Constructor places my book before a large circle of readers who have been practically active and energetic in the development of machine design, for no one of the technical professions has been followed by the English-speaking race with more activity and success than that of the construction of machinery. I therefore take pleasure in prefacing this book with a few words of special introduction. During the series of years in which my Constructor has grown from a small beginning to a large volume, the practice of machine construction has also been continuously developing, so that in every new edition changes and additions have been necessary. Much new matter has been added in this edition to the theoretical portion ; first, in the section on Graphical Statics, enabling many numerical calculations to be dispensed with, using in their places graphical meth- ods ; second, by the introduction of the methods of Kinematics, or the science of controlled movements, a science which reduces the apparently inexhaustible complexity of machine forms to a few simple and fundamental principles, the command of which may be of extraordinary "Value to the engineer. I am still constantly engaged with the subject of Kinematics, especially "with its practical applications, but on account of the pressure of other occupations I have not as yet been able to carry out my intention of treating this portion of the subject in a separate work, •corresponding to my work on Theoretical Kinematics. The work already published on this subject I have therefore characterized as an " outline " of a theory of machines." * The simplification of the conceptions concerning machines to which these kinematical studies led me, was of such importance that I have introduced the kinematical treatment into the Constructor in various places, especially in the latter portion of the book. Even where no special reference has been made to it, the theory has been followed, although the proof has been omitted in order to avoid burdening the non-theoretical reader with details not absolutely neces- sary for the practical application. It is in this manner that kinematical axioms have been intro- . Kinematic Analysis has as a necessary coun- terpart Kinematic Synthesis. This has been already seen (cases 19, 21, 30) in the application of pairs, chains and mechanisms to given machinal purposes. Kinematic synthesis may also be called a theory of the invention of mechanisms. This it can only be, how- ever, in a limited sense. It can in no case enable the genius of the inventor to be dispensed with, but by the aid of this theory his scope can be greatly extended. The application of synthesis to problems which have already been solved may also point the way to the so- lution of others as yet undetermined. In discussing this synthesis, I have grouped the pairs of kinematic elements into 21 orders [538-544] by means of which the determination of the greater num- ber of kinematic chains and dependent mechanisms may be made ; also eight classes of simple chains. The application of synthesis may be made in two forms, the direct and the indirect, and these again into general and special synthesis. Of these the indirect synthesis is the most useful [529]. It is my expecta- tion that this theoretical exposition of the subject, which I cannot expect to extend further, but by means of which I have been able to devise a number of new mechanisms, may find many successful applications by others. b. Applied Kinematics. 39. Applied Kinematics is not so much to be con- sidered as standing in opposition to theoretical kine- matics as it is included in it. In fact, applied kinematics has existed as a study for a long time, as in the treatise of Monge, without the existence of any theoretic foundation. That such a treatment of kinematics may be very useful for a time is readily admissible, but an ex post facto theoretical discussion may seem of little value to the practical man. Indeed my highly es- teemed former preceptor, Redtenbacher, considered an actual theoretical treatment of the movements of ma- chinery to be an impossibility. Under these circumstances I did not feel inclined to follow the ' ' Theoretical Kinematics " hastily with a treatise on the applied science. For this purpose it was not possible to arrange all the various forms of machines under the new classification hurriedly and properly in permanent form. Notwithstanding the simplicity of the preceding system, its application de- veloped many difficulties and required a succession of researches with which even my immediate pupils are not fully acquainted. A not inexcusable impatience on their part has led me to have my investigations in applied kinematics multiplied for a limited circulation although the matter was incomplete. I gave this per- mission reluctantly and with the condition that only a limited number of impressions, to be considered as " manuscript, " should be circulated. In this manner INTRODUCTION. four parts of the work have appeared, the last consist- ing entirely of the application of the symbols to lecture room models. The result of such premature publica- tion cannot always be foreseen by those who have urged it, but for the misunderstandings which have arisen from this source I can only express my regret. In the meantime I have since 1882 been engaged in the partial application of the principles of kinemat- ics to this book in such a manner as to avoid burden- ing the reader with theoretical matter, which would be contrary to the purpose of the work. The most im- portant subjects to which the kinematic method has been applied are here briefly noted. 40. With the great extension of modern mechani- cal engineering we find that the various mechanisms, (the number of which as we have seen is not great), are given a great variety of applications. It is the object of applied kinematics to furnish a clear distinc- tion between the various methods of practical applica- tion. It is apparent that the preceding analysis does not extend to this point, since it does not include the subject of the method of constraint, but only treats of the combination of the elementary parts which are involved. We may therefore properly term it the Elementary Kinematic Analysis. As a counterpart for this in applied kinematics we may place the subject of another analysis which relates to the conditions of motion in a given train, and which may be called Train Analysis, or the Analysis of Trains. This anal- ysis is not intended to solve anev^^ the construction of the various trains, but rather to elucidate clearly their method of action ; a train consisting of a closed group of elements and bearing the same relation to a machine as an atom does to a molecule. 41. Train analysis does not admit of an arrange- ment logically similar to the elementary analysis, but possesses a new and different order. This is due to the fact that the elements of which trains are composed occur only in pairs, while the trains of which machines are composed are considered singly. In Vose's pump, for example. Fig. 979*, there are two ratchet trains combined in one machine, vv^hile in Downton's pump, Fig. 979^ there are three trains. 42. The various methods of tain action may be divided into four principal kinematic divisions, viz. : Guiding, Storing, Driving, and Forming, §333.* The first three divisions are " Place-changing " and the last is "Form-changing." 43. Various forms of guiding devices may be mentioned ; linkages by means of which curved paths are obtained, parallel and straight line motions, also "position motions," as I have termed those by means of which a system of points may be transferred to another position parallel to the first. Guiding devices can be constructed from kinematic chains of every * 5" § 333, tl^s second of these has been translated "Supporting," and the English language lacks a suitable equivalent for '' Haltung" but in a corres- pondence with the author, the above has been adopted Trans. kind. It was by means of examples with chains for this purpose that the general conditions of motion in theoretical kinematics were illustrated, and the same conditions belong also to applied kinematics. 34- Storing includes those especial machine organs by means of which work can be accumulated and the supply drawn upon for later use. This, until now has not been considered as a special mechanical concep- tion, although it has had numerous applications. Stor- age of power may be accomplished in three quite different ways. a. By means of rigid elements, this being statical or dynamical. Examples of statical storage are found in elevated weights, compressed springs, etc., and of dynamical storage in fly wheels, or pendulums. One of the oldest forms of dynamical storage is the old- fashioned spindle [216]. b. By means of tension organs, acting by winding the tension organ upon a drum or pulley. Examples are seen in tower clocks, etc. c. By means of pressure organs. These are the most frequently used, and examples include tanks for water, oil, gas, air, stearti, also hydraulic accumulators. 45- Driving. In this term I include the transmis- sion of motion within a single train and also from one system to another. As "guiding " includes the control of the path of a point, "driving" considers the control of the velocities of various points in their paths. Ex- amples in this branch of applied kinematics are those which take into consideration the velocity of the var- ious parts of a mechanism. (See the close of Case 38). 46. Forming, includes the working of materials by means of machine tools. This fourth division is the richest of all, and offers the widest range to the genius of invention. This operation takes place by the action of the tool upon the material, or as I have called it, the '' work piece " [495]- In form changing machines, the work-piece is a part or the whole of a kinematic link, and is paired or chained with the tool by so ar- ranging the latter that it itself changes the original form of the work-piece into that of the envelope cor- responding to the motion in the pair or linkage em- ployed [495]. We can distinguish between three forms in which this action can occur. a. The tool is hard and operates by cutting the material from the work-piece which lies without the envelope of the desired form. Examples are found in lathes, planers, grinding machines, etc. h. The tool is of high resistance so as to be able to maintain its form, but does not act by cutting, but by pressure upon the yielding work-piece. It follows that the material which lies outside of the desired form is forced into another part of the work-piece without being removed from it. Examples are found in coin- ing presses, rolling mills, wire drawing benches, etc. c. The tool and the work-piece are both alike yielding, and act each upon the other, each being the INTRODUCTION. tool for the other piece. Examples are the various kinds of spinning, weaving, and other textile machin- ery. All three forms are described in this volume, many examples being given among the pressure organs. 47. It may appear from the preceding as if the theory of the action of the tool breaks through the logical arrangement given in the theoretical kinematics, since in Case a, one of the elements, the tool, cuts away and destroys its partner because it is enough harder to cut it. We must here distinguish between yielding and unyielding elements. This looks like a return to empiricism. The defect in the logic, how- is only apparent. All elementary pairs without excep- tion involve the idea that both of the partners evoke the latent forces by the action of deformation ; and at the same time the friction between the moving parts induces wear. Applied mechanics takes friction into account iii considering elementary pairs and investi- gates and provides for the consequent wear. The machine constructor endea-vors by all means within his power to reduce the alteration of form at points where it is not desired, but where it is the end to be accomplished he takes every opportunity to increase it. The forro.-changing action which occurs between the tool and the work-piece differs in degree only and not in kind, from the action taking place between the elements of every other pair in the machine [503]. 48. A similar idea may arise in connection with the method of form-changing given above Under (b), in which an alteration of form takes place without an actual removal of any of the particles. In this case the the correspondence of the kinematic to the mechanical action is evident. In case 8, as already noted, the de- formation which takes place in non-rigid bodies makes it only practicable to obtain approximate solutions. This only involves a quantitative, and not a qualita- tive distinction [502]. Examples of this occur in the construction of in- struments of precision. It is not possible to construct even a simple cylindrical pair (case 19) such as a cen- tre for a theodolite, or for an astronomical telescope, entirely free from error. By the use of a variety of methods the errors are kept as small as possible, and then by other methods, nearly always kinematic, the residtial errors are determined and the proper correc- tions made. 49. In other instances the designer may utilize the elastic yielding of the members of a kinemati: chain, as for instance in the method of Adolph Hirn, by which the springing of the beam of a steam engine is used to produce the indicator diagram of the steam pressure ; or the torsional deflection of a large shaft to measure the power transmitted.* This method is also found in Gidding's device for measuring valve friction (p. 285), and also in the * See Berliner Vtrkandlungen. Emery scale, in which a very small deflection of a diaphragm measures accurately weights of many tons. Although in many instances the deformations of material may be neglected, yet we should never per- mit ourselves to forget that they have been neglected. Otherwise important errors may creep into theoretical deductions, as well as in practical construction. This subject of the yielding of materials is receiving more attention at present than formerly. 50. The "order" of a system of transmission is a subject of importance since there are several meth- ods by which the various parts may be kinematically arranged. I have applied the term "order" to the method of arrangement, and distinguish between three different methods. a. "Series Order." This " order " exists when a number of transmissions are arranged in series, so that each acts upon the following one. If in a single machine, two, three, four or "n" transmissions are thus arranged in series, I call the whole a system of the second, third, fourth or n*'> order. Examples are found in Figs. 766, 767. A transmission can return upon itself. This I have called a " ring " system of transmission. (See p. 208). This return to the original must always occur in the kinematic chain of any mechanism since the elements exist only in the relation of pairing (Case 5). In the system under consideration (Case 41), the groups of elements follow each other in a series, or line as it may be termed, whence I have termed such a series a " line " transmission (p. 257). Ring transmission may also be combined with line transmission, the line being divided into two or more parts. An example of the first kind is seen on page 229, in which the pump mechanism is combined with the steam mechanism, as a line with a ring system. An intermediate form be- tween ring and line transmission is referred to on page 208. b. Combined Order. By this title is meant a com- bination of transmissions in which each transmission is connected to the next, but in which any one can be stopped without stopping the others. An example of this is shown in the ring transmission in Fig. 917. Under certain circumstances a number of the driven pulleys T^, T,, T^ - - Tn, may be allowed to run empty, in which case they become merely supporting sheaves (Case 43) ; as soon, however, as any load is thrown on any of them, the entire system is influenced by the increased stress upon the rope. Another example of "compo^md" order, is the multiple expansion steam engine. Here each engine of the compound, triple, or multiple expansion engine may be considered singly as a separate chain, and the entire machine as a series of transmissions. Each en- gine, Tj^, Tj, Tg, etc., exerts an influence upon the action of the others, but is not indispensable to their action, as would be the case if arranged in "series" order. Compound, Triple, Quadruple expansion en- XIV INTRODUCTION. gines are therefore, respectively of the second, third, and fourth order, but should also be considered to be- long to the class of "Compound order." c. "Parallel Order." This arrangement is the oldest and the one which occurs most frequently. It occurs when a number of different machines are all driven from one and the same transmission, this being the usual arrangement in manufacturing establish- ments. Any of the machines can be stopped or started independently of the others without affecting the motion, a suitable regulator being assumed. This principle may also be applied to the motors by which the transmission is driven, automatic couplings, such as shown on page loi, being used. A "parallel" order occurs in rope transmission when a number of ropes are used on the same pulley ; another instance is that of a train which is pulled by two locomotive engines. The three different "orders" are not always sharply defined, but the distinction' will be found of material assistance in the study of transmissions. An example in which all three, "orders" are used is found in the engine shown in diagram in Fig. 1023. Here the cyl- inder, piston, valve and steam form an escapement : the connection c 1 r being driven, and in turn operating a second r^ 1^^ b, and thence the valve. These three transmissions therefore form a "series" order, this also, returning to itself and being thus a ring system, and of the third order. The fly-wheel and its bearings form a dynamical power storage system, absorbing and giving out power in response to the irregularities of the action of the piston, this being of the ' ' com- pound " order. Frequently such an engine is made with an additional cut-off valve gear, with governor, also of " compound" order, also possibly a feed pump, ("parallel" order) and the engine usually drives an extensive transmission system by which a number of machines are operated (" parallel " order). In § 260 is shown the manner in which physical and chemical trains are arranged in series, the action of heat, of gases and electricity being considered ; the steam engine being the most notable example. 51. The magnitude of the exponent of the order of any train has an important influence upon the hurt- ful resistance of a machine, especially in a series order of a high degree. In such cases the injurious resist- ance increases at least directly as the exponent, and frequently more rapidly. It is therefore important in machine design to keep the degree of the order as low as practicable. In the system of pneumatic clocks of Mayrhofer (p. 171) the mechanism for several years was as high as the 17th order, but the degree subse- quently reduced to the 8th order. It may safely be affirmed that the simplicity of a machine may be measured by the closeness which the exponent of its order approaches unity. Examples are found in the Giffard injector, in which the guiding and driving mechanisms are united in one, and exponent becomes ^i ; the same is true of Siemens Geyser pump, Fig. 971a. The apparatus of Morrison & Ingram, Fig. 1 181, is a device of the 2nd order, which acts by a combination of guiding and driving. 52. The preceding pages have shown that applied kinematics, by means of the separation of the con- trolled motion into the forms of Guiding, Storing, Driving and Forming, and by means of the division of the various "orders, "has enabled the machine prob- lem to be solved as a whole. Theoretical kinematics has assisted in this solution by enabling the various problems to be investigated in a purely scientific man- ner. Without such a theoretical investigation, a sys- tem of applied kinematics would be an impossibility. At the same time practical instruction must be given by actual daily work as well. A clear understanding of the principles of the applied science cannot but be useful to the practical man, and as I believe, welcome also. The fundamental principles of machine construc- tion as I have sought to lay them down in the preced- ing pages, coincide in many points with the practical methods already in use. The practical mechanic is well acquainted with crank trains, gear trains, and the like, or if he is not familiar with them he is readily taught, but in combining these and arranging them so as to act upon each other the theory comes into play and shows clearly the best arrangement for the end in view. This is well shown in the case of the various valve gears, which have been in fact developed inde- pendently, instead of being the result of a theoretical analysis of various combinations of kinematic chains. The application of the kinematic analysis will facilitate work of this sort, making it clearer and simpler the more fully the fundamental principles are understood. For this reason I have introduced the kinematic princi- ples into this work, not to reduce invention to an art to be taught, but rather to bring the principles of science to its assistance. I am ready to admit that the general view of theo- retical kinematics which I have placed before the prac- tical man, may not be accepted without further proof being demanded. It may be considered only as an ingenious form of theorizing, of but little practical value. For the present I must ask my readers to prove by the test of practical application how far the princi- ples of kinematics may be made of genuine practical value. The principles included in cases 40 to 51 are practically applied in the latter half of this volume. The application of the analysis to the subject of ratchet gearing has produced an extensive series of results. Storage is clearly shown to be a form of ratchet gear; the discussion of the degree of "order" of ratchet trains will also, I believe, be found very useful. In the discussion of pressure organs (Chapter XXIII. and following) the subject of storage is highly developed. The notion of the two divisions of guiding, and driv- XIV INTRODUCTION. XV ing will also be found most useful. In like manner the methods of analysis as applied to ratchet trains, are found capable of equally prolific results when applied to pressure organ trains, not, to my knowledge, oth- erwise attainable. The great number of applications in this direction will be seen in § n},, these being the re- sult of the application of the theory sketched under Case 46, above. Since the subject of friction was considered in con- nection with rigid elements, it was also necessary to to take into account this resistance to the motion of fluids (§ 340), as also the loss of heat .in steam pipes (briefly discussed in § 338). In § 362 the very import- ant subject of boiler design is only generally consid- ered. The closing chapter relates to valves. These are treated as ratchets, not oniy from the theoretical standpoint, but also practically, and much more fully than in previous editions. The section on "fluid valves " will, I trust, be found of use to the practical man, as a subject worthy of further investigation. In closing, I may refer to the increasing size of this volume. In spite of my earnest efforts, it has not been possible to reduce its bulk. In many places evidence will be found of attempts at condensation, but nevertheless the work can hardly be called properly a "hand book" any longer. When discussing purely technical matters I can be brief, but in a practical work, it is above all things necessary to be clear and intelli- gible. In this I have endeavored to be guided by the dictum of Boileau : " Lhi ouvrage ne doit point para'lre trop travaille, niais il ne saurait Utre t?-op travailli.'' FuNCHAL, February, i88g. F. REULEAUX. T^BLE OF COIsrTE]SrTS. SECTION I. STRENGTH OK MATERIALS. Introductoi-y i Co-efficients of Resistance i Resistance to Tension and Compres- sion 2 Bodies of Uniform Strength 2 Resistance to Shearing 2 Resistance to Bending 2 Table of Sections 5 Value of the Quantity S 8 Sections of Uniform Resistance.... 8 Bodies of Uniform Resistance to Bending 8 Resistance to Shearing in the Neu- tral Plane 10 Beams with a Common Load 11 Resistance to Torsion 11 Polar Moment of Inertia and Section Modulus II Bodies of Uniform Resistance to Torsion 13 Resistance to Buckling 13 Columns of Uniform Resistance. ... 13 Compound Stresses 13 Resistance of Walls of Vessels 15 Calculation of Springs 18 SECTION II. THE ELEMENTS OF GRAPHOSTATICS. Introductory 22 Multiplication by Lines 22 Division by Lines 23 Multiplication and Division Com- bined 23 Area of Triangles 23 Area of Quadrilateral Figures 23 Area of Polygons 24 Graphical Calculation of Powers. ... 24 Powers of Trigonometrical Func- tions 25 Extraction of Roots 26 Addition and Subtraction of Forces. 26 Isolated forces in One Plane — Cord Polygon 26 Equilibrium of External Forces of Cord Polygon 27 Equilibrium of Internal Forces of Cord Polygon 28 Resultant of Isolated Forces in One Plane 29 Conditions of Equilibrium of Isolat- ed Forces 29 Force Couples 29 Equilibrium between Three Parallel Forces 30 Resultant of Several Parallel Forces 31 Decomposition of Forces 31 Uniformly Distributed Parallel Forces 32 Twisting and Bending Movements. - 33 Determination of Centre of Gravity 33 Resultant of Load on Water Wheel. 34 Force Plans for Framed Structures. 35 Force Plans for Roof Trusses 36 Graphical Determination of Wind Stresses 37 Force Plans for Framed Beams. ... 38 Remarks 38 SECTION in. THE CONSTRUCTION OF MACHINE ELEMENTS. Introductory 39 CHAPTER I. RIVETING. Rivets 39 Strength of Riveted Joints 40 Table and Proportional Scale 40 Riveting disposed in Groups 40 Steam Boiler Riveting 42 Table for Boiler Riveting 42 Table of Weights of Sheet Metal ... 43 Especial Forms of Riveted Joints... 43 CHAPTER II. HOOPING. Hooping by Shrinkage 45 Cold Hooping 45 Examples of Forced Connections. . . 46 Dimensions of Rings for Cold Forc- ing 47 CHAPTER III. KEYING. Keyed Connections 47 Cross Keyed Connections 48 Longitudinal Keys 48 Edge Keys 49 Methods of Keying Screw Propel- lers 49 Unloaded Keys 49 Methods of Securing Keys 50 CHAPTER IV. BOLTS AND SCREWS. Geometrical Construction of Screw Thread 50 Whitworth Screw System 51 Sellers' Screw Thread System 52 Metrical Screw Systems 52 New Systems 53 Nuts, Washers and Bolt Heads 54 Table for Metrical Bolts and Nuts . 55 Weig;ht of Round Iron 55 Special Forms of Bolts 55 Wrenches 56 Nut Locks 56 Special Forms of Screw Threads. . . 58 Screw Connections, Flange Joints. . 59 Unloaded Bolt Connections 60 CHAPTER V. JOURNALS. Various Kinds of Journals 60 A. — LATERAL JOURNALS. Overhung Journals 61 Example of Table of Journals 62 Neck Journals 62 Fork Journals 63 Multiple Journals 63 Half Journal 64 Friction of Journals 64 B. — THRUST BEARINGS. Proportions of Pivots 65 Friction of Flat Pivot Bearings. ... 66 Collar Thread Bearings 66 Multiple Collar Thread Bearings. . . 66 Compound Link as Thrust Bearing. 67 Attachment of Journals 67 CHAPTER VI. BEARINGS. Design and Proportion 68 A. — LATERAL BEARINGS. Pillow Blocks 68 Proportional Scale for Pillow Blocks 68 Various Forms of Journal Boxes ... 69 Narrow Bases— Large Pillow Blocks 69 Adjustable Pillow Blocks 70 Bearings with Three Part Boxes 70 Wall Bearings 71 Yoke Bearings 72 Wall Brackets 72 Hangers 73 Adjustable Hangers 74 Special Forms of Bearings 74 B. — THRUST BEARINGS. Step Bearings 75 Wall Step Bearings 75 Independent Step Bearings 76 Thrust Bearings with Wooden Sur- faces 76 Multiple Collar Bearings 77 Examples of Thrust Bearings 78 CHAPTER VII. SUPPORTS FOR BEARINGS. General Considerations 79 Simple Supports 79 Multiple Supports for Bearings 80 Calculation for Iron Columns 82 Forms for Iron Columns 84 CHAPTER VIII. AXLES. Various Kinds of Axles 85 A. — AXLES WITH CIRCULAR SECTION. Simple Sy metrical Axles 85 Non-Symmetrical Simple Axles 86 Graphical Calculation of Simple Loaded Axles 86 Proof Diagrams 87 Axles Loaded at Two Points 78 Railway Axles, Crane Pillars 88 Axles with Three or More Bearings. 89 Axles with Inclined Loads 90 B. — AXLES WITH COMBINED SECTION. Annular Section 90 Axles with Cruciform Section 90 Modified Ribbed Axle gi Compound Axles for Water Wheels. 91 Construction of Rib Profiles gi Wooden Axles , 92 CHAPTER IX. SHAFTING. Calculations for Cylindrical Shafting 92 Wrought Iron Shafting 93 Line Shafting 93 Determination of the Angle of Tor- sion 93 Journals for Shafting — Round Rolled Shafting 94 Combined Sections — Wooden Shaft- ing 94 Shafting Subjected to Deflection, , . 94 CHAPTER X. COUPLINGS. Various Kinds of Couplings. ....... 95 I. Rigid Couplings 95 II. Flexible Couplings 96 Various Kinds of Flexible Couplings 96 Couplings for Lengthwise and Par- allel Motion - 96 Jointed Couplings 97 III. Clutch Couplings 98 Toothed Clutch Couplings., 98 TABLE OF CONTENTS. xvn Friction Clutches 99 Automatic Couplings loi CHAPTER XI. SIMPLE LEVERS. Journals for Levers loi Cast Iron Rock Arms 102 Rock Arm Shafts 102 Lever Arms for Rectangular Sec- tion 103 Lever Arms for Combined Section . . 103 CHAPTER XII. CRANKS. Various Kinds of Cranks 104 Single Wrought Iron Cranks 104 Graphostatic Calculation of Single Crank 104 Cast Iron Cranks io5 Return Cranks io5 Graphostatic Calculation of Return Crank 105 Simple Crank Axle 106 Multiple Crank Shafts 107 Locomotive Axles 107 Hand Cranks 109 CHAPTER XIII. COMBINED LEVERS. Various Kinds of Combined Levers, no Walking Beams no Scale Beams m CHAPTER XIV. CONNECTING RODS. Various Parts of Connecting Rods. 112 Connections for Overhung Crank Pins 112 Stub Ends for Fork Journals 114 Connections for Neck Journals 114 Round Connecting Rods 116 Rods for Rectangular Section 116 Channeled and Ribbed Connecting Rods 117 Cast and Wrought Iron Rods nS CHAPTER XV. CROSS HEADS. Various Kinds of Cross Heads 118 Free Cross Heads 119 Cross Heads for Link Connections.. 119 Cross Heads for Guides 119 Guides and Guide Bars ■ 121 CHAPTER XVI. FRICTION WHEELS. Classification of Wheels. 122 The Two Applications of Friction Wheels 123 Friction Wheels for Parallel Axes . . 123 Friction Wheels for Inclined Axes- ■ 124 Wedge Friction Wheels 125 Special Applications of Friction Wheels 126 Roller Bearings 126 CHAPTER XVII. TOOTHED GEARING. Classification of Gear Wheels 127 A. The Construction of Spur Teeth General Considerations 12S Pitch Radius, Circumferential Divis- ion 128 Table of Radii of Pitch Circles 128 General Solution of Tooth Outlines 129 The Action of Gear Teeth 129 The Cycloidal Curves 130 The Generation of Cycloidal Curves 130 Tooth Outlines of Circular Arcs. . . . 131 Evolute Teeth for Interchangeable Gears 131 Pin Teeth 132 Disc Wheels with Pin Teeth 133 Mixed Tooth Outlines, Thumb Teeth 133 Tooth Friction in Spur Gearing 134 General Remarks .'i 135 B. Conical Gear Wheels General Considerations 135 Construction Circles for Bevel Gears 135 The Plane Gear Wheel 136 C. Hyperboloidal Gear Wheels Base Figures for Hj'perboloidal Teeth for Hyperboloidal Gears 138 D. Spiral Gears Cylindrical Spiral Gears 13S Approximately Cylindrical Spiral Gears 139 Spiral Gear Teeth and their Friction 140 Spiral Bevel Gears 141 Globoid Spiral Gears 142 E. Calculation of Pitch and Force of Gearing Pitch of Gear Wheels, Tooth Sec- tion 144 Pitch and Face of Hoisting Gears. . 144 Table of Cast Iron Hoisting Gears. 145 Pitch and Face of Gearing for Trans mission 145 Examples and Comments 147 F. Dimensions of Gear Wheels The Rim 147 The Arms of Gear Wheels 149 Table of Gear Wheel Arms 149 Gear Wheel Hubs 150 Weight of Gear Wheels 150 CHAPTER XVIII. RATCHET GEARING. Classification of Ratchet Gearing. . . 150 Toothed Running Ratchet Gears. . • 150 The Thrust upon the Pawl 152 The Sliding Flanks 153 Spring Ratchets, Quadrants 153 Methods of Securing Pawls, Silent Ratchets 153 Special Forms of Ratchet Wheels. . 154 Multiple Ratchets 154 - Step Ratchets 155 Stationary Ratchets 156 Ratchets of Precision 157 General Form of Toothed Ratchets. 158 Dimensions of Parts of Ratchet Gearing 158 Running Friction Ratchets 15S Release of Friction Pawls i5i Stationary Friction Ratchets 161 Releasing Ratchets 162 Checking Ratchets 163 Continuous Running Ratchets 164 Continuous Ratchets with Locking Teeth 165 Locking Ratchets 166 Escapements, Their Varieties 167 Uniform Escapements 167 Periodical Escapements 169 Adjustable Escapements 170 General Remarks upon Ratchet Mechanism 171 CHAPTER XIX. TENSION ORGANS CONSIDERED AS MACHINE ELEMENTS. Various Kinds of Tension Organs.. 172 Methods of Application 172 Technological Applications of Ten- sion Organs 177 Cord Friction 177 Ropes of Organic Fibres 178 Wire Rope 179 Weight of Wire Rope and its Influ- ence 180 Stiffness of Ropes 181 Rope Connections and Buffers 181 Stationary Chains 182 Running Chains , 182 Calculations for Chains 183 Weight of Chain 183 Chain Couplings 1S4 Chain Drums and Sheaves 185 Ratchet Tension Organs 1S5 CHAPTER XX. BELTING. Self-Guiding Belting 186 Guide Pulleys for Belting 1S6 Fast and Loose Pulleys 188 Cone Pulleys iSg Cross Section and Capacity of Belts. 190 Examples of Belt Transmission 191 Belt Connections 191 Proportions of Pulleys 193 Efficiency of Belting 194 CHAPTER XXI. ROPE TRANSMISSION. Various Kinds of Rope Transmis- sion ig4 A. Hemp Rope Transmission. 194 Specific Capacity, Cross Section of Rope 195 Sources of Loss in Hemp Rope Transmission 195 Pressure and Wear on Hemp Rope. 196 B. Cotton Rope Transmission. 196 C. Wire Rope Transmission.... 196 Specific Capacity, Cross Section of Rope ig6 Influence of Pulley Diameter 197 Deflection of Wire Ropes 198 Tightening Driving Ropes 200 Short Span Cable Transmission.... 200 Transmission with Inclined Cable.. 200 Construction of the Rope Curve. . . . 202 Arrangement of Pulleys 202 Construction of Rope Pulleys 202 Construction of Pulley Stations.... 204 Efiiciency of Rope Transmission . . . 205 Reuleaux's System of Rope Trans- mission 206 CHAPTER XXII. CHAIN TRANSMISSION. STRAP BRAKES. Specific Capacity of Driving Chains 211 Efficiency of Chain Transmission. . . 213 Intermediate Stations for Transmis- sion 213 Strap Brakes 214 Internal Strap Brakes 316 CHAPTER XXIII. PRESSURE ORGANS CONSIDERED AS MA- CHINE ELEMENTS. Various Kinds of Pressure Organs. 216 Methods of Using Pressure Organs. 216 Guiding by Pressure Organs 216 Guide Mechanism for Pressure Or- gans 217 Reservoirs for Pressure Organs.... 2i<8 Motors for Pressure Organs 2 it) A. Running Mechanism for Pressure Organs Running Mechanism operated by Weight 219 Running Mechanism Operated by Impact 219 Running Mechanism Operated against Gravity 221 Running Mechanism in which the Motor is Propelled 222 B. Ratchet Mechanism for Pres- sure Organs Fluid Running Ratchet Trains 223 Fluid Trains with Stationary Rat- chets. 225 Escapements for Pressure Organs.. 226 A. Unperiodic Escapements for Pressure Organs Fluid Escapements for Transporta- tion 227 Hydraulic Tools 228 Pressure Escapements for Moving Liquids 228 B. Periodical Pressure Escape- ments .... Pumping Machinery 229 Fluid Transmission at Long Dis- tance 233 Rotative Pressure Engines 233 Valve Gears for Rotative Engines . . 234 C. Adjustable Power Escape- ments Adjustable Pump Gears 236 xvm Adjustable Gears for Rotative Mo- tors 237 D. Escapements for Measure- ment of Volume Running Measuring Devices 239 Escapements for Measurement of Fluids 239 Technological Applications of Pres- sure Organs 240 CHAPTER XXIV. CONDUCTORS FOR PRESSURE ORGANS. Formulae for Cast Iron Pipes 242 Weights of Cast Iron Pipes 242 Pipes for High Pressures 242 Wrought Iron and Steel Pipes 243 Steam Pipes 245 Pipes of Copper and other Metals . . 246 Resistance to Flow in Pipes 246 Methods of Connecting Cast Iron Pipes ^48 Connections for Wrought Iron and Steel Pipes 249 Connections for Pipes of Lead and other Metals 251 Flexible Pipes 252 Pistons 252 Plungers and Stuffing Boxes 253 TABLE OF CONTENTS. pistons with Valves 255 Piston Rods 255 Specific Capacity of Pressure Trans- mission systems 255 Ring System of Distribution, with Pipe Conductors 256 Specific Capacity of Shafting 257 Specific Value of Long Distance Transmissions 259 CHAPTER XXV. RESERVOIRS FOR PRESSURE ORGANS. Various Kinds of Reservoirs 260 Cast Iron Tanks 260 Riveted Tanks 260 Tanks with Concave Bottoms 262 Combination Forms for Tanks 264 High Pressure Reservoirs, or Ac- cumulators 264 Steam Boilers 265 Boiler Details subjected to Internal Pressure 266 Boiler Flues subjected to External Pressure 269 Future Possibilities in Steam Boiler Construction 270 Reservoirs for Air and G-as 272 Other forms of Storage Reservoirs. 273 CHAPTER XXVI. RATCHETS FOR PRESSURE ORGANS, OR VALVES. The two Divisions of Valves 273 A. Lift Valves 274 Hinged or Flap Valves 274 Round Self- Acting Valves 275 Unbalanced Pressure on Lift Valves 277 Closing Pressure of Self-Acting Valves ■ • ■ , . 278 Mechanically Actuated Pump Valves 27S Valves with Spiral Movement 279 Balanced Valves 279 B. Sliding Valves Rotary Valves and Cocks 281 Gate Valves for Open and Closed Conductors 282 Slide Valves 282 Balanced Slide Valves 285 Fluid Valves 287 Stationary Valves 289 Stationary Machine Elements in General 2S9 SECTION IV. Mathematical Tables 291-301 ERI^^T^. Page 14, Case IV., first panel of table should read P=4 it'^ ^ Page 15, line 13 from bottom, second column, omit words "%" thick." Page 53, line 31 from bottom, second column, read "interpolated diameter/' instead of "interpolated meter." Page 61, formula (89) substitute P, instead of p. Page 61, line 11 from top, second column, after "Proportions of Journals," insert the formula number (93). Page 63,, line 39 from top, first column, after "Formula for Fork Journals" insert the formula number (98). Page 64, the formulae on lines 12 and 14, of § 96, should be numbered respectively (99) and (100). Page 64, line 33 from top, second column, for " Prow^ny " read "Prony." Page 89, line 17 from bottom, first column, for 85 mm.^8%" read 85 mm. =3%". Page 89, illustration at the bottom of second column, the diagram to the left should be Fig. 409, and that to the right. Fig. 410. Page 97, line 16 from bottom, second column, for "drawn '' read "driven." Page 103, the last formula on first column should be numbered (154) instead of (155). Page 144, formula at bottom of first column, the cube root sign applies to the vifhole of the second member and not to the numerator only, as printed. Page 175, line 17 from bottom, second column, for Harturcn, read Hartvi'ich. Page 195, line 29 from top, second column, for " can only be given by indeterminate results," read " can only give approximate results." Page 206, title of § 301 read Reuleaux's instead of Reuleux's. Page 255, example in second column, for 4 in. stroke, read 40 in. stroke. Page 263. The following revisions of formulas (385) and (386) have been commumcated bv Prof. Reuleaux and should be inserted : 2 ( R ■Tg^Y— ■C385) •(386^ THE CONSTRUCTOR: A HAND-BOOK OK MACHINE DESIGN. BY K. REULEAUX. Section L— STRENGTH OF MATERL\IvS. Introductory. The study of the strength of materials ultimately depends upon the question of the resistance which rigid bodies oppose to the operation of forces, and the following definitions are to be noted : SuPERFiciAi^ Pressure is the pressure upon a unit of surface. Tensii,E Strength is the resistance per unit of surface, ■which the molecular fibres oppose to separation. MoDUi,us OF RESIST.4.NCE is the strain which corresponds to the limit of elasticity, compression and extension, each hav- ing a corresponding modulus. Modulus of Rupture is the strain at which the molecular fibres cease to hold together. Modulus of Elasticity is the measure of the elastic exten- sion of a material, and is the force by which a prismatic body would be extended to its own length, supposing such extension were possible. Theoretical Resistance is the force which, when applied to any body, either as tension, compression, torsion or flexure, will produce in those fibres which are strained to the greatest extent a tension equal to the modulus of resistance ; or, in other words, it is the load which strains a body to its limit of elasticity. The Practical Resistance often improperly termed merely Resistance, is a definite but arbitrary working strain to which a body may be subjected within the limits of elasticity. The Coefficient of Safety is the ratio between the theo- retical resistance and the actual load, or, what amounts to the same thing, the ratio between the elastic limit and the actual tension of the fibres. The Breaking Lo.\d is that load which causes in those fibres ■which are subjected to the greatest strain, a tension equal to the modulus of rupture ; in every case this is equal to the force necessary to tear, crush, shear, twist, break, or otherwise de- form a bod}'. The Factor of Safety is the ratio between the breaking load and the actual load. As a general rule, for machine construction, the Coefiicient of Safety may be taken as double that which is used for con- struction subjected to statical forces. Circumstances may also require it to be taken as either greater or less than the custom- ary value, sometimes even narrower than is permitted for stati- cal forces. Care must be taken never to permit a material to be strained in use to its theoretical resistance ; although, indeed, there are some materials, such as wrought iron, which have been strained slightly beyond the limit of elasticity, without re- ducing the breaking load, or causing any apparent injury. (See The determination of the breaking load, and consequently the use of the modulus of rupture, is limited to those cases in which the actual breaking of the structure must be considered ; but for the actual calcvdations of working machinery the modu- lus of resistance, or limit of elasticity is of primary importance. ?2. Coefficients of Resistance.* The coefficients given in the following table are selected as the mean of man}' experiments upon the various materials named. Under the title " Wood ' ' is given an average value from ex- periments made with oak, beech, fir and ash. Those materials which show the greatest difi'erence between the modulus of rupture and the limit of elasticity also possess in the highest degree the property of toughness. * Throughout the original work all dimensions and quantities aie given in the inetric system, but these have been transformed into English units for English readers, except in the following table, where both are given. — Trans. Experiments upon wrought iron show that a strain beyond the limit of elasticity, if not carried too far, although it will cause a permanent deformation, will not lower the modulus of elasticity, but will raise the modulus of resistance. For example, a rod of wrought iron, subjected to a tensile strain of 28,400 lbs. per square inch, was subsequently found to have its limit of elasticity raised from 21,300 lbs. to 28,400 lbs. (This property is utilized in drawing wire). Tenacity is a particularly desirable property for a material of construction, and it may generally be approximately meas- ured by the ratios K : T and Ki : Tj. If the rod above referred to be subject to compression it will return to its former limit of elasticity. Table of Coefficients.* M-iTEEIAL. Wrought Iron Iron Wire Sheet Iron Cast Iron Spring Steel (hardened) Cast Steel (not hardened) Cast Steel (spring temper) Copper (hammered) . Copper Wire Brass Brass Wire Bell Metal (bronze) . . Phosphor Bronze . . . Aich Metal Lead Wood Hemp Rope (new) . Hemp Rope (old) . . Belting Granite Limestone Quartz Sandstone Brick Limestone Masonry Sandstone Masonry Brickwork Modulus of Re- Modu us of sistance. Rupt ure. Modulus of plasticity. Com- Ten- sion. Com- E. Tension. T. pres- pres- K. sion. Tj. Kj. 20,000 15 IS /; 22 28.400,000 21,300 21,300. 56,800 3",24a 20,000 30 70 28,400,000 42,600 99,400 17,000 32 24,140,000 45,440 10,000 7.5 15 11 63 14,200,000 10,650 21,300 15,620 88,460 20,000 50 to 70 80 28,400,000 70 to 90,000 113,600 20,000 25 So 28,400,000 35,500 113,600 30,000 65 to 150 100 42,600,000 go to 200,000 142,000 11,000 15,620,000 2.5 3,550 30 42,600 70 99,400> 13,000 12 40 18,460,000 17,040 46,800 6,500 4.S 12 no 9,230,000 6,Si6 17,040 i56,2oo> 10,000 13 50 14,200,000 18,460 71,000 3,200 9 13 4,544,000 12.7S0 18,460 15 21,300 36 51 120 15 21,300 75 106,500 500 I 11 5 710,000 1,420 1,846 7,100 11,000 2 1.8 9 5 1,562,000 2,840 2,556 12,780 7,100 250 (?) 5(?) 12 355,000 7,100 17,040 5o{?) M?) 5 71,000 1,420 7,100 15 to 20 1.6 2.9 20 to 30,000 2,272 4,iiS 8 11,360 5 7,100 17,040 7 9,940 0.6 85» 7,100 1-5 =,130 0.4 56S * The upper figures are kilogrammes per square millimetre, and the lower figures are pounds per square inch. THE CONSTRUCTOR. ?3. Resistance to Tension and Compression. A body is said to be under tension when the action of a force P, tends to extend it in the direction of its length. When the force acts in the opposite direction the body is said to be under compression ; but when the length is great in proportion to the cross section, a combined action occurs. (See ? i6.) I^et (7 be the cross section of the member : S, the strain due to the action of the force P ; then if we neglect the weight of the material we have : P=Sq (I) *EJ^aniple. A rafter exerts a horizontal thrust of 22,000 pounds, which must be borne by a rod of circular cross section, li we make 5" = 7100 pounds we have for the diameter of the rod d. Sq~- from which d ^= i.c 7100 — d- = 2 4 The principal action which the application of a force to a member produces is the consequent elongation or compression. A prismatical body subjected to the action of a force P, will have its original length / increased by in amount A, determined by the formula (2) and this holds good as long as 5 is not greater than the modu- lus of Resistance for tension T. This relation is also true for compression, in which case the limit depends upon the modu- lus of resistance T^ for compression. Example. Suppose the rod. whose diameter was determined in the pre- ceding example, to have a length of 114 ft. 10 in. or 137S inches, its elonga- tion under those conditions would be ,^_ i378 X 7'°° ^ 28,400,000 The preceding formula (2) is a fundamental one, and upon it is constructed the whole systematical study of the strength of materials. Formula (i) is of use when a section is strained beyond the limit of elasticity, as by it we may determine the force' required to rend or crush a material, using the proper Modulus of Rup- ture. Example. The force necessary to pull the above given rod asunder is P=Kq IT or /'=56,Soo X (2)' — = 178,442 lbs. I A. Bodies of Uniform Strength. By bodies of uniform strength are meant those in which the shape is so made that the cross sections at various points are subjected to the same strain S, and consequently a proportion- ally economical distribution of material secured. Such forms are not often employed in practice, although ap- proximate shapes may often be adopted, but they serve in many cases to determine the general style of a structure, and give it the effect of proportional strength without adhering too closely to the exact form. These forms will be found of value to the designer for both reasons : principally as a guide to the style of his work rather than for close determinations of economy. If a designer has become thoroughly familiar with the resist- ing capacity of various shapes, and can keep them so clearly in his mind that he can perceive the general form of the proper curve to be used in any particular case, he will be able to pro- duce, with an artistic freedom, designs which will approach the shapes indicated by mathematical analysis. The following forms are alike suitable for tension and com- pression. As examples of their practical use, the first two are applicable to cast columns, and .the third is suitable for chim- neys of masonry as well as for high piers of bridges and via- ducts. *In all cases the quantities given in the original examples have been converted into their English equivalents, which will account for the un- usual quantities chosen. {Trans.). SHAPE. EQUATION. REMARKS 4=V ;; p 5 cf p — X .s e = 2.718 = Base of natural logarithms. P log ^ = log — -1- o-434^-^ P, is distributed un- iformly throughout the whole length of the figure. Cross sec- tion circular. Profile parabolic. Approxi- mate form, a trun- cated cone with end diameter = — 2 P, is uniformly de- creased from above downward. Cross section circular. Form conical. The body is strain- ed by its own weight, y being the weight of a unit of volume. The cross section in- creases with the in- creasing load in the logarithmic propor- tion given. is- resistance to shearing. A body is said to be subjected to a shearing strain in any- cross section when the distorting force acts in the plane of that cross section. Let q, be the sectional area, and S, the force acting upon it, so that we have as in the case of tensile and compressive strains ^=S^ (3) The limit of elasticity will be reached when 5=5 0/ the lesser of the two Moduli of Resistance of the material, in the case of wrought iron, where T = Ti = 21,300, S = 17,040 lbs. while for cast iron T *- C 4- 2 \/2 I* = 0.638 .^ i/iS—{i — d{)Ai3 ^ (/i3 — /tl3) + ^^l (/;i3 — /jqS) */i3 + ii/ji! ili> — (i — i^)!l-fi + iih^^ i m + (/ii — b) h^ +(h — hi) 36 36(^ + ^1) /l3 l[^3(a'3-/3)+^,(/3 + a"3)] 0.9243 0.677 ^ 2.828^2 3 Ifi — (6 — ^0 /;i3 6/; bh—(h — bi)h-i b {Jfi — h-f) + ^1 {h-f — /z„3) &I1 b(h-h{) + bi{h-,-h^ b B + ^1 h{' oh ih+iihi b tfi —ib — i.,) h-f + bj hj' th bh — (i — i„)hi+iih. b /;3 + {hi — b) }i{! + (/; — /i{) Ifl 6h , >^ a^ ^ — 3 a = — ] Z": bIP bifl i + ibi h i + h i 2 (5 + ^1 h 2". b'- + iiil + ^1= b'- 12 + 4 3 ^1 + h' 12 (2 3 + bi) , _ b hr + ^1 hi (h + /, 1 a° 1 1 ' %i< («— — b-- — Ji \l\~ -..> 1 -* A.J "T- XIX. XX. XXI. XXII. XXIII. »..-Jf — It l.-*-.J ^.....-1) fc XXIV. Moment of Inertia y. 7 XAJ^i"'"'-^''^ + '' (""-■^'' +h{/'+ <^"^)\ Determined graphically or by experiment. - ^■1= 0,0491 (/* -r- (^-1— — then M = (?i 71 1)1 a a T Ti J When -< — then 3T = Ci 71 in ai a T TJ Tx J When-- = then M= °^ a^ 71 m a m a\ T I Example. For Cast Iron — = — . Tx 2 Taking the parabolic section No. XXXV. CE=g h, ai \ h. This gives -^ — ~ \, so that — — ^ ""t^i ^^^ ^^'^ -^f ^"^ ha^'e ■10,650 10,650 and M = 0.114 ^« ■ "1 71' With wrought iron, in which T^ T^ no investigation is necessar;^^ Sections of Uniform Resistance. In order to use the material to the greatest advantage to re- sist bending strains, it is necessary to pay especial attention to its distribution, particularly in those portions which are furthest from the neutral axis. The best economy is attained in this matter when the section is shaped so that the strains on both tension and compression sides shall reach the elastic limit simultaneously. For this purpose it is necessary to make «, T ^^ Sections which are thus proportioned are known as Sections of Uniform Resistance. Wrought iron sections which are sym- metrical about two axes fulfi' these conditions, since 7^= T^. For cast iron, when the bending strain is exerted constantly in one direction, it is best to make ffj = 2 a, for 7] ^ 2 71 Taking these conditions into consideration, the following sections (Figs, i, 2, 3) have been drawn, in which b and b^ may tave any desired proportion to each other : FIG. I. FIG. 2. For these sections, when b^ = b, we have : / = 27Si* 440/;* 992i5* Z = 34. 8i' 55 i' 02.45' F ^= igb'' 25(5- 40.86^ = I 0.97 .04 The tension side is nearest to the neutral axis. section modulus is determined from the value of / Since the , .S is always T to be taken as — L. F is the area of the section, and ^ is the in proportional economy of material, the cross section of Fig. i being taken as unity. The value of 9) may be determined thus : I? = /3 (8) in which the sub-numbered letters belong to the required sec- tion and the un-numbered letters to the given section whose economy is to be taken as unity. In this equation F^ pb'', Z = ah^ and .? is taken equal to S^ except when the ratio of a to a-x is not the same for both sections. It will be seen from an examination of (8) that a slight variation from the exact proportions is not very material. When the bending force acts alternately in opposite directions, so that the strains are re- versed, the sections which are symmetrical about two axes are the best for cast iron as wcli as for other materials, and the smaller value for 61 should always be taken under such circum- stances. If the force is constantly changing its direction, so that the neutral axi; passes through the centre of gravity, the most economical section is that of a circular ring, its resistance being greater than the cruciform or star-shaped sections, such as X., XII. and XX 7., Table ? 7, since there is in the former case a constant prop jrtion of the section and the greatest dis- tance from the plane of the bending. Example. A projecting beam of cast iron loaded as in No. I., g 6, carries a weight P-- 5,500 pounds at its extremit.v, the length being 78.75 inches. Taking the cross section of the shape Fig 2, we have by equation (4): .fl/= 5,500 X 78.75, Z=isb'^ 5,500 X 78.75 = 10.650 X 55*' ^= V 5,500X78.75 ,^ 0.3V • 10,650 X 55 The sectional area will then be 25 (0.9)' = 20.25 sq. in., as de- termined by the constant given for the section Fig. 2. If the security be taken at ij, 21,300 .5'= = 14,200. This gives a lighter beam, and according to equation (8) its weight would be (— — )' = 0.825 of the preceding. ? 10. Bodies of Uniform Resist.ance to Bending. A body is said to offer uniform resistance to bending when its shape is so chosen that in all sections of its length the maxi- mum strain, 5, for tension or compression has the same value, and the general form of equation (4) for such bodies is 1}U J = Constant. (9) Bodies shaped so as to oppose a uniform resistance to bending are frequently used in machine construction, approximations to the exact forms being often adopted, examples having already been shown in § 4. A variety of such shapes are given in the following table. The deflection in bodies of uniform resistance is of necessity greater than in prismatic bodies of the same strength. In many of the examples of the following table the deflection, /, is given, and in I. it is double, and in V. iJ- times what it would be in prismatic bodies similarly loaded. The elastic line for the following bodies, when exactly formed, is determined from the following equation : d^y __ Mo ao (10) dx'^ EJo ax in which Bio := the moment of the bending force for any given sec- tion, Jo = its moment of inertia, ao = its greatest fibre distance, ax = the greatest fibre distance on the same side as ao for any other section at a point x . For the radius of curvature, p, of the elastic curve at a point whose co-ordinates are .r, y, we have : P= ^J^a^ (") 71/0 ao which value is constant, and represents a circular arc when ax = ao ; that is, when the section is of uniform height at all points, as in V., X., XIV. THE CONSTRUCTOR. Ko. n. HI. IV. V. VI. VII VIII. iX. XI. Fori y^— -X — -a „..}. — Application of Load. Equation. For rectangular section - f- _ £ b Jfi " I ' In Cases I. and II,, z~6. Parabolic truncated wedge. Approximation to Form I,, Truncated wedge. - -1- •; Sustaining Power. S t Ifi 61 _ S /, !fl 6 I 61 Approximation to Form II., Truncated wedge. Normal wedge. Cubic parabolic truncated pyra- mid. Approximation to Form VI., Truncated Pyramid. Truncated cone ; approximating to the form given by equa- tion y /^ y -I For rectangular section, z y~ X- , y X z=-0; rL ^ — /: / Wedge. Parabolic-sided wedge. ' y _if^ Truncated pyramid on semi- cubic parabola. _ S i !fi 6 1 Volume. 5 i Ifi S i Ifi 6 I Sir d~' 3= ^T' S b Ifi 3l S blfi S b h"- 3/ i- bhl 4 — bid J- bhl 5 19 bhl 27 i9 ^IcT- loS Lbhl I. bhl 3 J. bhl 7 Remarks. Deflection of the free end ; Tile elastic curve is a parabola. Weaivest section at the bas6. In normal conditions the elastic curve bisects the angle of thfi wedge. Elastic curve a circular arc. Jo : JoE' b 7:3 Equatio: y 3/ ^ h yi I is of great- est importance when all the sec- tions are similar. Weakest section at the base. To obtain the same strength as in Forms I. to VII., make s r i6 b y r- /T A very useful form when the sharp end is removed. Elastic curve a circular arc. f=}- ^^^ aJoE'' Jo = Applicable to stone brackets, etc., in architecture. lO 777^' CONSTRUCTOR. No. XII. XIII. XIV. XV. Application of Load. Equation. Approximation to Form XI. For rectangular section -'■Hi Wedge, on semi-cubic parabola. 1„ 3/"" ' f 1 Sides on cubic parabola. z b y ~ h y h X ~1 Py ramid. Sustaining Power. 5 /' m S li h- S i /i' Si A2 Volume, 1^ bill 27 i-ihl 5 Lthl i bkl 3 Remarks. Weakest section at the base. Fundamental" shape for archi- tecture. Elastic curvea circular arc. h = bh^ Value depends upon the sim- plicity of the form. The preceding are only a few of the simpler forms which may be used, and it would be easy to multiply examples. By altering the breadth, or height, the relations become more or less complicated, as the case may be. For instance, in Case I., which is based on the parabola z rx~ ^' 4/'r r-= «/ i, it may be made the biquadratic parabola, —^1/ ' ' etc. Combination sections give rise to new forms, and a great number of combinations may be made. Bxamples will be found in the chapter on axles and shafts. The following discussion of springs will also give some in- stances of special forms, in which the neutral surface is irregular. Resistance to Shearing in the Neutral Plane. Since in a deflected beam there is on the tension side a con- tinual tension, and on the compression side a continual com- pression of the respective fibres, it follows that the neutral plane is subjected to a shearing action, and this must not be neglected in determining the width of the beam. The lower limit permissible is indeed a matter not likely to be reached, but at the same time it should be investigated. Calling the least permissible width Zo, and the mean force on either side of a given section R, then in order that the shearing strain at the neutral plane shall not exceed a value So, we must have : >^ U ■ (14) ''So 2/ in which So should in no case exceed f of the lesser modulus of resistance of the material under consideration (see ? 5). J, as before, is the moment of inertia of the section, i.e., the summation of the products of the elements of the section by the square of their distances from the neutral plane, while 1/ is the statical moment of the section, i.e., the summation of the products of the elements of the section by their distances from the neutral plane. For the rectangular section No. I., Table (? 7), '^0 =-=- U=' 4 ■ and for the double T section. No. VIII., l_j_ b h■'■ — {b — b^) /n^ 4 R is to be chosen according to the case under consideration, as, for example, in No. II. Table (§ 6) for all sections between B p and C, it is equal to the reaction — , etc. 2 Equation (14) is not so much used to determine a value for Zo, as to find out in any case whether the breadth of the neu- tral plane has been taken too small. As a matter of fact, this is a question which very seldom arises in ordinary construc- tions, especially in machine construction. 4 If in (14) we give zo any desired value, and make So ^ — s we obtain 4 Zo 2/ and substituting this in equation (4) we get : M^5 U (15) R % zoa M ■ — -^ is the lever arm of the force R ; this we may call ^. U : Zo a contains one of the height dimensions of the section ; hence equation (15) expresses a relation between two dimen- sions of the body under consideration. For a simple rectangu- lar cross section, taking the value of U, given above, in which h , h 16 A greater value for h must not be taken if we do not wish the shearing strain to exceed the extension or compression in the tension and compression sides of the beam. These considera- tions are often of importance for the danger section, as, for ex- ample, in No. II., Table {\ 6) for the point B. In this case / / = 8 A = — and we make — ^— . This limit of height, however, 2 . A 5 is so great that it is very rarely reached in practice. The most important application of this principle is found in the case of notched beams of wood, such as often occur in building construction. In such cases the resistance of the neu- tral plane is often ven,' much reduced by the cutting of the notches, sometimes to one-half what it would be in the solid beam, and making a corresponding reduction in the value of 7- For the double T section we have : h 16 Zo = b, and a = . [f-a-oay] If the brackets in fraction the value of - the denominator contain an improper — will approach the upper limit, but lor THE CONSTRUCTOR. II all ordinary cases this value is very great. The nearest ap- proach to this shearing action probably occurs iu T beams where the flange joins the web, but examples are very rare. BEAMS WITH A Common Load. When two prismatic beams are united iu the middle, and at that point subjected to a force P, the beams being supported at the ends, they will both be deflected, and the sum of their re- actions P' , and P" , enter into the support of P. The double reactions are found from the formula in Table (I 6), No. II., column 2, as follows : P' _ J' E' l"^ p77 — J" E" 17^ and since S"J'' P' = A J"E' S' J ' a' I' and P' I" we get S' '" ~ E" a"\l' ) ^ ' If the two beams are of the same material (E' = E'^), to ob- tain equal security, the product-^ ( \ If the beams are not the same length then a' = a", i.e., the heights must be the same unless the breadths are equal to each «ther. "*•--.. PIG 4. Kxample. A cast 5ron support shaped like a cross. Fig. 4, must support a weight, P, at the intersection. The lengths of the arms are to each other as 3 : 2. In order to obtain equal security in the four arms, which are of prismatic shape, we have from (16) Hence the cross section of the short arms must be to that of the long arms as 4 ; 9, and if the arms were of the same section the supporting power of the short arms would be to that of the loug arms as 9 ; 4. It also follows from the preceding, that rectangular sheet metal plates carrying a uniformly distributed load are stronger parallel to their shorter axis than parallel to the longer axis. For given loads and materials formula (16) may be used to govern the choice of dimensions and the relations of length to breadth. For beams of cast or wrought iron resting upon each other, a suitable proportion may be secured by taking the sum of their several supporting powers as the supporting power of the combination. This is often a matter for consideration in strengthening existing structures. Resistance to Torsion. Resisting Power and Angle of Rotation. A prismatic body which is subjected to the action of a force couple tending to rotate it about its geometric axis, opposes to such action its Resistance to Torsion. Under these conditions the elements in a normal section are subjected to a shearing strain, and until the elastic limit is reached there exists an equilibrium between the external rotating forces on the one hand and the strain moments of the various elements of the section on the other hand ; both being taken with regard to the polar axis of the centre of gravity of the section, i. e., the axis passing through the centre of gravity of the section and at right angles to it. Resistance to torsion may properly be con- sidered a higher species of resistance to shearing, to which it bears the same relation that resistance to bending holds to ten- sile and compressive strength. Let: ■ M = the statical moment for any given section of the rotating force, Jp = the polar moment of inertia of the section, /. e., its moment of inertia taken with regard to its polar axis (see \ 14), a ^ the distance of the farthest elements of the section from the centre of gravity, .S =: the shearing strain in the elements at a distance a, then a Jp (17) If the body is of a uniform section, then =^ is constant. Now (iS) if A be the lever arm of the rotating force P, for a moment M, the weakest or danger section will be that for which Af is a maximum, and for it we have /'=_£. J± Am a in which Am, is that value of A, which gives JIf, a maximum. The limit of elasticity is reached, as in the case of shearing action, when .S ^ — of the lesser of the moduli of resistance 5 . for tension or compression (see § 5). This is plainly visible by a comparison between the action of bending and twisting. The relative rotation which two sections of a prism at a given distance apart make with each other is called the angle of tor- sion. It is represented by the letter 1? ; and for two sections separated by a distance ^, we have in general terms : J^ = 7^ (19) a.v Jp G in which G is the modulus of torsion for the material used, and is equal to — of the modulus of elasticity E. In the following table will be found the values for : The moment M, at a given point x, of the prism, The force P, according to formula (18), and The torsional deflection in terms of angular measure, or in other words, the angle of torsion 1?. These quantities are given for a variety of cases, as shown in the cuts, and from them total moment, PR, of the twisting force may be determined. In case IV., .Sis the point of appli- cation at which the collected forces, with a lever arm R, would act, if concentrated to produce an equivalent result to the sum of the separate efforts, lo being the distance of the point S from the immovable end of the prism. Questions relating to torsion are of varying importance in machine construction, and come especially into consideration in calculations relating to springs. Case IV. illustrates the condi- tions which occur in determination of mill shafting. Cases V. and VI. occur in machine framing. ?I4. P0I.AR Moment of Inertia and Section Modueus. The polar moment of inertia, Jp, is easily determined, since we have //=/i + /= (20) in which _/, and J„ are the equatorial moments of inertia taken with regard to two axes at right angles to each other, and whose values are given for a variety of sections in the table of (§ 7). From this may be obtained the polar section modulus — =^ Zp for use in the preceding cases. An exception must be made for those sections in which we have not _/, ^ J„, as in cases III., VII., XII., XX., XXV., etc., I 7. For these it will be necessary to make a special correction in the values of Jp and Jp ■■ Zp, to provide for the warped surface which is assumed by the section under a heavy torsional strain. For a rectangle, which is a section of frequent occurrence in machine design, the corrected value of Jp and .2}* = — is given in the following table, while for the circle and the square no corrections are necessary for the values obtained from equa- tion 20. Example. A cylindrical prism of wrought iron is subjected to a torsional strain applied as in case I. of the following table. The force P ^^ 1,000 lbs., and the lever arm R = 24" ; while the bar is 4" in diameter and 4S" long. These quantities give for S, the strain at the circumference Jp 16 16 PP 1,000 X 24 1,909 lbs. 3.1416 64 and to get the angle of torsion we substitute this value in the formula : i.qcq 48 = ' r ^ ■ ^^ = 0.004 11,360,000 2 whicli corresponds to an angle of about o° 14'. 12 THE CONSTRUCTOR. No. II. III. IV. V. VI. Application, •I it*' 4 *^ ' ^ T p •..;y L: ffffff^tf^': i3 Moment j1/. M = P R for all points between A, andB, M=PR :L^ r- P I! = the collected moment of all the twisting forces. Jtf= the sum of the moments within jr. In the portion c: M=P R^l I In the portion C\ : M=P R L M=P ^(7-t) Twisting Force P, P = SJp _SJp SJp aR p^SJp aR When C\<^c p^SJp I aR c P— aR Angle of Torsion i9. iS = PRl Jp G _s_i ~ G a 2 Jp G 2 G a i Jv G 3 Ga ■ JpG _SJo ' G a PR ccj "jp G I G I tJ I PRl i JpG I 5- / ^ iG a Remarks. All sections between A^ and B, are equally strong. Weakest section at B. Twisting forces decrease uni- formly from 5, to ,<4. Weak- est section at B, General form of Cases I., II. and III. Weakest section at B. The value of ^ in III. will be reached in IV., when /o = — • 3 The shorter portion cx is the weaker. Weakest points at A\ and B. If we wish to reduce d, so that ^ shall be equal to one-half the modulus of resistance for torsion, i. e., = — ■ -— - • 21,300 = 8,520 lbs., we make P/$ ■■ d^ ^^ ttS or about aj^ inches. In this case the angle of torsion would be 11,360,000 1.2; which gives an angle of about i° 39". -,^ 16 X 1,000 X 24 = 2,^2'/ 3.1416X 8,520 = 0.02S8" SECTION TABLE. No. Section. Polar Moment of Inertia yij. Polar Section Modulus, , Jp Zp=.^ I. #} 32 16 No. II. III. ■—i-v Polar Moment of lnertia_/^j. 6 Polar Section Modulus, 7 _ Jp 33 3 n/ : 3\/ i°' + hi Approximately 3 (0.4^5 + 0.96A) THE CONSTRUCTOR. 13 I 15. Bodies of Uniform Resistance to Torsion. In order to make a body of uniform resistance to torsion it is necessary to take such sectional areas at various points as shall make in equation (17), So. constant, and also to take — ? =: constant. (21) Jp In case I. of the table in \ 13, for all sections M ■= PR, and hence in this case the body should be prismatic in shape. For cases II. and III. the necessary formula are given in the follow- ing table. For such bodies the angle of torsion is greater than for those of prismatic shape. The angle for each is given in the table, and is derived from the following : i± = ^L (22) in which y^ is the polar moment of inertia for the section taken at the point x. Form. tion. Equation and Angle of Torsion. Circular section y :^: 16 a SI G a Appro.\imate form = a trun- cated cone, with extremity = — d. Circular section ,? = 6. S— d^; 16 / Approximate form = .a trun- cated cone, with extremity d For other bodies of uniform resistance to torsion, see Torsion Springs (§ 20). ?i6. Resistance to Bucki7^i From these equations a limiting curve may readily be found, whose abscissas are those of a cycloid, and whose ordinates are those of a sinoide, and which may be called a cycloidal sinoide. A method of drawing this curve is given here- after, in the discussion of connect- ing rods, and the approximate shape is also shown in the second form of Fig. 5, in which the outline is a circular curve, or at least a line of very slight curvature. The strength of these columns ma}' be taken as 34 that of a cylindrical column of a diameter h and length /. I 18. Compound Stresses. fig. 5. It very often occurs that a variety of forces act upon a body at the same time and in a variety of ways, so that, for instance, a sec- tion is subjected at the same time to tension and bending, or to torsion and bending, etc. The resistance and the maximum strains are then to be de- termined in a different manner, according to circumstances. In the following table ip. 15) are given the principal formulsi for some of the more commonly occurring cases. Let: ^ = the greatest strain at the weakest section ; Z = the section modulus at the weakest section, which latter is indicated at B in the figures ; F ^ the area of the section ; J ^ its moment of inertia (2 7) ; Mb = a bending moment ; Md = a twisting moment ; Jlf; = an ideal moment, so that (Mb)t = an ideal bending moment, and {Md)i = an ideal twisting moment. An examination of these formulae will show that in many cases the combination of strains is a matter of importance. H THE CONSTRUCTOR. BUCKLING STRAINS. Application. Theoretical Support- ing Power. Remarks. The column is to be considered under compression, when : No For circular section -^- is less than For rectangular section —r- ib = lesser side). Material. II' 5 sV* Cast iron. I. A * p ^- JE 4 r- Freely loaded post. The lower end fixed. Weak- est section at the point of attachment. 6 8 Wrought iron. Wood. ^., j,^ ,„,, i; / lo iiK Cast iron II. \ ;J -1 /2 Post free at both ends, but kept in the line of the axis. Weakest sec- tion in the middle. 24 28 '3J< Wrought iron. Wood. .A' i 1 14 16 Cast iron. III. i| /3 Post fixed below, and held in the line of the axis above. 33 33 Wrought iron. l6 19 Wood. f '' ,.,, 1/-:^ 1^ \ 20 23 Cast iron. IV. t: ^ /2 Post fixed at both ends in the line of the axis. Weak points at the ends and in the mid- dle. 48 56 Wrought iron. i ' 23 27 Wood. t„ .- For example, in case I., if j*? ^ — ,''■£■, if the load is hung at 2 tlie edge of the section, P - Sbh , and hence is only one-fourth as great as it would be if applied centrally. If th^ section is circular (d), we have P = S~d^ 4 J? making Jf ■- P=—— d\ 5 4 and the sustaining power is still less than with a rectangular section. Case III. is derived from I. and II., and may be changed into either by making a, ox R ^ o. The so-called ideal moments are especially useful in these calculations. It will be noticed that in the case of elliptical and rectangular sections, li is taken in the plane of bending. These dimensions being known in advance, since the choice of profile is frequently permitted, it is possible by the use of the ideal moments to consider the question of combined strains, eince the quantity in the parenthesis to the right is the expres- sion for the lever arm of the force P for each case. This can generally be readilj' determined graphically, and so determined just like any case of ordinary bending. For example, in case II., if o^ 45°, we have cos a = sin a = 0.707 for the value of /i, and the section at B is to be calculated as if acted upon by a force P, with a lever arm o. 707 / (the pro- jection of / on the plane of attachment) + 0.707 — 6 In case I., making R = o, for a circular section {3Tii)i = P ~- o and substituting 6" — d^, we get P^= S — rf^, as we should since the stress is now purely tensile. In this case — - is the lever arm 8 ■which, if acting with a bending force P, would produce a strain of the same amount as that in the line of the axis. This is only rigidly exact when the shearing action which occurs in bending is neglected. Many useful applications of cases IV. and V. are found in discussing axles and shafts. III. IV. THE CONSTRUCTOR. COMPOUND STRAINS. IS Application. 't:l^\ B/ Sustaining Power. I +yf ■ for rectangular section, {bJt) Sbh P= ■ 1+6- for rectangular sectionj {bh) Sbh for rectangular section, (Bh) Sbh I R cos a -\- d —r-{sin a H — —cos a) PI is a bending moment Mb, PR is a twisting moment Md. SZ Mfflill \/ Ml- + il/o- -\- "2 Ml Mn COS a in which Mi is the bendiJig moment of Pu Mo that of P2. Ideal Moments. Ideal bending moment for stress 5: [Mi, \i = P [R -\ j. For rectangular section {Ih) : For circular section (d) d Mb >=-(-+4) For elliptical section {bh) : (-0'=-(^+4) Ideal bending moment for stress S: ( Mb For circular section (d) : (Mb ]i =P I i sin a + ~^cosa^ For elliptical section {l>h) : (Mby^P 1 1 sin a + -^cosa'j For rectangular section {bh) : I Mb )i —P (l'"' « + -^cos.'j Ideal Ijending moment: [Mb ]i = P (R cos a + I sin a + - For circular section {d) : yMb \i = P (r cos a. + i S2n a -i — cos a 1 For elliptical section (S/i) : {Alb ji = P [R cos a. + I sin a H — cos a. 1 For reotangular section {bh) ; I Mb \i = pIr cos a + Ideal bending moment for the stress 6" : (^Mb y --^Mb + Ideal twisting moment ; V V 'V + ^'i -Mb+ . 7lV + M^' rdeal bending moment for the stress S: (Mb ) i = I /"^l/r + -^^2- + 2 ^1-4 ^l^a ^-^-s- a In cases IV. and V. it is supposed that the section is arranged in four symmetrical portions about two lines of gravity, perpendicular to each other. i 19- Resistance of Walls of Vessels. Boilers, Cylinders, Etc. The fallowing table will serve to determine the resistance of the walls of cj'lindrical vessels subjected to pressure for the cases which usually occur in practice. The theory of resistance under these conditions is not fully settled, especially in the case of comparatively thin shells sub- jected to external pressure, for which the corresponding formulae do not give satisfactory results. In the following cases, let : p = the unbalance"d pressure upon the walls of the vessel ; .S = the maximum stress for the material used ; Ji = the modulus of elasticity of the material : r = the radius of the vessel ; (5 = the thickness of the walls. Although only approximate, the formulse for cases I. and II. hold good up to the limit of rupture. Examples: i. Given a wrought irou cyliuder, 40 inches diameter, %" thick, with a stress upon the material of 11,500 lbs. Under conditions of case I., the internal pressure permissible would be P—11. 2. A spherical vessel of the diameter and thickness griven above, according to case II., ■would have a safe resistance P= 23,000 — — ^ = 431 lbs. 20 3. A plate held as in case IV., 40" dia., ^" thick, and a pressure of 212 lbs., with a maximum stress 5" ^ 11,500, would have a thickness Y 3 y 11,500 ' 20 X 0.S16 X 0.136 ^= 2.22" or about 23<( inches. The deflection 7^ which a circular plate gives under a force p, may be determined, according to Grashof, by the formula for case III. : and for case IV. : 7_ (J A. E (24) (25) = 11,500 X 0.0185 = 212 lbs. Example : The plate of Example 3 preceding, with a value of £ = 28,400,- 000, would have a deflection of \2.25/ Y 2" i6 THE CONSTRUCTOR. RESISTANCE TO PRESSURE. No. > Application. Pressure p. /=s(/".+^-. p = S{ ■(4-r ^=^(-^) 25 {■ --r^f For vessels wiiose walls are required to be made very thick, as in the case of the cylinders and pumps of hydraulic presses or for cannon, etc., the preceding formula do not apply. Under these conditions the relative radial distances of the various por- tions of the thickness of the metal vary greatly, and their relation has an important influence upon the resistance. It is the relation which exists between the various stresses at different points which governs the various formula for the thickness of the walls, which are given below. Brix calcul^es the stresses at different points on the radius upon the supposi- tion that the internal diameter is not altered by the pressure ; Barlow admits such an alteration by pressure that the area of the annular section of metal is not reduced ; Lam6 makes neither of these assumptions, but calculates very closely the changes in the various stresses which are caused by the internal pressure at each point, and in this way has obtained the most reliable data as to the real behavior of the particles of the material, accord- ing to the modern theory. The results of the three theories are given in the following table : Quantities. / = P = ^lognat eS — i 25-- r 2 S Barlow. 5 'V-' S-p iS p ^ 2 S — p Lame.' {r + ^Y + >-■' i: S+p S-p' (r 4- 6f — 1^ II HS+P) 2S — P For the stress 5' in an annular ring lying between the radii r' and r, Lame gives [-(^•)']-^[-(,0'] If r' is the external radius of the vessel, so that r' = r -^ S, we have : or if we put ( i + - S' S /m' + I p //' — I 2 /i'^ 2 ji^ Example : If 5 = ^, that is, /x = 2, S' = —5— ^ ^— p, and as in the px^ ceding formulee, taking/ = iT, we have .y^- 8 40 5 This shows that the material is not used in an economical manner in ves- sels witli excessively thick walls. All three theories admit that the inner portion of the wall is strained the most, and hence it is for the inner wall that 5 should be chosen. The formula; of LamiS, as well as those of Barlow, show that beyond certain limits an increase in the thickness is not attended with any increase of strength. With, a given resisting power S, this limit will be reached when p^Si the theoretical resistance will be attained when p = the modu- lus of resistance of the material. At this point the internal pressure begins to stretch the inner fibres of the walls, and any increase in strain will cause rupture. The theoretical limit in. this case is reached when p ^ T, which is For Cast Iron = 10,650 lbs. " Wrought Iron =: 21,300 " " Cast Steel = 36,000 " Lack of homogeneity in the material may cause the danger pressure to be reached far within these limits, the material breaking without previously stretching. Since stresses exceeding 36,000 pounds are reached in guns of large calibre, it is evident that ordinary bronze is unsuitable for such conditions, and even homogeneous steel is often unequal to the pressure. The erosion of the chamber in the case of ordinary bronze cannon also acts to weaken the inner ring of material, and must be considered as a chemical deteriorating action. Various methods have been devised for strengthening guns by giving the various layers different tensions. Of these methods the principal is that of hooping. The principal result of this construction is to produce a compression in the inner layer. The pressure of the gases of explosion must then first overcome this compression and restore the normal condition before it can produce any extension of the fibres, and as a result a much higher degree of resistance is secured than when the metal is left in its normal condition. The calculations of the resistance of hooped guns offer many diificulties. If we have not only the inner pressure, but alsc» the outer pressure, p' , to consider, we may take the following formula, after Lame : ,1-^-^, = , ^+.fi .. (.7> (^+4r= S—P+2P' Putting I -| =: /i, as before, and solving with regard to p. we have : p = S II- + 2/'; /Z^ -f I ^ /i'^ -f I in which 6' will become less with regard to/, the greater))' be- comes. In the case of hooped guns /' is not constant and iuvari- ble, but depends upon the effect which the internal pres- sure /> has through the walls upon the hoops. Referring to Fig. 6, let it first be considered that under normal conditions the inner ring is under no strain, that is, J* = o, and also S\ = S"^ Now when the inner pres- sure j* becomes sensible while the external pressure/" = o, or at least may be neglected, then the layer at r' will be- come extended, and the ten- sions will be 5/ = S/. The stress S/ in the inner side of the hoop reacts with a pres- THE CONSTRUCTOR. 17 sure/', aud substituting this in Lame's formula, making i ^ — -^ = //', will give Making S' = S/ = S/ and substituting this value of/' in (28), gives m'+ I H"+ Ifi'+l According to (26), S' is dependent upou p aud 5, and by sub- stituting and transforming, we get /=5 '+1 . sii±±mM=^ ,30, (■ + t)"(- + ^)' + - In this case the stress S upon the inner ring is always greater than p, but the ratio approaches much nearer to unity than before, as the following table shows : IWhen We have And also r 1^ Z' /*' P S 5 P p 5' 5 I 2 1 0.600 1.667 0.667 0.400 I 0-5 2 1-5 o.Soo 1.250 0.406 0.325 2 I 3 2 0.905 1.05.7 0.143 0-135 It will be seen that the mere hooping of a gun with a ring of the same material as the inner tube adds very materially to its strength. If, however, the ring is forced on in any manner so as to produce an initial strain p' upon the tube, a still greater advantage will be the result. If we insert the value oi p' from (29) into formula (28), we have / = 5 + 2 5, , /<" — I (31) In this formula S.,' is partially a function of p, and also de- pends partly upon the extent to which the tube reacts. This latter condition exerts a most important influence upon the strength, as we shall see hereafter. If we assume that the hoop is under such an initial strain that, for the maximum value of />, the value of S^' = S (which is doubtless the most desirable condition), we shall then obtain from (3 1) If (5 = r this gives -j, and the radius of the hole in the ring r^, while afterwards they both have the same radius r. Under these conditions the shaft B will be subjected to a uni- formly distributed compression 6",, while the inner surface of the ring will be under a similar tension S.,- Taking the correspond- ing moduli of elasticity E^, and £^, we have from formula (2) ; Adding these together, we get : _ 5, 5, £. "- E, It is most important for the designer to know the best values for 7\ and r,. If we call T^ = — =-, we have ^ _ il :^ I :^ _ E^ E^ ^1 (36) i8 THE CONSTRUCTOR. 5i and 5, are dependent upon each other, and their relation is expressed by Lamii's formula : S^ = S., P-- = 5, + 7J-^ i + J+i which may be abbreviated bj' putting 5, = 5, p This gives : ^■. El n E.^ E% El (37) The difference between the value of the denomiuat6r and unity is so slight that in practice it may be neglected, and for a practical and useful formula we have 5, ■Si , V' = IF + c- El p E., E., El (38) In this formula we have for the following : = 0.5 0.6 0.7 o.S i.o 1.5 2.0 3.0 T p = 0.385 O.43S O.4S6 O.52S 0.600 0.724 0.800 0.882 We also have from equation (38): V' ^1 „„., c _ V- E^_ (39) Si = I + E, and S„ = 1 + E^ P El This value of i' is generall}' so small that great care is neces- sary, in turning and boring, to secure the correct sizes for r^ and ?'2- Example : With a wrought iron shaft and a cast iron hub we have ^l = 28,400,000 ; £0 =- 14,200,000. If 6 ^ 2r, then p = 0.8 by the table above ; and we may also assume that the stress, S-_>. in the interior of the ring due to the forcing should not exceed 7200 Ibs.^ This gives from equation (38) '/' = 7200 + 7200 X o 8 14,200,000 28,400,000 1408 .00071, or, in other words, the increased diameter of the shaft over that of the hole must be 0.00071 times its own diameter. If we make i/f = -, we shall have a stress in the ring of 1 + — X 14,200,000 14,200,000 X 0.8 : 16,950, 20,400,000 or nearly 17,000 pounds, which would be too great for the ring to stand. ?20. The Cai:'- P = R 3 (o.4(5 + 0.96/1) _ ^S f J^L_ 3 R ^Ifl. + hr- Approximately when h>b. s m m p = R 3 (0.4^ + 0.96A) Deflection. /=R^ = ii : /=i?^=I2- /=R9- f-^ 32 PF"-l Gdi f= Approximately 16 PR'-l /=- Gdi Elasticity. R S E JL R _S_ E R S I E ~d R S I G d R G Jh f s I G d R ^ G ' ly/ b"- ■\- ffi R S I G d Approximately __3_ PR"~l h"- + h'2- ~ ~2 'G~ ' i^ B R Kemarks. / = the devel- oped length of the spiral. All three forms of uniform re- sistance. The va.ue -r- R is the a- gle of rotation -^ , pro- duced by die load P. Cases VII. to X. are bodies of uniform resist- ance to torsion. Springs of the form of VII. and VIII. may also be combined into compound forms. In cases IX. to XII. / is al- ways the devel- oped length of the spring. It is immate- rial whether the breadth of the plate is parallel, normal, or ob- lique to the axis. Here, as in case XII., also the spring is measured to the apex of the cone. The weakest point is at B. By making a gradual reduc- tion in the value of /:, from B to the end, this may be made a form of uniform resistance. 20 THE CONSTRUCTOR. The quotient —^, shows that a small modulus of elasticity, ■when combined with a high modulus of resistance, indicates the best material for the construction of springs. According to the table in \ 2, we have : Hardened and tempered steel -t^ Ordinary steel (not hardened) Brass Wood 42,600,000 (go, 000)^ 28,400,000 (35.500)' 9,230,000 (6,816.'^ 1,562,000 (2,840)^ .00052 .00223 =: .01986 = .01936 This shows that hardened and tempered steel is theoretically the best material of springs. It is also worthy of note that in all the examples given, the deflection is proportioned to the load. It follows from this fact that the time of vibration which any of these loaded springs possesses, is of the so-called "sim- ple" character, of the same nature as that of a pendulum. Neglecting the weight of the spring itself, we have for the vibra- tion of a loaded spring the same rate as that of a simple mathematical pendulum of a length equal to the deflection of the spring/, which is in which ^ is the acceleration of gravity = 32.2 ft. (41) Examples on the theory of springs : i. Given a simple triangular spring, as in case II., for a load P = no Ib's., and a deflectiony"= 0.7S". Taking the material as cast steel, with E ^ 42,600,000, and making S, the greatest permissible stress, =^ 56,800 lbs., and also taking the length I ^= 15.75", we then have t E ' h from which Substituting this we get ; The 0.78 "5-75 56,! j^ ^ 56,800 X 15-75 X 15-75 0.78 X 42,600,000 I the formula" Eilfi or & = t !,6oo,c E/lfi 15-75 h olume 7 6 X 110 X (15-75)" 42,600,000 X 0-78 X Co-424)' bill 1. 018 X Q-424 X 15-75 = i.oiS" Example 2 : If we keep the same conditions, but make the length 11. 8", we shall have /i = 56,800 X 11-8 X ii-B 0.78 X 42,600,000 6 X no X (ii.8)» 42,600,000 X 0-78 X (0.238)3 ■ 0.238" 3-39 cu. in., thuscon- rp, , . ^u . ,r l^Ill 2.42 X 0.238 X II S The volume m this case ^= V ^ = ^ ^ ^ '^ ^ 2 2 firniing the remarks on formula (40) by showing that the volume depends upon the load and the deflection, and is independent of the proportional dimensions E.\ample 3 : Let us now suppose the same conditions to be applied ti a helical spring such as No. IX , also made of cast steel Since this is a torsion spring, in order to obtain the same security we must make 5— -i- of its preceding value, or — — - 56,800 = 45,440; and the wire may be taken as 0-24 in diameter. We then have from the fable P-^ — —, or 110 = from which we get 45,440 16 (0.24)3 R. ^ 45,440 X 3 1416 X (0-24)3 „, , 16 X no ^ "'' The length / is obtained from column (6) f SI /Gd = 31-3" in which G ^ ~ E - 5 /^ °-7^ X 17,040,000 X 0.24 2-X 1-121 X 45,440 This would make the number of coils 31.3 ^ 2XTX1.121 '■■'" or about 4K coils. If more colls are preferred, the diameter, d, of the wire mu^t he reduced and the calculation repeated. ' ' °' ■ 17,040,000. ^ ^itR~ The volume l\ This gives the ratio 4 l\ _ 1-416 "7 ^ 0-410 or 3-4 V 31-3 X 0.7854 X (0.24)2 = 1. 416 cu. in. as given in the table. Example 4: Torsion springs have recently been applied to railway c 1k™±°I" '":.l?'5:;':h'=iLl'.'^? design of an American, Mr. DnX; / cars in the shaped spring is belit'at the ends into two°eibmvsr.^'srwhi'ch "are ^Urhed bl'^K^lV to a block which rests on the axle bo.x. A saddle A transmits the n^H„f, I to the spring, while the other end is supported at C by a hoTk °^ *" "^"^ In prder to determine the stress S, in one branch A C, of such a spring, let us takfr the diameter of = 1.14", and the lever arm, R, which is the horizontal projection of A B, as 4". The load on the spring is one-fourth the load on the car, 22,000 lbs. + one-fourth the weight of the car itself, 18,000, and one-half of this is borne by each- branch of the spring, making the load at the end of the lever R in this case to be- ;,3oo lbs. In the preceding table, under case VII., column 4, 16 PR 16 5000 X 4 ~ "'',750 IT d^ IT (1.14)^ If this spring is made of Sheffield steel which has a modulus )f elasticity E =^ 24,140,000, then the modulus of torsion G '= ~ E ^ 9,656,000. If the length /= 33.5", the deflection, according to column 6 in the table, will be /= ^-^■^'^ ^ 2 X4X 68.750 X 33-5 _ J 6 ,/ Gd 9,656,000 X 1-^4 The above described spring weighs 24.2 pounds, which is about -^l-^ of its gross- load, or about n^r of its net load. Fig. 9, A double armed plate spring of the form No. III., to have the same supporting. pc vver would weigh about a hundred pounds, or jj ^ its gross load and '-^ its net load. As long ago as 1857 1 called attention to the superior economy of torsion springs- over plate springs for railway use. The principal reason for the tardiness of railway- men in appreciating this fact may have been partly due to the difficulty of securing- a proper temper in the round steel, although this seems to have been entirely over- come in the case of the Dudley Spring. In the little pamphlet on "'The Construction and Calculations of Springs," which I published at that date, the comparative weight of the torsion springs VII. and IX., and the triangular plate springs 11. and III., is given as /j^, instead of -{v: as in the^ preceding table, but the latter is snown to be more nearly correct in practice. In Figure 10 is shown the manner in which a helical spring may be applied to the bearing of a goods wagon in the place of a plate spring. The box is guided in the frame B, B, and the spring D is interposed between the sill A of the wagon and the journal box C The form of the spring is a single helix with the lower end flattened for about y^, of a turn in order to give a fair bearing in the cap E of the box. The upper end sc^e^\'s for about ij^ turnsinto the cap /'.where it is clamped by thescrew G after the load has been equalized, and in this way any desired adjustment may he- secured. An example will probably be the best method of showing the manner of calculat- ing such a spring. An ordinary German four wheeled goods wagon weighs about 11,000 pounds, and carries about 22.000 pounds load. This gives about 8,250 pounds to be supported by each spring. We will assume a deflection of 1'%", with a permissible fibre stress S of 6S,ooo lbs., and take G = 9,656,000, as before. Since it is desirable to use such diameters of spring steel as correspond to commer- cial sizes, it is better to select a diameter ^for the steel, and deduce a corresponding radius R for the helix, according to the formula for case IX., col. 4, page 64, IT d^ 10 P 777^ CONSTRUCTOR. 21 We will take the successive cases in which the diameter d of the steel is i", ij'b"* tYs" and i^V'- Now for these respective values we must select such a number of coils, «, that ■with a load P= 8,250 lbs., we shall get a compression^"^ i-75"- We have for n the fact that 2ttRh = the length / of the uncoiled spring. Substi- ituting this in the formula fory", case IX., we get ^~ ~ ~G 'd" som which /G d Now the least possible distance between the cap F and the socket E is nd -}- r, -and we must also provide for a space u between the coils, say 0.3". This gives the -distance ^ from centre to centre of coils of the unloadea spring nd +y + Iter n The tot^.l height of the spring, however, will be greater than ns by 1.5^ + d, since 3J4 coils enter into the cap F, and one-half the diamater of the steel in the last coil Tnust be added both top and bottom. Adding to this the thickness 5 of the cap and the socket, as J4". we obtain the entire height occupied by the spring and its fittings. ■Of course this height is limited by the space at our disposal between A and C, and in this case it may be taken at 14 inches, and in any case it will generally be depen- -dent upon other circumstances. Substituting these values in the several formulae, as ^iven above, and tabulating the results, we have: d = i" ifs" iKs" iiV R = 1. 610 1.94 2.30 2.71 or say = m 2.00 21% 'H A = 7-5 5.6 4.18 3- 19 K = ir, erect from C a perpendicular, and prolong O D to F, and O /^will be the desired product, x = a b c. Or we may make, as in Fig. 25, after having found O C = a b, draw E D = c (Case IV.), and prolong O D until it intersects at jpa perpendicular from C , when C F= x, i 23. Di'visioN BY Lines. Division may readily be accomplished by reversing the methods employed for multiplication. To divide a line a by a line b, we must find a third line x, which must contain the unit of a and b, -^ times. From the previous examples we may de- rive the following methods of division. I. Fig. 26. Make O E = unity, erect at E a perpendicular or inclined line, intersect it with the divisor O B = b, prolong A c/ X 1 I ^ :! .h Fig. 26. Fig. 27. Fig. 28. O B, and make O A = the dividend a. Draw from A a paral- lel to B E, and its intersection with O E prolonged will give the quotient x. For we have O C : O E = O A : O B, that is, , a X : 1 = a : b, or ;ir = -r-' II. Fig. 27. Make O E = unity, also lay off on O £■ the dis- tance O B ^ the divisor 5, erect at i? a perpendicular, and in- tersect it from O with O A ^ the dividend a. A perpendicular, erected from E, will then intersect O A at C, and (9 C^ .i', for we have again O C : O E = O A : O B ox x : i ^ a : b. III. Fig. 28. Make O ^= the divisor b ; on O B lay oS O E = I ; at j9 erect a perpendicular A B = the dividend a ; join O A. Erect at ii a perpendicular, and it will intersect O A at C. Then E C = x,{ox E C : O E = A B : O B, ox x : i = a : b. §24. Multiplication and Division Combined. When it is desired to multiply a number a into a fraction — , the operation really consists in mnltipljdng a by b, and dividing the product a x b hy c, in order to obtain the result x. If we recollect that for x = , we may write x : a = b : s, we will see at once how the combined operation maj' be performed by making the distance O E equal to the denominator r, instead of unity, as heretofore. We will then be multiplying the line a by the ratio -^.instead of . The following illustrations will make the operation clear. , I. In order to multiply a quantity a by a fraction — , we make, in Fig. 29, O A ^= a, lay off on O A, O E ^ c, erect at ii a perpendicular, and intersect it at B, with a distance from O c Fig. 29. Fig. 30. * See Culmann's " Graphical Statics.' equal to b\ then prolong O B until it intersects at Ca line drawn from A, parallel to E B. Then O C will equal x, for we have O C : O B ^ O A : O E, ox x : b ^ a : c, ox x ^ — . a b '' II. If we wish to find the product , we make, Fig. 30, OA == a, and make the distance O E ^ twice the unit of measure- ment, draw E B = b perpendicular to O E ; draw a line from A parallel to E B, and prolong O B until it intersects this last line at C. Then A C will be the desired product x, for A C : O A = B E : O E, ox X : a = b : 2, ox X = '!^. ' 2 These methods, which may be extended much in the same manner as the various methods of multiplication given in 'i 22, will be found of great service in the graphical calculations of areas, as we shall see. ?25- Area op Triangles. Since the area of a triangle is equal to the half-product of its base and altitude, it is readily calculated by the method given in the preceding section. I. Fig. 31. Selecting the side O B ^ b cf the given triangle O ^ i? as a base, which gives the perpendicular A A' = tU2 24 THE CONSTRUCTOR. height //, although this line need not be drawn, we mark off the distance O E ^ 2, units (inches, decimeters, etc.), and draw III. Fig. 37. The diagonal A C=b divides the figure ABCO into two triangles, the sum of whose heignts = O O', which is Fig. 31. Fig. s2. from B a line B C, parallel to an imaginary line A E. This line B (Twill intersect the side O A prolonged at C, and a per- pendicular dropped from (T to OB, will give C C ^ = the desired area/ (see VII., ? 22, and II., \. 24). II.- Fig. 32. From the end of the base line OB draw the perpendicular O E ^2 units, draw the altitude A A' ; also draw from A the line A C parallel to E B. This will cut off on the tase line the distance A' C, which is the product / = — . (? 22, VIII., and I 24, II.) _ _ ^ III. Fig. 33. Prolong the base line iJC and the side BA until the vertical distance between them O E ^ 2 units. Join Fig. 33. E to C, and draw from A a. line parallel to E C, intersecting the base at B,andB£> = — =/. (? 22, IX., and § 24, II.) IV. Fig. 34. From the vertex O, with the dividers open a distance equal to 2 units, intersect the base at E, and make the anti-projection of the base A B by drawing B Cparallel to O E, and A C normal to B C. Then A C= the product of the base b, and one-half the altitude O O' ^ It, and hence is the desired area/of the triangle. (§ 22, X., and § 24, II.) If the unit is taken as one inch, the value of the area y"%vill be given in square inches, or if a decimeter is taken as the unit, the area will be in square decimeters, etc. If we findy= y, the area of the triangle is seven-eighths of a square inch ; or if it measures 72 millimeters, the area would be 0.72 square decimeters, or 0.70 x 10,000 = 7200 sq. mm. §26. Area of Quadrhaterai, Figures. In determining the area of a quadrilateral figure, it is either obtained directly, as in the case of a parallelogram ; or it may Fig. 35. Fig. 36. be separated into triangles, which are measured separately ; or the figure may be reduced to its equivalent triangle. I. Required the area of the parallelogram ABCO, Fig. 35. Taking the side O A &s a. base line, lay oS O E = unity, and erect the perpendicular E E' = h. Prolong O E until it inter- sects a perpendicular from A at D, and the distance A D will be the area of /= A A. (g 22, IV.) II. The quadrilateral figure ABCO, Fig. 36, may readily be replaced by a triangle of equal area by drawing the line O A' parallel to the diagonal O B, for since the triangle O A' B is equal in area to O B C, we have the area of the triangle O A' A is equal to the area of the figure ABCO. Now, according to IV., I 25, we make O E = 2, and draw A D, the anti-projectioa of A A' and A D = f, the desired area, Fig. 38. the anti-projection which O B makes on A C. The multiplica- tion of O O' \>y — may be made according to XI., \ 22, and II., ? 24. Draw O' B E parallel to A C, making O E = 1, also draw A D parallel to E O, and C D normal to A D, then C D =/= the area of A B C O. IV. Fig. 38. The figure ABCO may be converted into a triangle whose altitude = 2, when the base will be equal to the h b product — . From O describe an arc with a radius O E ^= 2, and draw a tangent passing through an angle of the figure at B, opposite the angle O. From the other two angles, A and C, draw lines parallel to the diagonal O B, intersecting the tangent at A' and C A' C will then be the base of a triangle whose altitude = 2, and whose area is the same as the figure ABCO, and the area_/'= A' C . Many similar methods may be deduced from the preceding examples. ?27. Area of Polygons. The area of a polygon is measured by reducing it to its equivalent triangle. This may be done in the following manner : From the angle O of the polygon O A B C D E, Fig. 39, draw a diagonal O B to the next angle but one, and then from the Fig. 39. intermediate angle A draw A B' parallel to O B, prolonging the third side B C to B' . If we join G B' , we have the triangle O B B' = O B A, and hence the figure O B' C D E will have the same area as the original figure, but will have one less side. Then join O C, and draw B' C parallel to O C, and so we may proceed until we have obtained a triangle O C D' of equivalent area to the original figure, and whose area maj' be determined by any of the preceding methods. Regular polygons, such as the hexagon. Fig. 40, only require half the operation to be performed, and then the area measured as a parallelogram. ?2S. Graphical Calculation of Powers. A line a, raised to the «"' power, reall}' means the determina- tion of a line x whose length shall contain the unit of measure- ment rt« times. The following methods are applicable when a is a positive or negative whole number, and the process is really a repeated application of the multiplication of a by a. As in the previous cases, this operation may be performed in various ways. I. (See \ 22, I.) In Fig. 41 make O E ^ unity, erect at E a perpendicular, and intersect it at A^ with the distance OA^^a, the original factor. Carrying this distance O A-^ down to B^, and erecting a perpendicular at Bi, we get O A^ = a' (see I., ? 22). This again carried down to B,, and a perpendicular erected at B,, gives O A^^ a^, and so O A^^ a*, O A^ = a^, etc. If we lay off O Bm, equal to any power of a, say a"', and erect perpendicular at Bm, the intersection with O A^ prolonged will give the value of «'" -F '. Again, if we drop a perpendicular from the end point Am + i of any power of a to the axis O E, it will cut off a distance Bm, which will be the next lesser power of a (see I., I 23). THE CONSTRUCTOR. 25 The perpendicular A-^ E, from A^ upon O E, gives the first jower a'. If we now make O Aa = O E, and drop the perpen- Ani« Fig. 41. •diculary^o j? — i, we have OB — i = a — ■, which = — , which is the reciprocal of O A-^ ; in the same manner we get O B —z^^ a' ' a' II. By combining the methods of multiplication I. and III. ■of § 22, the following method for powers is derived. In Fig. 42, jnake O E = \, O Ay ^ a, E Ay perpendicular to O E, and draw irom A^ a perpendicular to O Ay, cutting O E at A, ; then O A^ = a'. From A.^ ^ perpendicular to the base will give A, and O A^ = a^ \ another perpendicular to O A^ gives A^ and O A^^ a*, and this may be continued indefinitely for positive powers of a. By working backward from E, we get OA — -^ as the recipro- cal of a, O A—-^ = — J-, and so on for negative powers of a. Both the preceding methods assume that a is greater than i ; "the following may be used when a is less than i : III. In ]?ig. 43 make O E ^ i, and draw O A = a at such an angle that A E is perpendicular to O A. Erect the perpendicu- K Fig. 43- lar E I, and continue with the alternate perpendiculars i 2, 2 3, 3 4, etc., and we have : O i ^ — , 2^ — =-. O x = — r-, etc. Working to the left from E in a similar manner, we get O — 2 ^ a^, O — 3 ^ a^, O — 4 ^ «*, etc., the positive powers "being to the left, and the negative powers to the right. The zigzag lines which are thus drawn back and forth between the two axes have a relation to the powers of a which may be utilized in the following manner : IV. Make, in Fig. i\d„OE=j, OA=a, and the angle O A E^, 26 THE CONSTRUCTOR. .-^ Fig. 47- Fig. 48- O — I ^ cot i, but in the following case a < i. II. Fig. 50. Make O E = s, O A ^ a, describe a semicircle on O E, erect a perpendicular at A, and join O C, then vfill o c=.ir = v/;r. III. Fig. 51. Make O E ^ 1, and mark off on C iT prolonged E A ■= a, draw on O ^ a semicircle, and erect a perpendicular at E, intersecting the circumference at C; then will E C^ X = v rt. The extraction of the fourth root may be performed by re- peating the method for square root. The graphical extraction of the cube root, fifth root, etc., is not so simple. Culmann uses for this purpose the logarithmic spiral, and Schlesinger con- structs a curve according to the method in § 28, but the advan- tages are not sufScient to warrant a further examination of the subject at this point. I 31. Addition and Subtraction op Forces. In all the preceding operations we have only considered the lines to represent absolute quantities, and paid little or no atten- tion to their direction or position in the plane of the diagram. The principal advantages of the graphical method are those which are connected with problems relating to the equilibrium of forces, and it is the application of the preceding methods of graphical arithmetic to the calculation of forces which really constitutes the method of graphostatics. When several forces are acting upon the same point, their resultant may be obtained by the addition of the lines repre- senting the forces when projected upon the co-ordinate axes. This addition of the projection of forces is known as graphical addition. This addition is performed by placing the lines repre- senting the forces end to end, forming a polygon, care being taken to avoid repeating any of the lines. If the forces, I) 2, 3, 4, 5, 6, Fig. 52, acting at O are in equilibrium, the sum of their projections will equal zero, and the polj-gon formed by the lines, as shown in Fig. 56, will close. The figure thus con- structed is called a force polygon. It is immaterial as to the order in which the lines are taken, as in Fig. 53 the result is the same whether taken in the order, i, 2, 3, 4, 5, 6, or i, 3, 4, 6, 5, 2, although the shape of the polygon will be different. As in arithmetic, the graphical subtraction of forces is the reverse of addition, and practically amounts to a separation of the sides of the force polygon into their respective forces. In graphostatics the forces are all taken in one plane, by projecting upon the plane of the diagram those forces which may be without it. The Fig. 52 preceding method of addition and subtraction of lines, which here represent forces, but which may be taken to represent any- thing, is called geometrical addition and subtraction. They bear the same relation to geometrical multiplication and divi- sion as the corresponding arithmographical methods do to each other. These little used methods, which are of the greatest in- terest to geometers, we cannot discuss here. ? 32- Resultant of Several Forces. In the preceding section we assumed that the given forces held eacli other in equilibrium, from which it followed that the diagram formed by the lines representing the forces returned to the starting-point and formed a closed polygon. If, however, the force polygon for a group of forces, such, for example, as the forces i to 5, Fig. 54, does not close, it follows that equi- librium does not exist at the point O. In order to obtain equi- librium it is necessary to apply a force 6 to the same point, whose direction and extent correspond to the line 5 6 of the polygon. This is the force necessary to bring the other forces into a state of equilibrium, and from it we also obtain a result- ant force /?, which is given in direction and absolute extent by the closing line of the polygon, but acts as an expression of the algebraic sum of the other forces, as shown by the arrow-head. From this it follows that in ever}' closed force polygon each single force represents the resultant of all the others in absolute extent and direction, except that the resultant tends to produce motion in an opposite direction from the corresponding force ia Fig. 54^ the polygon. In an unclosed pol}-gon the line necessary to close the figure gives the direction and extent of the resultant of the other forces, alwa3'S tending to produce motion opposed to the closing force. For example, in Fig. 54, A^ is the resultant for i and 2, and in. a similar manner the resultant for any of the other forces iu combination may be found. The method of representing the properties of forces by lines is also applicable to other quantities which possess the attributes of magnitude and direction, such as velocities ; also to the deter- mination of the path of the line which passes through the cen- tres of gravity of the stones of a vault, for instance ; and iu a figurative sense it may be applied to scientific discussions, in which the final result acts as a closing line to the force polygoa of argument. ?33- Isolated Forces in One Plane- Cord Polygon. If lines which represent forces, and hold a bod}' in equilibrium, do not intersect in one point, a condition which frequently occurs, but have a number of intersections from « to — ( // — i ) 2 ' in number, the foregoing solution can no longer be used ; but at the same time this more complicated case may readily be re- duced to the simpler form. For this purpose we assume the existence of a system of rigid, straight lines which, extending from each force to the next, form a polygon capable of resisting both tension and compres- sion iu the direction of its sides, and in which each single force is in equilibrium with the two forces which act along the sides intersecting it. A polygon formed in this manner is called a THE CONSTRUCTOR. 27 Cord Polygon, or in arch construction a thrust line, because all the sides are in compression, and in general such a figure may be called a link polygon. The angles of the cord polygon are called "knots." The link polygon m?y be used for the investigation of forces accor- ding to the preceding methods, when at each knot there exists Fig. 55- an equilibrium between the external forces and the stresses in the sides of the polygon ; for example, when the forces S^..-,_ and S^..^, at the knot K.,, have a resultant equal and opposed to P.,_, in extent and direction. The forces in the sides of the polygon may be called the internal forces of the link polygon. We have, then, for any given case two sets of forces to investigate : (i) the external forces, (2) the internal forces, since for each set there exists an equilibrium. I 34- eouii,ibrium of the externai< forces of the cord Polygon. If we take the forces /^ and P.,, find their resultant, combine this with ^3, find a second resultant, combine it with /*„ etc., we will find that in order to obtain equilibrium, the resultant with the next to the last force Pn — i, of the polygon, will be equal and opposed to the closing force Pk. This holds good so long as the direction and extent of the forces remains unchanged. From this it follows that the co-ordinate distances of the point of application of any force may be made equal to zero, without aifecting the equilibrium of the external forces. The combina- tion of these latter forces may then be effected in the same manner as if they acted at a single point. In this way the force Fig. 56. Fig. 57- Fig. 58. polygon can be used to determine the equilibrium of several independently-acting forces. If equilibrium exists, the polygon closes, and if it does not close, it shows the extent and direction of the force necessary to maintain equilibrium. It is practica- ble in this way to determine two unknown quantities in a force polygon. These may also refer to two forces, and may be either direction or extent, or, as sometimes occurs in practice, the direction of one force and the extent of the other. The following cases will serve to illustrate : I. Both directions given. In Fig. 56 we have the directions of the force lines 4 5' and A 6', and by their intersection at 5, we determine at once their length 4 5 and A 5. If their directions are interchangeable we have two solutions possible, the second giving the directions A VI' , and 4 V , and hence the forces A F"/and4 V. II. The extent of both forces given. Fig. 57. With the dis- tances equal to the extent of the two forces, we describe circular arcs from A and 4, and the intersection of these arcs determines the direction of the forces. Since the arcs intersect at two points, two solutions follow, giving the lines ^ 5, 4 5, and A V, 4 V. III. The direction of one force and the extent of the other- given. In Fig. 58 let the line 4 5 be the given direction of one force. With a radius A 5, equal to the extent of the other force, describe the arc shown by the dotted curve, and the two inter- sections give two solutions of the problem, as in cases I. and II. If the arc failed to intersect the line at all, it would prove the case to be impossible. Fig. 59. Fig. 60. The following examples will show the practical applications of the preceding principles : Example I. A crane A B C, Fig. 59, carries a load Z- at ^ ; it is of a cylindrical sliape at B, and held in position by a roller bearing, and at C there is also a pivot step. Required the forces /*! and /a <*t i? and C The centre of gravity of the crane itself is at 5, and its weight is equal to (?.* Both Z- and G act in a vertical direction, and the force at P^, if the bearing is smooth and we neglect its friction, acts in a horizontal direction. Combining G and Z. into one force Q ^= G + L, the position of whose resultant is T Q, we have the intersection (7 of a vertical through T Q, with a horizontal through Pi as a. point in the direction of the line of the force F^. This force must also act through the centre of the pivot C, since this is restrained from lateral motion by its bearing. This gives C O for the direction of the force /*.i. We can now draw the force polygon, Fig. 60, drawing L + G vertical, G Px parallel to O P-^, and F-i Pi parallel to O C. This determines the extent of both Pi Po, and by further analysis the entire load on the pivot C may be found. Fig. 61. Fig. 62. Example II. A crane constructed as shown in Fig. 61 carries a similar load to thft preceding. It is arranged with a cylindrical bearing at B, and at C there is a conical roller bearing upon a conical surface on the base of the column, the axes of both cones intersecting in the middle of the bearing B. We have, as before, the mean load Q ^ -L -\~ G ; we also have the direction of the pressure Pi, as it must be normal to tne surface of the cone at the point of contact. The intersection of /\ and Q determines O, and a line from O through the centre of the bearing B gives the direction of Z^- The force polygon can now be drawn, as shown in Fig. 62, and by making the vertical equal to Q, and the other two sides parallel to Pi and Pn, we determine at once the extent of the two latter forces. The vertical component of P^ will, in this case, be less than the total load Q, while in the previous example they were aJike. Hence it follows that the conical roller supports- a portion of the load. Fig. 63. Fig. 64. Example III. The crane shown in Fig. 63 is similar in construction to the pre- ceding one, except that the axes of the conical rollers intersect at a point D below the bearing B, If we now draw C O normal to the surface of contact CD, to the point O, and construct the force polygon, Fig. 64, we see that the change in the posi- tion ot the apex of the cone V causes the force /o to act from below instead of from above, as in the previous case. It will therefore be necessary to provide the bearing B with a collar to oppose the upward pressure, f * In ordinary wharf cranes the value of G, which mainly depends upon the capa- city and overhang of the crane, may be taken at ^ to J the load. f This defect may be seen in numerous existing examples of crane construction. In a case which came under the author's observation, a crane intended to have a capacity of thirty tons gave way under a load of only about twenty tons, because the proper provision was not made for the direction of a force upon a bearing. 28 THE CONSTRUCTOR. Example IV. Three forces of 70, 50 and So pounds act, as shown in Fig. 65, upon "a, body A B\t\ such a manner that tlieir resultant passes through the point A. Two ■Other forces of 05 and 60 pounds also act upon the point A, and hold the preceding -forces in equilibrium. Required the angles which the latter forces make with the former. Lay out the forces of 70, 50 and 80 pounds, as shown from C to Z) in Fig. 66, by the "heavy lines, then describe from Cand D circular arcs with radii of to and 95 respec- tively, and thus obtain the intersections E E' or F F' , which, when juined with C and Z>, complete the force polygon. Both solutions are given in the diagram. Fig. 65. Fig. 66. Example V. An obelisk is to be raised upon its base, Fig. 69, by turning it about ! D and P3, the direction DC; the other giving P„ the value P^ Z>i and' P3, the direction Z>i C " - If P3 should just equal the perpendicular distance from Cto P^ Pn, then but one -solution exists. The two results for the example given are shown in Fig. 67 at A Pj and A P3. Examples of this character seldom occur in actual practice. BQUII,IBRnjM OF InTERNAI, FORCES IN THE CORD POI,YGON. As already stated, we mean by the internal forces of the cord ■or link polygon the tension or compression ■which may exist in the different sides of the figure, as sho^wn at 5,.j, S.,.^, etc.. Fig. 69. These forces are of such an extent that they hold each other in equilibrium at the knots A', A', A'3, etc. Any t-wo of these, for example, S\.,, S.,.^, may be determined from their resultant P.,, when either their ^® direction, their magnitude, or one direction and one magnitude are given (see ? 34). This is done in the following manner : Construct the force polygon. Fig. 70, of the Fig. 70. external forces A",, A, Aj, which, if equilib- rium exists, will form a closed figure. From the extremities of the sides corresponding to the force f^ draw two lines parallel to the sides 5,.,, S.,.^, intersecting at O ; then the length of the lines C, and O., will represent the magnitude of the stresses in the sides S^.,, S,.^. In a like manner we may draw lines connecting the several cor- ners of the polygon. Fig. 70, with the pole O, and deter- ■mine all the internal forces of the link polygon, both in mag- nitude and direction ; so that when the external forces are known, and also the direction of two of the internal forces, the direction and magnitude of the others can be determined. This assists greatly in the construction of the link polygon, for by selecting one knot and determining the pole O, the sides of the link polygon can be drawn parallel to the respective rays. Fig. 69. The actual lengths of the sides of the link polygon are deter- mined by the positions of the lines of the external forces, from which the positions of the internal forces are also determined. The cord polygon will vary in its form according to the choice of a starting-point from which it is drawn. In Fig. 6g two forms are shown in dotted lines within the cross-hatched figure, their sides being parallel to those of the first polygon. Another solution of the same problem (the combination of the external forces into a link polygon) may be obtained by an application of the double solution of Case I., § 34. In Fig. 72 we have the directions 5]. 2 and ^'2.3 drawn from the extremity of the force A,, giving a new cord polj'gon, Fig. 71, of a verj' different form from the preceding one, which is also included in Fig. 71 for purposes of comparison. With the ex- ception of the first two sides, we have an entirely diflerent Fig. 71. figure, showing the variety of polygons which may be 1: onstructed from a given set of forces. THE CONSTRUCTOR. 29 The cord or link polygon, when taken in connection with the force polygon, forms what has been termed the graphical plan of forces. In most cases the entire subject can be discussed by the construction of one figure which may then be called the Force-pan, and of which examples are given in § 4S. Resultant of Isoi^ated Forces in One Plane. If we assume two of the sides of a cord polygon to be divided, and insert at the points of division forces corresponding to the stresses in the divided sides, the equilibrium will remain undis- turbed, as, for instance, in Fig. 73, the sides jTj A'^, and A', A'5, are cut and sustained. It will then be evident that the resultant of the forces, S^.g and St.-,, either on the right or the left will hold the remainder of the forces of the polygon in equilibrium. The position of this resultant force is determined by prolonging the sides until they intersect at H/. The direction and extent of this resultant is determined in the force polygon. Fig. 74, by the diagonal 4.6, which is the closing line of the forces S\.^ = Og, and Si3 = O4. This force is also on the one side the resultant of the forces /s and P^, and on the other side, of the forces P^, P,, jPj and P^. /k general it may be stated that the point of intersection of any two prolonged sides of the polygon is a point of the resultant of all the external forces beyond these sides, from which the direction and extent of said resultant way be determined. This principle is of great utilit}', as many examples will here- after illustrate. By reversing the above rule, the cord and force ....,.j«....-» Fig. 73. polygons may also be used for the decomposition of forces, as well as their resolution. For instance, if it is desired to decom- pose the force 4.6 into two others, /a and i°j, of given direction, draw one of them (for example, Pn ) in the cord polygon until it intersects 4.6 in the point N, and through this point draw P5, parallel to the side 4.5 of the force polygon. The first chosen line, A'e N, may be drawn either forwards or backwards on M N, without disturbing the equilibrium. ? 37- Conditions op Equilibrium for Isolated Forces in One Plane. In the preceding discussions it has been assumed that the forces whose equilibrium has been investigated were so situated that equilibrium really existed, so that according to the rule in the preceding paragraph it would be possible to reduce them to Fig. 75. two equal and opposing forces. This is, however, not neces- sarily the case when the force polygon is a closed figure, but it must follow when the cord polygon is also a closed figure, /. e., the actual positions of the forces must also be taken into account. If the positions are not correctly taken, the cord polygon will show what modification must be made in order to secure equilibrium and avoid the formation of rotating couples ; which will be discussed in the nest section. For this purpose one of the forces should lie left to be determined in position a. the last. I. Let this force be /'„, Fig. 75. Itsmaguitude is known, and its direction is parallel to the given line Z Z. After construct- ing the force polygon. Fig. 76, choose a pole O, and draw the rays to the angles from i to 6, so that A'l K, is parallel to I O, A'2 K, to 2 O, A'3 A'j to 3 O, etc., until A'5 A'o is reached. Then the closing line of the cord polygon must have the direction 6 O, and must also pass through K^. This determines its posi- tion entirely, and its intersection A'g with A'5 A'g is a point of the force A'5, which is now drawn parallel to 5.6. Fig. 76. Fig. 7S. If the final force is not given either in direction or magnitude^ it may be determined from the direction and position of the other forces as follows : II. Let the yet indeterminate force be P^, Fig. 77, while wa have given the direction of the force P^, which is A'^ A",, and its position A'l. We can draw the force polygon from the points I to 5, while from the point i we have only given the direction A I. The cord polygon may also be commenced by starting from A'l, and continuing through the points A',, A'3, A'j, A'5 and ■ Pi' Fig. 77. IC'^. We may then select any direction for its closing side A' L, and its intersection A'j with K-. A''5 will be a point in the line of the desired force P^. In order to determine its magnitude and direction, draw, in Fig. 78, O 6 parallel to A'l L, and join the point 5 with the point 6, when the line 5.6 will give the desired magnitude and direction of the force P-^. ?38. Force Couples. When a plane figure is subjected to the action of forces in couples, acting in its plane in such a manner that, while equal in magnitude and opposite in direction, they fall upon parallel lines, and do not oppose each other in the same straight line. Fig. 79. Fig. So. the force polygon will close without necessarily proving the existence of equilibrium in the figure. The conditions which obtain under these circumstances may be examined as follows : The forces P^ P, and P^ P^, Fig. 79, form a closed force poly- gon 1,2, 3, 4, Fig. 80, but at the same time equilibrium does not 30 THE CONSTRUCTOR. exist in the figure, but instead, a tendency to rotate about a common point with a statical moment which is equal to the sum of the moments of the couples {I\ — P-^ and ^I\ — P^. In order to secure equilibrium it is necessary' to introduce an addi- tional couple (/-"s — Pf,], whose tendency shall be to cause a rotation in the opposite direction, and whose moment shall be equal to the combined moments of the previous couples, and whose direction shall be parallel to the lines V V and VI VI, Fig. 82. Let us take. Fig. Si, the force polygon A i, 2, 3, 4. This is not yet complete, for we still lack the forces 5 and 6. We know that they must act through A, in opposition to the other couples, tut their magnitude is net yet determined. As already said, the two forces must be equal and parallel in order to be in equilibrium %vith the other couples, and only two forces can fulfill the conditions. Their direction is given, and can be laid off as at A Z. We choose any pole O, and join the rays O A, O 'i, O z, O i, O i, (= O A), and can then proceed to construct the cord polygon. Fig. 82. For this we have lines of direction //, Till, etc., up to VI VI, given from Fig. 79. Starting from any point K^ on //, we draw lines parallel to the rays O A and O 1 (their resultant being the force P^] until they intersect VI VI in K^, and II II in A', ; then draw A', A', parallel to O A, intersecting III III at A'3, A'3 K^ parallel to (9 3 until it intersects IV II', and K^ K^ parallel to O 4, intersecting V V at A'5. Onl3- the closing line I E, 5- n / My X m Ka v/\ 'y Q y K4, Fig. 81. Fig. 82. of the cord polygon is now lacking, as 't is the line joining K^ with A'5. which latter point has already been determined. We can now (see J 37, II.) draw the ray O, parallel to A'j A'g, com^ pleting the force polygon and the line A 5, will give the magni- tudes of P^ and I\. The path around the force polygon may be taken as A i, 2, 3, 4, 5 A, the sides 4.5 and 5 A being sup- posed to make an infinitely small angle with each other. The previous examples upon the force and cord polygon serve to show how geometrical addition and subtraction may be used to determine the equilibrium of diverging forces in one plane. Forces acting in intersecting or parallel planes may be examined in the same manner, and in many cases without a great degree of complication, as some following examples will illustrate. It is not intended, however, to undertake a general discussion of the subject here, but rather proceed at once to practical appli- cations of the special case of parallel forces. I 39- Equilibrium Between Three Parai,i,el Forces. In discussing the equilibrium between parallel forces, we may use purely arithmetical methods, or use geometrical addition and subtraction (force and cord polygons), as may be found most convenient. The present problem may be stated as that in which a force Q acts upon a body, and is to be held in equilibrium by two un- P. Fig. S3. Fig. 84. tnown forces, P, and P„, acting parallel to it and to each other. Drawing the line ABC, Fig. S3, normal to the given direc- tion of the forces, we must have, in the existence of equilibrium, J'l . Ali = P. . B~C, or P, rti = Pj a^, and also Ai -f Pj = Q. P, — ~ graphically, we may follow In order to determine A\ tie method in ? 24, and in Fig. S4"make O E ^ the divisor a^ O A = the factor a,, and taking E P to represent temporarily the force P.^, draw A C parallel to E B, which gives the propor- tional value of A,. By placing the triangle C A O in the dotted position O' B A', we have A' E = I\\ P^= O. This gives a figure in a form well suited for application to Fig. S3, as will be shown in the following examples : I. In Fig. 85 draw A D equal in value to O, join D with the third point of application C, and prolong Q until it intersects Fig. 85. Fig. 86. at A" a line drawn through A? parallel to A C. Then will we have the following relations, B E= P^,E F= P,. In Fig. 86 is shown a similar case, but with O inclined to A B C, and in Fig. 87 O is beyond AC II. By resolving the force O into two components applied at the points A and C, Figs. 88, 89, 90, we obtain inclined forces I^G. 88. Fig. 89. Fig. 90. ■whose components parallel to O are the desired values for Pj and P.,, while the components which are parallel to A B C neu- tralize each other. In all three figures B F= A, and F D = A,. III. By constructing the force pol3'gon, making A D ^ Q, and using any pole O, Figs. 91, 92, 93, and drawing the sides of the link polygon, so that A b\s, parallel to A O, b c parallel to Fig. 91. Fig. 92. Fig. 93. D O, and joining the closing line c A, the parallel to the latter in the force polygon O E will give E A = P^, and D E ^ P„. If it is desired to make the closing line fall upon A B C, or lie parallel to it, the cord polygon A. b Cmust be first drawn, and the pole O, determined by the intersection with A B oi 3. line D O parallel to b C, P> A having first been drawn equal to Q, O E may then be drawn parallel to A b, and we have E A = P^, and E D = P.,. _ In these cases Q is equal in magnitude to the resultant of P^ and P.,, and opposed to them in direction. If Q is to be deter- mined when Aj and P^ are given, similar methods to the fore- going are to be followed. Returning to the diagram O E A C B, Fig. 94, which we have already used in case A, we construct the triangles C A O and 9' ^ B c -i Fig. 94. Fig. 95. Fig. 96. B A' O', and draw B' C parallel to O A ; O' C and O B' par- allel to A' B, giving B' B = a^, B C = a^, B' O = P.„ O' C = ^1- . . From this we obtain the following solutions : IV. Transfer one of the forces to the opposite side of A C, P Fig. 97. Fig. Figs. 95, 96, so that A D =^ P.^ and E C= Pi, join D to E, and THE CONSTRUCTOR. 31 the line D E will intersect A Cat B, vchich -will be the point of application of the resultant Q, whose magnitude = £ D' -= P^-\- P^, since D D' is drawn parallel to A C- In Fig. 96 jP, and P., act in opposite directions, and their alge- braic sum D' E must be taken, and, as shown, the resultant Q acts bej'ond A C. V. The method shown in Fig. 97 follows from (II) : From the extremity a oi a A ■=■ P^ draw a line A' a of any length, making it parallel to A C. In a similar manner draw c C from the extremity oi c C =^ P.,. Draw A' A and C C, prolonging them until they meet at E, which latler will be a point in the line of the resultant E £, and the value of Q will be P-i + P^, which is also the resultant ol D E =^ C C and EE=A' A. VI. Following the method in (III), we may proceed as fol- lows, Fig. 98 : Make D E = P„, E A = P^, choose a pole O, and join the c'osing line O E oi the force polygon. Draw A c parallel to E O, c b parallel to O D, and A b parallel to (or, as in this case, the prolongation of) A O, and the intersection b will be a point in the line of the resultant O, whose maenitude = jyA. ?4o. Resui Qi -'] I i -jfl' "E Fig. 99. I. In order to combine the forces Qi to Qi, intersecting a common normal A E, Fig. 99, we first combine O^ and O. by transposition, as in Fig. 96, and obtain the resultant, Qi + O2 = b c. This may then be combined with O^. giving ^t^' = !2i + ft + Qi< snd this result with Qi, which finally gives the resultant, 5= !3i + {?2+ ft+i passing through 3T. This solu- tion is one which is sometimes desirable in machine construc- tion, as, for example, in the distribution of the weight of a loco- motive engine vipon the various axles. The method of deter- mining the resultant of several parallel forces in this way by the successive combination of pairs is very tedious and of limited application, and the method given below of using the force and cord polygons is much simpler. Fig. 100. II. Fig. 100. Form the force polygon of the given forces ft to Of,, by laying off lines successively from A, equal in length to the magnitudes of the several forces A i, 2. 3, 4, 5, 6, as shown in the left of the figure. The magnitude of the resultant will then be equal to the length of the closing line 6 A. To determine its position, proceed as follows : Select any point beside the line A s, as a. pole O. and join the rays O A, Ol, O 2, O 3, etc. Starting from a point b under Oi, draw b b' parallel to A O, and b c parallel to i C, and continue by drawing r^ parallel to 2 O, d e parallel to 3 O, etc., and finally reaching the closing line of the pol3'gon g g' parallel to O 6, intersecting b b' at q, which determines the position of the resultant Q (see §35)- The method shown in (J 36) may also easily be applied to the resolution of such forces, as in Fig. 100, the intersection of d c, prolonged to c' , gives the position of the resultant of O^ and ft, and its magnitude is shown at ^4 . ., in the force polygon, and in a like manner c' is the position of the resultant of £?4 and Q^. Decomposition of Forces into Two or More Parallel Forces. The methods of resolving forces by means of the cord poly- gon will also serve for their decomposition. If, for example, in anj' portion of a cord polj-gon a q b c d. Fig. loi, it is desired to substitute for a force Q, two forces Q-^ and Q., passing through e and/", we have oulj- to join the points e and f to obtain the form of cord polygon for the new forces. Fig. 101. Fig. 102. and determine their relative magnitudes by drawing 0\ parallel to ^y in the force polygon below. If the required force O-^ and (Jo both lie on the same side of O, Fig. 102, the solution is similar. We now prolong a q to its intersection e with O^, and join e f. Also mark the intersection of (2i with q b, and O, with q a. In the force polygon below we have Q-^=^ A \, 0.,^= \ . 2, or A i' = Q^anAV '2 = O^. If we have a beam A G loaded with parallel forces Q^ to ft, Fig. 103. Fig. 103, whose load is to be opposed b)' reactions P^ and P^ at A and G, we may first determine a resultant f) of all the forces, as in [f. 40), and then decompose this into values for Z', and P, by the method just given. We also omit the determination of O altogether,' and proceed to determine P^ and P^ directlj' as follows : Choose an}' pole O, and form the force polygon .A'l . 2 . . . . 5 ft and construct the cord polygon, making its sides parallel to their respective ra5'S, and draw b a parallel to A' O and/g, par- allel to O 5, their intersections Viiith the lines of the forces P^ and P, beings and 0-. Join ag, which will be the closing line of the polygon, and its parallel 06 in the force polj'gon gives P., = 5. 6 and /"j =: 6 . 7. If the sides a b and/g of the cord pol3-gon are prolonged in the other direction we obtain a' and g', giving, however, the same result, since n' g' is parallel to ag. The cord pol5-gon would then be the figure a' g' m b d c efiii a', and 111 indicates the position of the resultant of the forces Q^ to ft, or of Pi and P„. ■When a loaded beam is supported by three or more bearings it is necessary to take into account the resistance of the beam itself with some degree of accuracy, or else the problem be- comes indeterminate. This indeterminate character may, how- ever, be eliminated by the introduction of an equalizing lever. THE CONSTRUCTOR. Suppose we have, Fig. 104, a beam BCD, the resultant Q of whose entire load acts at M, and is opposed by the reactions of three supports at /"i, P.,, P^, at right angles through the points B, CanAB. We may now assume, temporarily, an approximate ratio be- tween two of the forces, e.g., P^ and Pn, and permit them to act at the extremities of an equalizing beam B^ C^, which in turn supports the main beam at ^ £■! ; makingthe ratio of ^1 C^ :E^By the same as has been chosen for P^ : P.,_. Now decompose Q into the components acting at E and D by means of the cord and *^3 Fig. 104. force polygons e m d and A O \ 2. This gives y4 1 = 5, i . 2 = P^, 2 Ji =: Pi-\- P.,, which last sum may be then divided between Bi and Ci by any of the above methods. p Each different approximation of the ratio -J, will give a dif- ferent value for P^. If P, and P^ are made equal to each other, £■ will be in the middle of B C, and the equalizing .lever will be of equal arms. The distribution of the load of locomotives and cars upon their spring is usually made with such equalizing levers. If the load is to be supported upon more than three or four points it will be necessary to use several equalizing levers, and Fig. 105. examples of this will be found in some locomotives. If, for example, we suppose M, Fig. 105, to be the point of application of the total load ^ of a locomotive, supported upon three axles B CD in such a manner that the weight shall be transmitted to the axles through the springs as shown, and also that the ■weights upon the wheels Caud D shall bear a determinate rela- tion to each other. This can be accomplished by the use of three springs and one equalizing lever upon each frame of the locomotive, the whole weight being thus supported upon eight points. Taking the relation between the forces P, and P-^^p : q, we erect a perpendicular E e, whose distance from the axle C and i? is in the proportion q : p. From any point e' on this for the equalizing lever. The distances of the points c and d from the verticals through Cand D give the length of the arms of the springs fj c.^ and d^ d^. These springs are made with arms of equal stiffness, since they are to support equal loads at both ends. For any chosen ratio p : q, and given distance be- tween the axles, the actual length of the equalizing lever will not affect the ratio of P^ to the sum P^ ■\- P^, as an inspection of the cord polygon b m c d will show. The springs which are attached to the ends of the equalizing lever must, of course, be made of sufficient stiffness to support the load which is thrown upon them, and the length of the sup- ports and their proportions chosen according to the previously determined distribution of the weight. Many similar examples to the preceding might be given, as they are of frequent occurrence in practice. The two springs which are attached to the equalizing lever may be replaced by a. single spring, as in Fig. 106. In this case the axes C C are con- nected rigidly to the lever dec, and the lever itself rests upon a spring b^ e^ c^, whose extremities are fastened to the frame. The arms b^ e^ and c^ e^ of the spring are of unequal length, and have the same relation p : q as that which exists betweeiL the arms of the lever bee. If the arms of the lever are not properly proportioned, or if any error has been made in the dis- tribution of the load, it will be made apparent by the inclined position which will be assumed by the equalizing lever. ?42. Uniformly Distributed Parallel Forces. When a beam is subjected to a uniformly distributed load, the force and cord polygons cannot be determined by the preceding methods, since in such cases the cord polygon becomes a figure Fig. 106. line draw lines to the bearing points of the wheels upon the rails, and any horizontal line will intersect these inclosed lines in points which will give the proportional length of arms c e d . I 2 1 r 3 r r 6 7 8 9 E n \ ,b C < X ^ t-^^ h / 111 ;| i ■R — - t: -" 'J 7 \ \ ^ X 1 f V V it Fig. 107. H. Fig. 108. of curved outline. The character of the curve may be deter- mined in the following manner : If we assume the load to be concentrated at a number of equidistant points, as in i, 2, 9, Fig. 107, and construct the cord polygon for these conditions, it will be evident that the sides a 7)/ and b c will intersect midway between i a and 2 b, and also midway between a b' , since the forces I and 2 are equal to each other. In the same manner cd and a i)/ intersect midway between 3 c and i a, which is also in the line of 2 b, that is, at b' , and likewise d e and a yJ/ intersect midway between b' and c' . In this way it may be shown that the intersections of the prolonged sides of the polygon from a Jl/ to iM are at equal distances from each other. This indi- cates a known property of the parabola whose vertex lies on, E J>/ line E M, and whose abscissa e E ^ . This parabola is the form assumed by the cord polygon when the load is uni- formly distributed, as was previously assumed. If we note that the triangle A M B represents the entire load collected at E, it "will readily be seen how the curve may be drawn in any case. If the chord A E B is inclined, as shown in Fig. loS, the divi- sions of A 77/ and MB will be equal in number, but the divi- sions of A /)/will be of different size from those of 71/ B. The point e lies in the middle of E M, but is not the vertex of the parabola. Link polygons which assume the form of curves ma}' also be used to show the effect of moving loads, and are then the figures which are contained within the successive sides of a regular polygon. Many examples are to be found in the case of railway bridges, traveling cranes, engine guide bars, etc. § 43- The investigation of the action of parallel forces, such as Qj to Qi and /\ P.,, Fig. 109, whose direction is normal to a beam, requires a knowledge of the statical moments of the external forces. These can best be obtained by use of the force and cord polygons. After constructing the force polygon A O /^, and cord polygon abed ef, let it be required to find the statical moment for any point .S upon the beam. This moment is the product of the resultant of all the forces upon one side or the other of the line 5' S^ into the lever arm / of this resultant from 6" .S,. The magnitude of this resultant is obtained from the distance A J ^ 1 , 5 in the force polygon, cut off by the rays O l and O 5, THE CONSTRUCTOR. 33 which are parallel to b c and/ 3, and its point of application is determined by prolonging these sides until they intersect at g. By drawing the perpendicular ^ o-^.,the lever arm I of the re- sultant Pz= h i is determined, for the force acting at the point S, and hence we have 7)/ = P I. This multiplication may also be performed graphically. By drawing the perpendicular O k va. the force polj-gon, we obtain the altitude of the triangle O h i from the base h i, and this tri- angle is similar to the triangle g s So, whose altitude is /. Call in O k = H and s Sa ^ t, we hava P:H=t:l, or M=Pl = Ht. This proves that the statical moment at any point in a beam is proportional to the corresponding ordinate of the cord polygon, parallel to the direction of the forces, since // is a constant. By making H equal to unity the conditions become similar to Case I., ? 22, in graphical multiplication, and the moment M becomes equal to the ordinate t. It is not necessar}' to deter- mine the position of the point of application^ of the resultant, since it is the relation between the statical moments which is most desirable, whether //"be chosen as a unit or not. This property of the cord polygon for parallel forces is most useful, and an example may be found in the case of axles. For such cases as for many others, it is most useful, since no modification of the diagram is necessary, the moments being found by the same construction which is required for the deter- mination of the forces. It is often convenient in practice to cover the figure containing the moment ordinates with section lining or with a light tint of color. ?44. Composition and Decomposition op Staticai, Moments. As shown in the preceding section, statical moments may be shown by means of lines of definite length and position in the same manner as simple forces. When two statical moments act in the same or in different directions, they may be combined by means of graphical addition in the same manner as has already been shown in § 31. li A B Cand ADC, Fig. no, represent the cord polygons for two sets of parallel forces which act nor- mal to the axis of a revolving body A C, in the directions A' B' and A' D' we have the following method : For a point ^ on the axis of the body we have the triangle T-^ S T/, in which the angle \ p. 5 »i X Fig. 117. expected from the symmetrical form of the structure, and an investigation of one-half is practically sufScient. If the load 2 P is taken as uniformly distributed over the entire distance ABC, instead of being concentrated at B, the P reactions at A and B will each be equal to — , and the load at i? = /*, so that )^ of the load on.ABa.n.d.B C" is referred to the knots A, B and C. From these conditions we obtain the force plan b, which is geometrically similar to the other, but only half as large. II. Double-trussed Beam (much used for constructions of all sizes). Fig. 117. In this case take vertical forces P, at B and C, and corresponding vertical reactions at A and D. In the first force plan, a is drawn equal to P, and I and 2 parallel re- spectively to A B and A E, thus determining the forces i and 2; I, being compression and 2, tension. Lines now drawn par- allel to B £ and £ P, determine the compression in 3, and the tension in 5, while the compression at 4 is the closing line of 3, l,andP; and the other half of the diagram is similar. If the vertical forces at A and B are not of the same magnitude, which is often the case in practice, the structure should be strengthened by the introduction of the diagonals £ C and B P. The second diagram shows the construction in this case. Let Pi = Hi bi be the force acting at A, and Pj = "2 ''■2 ^t B. Draw a vertical line from I to a horizontal through Q, which gives the length 3, of the vertical force at B, and by drawing the 3P 2 A ^ L - P 2P B 4 C 8P 2P 8 D 12 E F 7 ^1o ^ f^*^- - — — -: 8^ ><^^ Fig. 118. dotted diagonal line their resultant is found. If any of the ten- sion members are omitted the framework will tend to take an in- clined position until the various parts are at such an angle with each other that both constructions will give the same value for 3. For this reason it is best in nearly every case to use the di- agonal counterbraces. III. Triple Trussed Beam. Fig. 118. The uniformly distrib- uted load upon the framework gives the following distribution of forces. The force 3 P =: a 5 c is first decomposed in 2 and i, OT c e and e a ; then i, is connected toa 6 = 2 P, hy the line b e, and this latter decomposed into 3 and 4 or ^ 7^ and J^ b ; 2 and 3 are now joined by /" c, and the components at 5 and 6 or /£■ and o- c found. Since 5 and 10 are equal to each other, we may draw c h parallel to G H, and equal to c;^, which gives g h ^j \ the rest of the force plan is similar to the first half. ^' This distinction has been suggested by Cuhnann. Fig. 119. IV. " Another form of Triple Trussed Beam is shown in Fig. 119. The space between j? and C" is twice as great as between A and B, and the uniformly distributed load is equal to 12 P, act- ing at the various knots as shown in the figure. In the force plan, make a b c =^ $ P, and draw parallel to i and 2, the lines a e and e c ; then join, i with j,p (for the knot at B), and decompose into 3 and 4, or e _/ and f b. Now com- bine 2 with 3, giving c f, and draw 5 and 6 parallel to P (Tand F G, respectively. This case differs from the preceding, in that 5 is now compression instead of tension. The equality of the forces 6 and 10 gives g h = J, and the similar half of the dia- gram need not be drawn. V. Multiple Trussed Beam. Fig. 120. The beam A J is divided into eight equal parts, which are represented as being uniformly loaded, the load at each knot being shown in the figure. In constructing the force plan we make a e ^= j P, and by drawing the lines parallel to i and 2, we obtain a f anA f e ; then lay off a b =^ 2 P, and join the resultant bf. This decom- poses into 3 and 4, or fg and^ b. The forces 2 and 3 combine to give the resultant g e, which, by drawing lines parallel to K Cand K L, gives g h and k e for the values of 5 and 6. We now find that to proceed further we have three forces of given direction onlj', and since this is indeterminate, we must obtain one magnitude as well. This, for example, may be done for the force 7 as follows : The strut C L sustains the vertical com- ponents of 5 and 9, as well as its own direct load 2 P. Now 5 and 9 are equal to each other, since they are placed sj'mmetri- cally, and carry equal loads from the struts i? A' and KM, Hence in the force plan we maj' make h i, which represents the 36 THE CONSTRUCTOR. force 7, equal to twice the projection of 5 upon tlie vertical ■\- 1. P. This we can now combine with 6 = // e, giving i e, which in turn decomposes into i in and m c, or 10 and 11. Returning to the knot C, we may now take the line // /, and by drawing parallels to CL, CM and CD, obtain the figure h i k c. Fig. 120. ■which determines the forces S and 9. In the same manner pro- ceed from 12 to 15, which will complete the half plan. It may be noted that the principal beam A J is subjected to a uniform compression throughout its entire length. The force plan will, of course, be modified by various dis- tributions of the load, as in the case of simple beams, as shown in cases XII. and XIII., ? 6. ? 5°- Force Pl.^ns for Roop Trusses. Roof trusses furnish many and varied examples of frame- work.* In the following examples a uniformly distributed Fig. 121. vertical load is assumed, so that the burden upon any portion of a rafter may be considered as proportional to the length of that portion. I. Roof with Simple Principals. Fig. 121. A uniform load 2 j°upon each half gives as the external forces P, 2 /"and P a.t A, B and C. Lay off in the force plan a b =^ P, and draw a c and b c parallel to A B and A C, determining the forces i and 2 ; I being compression and 2 tension. Then dravi' the vertical c e, and also draw b e parallel to CD. thus giving both 3 and 5, and the diagram is completed by drawing d e. resultant into d e and e b respectively parallel X.o E C and E .S^. giving the forces 3 and 4, both being compression. By repeat- ing 2 and 3, in drawing 7 and S, we obtain the figure c d e f g, in which eg gives 5. This latter force might also have beett Fig. 122. II. Roof with Single-Trussed Principals. Fig. 122. This form is similar to the preceding, with the addition of the struts C E and C F. The distance ^ i? is to E B, as 2, is to 2 ; and the loads upon the respective portions are 6 /'and H P, which give the forces at the various knots as shown in the figure. Make a c \n the force plan equal to 7 P, and by drawing lines parallel to A E and A C, obtain the forces i and 2, or a d and d c ; then combine i with 5 P^ « 5, and decompose the dotted '^■- Many subjects for Graphical Analysis may be found in Ritter's " Roof and Bridge Construction," Hannover, 1863, in which the forces in the vari- ous members will also be found carefully determined numerically, thus affording convenient proof. Fig. 123. found by combining 4 and 4 P, and decomposing the resultant by lines parallel \.o B C and B F, an illustration of the various methods in which the force plan may be used. III. Another form, with Single Trussed Principals. Fig. 123. This roof is similar to the preceding except that the struts E Cand C F are placed horizontally. In this case A E ^ E B^ and the external forces at A and D are both equal to $ P. 2P Fig. 124. The forces from a to ^ in the force plan are determined as be- fore, giving da and cd for the forces i and 2, and the com- bination of I with 2 P gives the resultant d b, from which the thrusts 3 and 4, or d e and e b, are obtained. The value of i is the same as 3, and 8 is the same as 2 ; while 5 is the closing; line oi c d e d f, or of c df. The force 5 must also be the com- bination of the equal forces 4 and 6 with 2 P, which the dia- gram shows to be the case. If the rod C B is omitted, as is- frecpiently done, the strut E C F, if there is no joint at C, will, oppose its resistance to bending to the force 5 ; but there will be a tendency to rise at the apex/?, if the fastening be not made sufiBciently strong. 2P Fig. 125. IV. Third Roof with Single Trussed Principals. Fig. 124. In this form of truss, frequently known as the Belgian or French truss, the single vertical rod of the preceding form is replaced by a triangle BCD. The struts are placed in the middle of the rafters and the external forces are distributed as shown in the figure. In the force plan « 5 c = 3 /", and i, and 2 are determined as before. By the decomposition of the re- sultant of I and 2 P, we obtain the forces 3 and 4, or d e and b e, and from the resultant e c, of the forces 2 and 3, we get the tensions 5 and 6, in c f and e f. The second half of the dia- gram is the sj'mmetrical counterpart of the first. V. Roof Truss with Double Trussed Principals. Fig. 125. This construction does not differ greatly from that shown in Fig. 124, except that the struts employed to strengthen the THE CONSTRUCTOR. 17 ■rafters are divided into two. The spaces are equal to each other and the load uniformly distributed. As shown in the figure this gives a reaction of 5 P, or A and D. In the force plan a d = ^ P, and lines parallel to A E and A C drawn, de- termining the forces i and 2, or d e and e a. We then combine £ a with a ,b = 2 P, and decompose the dotted resultant e b, into the thrusts ^yaud f b, or 3 and 4, by drawing these lines parallel to jE Cand £ F. Again we take the resultant of the forces 4 and 2 P, and decompose it into 5 and 6, or f g and ^ c, which brings us to the middle of the symmetrical figure. The force 7 is the resultant of 6, and its counterpart S, and the load 2 P, and the half of this force is therefore equal to the pro- jection of 6 upon the vertical, less P, or in the diagram, to d li. Fig. 126. VI. IJnglish Roof Truss, with Multiple Trussed Principals, yig. 126. Here we have inclined struts, with vertical tie rods. The load is again uniformly distributed, each space bearing the load of 2 P. The reactions at A and D are each = 7 /*. In Ihe force plan we have ab-\-bc-\-cd-\-de = z X 2 P -\- P=^ 7 P, which gives the length of a e. The forces i and 2 are found by drawing/ « and e f, parallel to A E and A L. Now consider I as combined witli a b ^=1 P, and the resultant /' b, ■decomposed into f g and g b, giving the forces 3 and 4 ; again, •combine 2 and 3, and then decompose the resultant g e, into 5 and 6, or g h and h e, by drawing these latter parallel to L F and L M. In this manner we continue until we reach 12, or I d, which we then project upon the vertical. Now taking from d m, one-half the load P= d e, we have m e for one-half the stress on the middle rod B C. The remaining half of the force plan is similar. D___K^^^ Fig. 127. VII. Polygonal or Sickel Shaped Roof Truss. Fig. 127. This roof may be considered as a modification of the preceding form, and is used for higher and wider spans. It is hardly proper to assume that the load is here uniformly distributed even if the spaces are equal, for in the case of snow, much less weight would be carried by the steep portions A B or G H, than by the flatter surfaces C D or D E. We must therefore estimate the forces P-^, /\, P-^, acting as B, C, D, E, F, G, and make the reactions at A and B equal to Q ^ P^ -{- P., -\- P-^. In the force plan a b ^ P^, b c = P,. c d =P,,, and a d ^ Q, "which is first decomposed into i and 2, by drawing e a and d e parallel to A B and A J. Then combining i with /\, and de- composing the resultant, as before, we get 3 and 4, or ^ y and f b. Having 2 and 3, we get in like manner 5 and 5, ox g_f and d g ; then combining 4 and 5 with P^, and decomposing with parallels to C A' and C/?, we obtain the forces S and 9, and so proceed until we reach 12, which is the middle of the symmetrical figure. The members K L, D L, E L, and 3'T L are all subject to tension. ? 51- The Graphicai< Determination of Wind Stresses. In designing large and important roof trusses it is important to investigate the stresses due to wind pressure, as well as those due to the weight of the roof and of snow, and indeed, in some cases, the resistance to wind is the most important of all. As an illustration of the applicability of the graphical method to the determination of wind stresses, we will take the English roof truss. Fig. 126, whose conditions under a vertical load have already been examined, and consider it as also subjected to a wind stress JV, as shown in Fig. 128. We have first to determine the forces Q-^ and Q^, acting at the points A and D. The wind pressure will be taken as acting on the surface of the roof from A to B. l,et IV be the resultant of the entire wind pressure acting normal to AB, and let Phe the total vertical load upon that half of the truss. By combining these two forces we obtain the direction FS of their resultant, and also its magnitude, which we then lay off on the force plan at cc-i- Upon the other half of the truss we have only the verti- cal load, which may be considered as acting aty, and equal in magnitude to P. By prolonging its direction until it intersects the previously determined line at S, we have at 5 a point in the resultant of the entire load upon the roof, including wind pres- sure. By making c^ a^ in the force plan equal to P, we have a c for the direction of this resultant, which may then be laid off at 5 T'in the drawing. In order to determine the forces Q^ and 0.^ we must recollect that, according to \ 34, when we have two Fig. 128. closing forces to determine, we must also have at least two con- ditions given. In this case, then, we must first find the direc- tion of 0■^ and Q^. The wind pressure produces a horizontal thrust which must be met by the stability of the walls or columns upon which the roof rests. In each case it must be determined whether this horizontal thrust is borue equally or unequally by both sup- ports, and in what proportion it is divided. To this end we first find (according to \ 39) the proportion of the vertical com- ponent of the force a c, which comes upon each support (as found by the intersection of 5" T, prolonged with A £>), and then combine these vertical forces with their respective hori- zontal components. It often happens that all the horizontal thrust is borne by one of the supports, which it must of course be prepared to resist. This often occurs in the case of railway stations, and under such circumstances the direction of each force must be determined separately. First prolong the vertical at D downward until it intersects .S T, and join the intersection with A (the lines are only indicated in the figure). This gives the direction of the force at A. We have now both the direc- tion of the reaction at D and the direction of that at A. We must also consider the distribution of the forces at the various knots between A and B, and between B and D. We have for the points between A and B the resultants between the propor- tional parts of P and JV, while from B to D we have simply the proportional parts of P. This gives at A the force P^ ; aX E,F and G, the force P„ ; at the peak, the force P-^ ; aX H,J and K, P ' P the force />,= — , and at D, the vertical force P^ =--. 4 f' Returning now to the force plan, we make c d =^ P^, d e ■= e f=fg = P,, gh= P.Jii = ik = k 1= P^, and la = P^. We now have finally the length b 1, for the value of the reaction Q.^ at the point D, and a line (not shown) from b to d, gives the magnitude of the force Q-^ acting at A. The determination of the stresses in the various members can now readily be made. The decomposition oi bd hy drawing bin and 7« ^ 0.66 0.51 0.60 4X X A A 1. 00 A iX iX iX 2X 0.63 0.48 0-59 6X t\ H n iX X iX >x iX 2^ 0.61 0.48 0-59 13X y% W A iX A 1% 2 iX 2% 0.60 0.47 0-59 20 A If r^ iX % iX 2X 2 3X 0.60 0.47 0-59 29 % if A -S-% X iX 2X 2X 3X 0-59 0.47 0-59 36X A X n T% if 2.00 2X 2X aX 0.5S 0.47 0.59 50X Y^ lA w 2.00 X 2X 3 2X 4X 0.5S 0.47 0-59 66 d= i.5, so that the rivets and the perforated plate have not the same degree of security. The values of 0, for the rivets and for the plate should therefore be determined separately, and the smaller value taken for that of the completed joint. Let : 'P' = the modulus for the perforated plate, ' for the perforated plate, and is nearly always less than j^. It follows that for single riveted joints of steam boilers we should never assume a greater strength than one-half that of the solid plate. By the adoption of double riveting, while retaining the same pitch, rz = arf + 0.4'', we ought to obtain, according to the formula of \ 5S, a value of <\i" , twice as great, which in the case of very light plates would exceed unity. In that case, however, the value of 1^' is the lesser, and it determines the efficiency of the joint, so that the only gain due to the double riveting in that case is the increase in the value of ' = 3-3 „.. f - , , —0.71. It may be remarked that American prac- 1.65X0 3125 tice gives wider pitches than are generally used in Europe. f Three rows of rivets are used in this form of joint, and the outside rows of wide pitch make this method more trouble- some of execution than the group riveting shown in Fig. 144, which has a modulus of o.So. This is a point which should be borne in mind. The joints of gasometers exhibit but little variety in plates or rivets. The rivets are usuallj- about '4 " to yV" in diameter and i" pitch, with a lap at the joint of about j4^', the rivets being closed cold and the joints caulked with red lead. Table of the Weight of Sheet Metal. Thickness in Inches. Weight in Pounds per Square Foot. I Wro't Iron! Cast Iron, Coppc Zinc. tV 2-53 2-34 'A 5-05 4.69 z\ 7.5s 7-03 '/ 10.10 9-3S A 12.63 11.72 Vi 15.16 14.06 A 17.68 16.41 'A 20.21 1S.75 A 22.73 21.09 A 25-27 23-44 H 27.79 25-7S H 30-31 28. 13 il 32-S4 30-47 A 35-37 32.S1 if 37-90 35-16 I 40.42 37-50 2-73 2.89 3-71 5-47 5-73 7.42 8. 20 S.67 11.13 10.94 11.56 14.83 13-67 14-45 18.54 16.41 17-34 22.25 19.14 20. 23 25.96 21.88 23-13 29.67 24.61 26.02 33-38 27-34 28.91 37-oS 3.x oS 31.80 40.79 32.81 34.69 44-50 35-55 37-58 48. 21 38. 28 40.47 51-92 41.02 43-36 55-93 43-75 46.25 59-33 2-34 4-69 703 9-3S 11.72 14.06 16.41 18.75 21.09 23-44 25-78 28.13 30-47 32-81 35-16 37-50 I 61. Special Forms of Riveted Joints. Junction of Several Plates.— In Fig. 156 is shown the junction of three plates. In this case the corner of sheet No. 2 is bev- eled off and No. i worked down over the lap. ljfiSS^?l*'V-Ji !•««,* K.'A I 3 i Q J Cp 2 ■ef-e-^- -0--0- 1 3 i „.-' 9 i 2 s ■-Q-i-QrX^- -.e--0-| 3 if Fig. 156. Fig. 157. In Fig. 157 the junction of four plates is shown. Here the angles of sheets Nos. 2 and 3 are beveled and Nos. I and 4 are left unaltered. In the construction of steam boilers the shell may be formed either in cylindrical sections, as shown in Fig. 158, or in sections of a conical shape, the taper of all the sec- * The two rivets -^vhich lie between any pair of rivets in the main joint each bear a stress of \ P, and the rivets of the main joint also sustain a P- The flap does not transmit an^- stress to the rivets of the main seam For the stress on rivets P^= ^ T^d- S'-^, for the solid plate P^ Si 2aS. The modu- lus 0", taking S^ for shearing stress = . }., , is found from <}>" = -^-^^-5 — ' For the main seam we obtain tf)', from P= ^ (■2it — ^d) S S"^ = zad Si 2a8, whence q 3 ('^ - '^) ■which is greater than - t For example, in single riveting ^s" plate and fj" rivets have j%" pitch (the formula would give about i^4")> A" plate and Vs" livets have a pitch of zfs" (the formula would give about 21^5'') ; in double riveting, for %" plates with y^" rivets, the pitch would be ^yi" , while the formula would give about 3". Fig. 158. Fig. 159. tions bearing the same relation to the direction of the flame as shown in Fig. 159. This latter method requires that a slight curvature should be given to the sheets in order to secure the required taper. The determination of the taper and curvature of the sheets and lines for the rivet holes may be made in the following manner : 44 THE CONSTRUCTOR. Let— D = the diameter of the shell, as in Fig. 159, J) = the breadth of the sheet, Fig. 160, on a circumferen- tial seam, /, ^ the length of the sheet between pitch lines of rivets, /== the versed-sine of the arc £ ; we then have : / 6 ~D L (61) Example, we have B - Fig. 160. In a riveted tube where each section is made of an entire sheet = -nD. If the breadth B is twice the length /-, we have - = 0.7854 X 2 = 1.570S, so that f will be a little greater than i>^ times the thickness of the plate. In arranging the junction of sheets when the flap joint is em- ployed, care must be taken to avoid complicated intersections. This is best accomplished b}- making the flaps on the longitud- inal and circumferential seams come on opposite sides of the plates. Where the flaps are both on the same side, they are sometimes let into each other. Rciiiforcevient of Plates. — This may often be done verj' readily by the use of angle and T iron. In Fig. i6i is shown X.1* it'jL>=ll-r«,S tfx Fig. 161. -}. ,1 Fig. 162. -^1 = 13 T 8 rf — Fig. 163. J.I J an internal angle iron, and in Fig. 162 an external, and in Fig. 163 a simple! iron. The proportions for angle iron given by Redtenbacher are as follows : h = height of angle arm, (i = thickness. h = 4.5 '' + I". For T iron h^ = the base := 8 d + 2", and the height of the rib = ){/ii. In practice a great variety of proportions are made to suit all possible cases, examples of which may be found in the illustrated catalogues of the mills where they are rolled. tected from corrosion by being incased in copper. Screw stay- ■ bolts are now often made of soft wrought iron or mild steel, but copper bolts are still preferred by many. Fig. 167. Fig. 16S. Fig. 169. jdiG. 170. Construction of Angles (Figs. 167-170). — Angle junctions in riveted work are made either by flanging the plate or by the use of angle iron. In Fig. 167 the flange is turned inward, and in Fig. 1 58 it is turned outward. In these cases h is made the same as for angle iron of the same thickness. Figs. 169 and 170 show the use of internal and external angle iron. Fig. 171. Consti-Hction of Solid Angles. — These are the most difficult forms of riveted work, and may be made in several manners, the most important being shown in the illustrations. In Fig. 171 the vertical angle is made as in Fig. 167, and the horizontal angles as in Fig. 169, sheet No. 2 being beveled under the angle iron. In Fig. 172 all three angles are made as in' Fig. 169, the S^ U Fig. 173. Fig. 174. vertical angle iron being cut and bent over the horizontal, In Fig. 173 the angles are all made as in Fig. 169, but the angle irons are welded together at their junction. This makes an ex- cellent piece of work, but is difficult and expensive, and re- quires iirm support for the work, and is only applicable for important constructious. In Fig. 174 the vertical angle is made like Fig. 169, while the lower joint is made as in Fig. 170, mak- ing simple and substantial corner. Fig. 164. Fig. 165. Fig. 166. The strengthening of parallel plates which are near together is best done by the use of staybolts. In Figs. 164 and 165 is shown 3 copper staybolt after and before riveting, this form being used in locomotive fire boxes and marine boilers. The central hole affords a warning of the corrosion or weakening of the bolt by the escape of steam. It is best to remove the screw thread from the projecting portions before riveting over the heads. Fig. 166 shows a form of iron staybolt for the same purpose. The short piece of tube between the plates prevents them from being drawn out of shape while riveting, and the opening permits a free circulation of water. The bolt is pro- THE CONSTRUCTOR. 45 CHAPTER II. HOOPING. I 62. Hooping by Shrinkage. The use of hoops or bands is a very e£&cieiit method of unit- ing some combinations of machine elements, and also for strengthening existing combinations. The hoops or bands are arranged so as to encircle the portions to be united, and caused to exert sufficient pressure upon them to create such friction between the surfaces as to prevent any relative motion. It fol- lows that the material in the band is subjected to tension while the parts which are held together are under compression. The bodies to be hooped are nearly always either cylindrical or conical in shape. The pressure required to secure the hoops maj' be obtained either by shrinkage, a method formerly used very extensively, or by cold pressure, a modification being described in the latter part of I 64. The elongation which is produced by elevating the tempera- ture to a red heat may be taken for steel and wrought iron at about Tj^ij, while to keep within the limits of elasticity the re- sistance to contraction should be, for Cast or wrought iron TiVo Cast steel ^k Hence the allowance for shrinkage to be made in boring for a cast iron hub to fit over an unyielding centre should not be greater than and is best made from to T3B0 Lo 1 S50, especi- ally if the centre is very heavy. The ring can then be fitted to its place when at a dull red heat. For wrought iron or steel, rings, such as wheel centres, such precautions are not so essen- tial, since these materials permit of a slight extension without injury (see ■ If the centre possesses but very slight yield- ing elasticity, there may be danger, however, that the contrac- tion due to excessive cold may overstrain the material. Fig. 175. When wrought iron bands are to be used to secure iron jour- nals to wooden shafting, as shown in Fig. 175, the end of the shaft is made slightly conical, so that the bands, being raised only to a dull red heat, may be driven on with the hammer. The rings may be forged tapering, but the 'taper may be also readily produced by Clerk's method by repeated heating and cooling.* The red hot ring is immersed in the cooling tub for i<— d;-- -- ^-: kei i N'fii 1 1 h I Fig. 176. about half its axial height. The rapid contraction of one por- tion of the ring deflects the warmer portion towards the centre, and by repeating the process the taper may be produced to almost any extent which may be required. The following experiments, made in the Royal Technical Academy, will serve to illustrate the process. The ring shown in Fig. 176 had the following original dimensions : n = s'/y^, S = //', D = 8)4". After the first immersion the contraction was /^ " second " third " fourth " fifth " sixth 5 SB" 7 "J5 1 6 ST After the last immersion the dimensions were found to be /? == 7jf at the upper edge, and at the lower edge = S/'j". A method of connecting two flat bars by shrinking on a hoop is shown in Fig. 177, and has been used at Seraing with good results. The hubs of gear wheels or revolving cylinders are advanta- geously strengthened by bands if they are cast in several parts, as in this way they are firmly united into a compact whole.* Fig. 177. §63. Cold Hooping. In the place of shrinking bands to their places, the more re- cent method of forcing them on cold has come into use for bauds of moderate size, such as for hubs of wheels, cranks and levers. In this case the ring and its seat are both made truly cylindrical, with merely a slight bevel for entrance, and then by means of a press forced together.f The difl'erence in diam- eter between the ring and hub is very small, and may be calcu- lated as described in ? 19. An investigation of the force required to push a ring on may be found desirable. The force which is necessary to press a cylindrical pin into a hub by continuous motion may be taken as nearly proportional to rate of progress, since it has to over- come the resistance of sliding friction between the surfaces. The pressure p, per unit of surface, is equal to the initial radial stress S], which exists upon the pin. If we make r the radius of the pin, / the length of the hole, y the co-efficient of friction, we have for a maximum value of the forcing pressure Q : Q= xrizlS^f {62)- Taking /= 0.2, as indicated by experience shown in the fol- lowing cases, we have 5Q__ 27rr/ P = S,: (63) For the tangential stress S.,, in the metal of the ring, we obtain from formula 37, I 19 : (64) And taking the thickness of the metal of the ring as (5, we get the value of p : I +■ i + - (65) + I This gives for the following ratios of thickness to radius, cor- responding numerical values : — = 0.50 0.55 0.60 0.65 0.70 0.75 p = 0.3S5 0.415 0.43S 0.463 0.4S6 0.508 — = o.So 0.S5 0.90 1. 00 1. 10 1.20 ' i p = 0.528 0.54S 0.566 0.600 0.630 0.65S '■ — = 1.30 1.40 1.50 1,60 1.70 i.So 1.90 2.00 r p = 0.682 0.704 0.724 0.744 0-759 °-774 0-787 0.800 The following table shows examples of the practice of many of the leading Continental railways. In the table, 2r = diam- eter, / = length, <5 = thickness of hub, Q = total forcing pressure ; also, \V. I. ^ wrought iron, C. I. = cast iron, S = steel, C. S. = cast steel, B. S. = Bessemer steel. * See Procedings of the Royal Society, March, X., 1864, p. 238. 1S73. Civilingenieur, Vol. * For an account of the strengthening of a piston head by shrinking ott bands, see the Berliner Verhandlungen, 1876, Sheet XVI. t Soxuetimes these surfaces are made slightly conical, such being the case in Nos. IS to 17 of the following examples. 46 THE CONSTRUCTOR. i 64. ExAMPi,ES OF Forced Connections. 7 8 9 10 II 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 DESCRIPTION. EASTERN RAILWAY OF PRUSSIA. Locomotive Driving and Coupled Wheels, Locomotive Trailing Wheels, Tender Wheels Car Wheels, with Spokes Steel Plate Wheels, for Cars, Steel Plate Wheels, for Locomotives, . . UPPER SIIs 5>s 5>s 7 7 7H 6 5% cl5 Ol 6 5% 5% 5% 7 5 'A (>A c 9 OTti aU s'A 6to8 6|-8 4to6 7'A 7A 5A 7A TA 7 6H 7A 7-h 6U 7i\ 7 614' 7A 6J-i 7% 8 /IS 6 7fs OTiJ 7'A &A 7% 7A 7A (>/z 7 S/s 7 6/2 loX SA 6J-7 7to8 7toS 7A 6U ,15 3 2A 2^ 2 2lV 3% 3A 3/s ■"16 2A 3^ 2A A'A 3X 2^ 314' 2fl % 2X 2 2tJ 4>s 2A 3/s 3% 3 3X 2i-3 3to4 -2} 3^ 3 2A Material- Hub. Axle W.I. W- I- W. I. W. I. cs. C-S. W- I- W. I. W. I. CS- W- I. W. L W. I. W. I. W.I. cs. W. L C.I. W. I. W. I. W. I. C. I. W. I. W. I. W. I. C. I. W. I. C. I. W. I. W. I. W.I. W. 1. W. I. W. I. W. I- W- I. W. I. W. I. W. I. W. I. W. L W- I. W.I. W. I. W. I. W. I- C.S- W- I- C.S. CS- C-S- C-S. cs. C-S. CS- C-S. C-S. CS. CS- cs. CS. W. L W.I. CS. cs. W. I. W- I- W. I. W. I. W. I. W. I- W- I- W- I. W. I. W. I. B-S- Q Pounds- 160,600 160,600 118,800 110,000 110,000 154,000 220,000 to 330,000 165,000 to 220,000 iio,oooto 132,000 110,000 to 132,000 132,000 to 154,000 165,000 to 176,000 143,000 to 154,000 88,000 to 110,000 110,000 to 132,000 1 10,000 to 132,000 1 76,000 to 198,000 137,940 247,588 247,588 198,000 136,840 i6S,o8o 198,000 165,000 to 193,600 165,000 193,600 II 0,000 to 165,000 90, 200 go, 200 85,800 90, 200 85,800 68,200 77,000 88,000 W.l.orS. 155,000 to 220,000 W.l.orS. 220,000 to 330,000 W.l.orS, iiOjOOOto 165,000 W.l.orS, W.I,.orS W.l.orS W. I- W- I. W- I. W.l.orS, W.l.orS, CS. 220,000 220,000 132,000 176,000 176,000 176,000 33,000 77,000 to 88,000 55,000 to 66,000 55,000 to 66,000 39,600 to 48,400 22,000 66,000 REMARKS. With Key. Data as given. No Key. Data furnished. No Key. Data furnished. No Key- Data furnished- No Key- Data furnished. No Key. Data furnished. With Key. Measured Dimensions. With Key. Measured Dimensions. No Key. Measured Dimensions. No Key. Measured Dimensions. No Key. Measured Dimensions. With Key. Data furnished. Without Key. Data furnished. Without Key. Data furnished. No Key. Measured Dimensions. No Key. Measured Dimensions. No Key. Measured Dimensions. With Key. With Key. With Key. No Key. No Key, No Key, No Key. No Key. No Key No Key, No Key. With Ke}'. With Key. With Key. With Key. With Key. With Key No Key. No Key. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measui-ed Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. Measured Dimensions. With Key. With Key. No Key. Measured. Measured. With Key. Measured- No Key. Measured. No Key. Measured. Data furnished. Data furnished. Data furnished. Data furnished. With Key. No Key. No Key. No Key. No Key. No Key. THE CONSTRUCTOR. 47 From example No. 12 we obtain in formula (63) the value 5 X 176,000 S, = \ = 5,336 lbs. 7-5X^X7 According to (65) ^ = 0.53, and substituting these values in (64) gives ^"2 = 10,679 ll^s. From example No. 10 we have : 5 X 132,000 hence a = 3.86" X 0.92 = 3.5s" The actual thickness of the hub -was 3-54"- The ring form is not the only form of construction -which may be used for joining members by forcing, since other forms may also be used. An example may be found in Erhardt's flange joint, Fig. 17S* In this case clamps of hardened steel ^1 = 5.125 X ^ X 7-3125 also p = 0.44, and hence S^ = 12,734. lbs. From example No. 37 we have : = 5,603 lbs. 9 — — '-'1 — ' -. 5 X 220,000 ; 6,970 lbs. 7-5 X - X 6.7 also p = 0.526, giving S^ = 13,250 lbs. From eyample No. 16, taking Q= 132,000 lbs., we get S^ = 4867 lbs.; /J =0.77; S.i = 6i2o lbs.; and in No. 17, we have i'j ^ ,6617 lbs.; p = 0.569, and52:= 11,629 lbs., neither of which are excessive. The force required to force a hub off an axle upon which it has been pressed, is not materially different from the force with which it was pushed on. The bore of such a hub may also be reduced when necessary by forcing rings upon it. Such rings, when used for car wheel hubs, are usually made of rectangular cross sections, the diameter ranging from 2"Xi", to i^'^X iX", etc. An inspection of the table will show that there is a tendency towards increasing pressures. For car wheels, where until quite recently, pressures of 60,000 to 90,000 pounds were used, we now find So,ooo to 110,000 pounds not infrequently; while for locomotive wheels, over 200,000 pounds is the rule. Midway between the methods of shrinkage, and of cold forc- ing comes the lesser used method of expansion by use of boiling water. This system secures a much more uniform action of the temperature than is practicable with a red heat, and has been used with excellent results upon the Russian railways for fitting tires to plate wheels. The tires are suspended by a crane, in a tank of water which is kept at the boiling temperature by a jet of steam (the allowance for expansion being a little less than -|-Jjf of an inch to the foot of diameter. An immersion of 10 to 15 minutes is required to obtain the desired expansion. Three workmen can in this manner fit 12 to 14 tires per day of eleven hours. This method may also be found applicable to the fitting of hubs. 65. Dimensions of Rings for Cold Forcing. Since the forms of the various hubs may be taken as cjdin- drical in nearly every case, the stress^niay be calculated by the formulaj already given. Itis, however, desirable to present these in such form that they may be used to determine the thickness of hub which, when forced on cold, shall resist a determinate force. In (62) instead of the radial stress 5'j, substitute the tangential stress S.,, giving = 2 - y'/y^Sj p, which combined with (65) gives : 4=Vs rl/S,+ Q In this, Q\b the maximum force w'hich the hub can oppose to turning, at the diameter of the fit. If we take the moment of the force tending to rotate the wheel as P R^ we must have Q ?^> PR' ^Y> """i^^ then be the factor of resistance against slipping in any such case. This mode of attachment is then only practicable when 2 tt 7'1/S.> > O. By choosing different values for 6*2, and Q, various thicknesses for the metal of the hub may be obtained. Example. The following data are taken from Borsig's Express Locomo- tive at the Vienna Exposition ; Two pairs of coupled driving wheels of 38.19" radius, without keys; bore of cjiinders 17 ; steam pressure 147 pounds ; crank radius J? = 10". If we suppose the entire force upon the i)iston to act upon a single wheel, we have : PJ?^ (17)2 X 07S54 X 147 X 10 = 33,366 X 10 The bore of the wheel is 7.72" hence r = 3.S6'' while / == 7-S7". P R 333,660 This gives 3.S6 ■ = 86,440 lbs. The moment 333,660 is that which the friction of the wheel upon the axle should be able to resist without slipping. Hence it follows that Q must neessarily be greater than 86,440. If we take a value of iQ = 154,000 lbs-, thus giving ample margin against slipping, and also use a wrought iron hub, making S2 = 7120 lbs., taking y"= 0.2 as before : /• 2 TT X 3-S6 X 7.87 X 0.2 X 7T20 + 154,000 2 IT X 3-86 X 7.87 X 0.2 X 71=0 — 154,000 Fig. 17S. are used to create the pressure. These clamps serve to press the light flanges together, and they may be forced on by use of a screw clamp or other suitable press. Tests of such joints under steam, pneumatic and hydraulic pressure have shown the joint to be tight and serviceable. The system of forced connections has grown into extensive use, and appears to be applicable to many forms of construc- tion, and it is to be hoped that the forcing press, for which the firm of Schaeffer & Budenberg have made suitable pressure gauges, may be found an indispensable tool in all large workshops. CHAPTER III. KE VING. 'i 66. Keyed Connections. The simplest form of keyed connection consists of three parts, viz. : the two parts to be connected, and the key itself. The key is made with a slight amount of bevel on both sides, or a greater angle on one side, according to the manner in which the connection is made. The trigonometrical tangent of this angle is called the draft of the key. In Figs. 179 and 180 are shown both forms of keys. For the latter form we will assume that both sides have the same angle. Let: o := the angle of draft, P^the force to be transmitted, Q = the driving force upon the key, normal to P, 0''= the opposing force, tending to drive out the key, y=/o-(j, the co-efiicient of friction between the surfaces of the three parts. For ke3-s with draft upon one side, we have : = P2tg(a-\-2 or say (69). If we make /;j=o.S d, h.^ = d, c!=: 0.5 d, we shall have good practical proportions. Iu Fig. 1S3 we have two wrought iron rods coupled by wrought iron keys. In this case a wrought iron sleeve is used, whose thickness ^ = 0.25 d. Fig. 1S4 shows a form similar to Fig. 1S2, except that the key passes below the boss, instead of going through it, while in Fig. 1S5 the key is let into the side of the rod. The pressure upon the base surface of the key in the case of Fig. I S2 may be taken as : P^ _ (0.7854 ^'-^«^) ^3 ' b d bd which gives / ^= 2.14 S^ (70) quite a high enough value, especially if we take, in Fig. 1S3, 0= 0.25 d. The pressure becomes yet higher for the method shown in Pig. 1S5, for which case the value of S^ shonld not be taken too great. If the connection is intended to be taken apart frequently, the value of p should not be allowed to be too great. This may be accomplished either by reducing the value of .Sj, or by providing an increased cross section of metal about the mortise for the key, or by extending the surface by means of cotters or gibs, as shown in Fig. 1S6. The key may then be made smaller than already given above. The forms of keyed connection shown are used for example in the rod connections of water wheels, and in similar cases. Fig. 1S6. Fig. 187. In Fig. 1S7 is shown a method of keying a foundation bolt. The gibs or cotters are used to increase the strength. Fol- lowing the calciilations of (; 12, the depth of the three pieces- may be made alike in the middle. The anchor plate in the foundation masourj' should be arranged so as to give access to a nut on the lower end of the bolt, and this can be tightened by hand until the bearing is thrown upon the key, and the driving iu of the latter binds all the parts firmly together. ?6S. LoNGiTUDiNAi^ Keys. Keys of this class are principally used to secure the hubs of wheels to their shafts or axles. For this purpose they may be considered as divided into three classes, as follows : Concave, or hollowed keys. Fig. 18S, 1 ; Flat Surface keys. Fig. 188, 2, 4, S, and Recessed keys, Fig. 18S, 3. The Concave key is only suitable for constructions involving small resistance, and acts merely bj' the friction due to the pressure which it causes. The flat surface key is capable of ^^ .&-«. Jzi-y. ^^^^ Fig. 1 88. resisting much greater force and vibration, and when used in the multiple manner shown in 4 and 5, it makes a secure and efficient fastening. The recessed key, shown in 3, affords a very secure method of fastening hubs to shafts to which they have been closely fitted, and is simply and readily made. Ke5'S of this kind are also used as an additional precautionary fasten- ing for hubs which have been forced on. In determining the dimensions of keys it will be found most convenient to use empirical methods, except iu cases of great vibration ; the following formulse will be found to cover the usual range of work. The material for the key is taken as steel, and distinction is also made between cases in which the hub is subjected merely to endlong pressure, and those where torsional stresses exist. The former may be called draft-keys, the latter torsion ke3'S. If we call the diameter of the shaft D, the breadth of the key S, and the middle depth of the key S-^^ we have : for Draft keys, 5= 0.24" ■\ ; 5, = 0.16" -| ^ D D for Torsion keys, S ^ 0.16" -|- ; 6'i = 0.16' -| — — The taper of such keys is made about xij (71) THE CONSTRUCTOR. 49 For the more commonly occurring diameters we have the following proportions : £> = !>' 2" 3" 4" 5" 6" 7* S" 9" 10" For Draft Keys : c a// •-> "8 Q" — 1// A'' _9_// S// a// T.¥ A// a// 4 X" 9 // For Torsion Keys. 3 // Til// J 9 // S// Z// 16 4 B diameter than 1 ' 4 -^ "^S // T_l_ tJL// ■'is ■'lu we mav make c 1// •-"l 4 C Z IS For shafts of less 5'= , S^-= If several keys are to be used, they maj' 3 5 be made the same dimensions as single keys. For hubs which have been forced on, and hence would be secure without any kc}', the dimensions for draft keys may be used. ?69. Edge Keys. When the pressure upon a key acts at right angles to the plane oi its height, the difference between the positive and negative direction of the forces is readily distinguishable. 1±-- Fig. 189. When the pressure acts as in Fig. 189, the combination is inse- cure, since the only binding action of the key is that due to the pressure, and consequ .*nt friction between the parts. If the base of the key is ro agh, and the inclined face smooth, the action of a force in the direction H' , tends to tighten the parts together. An application of this action is shown in the curved key of Keruaul, shown in Fig. 190. When the hub is rotated Fig. 193. Fig. 192. in the direction of the arrow, the action is the same as that of the force H' , in the preceding case, and the shaft is firmly grasped. A countersunk screw at a, is used to tighten the ke)', and a similar one at b, to loosen it. This principle will be dis- cussed later, under the subject of couplings. \ 70. Methods of Keying Screw Propei.lers. In securing the propellers of steamships the greatest care must be observed in the methods employed, and in their appli- cation. In Fig. 191 is shown Rennie's method of securing one of the blades of a Griffith's double bladed propeller. In this case a rectangular key is used, passing through a cjdindrical pin which is cast in one piece with the blade and which is in turn held firmly by the four smaller keys shown. These latter keys are held in their places by caps secured firmly by jam nuts. (See ^71.) The blade and hub are both of bronze. Example. In a propeller by Penn & Son, d = 15", ?»= 7>^", i = 2K". Fig. 192. This shows a method used by Maudslay, Sons and Field, Raveuhill & Hodgson, and others. Two rectangular Fig. 191. keys, passing through the hub of the propeller boss, and re- cessed into the metal of the shaft, act at the same time to receive the thrust of the screw and to prevent rotation upon the shaft. In this case the hub is made of bronze. Example. In the *' Lord Warden," the middle dianjeter Qfd= 19", /^52", /i= SJ^", d= 3^8" ; in the " Lord Clyde," d ^205^5", ^= y/^, h =^ 10", ^^3". Fig. 193. This shows a method of using two longitudinal keys. The hub is bored with a quick taper, and a heavj' bronze nut holds the hub upon the cone, while the longitudinal keys resist the action of torsion. Example. In the "Minotaur,' ter = ; pitch S, and making n _^ Introducing the pitch S, and making ns = d, we have : 4 ■ i (73) (74) In both equations the third member may be neglected.* The value of p should not be permitted to exceed 1440 lbs. and -J- : If « = 8, 12, we have, taking 5', as above, p := 3600 X I (i — T + tit) or about = 1000 lbs. In the consideration of this subject, the friction of a screw should not be neglected. Let: Q = the force acting at the mean diameter of a screw, normal to the plane of the axis ; 6' = the pitch angle of the thread at the mean diameter, /=: ig . The pitch is determined by the formula 5'= 0.24 y/d + 0.625 — 0.175, the result, as with Whit- worth's system, being so modified that the number of threads per inch shall be a whole number. The following table gives the adopted number of threads per inch for various diameters : rf=X l\ Vs A % t's % V^ 'A i = 20 18 16 14 13 12 II 10 9 d=i/s iX i^ iX 1% m i^ 2 T = ^ 7 6 6 5% 5 5 A% d=2'X ^% ni 3 Z% 3K 31^ 4 ^=.% 4 4 i% lY^ 3X 3 3 d = 5 5)4 5'A 5'/ 6 — = 2yi 2% 2^ 2^ 2X .J The Sellers System compares very favorably with the Whit- worth System, and notwithstanding the difference in profile, it gives almost the same depth of thread. The angle is very con- venient, and the simplicity of the profile is such that a suitable tool may easily be made and used in the shop. These facts explain the rapid introduction of the system in America. The progression of the pitch is also more uniform than in Whit- v^forth's System ; and the uncertainty about the thread of the Yz" screw, which was always a stumbling block in the original Whitworth Scale, is avoided. The values for ^-^" and ■^" are retained as in the Whitworth Scale of 1S57, and y\" is also pro- vided for, so that the requirements of the English system of measurements are fully met, up to 2" . § 77- Metricai, Screw Systems. Recognizing the advantages which have followed from the introduction of the Whitworth System, various attempts have been made to devise a system of screw-threads which shall be adapted to the metric system of measurements. The following fourteen systems have been suggested : Armengaud, Redtenbacher, Paris-Lyons-Mediterranean R. R., Northern Railwajr of France, J. F. Cail, the French Navy^ Bodmer, two systems proposed by Ducommun, of Mulhouse. Alsace ; the Engineering Society of Mulhouse, Reishauer & Bluntschli, of Zurich ; the Pfalz-Saarbriick Society of German Engineers, and two systems of Delisle. The formula; and tables given in the previous editions of this work have also been spoken of as systems, but they are not en- titled to any such position, as they were merely adaptations of the Whitworth system. The number of proposed systems may be taken as an indication of the difficulty of the task. Indeed, it is only by very carefully weighing the respective merits of the various plans, that it is possible to say which is the best. The following requirements should be kept in mind as essentials in considering any system : 1. The profile of the thread should he such as viay be readily made nnih requisite accuracy. In this respect Whitworth's system is deficient, and the profile of the Sellers thread is to be preferred. 2. The pitch should be given, so far as possible, directly by the formula^ without requiring any vtodif cation of its resi4lts. Both Sellers and Whit- worth are deficient in this point, since they both modify the results of their formula.* 3. The gradation of bolt diameters should be so disposed that fractions of niillitnetei'S sitould not occur in diameters, afid that their gradation should. conflict as little as possible with the decimal system. All three of these requirements should be attained within the limits of generally used sizes, and should at least extend to bolts of 80 mm. in diameter. The last three systems, viz. : the Pfalz-Saarbriick sj'stem and the two of Delisle, are the only ones which appear to have considered these points, and these we shall examine somewhat in detail. ?78. Metrical Screw Thread Systems. DEMStE /, Pf.\i,z-Saarbruck and DeuslE //. The following three diagrams show the gradation of pitch and diameter for the three systems, the ordinates representing the pitch being shown on five times the scale of the bolt diameters, and the values being also given for d and j in the annexed tables. In the first two cases the profile of thread is exactly the same as in Sellers' system, while in the third, the base angle is made 26° 34'. This has been chosen for the purpose of making the theoretical height of the triangle of the thread equal to s. The thread is flattened as in the Sellers system. All three of these systems are marked by simplicity and in- telligibility. These features have been attained by abandoning the idea of representing the relation of 5 to rf by a single equa- tion (such as that of a parabola), and using two or more equa- tions of straight lines. A noticeable irregularity exists in the Pfalz-Saarbriick system between the diameters of 26 and 28 mm., indicating that a somewhat finer pitch is used in proportion to the diameter below 26 mm. The second system of Delisle is rather simpler than the first ; there is also an important difference in the angle of thread, as will be seen subsequently. * Journal of the Franklin Institute, 1864, Vol. 47, p. 344. f Journal of the Franklin Institute, 1S65, Vol. 49, p. 53. *In the old Whitworth scale all 33 values were modified ; in the Sellers sys- tem this is done with 31 out of 34 sizes. THE CONSTRUCTOR. S3 Deusi,e /. Fig. 20S. d= 4 5 6 j 7 8 jio 12 !i4 16 18 20 i = o.S i.o 1.2 1.4 16 1.8 2.0 2.2 2.4 2.6 2.8 d = 24 28 32 36 40 J48 56 J64 {72 80 -f =•- 3-2 3 6 4-044 4.8 5-2 5-6 6.0 6.4 1 11 1 1 1 1 6.8 iI5G7S 1012141G1S20 24 28 32 3G <10 -JS Fig. 2o8. ao liO 61 Diameter and pitch both in millimeters. For any interpolated diameter the next lesser 72 i't, ordinate is to be taken, as for example d = 60. Pf.\i.z-Saarbruck System. Fig. 209. %(d-I)j.J d = 6 7 8 10 12 i4Ji6 18 20 22 24 s = 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 d = 26 3-2 28 32 36 40 48 56 J64 J72 80 s = 3 6 4.0 4.4 4.8 5.4 6.0 6.6 7.2 7.8 No interpolation to be made. C7S 101211101S20232a2G25 32 36 40 Fig. 209. DelisIvE //. Fig. 210. d = 6 8 jio 12 14 16 18 20 24 s = 1.0 1.2,1.4 1.6 1.8 2.0 2.2:2.4' 2.8 d = 28 32 36 40 48 56 64 72 So s = 3.2 3.6 4.0 4.44-85.25.6 6.o'6.4 « 6 8 1012U1G1820 24 2S 3i 30 40 Fig. 2IO. Ot) lil) Gl In all three systems the superficial pressure is quite satisfac- ~tor3f. According to formula (74), taking 6' = 3600 we obtain for lvalues of/ — Delisle I S600 to 11,500 lbs. Pfalz-Saarbriick . . 8600 to 11,000 " Delisle II ... . 7600 to 10,000 " ?79- New Systems. A thorough investigation of the proposed systems of the Ger- inati Society of Engineers failed to produce any definite results, and the whole subject of a metrical screw thread system is still unsettled. For this reason it has been thought advisable to of- fer a further discussion of the problem.- It might seem a shorter plan to adopt some one of the three preceding systems, yet they all seem capable of improvement. *This is especially necessary for use in technical instruction, which will ■afford the surest method of introducing a metric screw thread system into practical use. The advocates of the Whitworth sj-stem urge the desirability ■of an international standard, in view of the widespread use of the American system, which is indeed already in use to some extent in Ger- many. In this case the conflict between the two systems of measurement "has been met by proposing to take for any dimension in English units the next higher dimension in millimetres. Such a system would be impractic- able for educational purposes and would lead to many errors in actual practice. It also seems only to be practicable for the old Whitworth scale, and for the new scale, with its close divisions, its application would be im- possible. A comparison between the preceding diagrams will show that a close adherence to the Whitworth system would result in a complication of •dimensions -which would be most undesirable. For any interpolated meter the next greater ordinate is to be taken, as for example d = do.* The subject will bear further investigation in two main points one being the gradation of diameters and the other the profile of thread. The actual diameters and their gradation are of more practical importance than the gradation of threads. This is shown by the fact that the Whitworth profile has long been in use with the bolt diameters taken in Prussian inches, and more recently with dimensions in millimetres with Whitworth profile. One of the first requisites of such a series is that the diameters should follow the decimal divisions (see the third condition of § 77). This point is not met by the preceding sys- tems, since they lack the natural divisions 30, 50, 60 and 70. The removal of this objection introduces a new difficulty, but not an insuperable one. The critical feature of the screw thread system is really the relation which the diameter bears to the profile. A thread should not be said to be coarse or fiue, implying the ratio s : d, but rather should the depth of thread be considered, or the rati oi : d. This can best be illustrated by an example : If we select two equal sizes from the systems Delisle I and 1 1, we shall find that for the same pitch the threads are not alike. l{ d^ 60 mm. we shall have (see the dotted lines in Fig. 208 and 210) in both cases 5=: 5.6, hence the angle of thread is the same. The working depth /, however, is : in 1 : /^^io ^0.65 .5 = 3.64 mm. in II : if ^ 3^/0 ^ 0.75 i = 4.20 mm. ■^In both his systems Delisle has provided for the interpolation of inter- mediate diameters, but these have been omitted from the diagrams and tables to avoid obscurity. 54 THE CONSTRUCTOR. This gives for the diameter of the bolt at the bottom of the thread in I : d^= 57.70 — cross section 21S2 sq. mm. in II : fl'i= 51.60— " " 2091 " " ■which shows a difference in resistance of about 5 per cent, be- tween the two bolts, the second having the coarser thread. We see here that a choice of the relation of jt to d affects the pro- file of thread, and it is this which led Delisle to suggest two systems. Whether the angle of 53° S' is preferable to the Sellers angle of 60° is uncertain. Among the preceding systems may be noted two for the latter, five for the former, and three for ^ d still smaller angles ; and if the choice be given, it seems rather better to go below the Whitworth angle of 55° than above it. We prefer the angle as shown in Fig. 211 : 2 /3=S3° 8' or to =s \ (77) (7S) (79) and hence For sizes of d from 4 to 40 mm. the pitch may be .s^o.4 + o.i rf .... and for sizes of d, from 40 to So mm. and over* s=2 -\- 0.06 d .... with the following series of diameters : 4 5 6 7 S 12 14 16 iS 20 22 24 36 40 . 45 . 50 Formula (78) is the same as in Delisle II, from 6 to 40 mm. Interpolation for intermediate diameters seems unnecessary ; ,.(--tn-*i 9 10 26 28 30 60 70 80 ' ^ V . + .^ ll A .*t> 1 1^- — -*f -> < ■""""*'" ~ V t< Cl=.30... ; Fig. 212. should it be done, however, the formula should not be departed from, since the values in the second and third groups above ■will give round numbers, and offer no difiiculty for their pro- duction on the screw cutting lathe. If it is still desired to use the angle of 60°, and yet retain the other proportions, we may take for rf^4 to 8, 5' =0.2 d (as in Delisle I) 1 forrf ^ 8 to 40, s' ^0.8 \- o.\ d (as in Delisle I I . . (So) for a' = 4o to 80, .r'^ 1.5 4- o-oS d J in which arrangements the sizes 30, 45, 50, 60, 70 remain in the series, which may also be extended above 80 mm. The two plans may be compared to Fig. 212, in which the formulae are re- spectively applied to a diameter of 80 mm. The radii to the bottom of thread r\ and i\, are almost identical, as are also working depths, although the profiles differ, as shown by the triangles ABC and D E F. Instead of numbering the sizes arbitrarily, it seems preferable to use the bolt diameter for the number. Screw No. 20 would then stand for rf = 20 mm., No. 4 for rf ^4 mm. Any establishment could omit numbers not desired without impairing the system, while for fine work_ smaller numbers could readily be added. ?So. Nuts, Washers and Bolt Heads. The thickness of metal in a nut bears a close relation both to the depth of thread /, and to the pitch j. It is desirable that the formula to be used should give the dimensions readily ia order to avoid the necessity of approximating. Fig. 213. For the diameter D, of the inscribed circle of the hexagon we may take for finished nuts : Z> = .o4-l- rf-f 0.5.? ...... (Si) The maximum pressure upon the base of the nut in this case (for d = 3'') = about 2400 lbs. per square inch. Unfinished nuts are made somewhat heavier, and lor them we have A = o.i4"+'''+5-J J ..» Fig. 214. Fig. 215. Fig. 216. The use of the washer insures a better bearing for the nut in case the surface is not true. Its dimensions may be taken as diameter = U=d -\- 10s thickness ^ u = s 4 (83) *For sizes over So mm. -we have not yet established relations. If we take d= 150 mm. which is about as high as Whitworth or Sellers have gone, s^^ II mm., which seems a good proportion. See § 87. Bolt heads are often made square, but are preferable hexagon- al, and for them we may take D and Z)„ the same as for nuts^ and the height /« = 0.7 fl^. Fig. 213. For finished nuts the upper surface may be finished with a bevel of a frustum of a cone whose base =Z?, and a base angle of 30°, Fig. 214, or as a portion of a sphere with a radius of f D, Fig. 215, while unfinished nuts have the corners beveled off above and below, as shown in Fig. 216. THE CONSTRUCTOR. SS 0,064 0/Mrt Fig. 217. Bolts and Nuts. (Metric system). Bolt Dia. Pitch. Depth of Thread. Bottom Dia. of Bolt. dx Nut. Washer. Bolt Head Load. p mm. s t D A U u n kilos. 4 0.8 0.60 2.80 9 12 I 3 16 5 0.9 0.68 3.65 10.5 — 14 I 3 5 27 6 I.O 075 450 12 — 16 I 4 41 7 I.I 083 5-35 13-5 — is: 15 5 57 8 1.2 0.90 6.20 15 — 20 1-5 6 77 9 1-3 0.98 7-05 16.5 — 22 1-5 6 99 10 1.4 1.05 7.90 18 21 24 1-5 7 125 12 1.6 1.20 9.60 21 24 28 2 S 184 14 1.8 1-35 11.30 24 27 32 2 10 255 16 2.0 1.50 13.00 27 ,30 36 2 II Z7,'i 18 2.2 1.65 14.70 .30 33 40 3 13 432 20 2.4 1.80 16.40 33 36 44 3 14 .53S 22 2.6 1-95 18.10 36 39 48 3 IS 655 24 2.8 2.10 19.80 39 42 52 3 17 784 26 3-0 2.25 21.50 42 45 56 4 18 841 28 3-2 2.40 23.20 45 48 60 4 20 1076 ,S0 3.4 2.55 24.90 48 51 64 4 21 1240 32 3.6 2.70 26.60 51 54 68 4 22 1415 36 4.0 3.00 30.00 57 60 76 5 25 1800 40 4.4 3-30 3340 63 66 84 5 28 2231 45 4-7 3-53 37-95 70 73 92 6 32 28S0 50 5-0 3-75 42.50 76 79 I'X) 6 35 3613 60 ,S.6 4.20 51.60 89 92 n6 7 42 5325 70 6.2 465 60.70 102 105 132 7 49 7369 80 6.8 5.10 69.80 "5 118 148 8 56 9744 J Si. Table and Proportional Scale for Metrical Bolts and Nuts.-' The preceding table contains a summary of the preceding discussion, aud Fig. 217 is a dia- gram in which the relations of the parts are shown graphically. The value for j- is shown on a five-fold scale. The dotted line gives the value for s' of formula (80) . The diagram Fig. 217 shows the pitch of thread and the pressure upon a unit of area, for the dimensions of nuts and bolt heads for the preceding metric screw thread system for diam- eters from 4 to So mm. The pitch is shown five times full scale (line E) and ten times full scale (line F)\ the bolt diameter in its actual size (Hue/^), all measured from the base line A. The line i? is i mm. from D, and Cis 4 mm. from D while the dis- tance between A and G is 0.7 d. The various details may be summed up as follows : Between A and E = the fivefold pitch, " E and B = dia. of finished nut, " E and C ^ dia. of rough nut, " i^and D = dia. of washer, " A and G = height of bolt head. The tangent of the pitch angle ranges from 0.064 to 0.047, and the pressure per sq. mm. on the thread, from 0.46 to 0.67 kilogrammes. ?82. Weight of Round Iron. The weights in the following table are given by the Formula G = 2.6i'j d', the bars being one foot long and d = diameter in inches. For cast iron, multiply the values in the taole by o 93 and for bronze by 1.092. A hexagonal rod whose inscribed diameter = d is 1. 103 time the weight of a bar of wrought iron of the same diameter. Weight of Wrought Iron Rods. One Foot Long. I .■63 • 255 .368 .500 •654 .826 1.02 1.23 1-47 1.72 2.0 2.29 2.61 2.94 3.31 i>4 lA '^ iH lit IlTT 2 3-68 4.09 4-50 4-94 536 5.89 6.39 6.91 7-43 8.01 8.57 9.20 9-79 10.47 11.02 G 2^ 2tV 2^ 2tV 2H 2M ol 3 2tt 3 11.82 12.50 13-25 13-95 14.76 15-54 16.36 17.14 1S.03 19. 1 1 19.79 2061 21-63 22.52 23-56 ?S3. Special Forms op Bolts. Instead of being made with square or hexagon heads, bolts are sometimes fitted with special heads, instances of which are shown in Figs. 21S to 222 ; the last being countersunk. These are all furnished with means to prevent the bolt from turning when the nut is operated. *This table has been kept in the metric system for obvious rea.=:ons. Trans. 56 THE CONSTRUCTOR. In Fig. ?23 is shown an anchor bolt witli cast iron plate for brickwork the bolt being inserted from above and locked by being turneO. 90°. The area of the anchor plate should in no case be ) e jS 'ian 100 d{- In Fitr 224 is shown a form of anchor bolt for masonrj' with a cast ;:".~"i washer, secured by a key. The washer should be not less than 25 d^ in area. Such plates are often made of wrought iron. Fig. 21S. Fig. 219. Fig. 220. Fig. 221. Fig. 222. In Figs. 225 and 226 are shown bolts secviredby cross ke3'S and side keys. In these two figures the nuts are shown iu different positions, the latter being the more convenient to use the pro- portions shown in Figs. 214 to 216. Figs. 227 and 22S are forms C",6a -ZJ^- Fig. 223. ^vB^ Fig. 224. of stud bolts. Fig. 229 is a cap screw. For small work these cap screws are often made with cylindrical heads with slots for use with a screw-driver. ?S4. Wrenches. A wrench is a lever adapted to tighten and loosen nuts and bolt heads. The simple wrench, shown in Fig. 230, in two forms, consists of a flat or round handle fitted to the shape of the nut, the dimensions being based upon the unit D, which is the diameter of the nut as given in formula (Si). The double wrench, Fig. 231, is arranged to receive nuts of different sizes at the opposite ends of the handle. If the ends are inclined so as Fig. 230. to bring the corners of the hexagon at 15° and 45° with the axis of the handle the wrench will be able to operate in con- tracted spaces by j'j revolution of the nut.* Fig. 231. §85. Not Locks. For bolts made according to the preceding proportions, the angles of pitch are not steep enough to allow the pressure in the direction of the axis of the bolt to overcome the resistance of friction and cause backward rotation. If, however, there is much vibration, lost motion niaj- appear and gradually cause the connection to work loose. This is true of foundation bolts as well as of those in moving parts of machinery and in loco- motive and marine engines. For these and similar cases it is necessary to have some method of securing the bolt or nut from coming loose, and a variety of such nut locks are here shown. Fig. 232. Fig. 233. Fig. 234. One of the oldest and most useful forms is the jam nut. Fig. 232. Both nuts should be truly faced so that they will bear fairly upon each other. The thin nut is frequently placed under the thicker one, but this is immaterial since a nut of a thickness of 0.45 to 0.4^ is as strong as the bolt thread. The security obtained b}' the use of the jam nut is not ver}' great, and the form with right and left hand thread, as shown in Fig. 244, is to be preferred when greater securitj' is essential. In Fig. 233 is shown a split pin, often used in connection with a jam nut. Fig. 234 shows an arrangement with a key upon the nut, making a very convenient and secvire combination. In the three preceding cases the action is such as to tighten the nut upon the thread. The three following methods are made to hold b}' fastening the nut or bolt, or both, to the parts which they are intended to hold together. Fig. 235 is used in the spring hangers of Borsig's locomotives. Fig. 2j6 on an oil cup lid, and Fig. 237 on a set screw for a connecting rod end, arranged to lock at any 1-12 part of a turn. In the following methods the nut is held from turning by be- ing locked to one of the stationary pieces, the bolt itself being secured in a similar manner. The form shown in Fig. 23S is used for bearing cap bolts, the support at the middle of the * This idea is due to Proell. THE CONSTRUCTOR. 57 split pin keeping it from bending. The method shown in Fig. 239 is used for the bolts in a steam piston, while that in Fig. 240 is for a bearing cap. The latter form is arranged by means of the sever notches, to lock at every y'y of a turn, while the other two require Y(, of a turn between successive positions. ElG. 235. Fig. 236. Fig. 237. Fig. 241 shows a device for securing the nuts of stufBng box bolts as applied to locomotive engines. The ratchet wheels are attached to the nuts,. and similar notched nuts may be used to advantage in many places. Fig. 238. Fig. 239, A method of securing the bolts for locomotive springs, used by Borsig, is shown in Fig. 242. The tension of the spring keeps the bolt from turning, and the cap which secures the nut is fitted to the end of the bolt as shown ; this locks for every Yf, of a turn. Fig. 243, shows a nut arranged to be locked by a set screw. This method, used by Penn, is a very useful form Fig. 241. for bearings, spring hangers, and other situations, since it per- mits any fraction of a turn to be made. The nut, in this case, should be a little thicker than usual in order that the lower cylindrical portion may not be too weak. The diameter Z),, is in this case taken from formula (82). The small set screw should be made of steel and hardened. This form of nut lock is especially useful on marine engines. Fig. 242. Fig. 243. A different class of nut locks depends for its action upon the introduction of an elastic resistance between the bolt and the nut.* The elastic washer of Pagel and similar devices have found many applications. Parsons' bolts belong to this class.f * See l^udewig- Nut Locking Devices. Bavarian Industrial and Technical Journal, iSyo, pp. 17, 144, 283 ; also Journal of the Society of German Engineers. t Engineer, July, 1S67, p. 16 ; Nov., p. 391 ; Engineering, 1S67, Nov., p. 411 ; Railroad Journal, 1S6S, pp. 77, 117. In this form the body of the bolt is fluted, so that the cross section is reduced to about the same area as that of the bolt at the base of the thread. This increases the elasticity of the bolt and enables the nut to be tightened so that it is much less likely to come loose. Fig. 244 shows a modification of this form used by Gerber for bridge connections. The security is still further increased by the use of a left hand jam nut. Instead of being fluted, the body of the bolt may be flattened on four sides, or the reduction of area may be obtained by drilling a hole into the bolt from the head to the beginning of the thread. Fig. 244. Fig. 245. One of the most important instances of screw fastenings may be found in the construction of built-up screw propellers, in which the blades of the scre%v are bolted fast to the hub, a con- nection requiring the greatest strength and security. Fig. 245 shows the base of such a propeller blade, from the same example as shown in Fig. 192. The flange of the blade issecur- FiG. 246. ed to the hub by sixteen cap-screws. Four set screws serve to provide a small adjustment of the blade within the range of motion of the oval bolt holes. All of the cap screws are secured. Fig. 246 shows the method adopted by Penn. The bolts, which in the case of the Minotaur are 3J4" diameter, have a common ring washer under the heads. When the bolts have been screwed up as tightly as pos- sible, a ratchet washer with hexagonal hole is slipped over each bolt head. These ratchet washers are prevented from turning by the introduction of small locking pieces which are bolted fast to the large ring washer, being held down by the thin nuts shown. The ratchet washers have 11 teeth, and hence each bolt may be locked at -^^ P^rt of a turn. Fig. 247 shows a method by Maudslay. Here each pair of bolts is held by a flat key which permits looking at j'j part of a revolution. 58 THE CONSTRUCTOR. A continuous washer ring is not used with this method, but one washer is put under each pair of bolt heads, to which the lock- ing key is bolted. Another method by Maudslay is shown in Fig. 248. A double washer is placed under two adjacent bolt Fig. 247 Fig. 24S. heads, and each bolt is locked by a small block held against one of the faces of the bolt head by a small bolt. Three bolt holes situated 40° apart are tapped in the washer for each block, thus giving an adjustment of xV of 3. turn. The method by Penn gives the best opportunity for adjustment. Speciai, Forms of Screw Threads. Screw threads of siiuare or trapezoidal section may be used for bolts, but in their use it is desirable to use a deeper nut in order to secure a sufficient number of threads in the nut to keep the pressure per square inch on the thread within the pre- scribed limits. Trapezoidal threads are well suited for bolts, since the relation between 5 and d permits the use of the same proportions as those given for V threads in Fig. 211. In fact the thread in Fig. 250 may be given the same proportions as that in Fig. 211, for depth t, and pitch i, making the angles respectively equal to 0° on one side and 45° on the other. These forms of screw-threads are principally used for screw- presses and for similar uses. Fig. 24.9. For such screws the diameter (/,, at Fig. 250. the bottom of thread, If, however. is generally determined from formula (72). If, however, it is desired to make the diameter d^ a niimimum, we must consider the pressure to act only on one side of the thread in the nut and then take the pressure per square inch at double the previ- ous allowance, or 1 = 7110 lbs. We then have, rf, = 0.0134 v''/^) ,_ , Passes,/; I («^> The depth of thread, both for square and trapezoidal threads, is. and for square threads- 'L 5 and for trapezoidal threads- 8 4 — ^ i— '^' (85) Formula (84) is applicable to screws of locomotive springs, since in this case the conditions are well complied with. In order that the nut may not wear or grind out, the working pressure on the threads should not exceed say 700 lbs. per square inch. These conditions will obtain, according to (73), when the number of threads //, in the cast iron or bronze nut is not less than If / = = 0.0014 -d, we have 0.0003554(1-34) n = 0.00245 -5'= 0.00312 dl (86) (Sy) The depth of thread, from (S5) = rf = 3.9i2", or about 3- = 0.392", which gives Example. For a pressure of 55.000 lbs., we have, under the preceding for- mulEe, from (84) the diameter at the bottom of the thread «',= 0.0134 v^/'= 0.0134 X 234.5 = 3.14" S il// 16 ■ From (87) we have, makings ^ 7710 lbs , themimimum num- ber of threads in the nut n = 00245 ■5'= 17.4 which gives for the height of the nut for square thread h = ns = I7.4X.7S5^ 13.65", while for trapezoidal thread /; = 17.4 X .523 = 9. i". In many cases the diameter of such screws is made greater than the normal diameter indicated in the preceding discussion ! i r Fig. 251. Fig. 252 for the given load. Such screws may be called enlarged.screws, as compared with the normal dimensions as previously deter- mined. For such screws the same cross section of thread and the same height of nut may be given as for the normal screw of the same load, in which case the wear will practically be the same for both examples. Enlarged screws are frequently used for presses, where the diameter must be made greater than indi- cated by formula (84) in order to resist bending stresses. I 87. , Screw Connections, Fi.ange Joints. In screwed connections a distinction may be made as to whether the force acts parallel to the direction of the axis, or at right angles to it. The latter condition, which produces shearing stresses, is shown in the examples given in Figs. 251, 252 and 253. If we take d, as the diameter of the rod through which the force acts, we may call d', the bolt diameter, and 253. Fig. 254. then determine their relation for various cases. In Fig. 251, d'^d ; in Fig. 252, d'^\.\ d ; in Fig. 253, d'^^=d ; the increased diameter for Fig. 252, being given because it is possible in that case for the load to act so unequally that the greater portion may pass through one of the rods. Fig 254 shows a tumbuckle THE CONSTRUCTOR. 59 ■with right and left hand thread. In this it is desirable to make the nut somewhat deeper than d, as shown. A form of junc- tion piece for a point where four members meet is shown in I'ig- 255. Such examples as the preceding are of frequent oc- currence in bridge and roof construction.* Fig. 255. Fig. 257. Bolt connections which bring shearing stresses upon the bolts are of frequent occurrence in bridges built with pin-con- nections, the general method in use in America. These designs exhibit very fully the substitution of bolt or pin-connections for riveting, and the method has been carried to great perfection. Some examples are here given. Figs. 256 and 257 show an in- tersection of several members of the bridge over the Ohio, at Cincinnati. The top chord and the posts are double, and are tys i lyi Fig. 258. made of plate T and angle iron. The diagonal rods and braces to resist the action of the wind are connected to the upper chord by means of a bolt passing entirely through the beams and threaded at both ends. The nut on the left end is in the form of a fork to receive the ends of the braces, while the right hand end is fitted with a thin octagonal nut. The ends of the braces are held by a bolt passing through the fork, with a nut at each end. The pins are carefully turned and closely fitted ;t after years of service they show no signs of looseness. J The proportions are such that stress on the bolts does not exceed * other good exaiuples of similar work in roof construction may be found in E. Brandt's "Iron Constructions," Berlin, Ernst and Korn, 1S71, 2d Edition. t It is well known that variations in temperature during the borinar of the holes for the pins in the eye bars may make suiificient difference to mater- ially affect the fit. This has been overcome by the use of a double boring machine which the author saw at work in the notable bridg-e works at Phoenixville. whereby both ends are bored simultaneously, the distance being gauged by a wrought iron jig bar, which varied in length to the same extent as the eye-hars themselves. X See H. Fontaire. " llndustrie des Etats Unis," Paris.Baudrj', 187S. Rol- ler, Highway Bridge's New York, Vv'iley, 1S7S. 15,000. lbs, in most cases not more than 10,000 to 12,000 lbs. The e connection of the posts to the chords (in the illustration the riv- ets are omitted) is both simple and strong. The posts are provided with cast iron ends, which are fitted with square projections en- tering into the tops of the posts ; in these capitals are wrought iron dowel pins which pass through the lower angle iron and lower plates of the top chord. The diameter d, of the main bolts varies from 4 to 5;^ or 6 inches or even heavier, according to the load. Their dimensions are based upon as bearing stress of 8000 lbs., while the diagonal braces and the lower chord are proportioned upon a tensile stress of 10,000 lbs. (a ratio of o.S, see I 5). The compressive stress in the top chord is about 8,500 lbs., and in the posts, owing to the bending action, only about 5000 lbs. Fig. 25S shows an intersection on the lower chord of the Niagara railway bridge (9 spans over a total width of stream of about 1900 feet). In this case the posts and top chord are made of the ingenious Phcenix column of quadrant iron. The illustration especially shows the method by which the cross beams are connected to the longitudinal members. In this case the stress in the body of the screw bolts is about 8000 lbs., rather more than given for press screws in \ 86. A cast iron base, through which the large pin bolt passes receives the thrust of the post, and to it the cross beams of I shape are bolted. On these cross beams are wooden stringers to which the roadway is secured. It will be noticed that these examples of bolt work far ex- ceed the limit of size set by the Society of German Engineers for bolt dimensions, viz., 80 mm. or 3fg". Should such sizes be necessary the formulae in \ 79 should be reconsidered. Pig. 259. Fig 260. F7G. 261. Fig. 262. Fig, 263. Fig. 264. In uniting the various parts of iron constructions, flange joints are very frequently used. These are made in a great variety of forms for various conditions. The following figures show some examples of comer junctions with flanges. Fig. 259 shows three external flanges, with a dished base. Fig. 260, also three external flanges, with an external plinth on the base. Fig. 261 shows one external flange, and two which are half external and half internal. Fig. 262 has three half external flanges and a base as in Fig. 260. Fig. 263 has also three half external flanges, and Fig. 264 two external and one half- external flange. The last three examples produce a more pleasing external appearance than the preceding forms. If the form shown in Fig, 262 is used, with the flanges all turned inward, the bolts cannot be unscrewed from without. Proportions for flange joints are shown in Fig. 265, the bolt diameter d, being obtained from the thickness of metal ^. The distance between bolts is usually lyi to 3 /), i? being the width of the nut across the flat dimension. The width of flange is given in the illustration for metric sizes = 10 mm. + 2.8 & = Yi" + 2.Z6. _ If the flanges are finished on the planing machine, a ledge is left for finishing, as shown on the left of Fig. 265, in order that a' fair bearing may be secured. Flange joints which are to be bolted together without finishing are made as shown in Fig. 266, with a gasket of some form of elastic packing. Such flanges are sometimes made for vessels with very thin walls, and on the left of Fig. 266 is shown the method of assembling a cylindrical vessel, such as a water tank. The base has internal flanges for the bottom pieces, with an external flange for the connection to the body. By turning the flanges of the bottom inward a flat exterior base is obtained, well adapted to sustain the load of the water. The walls are very light, (5 = only about yi" , the bolts are Y%" diameter, and their distance from centre to centre, in the base, 13.5 d. and in the vertical joints of the walls 15 d, and in the circumferential joints 20 d. 6o IHE CONSTRUCTOR. i 88. Uni Stee. 75 14,220 0.17^/^ 0.0244 /? Vp 0.019 /? x/> Interinitteni Pressure. s = / _ d '~ d = 0.0273 Wrought Iron. 150 7000 o.o8v'i Steel. 150 11,840 o.iov';; sf' Vp 0.02 Y/4^/? If n > 150, the ratio of I : d, is first approximatad and the value substituted in the last formulas of the table. 62 THE CONSTRUCTOR. For hollow journals the following proportions may be adopted. Let (/(, = the external and d-^ the internal diameter of the equivalent solid journal, i/' d we have : * (94 the length of both solid and hollow journals being the same. If, however, the ratio of diameter to length is to be the same then d„ ^ I d f i> (95) from which the following series is obtained. rfi : rf(, = V ^ 0.4 0.5 0.6 0.7 0.75 0.8 ^ I :v/r -t I* = I.OI 1.05 1.06 1. 10 1. 14 1. 19 I.21 i.;,o .■^i = I.OI 1.03 I.Ob 1. 15 In both cases there exists a smaller superficial pressure for the hollow journal than for the solid one. A common ratio of internal to external diameter is 0.6, and such journals were fre- quently used in cast iron work and are again being used in con- nection with hollow steel shafting and axles. Bronze boxes or their substitutes, such as white metal or other combinations, belong more especially to the subject of bearings (§ 96), and their use permits a higher superficial pres- sure without creating an excessive increase in the coefficient of friction. For moderate speeds, boxes of cast iron give results which are as satisfactory as can be obtained with bronze. This is especially the case with machines which are actuated by hand. For heavier or continuous service cast iron boxes are only suit- able when the pressure is not great, and examples of such bearings will be given in a later chapter. Bearings of wood may be operated satisfactorily at a pressure double that which is used with bronze, if the journal runs in water, or is kept wet. For heavy mill shafting making from 60 to So revolutions per minute, wooden bearings lubricated with grease are often used. For mill spindles, boxes with bearings of willow wood are sometimes used with good results. In this case the speed some- times exceeds 100 revolutions per minute, but the pressures are light. § 91- ExAMPi^Es AND Tables of Journai^s. In the following tables are collected the results of the for- mula (93) in which the number of revolutions of the journal is assumed to be not greater than 150. 1. Example, a water wheel weighing 66,000 pounds carries a load of 212 cubic feet of water. The axis of the wheel is of cast iron, and the load is equally distributed between the two journals, giving -a load upon each journal of 33,000 4- 6605 = 39,605 lbs. The nearest value to this in the table is 40,05s lbs., which would give a diameter of .S^ inches, and a length of 12^ inches. 2. Example. A wrought iron shaft for a similar load, but subjected to alternating action, should have, according to the table, a diameter of about 57-^", and the same length. If in cast steel, with alternate action, diameter should be about 4^ inches, and length of 4.75 X i-s = 6.175". 3- Example.* The centres of the walking beam of the water engine at Eleyberg in Belgium each bear a load of 309.210 lbs. The journals are hol- low, with a ratio of external to internal diameter of 0.5. We have from (93) and (94) do = 1.02 X 0.043 v 309,210 ^ 24. 3S" and a length lo = 24.3S X 1.5 = 36.57" which gives a pressure of about 350 pounds per square inch of projected area. The actual dimensions of these journals are o'o^ig^^'", l^=^\%" , which gives a stress at the base of the journal of a little over 4000 lbs., but the actual bearingis only \S}i" long, which gives a pressure of nearly 1000 lbs. per square inch, whicli appears to be too great ; and in actual practice these journals are obliged to be kept cool with water. In actual practice there is very little uniformity in the pro- portions of journals. Sometimes the distinction between con- stant and alternate action of load is considered but often it is neglected. In the case of locomotive crank pins, for example, p is frequently as high as 1500 to 3000 pounds per square inch, and on the cross head pin, as high as 4500 pounds. On the other hand quite low values of p are sometimes found on the crank pins of marine engines, f In all cases careful lubrication is of the utmost importance. When the number of revolutions is very great the length of the journal should be made greater than is given above. Table of Journals. Value ot P. 1.25 T..50 1-75 2.00 2 25 2.50 ■75 3.00 325 3-So 4.00 425 4.50 4-75 5-o> 5-50 6.00 6.50 7.00 7-5° 8.00 8.50 9.00 9.50 10 00 10.50 It. 00 11.50 12.00 0.20 o 20 0.25 o 25 0.28 0.28 0.32 0.32 032 0.36 0.40 0.40 0.40 044 0.46 048 0.50 0.52 0.60 0.62 0.64 0.68 0.72 0.74 0.76 0.80 0.85 o.go 0.92 0.95 Direction of Load Constant. Wrt. Iron Castlro n / _ I ~d~^ Steel 1121 1752 2523 3434 4485 5677 7009 8481 10093 1184s 13738 17943 20256 22709 25303 28036 33924 40373 4738' 5495 63082 71773 81025 90838 101212 112141 123641 135696 148313 1 61 489 554 866 1247 1698 22l8 2807 346s 4193 4989 5856 6792 8870 10014 11227 12509 13860 16771 19959 =3424 27167 31187 34483 40058 44909 50037 55413 61 1 26 67087 73324 79838 1419 2217 3193 4346 5677 6870 8871 10734 12774 14992 17387 22709 25637 28742 32025 35484 4=935 51096 59967 69548 79838 102520 11491S 128097 141935 171741 187709 204386 Direction of Load Varying. Wrt. Iron C'tlron Steel I I I d d d " 1419 724 1833 2217 1113 2i83 3193 1629 4124 4346 2218 5163 5677 2896 7331 6870 3666 9278 8871 4526 11455 10734 5476 13861 12774 6517 16495 14992 7649 19359 17387 S870 22452 22709 29325 25637 33106 28742 37115 32025 41353 35484 45821 4293s 55443 51096 65982 59967 79260 69548 89809 79838 103097 90868 117301 102520 J32422 114915 148460 128097 165413 141935 183284 156483 202070 171741 221773 187709 242394 204386 263934 4. Bxample. An axle on a railway carriag^e makes from 200 10300 revolu- / tions per minute ; n may taken = 270, and from_(g3) we have —y o.i3\/ ; 270 ^ 2.14. In practice the ratio is made from j.8 to 2.0. The journals of fan blowers are often operated at more than 1200 revolutions'; hence we get, in / ,— . / ' . such cases — r- =0.13 \/i2oo ^4,5, or for steel ---^0.17 v 1200:= 5,5. The blowers made by Sturtevant, of Boston, have steel shafts, with the journals 5 to 6 diameters in length. ? 92. Neck Journals. When a journal is placed between two loaded parts of a shaft, as shown in Fig. 274, it is called a Neck Journal. f<.-l- FiG. 274. In such cases the diameter^-' is dependent upon other condi- tions than those of mere pressure. In order that the wear * Portfeuille de Johu Cockerill, I. p. 18 t See Marks, " Crank Pins and Journals," Philadelphia, Kildare, 1878, where the following values of/ are given : Swatara, 400 ; Saco, 412 ; Wamp- anoag, 725 ; Wabash, 470. The third of these engines had a cylinder 100" diameter, and crank pin 16'' dia., 27" long, and the stress in the preceding cases was respectively, 4039, 3071, 10,537, ^nd 2745 lbs. THE CONSTRUCTOR. 63 may not be greater than in the case of overhung journals, the conditions of speed, lubrication, bearing metal, being the same, the length should not be made less than the corresponding end journal. If it is practicable to make the length greater, it may be done to advantage, and the weai thereby greatly re- duced* In many cases, however, the lack of space limits the length, as for example, in the case of crank axles for inside connected locomotives. Such journals are properly considered merely as enlarged end journals. For hollow journals of this type formula (94) may be used. I. Example. Borsig's Express Locomotive at the Vienna^ Exposition.! The journal of the rear drawing axle of steel was loaded with i3,2cx> lbs. d = iy%" , V = 7xV " According to the table the ^corresponding journal is Intermittent Pressure. ■- 692.4 lbs. given &sd ^ 3H". ^= 3-^25 X 1-94 = 6.1", and_?> = ^:l = 3.125 X 6.1 In this case /' is much greater than /, and for the given values of/', and d* 13,200 ■we have p = = 253.3 lbs. 7125 X 7-3125 while if /' = /. the pressure p — '3,2oo : 303 lbs. 7.125 X 6.1 2. Example. In the same locomotive the forward axle carried the crank pin journal upon which the entire force of the piston was exerted. The total pressure on the piston was 32,120 lbs., and the dimensions of the pin -were if -= ^Yz". I' = aVz". The corresponding values from the table of the preceding section give (/ ^ 4,',4 ",/ = 4, 25 x 1.3 ^S/^"/! ^ about 1400 lbs. The actual value oi p, for the sizes used is — '^-- = 1730 4-125 X 4-5 lbs. In this case /' is less than /, on account of lack of room, which accounts for the increase in superficial pressure. I 93- Fork Journai^. A Neck Journal which is held at both ends in a 3'oke or fork, as shown in Fig. 275, may be called a Fork Journal. Such journals may safely be made of lesser diameter than those which are overhung. If we let /'= the load, /^length, andrf = diameter, and s, the maximum permissible stress, we have from case VIII. I 6, ^=/4/4 /' and if, as in the beginning of § 90, we put p =—p d 1 4P T / (96) (97) Proceeding as in J 90 we obtain the following collection of proportions. Formttlce for Fork Journals. Constant Pyessure. Wroughtjlron. Cast Iron. Steel. S500 4250 14,220 & g fe o f Po s I ~d po s I S500 4250 4250 14,220 I 0.0121%//' 0.0171 -v//" 0.0095 v^ /" Intermittent Pressure. Wrought Iron. Cast Iron. 8500 8500 4250 4250 I L d = - P = 711 V s = S500 v.. I d = 3 s L d = 0.0212 v//' o.oi2i\//* o.oi~}\y/ P Constant Pressure. Wrought Iron. Cast Iron 355 4250 steel. 14,220 14,220 I 0.0095 \/ P Steel. 711 14,220 4 0.029 V/' 0.0185 \//' * See 1 109. fSee Berliner Verhandlung, 1874, p. 3S9. Wro'.ght Iron. Cast Iron. Steel. p = 1422 711 1422 s = 7II0 3550 11,845 I 'd = 2 2 3-5 d — 0.0185^/ P 0.026 x//" 0.0158 %/ High Speed journals of this sort are seldom used, and need not be considered liere. It wiU be noticed that these Fork Fig. 277. Fig. 275. Journals are comparatively small in diameter and of greater length ratio than the preceding forms. Bxample. A Fork Journal of wrought iron bears a load /*= 4400 lbs., act- ing constantly in one direction and revolves at a moderate speed. We have then if =0.0212 \/44co = 1.4") /=■ 1.4" >C 3^ 4.2". For an overhung journal under similar conditions we have, from the table of ? gi. d ^2", / = 3". The product of the length and diameter is approximately the same in both cases. If the length 4,2" is found inconveniently long, it may be diminished, providing d be proportionally increased. The strength will then be un- necessarily increased and the resistance of friction somewhat greater. These are only examples of the many variations which are to be met among the many conditions of practice. I 94- Multiple Journals. In some cases the resistance of friction becomes so great that a modification of the fork journal is resorted to in order to re- duce it within practical limits. Such an arrangement is shown in Fig. 276, which may be called a multiple journal. If we as- sume the load to be equally distributed among the plates, this -M Fig. 276. arrangement' may be considered as a series of fork journals. If the number of members on each side be taken = /\, each pair will support a A'th portion of the load P, and d will be /i times as large as would be required for a simple fork journal. If/<'= 2 3 4 5 6 7 8 We have /J 0.7 0.57 0.5 0.45 0.41 0.38 0.35 Journals of this kind are generally of the slow-moving class, with a length ratio = i. The total length of journal is the^ 2 A" d. Journals of this sort will be found is some varieties of chain links, of which examples will be giveu later. * * Joints of this kind may sometimes be subjected satisfactorily to a greater pressure than the calculation would indicate. Engineer VoUhering has used such a joint in a system of levers to operate a hea\'y drawbridge. In this case the load was about 95,000 lbs. IC= 10, the thickness of each plate |", d = 1^5, both plates and journal being of steel. 64 THE CONSTRUCTOR. I 95- Hai,f Journal. lu those cases in which the reduction of the moment of fric- tion is of great importance, the length of a journal may be somewhat increased, if the bearing surface is limited to one- half the circumference, as shown in Fig. 277, which shows such a bearing, the load acting constantly in one direction and the movement extending only through a small angle. In such cases it is desirable to have a small supplementary journal as shown in the figure, in order to meet unexpected lateral pres- sures. In such half journals, provided the unused side of the material is proportionally increased, d is independent of P, and p only is to be considered. We have for Wrought Iron. Cast :rcwi. Steel. Po = S500 4250 14,220 p = 6700 3340 11,160 Example : For a pressure P"^ 220,000 lbs., acting in a constant direction upon a slow moving journal of wrought iron, we have from (93) d = o.oi-j s/'z-zo^ooo = 7.97", say 8", and / = 4"; for a fork journal, according to (98} f/ = 0.0121 \/22o,ooo ^ 5.67", and / is the same; for a multiple bearing with eight parts on a side d = 0.35 X 5.67 = i.gS", say 2", and a total length 7 = 2 X 16 -= 32". If now we take for a half journal the same conditions and make d == 2", we get /^ 2 X 8 ^ 16". We may, however, make d .= 1.5", iu which case / = T-*s " X 16 ^ 21 28". The journal friction will in this case be 5 that of the overhung journal, f^s that of the fork journal, | that of the multiple bearing journal, which latter is nearly 60 per cent, longer. Au application of this form of journal will be seen in Fossey's Coupling. Woolf has also used it ou the cast iron crosshead of a large pumping engine.^ The principle of the half journal may be seen carried to its extreme limit in the knife edge bearings of weighing machine in which the friction is reduced to a minimum. The superficial pressure upon these very small surfaces is correspondingly high, ranging from 15,000 to 150,000 lbs. per square inch. The hardened steel edges and bearings seem to be able to stand these pressures without injury. f ? 96. Friction of Journai,s. New journals show greater frictional resistance than those which have worn to a good bearing. At first the journal only comes in contact with the metal of the bearing in a limited number of spots until after a moderate amount of wear the superficial pressure is distributed over the projected area of the bearing, giving the value ofp, as indicated in § go.J For a diameter d, and load P, for a cylindrical journal, whose ccefficient of friction =y, we have for the initial force P, which the resistance of friction holds in equilibrium, for new, unworn journals F-- -fP, and for smoothly worn journals F-- ~fP The reduction in frictional resistance is equal to — ; or about o.Si times less in a smoothly worn bearing than in a new one. The actual value of F is, however, greatly dependent on f. This, however, is not only dependent on the lubrication and condition of surfaces, as according to the theories of Morin and Coulomb, but also upon the superficial pressure p, and speed of rubbing surfaces v. \ Additional researches upon this subject are yet greatly to be desired. 11 Rennie's experiments with cast iron journals in bronze bear ings, with copious Uibrications : When /> = 3.2 175 315 492 668 739 /" = 0.157 0.225 0.215 0.222 0.234 0234 no account being taken of v, in these experiments. Hirn experimented with cast iron ou bronze with full lubri- cation, the value of v being equal to 335 feet per minute : When / ^ 3 5.26 7.54 9.71 12 y= 0.0376 0.02II 0.0226 0.0199 0.0183 and these experiments showed that for small values of/, _/ diminishes as, p increases. Hirn also found that \i p remained constant, and equal to I2 lbs., that when Z' = 92 164 184 275 327 335 367 /^ 0.0086 0.0121 0.0128 0.0165 o.otSi 0.0183 0.0191 thus being at all times quite small, but still constantly increas- ing with the increase of velocit}'. Morin's researches gave with pressures of 14 to 20 pounds per square inch, values of y, from 0.05 to o.ii for journals lubri- cated with oil, and from 0.08 to 0.16 when lubricated with grease. The following results were obtained at the Royal Technical Academy from experiments after Morin, upon Clair's apparatus. The journal was of wrought iron in brass bearings, freely lubri- catecl with oil . First Test. Second Test. Bearing Surface ....... 12.800 sq. mm. 128 sq. mm Total pressure /■ 16.5 kilo 16.5 kilo Pressure pr. sq. mm 0.00129 kilo 0.129 " Observed friction 1.25 kilo 2.65 " Coef&cienty 0.076 0.160 The author's experiments with an apparatus resembling a Prowny brake with surfaces of wrought iron on bronze with good lubrication and velocities of 30 to 35 feet per minute, gave the following results : P= 50 122 192 335 484 624 711 y^o.ogoo.oS7 0.095 0.118 0.171 0.184 0.180 Here the value of y was doubled, while/ increased 15 times. If/ remained constant and equal to 470 lbs. we have for z' = 79 14.17 34.64 55.1 y^ 0.222 0.210 o 191 0.167 In this case the coefficient of friction diminishes for an increase in the value of n, contrary to the results in Hirn's observations, the value of/ being above 40 times greater than Hirn used. These latter results appear to be more in accordance with Morin's, in that the friction of rest is greater than the friction of motion, and hence for small velocities the friction should be greater than with higher velocities. This law appears to hold good only between certain limits for v, either side of which J increases for increasing velocity. Hirn's researches lay beyond these limits. Those of the author are, only preliminary to a fuller series of observations. The following table give some results of the wear on boxes of various kinds in railway service : * See Tredgold, " Cornish Bumping I^ngines." I In large track scales, pressures as high as 425,000 lbs. per square inch are found upon bearings less than j'l" wide. The knife edges on the large Werder Testing Machine at the Royal Technical Academy are 360 mm. long, and sustain a maximum pressure of 100,000 kilograms, or 277.S kg. per mm , or at i mm., in width is equal to 556.6 kilograms per square millimetre, or Sio.ooo lbs. per square inch, and this pressure has been sustained without apparent injury. X See Reye, Theorie der Zapfenreibung, Civ. Ing- VI., i860, p. 235 , also Grove, Trag-uudStuzzapfen, Mitth. d. Gen. Vereins fur Hannover, 1876. § See Hirn, Etudes surles frottements medints. Bulletin von MUlhausen, 1854. p. 1S8, also the researches of Reunie, Sella, Bochet, and others. \ Engineer, Nov., 1873, p. 312, contains a brief, but valuable discussion upon the action of railway axles in their actual conditions of operation. The following abstract gives the results ; The brasses were all , poured from the same crucible and consisted of a Distance. Km. for Wear on 4 boxes in Kind of Alloy. f a wear of i kilo- grammes for 1000 gramme from 4 boxes. Kilometres. Kilometres. Grammes. I. Gun Metal S3 Cu. 17 Sn. . . 90,390 11.06 2. " " S2 Cu. 18 Sn. . . 99,900 10.01 ^• White Metal 3 Cu. 90 Sn. 7 Sb. 72,280 13.83 4- " sCu.SsSn. 10 Sb 8S,i45 11-34 Lead Composit'n 84 Pb. 16 Sb 81,280 12.30 t Phosphorbronze 439,200 2.33 7- Parsons' White Brass . . . 385,275 2.60 8 Dewrance's Babbit Metal . 637,679 1.57 mixture of 7 parts copper and I part tin. They all worked under the same car and all had the same lubrication. In running 28000 miles the losses were as follows : 1. d = Journals. Boxes. 3j. /"Si, 3J, "-6*, 33, " = 7", loss = 3'," ; loss = 5 lbs " — 3 '• " _2j" Taking the journal load as 1 612, 554 and 427 lbs. [,000 lbs., the value of/ in 'the three cases is \ Nos. t to 6 are from the work of Dr. Kunzel on Bronze bearings, Dresden, 1875. The others are from The Engineer, Vol. 41, 1S76, pp. 4 and 31, all be- ing given in metric quantities as readily comparable. THE CONSTRUCTOR. 65 B. THRUST BEARINGS. I 97- Proportions of Pivots. A thrust bearing which is formed on the end of a shaft and bears the pressure upon its sectional area, is termed a pivot. For ordinary cases these are made in the form shown in Fig. 178. The pressure p is uniformly distributed over the area of the end of the shaft, and the velocity is proportional to the dis- tance p of any given element from the centre. A small oil chamber of a radius i\ is formed in the middle of the bearing. If the outer radius is ;-„, we have /' 0-5^ (n+n ) and for the elements on the outside radius 0-5/ (''i+''o) In the formulse for a uniformly distributed pressure /, we have taken t\ = J Va and the two diametral oil channels are made of a width = ^Jj d. We then have for a given load P: P=S,i6pd'> (loi) In order that there may not be too much wear for fast run, ning bearings (see § 90) we may take/ = ~, and have for high speed pivots : /'=Si6rf^ — (102) Alternating pressures do not occur in these bearings and need not be considered. The value of a may be taken for wrought iron on bronze as = 75. Bearings of lignum vitfe running in water may bear loads of 1500 pounds per square inch even at high speeds.* The following formula and tables will serve for the propor- tions for end pivots : FoRMuiv.5 FOR Pivots (103) Wro't Iron or Steel Cast Iron Iron or Steel on on Bronze. Lignum Vitae. 700 0-05 \/p 350 ^ 1422 _ O.C7 ^p 0.035 ^p /=1422 (!'= 0.035 v/7> on Bronze. { p^ 1422 Slow movmg Pivots | ^^^ 035 ^/-^ f/ = 70o M=:or 150 = 75 _ = 0.004 V'p;, FtI Fig. 278. Fig. 2S0. There is a general tendency in machine practice to use smaller diameters for pivot bearings, f in order to reduce the resistance of friction. In order to reduce the effect of higher speeds upon pivots bearing heavy pressure a series of disks is often used. If, in Fig. 279, the number of plates between the eud of the spindle and the step is i, 2, 3, 4, . . . i, we have for the proportion of turns between each pair of surfaces yi, ji, ji, times n. i-f/ This device has been used for steps of turbines, mill spindles, etc., by Escher, Wyss & Co., Reiter and others. But few ex- amples now remain of this firm for the thrust bearings of screw propeller shafts ; the disks bound together and were Fig. 279. overheated and injured. So far as experience indicates, such thrust bearings are capable of standing pressures of 1400 pounds per square inch or even more. The important point to be con- sidered is, therefore, the reduction of the superficial pressure /. The use of other materials than iron, wood or bronze, and their substitutes, such as white metal. Babbitt metal, etc. , has often been attempted. The subject of wooden bearings will be considered hereafter. Besides the use of hardened steel, which is of small value for great pressures, such bearings have also been made of stone, glass, J or hard burned clay,J but none of * Penn has used lignum vitae bearings with pressures oi p = pounds. (See Burgh.) ■ 7000 to Sooo t At the establishment of Gruson, in Madgeburg, a boring mill is made with cast iron spindle in cast iron bearings, with a superficial pressure of more than 20,000 pounds, without ill results. X Bearings of glass have been used for more than twelve years atthe works of E. Acker & Co., at Graggenan, near Rastatt. These bearings are very durable and cheap and require but little lubrication. g Shown at the Exposition of 1867, by Leoni, of I,ondon, with good results. 66 THE CONSTRUCTOR. these materials have come into general use. Girard used a pump to keep a film of water between the friction surfaces, and atter deductmg the power to operate the pump showed a very light resistance.* A similar device was shown by Girard at the Exposition of 1867, in which the water jet was 'operated by a blast of air. This apparatus was rather of the nature of a scientific apparatus, than as a practical application. There were also exhibited journals which ran in bearings in which water was inclosed. t The experience of general practice, how- ever, shows that the ordinary forms are sufficient, without re- quiring the use of any of these complicated devices. ? gs- Friction of Fi,at Pivot Bearings. If a flat pivot bearing with annular bearing surface, as in Fig 278, has an inner radius ')\, and an outer radius ?-„ with a load -r, we have for the tangential frictional resistance F= —fP 3 I — -' (103) in which f is the coefficient of friction. For rapidly running pivots we have F= -Lp ' + 7-V '0/ (104) The second value is rather less than the first, since, from the previous proportions r-^ = \ r^, which gives for running pivots p= '^fP, and the ratio of the two values is as 7 to 6, while if i\ = O, it is as 4 to 3. For values of _/"see § 96. ro 3) Example. In thecraneof example i, §97, P=39, 600 lbs. ro = 3H"- /■^o.is. This gives in {104) j^= 0.075- X 39,600. 3 = 39§olbs. The force required to overcome this resistance, if acting: at a lever arm 40 inches from the axis would be — ■ 3960 X 3-° 5 40 322 lbs. i 99- Coi,r,AR Thrust Bearings. The use of collars to receive thrusts on hocizontal bearings is similar to such use on vertical shafts, and a form is shown in Fig. 280. In this case the inner diameter 2r, cannot be less than the diameter D of the shaft. It is best to make it suffi- ciently greater to permit a small oil channel to be used as shown in the figure, and oil ways should also be cut in the bearing surface. Fig. 281. Fig. 282. Fig. 283. If ^0 — ?', is made ftie same as before, good proportions will be obtained, although the rubbing surfaces will move at a some- what higher velocity. For this reason such bearings are not to be recommended when high values of P must be carried. The resistance of friction may be calculated by the formuteof the preceding section. Mui,Tipi=Z)=i5". Breadth of collars =* = m—r, =2" Num- ber of collars ^=9. /^o n 2 . iMum Here/. 39600 = 40 lbs. The velocity z/, at a radius 9fXi7X2 " -275 ft. This gi%'es in (104) takingy"= F^ 39600 and-the friction horse power HP = 9-5 I = 3465 lbs. 3465 X 273 = = 29 H. 33000 Example 2. ^8. 2 ?-i = i? . Turbines on the Rhine at Schaffhausen. = 9". Collar width <5, =. ro — rj = i s.^", m = 30800 /'^ 30800 Its. « = This gives/ ■= V — 133 feet. 9 IT X 10,625 X l.6zs = 63 lbs. _ C.I / F= 30800 ( 9 12.2^ 2664. H 2664 X 133 „ P. = = 10.7 H. 33000 Example 3. Girard Turbine at Geneva. J P= 33,000 lbs. w = 16, 2 ri = ZJ ^ 9.8'' b = Tq — rii ^ %", nt =12 3^. coo This gives/ ^'^' 12 TT X 11.17s X 1.375 ~'' 1/ = 46.7 ft. From (104) we get F— 2970 and the friction horse-power is !97o2"= 17,600 lbs., we have for a wrought iron shaft with a constant direction to the pressure, from the table in J 91, d^ 4'', 1—6". If p _ 15 lbs. we have from formula {90) _ =11.9 and from (89) we have rf — n.="; hence 2 = II. 2 X ll-g = 133" ! ! II Sellers recommends a mixture of tallow and oil, which becomes more liquid should the bearing grow warm. if See Berliner Verhaudlungen, iS76,"p. Sg. Another form of adjustable pillow block is shown in Fig. 305. This is used by Sturtevant in some of his fan blowers. In this case the ratio of / to rfis very great (see example 4, I 91). The adjustability is obtained by pivoting the bearing A upon a A J Fig. 305. cross bolt B, which passes through the cheeks of the pedestal also ; the latter being adjustable about the axis BC. The bear- ing is lined with white metal, and the end thrust is taken up by a block of lignum vitse. If an adjustment in the direction A A IS required, the bolt C may be loosened and the required move- ment made. The provision for lubrication is especially note- worthy both in the manner of supply and in the collection of the overflow. i no. Bearings with Three-Part Boxes. In horizontal steam engines and in similar service, the pressure upon the journal is thrown first on one side and' then on the other, while at the same time there is a constant vertical pressure, such for instance as is due to the weight of a fly wheel. Attempts to remedy the tendencv to overwear by mak- ing the boxes inclined, have proved but a partial remedy, and the best method of construction in such cases is to make the box in three parts, one of which receives the constant vertical pressure, while the other two provide for the backward and for- ward thrust. Such a bearing is shown in Fig. 306. The modu- lus (/i = 1.15^-1- 0.4". The bottom box rests on two wedges which are tapped with screw threads and can be adjusted and locked at any desired point by the bolts shown. The side boxes are each held up by two steel set screws ; a wrought iron plate being interposed between the screws and the boxes. If it be- THE CONSTRUCTOR. 7f comes necessary to remove the side boxes the cap is first taken off, and the iron plates taken out, when the boxes can be sepa- rated far enough from the shaft to permit their removal without those cases in which an alternating up and down pressure is combined with a constant lateral pressure. The latter would not be provided for in an ordinary pillow block, but here it Iktk! ftMliJij, ilib.il, t^iaitHnA n p m % i itjutiltiltilii "!:^' Fig. 306. interference with the shaft. The body of the bearing is in creased in width in order to provide for the ^increased lateral pressure. Fig. 307. Another three-part bearing * is shown in Fig. 307. In this case there is no vertical adjustment to the lower box — and if necessary it must be raised hy packing underneath. The side boxes are set up by wedges which are adjusted by set screws through the cap. Each wedge carries a screw on its upper end, and the nuts for these screws are fitted so as to revolve in the cap, being turned by a wrench on the hexagonal head, and then clamped in position by the thin jam nut shown. The heavy inclined ribs stiffen the body of the bearing to resist the stock and thrust of the piston. It is often convenient (as in the case of the original of the figure) to cast the body of the bearing in one piece with the bed plate of the engine. A third, and simple form of three-part bearing (by Schultz Brothers in Mayence) is shown in Fig. 308. It is suitable for Fig. 308. is taken up by the small side box. This form is suited for small vertical engines in which the pull of the belt is toward one side. 'i III. Pedestal Be-4rings. Bearings which are not placed directly upon a base plate, but are raised upon feet or pedestal are called pedestal bearings 8=6.6-- FlG. 309. * From a steam engine by the Soc. Fives- Lille in Paris. That shown in Fig. 309 is similar to the one in Fig. 293, placed upon a pedestal. Such pedestals vary greatly both in form and height. The width of the foot is made equal to the height of the journal in the form shown, which gives the base and the legs a sufficiently slender appearance. I 112. W.\LIv BE.ARINGS. The wall bearing shown in Fig. 310 is the same as shown in Fig. 293, with the addition of the bracket. The base here is placed at right angles to the joint in the boxes and parallel to the axis of the bearing, the whole being made in the bracket form shown. The cap and the boxes are of the same form and proportions as for a pillow block for the same size journal. The bolts may either be tapped into the body of the bearing, or made as stud bolts, using the forms shown in Figs. 225 and 226 ^ S3, with key. For larger sizes the opening in the plate should be surrounded with a rib of a thickness cirfj and width = 0.4^/,, the latter being measured in the direction of the axis of the journal. Fig. 311 shows an adjustable wall bearing by Sellers. In this case the cast iron boxes are somewhat lighter than for pillow blocks and are made with a cylindrical cross piece in the middle, in which the spherical seats are placed. The especial feature is the method by which the vertical adjustment is made. The two plugs which support the boxes have cast upon them a very shallow screw thread, and the nuts in the sockets have also their threads cast in them. The thread only extends along 72 THE CONSTRUCTOR. a portion of the length of the plugs as shown, in order to per- mit securing them in position. This is done by the two self screws which clamp them firmly in their places. The opening through the upper plug gives access for the tube of a lubricator. ..J., Fig. 310. The projection from the wall a is made constant for bear- ings forjournals 2" to 4" in diameter and equals (>". The ele- gance of the form is noticeable in the principal elevation and also in the horizontal section. I 113- Yoke Bearings. The bearings used on vertical shafts may be considered as a variety of wall bearings. In situations where the space is lim- ited the forms shown are not always convenient, the first, be- cause it is not symmetrically disposed about the parting of the Fig. 311. boxes, and the second, because of the space it requires. For this service a compact, symmetrical bearing, whose base is at right angles to the parting of the boxes, is often very desirable. Such a construction is shown in Fig. 312, and may be called a Yoke Bearing. In this case the cap and body together form a rectangular 5'oke, in which the bronze boxes are placed in a transverse direction. In the illustration the wear can only be taken up in one direction, but if it is desired in both directions the cast iron block on the right may be replaced by a wedge as shown on the left. By removing the cap, the wedge and the block can be easi- ly removed and the shaft moved sideways to a sufficient extent to permit the removal of the boxes. The cap bolts are provided with collars forged upon them and serve also to fasten the bear- ing in place. The modulus for the dimensions is the same as (107), q'i = 1.150' +0.4".* l,5cl Fig. 312. § 114. Wai,Iv Brackets. " In Fig. 313 is shown a form of bearing similar to Fig. 293, which may be called a wall bracket bearing. The cap bolts are inserted from below, which permits their ready removal and replacement. If only two bolts are used in the wall plate, it is I/6S Fig. 313- desirable that it should be held from lateral motion between wedges, and should also be firmly secured against vertical mo- * For such a Yoke Bearing, see Engineers' and Machinists' Assistant, London, 1854, PI. I. THE CONSTRUCTOR. 73 tions by some of the methods given in the following chapter. Where it is not practicable to secure it in this manner, four bolts should be used. ^c-0,r)->K- 1.3 >'k-0,5>[<- 0,87->i Fig. 314. Another form of wall bracket is shown in Fig. 314. It is sim- ilar to the Yoke Bearing, and can often be of service, as for example in Fig. 350, § 126, although it is not of as general ap- plication as the preceding form. The bolts for the cap are made with heads, of the ordinary cap screw form. Various other wall and bracket bearings may be made b}- combination of a wall plate and pillow block in different po- sitions, and these may be grouped in the general class of Arm Bearings, each form being governed by the conditions of the special case under consideration. 115. H.^NGKRS. , According to the definition in \ 103 a pillow block by inver- sion becomes a hanger, the pressure of the journal falling upon the cap box. If the journal is one of wrought iron proportioned to bear the loads given in S 91, the bolts for the cap and base plate will not be strong enough if determined from the same Fig. 315- This is unit of proportion as alreadj' given for such bearings, also true for the cap, and feet of the base. For this service, good dimensions may be obtained by using for the boxes the previous modulus (/j = 1.15 rf + 0.4", and also E as before, and for all other portions the special modulus. /?■ 1.75 rf + 0.4'' (109) If a pillow block is to be used as a hanger for a neck journal, the cap bolts should be increased to such size as would be given by the use of formula (109), in which d is the diameter of the neck journal corresponding to an equivalent end journal. Example : A load of 17,600 lbs. would give, according to the table in J91 for a wrought iron journal a diameter of about 4 inches. If this load is carried on the cap of the bearing we use the modulus, This gives for the diameter of the cap bolts 7.4 s 0.2 = 1.48"^ say I }i,"- A neck journal of 6;4"' diameter to bearthe same load would have for its normal unit d ^ 1.15 s 6.75 -j- 0.4 = 8.15''', which is greater than the preceding value and hence may be used safely, even should the full load be carried by the cap. Sellers makes a short hanger which resembles in form and dimensions the corresponding size pillow block, with the boxes turned iSo° and the drip cups cast on the cap instead of the base. In most cases, however, a greater distance is required be- tween the shaft and the base plate for hangers than is given in pillow blocks, for which reason they are best considered as a separate form of construction. The hanger shown in Fig. 315 is called, from its form, a Ribbed Hanger. The boxes are carried in the hook-shaped por- tion below, their form bei ng the same as we have already shown. The cap is secured with a key and clamped in the de- sired position \>y the bolt shown. For journals of less than 2 inches diameter, but one bolt need be used in each foot, and in such case their diameter is ^ 0.3 d-^, the bosses on the plate be altered to correspond. Fig. 316. In the Post Hanger, Fig. 316, the general arrangement is the same as in the preceding form, the principal difference being in the frame. The column is made hollow and its internal diam- eter = 0.55 d^. For the larger sizes four bolt holes are made in the base plate, as shown in Fig. 315. Hangers are not generall}- bolted directly to the ceiling beams, but to strong pieces, or intermediate timbers, and by Fig. 317. Fig. 318. D'; 1.75 a' -f 0.4'' = 1-75 x 4" + 0.4'' = 74'' varj'ing the thickness of these pieces any desired amount of drop maj' be obtained. If the variation is too great to be se- cured in this manner a different depth hanger must be used. If the building is of so-called fire-proof construction, with 74 THE CONSTRUCTOR. ceilings of iron beams and brick arches, tlie form of the base of the hanger must be correspondingly modified. A practical method is shown in Fig. 317, in which hook bolts are used. The bolts, which are four in number, pass through sockets cast in the base of the hauger, and their method of attachment avoids weakening the beam. The base of the hanger is made with ledges which fit over the edge of the beam and permit the use of wedges on each side. The form shown in Fig. 31S, which is due to Fairbairn is in- tended to bring the shaft parallel to the beam, while the pre- vious form carries the shaft at right angles to the beams. The attachment of the hanger both to the beam and the arch makes a very secure fastening, but the inaccessibility of the bolt head is an objection. In this case also the beam is not %veakened by drilling, hook bolts and keys being used, as in the previous case. »....-.l„6 . « Fig. 319. I 116. • Adjustabi,e H.^ngers. EJThe most generally used of the Sellers' adjustable bearings is thejhanger shown in Fig. 319. ^ Thel method of holding and Fig. 320. adjusting the boxes by means of screw plugs is the same as shown in the wall bearings, Fig. 311. Especially to be noted is the attachment of the drip cup, which may be easily removed by withdrawing the small pin with enlarged ends. The drop, or distance from base to centre of shaft, ^= a ^ 3.5 rf/ in the illustration, but in some cases it must be made greater. These hangers, like all of Sellers' bearings, show very careful modeling and proportioning, which the small size of the illustrations can only imperfectly show. In Fig. 320 is shown Sellers' countershaft hanger. In this form the shaft is put in place from the side, and the amount of wear in the boxes is so slight that they are made solid, instead of in halves. The cap — which is secured by a bolt, holds the box in place, and the drip cup is cast i;i one piece with the body of the hanger and provision is made for a drip cock to remove the waste oil. The illustration shows also the arm for carrying the belt shifter. Sturtevant uses ball and socket hangers also for the counter- shafts of his fan blowers. These are somewhat different from the preceding. Fig. 321 shows the boxes in perspective and in cross section. The section shows the white metal lining and also the arrangement of double oil chambers, which, by means Fig. 321. Fig. 322. of wi eking, keep the journal lubricated." The outer ends of the box casting are formed into drip channels, and also receive the shoulders on the shaft. These shoulders, as shown in Fig. 322, run freely in the boxes without contact. The journal as shown is on the end of the shaft, and the pressure is so smal! that the wear is inappreciable. l 117. Speci.^l Forms of Bearings. In propeller shafts where the screw is arranged to be lifted it is necessary to design bearings which are to be entirely im- mersed in water. Penn's practice is to line such bearings with wood, which has proved especially satisfactory. In Fig. 323, is given an illustration of such a bearing as constructed by Ravenhill & Hodgson, the diameter of the shaft being about 19 inches. The body of the bearing is of bronze, the boxes are of cylindrical section fitted with strips of lignum vitse set in a Fig. 323. special lining metal. The pin, projecting from the bottom, enters into a corresponding recess in the stern frame, when the screw is lowered into place. On the Prussian State Railway there have recently been adopted two standard forms of bearings for use under cars — one form being for bronze, the other for white metal boxes. In THE CONSTRUCTOR. 75 Fig. 324 is shown details in partial section of the latter form, with a few dimensions. The bearing is made in two principal parts, the body and the lower portion, both being provided with oil chambers having openings and covers to keep out the dust. The joint between the two parts is in the horizontal plane passing through the axis of the journal, the parts being body of the bearing is a filling block, somewhat similar to that used in the bearing shown in Fig. 312, arranged so that its re- moval facilitates the changing of boxes. Fig. 324. kept in position by three dowel pins. A wrought iron yoke holds the lower portion up to the body of the bearing by means of the bolt shown, the head being secured by the internal hex- agonal socket shown. The white metal lining is cast in the body of the box by be- ing poured upon the journal. The inner end of the journal is provided with a wooden dust guard packed with a ring of felt. As will be seen, lubrication is provided both above and be- low. The upper chamber contains wicking and affords a means of prompt and copious lubrication in case the journal grows hot. The principal source of lubrication, however, is from be- low, the oil being wiped upon the journal by a brush, which is fed with oil by a wick reaching into the chamber below. The oil brush is shown, with its spring holders in the lower right hand corner of the illustration. In order to permit the boxes to adjust themselves to the journal when the axle assumes an inclined position with re- gard to the bearing a certain amount of play is given, as is shown in the plan view, where the ledges cast upon the bearing are made' parallel for a short distance and then diverge from below upward from a width of 34 mm. to 42 mm. All the dimensions in Fig. 324 are in millimeters, as this is a standard Prussian railway journal box. This construction is undoubtedly well adapted to meet the requirements, but it is a question whether the results might not be attained by simpler means.* The second form of standard bearing of the Prussian Railways differs from the first mainly in the boxes. These are cast of bronze with semi-cylindrical projections on the track, which enter into corresponding recesses in the bearing, and permit the boxes to adjust themselves to the journal. The guides for the bearings are given an amount of play similar to the previous form, and there is no change in the de- tails of the lower portion. Fig. 325 shows a form of American axle bearing. This is sim- ilar to the older pattern designed by Lightuer t It is only ar- ranged for lubrication from below and is designed so as to per- mit a box to be removed and replaced in the shortest possible time. The body is of very simple form and is cast in one piece and a large opening and lid renders it readily accessible from without. The box is made of bronze, and between it and the * This question of railway journal boxes is an instructive example of the importance of constructive simplicity as applied to machine elements. Since in the year 1877 in Prussia there were in use 315,000 axles or over 630,000 boxes. The cost of these represents an investment upon which every penny economized in construction foots up an important total. t See Heusinger, Schmiervorrichtungen {Lubrication), Wiesbaden, 1864, p. 83. Fig. 325. This filling block, which is sometimes rounded on top to provide adjustment, is held between two small projections, but can easily be removed when the pressure is removed by use of a liftingjack. The change of boxes can be effected in a few minutes. A brush or pad for distributing the oil is not used, but instead the vacant space in the bearing is packed with waste, which feeds the oil to the journal. This form of journal box has proved very efficient in service.* B. THRUST BEARINGS. \ iiS. Step Bearings. In Fig. 326 is shown a form of step bearing for vertical shaft. The bearing piece or step proper is made with very obtuse Fig. 326. point on the under side in order that it may be able to adjust itself to the shaft. In order to provide for adjustment in the position of the bearing the bolt holes in the baseplate are elongated in a cross-wise direction, while those in the bearing are elongated length-wise, thus permitting adjustment in any direction. I 119- Wale Step Bearings. The following is a modified form of step bearings, and is in- tended to be used with the wall plate supported on a key be- neath its lower edge ; this key may be made = o.S d-^ deep, so * A standard axle and journal box were adopted in the United States in 1873, and at that time there were over 1,200,000 axles in service. 76 THE CONSTRUCTOR. that by its removal the bearing may be taken from under the journal, without removing the shaft from its place. Fig 327. The recess in the step plate serves an oil chamber ; end- long wear may be taken up very conveniently by the adjust- ment provided by the set screw. ? 120. Independent Step Bearings. In many cases, as in examples by Belgian designers, the lower bearing of a vertical shaft is divided into two independent parts, a pure lateral bearing and a pure thrust bearing. For the lateral bearingmay be used a pillow block or yoke bearing of one of the forms already described, while the vertical thrust is taken by a simple step quite close to the preceding bearing. This makes the step bearing readily accessible and also readily adjustable in the direction of wear. The following example is selected from among a number of such bearings. Fig. 32S. The step itself is made of bronze. This is carried on the bluntly coned head of the stout set screw, a steel plate being interposed, while the prismatic form of the screw head pre- vents rotation of the step. The screw itself is kept from mov- ing by Penn's method within the bearing, and the whole is bolted down to a base plate. The modulus for the dimensions is the same as before. An application of this form is shown in a 126, I 121. Thrust Bearings with Wooden Surfaces. For bearings which are operated wet, the use of Lignum Vita has been found to give the best results. The wood is inserted in a similar manner to that shown in \ 117, the pieces being made in the form of plugs. In Fig. 329 is shown the step of a screw propeller shaft of this type. The plugs are inserted in a bronze plate, and the end of the shaft faced with bronze. A bearing of this form on the " Orontes " had 37 plugs each i}i" diameter, and on 50 H. P. nominal English gunboats the thrust plates have 7 plugs each 1" diameter. Both these examples are bv James Watt & Co. Collar bearings with surfaces of wood are often made ; these should be always worked under water. Penn, to whom the introduction of such wooden bearing surfaces is mainly due, has especiall}' used them in various bearings in the length of a screw pro- peller shaft, the lower half of the shaft running in a water trough. The usual construction of the thrust ring between the hub of the screw propeller and the stern post is shown in Fig. 330. A is the shaft with a bronze sleeve fitting into the wooden lining of the hole through the stern tube ; B is the hub of the screw pro- peller ; C the thrust ring with its wooden plugs ; D is the nozzle on the end of the stern tube showing the stiffening ribs which assist in receiving the thrust. The parts B, C, D and E are of bronze. Fig. 331. Fig- 331 shows a form of thrust ring used on the imperial steamships "Kaiser," "Friedrich Karl," "Preussen," "Vineta," " Frej-a," "Ariadne," "Nautilus" and "Cyklop." The ring is made in halves, and can readily be removed and replaced. THE CONSTRUCTOR. 77 The two axial projections enter into recesses iu the flauge ou the end of the tube, and prevent the thrust ring from revolving. The dimensions of the wooden bearing surfaces on the various ships above named are approximately as given in the following table : jKaiser. Friedr. Karl. Preussen. Vineta. Freya. Ariad- ne. Nauti- lus. Cyklop b' Surface sq. ft. 28" ('Y^" 4.078 24^'/ IW 0.740 2634:" 1%" 2.629 l(>yz" l%" I.ISS WA" A%" 1.840 I9X" A'A" 1.840 xoy^" 8K 0.476 0.238 In the "Wasp'' the thrust ring is made with 6 sectors of 3 1S6 sq. ft. surface , in the " Leipzig" there are So small sectors with a total surface of 2.422 sq. ft. The use of such thrust rings filled with blocks of lignum vitte has been most successful in vessels of the German navy, and the wear on the wood has been so slight that renewal is rarely necessary. I 122. MuLTipivE Coivi,AR Bearings. For thrust bearings which are subjected to heavy service, the multiple collar bearing is most valuable. These are very gen- erally used to receive the thrust of screw propellers, but are also used in other situations, as, for example, large turbines, also centrifugal machines of great size and weight, such as are used in sugar refineries. The forms which may be given to these bearings are quite varied ; but in every case the most important consideration is the pressure to which the various surfaces are subjected. For pillow blocks in which the shaft is made with several collars, the boxes may be cast iu bronze with internal collars , Fig. 332. as shown in Fig. 332.* For larger dimensions, the boxes may be strengthened by ring shaped ribs, let into recesses in the cap and body of the bearing. Example : — The thrust bearing on the " City of Richmond,'* built by Todd & MacGregor, of Glasgow, from the designs of Jaffrey.f has 12 rings ; inside diameter, 19''; outside diameter, 23"; total length of the bearing, ^3%'^ "The boxes are strengthened by three ribs of K" depth by 4" wide. The engines indicate 3340 H. P., and the speed of the vessel is about 1343 feet per minute James Watt & Co. make the boxes free in the bearing, and support them by set screws at the ends, as shown in Fig. 333. On the "Medusa" and "Triton " four set screws are used in each flange, the shaft being 7" diam- eter, with six rings. In the "Jason," by the same firm, there are six set screws in each flange, the shaft being 12" diameter, with eight rings, j Boxes of cast iron lined with white metal are 1^-4- FiG. 333- sometimes used by various makers, as, for example, in the "Mooltan" by Day & Co., in which the shaft is 13X" diameter, and has twelve rings. The design shown in Fig. 334, which is a French pattern, uses an adjustable bearing lined with white metal. In Fig. 335 is shown a form of thrust bearing in which the rings are made of bronze separately, and fitted to the body and cap. This form is the design of Ravenhill & Hodgson. Espe- cially to be noted is the arrangement of bolts. These are iu two sets, the first securing the body of the bearing to the sole plate, and the second being the cap bolts. The ledge or tongue which is let into the sole plate is arranged with a space as shown on the left, in which a key is fitted to provide for the take-up of the wear upon the rings. The cross section in upper right hand portion of the illustration shows the construction and application of the bronze rings. The arrangement provides for a constant distribution of grease, thus preventing the rust- ing of the journal by the application of water for cooling. Fig. 334- In Figs. 336 and 337 is shown a thrust bearing by Penn, as used on the " Kaiser." Here the bearing surfaces are made in separate rings of still simpler form than the preceding. These Fig. 335. rings, which are made of bronze, are in halves for convenience of construction. In the "Kaiser" d is equal to iSj^", and there are eight rings ou the shaft and in the bearing. The six bolts * See Armengaud, Vignole des Mecaniciens. PI. 13, Fig. 32. t Engineering, May, 1875, p. 403. t See Burgh. Fig. 336. are arranged so as to act both as cap bolts and fastenings for the bearing. The adjustment for wear is similar to the pre- ceding case. The dimensions are based on the same modulus as already given, viz. : rfj^ 1.15 i/-!- 0.4". A most noticeable form of thrust bearing is that of Maudslay, 78 THE CONSTRUCTOR. shown in Figs. 338 to 340, as used on the "Elizabeth." For each collar on the shaft there is provided a separate ring and support, with means for ample lubrication. The bearing rings are made of horse shoe form, and are of cast iron lined with white metal. The collars on the shaft dip into an oil trough. They are also pro- vided with oil cups" above, so that as in the case of the car axle box previously described, lubrication is supplied both above and below. Each ring may be adjusted by its own set screws, or all can be ad- justed together. The propor- tious are all based upon the previous modulus, d^=^\.i$d + 0.4", and the shape and dimensions give an excellent appearance. In the " Eliza- \ 123- Examples of Thrust Bearings. The following examples are taken from twelve of the most important vessels of the German navy, the data being furnished to the author with the approval and authority of the Chief of Admiralty. The power and speed of the engines and the velocity of the vessel are all most important data, and are obtained from of&cial tests. From these may be obtained, as -»- -1.01 -^1 beth' Fig. 337. the shaft is I2>'2" diameter. iiBlM "»J2» Fig. 339. in \ 100, the maximum pressure upon the thrust bearing sur- faces. It is important to observe that in only two cases out of the twelve was a thrust ring used between the stern post and -B — T V Jozii c) (|) -'(|) Fig. 338. Fig. 340. propeller hub. The elasticity of the hull of the ship may some- times cause the entire force to be thrown on the thrust bearing, and at other times much may be taken by the thrust ring. The data given in the table will also be found valuable for other purposes. EXAMPLES OF THRUST BEARINGS. No. Name OF Builder OF (LI 5 11 o'C t4-. .2 a 01 . -^ u5 . u §•5; u . II go tfl 3 Vessel Engines. ^1 £S3 01 2; a S« a 2 II V (5 JO 3 OJ p. ^0 OS on No I Armored Frigate Konig Wilhelm. Mandslay Sons & Field, London. 8325 1491 18" 63.86 6 Anti- mony. 8.467 24M" 18" Bearing cooled with Water. Worked well. Ran warm thrust ring in stern post. Thrust 3 Armored Frigate Kaiser. John Penu & Sons, Greenwich. 7803.3 ■457 18" 77.00 s Brouze. 7.104 23" 181V' Ditto. before the thrust ring was applied. Made with- out thrust nngm stern post, ^121. 3 Armored Frigate Friedrich Karl. ("Societe des Forges et ") < Chantiers de la Mediter- > (ranee, Marseilles. J 35°3 1328 15" 61.82 11 = 18^8" i=2cy," White Metal. 8.004 iS/a" 15^" Ditto. ring and ran warm. Since Ditto. its applica- tion, works Armored Frigate rstettiner Maschinenbau ") well. 4 Preussen. Decked Corvette \ Aktiengesellschaft Vulkan \ (in Bredow bei Stettin. J 43S6.7 1408 .6K" 64.5 8 Bronze. 5.371 20/s" ^6%" Ditto. Worked well Ditto. 5 l,eipzig. Decked Corvette Ditto John Fcnn & Sons, 35193 1437 16" 72.4 8 Bronze. 4.816 igys" 16" Ditto. Ditto. Ditto. 6 Vineta. Greenwich. (■Miirkisch-Schlesische Ma-~) 1359.3 1 120 loj^" 67.9 6 Bronze. 1.489 125-3" io;!s" Ditto. Ditto. Ditto. 7 Decked Corvette -< schinenbau, und HUtten > 2598.8 '.=i57 12K" S2.52 8 Bronze. 2-528 15" 12I/;" Ditto. Ditto. Ditto. Freya. (.Aktiengesellschaft. J Ran warmed first, after- 8 Decked Corvette Ditto. 1726.9 1282 iij-s" 80.24 7 Bronze. 3.391 14%" 1154" Ditto. Ditto. Ariadne. ed well. Decked Corvette Mazeline & Co., Anti- No 9 Augusta. Havre. 1127 1245 11'' 62.09 II mony. 5.177 14A' iiYt" well. thrust ring. Fitted Gunboat Moller & Hollberg Nautilus. in Grabow. 504.2 1047 7K" 109.30 6 mony. 1. 159 9K" 7%" Ditto. Ditto. with rstettiner IMaschinenbau ) II Cyklop. \ Aktiengesellschaft Vulkan \ (inBrcdow bei Stettin. J 245.4 894 ifi" 143.89 4 Lignum VitEe. 0.496 IV*" 5^3" Ditto. Ditto. Ditto. J2 ArmoredGuuboat Wespe. J Aktiengesellschaft, Weserl tin Bremen. j 799-7 1054 6%." 13S.83 i=io%" 8=9?'3" Bronze. 1.728 9^3" 7J^" Ditto. Ditto. Ditto. THE CONSTRUCTOR. 79 CHAPTER VII. SUPPORTS FOR BEARINGS. I 124. Generai< Considerations. The function of a support for one or more beariugs is to hold them iu a firm and definite position with regard to the frame or other parts of a machine. Such supports are nearly always made of cast iron, and in the following treatment of the subject this material is the only one considered. Simple supports are those which are intended to hold but one bearing, in distinction from those supports which are ar- ranged to receive several. In both cases the following consid- erations should be observed as closely as may be, when, as is Fig. 341. usually the case, the shafts which the bearings carry are fitted with gear wheels which should be near the bearings. 1. The bearings should be as near to the hubs of the gear wheels as practicable. 2. The pressure upon the journal should, in no case, act in the direction of the joint between the boxes. 3. The support for the boxes should be so arranged as to allow the easy removal of shafts and gear wheels. 4. The number of bearing surfaces should be made as few as possible, and all finished surfaces should be capable of being finished at one setting on the planing machine. 5. Whenever possible, and especially in situations of difficult access, the bearings should be so disposed that the boxes may be removed and renewed without involving the removal of the shafts from their position. Simple Supports. A simple Support for a single pillow block is shown in Fig. 341. It is intended ior a bearing such as is shown iu 2 107 ; hence the upper portion is made correspondingly narrow. The two legs which form the main portions are reinforced by a cross girth, D E. The position of the points D and E may alwa3'S be well placed by observing the following method : Taking the total height A -B as a diameter, draw from the centre E a semi-circle yl G B ; take the middle point of the arc AG B ^t tT/ join B G, and prolong it, making G H=A F ; then join H to A, and draw CC parallel to HA, and ^ Cis the height from the base to the cross girth. The dimensions of the various parts are dependent upon the pressui'e on the bearing, and must usually be governed hy the dimensions of the pillow block and by the judgment of the designer. In order to meet the requirements of Rule 5 of the preceding section, there should be under the pillow block a removable plate, which may be given a thick- ness of o.3fi',. Fig. 342 is a similar form of support suitable for heavier di- mensions. Fig. 343 is a support for a wall b.=,aring. This is arranged to be built into the wall, and forms an opening through which the shaft can pass, and resembling what a builder calls a bull's eye window. The pressure of the journal is received by the Fig. 343. bracket bearing, which is supported on the key beneath, and can be removed without disturbing the shaft. One point which should not be overlooked is the bearing plate in the wall, shown in tangential dotted lines below the cylinder. The di- mensions in the illustration are based on the modulus d^ of the bearing. Fig. 344. A wall bracket support is shown in Fig. 344. This is intended to carry a pillow block, and the T slot for the bolt heads ena- bles the distance of the bearing from the v,-all to be adjusted. This form may be used for bearings of various sizes. A simpler and lighter form of bracket is shown in Fig. 345. This is merely an arm attached to a wall and adapted for a horizontal shaft. Frequently the joint between the base of a bearing support and its foundation is made with cement. When this is done, the base is adjusted to its position, resting upon wedges,'and the joint being closed with clay, the liquid cement is run in ; this 8o THE CONSTRUCTOR. ■will harden iu a few days so that the wedges may be driven out and the bolts fully tightened. CiLtunrrri ._.,<2iii_ _ V "■■v. Fig. 345. 126. Multiple Supports for Bearings. " Fig. 346_represents a bridge support. The vertical shaft A B comes from below, as for example, from a turbine, and trans- mits its motion to the horizontal shaft CD. The journal pres- sure acts at E, at right angles to the plane of the two shafts, ^ Fig. 346. and at F it acts in an inclined direction downward, both from the pressure of the gear teeth, and also because of the weight of the wheels and shafts. These pressures are best received at E, by a yoke bearing as shown in \ 113, and at F, by a bracket bearing, \ 1 14, supported on an adjusting key. Fig. 347 shows a support for a step-bearing. Here the hori- zontal shaft A B runs in a bracket bearing at C, and transmits motion to a vertical shaft which is supported at D, by a step- FiG. 347- tearing, ? 119. The latter, as the illustration partially shows, is carried on an adjusting key in such a manner that it can readily be removed from below. The bridge which carries the step-bearing is bolted to the box-shaped base and the nuts for the foundation bolts are placed inside the base. Another form for similar service is shown in Fig. 348. The shaft A C, for the large gear-wheel terminates in the support and is provided with a small bracket bearing at C. On account of the position of the wheel, this is not very accessible. The bearings for the vertical shaft DEE, are intended to be of the form described in § 1 20, a yoke bearing being fitted into a space cast iu the upper part of the frame at E, while an independent Fig. 348. step at F is used similar to that shown in Fig. 328. The upper part of the frame is made cir- cular in shape, so that a cast- iron cover may be placed over the pinion, as shown in the dotted lines. The base plate is held down to the stone foundation by four bolts ; two of the bolts pass through the columns, as shown in the illus- trations, and so bind the two plates firmly together. The plan view shows how the col- % urns are keyed into the entab- FiG. 349. lature. ^ The base of the columns are let into the base plate as shown in Fig. 349, and an iron cement is used. Fig. 350. THE CONSTRUCTOR. 8£ In Fig. 350 is shown a support for two vertical shafts, A and i?, the motion being transmitted from one to the other by means of spur gears. The shaft A, for instance, may be that of a turbine wheel, and B, the main driving shaft of the mill.* At A there is a bracket bearing such as shown in Fig. 314, and at j5 a step bearing, with a removable block beneath it, so that the bearing may be removed or examined without removing the •wheel or shaft. Fig. 351 shows a frame for a vertical shaft A B, which trans- mits its motion to a horizontal shaft D E. At C is a yoke bear- ing and at .S a bracket-bearing. The horizontal bevel gear is Fig. 351- inclosed in the semi-circular frame, so that a cover may easily be adapted, as in the previous case. The removal of the vertical shaft is not quite so convenient in this form as in some others, but presents no serious difficulty. In some cases the lower part of the frame is entirely closed and the shaft inclosed in a sort of pilaster, to avoid accidents. For a shaft running parallel to a wall, as at A B. Fig. 352, and transmitting its motion to one D E, at tight angles, the frame shown in the illustration is suitable. The bearing for the Fig. 352. main shaft at C maj' be a pillow-block, while a bracket bearing is suitable at F. The distance of the pillow-block from the wall is adjustable (as in Fig. 344). If the gears are equal in size the form may be as shown in plan in Fig. 353. In this case the journal at Cruas in a bracket bearing. If the construction is Fig. 353. Fig. 354- intended to fit in the corner of a building, the frame is modi- fied as shown in Fig. 354; the bearings at G and Hare then the same. Both these forms are shown in Fig. 355 and 356 in pseudo-perspective. Very often a main overhead driving shaft is required to trans- mit motion both to horizontal and vertical shafts from one point, and the combination of Fig. 357 is an example. Here the frame-work is made a portion of one of the columns of the building and is really simple in construction ; at A should be Fig. 355. Fig. 356. used a bracket like Fig. 313 ; at E and E, wall Fig. 310, and at C, a step bearing like Fig. 327. brackets like Fig. 35S shows a wall frame for four bearings. A horizontal shaft A B, is to transmit motion to the vertical shaft C D, and. two horizontal shafts E and F, by means of bevel gears. At B: I \ Di *Such a frame is used in a spinning-mill at Chur, the frame and one-half of the ■ iarge gear-wheel being in an archway in the large end wall of the building. Fig. 35S. is a bracket, and at C a step bracket, as in Fig. 327, while th* bearings at E and Eaie wall-brackets, like Fig. 310. 82 THE CONSTRUCTOR. Bj a proper choice of journal diameters and clearances the seats for the four beariugs may be brought into one plane, and the other conditions of § 124 readily complied with. An examination of the fundamental principles of construction of supports for bearings will show that all forms may be repre- sented by a rigid piece adapted to hold in fixed relation two or more revolving bodies, in such manner as to permit the applica- tion of the various details of construction such as boxes, caps, bolts, etc. It is often desirable to sketch out in the first place a general scheme of the construction in order that the direction and manner of resistances and arrangement of parts may be examined more readily. The frame shown in Fig. 350 is simi- lar to the elementary shape of Fig. 359, which resembles a sim- j)le connecting rod ; which indeed the base plate really is, the ■.--.'/ -'. Fig. 359. variations being due to the especial conditions and not to any fundamental difference. The bridge frame, Fig. 346, is in ele- mentary form Fig. 360. The step supports of Fig. 347 and 34S may be shown in principle either in Figs. 360 or 361, since in these elementary schemes a bearing may be shown either by J^] Fig. 360. the journals or the reverse. The four-fold bearing support just described may be sketched in Fig. 362. To show how these elementary sketches may serve, the fol- lowing application to one of Lemielle's ventilators will indicate. Fig. 363. Here, Fig. 363, nine bearings are to be supported. Three of these are for the drum, which is fast to the driving crank ; it is carried by the two neck bearings at A and B, and the thrust bearing at C. The six bearings at Z), E, F, and G. H, I, are for the rods of the buckets ; the supports for all of these are then the beams A, A, , the masonry, and the cranked rod B, I, D,C. 'i 127. Cai,cui= 33,000 lbs., a solid cast iron column 157.5 in. high, the diameter rf— 0,024s Vi ^ /" —4.15'', or about 4,V'. Under the same conditions a wrought iron column would be 3j4" diameter. An inspection of the formula shows that the shorter / becomes, the smaller is the value of d. The cross section must, however' never be allowed to become so small that the limit of permis- sible stress shall be passed. In order that the stress upon the cross section shall not exceed 8500 lbs. for either cast or wrought iron (their modulus for com- pression in either case being 21,300 lbs.), d should in no case be taken as less than d = 0.0122 sf P or the load should not be greater than P=6397rf^ • (112) The following table for round solid cast iron posts is calculated from formulas (no) and (112), and gives the loads which may safely be put upon columns of the respective heights and diame- ters given. The quantities marked with an asterisk are calculated from formula (112) and are a marked reduction upon the loads other- wise obtained. T E ■'■Drewitz has tested cast iron columns with a load equal to tt^-^^ without obseri'ing perceptible alteration. Frbkam's Banzeibung, V., p. 534. THE CONSTRUCTOR. 83 STRENGTH OF SOWD CAST IRON COI^UMNS. d /==Sft. 10 ft. 12 ft. 14 ft. 16 ft. iSft. 1 in. 297 191 132 97 75 59 i>4 1.504 994 671 493 377 298 2 4,753 3,055 2,122 1,559 1,193 942 ^y^ 11,600 7,460 5,180 3,806 2,914 2,302 3 24,060 15,830 10,740 7,892 6,043 4,774 M 44,470 28,660 19,900 14,620 11,200 8,845 A 76,040 48,890 33,950 24,950 19,100 ■5,090 4% 121, Soo 78,310 54,380 39,950 30,590 24,170 5 154,400* 119,300 82,890 61,460 47,060 37,180 •SK 186,900* 174,700 121,400 89,160 68,260 53,930 6 222,400* 222,400" 171,900 126,250 96,680 76,400 Hollow Columns. — Cast iron Columns are gener- erally made hollow. The dimensions in this case may readily be determined from the formulje for solid columns. If the external diameter is «'(,, the internal di- ameter (/„ and the diameter of a solid column of equal strength, d, we have Y^=(l)' Fig. 365. • • • (113) d. The ratio of internal to external diameter — !- ^ 1/, is conve- niently made 0.7 to 0.8. We have for : 0.5 0.6 0.7 0.7s 0.8 0.S5 0.9 0.95 1. 10 1. 14 1.20 1. 31 1.52 i, = ~^= I.0I6 1.035 The limits of stress fall within the formula for compression and the above results are close approximations. It is to be ob- ("6) d,= d,\ i-0-000I5;t-J in order that satisfactory castings may be produced. E.vample 3. In a barracks in Berlin are hoUow columns of 142 inches height, bearing loads of 37.i3o lbs. These are made of diameter rfj = 6^" According to (115) this should give the internal diameter : di = 6.1875 Wi — 0. According to (116) we have di ^ 6.1875 ,00000036 37^l8o^X 142- = 5.S8" V I — 0.C0015 37,l8° (6.1875)2 = 5-71" This would give a thickness of metal of about ]4". The empirical thickness for such a column is about 'i", and the actual internal diameter was 4%". Example 4. A cast iron column of 1S5 inches height and gii inches outside diameter has to bear a load of 275,000 lbs., and was made with an internal diameter of 63^ inches. According to (115), for direct resistance to thrust we get: but according to (116) : ''i = 925 .. 275,000 X (iSs)- - 0.00000036 "' ^ ^J ^' = 7.92" (9-=5)* V^ 275,000 -0.00015 — = 6.65" '9-25)= or very near the actual dimensions. These examples show how important it is, to take all the conditions into account, in order to avoid errors, and a careful examination of the circumstances attending each case should always be considered. Fluted Col/ii/iiis.— The cruciform section may serve as an ex- ample of such columns. The thickness and breadth, b and li, of the ribs may be determ- ined by comparison with the diameter d, of an equivalent round solid column by making : k = 35 16 not be less than : from which the approximate thickness d, for any breadth h, may be obtained. In order to keep within safe limits the cross section should bli==- or the load more than 17000 I (US) /'= 17000 i/; . Example 5. To substitute a cruciform column for the solid one of E.k- ample i, we may take h = i.sd = 1.5 X 4-15 = 6 225", We then have from (117) . 4 = 4.15 X 0.59 ' 4-15 N ■ 0.72" \ 6.225/ The safe load according to (118) would be : P= 17,000 X 6.228 X 0.72 = 76,200 lbs. For a direct calculation of b and /i we may use the following : 30 PP PI' ' b = ^5 =: o ooo.oco 21 14,220,000 - 2 /;' A ' and hence : P^ 4,762,000 bh^ \ ■ ("9) Care should be taken that the load does not exceed the limit given by (uS). Example 6 In the new building of the sugar refinery of Waghilusel, built in 1859-60, are columns of cruciform section. Those in the basernent bear a load of 264,000 lbs., and are 78.74" high, the ribs being 2" X i4iV'- According to (119) these posts should sustain a load of 2 X (i4.iS75l' P = 4,762,000 —— '' ,„"' = 4,386,000 lbs. (78.74)- According to (118) P= 17,000 X 2 X 14.1875 = 482,300 lbs. which is much more than the actual load. Columns of Angle and T Iron. — These are much used in bridge trusses, especially in America. (See \ 87). The vertical posts may be considered as columns with jointed ends. Case I, Fig. 364, and the upper chord is in compression and may be con- sidered as Case III, Fig. 364. The following figures show many of the forms, in section, which may be used for this purpose. auHH LJ JL n '^v Fig 367. The first is the column of the Phoenix Bridge Works at Phoenix- ville, Pennsylvania. This is shown made of four segments, but six or more are used. This form may be strengthened by rivet- 84 THE CONSTRUCTOR. ing flat iron between the joints of the segments. The four fol- lowing sections are from the Keystone Bridge Works. The sectional distribution of material should be chosen so that the i equatorial i moment of inertia on both the principal ase= are the same (see {7). The fifth section shows a double T iron, in the middle in dotted lines. This is used in bridge chords, where two or more such shapes are sometimes intro- duced. The last form is a combination of four pieces of angle iron recently usel for pump rods in mine shafts. The resistance to thrust is here dependent upon the distance between the guides of the rod. Grouped Coliuiins. — It is sometimes a question whether, in the support of very important loads, as well as for economy of material, it is not best to use two or three columns instead of one. If we let m be the number u^ed, instead of one, we have, for the supposition that the columns are in compression, the re- lation for similar sections. Fi = v/;« y . (120) Tliis shows that grouped columns use vwT times as much material as a single column. It is also economy of material to use a small number of heavUy loaded columns to sustain a given load. Example 7. This subject may also be treated by the aid of the preceding table. ]ff we have a load of 2Sco lbs. upon a column iS feet high, the diam- eter for a solid round column would be 2%'', while for four columns of 2 inches diameter we have 4 X 740 = 2960, or about the same. The cross sec- tions are to each other as 4 X (2)^ : : (2.75)-, or as 16 : 7 56, or y/j, : i. Variations in the hei=;ht of columns affect the economy of material, other things being equal, to a marked degree, since the resistance to compression varies directly as the height (/). It is sometimes desirable to make a column in sev- eral portions, when a proportional reduction in height can thereby be secured. The triple central core of the column shown in Fig. 36S, is an ex- ample and is a form often used by architects in connection with columns of brickwork.* This is not as effective as a single column, since the volume ratio is yi ^m, i. e., _;< -v/ 3 = 0.865. In conclusion it must be remarked that the col- umns which are used in machine construction are usually made much heavier than the preceding calculations indicate. This is due to the fact that such columns are often subjected to bending and tensional stresses, as well as to much vibration and the additional material is needed to meet these con- ditions. Columns of cast iron which are subjected to tension, as in the framing of vertical engines, should be made at least double the section given by (ii2'i, ('114'), (1161, and (ii8). The security is also made greater in the case of buildings, as the result in Example 6 shows. § 128. FORilS FOR IRO:s COLUilXS. The columns which are used in machine construction must be held down to the iron base plates of the machines, or if used in connection with building construction are secured to foundations of masonry. Heavily loaded columns are often placed upon foundation stones with only a sheet of lead beneath, and no fasten'ng, but otherwise some form of anchorage must be used. Fig Fig. Fig. 370. Fig. 371. The illustrations show three forms of fastening. In each case the sole plate is placed beneath the pavement. In the first case a special form of sole plate is held down to the masonry by an anchor bolt ; in the second the flange which is cast on the column is bolted to the keys shown ; the third construction (by Eorsig'i is arranged with a short cylinder bolted to the faced sole plate and made so as to give a space in which melted lead may be poured after the column is set in its exact position. A hole is left in the side of the column to admit the melted metal. The portion of the base of the column which shows above the pave- ment is made to conform to the general style of the building. In Fig. 369 a simple moulding is used between the plinth and shaft; in Fig. 370 a bead is added; and in Fig. 371 a double moulding of more elaborate outline is used. PIG. 372. The capitals of such columns are made in many varied forms. Fig. 372 shows, in section and elevation, a capital arranged to carry a beam and also to support the base of the column of the floor above. A recess in the top of the column receives the main beam, and affords a good place for a joint. If iron beams are used, this recess is made proportionately narrower. The base of the upper column is seciu-ely bolted down as shown.* Fig. 373- Fig. 374- Fig. 375- The capitals of iron columns afford much opportunity for ef- fective decoration, which in many cases is neglected, although comparatively easy of execution. For the lower columns of heavj' buildings the simple cubic capital so often found in Ro- manesque buildings is most suitable, and a good example is shown in Fig. 373.+ Fig. 377. Fig. 37S * For example, the columns in the vestibule of the theatre at Carlsruhe. A somewhat lighter form is shown in Fig. 374, and for some situations the various Gothic capitals are suitable, Fig. 375. In * other forms will be lound in Brandt's Eisenkonstruktion, Berlin, 1463. t Shown among other places in the Osteu. Lloyd, in Trieste, and in the Ar.senal at Vienna. 777^ COXSTRUCTOR. all three examples the pattern making and moulding is not dif- ficult The form most used in machine construction is shown in Fig. 3/5, being something between the Roman Doric and the Tuscan orders, and having an echinus beneath the cap plate, and an astragal bead around the column a short distance below- Bj- Tarj'ing the distance of the latter from the former the effect can be modified for taller or shorter columns. The heavier form of the Grecian Doric is unsuitable for ma- chine construction and is seldom used. Jlore appropriate is the modified Corinthian capital shown in Fig. 377. The top is a cornice of overhanging leaves, terminating in an astragal on the shaft. By omitting the ornament the same form may be re- tained, as showu in the right hand half of the illustration, and also in Fig. 34.S. The fluting of the column is by no means ob- jectionable, at least in Germany. The fluted capital is readily cast by being made in a core box. Fig. 378 shows a capital of Renaissance form with octagonal abacus, well suited for slender columns. The support of beams, either iron or wooden, is best accom- plished by the introduction of a plate between the column and the beam, and this may be treated simply, yet in harmony ^vith the style of the rest of the work. Fig. 379 shows such a sup- port on the cubic capital already shown, and is adapted for very in which the solidity and substantial character of this fomi of construction- is well shown. Fig. 3S2. Fir. 3S1 hea\'3^ construction. Fig. 3S0 shows a lighter capital, in which the support for the beam is made of a box form ; Fig. 3S1 is a still lighter design. This illustration also shows the effect of a iigh stj-lobate or base moulding, suitable for tall slender columns. As shown in this example, such bases are usually made octagonal in section, which approaches the Gothic stj'le, but they are fre- quently made round. As in architecture, the columns are usu- ally made tapering from below upward, the upper diameter be- ing o.S to 0.7 that at the base. Fig. 3S2 shows a more elaborate form of capital and bearer. Columns of cruciform section, already referred to, are often used in the construction of industrial establishments. The}- are sometimes to be preferred to hollow columns, since the latter are often cast of such unequal thickness as to be unreliable. Figs. 3S3 to 385 show such a column. Fig. 3S3 is from the Rail- way of St. Germain. Here the flutings extend from top to bot- tom and the column is swelled slightlj' in the middle. The form shown in Fig. 3S4, from the Tobacco Factory at Strasburg, is more elegant in its appearance. Here a rectangular base is used for the lower floor, but is omitted above. The method of connecting the base and column, as well as the connection between capital, beam, and column above, is shown in Fig. 3S5 ; -:^ m / Fig. 3S3 Fig. 3S4- These examples will serve at least to show the varietv of forms of columns which may be used, and the manner in which a little oruameut may be introduced into machine construction. CHAPTER VIII. AXLES. I 129. Various Kixds of .\s;i,es. Axles may be considered as beams which carry revolving or oscillatiug loads, and hence are provided with journals at certain portions of their lengths ; they may be subjected to deflection or to compression, as in the cases of journals already discussed, according as the load act normal or parallel to the axis. Axles which are subjected only to thrust, are not often found, the far greater portion being those bearing deflecting loads, although, many are under combined stresses. These may be divided into two classes : those which have the load applied at but one place, and those in which several loads are borne at various points. The first are called Simple-loaded Axles ; the second. Multiple-loaded Axles. The sections of axles of cast or wrought iron may be either circular or varied, and this gives rise to another subdivision in the calculations. The methods of graphostatics are especially applicable to the subject of axles, and in the following pages both numerical and graphical solutions will be discussed. A. AXLES JriTH CIRCULAR SECTION. SiMPtE Symmetric.\i, Axles. Q Fig. 3S6. 86 THE CONSTRUCTOR. The load Q is in this case applied normal to the direction of the axis, midway between the two journals, upon a seat for a hub, as shown in Fig. 386. The portion between the hub-seat and journals is called the shank of the axle. The journals are proportioned according to the methods given in Chapter V, taking /'= ^ Q, and the axle then proportioned so as to give approximately the same strength as the journals throughout. Let: d = diameter, I = length of journals, e = height of shoulder or collar, D = diameter of middle, or hub-seat, i^=its breadth, D' = diameter of shank at the junction with Z), e' = }i [D — D') the shoulder at the latter junction, a = the length of shank, then we have : £y_ if a — 0.5 d d^f 0.5/ (I2I)' This will give the axle the same security as the journal, so that the approximate stress will be .S'=S5oo lbs. for wrought iron, and 4260 lbs. for cast iron. If a higher or lower stress is desired, the journal should be proportioned for the desired stress, and the corresponding dimensions for the axle deduced. The strongest form for the shank of an axle is that of a cubic parabola (see § 10, No. VI), and the student will find this a valu- able subject for investigation. In practice it is made a portion of a truncated coue whose larger diameter^/?', and the smaller diameter = (/= 2^. The value of c^ should only be made large enough to provide sufficient depth for a keyway. Non-Symmetrical Simple Axles. If the two shank portions of a simple loaded axle are of un- equal length, as in Fig. 387, the load on the two journals d^and d^ will be unequally divided, and we have the proportion a, P. P._ P., a 2 P, (122) Q 'Zi + «2 ' Q «i + "2 -• 1 "1 The hub-seat divides the axle into two parts, each of which may be considered as the half of a symmetrical axle, and so the Fig. 388. ■whole proportioned. The value of D' is determined for each shank, and the greater value taken for both sides, li a-^ = a^ the a.xle becomes symmetrical. If the seat for the load Q does not lie between the two jour- nals, but projects, as in Fig. 389 [a^ becoming negative), the load is said to overhang, and the journal /? becomes a neck journal (see § 92). We have for the relations of forces : 5. Q Q , + «. Px (123) ^\ Pi «1 + 0-2 The diameter of the journal di is first determined, then a diameter for an assumed journal d., for the loaded point, and then a value D for a neck journal, taking for D the greater of the values for £>' and D", as given for the two ends by formula (121), the length of journal being made /, = v /j' -j- //. The cubic parabola is shown drawn in the shank a.^, being the shape for uniform resistance in this case, and we have for the diameter 6, at the root of the hub-seat : 6 ^ (124) See (2 10, Case VI, Remark). Exampie.— 'L^i the load Q, acting in one direction, be 14,520 pounds, a-i = 47-25", ^2 = 23.625'', b-=i2,", material cast iron, number of revolutions, n^ 150. We have /'i = 0.5 ;2= 7,260 lbs. /'3=i. 5 ^=21,780 lbs. From the table of ggi we have o'x^ about 3^", tfa = about 614!", also /i = 5!^", 4 = 9^". "We then have: Z> = 6.25 (5.62; -6.25 ^^ I^Sts h = ^(5.625)= + (9.375)2 10.72, say io3;i". 3 9-375 10.93, say II . 6.97, say 7". 2 1^2. Graphical Calculation of Simple-Loaded Axles. The determination of the forces acting upon the journals may- be made according to the methods given in Cases I to V, of I 39. In a similar manner the cord polygon may be employed as in ? 43 and I 44 to determine the statical moments of the parallel forces at various points, and the polygon so constructed may be called the surface of moments. The simple method about to be given will serve as a general graphical solution of the problem. /. The Load Acts Normal to the Axis. Fig. 389. Fig. 390. (a). Hub and Load between the Journals. — Draw the line A C, equal in length to the distance between centres of journals, and upon it construct any triangle ABC, whose apex lies on the line of the load Q. Draw A 3 normal to A C, making -4 3 = O; draw 3 . O parallel to B C, and 2 . O parallel to A C ; then A . 2 ^7-^,2.3 = P.,, By dropping the perpendiculars from the ends of the hub-seat we may divide Q into two forces O-^ and Q.^, shown in the force polygon by O b, parallel to B^ B, ; giving A b^^ Oi, b . 2^ Qy The vertical ordinate /, at any point of the surface of moments is proportional to the statical moment Ji/v at its point of intersection with the axis, as for example the or- dinate /j, at the base of the journal forPj. We have in any case ; y'= -^--^Mj.. d,^ = ^'-^M, and hence ; = 4 (124} from which y can readily be obtained.* (b). The Hub-Seat between the Journals and the Load Over- hmig. Fig. 390. — Draw the line A C, parallel to the axis, con- struct a triangle with the points A, B and C on the lines of the directions of the forces, drop a perpendicular from the point D, where D d=Q, make O . i parallel to A C, and equal to CD, make A . 1 . t, normal to A C, also O . 3 parallel to C B, and i . 3 will = Q, A 1 = P^, 2 . A = P.,. The force O is decomposed into two forces at the ends of the hubs, and by dropping thf: perpendiculars, the points C, and C, are determined, and O c drawn parallel to C^ C,, giving the values c . 3 and i . C for the * If it is desired to determine a series of values of /, beginning from f^, it may readily be done by using a table of cube roots of numbers such as are given at the end of this volume; if the greatest value of r is the starting point, the table of cube roots of decimal numbers is useful, the space being divided into ten parts and the outline laid off correspondingly. THE CONSTRUCTOR. «7 forces at C^ and C.^ respectively. The diagram shows that at a point within the hub-seat the stresses are reversed and the bend- ing moment is zero. Fig. 391. Fig. 392. (c). Overhung Axle ivilh Load Outside the Journals, Fig. 391 — Construct the triangle A B C, as in the preceding case {b), and place D so that Dd= O, draw A . 3 normal to A C, make O ■ 2= CD, and parallel to A C, and draw O . 3 parallel to C B, and we have again A . 2 = P^, ^ . A=: P.,. Divide Q into Cj and C.2 and make Oc parallel to Cj C,, giving c . 3 and 2 . c for the forces at Q and C.,. The journal at B being uniformly loaded, its moment surface is outlined by a parabolic curve (see ^ 42). Fig. 394. (d). Overhung Axle, with Load bettveen Journals, V\g. ^^2. — Construct the triangle ^ B Ca.s in case (a), divide Q into Bi and B,, which gives the polygon A C, B^ B., (which is equiva- lent to the other one A C, jS'j B.^). In the force polygon, i . 3 = g, 2 . I = Pj, 3 . 2 ^ /^2i ^°, hence we get for its vertical component O =fif\ — e^ e,, which combined with Pi gives the resultant P.^. This is proved by the intersection of P/ and /■/ at S must fall on the line of the resultant of P^ and P,. If we neglect the compression in the direction of the axis, we may now draw the force polygon a 2 3 O of the forces P^, P.,, P3, P„ as shown at the left of Fig. 407, and thus obtain the surface of moments abed. Fig. 40S. A crane with swivel column, to which the jib or boom is rigidly attached, may be examined as shown in Fig. 40S. The position oi O = L -\- G \s taken as before, making q-^ q.^ repre- sent Q, draw A q^ normal to the axis, join 9, D and draw ^2 ?3 parallel to A q^ till it intersects with q^ D. We then have q^ q^ for the horizontal force P^ at A, and q., q^ the corresponding horizontal P^ at D. The step bearing at D will be subjected to an inclined thrust, the resultant of O and /'j. In a similar manner we obtain the horizontal forces P.^ and P^ equal and opposite, and acting at B and C, and the resultant of the force at B with O gives the inclined force due to the rod B E. The four horizontal forces have the same action as the load on the axle in Fig. 394. We may thus obtain the surface of moments abed, which shows a zero point for bending moments between B and D , and also indicates a forward bending above and a backward below. In the force polj'gon 2 a-= P.,, (Z 2 = P^, 21 = Pi and I 2 = /"j. V \ ■ 1a. bI ic yP; Fig. 409. §136. Axi,ES WITH Three or more Bearings. The number of bearings for an axle is often as great as four. In such a case the forces and moments ma)' be found as follows : 90 THE CONSTRUCTOR. Starting at a, Pig. 409,- with the given forces I to 5, we form the force polygon a^ O, and, according to J 40, the link poly- gon abed efg, and join the closing line g: a, parallel to O 6, in the force polygon ; giving 5 . 6 = the force P.^ at G,6 . a~ the force F^ at A. From P^ and P^ the journals d-^ and d.^ may be determined, and the ordinates of the cord polygon give the means of obtaining the axle diameter as before. ^ The intersection^, oi ab unAfe, prolonged, is a point of the line of direction G g, of the resultant of the forces i to 4. If it is desired to find the successive resultants of the various forces as they are combined (see I 40), it will be found convenient to choose O, so that a/ will be parallel to A F. The inclined link polygon may also be transferred to a closing line parallel to A F. If the shanks of the axle overhang the journals, as in Fig. 410, the procedure is similar to the preceding. Beginning at the point a, the force polygon « 5 C> is coustructed, and the first side of the cord polygon b a, drawn to the line of the first force, the second to the line C c, of the second force, and so on to the closing line eb. The first and nth line of the cord polygon in- tersect as before on the line H h of the resultant. Variations on these examples may occur, as when the loads act in inclined directions, or opposed to each other, the methods being similar in all cases. I 137. Axi,ES WITH Inclined Loads. The analytical investigation of axles becomes more difficult when, as in Fig. 411, the loads act in different planes, but the graphical method is readily applied. The force polygons A O^ Fig. 411. I, and D O., 2, are constructed for the forces Qi and O.,, re- spectively. Fig. 412, the polar distances G O^ and HO., being made equal to each other, so that the closing lines of the two cord polyjjous A b' £>, aud A c" D, coincide in A D. Then construct the second cord polygon with the inclined ordinates B B" = Bb", CC"= Cc", &c., making the angle u with the force plane of the ordinates of the first polygon, and inclined backwards as drawn. Then make B b = B" b\ Cc= C" c' , E e ■= E" e' , &c., and draw the cord polygon A befcD, fitem which can be obtained (according to \ 44) the bending moments Fig. 412. for the axle. The line b efc is a curve (hyperbola), A b and c D are straight lines. Draw O^ O/ parallel to A i. On O./ par- allel to £> 2, and drop the perpendiculars O/ J and 0/ /\, and A I will be the force on the journal /\, and D A^that at P.,, measured on the scale of the force polygon. Th'eir directions are determined by combining A G with H :!., and D II viith. G 1 at the angle u. B. AXLES WITH COMBINED SECTION. I 138. Annoi^ar Section. If it is desired to make au axle with annular section, or iiE other words, a tubular axle, the journals should first be calcu- lated, according to the method given in ^ 90 for tubular journals, and then, retaining the same proportional thickness, determine the dimensions of the other parts in the same manner as for solid axles. The most commonly used ratio of internal to ex- ternal diameters is o . 5. Instead of doing this, all the dimen- sions for a solid axle may be determined, and then having chosen a ratio for diameters, increase all the sizes according to formula (95). See also I 141. ?I39- Axi,ES WITH Cruciform Section. In cases where axles are made of cast iron the cruciform sec- tion, with circular centre and four ribs, is sometimes used. The- shanks are then usually made of the ordinary conoidal form. Fig. 413, and in some cases the ribs gradually swell into a junc- tion at the ends with the central core, Fig. 414. Fig. 413-414. In designing such an axle, first proceed as if drawing a solid ■ circular section as shown by the dotted lines, of the diameter corresponding to the 'portion A'' when the ribs join the head. Then for any point {x) of the shaft : y = the diameter of the assumed round axle, or equivalent conoid , h = height of ribs ; b = thickness of ribs ; k = diameter of core ; and the proportions are obtained from the following formula : + 16^ h J^ h (-1)1 (I27> This formula serves for the pure cruciform section, without core by making k = b. The results vary so slightly when k = 0.2 h, that the follow- ing table may be used for both sections : b h h k Values of — when — y k 0.80 0-75 0.70 0.65 o.bo 0-55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.05 1.30 1.40 1.50 1.61 1.72 1:84 1.94 2.04 2.1.S 2.18 2.22 2.26 2.27 0.06 1.30 I -39 1.48 t.s8 1.68 1.79 1.87 I -95 2.02 2.07 2. II 2.13 2.14 0.07 1.29 1.38 1.46 1.56 1.6.S 1.74 1 82 1.89 1.94 1.98 2.00 2.02 2.02 0.0S 1.28 1.36 1.4=; I.S3 1.62 1.70 1.76 I.«3 1.87 I.ql 1-93 '•93 1.93 0.09 1.27 1-35 1.43 i-.Si I ■,=59 1.66 1.72 1.77 1.81 1.84 1.86 1.87 1.87 o.io 1.27 '•.34 1.42 1.49 l.=;6 1.S3 1.68 1.72 1-75 1.78 i.8o I. So 1.81 0.1 1 1.25 l-,33 1.40 1-47 1.54 1.60 1.64 1.68 1.71 1-73 1.74 '•75 '•75 0.7 2 t.2S 1.32 1.39 1-45 LSI I..S7 1.61 1.64 1.67 1.68 1.69 1.70 1.70 0.13 l-s.'^ 1.3 1 l-.3« 1-43 1.49 I..S4 I.,S« 1.61 1.63 1.64 '.65 i.fs 1.65 0.14 1.24 1.30 1.36 1.42 1.47 I.";! i-SS 1-57 i-,S9 1.60 1.61 1.61 1.61 0.15 1.23 1.29 I.3.5 1.40 1-4.'; 1.48 1.52 '•54 1.56 '■57 1.5s 1.5s 1.58 0.16 1-23 1.2S I -.34 '.3« 1-43 1.46 1.49 r.52 1-53 1.54 i-SS 1^55 '•55 0.17 1.22 1.27 1-33 1-37 1. 41 1.45 '•47 1.49 1-50 i.S' 1-52 '•52 1.52 THE CONSTRUCTOR. 91 Exajnple i.— Simple Cruciform Section. — If the height of the ribs at any point is made double the diameter y\ of the ideal conoid, we have in the third line of the table, lirst and last columns, the thickness of rib b = 0.07 h. Example 1 — Suppose a core to be used and at any given place h = i.^y, and k = 0.6 /:, we have, according to line S, columns 6 and i, 6= 0.12 of the height h at the same place. ? 141. Compound Axi,es for Water Whe;e;i,s. In Fig. 417 is chown an axle for a water-wheel, made of cast and wrought iron. This was made to replace a broken axle of wrought iron, for a wheel 32. S feet (10 m.) diameter, 19. 6S feet (6 m.J in width.* The load is carried at four points, as shown Fig. 415- Fig. 416. giving a total of 82,104 Ihs.f The shaft consists of a drum of sheet iron yi" thick and 44" outside diameter, made in three sections riveted to the central spiders of the wheel. The two journals are fitted to the cast iron heads with a slight taper, the ends being prolonged into the middle of the drum, where they are drawn together by a right and left hand nut. The journals We may make b, constant and determine k, or let k be con- stant and d vary. The latter case is shown in Fig. 415. Here the shanks are also cruciform in section, and the hub-seats are made to receive keys, as shown in both sections, and the central one is strengthened by transverse ribs, A small auxiliary jour- nal is shown at the end of the main journal, and is very useful in erection. I 140. Modified Ribbed Axle. For heavily loaded axles the form shown in Fig. 416 is suit- able, the ribs being provided with flanges along the edge. Fair- bairn has used such axles for water-wheels, and Rieter& Co., of Winterthur have made them for the same purpose. The pro- portions are determined by taking the diameter y, of an ideal shaft of circular section, and calculating /;, as before. We may then make the flange thickness c = b, the thickness of the ribs, and then the flange breadth b-^ is obtained from the formula : b 1 + \(>\ hi h \ h I (128) from which the following table has been calculated : b A 0.05 0.06 Value of -'-, when — b y l.IO 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 , 3-64 2-75 7-94 6.17 4.81 6.99 5-38 413 317 2.34 1.07 0.07 6.70 5-12 391 3-45 2.24 1.61 I.OI 0.08 0.09 O.IO 6.82 516 3-91 3-45 2.24 1.61 1. 17 5-45 4.1 1 3.10 2-33 1-73 I.OI 6.00 4.48 3-37 2-53 1.89 1-39 0.1 1 5-05 3-77 2.82 2.1 1 1-57 IIS 0.12 6.56 4-34 3-23 2.42 1.80 1-34 0.13 5-73 3-78 2.S1 2.10 1..56 1. 15 0.14 5.06 3-34 2.4S 1.85 1.38 1. 01 The ratio between b-^ and b is never made greater than 6 to 7, and as it does not fall below unity the table is only given be- tween these limits. The profile is determined for a few points and these are joined by a continuous line. C D 11,924. 11,924. j -~Lf- -"-'^ Fig. 417. are ^Ys," diameter and 1 1 " long. The circumferential joints in the drum are strengthened by pieces of angle iron as shown. The stress in the shell of the drum is only 3100 lbs., and on the riveting about 6400 lbs. I 142. Construction of Rib Profiles. In drawing the curved outline of ribs such as shown in the preceding designs, the following methods may be employed. In the various diagrams A B is the geometric axis of the piece, ,S the highest point of the curve, and /C the lowest point, these both being already determined. I. Circular Arc. — This can only be used to advantage when on such a small scale that it can be drawn with compasses or trammel. Fig. 418. 2. Parabola.— V>rs.-^ S D and C K parallel to A B, divide 5 D into any number of equal parts, as for example, six parts, and divide Z) A' into the same number. Drop perpendiculars from I, II, III, &c., join the lines 5' i, 5 2, 5 3, &c., and the in- tersections of these with the perpendiculars I, II, III, &c., will be points in the parabola. 3. Sinoide. — Draw 5 Z> and (TA'^parallel to A B ; with a radius A S draw a circle about A ; divide the arc 5 £, cut off between S D and (Tiifinto six, or any number of parts ; draw from the points of division, lines parallel to A B, and from I, II, III, &c., perpendiculars to A B, and the intersections will give points in the sinoide. *This wheel belongs to the Societe des Faux du Rhone, at Geneva. Annales du Genie Civil, 1866 and 1S72. t See diagram in Fig. 409, where the loads are in this proportion. See 92 THE CONSTRUCTOR. 4. Elastic Line. — By bending an elastic rod of uniform pris- matic cross section, keeping it upon the points K^, S, and A'j, the elastic curve may be drawn directly from the rod, using it as a ruler. For large sizes the rod may be ;54" to iX"thick, and kept under water : for smaller sizes, about ){'' thick is sufficient. Let: &- 5. Cardiode. — The following method may be used for drawing the curve directly in the pattern loft. A wooden template S' K E C (Fi.g. 420) is made, in which E C and E S' are straight edges, and C S"^CS, and CE^C K. Guide points are placed at C and A', and the edge C E kept against the point C, and the edge S' E against the point K. The point S' of the Fig. 420. template will then describe a cardiode curve and by attaching a pencil point at S' it may be drawn directly for pattern makers' use. The most convenient method in practice is to obtain a few points by (2) or (3), and then join them by a flexible spline or ruler. ? 143- Wooden Axi,es. For some water-wheels axles of oak are still used, and these are made polygonal in section. They are made prismatic, the diameter being at all points equal to that necessary at the point of greatest stress, and the methods of attaching journals are shown in i< 102. The diameter may be obtained by multiplying the diameter for cast iron by 1.55 (the cube root of the ratio of the modulus of cast iron and oak). This must be the full actual diameter, as it is sometimes weakened by mortises cut for the arms of the wheel. Should this give a less diameter than required for the attachment of the journals, the diameter at the latter point must be taken for the whole axle. The choice between iron and wooden axles must be governed entirely by local reasons of cost and convenience. Example. — A. water-wheel axle with shanks 106.25" long is loaded so that the journals of cast iron require to be 3^" diameter and 55^" long. Accord- ing to the formula in ^ 130 we have D ■- 2.625 For corresponding strength in wood, the axle should be at a minimum: D' =|i2 X 1.55 = 1S.6". CHAPTER IX. SHAFTING. I 144. Calculations for Cylindrical Shafting. In machine construction those axles which are used to trans- mit twisting moments are called shafting. In order to fulfill this purpose two requirements must be met : first, the ultimate strength must be sufficiently great, and se- cond, the torsional spring must be kept within proper limits. In actual practice, shafting is subjected not only to torsional stresses, but also to bending due to the weight and pressure of gears, pulleys, levers, etc., which are carried. These latter in- fluences will not be considered at first, and the calculations made only for round, solid wrought- and cast-iron shafting. P^ the force acting to rotate the shaft ; R = the lever arm at which it acts ; 7V= tiie horse power transmitted ; n = the number of revolutions per minute; rf^the diameter of the shaft; L = the length of shaft in feet ; iJ = the angle of torsion in degrees ; 6'^ the fibre stress at the circumference; G = the modulus of torsion of the material = -| of the modu- lus of elasticity. We then have for strength : xf: ^\^PR. (129) and for stiffness : d=^. J2_ ff G • ('3°) In order to have the same security for the shafting as already given to journals the value of 6' should be only i the fibre stress (see 2 5), but in actual practice the stress is taken the same as for journals, viz.: for wrought iron 5=8500 lbs., and for cast iron 6' = 4250 lbs. This gives the following results for strength : For wrought iron shafts. rf= 0.091 ^^ Pi? = 3.33 ^ N For cast iron shafts, a' = o.ii4 ■^■^ Pi? = 4. 20 9°^ 0.075 ^, w'hich gives for stiff'- ness against torsion : N N and for cast iron .shafts : d = <:>.2S7<^~P^=5- jV The quotient of effect — is obtained from the relation to the n statical moment /■ J? as follows : 33000 X 12 ^<1 (133) (134) PR=- =: 63.020 11 II (135) From these formulae the following table for round wrought iron shafts has been calculated. An inspection of the table will show that it is quite possible for a shaft to be strong enough to resist permanent deformation and yet be so light as to be liable to spring under its load. For example, a shaft 26 feet long, with a twisting force of 220 lbs. applied at one end, and acting with a lever arm of 20", gives a turning momeut /'A'^4400 inch lbs., which wottld require a shaft only li inches diameter (see column 2). This, however, would permit far too much torsion, and in order that the angular deflection should not exceed the limit of 0.075° per foot, a corresponding value oi P J? \n column 4, must be found, and against it in column i will be given the diameter, in this case about 2f " ; which, by comparison with column 2, gives about five-fold strength. For short shafts this examination of angular deflection is un- necessary, as for example, in the short lengths between two gear wheels, for here the value of 1* will be small enough in any ca.'^e. With longer shafts, and in all special constructions, it is important to consider the angular deflection and keep it within the given limit. For shafting of cast iron the same table may be used by tak- ing double the values for P P, or for — For steel shafts, whose modulus of resistance is f greater than wrought iron, the diameters in both cases may be taken as V 0.6, that is, 0.S4 times that of correspondingh- loaded wrought iron shafts. Shafting which is subjected to sudden and violent shocks, as in rolling mills, etc., must be made much stronger than the pre- ceding formulas require, and these must be classed with the special cases which occur in every branch of construction. Example I. a crane chain carrying a load of 5940 lbs. is operated by a drum 01 7.3 inches radius, measured to the centre of the chain; required the 777^ CONSTRUCTOR. 93 diameter of the shnft to resist fhe torsion thus brought upon it. Here PR = 43,362, which by reiereiice to columu 2 gives a diameter between^ 3 and 3j^ inches, sa}' sj's". This would require to be somewhat increased for bending stress, for which see \ 150. Example 2. A turbine delivers 92 horse power to a wrought iron shaft, making 114 revohitions per minute, and of a length = SJ^ leet. In this case N = 0.S07, which, by column 2 in the table, would require about 35^ inches 11 diameter. If the deflection is not to exceed 0.075° per foot, we have, in col- N unin 4, a value of ~ = 0.S03, which gives a diameter of 4j^", and with this n diameter the angle of torsion would be 8.5 X 0.075 = 0.65°. ■^ similar case in practice has a shaft diameter of 5^", which gives a still smaller angular de- flection. ?. 145- Wrought Iron Shafting. FOR STRENGTH. FOR STIFFNESS (Torsional) d PR N 11 PR .'V n I 1,327 0.021 123 0.0019 iX 2,591 0.052 301 0.0048 ^y^ 4,479 0.071 625 0.0099 t-Ya 7,112 0.114 1,357 0.0183 2 10,616 0.168 1,975 0.0313 ^% 15. 115 0.239 3,t64 0.0502 2K 20,730 0.329 4,822 0.0765 2^ 27,600 0.433 7,061 0.1120 3 35.830 0.56S 10,000 0.1587 1% 56,890 0.902 18,520 0.2941 4 , 84,93° 1.347 31,600 0.5015 A% 120,900 1.919 50,620 0.8032 5 165,800 2.632 77,160 1.2240 sy 220,800 3-503 ir,ooo 1 .7920 6 286,600 4-548 160,000 2.5390 6X 364,400 5-784 220,300 3.4960 7 455,200 7.222 296,400 4.7040 VA 559, Soo 8.883 390, 600 6. 2000 8 679,400 10.780 505,700 8.0240 ' 8K 815,000 12.930 644,400 10.2300 9 967,400 15.350 810,000 12.8600 9K 1,138,000 18.050 982,700 15.6000 10 1,327,000 21.050 1,230,000 19.5900 loK 1,536,000 24.380 1,501,000 23.8100 II 1,766,000 28.020 1, 808,000 2S.6800 11;^ 2,0lS,000 32.020 2,159,000 34.2600 12 2,293,000 36.390 2,560,000 40. 6200 1 146. Line Shafting. In the previous discussion we have assumed that the bending forces upon shafting might be neglected. As a matter of fact, this is rarely the case, only occurring when the turning moments are those due to a simple force couple. Nearly all the shafting used for power transmission is subjected to bending stresses due to belt pull, pressure of gear teeth, weight of gears and pulleys, and to take all of these into consideration would make a very complicated calculation. In most cases ample strength will be given by taking the diameters according to the formulas (133) or (134). As already shown, these give ample strength, so that any ordinary bending stresses are provided for. These give reduced diameters for the higher speeds, shafting for high speed machinery running at 120, 140 or even 200 revolutions per minute. First movers run a lower speed and are proportionally heavier, and the line shafting generally is gradually reduced in diameter in the successive ascending floods of a building. Such line shaft- ing is only occasionally made of cast iron, when moderate power is to be transmitted. The practice in the proportion of shaft diameters is not alto- gether consistent. In many cases very high stresses are per- mitted, as in the case of locomotives, in which stresses of 12,000 to 15,000 lbs. are borne by wrought iron cranked axles; shafts of screw propeller engines usually carry 7,000 to 8,500 lbs., while in many instances the stresses upon line shafting are very light, when the high rotative speed is taken into consideration. This is particularly the case in England, the shafting running at higher speeds with a proportional reduction in diameter. The greatest difficulty to be encountered lies in the fact that the forces are rarely given with sufScient accurac}', the so-called "nominal" horse power which a shaft is supposed to transmit bearing no definite relation to the actual power. In most cases, however, the use of the formulas above given for stiffness, with a slight increase for very long shafts, will give satisfactory re- sults. A few examples will serve to illustrate the manner in which the methods given may be applied, and the remarks which have been made should be borne in mind in connection with the ap- plication. Example i. The screw shaft of a large war ship is driven by two cylinders, each exerting a total pressure of 176,000 pounds, on cranks of 21.75" radius, situated at right angles to each other The shaft is of wrought Iron, and between the crank shaft and the propeller it is 72 feet long, by 15" diameter. Calculating this for strength by forniula (131) we Jiave ; PR. = 2\/ 0.5 X 176,000 X 21.75 = 5,412,000 : 15.98", say 16" If it is desired that the torsional deflection shall not exceed 0.075^ per foot of length, formula (a33) must be used, giving: ' (f =0.3 i^5, 412, 000= 14.47". This is somewhat less than the previous dimensions, and consequently the deflection will be less than 72 X .075 = 5.4°. £.ra!J/ph' 2. In the mills at Saltaire there is a cast iron driving shaft mak- ing 92 revolutions per minute, and transmitting 300 horse power, the diame- ter being 10 inches. According to formula (134) the diameter would be : d = S-63 iJ/^take {M^~)^ =0.975 M^, -f 0.25 M^ . . (140) and when 7J/^ > 71/^ take (.^4); =0.625 M^ + 0.6 M^ . . (141) An examination will be made, first by the analytical, and then by the graphical method. I. Analytical JMclhod. — The axle or shaft ABC, shown in Fig. 421, carries'a gear wheel R at C, which acts tangentially to ■T"' Fig. 421. rotate the shaft with a moment Md ^ Q R, and also acts to bend the shaft with a force whose reactions are parallel to Q, and are Pi=— — at A, and P., = -^=-— at B. The greatest stress is a-\- s ' a-\- s at C, for there both bending moments are at their maximum Mb ^ P^_P, a , hence calculation should be made for this point. Example.— 'L&t Q= 5500 lbs. J\ = 11^^", a = iQ^^", J = 78^4'', then 78 7^ = —^ 0= 0.8 = 4400 lbs. 98.50 I9-7S , />„ = i.50 i2 = 0.2 iQ = iioo lbs. Also ^^d = 5SOO X 11-75 = 64,625- Mj^ = 4400 X 19.75 = S6,Qoo. M]^ ^^'^d and formula (140) is used. = 0-975 X 36,900 -1- 0.25 X 64.625 = 84,727 -I- 13,656 = 98,383" lbs. Hence We have (J>f/,)i From this the diameter at Ccan be calculated. If the shaft is of cast iron with cruciform sectiou, we have for the diameter D, and taking .S= 4250 -we have 32 oS3_X_32_ 4250 TT -.6%" The journal at .-i is found in the table of §gi, column 4, to be about 2^". For the neck journal at B, we have from the table of g 145, taking the double value for cast iron, d-i = ^Vz"- THE CONSTRUCTOR. 95 Graphical Method. — The same example may be solved graphi- cally, lu Fig. 422, with a horizontal closing line, construct the link polygon a b c, for the bending moments, and the force polygon a 10, giving the forces P^ and />,, and also a c c' , the surface of moments for the shank A C. The moment ISId is yet to be determined. In the force poly- gon with a distance R from the pole O, draw a vertical ordinate ; this will be il/rf. Lay its value off at c'r,, and bl\, and 's of these values give c' c^ b^ b for the parallelogram of torsion for the shank CB. Pig. 422. The combination of the bending and twisting moments may then be made by formula (139). Make cc.^^ '/% c c' and join c. b, then at any point of the polj'gon, as for example at/, the distance// ^ Ys/f'- Now transfer c' c^ to a b, at c' c^' ; then will the hypotenuse of the triangle C2 c' c/ divided by c^ cj = ■^{ficc')'' + (^iCtC^Y', and the sum cc, + c.^Ca'= c c, + ac^the desired moment {Mi,)t for the point C. In the same manner we obtain// -|-/2/„'=//,-|-//3 the moment {jVb)i for the point J^- The line ("3/ b^ is a curve (hyperbola) which may be taken approximately with sufficient accuracy as a straight line i\ b^,. The various dimensions may be obtained from the pol5-gou a c b b^ c^ c' in a similar manner as shown in the discussion of axles. Other discussions of this subject will be given when consid- ering rock shafts and crank axles. .. CHAPTER X. COUPLINGS. ? 151- Various Kinds of Coupi -"I _f"] • fore, viz., <5 = Fig. 449. fitted, each being provided with a bronze shoe. These are thrown in and out of action by means of a sliding collar £', which operates right and left hand screws by means of the lever d. The clamps slide in radial grooves and the details are fully shown in the illustration. The nuts for the right and left hand screws can be closely adjusted and clamped by set screws, so that a radial movement of less than y'^" 1== sufficient. There is no danger of wedging the parts fast in this form of clutch, as may be the case in cone clutches, as the elastic reaction of the cylin- der assists in the direction to release the parts. At the same tinJe the screws prevent the coupling from releasing itself and 2S the axial pressure Q, upon the collar B', can be transmitted so that the screws need not have too quick a pitch. If s is the pitch, b the length of lever arm,ythe coefiScient of friction of the clamping pieces, we have for the transmission of a given moment {P -R), neglecting the friction of the screws, P ^ s PJ? , . 7°'^=-i JR ('47> which gives a very small value for 0. If the parts are so arranged that B is the driven part, there will be no collar friction at B' , when the coupling is not in ac- tion. When the shaft is vertical, a weight may be used instead of a collar and lever, and by gradually lowering it the apparatus may be started with very little shock. The first clutch made by Koechlin was designed for the transmission of 30 H. ?."■' The above value corresponds to a minimum value of R. The modu- lus is the same as before 3 ^16 A verj' excellent form of this coupling was designed by Bod- mer, independently of Koechlin, f and a similar arrangemen.* has been adapted to mill gearing with success, j Fig. 450. Cylinder couplings in which the clamping pieces are operated, by toggle joints are also made. An example is shown in Fig. 450, which is a clutch by Fossey, as applied to mint machinery .§ This is a very compact design and is arranged with four clamps, which have no bronze shoes. The toggle links are as wide as the clamps and are fitted with half-journals to transmit the pres- sure outwards, while to draw the clamps back, light through bolts are used (see ? 95). If the toggle links make with the axis an angle 90° -|- o, we have for the axial collar-pressure : ^ Ptan a. PR tan a , „, 0= = (148) ^ f R f ^ ' The angle a may be taken very small, since there is no danger of clamping. The value may be as small as a = 2°, or even 1°. Fig. 451 For a = i^^° we have Q _ 0.03 I ~P " "a 15" ~~ T Another form of cylinder coupling using toggle levers, has^ * Bulletin von Mulhausen, 1S54, p. 138. t See Fairbairn, Mills and Millwork, Vol. II., p. 92. X See Uhland's Prakt. Masch Konstrukteur, 1S69, p. 97. I See Armengaud's Publ. Industrielle., Vol. XVII, PI. 10. THE CONSTRUCTOR. lOI been designed by Garand.* Jackson uses hydraulic pressure to force the clamps into contact.f Dohmeu-Leblauc uses springs to throw the toggles out of actiou.j Schurmann uses, instead of the separate clamps, a ring, which is compressed externally ;g Napier also uses a ring, expanded from within. || Becker ar- ranges the clamp blocks to be operated by centrifugal force.^ These are only a few of various modifications of the cylinder ■coupling. A form of axial friction coupling which acts with very slight pressure is the Weston clutch, made by Tangye.** This acts upon the principle of multiple plate friction jsee \ loi), as is shown in Fig. 4,51. The wooden discs are engaged with the case, and the iron ones ■with the shaft. In the form shown the plates are pressed to- Fig. 452. gether by the springs, and released by drawing back the collar B and releasing the spring pressure. A larger example of Weston's clutch is shown in Fig. 452. ^ is a winding drum, B the shaft driven by the engine. The outer disc C, and the inner •discs of the coupling are held apart by spiral sprinos, as shown at a. A light pull on the cord c holds the drum stationary ; a strong pull engages the clutch for winding; if the cord is left slack the load on the drum runs backward. I 158. AuTOM.\Tic Couplings. When power is transmitted to a shaft from two different sources, as from two independent engines, it is desirable to have one or both of them connected by a coupling which will auto- matically release or engage with the shaft, according to the dis- tribution of work. If one motor tends to overrun it will then be given more of the work, and so the resistance will be equal- ized. Such a device is the coupling of Pcuyer-Ouertier, gener- ally known as Pouyer's Coupling. Fig. 453- This is shown in Fig, 453. In this case the parts are so dis- posed that the part A, which is driven by one source of motive power, is loose on the shaft B, This part A maj' have gear teeth upon its circumference, for example, or may have a gear wheel mounted upon its hub, as shown by the dotted lines ; the hub being bushed with bronze. Upon the shaft B a ratchet •wheel is keyed ; the pawls a, a, being upon A, engage with the * Dingler's Polyt. Journal, Vol. 149, p. 22. fDin^ler's Polyt. Journal, Vol. 153, p. 251. t German Patent, 16Q52. ?Zeitschr. d. Vereins d. Ine;., Vol. V, 1861, p. 301. ll Engineer, 1868, July, p. 64. ^German Patent, 720:;. *»ln the U. S. by the Yale &. Tovvne Mfg. Co.! teeth when A drives B, but if B gains upon ^, or ^ stops while B continues to move, the pawls are thrown out of action. The direction of motion is shown by the arrow. The pawls are re- leased by the action of th- friction bauds b^ and A^, which are carried forward by friction upon B, whenever B gains upon A, the levers b throwing the pawls a out of gear. As soon as the limit of travel of the levers b, b, is reached, the friction bauds h^, b., slip upon B, being able to move no faster than A. When the speed of A increases and gains upon B, the pawls are again thrown into gear and A is automatically coupled to the shaft. In order that the pawls may not bind upon the ratchet teeth in releasing it is necessary that the angle 7, v/hich the pawl makes with the face of the ratchet tooth, must be less than the com- plemeut of the angle of friction ; iu this case y = 6o°. Pouyer uses only one friction band and makes both pawls engage at the same time. In the illustration the ratchetwheelis made with an odd number of teeth (13), and the pawls are placed so that a movement of only )4 the pitch will cause the parts to become engaged. The above proportion of the angle of the teeth is of importance, as otherwise the points of the teeth are apt cobe broken. The pawls also should be of hardened steel. Fig. 454. In Germany Uhlhom's Coupling is used for similar service, as shown in Fig. 454. Here A is the part connected to the motor, and B is fast to the driven shaft. A is an iuternal ratchet wheel into which the pawls 6 enter. The springs a serve to insure the entrance of the pawls into the teeth, which engagement continues so long as a drives B. If the speed of A is retarded, the pawls are retracted as shown in the lower part of the figure. In this case the springs act to keep them out of gear, being the reverse action to that of an ordinary ratchet gear. The pawls are fitted with half-journals (see ? 95), and are held in place bj' a plate ring, as shown. Ulilhorn originally used only two ratchet teeth in A, but increased the number afterwards to four, so that the parts would engage in a movement of one- fourth a revolution. It is better to use an odd number, as three, and by proper spacing of the pawls the greatest play will be one- half a space, or one-sixth a revolution with three teeth, as in the case of Pouyer's Couplings. B may be the driving part in- stead of the driven, but in that case the direction of the arrow must be reversed. CHAPTER XI. SIMPLE LEVERS. 2159- jouRN.\i,s FOR Levers. In machine design a simple lever, or rocker arm, is a lever arm which is mounted upon an axle or shaft, at the end, about which it moves, and carries a journal upon the other end. For the proportion of the journal see Chapter V. The forms which Fig. 455- may be given to such journals are shown in Fig. 455, and are single overhung, double, or forked. The manner of securing the pin in the hub or the lever is most important. The pin should not be driven in up to the shoulder on the taper, but sufficient space left to insure that the fit is tight in the taper. This clearance is shown plainlj' in the figure. The same result I02 THE CONSTRUCTOR. may. be attained by counter-sinking the collar into the hub on the lever. In the case of double overhanging pins, care should be taken that the load is equally divided between the two sides, so that the pressure upon each pin shall be equal to )A P. In the fork-ended lever the fit on both ends of the pin should be portions of the same cone. Example i. Vor P = 4400 lbs., we have from the table in ggo for alternating pressure and wrought iron journal, the diameter d= 0.027 v^44oo = i-S", and the length the same. For steel, we have 1^ = 0.024 -y/ 4400 = 1.6", and the length /= 1.3 X 1.6 = 2. oS". For a forked lever, a wrought iron pin with the same load the diameter, according to (9S) would be rf= 0.0185 s/ 44oo = 1.2", and the length / = 3.5 X 1.2" = 4.2"- All levers are not subjected to alternating pressure, but have the pressure constantly iu one direction, as for e.'^ample, the beams of single-acting pumping engines, etc. In such cases larger journals are needed. Example ■2. A wrought iron journal for a forked lever, under constant pressure of 4400 lbs., according to formula (98J, should have a diameter £/ = 0.0212 \/ 4400= 1.4", and length 1= 3rfj= 4.2''. If the material had been cast iron we should have h_ad^=o.29 \/440o = 1.92", say 2", and 1=6". For steel ■we have d = 0.0 185 \/ 4400 = 1.2", l^ j,d^ 4.8". 2.J_ Fig. 456. _ 2 160. Cast Iron Rock Arms. Rock arms may be either of cast or wrought iron. The hubs for wrought iron arms are given in the preceding illustrations, and in Fig. 456 are given some proportions for the various parts of cast iron arms. A fork-ended arm is shown below, among the walking beams, or if the fork hub is on the main axle, see the rules already given under Axles, Chapter VIII. §i6r. ■ Rock Arm Shafts. The axle upon which a rock arm works is usually subject both to bending and torsional stresses. The methods of calculation for all important cases are given in Chapters VIII and IX. The case which occurs most commonly is the overhung rock arm at the end of a shaft, and this is here given a special examination. If we have a, the distance between two planes normal to the axis, and passing through the middle of the pin and the middle of the bearing on the shaft, Fig. 457, there is an ideal bending moment with a lever arm J?, acting upon the bearing of the shaft, for a load P on the pin equal to {Mb i=:.Pa' = p( 8 -x/A'^ + / T X 4-3 > X 3.875 X 0.2 X 10,650 4- 50,000 X 4-3 X 3-875 X 0.2 X 10,650 — 50,000 ^ 1/ f I6I500 - , , The key is used as an extra precaution for security. Fig. 458. A special method of keying, especially adapted for the hubs of levers and wheels, has been designed by engineer Peters. It consists of two parallel systems of keys, as shown iu Fig. 458. THE CONSTRUCTOR. 103 The taper of the keys is t,',,. The arrangement shown at (a) is preferable, as it weakens the hub less than {b). The angle a may be taken = 135°, the thickness of ke3's b = y'j D', and mean width h= 2.b. The form {a) is especially suited for hubs which are made in two parts. Those hubs which are upon shafts subjected to bending, are cousidered under the heading of Combined Levers, in Chapter XIII. I 162. Lever Arms op Rect.\nguic, 7?' = 0.625 R+0.6C Rye. may be determined readily by the graphical method. Fig. The third case shows the method for inclined arms. or if and is R' 460. Example 2. — In the case of the lever of the preceding example, let C= I5-75''- This gives R ^ Cand we have from {154) ; R' = 0.975 X 24 + 0.25 X 15- 75 = = 23-4 + 3-94 = 27.34'' This gives for 6, 4400X27.34 b = 0.00072 1-7 (7-125)- Cast iron arms are sometimes made of cruciform section, see Fig. 456, in which case the ribs may be neglected. « 163. Lever Arms of Combined Section. The sections shown in Fig. 461 are designed to secure an economy of material. Their dimensions are readilj- determined by first calculating a corresponding arm of rectangular section, and then transforming it into an I section, or double II shape. If //q be the depth aud b^ the breadth of the equivalent rectan- gular arm, and li and b the corresponding terrcs to be fouud, as in Fig. 461, we have A — _'_ ba~ i-\-a in which I . . (155) (I-OCx-Kt)] B These formulas permit a choice of the ratios -7- and -7-, which may be left to the judgment of the designer. In (155) the angle Fig. 460. irons of the third example in Fig. 46; have been neglected, and may be considered as making up for the weakening of the rivet holes. The following table .gives a series of values for (155) which will simplifj' the calculations materially. The table will also be found useful for other purposes, as all sorts of beams^ crane booms, etc. -c ■ '^ -^ ■■ - Fig. 461. i 161. Table for Transforming Arm Sections. // Values of I I J- 2 c B 0.50 3 0.43 3-5 0.38 4 4-5 5 6 7 8 lO 6 0.33 0.30 0.27 0.23 0.20 0.18 0.14 7 0.52 0.45 0.40 0.35 0.32 0.29 0.25 0.21 0.19 0.15 8 0.54 0.47 0.42 0-37 °-34 o3t 0.26 0.23 0.20 0.16 9 0.56 049 0.44 0-39 0.36 0.33 0.2S 0.24 0.22 0,18 10 0.58 o.,Si 0.46 0.41 0-37 0.34 0.29 0.26 0.23 0.19- II 0.60 O..S3 0.48 0.43 0-39 0.36 0-31 0.27 0.24 0.20 12 0.62 O..S.S 0.50 0.44 0.41 0.37 0.32 0.29 0.26 o.2r 14 0.64 0.5S 0.,S2 0.47 0.44 0.40 0.35 0.21 0.2S 0.23 16 0.67 0-60 o.s'; 0.50 0.47 0-43 0.38 0-34 0.30 0.25 lH 0.69 0.63 O..S7 0.52 0.49 0.46 0.40 0.36 0.33 0.27 20 0.71 0.65 0.60 0-.S5 0.52 0.4S 0.42 0.3S 0-,34 0.29. 22 0.73 0.67 0.62 °-S7 o.,S3 0.50 0.45 0.40 0.37 0.31 24 0.75 0.68 0.64 0.59 0.56 0.52 0.47 0.42 0.3S 0-3,3 27 0.76 0.71 0.66 0.62 o.,=i8 0-5.') 0.50 04s 0.41 0-3.5 ,30 0.7S 0.73 0.68 0.64 0.61 0.57 0.52 0.47 0.43 0-37 ss 0.79 0.7s 0.70 0.66 0.63 0.60 0.54 0.50 0-45 0-39' ^6 0.81 0.76 0.72 0.68 0.65 0.61 0.56 0.52 0.48 0.41 40 0.83 0.78 0.74 0.70 0.67 0.64 o.,58 0-54 0.50 0.44 4S 0.S4 0.80 0.76 0.72 0.69 0.66 0.61 0-.S7 0-.S3 0.47 5° 0.85 o.Si 0.78 0.74 0.71 0.68 0.63 0-59 0.56 0.49 Example 1. A lever arm has a length i? = 78.75" and the journal pres- sure at the end = P = 5500 pounds. It is to be of cast iron of double Tsec- tion with a height /iq = i2^s' '■ According to (153) we have for a rectangu- lar section Bq ~ 0.00144 5500 X 78-75 -= 3-9 (12.625)2 This is also so thick as to be impracticable, and hence the double /'sec- tion may be compared. Here we may take c \ n =^ \ : \z.^ B \ b = ^, and we get from the table =^ 0.44 and h — 0.44 b^ =- i. 71", aud the flange r + J 104 THE CONSTRUCTOR. breadth ^ = o. 44^ = 1.71 X 0.44 = 0.732, the web thickness = c = ^ = /: = — '- — ^ " 1.05", all of which are practical dimensions. It maybe found de- sirable to have c= b, or any reasonable ratio for B : b, and ^ ; /* be chosen. Example 2. A wroug^ht iron arm lias been found to require b^ = 2%", n = 325-6". It is desired to make -j- =0.25 and in column 10 we find o 25 opposite — '—= 16. Hence i = 0.57" and .5= loX 0.57 = 5 7°" andi:= "^^ =o.S". CHAPTER XII. CRANKS. ? .65. Various Kinds of Cranks. Cranks are these forms of simple levers which are so arranged that they may, together with their various connections, make entire and repeated revolutions about an axis. These may be divided into the following four classes : 1. Single Overhung Cranks. 2. Return Cranks. 3. Double Cranks, or Cranked Axles. 4. Eccentrics. These will be briefly treated in succession. I 166. Single; Wrought Iron Cranks. These cranks may be proportioned according to the rules given for simple levei-s and rocker arms {\ i^(jeiseq). Fig. 462 shows the usual form; the arm tapers to two- thirds its base dimensions both ways, and is made slightly convex on the back. Fig. 463. The crank-pin is forced or driven in, and secured with a cap bolt. Fig. 463 shows a crank forged in one piece. In this case tue width of the arm at the base is determined by the necessary amount of shoulder on the shaft. The proportions of the pin are obtained from the rules in ^ 159. ? 167. Graphostatic Calculations for Single Overhung Cr.\nk. The crank is such an important detail of machine construc- tion that it demands a most careful discussion, hence a grapho- static investigation of the stresses in it is here given. The Crank r^.t'/t'.— Having calculated d and /, draw the skeleton diagram of the crank, that is, the neutral axis A B C D E, Fig. 464, in which B C represents the axis of the crank arm, which in this case lies normal to the axis of the shaft, and is placed in its proportional distance from the centre of the crank-pin A, and from the bearing D. Then lay oif the force /"from a, normal to E a, choose the pole O of the force polygon (this being best placed upon aline passing through the end of /'and parallel to the axis E a), draw the ray a d O, and line d E, also the ra)' O P^ parallel \.o d E \ then a d E will represent the cord polygon for the bending which P produces upon the axle a C E, and P P^ represents the force upon the journal E, and /'i a the force upon the journal D. Also make a /^ equal to the crank radius 7?, draw F G, and this latter will Tse the twisting moment I J 140) which /'exerts upon the axis. This moment Jlfj may be combined with the bending moment Jifb, to give for each point an ideal bending moment. Mi = f yi/s + f '^M,-' + Mi (see ? 45), from which the polygon curve c' d' e' and surface of moments Cc' d' e' E are obtained. From the latter, in combination with the pin diameter d, and ordinate t of the base of the pin, the diameter of the shaft may be obtained according to formula (124). The Crank Arm. — Prolong E a to fl„, and transfer the cord polygon Dad to the base line B C, that is, make the angle a„ B C= the angle £> a d, and then will B Uo C be, with hori- zontal ordinates, the surface of moments for the bending of the crank arm due to the force P. Also make C Co^^ B bo= C C, then will the horizontal ordinates of the torsion rectangle B ba Co C be 'the moments with which /"acts to twist the crank arm about the axis B C. This moment may again be combined with the bending moment to give an ideal moment as before ; {i7o a' = -| rto C, draw B a' , make at any point //, the space H i = \ B ba, and make H li = ha h' -\- h' i) which gives the surface of moments B b' h F C for the crank arm. From this and from the diameter d and ordinate t, we can construct the conoidal form of the arm / K T /)/, according to formula (124). From this, again, the profile S T U ^' oi axi arm of rectangular section may be derived, the width h being assumed for any point and the corresponding thickness b obtained from the value y of the conoid, according to the formula : b_ y 0.6 iM (156) in using which, the second table 01 numbers at the close of this work will be found useful. If the position of the axis B Cdoes not give satisfactory results, the operation must be repeated with a better relation of parts. By proceeding in this manner the dimensions of a crank and axle may be so determined that the}' will be equal in strength to the pin upon which the power is exerted. In the preceding diagram the crank arm was taken as normal to the a.'vle. A slight inclination may be neglected, but if the Fig. 46.5. angle is greater, as shown in Fig. 460, it should be so considered in the diagram. The procedure is then as follows (Fig. 465): The diagram for the crank shape is constructed as before, the portion under a b being used only for the shank A B of the crank-pin, and the portion under C E being combined as before with the torsion moment, to obtain the surface of moments Cc' d' e' E. The crank arm is again subjected to bending and twisting; the lever arm is now B' C, A B' being made normal to the axis THE CONSTRUCTOR. 105 B C oi the crank arm, the bending polygon being a portion of the triangle C B' C , in which the angle at B' is equal to the angle d a D. The twisting force acts with a lever arm A B' ; its moment is obtained b}' drawing an ordinate at a' normal to B C, B' a' being taken equal to B' A. The combination of moments gives the surface of moments B b" c" C in same manner, and of the same use as in the preceding case. I 16S. Cast Iron" Cranks. The crank-pin is sometimes made spherical instead of cj-lin- drical ; such a one is .shown in Fig. 466 on a cast iron crank. The sphere will be of suitable diameter if described from the middle of a normally proportioned overhung crank-piu without making allowance for shoulder. The crank-pin is secured by cold rivetting the end in place, an excellent method and one often used. The I formed section can be proportioned by the iise of the table in \ 164. When li is taken as equal to the hub -A- 1 54 Fig. 466. diameter, the cross section sometimes works out too light to be suitable for casting, and in such cases it must be increased according to judgment. Sometimes cast iron cranks are made simply by laying out the proper hubs for the shaft and crank- pin, and then joining them by an arm of rectangular section. If it is desired to employ the graphostatic method, the dimen- sions may first be determined for a wrought iron crank of rec- tangular section, and then doubling the depth (see | 162) for cast iron, and obtaining the proportions for I formed section according to \ 164. I 169. The Return- Crank. A return crank is one which is formeil upon the pin of an ordinar}' overhung crank, returning back toward and having rotation about the. same crank shaft as the main crank. Fig. 467 shows a wrought iron return crank otherwise similar in con- struction to the one shown in Fig. 463. Frequently the return arm is on the same line as the main crank, as shown in the illustration, but in many cases it is different! 5' placed. The arm and pin of the return crank are similar in shape and propor- tions to an ordinary overhung crank. The arm of the main crank demands no especial consideration, when, as is usually the case, there is but little pressure on the pin of the return •crank. The main crank-pin must be determined separate!}'. It is subjected both to bending and to torsion. For this purpose the formula (154) are to be used, remembering that when the return crank is driven by the main crank the moment of the return crank is greatest in the middle of the main crank-pin. I 170. Graphostatic C.alcui^.ation op the Return Cr.ank. The graphostatic diagram for a return crank, with both main and return crank inclined to the axis of the crank shaft, is shown in Fig. 46S. The skeleton ABCDEFGHIis first drawn, the dimensions A B C E and F G being taken to cor- respond with those chosen to meet the requirements of the cranks under consideration. The pressure i upon the return crank-pin is here taken as opposed to the pressure 2 upon the main crank -pin. Force polygon. — After choosing a scale for the measurement of the forces, the force polygon (on the right) can be drawn. The line o to i, measured upward, represents the pressure on the return crank-pin ; O is the pole chosen on a horizontal line drawn through o, and the line i to 2 represents the pressure on the main crank-pin, measured downwards. Draw the rays o O, I O, 2 O, also draw the line a d' parallel to i O, until it inter- sects at d' the line dropped from D (the line of direction of the i! Fig. 467. force 2) ; draw d' g parallel to 2 C until it intersects a perpen- dicular through C, the line of the force 3, which we know acts upward, but the magnitude of which is 3'et undetermined. In order to determine it, as well as the fourth force which acts at H, join g with //, giving H a as the closing line which is hori- zontal because we have chosen the pole C on a horizontal through o. Now draw in the force pohgon O 3 parallel to J{ g., then the line 2 to 3 is the third force acting at G upward, and the line 3 to o gives the downward force at U. Hence we have the figure a d' kg H as the cord polygon of the system of forces. At k is a zero point (see \ 132) and for convenience in showing the figure it is preferable to turn the triangle /■ g Ff over to the position (4 _^' //. The cord pol3'gon thus found will be of ser- vice in constructing the surface of moments, as will be seen .4& /»'' Fig. 46S. later. For the determination of the shank A B draw from A on the pressure i the triangle a b b', whose ordinates will serve to determine its dimensions. Crank-Fin C F D. — This is subject to bending, as shown by the surface of moments c d' e, and to twisting by the force I acting as a lever arm r^ C c — B b. In order to determine the twisting moment, take a I = r, and draw the ordinate / /', this latter will then be the desired moment, and the corresponding surface a rectangle on c e. Combining this, as before, with the trapezoid cd' e gives the surface of moments c c' d" e' e. Should it occur that the onlj' pressure acting is that upon the return io6 THE CONSTRUTOR. crank-pin, the surface will be modified as follows : prolong the line a d' to in' , and taking this bending polygon, obtain the corresponding surface of moments c' d" e, from which the crank pin C D E can be proportioned. The minimum length / of the crank-pin must be that due to the pressure 2, as given before, for overhung journals. Axle F G H I. — This is subjected to bending according to the polygon Ffg' H, and also to torsion by the moment of the force 2 less that of the force i. In order to find the first, we choose in the force polygon a second pole O' , upon a horizontal passiug through the starting point of the force 2, returning the same pole distance. Draw 2 O' and make d g" parallel to it, make d 11^ C c = R, and we have in the ordinate 11 n' the de- sired twisting moment. Make the abscissa of the ordinate at a' ^ A a^= R — r, and this ordinate will then be the moment with which the force i twists the arm backward. Taking this from n n' gives the height Ff of the torsion rectangle F I i' f which we may combine with the bending surface in the manner already given, and thus obtain the surface of mo- ments Ff"g" h" i" I. Should the case occur in which. the force I becomes zero, as is the case at some points in steam engines when the return crank operates the valve motion, we have for a bending surface F fog" //, and for a torsion surface F Fo i I, which gives a surface of greater ordinates to be used. Such a case is given in unl ttered dotted outline shown upon the base F I. It is assumed that the portion H I is subjected only to the action of a torsion couple, heuce the polygon there becomes a rectangle. Return Crank Arm B C. — This is subjected to torsion by the force I, with an arm A A^ perpendicular to C B prolonged (its moment being equal to the ordinate at a„ ), and to bending by the arm A^ C, whose polygon is a triangle on C A^ and angle at Ao equal to I a a^. The reduced surface is shown 2X C B Co c". Main Crank Ann E F. — This is subjected to bending for- wards with a moment surface Do F F" , the angle at Do being equal to edg'', and to forward twisting with an arm D Do, which is perpendicular to FF prolonged; it is also subjected to backward bending by the force i, with a surface EoFF', and backward twisting by the arm A Fo normal to F F. The combined bending moments give the surface E d, Co F'" F, and the combined twisting moments the rectangular shown upon F F, the combination of both resulting in the final surface E e"' f" F. Should the force i become zero the figure will be increased to that shown by the dotted lines. The Simple Cr.\nk Axle. Crank axles may be divided into simple and multiple cranks. A simple crank axle is shown in Fig. 469. to P'\i\ magnitude and also parallel in direction, and at A' is a normal pressure, which is iV= C? tan oc and is a maximum for the position A'l Zj Bl. Hence we may safely assume that the moments with which the crank arms and the axle are bent attain the maximum at the same time, and are those due to the force P. In the example the crank pin is at E, at B and H are: bearings, at ^ is a couple by which the .shai^t is subjected to torsion due to the force P acting with a lever arm R. This problem is very similar to the preceding, the portion H G taking the place of the return crank, with the difference that the force at H is variable and indeterminate, but is dependent upon the pressure P aX E. Force Polygon. — In order to make the closing line of the polygon horizontal, draw the line B e' to any desired point e^ on the normal E e' , join e' with BI ; then on any convenient scale draw the force P, from O, in the diagram on the right, and make o O parallel to H c' , i O parallel to B e' and O 2 normal to P. Then the distance i to 2 is the upward force /", acting at B, and 2 to o the force P^ at H, O.^ being the pole distance. K, K Fig. ^^o. Fig. 469. The analytical discussion of such a crank axle is such a com- plicated matter, and the practical results are so readily obtained with all needful accuracy by the graphostatic method, that the latter is only given here. In Fig. 470 is shown a skeleton dia- gram ABCDEFGHci a crank axle with both arms in- clined. If we make the value of the force P, which acts upon the crank pin, equal to Q when it acts in the direction K M, it will be equal to — =^— when the connecting rod is in any inclined position K L\ CA. being the angle of the rod with the axis KM. For a constant force O the pressure P will be a maxi- mum when K L acts normal to L M, and this is so nearly the same as the value for the vertical position ML, of the crank, or Q . ■ — , that this latter may be taken for the graphical exami- COS C\ I nation without a closer determination. The force at M is equal Axle S/iank H G. — This is subjected to bending by the force P^ at H. The triangle H G g is the surface of moments, and the ordinates may be used to determine the dimensions of the journal at H. Axle Sliank B C. — The surface of moments for bending is the triangle B C c. In addition to the bending is the twisting moment P R ; in order to determine this make O' i normal to /'and equal to O 2, and also make Eoe„ parallel to O, and equal to R, then o Fa is the desired moment, which laid off at C c' and A a' and combined with B C c in the manner already described, gives the surface of moments A B C c" b" a". Crank Pin D E F. — The surface for bending moments is the figure d ff e' d'. For twisting we have the force P^ at H, with a lever arm oi E e = R. Make H g = E e ^ R and the ordinate g g' is the desired moment, which transferred to f" d'" and combined with the preceding surface gives the surface dff" e" d" . The greatest ordinate e e" should be used if the piu is to be cylindrical. Crank Arm G F. — Draw E Do parallel to H D' normal to C D. We then have forward bending by the force P at Do ; backward bending by /^, acting at D' . The cord polygons for these are, the triangle Do C i (with C i = o Ho in the force polygon, where //„ /;„ ^= C Do), and D' C i\ which when combined give the surface C i" i'" for the bending of the arm D C- We also have a forward twisting from the force /*with the arm E D- =^ k koin the force polygon, and the moment o k acting backward from the force P-^ with a lever arm H D' = H I in the cord polygon and a moment / /'. The difference be- tween these moments laid off at D do and C Co and the resulting torsion rectangle combined with the bending triangle gives the surface C D 1 1\ so that all five portions of the diagram now have their moment surfaces determined. The method of using these for the determination of dimensions is the same as before. THE CONSTRCTOR. 107 The figures show clearly the various stresses at the respective portions of the crauk aud throw light upou the mauuer in which breakages occur. If both crank arms are normal to the axis, the solution is greatly simplified, and the diagram assumes the form given in Fig. 47'- In this we have again A B CD E FG H as the skel- eton, and at ^ a torsion couple whose moment is equal to PR. Force Polygon. — In this case the altitude e e' of the triangle Be' H is taken as the measure of the force P. B b" is made equal to ee' , b" O drawn parallel to e' H, and Ob made normal to Bb" , thus giving b" b as the force jP, at H, b B that at i?, and 06 is the corresponding pole distance. Axle Shank H G. — This is only subjected to bending, aud the surface of moments is H Gg. Fig. 471. Axle Shank A B C— This is subjected to bending, as indi- cated b}' the triangle B Cc, and also to torsion by a moment PJ?. Make e' O' parallel to C B and equal to the pole distance b O, diaw e'" p parallel to e' O' and equal in length to E e^= R, then e e'" is the desired twisting moment, giving for A C the torsion rectangle whose altitude A' a' = Bb'" = e e'". The combination of bending and torsion moments gives the moment surface A B Cc"' b' a. Crank Pin D E F- — This is subjected to bending according to the surface of moments CG g c, and to torsion by the force P, at H, with a lever arm R = CD = H f, and a moment ff = Gf" = Cf". By combining the twisting and bending mo- inents the surface C G g' e" c' is obtained, and for cylindrical crank pins the rectangle of a height G g" = Cc" = e e" is to be substituted for the irregular outline. Crank Ann F . G. — This is subjected to bending by the force Pj acting at G. The surface of moments is G Fft^, the angle at G being equal to f Hf ; it is subject to torsion by the same force acting with a lever arm H G, giving a moment G g^= G h ^= Fi. The combination of twisting and bending moments gives the surface FG h' i'. Crank Ann CD. — Here we have bending with the force P, and an already known moment ee"' =s Ck at C. Twisting is due to the moment Cc= Cl = Dl'. For the combined mo- meuts these give the surface C D dk'. Fig. 472. For the same given distances of E from B and H the torsion stresses on the crank arms are greater for arms normal to the axis than for inclined arms, so that in the former case heavier arms are required. The torsion in the crank arms grows less and less the nearer the points C and G approach B and H. which is a point to be considered in the interest of economy of material. It is also to be noted that the total length of crank axle FG H or D CB is less for inclined arms than for right- angled cranks. In many cases a crank axle is so situated that it is subjected to torsion at either one end or the other. In such cases the dia- gram should be constructed for both sets of conditions, and laid upon each other, the greater value in all cases being taken. Of course, care must be taken to use the same pole distance and same scale for mearuring forces in both cases. An example of such a case is found in the paddle engines made by Penn, with oscillating cylinders, the air pump being worked from the mid- dle of the crank pin. The conditions in this case are somewhat different from the preceding, and may be examined with the help of the following diagram (Fig. 472) ; Here we have the skeleton ABCDEFGH, and not taking into account the force at E, the force couple gives by means of the cord and force polygon the moment values B b=: Cc = Gg = Hh, from which the following results are obtained : Axle Shank A B C. — Pure torsion, which, converted into an equivalent bending moment, gives B b' = C c' =-.-\^Bb (see IV., § 16, when Md = O}. Axle Shank G H. — This is the same as the preceding, and Hh'=Gg'=Cc'. Crank Pin DEE. — We have here the same twisting moment as in the axle shanks Dd = Ff^ S b and Dd„^ Ff„-=B b'. Crank Ann CD. — We have in this portion a bending moment of the magnitude Cc" ^ Dd' ^ Cc, of w-hich the plane stands normal to the plane of the surface of the crank arm. The sur- face of moments is in this case equal to a rectangle of the height Bb = Cc. Cfank Arm EG. — In this case we have both torsion and bending. The couple is decomposed at G into two parts, one acting normal to the axis of the crank arm, and the other in the direction of the arm. The first gives the torsion rectangle G Ff" g", the latter the bending rectangle E Gii' , which com- bined give the moment surface F G g"' /'", in which we again have pq = lG /, /> r = | G g" , pt=Gi --qs+qr. Thus far we have proceeded as though there were no force acting at E. When such exists, however, first determine the bending and twisting moments as shown in Fig. 472, add or subtract, according to direction, the twisting moments, taking into account the position of the planes of bending action, and finally combine the bending and twisting moments so found, according to the method of Case IV., 'i i5. The amount of work which this investigation requires of the drawing-room of any machine-shop is small compared with the importance of a thorough determination of all the stresses which act upon such a piece of work as a crank shaft forging. Fig. 473- J 172. MuT,Tipi.E Crank Shafts, Locomotive Axi,es. One of the most important forms of crank axles made of wrought iron or steel is that used for locomotive engines. As an example of this subject, the crank axle for an inside con- nected locomotive is given in Fig. 473. In drawing the diagram of moments it is necessary to take into account tne diameter of the driving-wheels, as will be shown in Fig. 474. Cj and C, are centres of the steam cylinders. Ay and A., are the journals, aud By Dy and B., D^ are tlie hubs of the respective driving-wheels. The cranks at Cy and C, are placed at right angles with each other, taking the position which the axle shows in Fig. 473. An inspection of the figure shows three distinct loads acting upon the axle : l, the pressure in the vertical plane due to the weight of the locomotive and to the lateral action upon the wheel io8 THE CONSTRUCTOR. flanges ; 2, the horizontal pressure of the piston against the crank C, opposed by a corresponding adhesion at the circumfer- ence of the driving-wheels ; 3, the oblique pressure of the con- necting rod acting upon the crank C^. Other small pressures, such as those due to the eccentrics, etc., may be neglected. H So Fig. 474. Forces and TiToments in the Vertical Plane. — Fig. 474. From the point Sf, of the height of the centre of gravity of the loco- motive laj' off the force Q, to represent that portion of the weight which is borne by the axle under consideration. The oscillations and action of centrifugal force upon curves also produces a horizontal force H, which may be taken as equal to 0.4 Q. The resultant R of the two forces O and H is the load upon the axle. This may be decomposed into the pressures P^ and P, upon the journal at A-^ and A,_. and into the pressures (2i and O., upon the wheels at E-^ and E.^, which pressures, with their reactions, produce the stresses on the axle. The forces Q^ and O, can be decomposed into two others referred to the wheel hubs i?i Z?! and /?, D^- This gives six vertical pressures acting to bend the axle, viz. : i, 2, 3 and 4 acting downward at D^, A-^, A2 and D.,, and 5 and 6 acting upward at B., and B^. From these forces, by choosing anj' desired pole distance, the force polygon E, 4, O may be constructed, and also the cord poly- gon or surface of moments d^ a^ a„ d^_ b„ b^, and this surface gives hy its ordinates the proportional bending moments in the verti- cal plane for each point in the axle ; this entire surface is desig- nated by the letter /'. Fig. 475- Forces and Moments in the Horizontal Plane. — Fig. 475. As already shown in a preceding paragraph, the pressure P ovi the crank pin for the position L yl/of the crank is somewhat greater than the pressure P^ on the piston ; its moment of rotation about the shaft is P. . R cos n, which = P^ R, so that upon the as- cos a sumption that the wheel on the left slips on the rail, the other one must oppose a resistance whose moment equals Pf, R and the frictional resistance 3 at £'2 = /o — Combining this force 3 at E., and also the force 4 =: P^, and the resistances i and 2 at the journals, we are enabled to construct the force polygon .^j 2 (9 and the corresponding cord polygon // for the horizon- tal forces, as shown in the light sectional portion of the diagram. The forces 1 and 2 are found bv taking tlie position of the re- sultant of the two forces 3 and 4, as shown in the figure, and decomposing their sum into the portions which would go re- spectively to x4^ and A.,, as shown by the construction given in the dotted lines. Forces and IMoments in the Inclined Plane 0/ the Connecting' Rod. — The force Q = S acts at (Ti, making an anale with the horizontal equal to 3/ K L. As shown in the illustration, this maj' be decomposed into the two opposing forces 6 and 7 at A^ and A.,, and by taking the same pole distance as before to con- struct the force polygon we obtain the cord polj'gon 5, shown by the dark section lining, and giving the surface of moments for bending in the inclined plane of the connecting rod. Fig. 476. Combination of the Three Preceding Cord Polygons for Bend- ing of the Axle. — Fig. 476. Since the three preceding sets of forces are acting at the same time to produce bending in the axle, it is necessary to combine the diagrams in "order to obtain the final result. For this purpose we can treat the respective ordinates in the same manner as if they were forces, as in \ 44. Taking the successive points upon the axle, we construct the corresponding ordinate polygons, whose closing lines give the resulting moment both in direction and magnitude. One of these ordinate polygons is shown in the upper portion of Fig. 474, to the left: it belongs to the point Q. The vertical ordi- nate /'in this case acts upward, the horizontal ordinate //^con- tinues toward the left, and the inclined ordinate 5' also continues to the left, thus giving the resultant T as the line joining the origin of V with the termination of 6". We thus obtain for the entire axle the surface of moments ZJ.j /?i flj f , fj ^22 ^^27 which gives the proportion of bending stresses of the axle, as distin- guished from those of the crank arms. The Torsional ^Foments for the Axle. — The position of crank described above and selected for this investigation gives a tor- sional moment only upon the crank to the left, and also one of the magnitude PR upon the axle extending to the point D„. If both cranks stand at an angle of 45° with the horizontal, there will be produced in both end shanks Cj D^ and C^ D., moments equal to \/ 2 PR, or about 1.4 PR. Under these circumstances the moments at the ends become Z?, (// ^ /?, a'./, while in the body of the shaft (T, C, we have the moment C^ <"/ ^ C", c' = PR, always keeping the scale of forces and the pole distance the same in all of the diagrams. It must be remembered that in this position of the cranks the bending moments are somewhat different from those shown in the preceding diagrams. Combination of Bending and Tivisting liloments. — The bending and twisting moments can now be combined accor- ding to the formula of §45, and thus the surface of moments D^D^d^b^ . . . . dr," obtained, b}^ the help of which the shanks C, D^ and C, £>., and body of the axle C, C, can be pro- portioned, after the diameter for any one of the ordinates, as, for example, that at B^ 5,, has been determined. The half of the diagram which gives the greatest ordinates should be used for both halves of the axle. Crank Pin at C^- — The two crank pins are treated separately in Figs. 477 and 47S, since the moments can be laid out more conveniently in that way. For the pin EG at Cj we have, iu addition to the bending moments obtained from Fig. 476, and shown by the surface EG c^, the combined forces on the left, up to the point E, acting to twist the pin. The resultant of these forces is j^et to be found. The vertical forces are those shown at I, 2 and 6 of Fig. 474, their algebraic sum being shown at /, in Fig. 477. The horizontal force acting backwards is //, repre- sented as I, iu Fig. 475. The inclined force acting downwards 777^ CONSTRUCTOR. 109 and back'n-ards, shown at ///, corresponds to the force 6, of Fig. 475. The closing line (not shown) from /// to C^ would give the resultant, and its horizontal component //•''acts to twist the crank pin FG, with a lever arm EF-^R. In the force polygon (above, on the left) we take /, remembering that the crank J IC is taken in the horizontal position. The moment of this vertical component has the magnitude kk'. Also we have act- ing to twist the pin the couple shown on the left (as discussed in connection with Fig. 472) with a moment already determined and shown at C-^c/ in Fig. 476, and here laid off at K k, from which, since the previously determined twistiug moment k k' acts in the opposite direction, we must subtract kk', giving finally for the crank pin K L the twisting moment Kk', which, when combined with the bending moment, gives the surface KLc^'. Crank Ann J K. — This is subjected to twisting by the moment A'(/=the vertical component v„ of the ordinate polygon V^H^S-^T^. For bending in the vertical plane we have the moment Kl^Kk, as already shown in Fig. 472 ; also in the same manner and direction by the vertical component of the forces V, VI and VII with the moments b b^ at K (see the dia- gram of these moments in the upper left portion of Fig. 477). It is subject to bending in the horizontal plane by the horizon- tal component /i^ of the ordinate polygon, the moment being b b^. The combination of bending moments gives the surface / A'b/ b./, and the final combination with the twisting moment Kd gives the surface / A' i.^". Crank Ar»i L 3J. — The twisting moment is L d, = the verti- cal component i\ of the ordinate polygon for the point M. The bending moment L b.i = K k, also b, b^ due to the vertical force at I\I, and also the bending moment i-, 65 = the horizontal com- ponent h^ of the ordinate polygon. The combination of bend- ing moments gives the surface ill Lb/, and the final combina- tion with the twisting moment gives the surface ML b/' . Of the four crank arms, J K is subjected to the greatest stress at the pin, and G H at the axle. In practice, therefore, the surfaces y A' A/' and C/Zi^,"' should be drawn upon each other and the greatest ordinate used. The resulting dimensions, with possibly slight modifications, should then be used for all four arms. Although the construction of such a graphostat'c diagram of moments involves some labor, the result is most satisfactory, since by assuming a stress of say 3 the modulus of working stress (about 17,500 lbs. for wrought iron, 25,000 lbs. for steel) the design can be properly proportioned without further care. The calculations for locomotive axles with outside cranks is similar to the preceding, although the diagrams are necessarily somewhat different, although laid out in the same general manner. Fig. 479. Fig. 480. i 173- Hand Cranks. The chief peculiarity in a hand crank lies in the adaptation of the crank pin to be operated by hand. In Fig. 479 is shown a crank for two men, and in Fig. 4S0 for one man. The dimen- sions for the parts indicated by the letters are as follows : For 2 men. For i man. /?= 14" to \W' 12''' to W I' = 16" to 19'' 11" to 13'' D~ I J" to If" ii"to I*' The other dimensions figured in the illustrations are in milli- metres. When placed at opposite ends of the same shaft, hand cranks should be set at 120° with each other. Fig. 4S1. Fig. 482. Fig. 483. Fig. 4S4. ? 174. Eccentrics. An eccentric is nothing more than a ciank in which (if the crank arm is R and the shaft diameter D\ the crank pin diam- eter d' is made so great that it exceeds /? -(- 2 j?, or is greater than the shaft and twice the throw. The simpler forms of eccen- tric construction are shown in the illustrations. The most prac- no THE CONSTRUCTOR. tical of these is that shown in Fig. 4S3, the flanges on the strap, as shown in the section, serving to retain the oil and insure good lubrication. The breadth of the eccentric (properl)- the length of pin /) is the same as that of the equivalent overhung journal subjected to the same pressure ; for the depth of flange a we have a = 1.5 ^ ^ 0.07/ -j- 0.2 (157) from which the other dimensions can be determined as in the illustrations. For some forms of shafts with multiple cranks or other ob- structions the eccentrics cannot be made as shown above, but must be in halves, bolted together. CHAPTER XIII. COMBINED LEVERS. I 175- Various Kinds of Combined Levers. Two simple levers with the same hub form what is termed a Combined Lever. AVheu both arms have a common centre line they form a Beam, or so-called Walking Beam ; and when they form an angle with each other they are called an Angle Beam, J,=OB Fig. 4S5. or frequently a Bell Crank. The pressure {?, upon the axle of an angle lever A O B, Fig. 4S5, is determined by the relatio- Q = V/'i- + P.^ — 2 P, P, cos a if P^ is the force acting at A, and P„ that at B, both acting at right angles to their respective arms"; a being the angle between the arms. This may be shown graphically by making P^ = OB and P.^ = A, when Q will = A B, the, third side of the tri- angle. If the forces /*, and P., do not act at right angles to the arms, the triangle must be constructed bv drawing lines from O, normal to the directions of the forces. The variety of combined levers is very great, and only a few of the principal forms are here given. Fig. 4S6, ?i76. Walking Beams. One of the principal forms of combined levers is the walking beam, for use in some forms of steam engine. These are usually made of cast iron, with journals and pins similar to those given in Fig. 456 ; and other forms of journals are also shown in the following figures. Fig. 4S6 a shows an ornamented beam-end, with the pin keyed fast Fig. 4S6 b shows a beam-end with a bored cross-head and pins cornbined, fitted on the turned end of the beam and secured by the pinned collar shown. This construction requires careful fitting, and is somewhat expensive. Fig. 4S7 a. This is a fork journal ; the fit is made with a very slight taper, secured by cap bolt and large washer at one end. The pin is kept from turning by a projection under the head, let into the boss on the beam. Fig. 487. Fig. 487 b. This is a spherical bearing with its shank driven into the end of the beam and keyed fast, this form .giving great freedom of motion to the connecting rod.. The diameters of pins are determined as already given in I 90. The load is to be considered as acting continuously or intermit- tently, according as the engine is single or double acting. Fig, 4SS shows a form of beam which has been extensively used. In order to secure lateral stiffness, the beam centre should not be made too short. A good proportion is that given in the figure, in which the distance between centres of bearings is .made equal io 6d + ^\ A. The distance between centres of journals for the ends of the beam is made from 4.6 a'j to 5.5 d^ ; Fig. 4SS. <^i being the journal diameter, as shown. The depth h of beam m the middle must not be made less than k = Ad + A (15S) in which d is the diameter of the beam centre, and A the half length of the beam. If the two arms are of unequal length their mean should be taken.* The curved outline of such beams is drawn according to the methods given in g 142, starting from the crown of the beam to the hub for the pins at the ends. The ribs in the middle of the beam are given the same thickness, c, as the flange at the edges, and the breadth of flange is shown in the plan at B (see I 163)' Another form of beam is shown in Fig. 4S9. This is made double, and in such case each half is calculated separately. In Fig. 490 IS shown a section of such a double beam in which the parts are somewhat widely separated. The two plates are firmly bolted together, the bolts passing through tubular sti ts, as .shown, and the parallel motion rods are hung between the two parts of the beam. * In the United states much greater depth is given to beams of this sort sometimes 2 to 2}^ times that given by the formula. Skeleton beams w°th cast-irou centres and wrought-iron bands are also much used. 777^ CONSTRUCTOR. Ill A beam of somewhat unusual form is shown in Fig 491, being a poition of the hydraulic riveting machine of Mackay & Mc- George, built by Rigg." The beam centre is at A, the rivet die at B, the hydraulic pressure is exerted by small and large cylin- ders at D and C respectively. The water pressure is taken from au accumulator and discharged into an outlet pipe placed some- what higher than D. By lueaus of a suitably arranged valve bolts. Fig. 494 shows the form used on American locomotives. The example is from a passenger engine, and extends between Fig 4S9. gear the high pressure water is first exerted upon the small cj'l- inder, and water from the discharge pipe delivered to the large cylinder, thus closing the die upon the rivet at B. Then the high pressure water is also delivered to the large cylinder, making a still greater pressure upon the rivet, with practically Fig. 490. no expenditure of water, as that cylinder is already filled. The pressure upon the rivet is 60 tons. The beam is made of a sec- tion of uniform resistance (see 2 9). At .£" is a short shear for cutting beams, angle iron. etc. The distance B Cis 12 feet. Wrought iron beams are not uncommon, and for moderate Fig. 492. loads and dimensions are conveniently made in the double form, as shown in Fig. 492. The depth /; in the middle may be taken at 0.8 times the value given b}' formula (158). For larger beams' of wrought iron, the girder form shown in Fig. 491 is to be pre- ferred. Fig. 493. Another form of beam is the equalizing lever, used to distrib- ute the weight among the springs (see Figs. 102 and 103, ? 41). In Fig. 493 is shown a lever of ViTought iron for a heavy engin-^ {the Prussian standard freight engine). The length A B is iiSo mm. = 462", and the connections at A, O and B are made with * See Engineering, March, 1S75, p. £23. Fig. 494. the springs of the driving-wheels, being jl feet long. At O, A and B are half journals, and the connections at A and B are not rigid. The bearings are not on a straight line, as in the German form, but the variation is trifling. Fig. 491. Sc.il^E BE.IMS. In scale beams the bearings are usually made upon knife edges (see ? 95), generally with an angle of 60°. A special form is here given, Fig. 495, which may serve as an example, showing the main supporting beam of a bridge-scale, in triangular form. In the construction a, the main bearings are at OO; the bear- ings A A form a double journal analogous to Fig. 476 ; at j? is the end journal, here set in a cast-iron head. In the form shown at b, we have two separate bearings at O O, the parts being held together by a bolt C* * For similar examples see E. Brauer's ' Construction), Weimar, Voss, iSSo. Konstruktion der Waage " (Scale 112 THE CONSTRUCTOR. Scale beams should show very little deflection under their load. They are therefore made very deep in proportion to their total section, aud the stresses taken at 4250, S500 and 14,220 lbs. respectively for cast-iron, wrought-iron and steel. Fig. 495- CHAPTER XIV. CONNECTING RODS. I 178. Various Parts of Connecting Rods. Connecting rods are used in various forms for transmitting the motion of various reciprocating parts of machines to levers, beams or cranks, or vice versa. It is necessary to consider sepa- rately the ends or heads which contain the bearings for the crank and cross-head pins, from the body of the rod. The dimensions and proportions of the ends are governed, to a greater or less extent, by the dimensions of the bearings, the latter being either forked, overhung or necked, and their size determined by the pressure to which they are subjected. I "79- Connections for Overhung Crank Pins. The strap and key connectiou shown in Fig. 496 is widely used. The boxes are surrounded and drawn together by the liHil |ii'pf?-':'-I|| "iS?..^... Fig. 496. strap and key, and by driving up the latter they may be closed together to take up wear. In determining the dimensions, the boxes and their surrounding parts -vfrill be considered separatelj-, as in the case with other bearings. The unit or modulus for the boxes is e ^= 0.0"] d -\- o.wW (159) being the same as used or other bearings, d being the diameter of pin. Fig. 497- Fig. 497 shows two views of the brasses, the dimensions of the other parts being based on the following modulus : rt'i = 0.0267 \/P+ 0.2'' (160) The breadth b may be made equal to 0.8 ^j, or if the length of the journal is made equal to its diameter b becomes = d — • 2 fi.;K.^r=.irf.^35g^.^jjgl^a^H Fig. 502. is combined with its own locking device. The boxes are made of wrought iron lined with white metal, and an oil chamber is formed in the one shown on the left. Fig. 503. Fig. 503 shows an end of cast iron, also made solid, and with the key acting to take up the wear from below, much as in the design of Sharp, Fig. 49S. Cast iron rods were formerly much used on the parallel motion connections of beam engines. 114 THE CONSTRUCTOR. I I So. Stub Ends for Fork Journals. Fork journals designed according to the method previously given, are made much smaller in diameter than the correspond- ing overhung journals. On this account the breadth b' cannot be determined in the same proportion to the diameter of pin d, as with overhung pins, as the pioportion will vary somewhat for various conditions. In order to take this difference into account, we may take for such rod ends, instead of the modulus ^iven in (i6o), the following : ■ . S+v>v^ '-'^^ in which b is the breadth corresponding to the length of the normal pin, and (/jits modulus, calculated according to (i6o). This enables us to use the proportions of all the preceding ex- amples for fork journals as readily as for overhung crank pins. The thickness of metal c in the boxes may be made the same as before, using in every case the actual diameter d' of the jour- nal. The formula (i6i) assumes the same material to be used \fi' /^ Fig. 504. in both cases, and gives the rod end for the fork journal approximately the same strength as one proportioned for a nor- mal pin. It is not, however, possible to make an empirical formula cover every case, and some examples will be found much heavier, such as, for example, would give a modulus of One of the portions whose dimensions will not bear much reduction is the key, since it is subject to shearing action and its limited surface must not be subjected to too great pressure. /i \b' ' d ) 0,5d r*--*?- !-l— *, m^i 1 .] — Eljf^ ^ 1 "- t-~=3 ^ i^h J ■--_=%_ f^"! — '■■■ -- -^m% Fig. 505. For this reason the dimensions of the key should in no case be made less than those given for stub ends for overhung pins. Example. Given an alternating pressure ol 7920 pounds, let it be required to de- sign a strap end similar to Fig. .196. We have for an overhung pin according to (93), 2-375 For the boxes we have e = 0.07 X 4-75 + 0.125 = 0,45'', say /g". In the following examples are given modern designs for rod Fig. 509. ends for neck journals, and others may be obtained by modifi- cations of the preceding forms. Fig. 510 shows a solid end connection for a spherical journal. ITlie sphere in this case is made 1.5 times the diameter of the Fig. 511. Fig. 512. Fig. 510. corresponding cylindrical journal, and an example of this form d' may be seen on the beam in Fig. 4875. This gives — ~ = 1.5 ; a and if, as before, we make b' = 6, we have d\ = di y i-S = 1.225 d^. If, again, d = 2.375'', we have d' = 3.56", tfj = 2.56", and d\ = 2.56 X 1-225 = 3-i3''. say 35". The boxes are made without side flanges, so that they can be removed by backing out the key. The key may be arranged to be fitted above or below the boxes, as may be desired. When used upon locomo- tive engines, this form is sometimes strengthened as indicated by the dotted lines. For the connections of crank axles, return cranks and similar situations it is necessary to use a form of rod end which can be opened. The following forms are designed for this purpose, being made with blocks which are firmly bolted in place, but readily removable. Fig. 511 shows a form similar to Fig. 500. The block is fitted between two shoulders and also secured by two through bolts. Fig. 512 shows a design by Krauss, in the same stjde as Fig. 502, and used with it on a locomotive connection. The block is here made of bronze, and also forms one-half of the bearing ; it is held in place by a through bolt, which is omitted in the draw- ing. A cross-section is shown above, the offsets serving to keep the block from twisting on the bolt. The gap between the boxes is filled with slips of copper. The rod and bolt are both made of steel. Figs. 513 and 514 show two forms of eccentric straps, both intended to be made of bronze. The breadth // is equal to /, the length of the corresponding cast-iron journal (see | 92). If d = lyV'i '^i = i-^". if = * = 2.375", we have, if rf' = 15.75, b' = l= 2.375'', d' -^■^il 5-75 5625 : 5.71"- The diameter, d, of the bolts of these eccentric straps is determined from the following : tS ^ 0.33 fi?j -(- 0.06 o'l' (162) in which d-^' is the modulus for a neck journal and d^ the mod- Fig. 513. Fig. 514- Fig. 515. Fig. 316. ulus for the corresponding overhung pin. If we take the values above given, rf/ =: 5.71" and d-^ = i.S", we get (! = 0.33 X 1-8 -1- 0.06 X S.71 = 0.9S66", say i". If we make d' = d and d-/ = t/,, we obtain from (162) the same dimensions as on a capped and bolted rod end. ii6 THE CONSTRUCTOR. In Fig. 515 is shown a design for a cast iron strap, with bronze lining, although this latter may be omitted. The eccentric rod is secured by means of a key, and if two eccentrics are placed close side by side, the keys should be placed at 45° from the position shown. Fig. 516. This is a wrought-iron strap, also lined with bronze. In this, as in the preceding example, the joints between the two halves of the bronze lining are close, and those of the strap are open, and by filing the ends the halves may be closed together to provide for wear. Instead of forging the rod in one piece with the lower strap, it may be made with a T head and bolted fast, as shown by the dotted lines. Example The eccentric straps on the engines of the "Arizona," 6600 H. P., by John Elder & Co., of Glasgow, are made as in Fig. 515, but with the rods attached by T heads, as described above. The diameter of eccentrics d' = 54", the breadth I = 5", and the shaft diameter = 22 J^". rod is made with a forked end, and two bearings, its lateral stiffness is thereby increased, and m may be made as low as 4, If w = 20 we have for wrought iron or steel, C^ 0.0346. Example i.— For a wrought iron connecting rod 118.11" long, acting under a pressure of 31,680 pounds, taking m = 20, and C = 0.0346 we have a diameter D = 0.0346 J II8.Il^/3l68o = s". This gives the diameter in the middle ; it may be somewhat reduced at the ends, these latter being made of a diameter = 0.7 D, giving a cycloidal sinoide as in Fig. 5, formula (23). The ends of the rod should be worked off into the body m such a manner as not to make too abrupt a change of cross section. This becomes more important in high speed engines. In the case of locomotives there is sometimes a marked bending action upon the rod, there being a so-called "whip action" at every revolution of the crank, dependent upon the rotative velocity §182. Round Connecting Rods. The body of a connecting rod may be made of wrought iron, cast iron, steel, or even wood. In the latter case it is usually only subject to tension. If the rod is of circular cross section, of diameter D, and the force of tension be P, we have the following relations : Wrought Iron Steel Cast Iron D :0.014s VP D Vf D = 0.0117 Oak ^P D Vp = 0.0578 (163) These give stresses of 5600, 9500, 2S00 and 400 pounds respec- tively, or above two-thirds the value given for ordinary condi- tions. These formulce ma}' also be used for short rods which are subjected to compression, but if the length L, of the rod is so great as to permit bending, the diameter must be made some- what greater. From an examination of case II, \ 16, and also I 127, we should not permit P to be greater than — i^^— , in v^hich/ is the moment of inertia of the cross section of the rod, and E the modulus of elasticity of the material employed. In order to determine how small P must be, or rather how large the co-efiicient of safety in, must be taken so that we shall have P = — . i-^ , there are various conditions m L'- to be considered ; the requirements being almost as varied as in the case of columns. Leaving then the value of ?«, to be subsequently determined, V7e have / = -^ D'' and E = 28,400 000, for wrought iron and &4 steel, 14,200,000 for cast iron, and 1,562,000 for oak, and hence the following formula for the diameter of rod. Wrought Iron or Steel D = o.oi 64 ^» Cast Iron Wood We have for s]lVP -^lVp D = 0.034 ■^'« sj -L ^ P D: ; 0.0195 V ;k (164) ^« 1.5 I. II 3 1.32 4 1.41 10 15 1.7S 1.97 25 2.24 3° 2.34 40 50 60 2.51 2.66 2.78 If we represent the entire co-efficient of a//, \^ P by C we may write for the above formulce and may then determine values for C according to the degree of security required. As already stated, there is a wide variety of values of in to be deduced from practice. For stationary engines of moderate size we find in, very high, often 50 to 60. These however are not to be taken as standards because they are rarely designed for economy of material, but rather for per- fection of action. For medium and large stationary engines we find m from 5 to 25, probably averaging about 20. If the and the weight of the rod. This action also occurs in a lesser degree in slower running engines, and is greatest at a point between the middle and the crank end of the rod. For this reason it is sometimes thought desirable to make the greatest diameter of rod, not at the middle, but somewhat nearer the crank end, as shown in Fig. 5'7. For moderate piston speeds this point need hardly be con- sidered as it is amply provided for in the co-efficient of security, but for high speeds and heavy ends it should be given due con- sideration. In the high speed type of engines such as the Porter Allen, the greatest strength of rod will be found at the crank pin end. At the same time, as will be seen, the value of ■m, for high speed locomotive engines, is usually made small. For marine engines, m is usually taken quite high, viz.: 30, 40, 60 or even So, and the ratio — =^ proportionally smaller. In such engines the rod is generally made proportional to the cylinder diameter, being about 0.0S5 to 0.095 times the bore. It must be remembered that in marine engines the stresses due to flexure of the hull, and general lack of rigidity, demand a higher co-efficient of security than for stationary engines. Fig. 518. In Fig. 518 is shown a rod for a screw propeller engine. The body of this rod is truly cylindrical, and the ends are similar to that shown in Pig. 500. Let E = 94,600 lbs. L = 60''. Taking, as before. Example 2.- have , w ^ 20 we D , — ^ = 0.0346 i L D = 0.0346 v/94,600 fi = 4.67". \/94,6oo In a similar case, executed by Maudslay, the rod was made 6" in diameter, which corresponds to a value in = 54.7. The diameter S, of the bolts in this case was 3", and according to the rule given for Fig. 499, they should be sji^'. ^183. Rods of Rectangular Section. If it is desired to make the body of the rod rectangular in cross section, it is first necessary to determine the diameter for circular section by the methods of the preceding section, and then determine the equivalent rectangular section. THE CONSTRUCTOR. 117 Let: ■ h, be the larger side of the rectangle, b, be the shorter side, (5, the diameter of the equivalent circular section at the same point ; then for a given value of /;, we have : ('65) h A=^liLi = o.84^/- and for a given value of b : h 3 51- / rf \' / "J >' (166) and for a given ratio . 4=i 1 '1 F B [ 1 -::> C G -i.' 41 D cases it is best to determine an ideal round rod, according to Fig. 5, from which the desired section can be derived. For any given case, let : (5 = the diameter of the ideal rod, n, and b, the width, and thickness, respectively, then for any selected value of b, 6 b '>\i6 ]( b\ h from which we get the following table ; (171) i b & b ^. When the connecting rod acts directly upon a crank the angle a is usually 20° or more, but when the connection is to a beam it is seldom greater than 10°. Another form of wrought iron cross head for link connections is shown in Fig. 531. Thisfoim is especially convenient when occasion requires that the piston rod be disconnected readily, and is especially adapted for direct- acting steam engines. ? 189. Cross Heads for Goides. Cross heads for use with guide bars are made in many varied forms for steam engines and pumps. The form is modified to a great estent by the number and arrangement of the guide bars. Fig. 532 shows a much used form of cross head for four guide bars. If the engine runs constantly in the same direction, and the pressure upon the piston acts alwa}-s in the direction of its ■t-^ Fig. 53 motion or in the opposite directiou, the pressure will be almost entirely confiued to one pair of guide surfaces, the other pair only coming into action in the case of extraneous forces. If the pressure acts sometimes with the direction of motion and sometimes against it, the result will be to cause the pressure on the slides to alternate. In most steam engines the pressure changes not only in directiou but in magnitude, especially near the end of the stroke. The slides should be made of a softer material than the guide bars in ordei that the greater wear may- come upon those parts which are most easily replaced. In order to reduce wear it is also desirable that the surface of each slide should not be less than 2.5 P\ /^ being the total pressure on the piston in kilogrammes, and the area thus obtained being in square millimetres. This is about equivalent to 0.0018 P\ P be- ing the total pressure in pounds, and the area given in square inches. Many use double this area, or 0.0036 P, with corre- sponding reduced wear on the parts. The pressure on the sur- I20 THE CONSTRUCTOR. face of the slides, witii the ordinar)' ratio of connectiug rod to crank arrc, will then be about i3o pounds per square inch in the first case and about 60 pounds in the second. If we represent the superficial pressure, rubbing Velocity and coefficient of friction for slide and crank pin respectivelj' by At A. ^'i' ^'2>/i-fi< '^^ have for the lineal wear per second : U^ '= fiPi ''■'\f\ ^'^'i ^i ^^ I'-iPi ^2/2. ill which y-i and ji^ are coefficients due to the materials used. Some of these values vary at differ- FiG. 533- ent portions of the [stroke. If, however, we take them at the same instant, we have the ratio of wear for that point, U\^ ^ I'lPiA'^'i U-i HPifi ^2 The point of maximum wear upon guides is near the middle 2 Tt R n -K d n of the stroke, where v, = and zu = 7 1 60 X 12 60 X 12 Taking the values of /^ and U the same in both cases, we ob- tain, by substitution in the preceding equation, A d_ p". ^ R which gives an average ratio of about yV, and taking /j at 1420 pounds gives about 120 pounds for/,. If we consider the pres- sure on the pin to be alternating and that on the slides contin- — ,. " -^\ a. ^ 1 ' 1 ? [ :f i'' -o>--~- —~ 1 V/ - " 1 ■' Fig. 534. nous, p., becomes onlj' 710, making />, about 60 pounds. If the ratio of connecting rod to crank arm is unusually small, the pressure Q on the slides at mid-stroke should be calculated, and It may be taken as C? = j The cross head shown in Fig. 533 is arranged for a fork jour- nal, the latter being also in this case made spherical. The fork, which is keyed to the piston rod, is intended to be made of wrought iron ; should it be made, instead, of cast iron, the thickness of the metal about the hub should be increased to o.2Srfi, and its length to \.T$d^. This form permits the slides to be brought closer together than in the preceding design. A very simple form of cross head for four-bar guides is used on man}' American locomotives, as shown at a and b^ Fig. 534. For constructive reasons, to obtain the necessary clearance, this form is sometimes made as at b, with the middle plane of the guides above the axis of the piston rod. The cross head is of cast iron, with the pin cast in, and finished by special machin- erj'. A similar form of cross head to that shown at a is used on the Porter-Allen engine, except that a steel pin is inserted as shown at c. The flattening of the top and bottom of the pin serves to assist in the distribution of the lubricant. "■■ The area of slides in America is about that given by the fore- going rule. Example. A wood-burninjj passenger engine has cylinder 16" diameter, at no pounds pressure, giving P= 22,110 lbs. The surface of each slide measured 79 square inches, or about 22,110 X 0.0036 = 79. 59 sq. in. The forms of cross head shown are generally fitted with slides of white metal or bronze, and in some instances bearing surfaces of glass have given good results. There is one form of marine engine which requires a special form of cross head. This is the so-called back-acting engine, in which the crank shaft is placed between the cjdinder and the cross head, and there are two piston rods, passing above and Fig. 535. below the shaft. There have been man}- varieties of this tj'pe constructed. In Fig. 535 is shown a design by Maudslay. The body of the cross head is formed like an axle, with two project- ing bosses for the attachment of the piston rods. The distance E is governed by the diameter of the crank shaft, and A by the clearance space required for the crank arms. In this design the slides are placed outside of the piston rods ; other builders, as Ravenhill, place them between the rods and the journal d' , where, as will be seen, there is sufficient room. The lower por- tion of the slides are made of bronze and fitted with adjusting keys. The dimensions of the body are obtained by considering it as an axle, remembering that the forces act to produce twist- ing with the arm E as well as bending with the arm A. The length I,' is to be taken in connection with the diameter d' , so as to keep the pressure on the journal within practical limits. English practice in such construction gives pressures ranging from Soo to 1800 pounds per square inch. The diameter 6 of the threaded ends of the rods is the same as given for Fig. 499. Fig. 536. In Fig. 536 is shown Stephenson's cross head. Here the guides are brought so close together that each pair merge into one, and there are but two guide bars. The middle piece, ot wrought iron, is made with two journals, for a forked connect- ing rod. The slides are best made of bronze, the area being as before = o.oojb P, except in the case of locomotives, where the limited space often causes it to be reduced to o.ooiS /". Another design for double guide bars is that of Borsig, shown in Fig. 537. This contains a fork journal, whose projected area /' X d' should not be made too small. Sometimes this is made so small that the pressure reaches 3000 to 4000 pounds, and hot bearings and cut boxes are apt to lollow. Judgment in this re- spect is most important for all bearings. The slides are made of cast iron, with bronze shoes, which are packed out with thin slips of copper or zinc. * This is also done on a horizontal engine built by Brown, of Wintherthur See Engineering, Jan., 1880, p. 70. THE CONSTRUCTOR. 121 Fig- 53S shows a noteworthy form of cross head used ou thft Western Railway of France. The body is of wrought iron, the slides and piston rod connection are of steel. The manner in Fig. 537- which the reverse taper of the rod is secured, by means of a key and conical ^hell of steel, is of peculiar interest. Since this is a special construction a few dimensions are given in the figure (in millimetres). The area of the slides does not appear to be ^ ^ large. The rather complicated form of the head of the pin is shown in the lower right hand corner of the illustration. Fig. 539 shows a cross head of the so-called "slipper " tj'pe, for single guide bar. This is well adapted for situations in which the direction of rotation is constant and the pressure al- ways downward. In order to provide for possible lifting forces, and to meet the reverse action of compression and inertia, the beveled shoes are used, although a square shoulder is to be pre- ferred. The area of slide should not be less than 0.0036 P, pref- erably more.'- Another form of cross head for single guide is given in Fig. 540. This is from a marine engine by Humphrey's, Tennant & Co., and is intended to serve for pressure in either direction. In this case the bearing is in the cross head, and the pin is intended >. 0'<5a>-t US: Fig. sag- to be fast in the connecting rod, being attached as in Fig. 537. The wear on the bearing is taken up by the removal of thin slips of copper, originally placed in the vertical joint ; and wear upon the guide, by the insertion of similar slips between the cross head and slide. The whole construction is applicable to many situations. The middle portion is in this case made of bronze, but may be of cast iron, when the bearing is lined with white metal. The modulus for the dimensions is the same as in formula (160), and the bolt diameter rf, as in Fig. 499, using for d the diameter of the equivalent normal wrought iron overhung pin. A somewhat similar cross head has been designed by Napier for use with the horizontal back-acting marine engine, Fig. 541. This is intended to be used with a forked connecting rod. The Fig. 540. middle block is made of cast iron, and the distance B is kept as small as possible, in order to reduce the size and weight. The depth of arm h is determined as in Case I or II, ^ 6. The bolts, whose diameter 6 is calculated as for Fig. 499, are secured by jam nuts. In Fig. 542 is another excellent design by Maudslay for simi- lar service. This is for an ordinary connecting rod, as in Fig. 5 1 8. The pin is formed in the crooked wrought iron piece which also forms the arms. The thickness h' of the latter is deter- mined from the corresponding moment after having selected the depth /;, which in this case is made equal to d' . The value Fig. 541. of d' is calculated as in the case of an axle. The screw diam- eter i' is calculated as before, and should be made full}- as large as the formula gives. The small lug on the lower part of the right arm is for the attachment of the pump rod. Such attach- ments are frequently made to the cross heads of marine engines, of which this is a good example. On the left the slide is shown in section. This is cast of bronze, with the channels shown * For a similar cross head, designed by Stroudley, for locomotive sen-ice, see Engineering , Feb., 1867, p. 65. Fig. 542. filled with white metal. The small shoe on the right, which is secured by screws, can be removed, so that slips of thin copper can be inserted to take up for wear. These last two cross heads, although unusual in appearance, show how a difficult construc- tive problem can be solved completely, and may be regarded as types. ? 190. Guides and Guide Bars. Guides are made of wrought iron, cast iron or steel. If the entire pressure comes upon one guide, as in the designs just de- scribed, and the guide is supported onlj' at the ends, which are 122 THE CONSTRUCTOR. separated by a distance = jj + i,, it must be calculated to resist bending. Taking the crank at right angles to the guide, as the most unfavorable position, and calling the pressure Q, and the distances of the two points of support from the centre of the i" TlflB iillllllliilliilMM Ii^^ Fig. 543. cross head as s-^ and s.^, Fig. 543, we have the bending moment of the bar = Q — ~= — , and for the relation between the depth and width of bar : Jl + s., ■'= /I ■'1 + -^2 (177) Q S 6 s^ The permissible value of stress 6" for wrought iron or steel should be small, say 7000 pounds, in order that but little deflec- tion shall occur. Any springing is especially hurtful in this case, since it prevents the entire surface of the slides from bear- ing fairly, and thus causes greatly increased pressure upon the Fig. 544. points vfhich are in contact. Deflections of ^V" or more are sometimes found, w^ith corresponding irregular wear upon the slides. This subject can be tboroughlj' investigated graphically by taking the various positions of the load. In Fig. 544 is shown a form of cast iron guides, intended to receive pressure only upon the lower guide. This is only sub- ject to compression, and hence very little deflection can occur. Fig. 545. The sectional view on the left shows the disposition of the ma- terial, and it will be noticed that the flanges on the cross head are arranged so as to retain the oil. The upper guide is bolted to the lower, and should the motion be reversed, throwiug the pressure on the upper guide, the bolts must be made pro|3ortion- ally stronger. A form of guides which is coming more and more into use for stationary engines is that shown in Fig. 54.5. Here the flat Fig. S46, guide surfaces are replaced by portions of a cylinder. An espe- cial advantage of this construction lies in the possibility of bor- ing the guide surfaces in exact alignment with the cylinder. Any twisting of the cross head is prevented by the connecting rod and crank pin, or, if necessary, a tongue on the lower slide may fit into a groove in the guide. The cross head for such guides may be similar to Fig. 537, the lower guide being adjusted b}' a key. The single guide bar has been used in locomotive practice. Fig. 546, which was shown both on American and Belgian en- gines at the Paris Exposition of 1S78. The guide is bolted to the cylinder at C, and to the yoke at _/. The cross head is a. simple modification of the form in Fig. 5346. Engineer J.J. Birckel has shown that there is a heavy lateral stress on such a guide bar, due to the necessar}' end plaj' in the driving axles, and a wide bar is therefore necessary. He makes the width. f> = 2-< /;, and makes // = Const 7 in which G is the weight of the parts subject to lateral vibra- tion, O the normal component of the piston pressure, X the length of guide bar, and // the distance from centre of bar to centre of rod. In the case of a cylinder iS" diameter at 100 lbs. steam pressure, G = SSoo lbs., /, = 51.2 and // = T-S''^ the values obtained are : b =^ S'^, h = 3". Fig. 547- Fig. 547 is a cast iron guide for horizontal marine engine, suitable for a cross head such as is shown in Fig. 540. This is especially arranged to retain the lubricating oil, and as the cross head moves between the positions i' — i and 2 — 2', every stroke, it dips in the oil at each end and carries it over the guide. Example. The steamship " Arizona " is fitted with single guide bars and automatic lubrication. The pressure on one slide is 64,000 lbs., the area being 47" X 27" = 1269 sq. in., or a pressure of about 50 pounds per inch. CHAPTER XVI. FRICTION WHEELS. I 191. Classification of Wheels. Wheels are used in many varied ways to transmit motion in machine construction. They may be divided into two great classes : 1. Friction wheels, 2. Gear wheels, according as they transmit motion by frictional contact, or by the engagement of gear teeth. Each of these classes may again be divided into : (a) Direct acting, and (5) Indirect acting wheels, according as the force is transmitted directly from one wheel to another, or indirectly, by means of belt, cord, chain, or similar device. This gives four divisions for consideration, as follows : I. Direct Acting Friction Wheels, or friction gearing, pure and '"•imple. II. Direct Acting Tooth Gearing, otherwise called simply, gearing. III. Indirect Acting Friction Wheels, such as Pulleys, Cora Wheels, &c. IV. Indirect Acting Tooth Gearing, such as Chain Wheels. The first three forms exhibit the greatest variety, and will be given the first consideration. The relative position of the axes has a most important influ- ence upon the form of a pair of wheels. The positions may be grouped as follows : 1. The axes geometrically coincide, 2. They are parallel, 3. They intersect, at an angle, 4. They are at an angle, by pass without intersecting. This gives four groups under each of the preceding main divi- sions. THE CONSTRUCTOR. i 192- Thb Two Applications of Friction Wheei may be called driving friction wheels, or commonly simple friction wheels, and the second application includes all the various forms of friction rollers, roller bearings, ball bearings, and the like. The two kinds have also been termed friction wheels and anti-fi-iction wheels. 5 I93' Friction Wheels for Parallel Axes. The surfaces of a pair of friction wheels in contact are almost always of circular curvature, and when a pair of such wheels roll freely upon each other the number of revolutions will bear an inverse relation to the radii of the respective circles. This Fig. 54S. ratio is called the velocity ratio of the wheels. If we call the revolutions per minute of each wheel n for the driver and n^ for the driven wheel ; and the corresponding radii R and R^, we have for the velocity ratio : ?=| (-»> Friction wheels for parallel axes are made with cylindrical surfaces. Fig. 54S. In order that there shall be no slipping be- tween the surfaces we must have a pressure O, which, to transmit a force P, at the periphery of the wheels, must not be less than Q = J ('79) f being the co-efficient of friction. Ihe value of_/for various materials may be taken as follows : For Iron on Iron o.io to 0.30 " Wood on Iron o.io to o.5o " Wood on Wood 0.40 to 0.60 Friction driving is often very simple and practically effective It had been almost neglected for general uses, when it was very successfully applied in various forms of saw mill machinery. This was especially the case in the lumber regions of America.* The best results are obtained in practice from surfaces of wood on iron, the wooden surface being preferably the driver, so that any stoppage on starting shall not wear hollows in the softer material.! The rim is built up in such a manner as to place the grain of the wood as nearly as possible in the direc- tion of the circumference. The best wood for the purpose is maple, but linden, poplar and pine have been used with good results. Great care must be taken to make the wheels truly cylindrical, and they should be keyed upon their axles and fin- ished while running in their own proper bearings. Under these conditions a wheel of maple can transmit a circumferential force * See Wicklin, "Frictional Gearing," Sci. Am., vol. 26, p. 227; also Apple- ton's " Cyclopedia of Mechanics," vol. 2, p. 36 ; also Cooper's " Use of Belt- ing," p. 288. . t Surfaces of compressed paper against iron are now m general use.— Trans of about 2S pounds per inch of face width, or from 15 to 20 pounds for the other woods above mentioned. This gives for maple face : ■ 28" iiSo(///^) (180) and a width i >< to 2 times greater for the other woods, II P beingthe horsepower transmitted, and v the circumferential velocity in feet per minute. Substituting for v its equivalent value, ■ Rn we have 2414 HP R (.81) Such wheels are made in practice up to 6 feet in din meter and 30 inches face, transmitting upwards of 60 horse power. According to the experiments of Wicklin, the coefficient of friction is about 0.30 to 0.32, from which the pressure of contact must be (9 = 3 '-3 P. The ease with which these wheels can be thrown out of gear is a very convenient feature. Example I. Let 10 H. P. be required to be transmitted by friction wheels, the speed of shaft being 80 revolutions per minute, and a circumferential velocity of iiSo feet per minute given. AVe get from (180) b = — ^ . 10 = 10" face, and from (iSi) R = 2414 X 10 If the driven shaft is run 100 rev- 10 X 80 olutions per minute, the radius of its wheel will be R\ = 30'' X o-8 = 24". Example 2. Required to transmit i H. P., the given value being n = 90^ «i = 75, y? = 12", R = 13.66". From (181) we have 2414 *= . : X90 = 21/". If pine is used, this should be doubled, giving b = 4j^". The method of construction of these wheels is as follows : For large wheels, 4 to 10 feet in diameter, the rims are made from 6 to 7 inches deep, built up of wooden segments ij/in. to 2 in. thick, forming "a to yj the circumference, and so placed that the direction of the fibre shall follow the circumference of !< Fig. 549- Fig. 550. the wheel as nearly as possible. These segments are firmly clamped together and secured by bolts or nails. The actual face is made about 2 in. narrower than the working face b. This rim is then securely fastened to the arms, which are ver}' strong and made with feet or pads which are mortised into the rim and both keyed and bolted fast. The number of arms varies from 6 to S, and for very wide faces two sets are used ; see Fig. 549. An additional ring of wood is then put on each side, bringing the width up to the full value of b, and these outer segments are deeper than the others, so that the ends of the keys are en- tirely covered ; the completed wheel is then turned and finished in place, as before stated. Smaller wheels are built upon iron drums, the segments being screwed together and clamped between the outer rims. Fig. 550. Projections on the iron rim, let into wood, prevent the latter from turning. The total thickness of rim is about 4 in. Care must be taken that the wood is thoroughly dry. The driven wheel of iron is made similar to a belt pulley, but with a much stronger rim and more and heavier arms ; when a wider face than 16 in. to iS in., double arms are used. Both wooden and iron wheels should be carefully balanced, in order to avoid vibration. , An important and ingenious use of friction wheels is in con- nection with a drop hammer, the wheels being used to raise the drop. Merrill's drop hammer, Fig. 551, is operated by two iron friction wheels A and C, which together act upon the oak plank B, to which the hammer drop is attached. The roller A is the driven one, and its shaft runs in eccentric bearings on each side, which are operated by levers D and press the parts to- 124 THE CONSTRUCTOR. gether. When the parts are in the position shown, the plank and hammer are raised, and when the lever D is lifted, the wheels separate and the hammer is allowed to drop. In some I Fig. 551. similar designs both rollers are driven, as in the hammer of Hotchkiss and Stiles,* and also in the so-called " Precision Hammer," of Hasse o: Co., of Berlin. f I 194. Friction Wheels for Inclined Axes. When the axes are inclined to each other, the surfaces of the •wheels, unless they are very narrow, become portions of cones, •with a common apex at the intersection of the axes. Fig. 552. Each pair of circles in the surfaces then roll together as if cyl- indrical. Wheels of this sort may be constructed in a similar Fig. 552. •manner to those described in the precediug section. In Fig. 553 are shown, at a and b, t-n'O sizes of conical wooden friction wheels. The outer disk is placed with the fibres in a radial' direction, but the others have the grain of the wood arranged as nearly as possible circumferentially. These disks should be most carefully fitted, glued and bolted together. Especially im- portant is it that conical surfaces should be turned to the cor- FiG. 553- rect angle. The pressure is applied from the end of one of the two shafts in such a manner that the force may be applied or removed at the thrust bearing. The most extensive application of friction driving, both with cylindrical and conical su'cfaces, is found in locomotive engines. The high pressures necessarily used compel in this case the use of iron or steel tires. The force Q here exceeds 6 tons.j In some cases a combination of one conical wheel and one narrow wheel with rounded edge, as in Fig. 554, may be used for the transmission of small po-s\'ers. In this case both wheels are made of iron. The pressure is easily applied to the disk wheel B, and the mechanism is so arranged that it can be shifted along its axis, so that a variable speed motion is obtained. It must be noted that in this form the surfaces in contact are ne- cessarily very limited, and hence it is desirable, as in the case of friction couplings, to have the diameters as large as possible. * See .\ppleton's " Cyclopedia of Mechanics," vol. 2, p. 85. \ German Patent 2685. In this hammer the lower part of the plank is re- duced, and the whole design very ingeniously worked out. \ The surfaces in contact are sensibly flattened. Krauss' experiments showed that with a pressure of 12 000 pounds, a steel tire on an iron rail gave a surface of contact of 0.309 sq. in., and with a pressure of S250 pounds, a surface of 0.24 sq. in. In the Foutaine locomotive the pressure of contact ■was about 8 tous on each wheel. and the linear velocity high, in order that the driving force may be kept as small as practicable. The most convenient modifi- cation of this form is that in which the angle B of the cone is made 180°, when we obtain a pair of friction disks. Fig. 555. The velocity ratio, when A is the driver and B the driven, and X is the distance from the axis of «, is expressed by : i=^^i^, which = r (1S2) when /3 ^ 180°. The change of velocity is expressed hy the line O N. \i B\s the driver and A driven, we have X sin /3 , which n (183) when ,9 = 180 ; n being the number of revolutions of B. These are the equations of an equilateral hyperbola; see Fig. 555. When the value oi x approaches near zero, the driving oi A by B becomes impracticable.* Fig. 554. In Fig. 556 is shown a form of variable speed gear in which one disk is placed between two others. The disks A^ and A.,_ revolve with the same velocity in opposite directions, and the driven disk B is placed between. The velocity ratio can be varied from o to ^- proportional to .-tr.f The pressure is applied at the ends of both horizontal shafts. This arrange- -" ^■— i'Ea] Al -j- j- Fig. 555. Fig. 556. ment has been used for driving centrifugal machines, and more recently for potters' -n'heels, the control over the speed being especially useful in the latter case, the position of the variable disk being controlled by a treadle. Another arrangement of disk friction wheels to produce a variable speed is that of Rupp, shown in 557. A is the driver, B the driven, and C the intermediate, the latter being ad- justable on its axis. The variation is bet-sveen the limits a—R -, R and R a—R according to the relation ?/[ X 11 a — X {1S4) which gives the equilateral hyperbola shown in Fig. 557, inter- secting the axis of ordinates when jr ^ o. Rupp recommends especially that the intermediate wheel be made of a number of * In the variable speed gear of Lecoeur (Gernian Patent 17,078) a loose disk is filled ia the centre of --/, so that if B approaches too near the centre the motion ceases. t See Berliner Verhandlung, 1S66, p, 39. This arrangement has been used especially for regulating" the speed of cotton-spinning machinery. THE CONSTRUCTOR. 125 thin disks, all loose upon the shaft. This does not appear to be advantageous in view of formula (1S4), since there is a different ratio for each disk, and hence some of them must slip. A similar device is that of Barnhurst, Fig. 558, in which the disk is placed between two cones,* By making two of the disks fast on one shaft, and placing the driving wheel between them, with sufScient clearance to enable either to be brought in contact with the driver, the driven shaft may be operated in either direction or allowed to remain Fig. 558. at rest. Fig. 559. A-y A^ are the driven, and B the driver. This is ingeniously applied in Cheret's Press, in which the screw of the press is on the axis of B, and is turned in either direction by the friction wheels. Fig- 559- I 195. Friction Wheels with Inclined Axes not Intersecting. In the case of friction wheels whose axes are rigidly held, and, while inclined, do not intersect each other, there is always more or less lateral slipping. The figures which, under these condi- tions, exert a maximum amount of rolling action and a mini- mum of slipping are a pair of h3'perboloids of revolution (see ?2i8). If, however, the axes are so arranged as to permit longitudinal motion, either with the bearings or in them, the ■wheels will be relieved from slipping. Such an arrangement, by Robertson, is shown in Fig. 55o.-(- The disk A acts upon a cyl- inder B, the axis of which makes a small angle with that of A. When the disk A is revolved, it rolls a helical path upon the cylinder, and also moves in the direction of its axis. The angle a corresponds to the angle of the screw thread. Robertson has applied this device as a feed motion to a wood lathe. This ar- TZl" Fig. 560. rangement ma}' also be reversed, A being held in its bearings, and B, with its bearings, permitted to travel. The same principle may be used with cones on disks, but these devices appear to possess limited practical application. Friction wheels, the axes of which coincide, are the same as friction couplings. ? 196. Wedge Friction Wheels. Wedge friction wheels are those in which the cross section of the rim is wedge-shaped. They were designed in Italy by Mi- notto and in England by Robertson, and hence are known by both names ; in both cases being applied to wheels with parallel axes. Two forms of rim section are given in Fig, 561. In this case the radial pressure Q is much less than with cylindrical wheels, and for any wedge angle Q it is equal to Q=P sin \- f cos 7 (185) A disadvantage of this form is the fact that true rolling action only takes place in one cylindrical section through each rim, and hence there is much hurtful friction from the slippage at other points ; this defect becomes less as the ratio of the wedge depths k, k-^ to the radii Ji, J?^ diminishes.- In order that the k k^ ratio -77 and ^ may be kept as small as possible without re- ducing the surface of contact, the rim is made with multiple grooves, as in the form on the right. The angle 8 is generally made ^ 30° although Robertson used much smaller angles. krJi-l], Fig. 561. These wheels grow warm and wear rapidly when operated con- tinuously at high speeds. Minotto has also made especial ef- forts to design bevel wedge friction wheels ; he uses only one groove, and adjusts the position so that wedge profile shall al- ways act at the same point. Robertson makes the grooves non- adjustable, as in spur wheels. Wedge friction driving has been proposed for locomotive driving, and models made on this plan have ascended steep grades ; the wear in this case comes mainly upon the track. Wedge friction wheels have been used in America for many years on winding engines ; and they are especially useful in driving ship's windlasses, on account of the ease with which the}' can be thrown in and out of gear.t More recently wedge friction wheels have been used by Gwyune and also by Weber in Berlin, at high speeds, and apparently with good endurance, * See Engineer^ June, 18S0, p. 404 ; also H. Kdnig, German Patent No. 9365. t See En^neer, 1867, p. 410, in which many interesting designs by Robert- son are given. * Hansen, in Dingier^ s Journal ^ vol. 137, 1855, p. 1, shows that the actual rolling circle is always on that portion of the wedge surface towards the driving-wheel, and chano;es its position when the driver becomes the driven. See also Ad. Ernst, in Zeitschr. d. V. deutscher Ingenieure, xxvi, p, 243. t H. D. Andrews' steam windlasses are made with wedge gear of from 4 to 12 grooves. The diameters of the friction wheels are as follows; H. P. Slow speed. Fast speed. Drum, Diam, Length, 5 4 — 30" 8 — 26" 6" 27" S 4—30" 8—26" 8" 27" 10 6—36" 12—30" 8" 30" 15 6 — 36" 12 — 30" 8" 30" 126 THE CONSTRUCTOR. driving centrifugal pumps at 700 revolutions per minute. These wheels are with single groove and wedge, the wedge being of curved profile, and hence acting somewhat like the adjustable device of Minotto.* Fig. 562. Single-groove friction wheels have also been used in America for mill gearing. Sellers has devised an ingenious form of wedge friction gear for changing the rate of feed on engine lathes. This is com- posed. Fig. 562, of two simple disks and a pair of ver}' obtuse cone plates, the latter being pressed together by springs. The axis of the cone plates is movable, thus giving change of speeds. The ratio of change is similar to Rupp's gearing, formula (184). §'97- Speciai, Applications of Friction Wheels. The previously stated condition of wedge friction wheels, that there is but one line at which rolling action takes place, and that slipping occurs at all other points of contact, is utilized in vari- ous methods in machine design, as for example, in rolling mill machinery. In this case a third piece is driven, compressed and altered in form between two friction rolling members. The rolls and the metal may be considered as a train of friction gearing. In the case of a plate mill, the plate may be considered as a pair of friction wheels of infinitely great radii ; this is also the case in rolling bars. In a tire mill one surface is an internal and one an external wheel, of variable radius. The three-high mill may be similarly compared to a train of friction gears. Fig. 563. A very interesting application is that referred to in ? 14S, as in use at the Kirkstall Forge, and sho%vn in Fig. 563. A and B are plane friction disks. The round bar C passes between them, slightly above the centre and partlj' rolling, partly sliding, re- ceives both an endlong motion and a motion of revolution upon its axis. The disks revolve in the same direction, and of the opposed forces which tend to cause revolution of the bar those which act in the portion of the disks between their axes, i. e., between the vertical dotted lines in the figure, preponderate, and determine the direction in which the round bar revolves. The horizontal components of the sliding forces at all portions of the disks, act to carry the bar forwaril, so that it receives a combined spiral motion and is at the same time rolled and straightened. The earlier method of rolling round bars was by means of semicircular grooves, but this does not give either as round or as straight a product. Many similar examples in roll- ing mill machinery will be found, resembling friction driving In the same way, various forms of grinding mills are made upon the principle of friction combinations, as in the case of the Eogardus mills, with flat grinding disks, and also in the case of grinding rollers, Fig. 564. Here the round trough A revolves, * See Engineerings iS68, pp. 502, 593, and 1869, p. 353. Engineer Brauer, assistant in the Royal Technical Pligli School, lias attempted to adapt the principle of the Weston Clutch (g 157) to friction wheels. The wheels are made of a number of thin plates, with rubber washers between them, and a slight axial pressure is sufficient to cause them to grasp each other with much friction. A description will be found in Berlin Verhandlung, 1877, p. 295. and in it act the rollers B^, B^, and the width of face of the rollers compels a sliding action, forward on the outer edge and backward on the inner. The trough may be stationary and the shaft a, carrying the rollers, revolve. Rollers with inclined axes are also used for grinding, and a similar device has been made for straightening round rods. Fig. 564. ? 19S. Roller Bearings. Roller bearings, sometimes called anti-friction rollers, maybe used in either of two forms : ((?), in such manner that the rollers are carried in their own bearings, the latter receiving the load ; (b), or in such a manner that the rollers are placed between two moving surfaces and act with a rolling motion upon both of , them. Roller bearings are used in connection with surfaces which are flat, round, or even spiral. Examples of rollers upon cylindri- cal surfaces are given in Fig. 565, in which a and b are forms used on pillar cranes, and b-^ is the more general form of b. Roll- ers are also used in axle bearings, and in heavy pulley blocks, where indeed the sheaves themselves are a form of friction roller. Fig. 565. A form of roller bearing which is subject to very heavy loads is that used to carry the ends of bridge beams and trusses, to provide for expansion and contraction. These are made either with round rollers, as at a, Fig. 566, or with double segments, as at 5. For round, solid rollers, the load may approximately be in- vestigated as follows : — Let / be the length, r the radius of each I roller, and P the load. This load will be carried by a surface of a width b, included in the angle (measured at the centre of the roller) /J = 2?>. We have for the relation of these elements : P = Elr 48 and S- 16 /3^ £ being the modulus of elasticity, and 5" the fibre stress upon the material. Also : 5=0.83 ^/£: ^(r and It will be seen that for any given material the relation - — can be so made as to keep the stress within practicable limits. THE CONSTRUCTOR. 127 These may be chosen as follows, both surfaces being of the same material : Cast Iron. U^roit^ht Iron. Steei {hardened) E^ 14,220,000 28,440,000 42,660,000 p — = 425 to 500 340 to 400 1000 to 1400 / r 5'^ 11,000 to 12,000 11,000 to 13,500 25,000 to 32,000 - -<-x— X--)(-°-X-°)(-°->---^§ Fig. 566. Example I. The bridge over the Elbe at Hohnstorf has spans of 330 feet* The bearings are made of cast iron of the form shown at /'. The pressure is 792.000 pounds on six rollers, the dimensions of the latter being, / = 53", »-= 4,125". We have therefore — ,= — ^ = 603.8, rl 53X4.125 hence p= ^/ i? A /603.8 = 0.126. ^ 14,220,000 ^ This gives for the breadth h of the contact surface under this load. S = fi r = 4.125 X 0.126 = 0.522, and 5 = 14.220,000 (0.126)2 = 14,280 lbs. Example 2. Bridge over the Rhine at Wesel ; span 125.7 feet, rollers and bearings of hardened steel. The load is 770,000 pounds ou si.x rollers, as P shown at a, and /= 27 75" : 3.875". These values give r / and h =0.43", and 5= 32.450 lbs. Example s Clifton Bridge at Niagara. The load of 171,600 lbs. is carried upon II steel rollers, on bearings of the same material, their dimensions be- ing/ =6.3", r=o.6". This gives a high value for -^ — = 4127; )3 = 0.1 7, hence d = 0.102, and 5= 74,210 lbs. Ball bearings are frequently used instead of cjdindrical bear- ings, and for some ., a.,, so that they may be moved to or from a, or may be pressed against the latter. The length of these links is governed by a screw adjustment. The rollers are pressed together by the ring- roller 7?,, acting upon the planet roller r, and rollers r, and ^'2, the latter bein,g loose upon the axes of the main rolls j?; and /t'j. The planet roller r.^ acts a,gainst the roller 7; which is fast on the axis of /?. If it is desired to exert greater pressure upon the rollers, the roller 7?, is forced towards A'3 by means of the lever combination a, 6, f, d^, the lever 31.67 200 3 1 -S3 31-99 32.15 32-31 32-47 32.63 32-79 32-95 33-10 33.26 210 33-42 33-58 33.74 33.90 34.06 34.22 34-38 34.54 34.70 34.85 220 35-01 35-17 35.33 35-49 35.65 35-81 35-97 36.13 36.29 36.45 230 36.61 36.76 36.92 37.08 37.24 37-40 37-56 37-72 37-88 38.04 240 38.20 38.36 38.51 38.67 38-S3 3S.99 39-15 39-31 39.47 39.63 250 39-79 39,95 40.11 40.27 40.42 40.58 40.74 40.90 41.06 41.22 260 41.38 41.54 41.70 41.86 42.02 42. iS 42.34 42.49 42.65 42.81 270 42.97 43-13 43.29 43.45 43.61 43-77 43-93 44.09 44.25 44.40 280 44.56 44-72 44-88 45-04 45.20 45-36 45-52 45-68 45.84 46.00 290 46.1s 46.31 46.47 46.63 46.79 46.95 47-11 47.27 47-43 47-59 ? 203. Generai, Solution of Tooth Outunes. In a pair of gear wheels, the two tooth outlines which work together lie in a section at right angles to the axes of the wheels and in the plane of this section the construction and action of the teeth is to be considered. The so-called general solution of tooth outlines is that by which, if a form of tooth be given for one wheel, the proper form of tooth for the other wheel may be drawn so that the motion will be transmitted with a uniform velocity ratio. Several such solutions will be given. I. The Author's First Solution. Fig. 570. Given the tooth Fig. 570. Fig. 571. profile a S b c, also the pitch circle T, of the wheel O, and the pitch circle 7", of the wheel Oj ; required the tooth curve a^ S for the wheel Oj. Place the given curve so that the point S, where it crosses the pitch circle, lies on the line joining the centres O O,, thus mak- ing 6", a point common to both profiles. In order to find a second point a^, which shall work in contact with a point a, draw a 1 normal to the given curve at a, make the arc S i' = arc 5' I, and the distance i s^ ^ S l', and S s-^ = 1' i. Then with ^as a centre strike an arc with a radius S, (/, and from i', an arc with a radius i a, and the intersection of these arcs will be the desired point a^ of the required curve. For such points as <-, where the normal to the curve does not intersect the pitch circle the given pitch circles cannot be used, therefore if these points are required the pitch circles must be transposed (exaggerated in the figure). The curve thus found sometimes assumes an impracticable form without being geometrically incorrect. II. Abridged Solution. [Poncelet.) Fig. 571. Mark off ou the pitch circle T^, the points s^, t^, n^t'i , which roll into contact with points j', t, u, v, . ... of the given circle T, draw from s^, t-^, «i, &c., arcs with radii respectively equal in length to the normals to the given tooth outline va, u c, etc., then will a curve drawn tangent to these arcs be the re- quired outline. The points s, t, u, v, should be taken close to- gether. If the lengths of the normals va, u c, etc., are taken backward from the points 5j, /j, //,, &c., instead of forward, the outline for an internal gear tooth will be obtained for the wheel Fig. 573- III. The Author's Second Solution. Fig. 572, The tooth out- line a 5 c Srf ^ the profile of the tooth for the wheel T, (7, 5, b^ that for the wheel 7",, a^ a^ the prolongation of the flank outline for the latter tooth, and I S II the line of action between the limits of the outside diameter circles K aud Ky Lay off from S, on both pitch circles the corresponding spaces 5 i, i 2, 2 3, &c., 5" i', i' 2', 2' 3', &c. Take in the dividers successively S a, 1 a, 2 a, 3 a, &c., and describe arcs from i', 2', 3', &c., and the envel- ope of these arcs will give the path (7 n-j g, or so-called theoreti- cal profile of the flank. The actual profile of the flank a-^f is drawn tangeut to the theoretical curve, to the point where it crosses the clearance circle F^. The theoretical curve is a pro- longed or abridged C3'cloidal curve (see ? 205). In the figure, in which 7" is a straight line, or rack, the curve is an abridged evolute. ^204. The Action of Ge-^r Teeth. In solution III, of the preceding section, reference was made to the li?ie of action * of a pair of gear wheels, and this line bears an important relation to the theory of the action of gear wheels. The line of action intersects the pitch circle at the same point as the tooth profile and cuts the latter at right angles, so that the tangent TV iV of the line of action (Fig. 572) is normal to the tooth profile. Each point of action corresponds to a point of contact of the teeth and also to a point of contact of each of the pitch circles ; so that, for example, the point II of the line of ac- tion corresponds to the point 2 on T, and 2' on T'. That por- tion of the pitch circle between the pitch point of the line of * First discussed in Moll & Reuleaux's schineubau." Konstruktionslehre fiir den Ma- 130 THE CONSTRUCTOR. actiou aud the initial poiut of contact is called the rolling arc for the given point. For example 5' 2 is the rolling arc on T', for the point II, and S ^' on 7'j, for the same point. The sum of the rolling arcs between the two extreme points (arc I 5-1- 55, or arc i' 5 -|- .S5') is called the arc of actiou, and its length indicates the duration of the action of the given pair of teeth, which is easily determined graphically. It depends upon the length of that portion of the line of action which it is desired to use. This is usually taken between the limits of the circles of the outside and the base of the teeth, which gives in Fig. 572 the line of action VI. For any wheel of given tooth outline and pitch diameter there is but one line of action, and lor a given line of action but one tooth profile. This latter can only be determined from the line of action when the rolling arcs for the pitch poiut of the Hue of action are also given. For cycloidal teeth the rolling arc is also the line of actiou aud for this reason the geometrical discussion is much simpli- fied. In order that a pair of gear wheels should work properly together, their lines of action should correspond and their roll- ing arcs be of equal length for homologous points of action. By conforming to these conditions anj' number of gear wheels miy be made to operate with a given wheel. Such wheels are said to be interchangeable or series wheels, since the common line of actiou is symmetrically disposed on each side of the pitch circle, as well as on each side of a radial line passing through its pitch point. The ray drawn from the pitch point through any point on the line of action (as SI, in i'ig. 572) gives the direction of the pressure between the teeth for that point. \ 205- The Cyci90 The value of the coefficient of friction/" is in no case small, even when the teeth are well lubricated, on account of the usual high pressures ; a usual value maj- be taken, /^ 0.15, while for new and dry wheels it reaches 0.20 to 0.25 and even higher. The minus sign in the formula is to be used when one of the wheels (Z^) is an internal gear. ExajiipU I. In a pair of epic^-cloidal gears, of seven teeth, the value of e = 1.225. TakingX= 0.15 we have according to (191 a) for the loss by tooth fric- tion; 2 1.225 ^— 3.14X0.15 X — X =0.0824, or about 8J^ per cent. Example 2. Epicyctaidiil Teeth. ^^^^^40,6 = 1.44 andweget: = 0.0169, or about 1.7 per cent. Example 2,. Epicycloidal Teeth. .2=7, Z\ = — 60 (internal gear). £ = 1.40 and we get : A = 3.14 X 0.15 X — X ■ 40 /r=3 14X0.15 (J — sV) - Example 4. Epicycloidal Teeth. Z= -^ — = 4.2 percent. 1, Z\ = K (rack), e = 1.37 4.6 per cent. 1-37 Example s. Pin-tooth Gearing. Z^6, Zj = 40. We have, as determined by construction, as in Fig. 5S3, e = 1.166. Hence we get from (191 c) : A=3.i4Xo.ls(i+^5)Xi.66 Example &. Evolute Teeth. ^=^1 = 40. e 3X 1.92 ■ 2.6 percent. = 1.92. We have from (igi <5): />r = 3-14X0.15 X — X 40 = 3.4 per cent., or double that in Examp. 2. It will be seen that the tooth friction is least with epicycloidal teeth and greatest for pin gearing ; evolute teeth being midway between. The wear upon gear teeth is aifected by other considerations besides that of the coefficient of friction, the pressure of the teeth upon each other, and the relative rubbing movement of various portions of the profile also entering into the problem. The wear is therefore not constant for a constant pressure, and it is an error to assume, as is sometimes done, that the form of evolute teeth is unaltered by wear. These teeth usually show the greatest proportional alteration by wear, since the flank of the tooth below the pitch circle has a very much less rubbing movement than the portion of the opposing tooth which rubs against it and hence the wear is unequal. ^ Approximately. THE CONSTRUCTOR. 135 The effect of this may frequentl3' be observed in practice, where the smaller of a pair oi evolute gear wheels -will be no- ticed to be worn into deep hollows below the pitch circle. The conclusions given above about the percentage of loss may also be determined geometrically iu the following manner : Take the two portions of the tooth profiles ivhicli wor/i together and divide each by the chord of the corresponding portion of tlie line of action, multiply each result by the ratio of the length of its poiiion of the line of action to the entire length of the line of action, and then multiply the sum of the two quotients by the coefficient of friction. The result will be the percentage of loss, pr. The chord re- ferred to becomes the line of action itself in the case of evolute teeth. This method serves also for pin teeth, and is verj' useful for the designer, as the data can all be taken off the drawing with the dividers. § 214. Generai, Remarks on the Foregoing Methods. Each of the preceding methods possesses its merits and dis- advantages. Epicycloidal Teeth. These possess the great advantage that they -will work together iu any series with as few as seven teeth, while for evolute teeth the lowest in series is 14 teeth, and in no case fewer than 1 1. The loss from tooth friction is a mini- mum with this form, and the wear less injurious to the shape of the tooth. The minor objections which have been raised are that the double curve increases the diiHculty of construction, and that any variation of the distance between centre causes im- perfect action to follow. Evolute Teeth. The advantages of this form are that the simple shape is readily made and that any variation of the dis- tance between centres does not affect the action. Against these must be set the fact that for low numbered pinions the flanks must be altered to avoid interference, or the tops of the teeth must be taken off. The fact that the distance between centres may vary is rather an objection iu many cases, as the arc of action is reduced, and iu transmission of heavy power the shocks upon the teeth are liable to be increased. Evolute teeth are well suited for interchangeable gears, if low numbered pinions are not required (30 teeth being the minimum), and where but small power is to be transmitted they are excel- lently adapted. For wheels which run only in pairs, and hence for bevel gears, this form is excellent, since it is so readily made. See \ 222. Pin tooth gearing and the mixed outlines are only used for special work, such as in hoisting machinery and the like, and in such cases the wheels are often made of wrought iron or steel. Disc wheels have a very limited application, but iu some spe- cial forms of mechanism they are very useful, and will be dis- cussed further. See Chapter XVIII. If bevel gears are required to interchange (see \ 200) they" must not only be of the same pitch, but nmstalso have the same length of contact line, A S, Fig. 596. Since these conditions are very infrequent, it follows that bevel gears are generally only made to work in pairs. In practice it is found that a vari- ation of less than 5 per cent, in the length of *he contact line may be neglected. Gears of the same pitch and same angle of Fig. 596. axes, but with a small variation of contact line, are called "bastard gears.'' A pair of right angled bevel gears of 80 and 45 teeth, might be altered in practice, if required, into bastard gears of So (i ±0.05), /. c., 84 to 76 teeth, which would work with the other gear of 45 teeth. I 216. Construction Circles for Bevei. Gears. The geometrical figures which are formed by one cone rolling upon another, require that both cones should have a common apex. The surface thus developed is called a spherical cycloid. Of these there are five particular forms, as with the plane cy- cloids, the latter being really those for a cone with an apex: angle of iSo°. The spherical cycloid is very similar in form to the plane cj-cloid, as are also the corresponding evolutes ; the branches of the curves assuming a zig-zag form.* B. CONICAL GEAR WHEELS. ?2IS. Generai< Considerations. In the case of conical gear wheels, or as they are generally termed. Bevel Gears, the working circles of a pair of gears which run together, lie on the surfaces of a pair of cones, the apex of each cone being at the intersection of the axes of rotation. In such case the pitch circles are taken at the/> base circles of the respective cones, as S D, and S E, Fig. 596. The length of the teeth is measured on the supplementary cone, to each base cone, SB being the supplementary cone for S D, and 6' C that for 5' E, B C being at right angles to A S. The length of teeth is laid off on SB and SC, and the width of face on SA; the tooth thickness being spaced off on the pitch circle and all the teeth converging to the point A. The respective radii S D and S E of the two cones are found by dividing the angle a of the axes, in such a manner that the perpendiculars S D and S E let fall from Sto the axes, bear the same ratio to each other as do the numbers of teeth, or inverse- ly as the number of revolutions : thus S D : S E = Z : Z^ = «, : n. There are, therefore, two solutions possible, according as the pitch line S A is taken within the angle o, or in its sup- plement ; or what is the same thing, according to which angle is taken as the angle of the axes. The difference between the two consists iu the fact that for a constant direction of revolu- tion of the driving shaft the driven gear revolves in one direc- tion for the first solution and in the opposite direction for the second solution. One of the solutions gives an internal gear, when «i : n <^ cos a. Fig. 597. The use of the spherical cycloid for the formation of bevel gear teeth would involve many diificulties. In order to construct such teeth, it is therefore common to use the method (first de- vised by Tredgold) of auxiliary circles, based upon the supple- mentary cones, and enabling the teeth to be laid out iu a simi- lar manner to those of spur gears. The auxiliary circles for the bevel gears R and 7?,, Fig. 597, are those of the spur gears hav- ing the same pitch, their radii being respectively r and }\, the elements B S and CS of the supplementary cones. For any given angle a between the axes, the radius r, and number of teeth 3, for the auxiliary circle can be determined *See Berliner Verhandlung the Spherical Cycloid. 1S76, pp. 321, 449, Reuleaux, Development of 136 THE CONSTRUCTOR. from the radii R and A",, and tooth numbers Z a.niL Z^, by the following formula : r _ y/R'- -^ R^- -Y 2 R R^ cos g R "^T^ cos~a; z^ _ VZ' + Z^ + 2 ZZ^cosfl ^ Z^-\- Z cos a If the axes are at right angles, we have r \/R' + R'' z \^Z-+Z;'^ R r^m Z Zy (192) Example. — A pair of bevel gears have 30 and 50 teeth, and an angle between axes a. = 60°, hence cos a = J^, and we have for the auxiliary circle of the 30 . /.T^^ tooth gear : . V. V^3°^ + 50^ + 2 . 30 . 50 . OS 50 + 30.0.5 ~" \/ 4900 For the 50 tooth gear we have also ; ^i = 50 ■ =32-3> say 32. = 64. 30 + 50 . 0.5 From these numbers and the given pitch, the auxiliary circles can be laid off and the teeth drawn. Low tooth numbers are not available for bevel gears, since the errors which are involved in the method of auxiliar}' circles be- come disproportionately great. By using not fewer than 24 teeth for the bevel gear, a minimum of 28 for the auxiliary cir- cle is obtained, and the evolute system can be used to advant- age. This form of tooth is best adapted for this purpose, on account of its simplicity of form, notwithstanding the minor defects which have already been noticed. The loss from tooth friction in bevel gears is approximately equal to that of their corresponding auxiliary gears. Fig. 598. l 217. The PL.4NE GE.A.R Wheel. Internally toothed bevel gears are not used, on account of the practical difiiculties involved in their construction. There is, however, an interesting form of gear wheel which lies interme- diate between the external and internal forms. If the numeri- cal ratio between a pair of bevel gears is = cos a, one of the so- lutions for the base cone gives for the latter a plane surface, 5 E, Fig. 598. Fig. 599. The supplementary cone in this case becomes a C3dinder, and the radius of the construction circle becomes infinitely great, hence the tooth outlines are similar to those used for rack teeth. If the evolute system is used the teeth are very simple, and the plane gear in some cases becomes a very convenient form of construction . As already stated, the ratio is — ' = cosa (193) from which, if for example a ^ 60°, we have -~ = ^. If the angular relation of the axes is given it follows that but one ve- locity ratio can be obtained. This is determined from the angle 72, which is one-half the apex angle of the cone R2, and from the ratio ---? = sin y^. It is sometimes very convenient to arrange a plane gear so that it may work with both of a pair of bevel wheels. This is shown in Fig. 599, in which the gears j?,, R^ have the semi-apex angles y.^, J's, and have their axes at right angles. We then have : R, , ~ = tan y,, = cot 73, R3 from which we obtain the following values : -^ = tan 3-2 = 1 i i f I I 2 3 4 72^ J?, = Sm }'., : 14° i8°3o' 26°4o' 36°5o' 45° S3°io' 63°2o'7i°30 76° = 0.2420.317 0.449 0.6000.7070.8000.8940.9480.970 Either of the wheels 7?,, .^3, can be used with the plane gear Ri if the number of teeth have the ratio given by the value of sin 7.,. Although this limits its application, yet the plane gear is frequently found ver}' useful for angular transmissions.* C. HYPERBOLOIDAL GEAR WHEELS. I 21S. B.\SE Figures for Hyperboeoidae Wheees. Hyperboloidal wheels are used to transmit motion between inclined, non-intersecting axes. The figures upon which they are based are hyperboloids of revolution having a common generatrix. These may be determined in the following manner. Fig. 600. In Fig. 600 is given a projection normal to the line of shortest distance between the two a.xes. The angle a is divided into two parts /3 and ft, in such a manner that the perpendiculars let fall from any point A, of the line S A, upon the two axes, shall be inversely proportional to the revolutions of the gears. S A \% then the contact line of the hyperboloids \ A B = R' and A C *The so called "Universal Gears" of Prof. Beylich, introduced in 1866, should'be considered as a variety of conical gears in which the angle of the axes may be conveniently varied. These may be used for axes of angles varying from 0° to 180° As shown in the illustration, these wheels are formed of globoids of the III Class (see g 224), the meridians ibrming the teeth and spaces. They have found but limited application. A model of these gears is in the kinematic cabinet of the R03'al Technical High School. THE CONSTRUCTOR. 137 = ^'1, are projections of the radii of the hyperboloids intersect- ing at A. We have (194) The actual radii R and R-^ are yet to be determined, as well as the radii S D =: r, and S £ = r^ of the gorge circles. For the latter we have : also : tan/3 tan ft (195) 1" cos a that is, r and i\ have the same relation to each other as the por- tions A F a.nA. A G of a perpendicular to the line of contact. If we call the shortest perpendicular distance between the axes = a, we have ; 1 + -Hi)" a I -| cos a n 1 + 2 —^ cos « + I — i 1 n \" ) } (196) The radii R and R^ are hypotenuses for the triangles whose sides are R' and ?', R^' and 7\ (see the left of the figure) or : R, = VR'^- + r{- (197) R' and R/ being determined as above, when the distance S A = / is given. For the angles /3 and /?, we have the general ex- pressions : sin a tan/3: tan /3i = 4- cos a "1 1 — - -{- cos a (198) As in the case of bevel gears, two solutions are possible ac- cording as the angle a, or its supplement, is taken in determin- ing the line of contact S A, Fig. 601. The choice of solution Fig. 601. governs the direction of rotation of the driven gear, and one of the solutions renders it practicable to make an internal gear ; although this construction has been little used, and has but little practical value. If the angle of the axes a = 90" we have = tan^ /3 = (?)" (199) a tan/3 = n^ -( «^ -I- «i^ ' n"" -\- 11^ ' "1 (200) In the construction of the wheels, corresponding zones are chosen on the two hyperboloids. If the distance between the axes is small, the zones Ij'ing in the gorge circles are generally unsuitable, but when the distance is greater they may be used and the figures approximated by truncated cones. Example I. a = 40°, Fig. 602. = Vsi (see Exaraple i, in g 221), a = 4". Also tan 6 = f 0.5 + COS 40° _ ri 2 + cos 40° ~ g _ 1+2 co s 40° r 1+2X2 cos 40" r = 0.31398 X 4= 1-256", ri = 4 — 1.256 = 2.744". sin 40*^ 0.642S 2.766 8.064 = 0-31398, : 0.232393 = tan 13° 5', and 2 -f cos 40° 2.766 f 1 = 40° - /3 = 26° 55'. Ifwetake5'y« = /= 8" we have .A" = /sin 13° 5' = 3 X 0.126368 = 1.81" J?l'= 8 sin 26° 55' = 8 X 0.452634 = 3.62"; finally :" and — - — — or say tlie number of teeth Z = ^6, and .S' = v/ (i.Si)= + (1.256): R\ = \/ (3-62)= -I- (2.744)= = 4-54". Exa7nple 2. a ^ 90' ^1 = 20 ; a = 0.75". We have from (197) r / g \- Si — = — = = 3.24 and from (200) ri \ 5 / 25 ^ ^' ^ aX 9 - ^ o 75 X Si S- + 9- 106 0.573 = 0.573" and j'l = For ^, we have tan ^ = ■ — - 3-24 ■ i.S, hence (S = ■ 60° 57', and ft = 29° 3'. If we make 2? = 2' , we have from (197); 7;" = .^ ^- . — r- = \/-2- — 0.5732 = 1.916'', and hence .^i', according" to (194) is = g J?i = 1.063", hence Jii = \/ 1.0632 + 0.1772 = 1.07S". The appearance of such a pair of gears is shown in Fig-. 602. According to the table in ? 202 the pitch for the larger gear is : / ^ = = o.-sk", 5-73 5-73 and for the smaller gear fi = -^ — — = 0.339". 3.18 Example 3. a = go°, -^ = 1, ^ = 45°, r — r\, R — R^. In this case the hy- perboloids become similar (see Example 4, \ 221.) -Exainple 4. In the special case in which — = cos a, and the position of the contact Itlie, which is determined byiS, lies in the supplement to a, 'so that — = cos a, the base figures become, the one a normal cone and the other a plane hyperboloid, see Fig 603. This construction is similar to the preceding forms of plane and bevel gears, and may be conveniently used to work with a train of common bevel gears, although but few practical appli- cations occur, partially owing to the fact that the prolonged axis of the bevel I3S THE CONSTRUCTOR. gear passes through the plane gear. For a — 60" obtain the plane gear. We have tan (3 = R' sin 30° Also -jr-T- = — :— — ^ = 0.5 ; r = o,r\=a,R i \/3, f- = R', R\ -i = — Vz = — cos 60^^ we n = 30°, tan ^1 = 00 , ^1 = 90°. If — — be negative and less than cos a. we obtain an hyperboloidal internal 71 gear. Fig. 603. Rack teeth may also be constructed to work with hyperbo- loidal gears. In this case the teeth of the rack are inclined while the pinion becomes an ordinary cylindrical spur gear, since in order to satisfy equation (195) with »'; = 00, the angle /? = (7, and /3i = a, see Fig. 604. Applications of this construc- tion may be found in various machine tools. Fig. 604. \ 219. Teeth for Hyperboi,oidai, Gears. The construction of the exact forms for the teeth of hyper- boloidal gears is a very difficult operation, and in practice an approximation is used similar to that employed for bevel gears. The method adopted is to determine the supplementary cone to the hyperboloid used, and as in the case of bevel gears, use the corresponding construction circle. I ■ Fig. 605. The apex // (Fig. 605) is determined by drawing ^//"per- pendicular to the generatrix 5 A, which, as before, is taken parallel to the plane of the drawing. The teeth will be formed with sufficient accuracy' if two construction hyperboloids are taken with the same angle of contact as the base hyperboloids, according to the conditions in (19S) and (199), and the teeth are formed on the surfaces, which are described by the edges of the construction hyperboloids upon the base hyperboloids.* If it is desired to approximate to the hyperboloidal zone by the use of a conical surface, the apex must be determined. In this case the generatrix S A \s rotated about the axis H S until A falls on the pointy of the circumference, when the new pro- jection of the generatrix will pass through the apex M of the cone. The tooth friction of hyperboloidal gears is necessarily great. This will be considered later, in connection with the speed of the rubbing surfaces, which is similar to that of the spiral gears which are tangent at the gorge circles (see ^ 220.) D. SPIRAL GEARS. Cyi,indricai< Spirai< Gears. Cylindrical spiral gears may be used in the same m?nner as hyperboloidal gears for the transmission of motion between in- clined axes, and in some.cases possess advantages over the lat- ter. There are a number of useful variations of spiral gears. Fig. 606. In Fig. 6o5 is shown a pair of wheels, A and B, both with left hand spirals and corresponding tooth profiles. The pitch angles y and ;'i are so chosen that at the point of contact the pitch cy- linders have a common tangent, so that if a be the angle, of in- clination of the axes, y + jj -|- a ^ 180°. If we indicate by w and v^ the circumferential velocity in the direction of the tan- gent and normal respectively, we have : v^ sin y , iu R sin y Z , . — !- = — = whence — -^ i — =: — . . . (201) V sin 7j !i J?i sin 7; Z^ The normal pitches, J = / sin 7, and ij = /^ sin 7i must be equal to each other, whence — ^- ?. fi sm y As indicated by the components of velocity v' and v/, there is an end long sliding action of the teeth upon each other, with a velocity : c'= z/-}- i// = <; (cot 7+ cot 7i) (202) This sliding consumes power and causes wear, and will be at a minimum when z.' and z';' aje equally great, that is when 7 = 71- With regard to the choice of y and 7, the conditions may be so taken that the position of the coinciding tangents of the two spirals shall be slightly before or slightly after the actual line of contact, but as close as may be possible. This is similar to the position of the line of contact of hyperboloidal gears (g 21S) and may be stated as follows : "' R cot 7 R-^ cot 7i ■ -\- cos a (203) - - -|- cos a as also cot y =r. (204) * See Herrmann's Weisbach's Mechanics, II. ed., Ill, i, p. 418 ei seq. --{• cos a THE CONSTRUCTOR. 139 For a = 90° we have cot y =: — 1-. Such spiral wheels, when 11 the teeth are well made, transmit motion very smoothly, but the surface of working contact is very small. When the axes are at right angles and the wheels the same size, it is often incon- venient to use spiral gears on account of the large size required. ExafiiJ^le. Fig. 607. Let - Fig. 607. : 3, and a = 90°. 'vVe have from (203) R m^ ■ 9 and from (204) cot 7 = = 3, whence y =^ 18° 26' and vi ^ 71'^ n 34'. The sliding velocity is c' = c (3 + 0.333) = 3j ^- The small value of the angle y makes it undesirable to use the smaller gear as the driver. These objectionable features are of increasing importance and for example, — ^ := 5, and ■ = 10, we get — rj— = 25, and 100, and y about iij° and 53°. The difficulty of cutting the teeth on the lathe also increases, as may readily be seen. ? 22r. Approximately Cylindrical Spiral Gears. If, of the preceding conditions, only those of formula (201) and (203) are strictly observed, the difSculties of construction are much reduced and at the same time satisfactory wheels ob- tained. Three methods may be employed : (a) a slight modification from the correct spiral form may be given to both wheels, (d) one gear may be made a true spiral, and the variation all thrown Fig. 608. In many cases the worm is made a true spiral and the conse- quent wear disregarded, but in more careful work the method (b) is adopted and the worm wheel cut with a hob, which makes the proper modification in the shape of the teeth. The friction between the worm and teeth of the worm wheel is very great, as the thread slides entirely across the teeth. We have for the coefficient of fraction y, for the ratio between the actual force /" and a force P acting at the same lever arm on the screw, but free from frictional resistance, approximately : ~P'' I +/. 2Tr J? ft ZTT R Fory = o.!6 we have practically P' , R It follows that to obtain the minimum of frictional loss. (205) R must be made as small as practicable. Morin gives the rule R = ;^ t, which makes P' P^ ~P 4; Red- tenbacher makes R = 1.6 /, whence — - = 2.6. If we make P' 7? = i', we get — ■■ 2, and this is as low as R_ ~T can well be made. In this case it will be seen that a higher efficiency than 50 per cent, cannot be obtained, and it is also ajaparent that the worm must be the driver, since the resistance of friction would just balance the reverse driving action. The ordinary tooth friction and the journal friction must of course be added. Fig. 610. Fig. 611. Fig. 612. The tooth outlines for both worm and wheel are the same as for a rack and gear wheel, taken on a longitudinal section through the axis of the worm. The evolute tooth is especially applicable, and Z-^ must not be less than 2S (2 209). The surface of contact is theoretically only a mathematical point, but in practice there is a small flattened surface of contact, and if a larger surface is desired the wheel must be cut with a hob of the same form as the worm which is to work with it. Wheels which have a contact bearing of a point only, may be called precision-gears, as distinguished from power-transmitting gears. The difference, however, cannot be sharply maintained, for as already shown, worm gearing is used for the transmission of both large and small forces. The possible variations of the pitch angle permit a great va- riety of spiral gear combinations, as the following examples show : into the other gear, or (t) the wear which is at first caused by running the approximate forms together may be disregarded until the parts have worn themselves iuto smooth action. From these reasons a widely varying practice in the construction of spiral gears will be found. One of the most important applica tions is that of the worm and worm wheel, Fig. 60S. In this case a = 90° and .2' = i , the teeth of the wheel R^ being inclined at an angle 7, with the edge of the wheel, whence tan y = - In the arrangement shown in Fig. 609, we have = 0.15916 R o ^ 90 — y and the teeth on R^ are made parallel to the axis. The pitch of the screw is here made = — ! — cos 7 ■wheel. The velocity ratio of transmission, according to the fundamental formula (186) is it^:n ^ Z : Z^, or this case it equals — -.* Example I. Given — = i, the perpendicular distance between axes a = R -\- Ri, and the angle between axes a = 40^. If we make y = 60°, we have from (3 220)71=1 So — 40 — 60 = 80° (see Fig. 610), and from (201) -5- — ^ - sin 80° 0,5 X o. ^' sin 69° 0.S660 determined. Ifwemake«== Sill y n = 0.56S6, from which J? and .^i may be readily l" we have -ft 1.5686 2.55 and R = i»45"- For Z= 20, Z\ = 40, the normal pitch r = ^ sin 7 - 2 X T X 1-45 X 0.866 ■ = 0.272 X 1-45 = 0-394 ■ for a pitch t^ of the xhe circumferential pitch t = -^^ 0.394 ^ sin 7 ■ °.454", h = °.394 tR sin-) : 0.400'. o 9S4S c (cot 60° + cot 80°) = c (0.5774 The sliding velocity c\ according to (202) -1-0.1763) =0.7537. ... Example 2. In order to make c' a minimum, we may make 7 = 71 iSo — a = 70'- , see Fig. 611. <..t(,(,. 2 X IT X 1-333 X 0.9397 * In the illustration Z-^ = 30, which in (203/ for a true spiral would requi .ffl — gocff, and y = 88.1°. We then have -=- = J, i?i .'n o. 394 = 0.394", /= i*! = = 0.419, ana 20 " 0.9397 c' = 2 cot 70° X <; = 0.728 c. It will be seen that the value of c' in Example 1 approached very closely to the miuimum. I40 THE CONSTRUCTOR. Exmttpls 3. If so desired we may make 7 ^ 90°, when one wheel will be- come an ordinary spur gear. Fig 612, and we have 71 = 180 — 40 — 90 = 50°. -— - = 0.5 X 0.7660 — 0.383, iVi = 2. So", R = I. II'', T = 0.348 ' , t ^ r, /i = 0.454", C" = 0.8391 c. If instead of a, the normal pitch t is given, as is generally the case with hobbed -worm wheels, we choose ;- and j-j and then have Ji sin 7 = '—, whence : 2 IT 2 5r sin ) R,- Z,: (206) 2 IT sm y^ Both R and r may be given, when }■ must be determined, and we have ; sm }•=■ — Fig. 613. (207) Fig. 6t5. The following examples illustrate a variety of cases : Example 4. 180 — 90 45" : 90°, Z ^ Zy The sliding to be a minimum, hence v — Vi The two wheels are similar, both being left hand or as in Fig. 613, both right hand. The sliding velocity is c' = 2 col 45^ X c = 2 c. Exaniple s- In the arrangement shown in Fig. 614 there is added to the right angled pair A B. a third wheel C, also right angled, when the wheels ^ and C will revolve in opposite directions. The middle gear B reverses the motion, as in the case of bevel gears. Example 6. When a = o, the axes are parallel and a pair of spur gears with spiral teeth is obtained, this form being called Hooke's or White's gearing, Fig. 615. y and y^ together include 180°, and one gear is left, and the other right hand. In this case the teeth are formed in true spirals. In this case the sliding velocity c' — o. For the wear on this form of gear see 1 222. When (x = o andy=s o the wheels become spur gears. Fig. 617. Py-i- • • Fig. 61S. Fig. 619. If the other limit of spiral gears is reached some noteworthy forms are obtained. Example 7. a = 90°, 7 = 10°. 71 = 80°, i?i = w. This gives a rack and screw, Fig. 616. If 7j ^ go° and the teeth normal, 7 = 10*^ and a = So°, and the teeth of the rack correspond to the section of a nut. In the Sellers planing ma- chine the rack teeth are placed at sucli an angle that the lateral pressure just balances the opposing tooth friction. Example 8. j? = 7?, = co. This gives two racks, sliding in each other, Fig. 618. We have, as before, z'l : v = sin 7 : sin yj. If a = 90°, as in Fig. 619, and 7 = 71=45°, we have v^=vi. This construction is used in some forms of boring machinery for cannon, and in screw cutting machines. Example 9. a = 90°, yi= 90°, also y = o, both radii of indefinite magnitude. Fig. 620. This is the so-called revolving rack, used on governors and similar apparatus in which endlong motion is to be transmitted from a revolving piece. The velocity ratio of .-1 to ^ = o. Example 10. The worm, or endless screw, as already stated, is a form of spiral gear wheel. These are two special forms of worm gear which although seldom used, are of interest. 1 here are the forms of internal gearing shown in Figs. 621 and 622. In the former the worm wheel is the internal gear, while the latter shows an internal worm, with external or spur worm-wheel. Fig. 620. Fig. 621. Fig. 622. I 222. Spiral Ge.4r Teeth and their Friction. Spiral gears are cut in a similar manner to screws, the tool being carried in the slide rest of an engine lathe, and set at the proper angle. The pitch of the screw thread is : s ^=2- R tan 7, and the travel of the rest is effected \>y proper change gears, according to the selected values of 7 and )'j.* The tooth outline to be used is determined according to the radius of curvature of the supplementary spiral, that is, to that at right angles to the spiral to be cut. The radii of curvature r and r^ to be used are : R R\ I o\ r=—.^ — ,7'. = ^-J — (208) sin - 7 sin - 7i These give the radii for the construction circles to be used with the pitch r ; the shape of the tool with which the teeth are cut is then determined. Exajnple I. For the wheels of the first example in the preceding section , we have : — '-15 _ » , ^ 2-55 ''~sin=6oO -'^ ' ' sin = So°' If it is preferred to determine ?', graphically from formula (20S) the method given in \ 29 maybe employed. The frictional resistance of spiral gearing is often a matter of much importance. If the frictional resistance is assumed to be zero, we have for the relation of the force /"applied to the driv- ing wheel, to the force Q delivered by the driven wheel : 2.58". P sm 7 Q sm (209) The ordinary tooth friction, which is the same as that of the construction gears (see J 213) to which must be added the fric- tion due to the sliding of the teeth, whenever a. is greater than zero. The value of the latter friction is governed by the sliding velocity c' . For the calculation of the loss of useful effect we may use the formula : P' sin -/i sin (/ -I- ^) , P"'~sinT' ^hItT— <) in which 1^ is the angle of friction for the coeiBcienty] whence tan =y. Fory"= 0.16 we have (6 = 9°. Example 2. For the wheels in the preceding example we have P' _ sin 80° sin 69° _ 0.9S48 X 0.9336 _ P sin 60° sin 71*-" 0.S660 X 0.9455 To this must be added the ordinary friction of the equivalent spur gears. Another source of loss is that due to the lateral forces K and K-^, acting in the direction of the axes. For these we have -^, = cot(7+rt, ^' = cot()-,-iJ). . . . (211) Example z- For the preceding- .erears we have A'= P' cot 6g° =0.3839 /^^ Kx = Q cot 71'^ —0.3443 Q from which values, in connection with the known dimensions of thejounials the corresponding resistances can be determined. When a =£?, that is, for parallel axes, the sliding action of the teeth is zero, and the value of P^ in v-'^) is the same as P\ hence spiral gears for paraiiel axes work without the tooth fric- tion due to lateral sliding, the ordinary tooth friction alone re- maining, as well as the forces K and A'j. * Brocot's Tables will be found of service in arranging change gears, (Calcul des Rouages par Approximation. Paris, 1862). 777.5' CONSTRUCTOR. 141 The tooth friction may be reduced to a very small amount by reducing the beariug surface of the teeth of one gear to a point of contact, or practically to a knife edge. Such gears (devised by Hooke) are only of use for purposes of precision, but in some cases are found serviceable.* Fig. 623. Fig. 624. Instead of the edge bearing, a rounded surface may be used, with its highest part corresponding to the lineal bearing as al- ready shown by Hooke and by Willis. The tooth outlines for both gears are determined as usual, and then one or both pro- files are redrawn within the original curves, Fig. 623, and the modified outline.'^ used to form the tooth spiral ; teeth so con- structed running nearly free from friction. In such cases the length of flanky, and face k may be reduced as shown. Such forms are more properly to be considered as screw thread pro- files than as gear teeth. Willis has shown that in both gears the flanks may be made radial and the crown of the teeth semi- circular, Fig. 624. Since such teeth are weakest at the base, it is preferable to use a modified form of the evolute tooth. Fig. 625. This may be approximated to by using a circular arc of smaller radius than B S =: R cos a, the centre B' being taken on the normal N N, through the point of contact. B..- H..,,--v Fig. 625. A similar form to the preceding gears is the so-called step- gearing, Fig. 626, frequently used in planing machines (by Shanks, Collier and others). The tooth profiles may be modi- fied as above, to reduce friction, but the gradation .j should be as great or greater than the pitch t. Fewer than four sections should not be used. An objection to the use of spiral gears is the axial pressure K, this, however, can be eliminated by the use of double gears of opposite inclination. Such gears have been known for a long time (White, 1808) and for moderate service, have been frequently used, as in spinning machinery, tower clocks, etc., and more recently they have been applied to heavy work, nota- bly for rolling mill gearing, both in Germany and America. The pinions used in rolling mill work are made with 9 to 16 teeth, with pitch diameters from 4" to 24" and over. Evolute teeth are used, with a base angle from 62 to 69°. The face length of the teeth is made about 0.22 t. If the evolute curve is accurately made, the tooth contact is practically the same as with ordinary spur gears, and the surfa- ces of contact can readily be discerned, extending diagonally across the teeth. When such a surface of wear is visible, of course the teeth are not free from friction. Fig. 627 shows a cast steel pinion of ten teeth, for rolling mill service. This gear is cast in one piece with its shaft and coupling ends, although in many cases the shaft is made separately. The space .? between teeth at the middle of the gear, is called in the Westphalian shops the "spring'' ,of the teeth. If it is desired to approximate to the frictionless action of the teeth, this " spring " must be slightly greater than the pitch. Fig. 627. For very large transmissions the gears may be made in two parts. Fig. 628 shows a pair of such gears for a reversing roll- ing mill by the Hagener Steel Works. The pitch diameter is 43-3", the pitch S|<", the face of each gear 20", and the total weight 24,200 pounds. The teeth are made with double reverse angles on each gear, so that the conditions are the same when running in either direction, and the whole is a masterpiece of machine work in steel. Fig. 628. 223 Spirai< Beveiv Gears. Spirally formed teeth are sometimes used on bevel gears, and in this case the distance a, between the axes becomes zero, while the angle a remains to be given. For the curvature of the teeth it is best to use a conical spiral of constant pitch, the projection of which on the base of the cone is an Archimedean spiral. Frequent applications of such wheels are to be found in spinning machinery, and they are operated successfully at quite high ve- locities."^ Fig. 629. The same varieties may be made in bevel, as in spur gears, and in Fig. 629 is shown a reverse spiral bevel gear of cast iron, as made by Jackson & Co., at Manchester. Similar gears are made of cast steel by Asthover 6c Co., at Annen in Westphalia. Stepped teeth are also used in bevel gears, and in Fig. 630 is shown such a wheel by A. Piat fils, of Paris. * These gears have been used in physical apparatus by Breguet for speeds exceeding 2000, or according to Haton, as high as Sooo revolutions per second or 480,000 per minute. * For a machine for the correct construction of the teeth of spiral bevel gears, see Genie ludustriel. Vol. XII. p. 255. 142 THE CONSTRUCTOR. Fig. 630, I 224. Globoid Spiral Gears. If a circle is revolved about an axis A A^ coinciding with one of its diameters, and at the same time a radius C^" is moved about the centre C, with an angular velocitj' proportional to that of the circle itself, the circle will generate a sphere and the point of the radius which is at the surface of the sphere will trace a form of spiral curve. This may be called a spherical spiral,* and adjoining lines of the spiral on the same meridian are equidistant. Fig. 632. If the radius C 5" passes the axis of rotation, the new spiral will intersect the one previously traced, as at A^. Instead of a mere radial line, may be substituted a point which at the same time traces the outline of a tooth space, so that a spherical screw thread is generated with which a spur gear will engage at any point. Fig. 632. If the axes A and B are maintained in their proper positions, the spiral when driven, will operate the gear in the same manner as a worm and worm wheel, | 221. The practical value of this especial form is extended by the fact that the axis of rotation need not coincide with a diameter of the circle. Under these conditions there may occur a num- ber of forms of bodies of revolution bearing an affinity to the sphere, and to which the writer has given the general name of globoids. The corresponding spirals may be called globoid spirals and the resulting gears, globoid spiral gear wheels. Many of these may be made of practical use. (See Fig. 633.) There are numerous forms of globoids according to the posi- tion which the describing circle holds to the principal axis. The axis about which the radius C S turns is called the counter-axis. It stands at right angles to the starting position of the describ- ing circle, and either intersects the principal .axis, or is inclined to it without cutting it. We have then r, for the radius of the describing circle ; a the shortest distance between the axes A and C, c the distance of the centre of the describing circle from the plane of the principal axis, (5 the angle which the principal axis makes with the plane of the describing circle, extending from 0° to 90°. This gives four classes of globoids, as follows : I. a^o, c = o. II. a^ 0, c chosen at will. III. a chosen at will, c = o. IV. a and c chosen at will. A right globoid is one in which 6 = 0, and when (5 is an acute angle we get an inclined globoid. The first class is represented by the globoid Fig. 634, giving a symmetrical conical section ; if J = o we obtain the previously described sphere. The second class gives the inclined globoid. Fig. 635, with unsymmetrical conical sections, with regard to the equator, the spiral being on the zone mantles. If c! = c we obtain a sym- metrical, cylindrical hollow section of a sphere, Fig. 636. The spiral, when a^o, becomes a spherical cycloid. If (5 = 90° the figure becomes a plane cone, or plane ring, and the curve be- comes a plane cycloid. Fig. 633. We have the third class when (5 ^ 0, and a > r, giving a so- called cylindrical ring, or right globoid ring. Fig. 637 a, and when a < r, the apple shaped globoid, Fig. 6373. If 6 is an acute angle, the globoid is flattened. Fig. 638 ; the globoid of Class I is the limiting case. The spiral curves are globoidal cycloids, which become plane figures when ^ -= 90°, and the globoid becomes a plane ring or plane cone. Fig. 635. Fig. 636. The fourth class gives the highest forms. Fig. 639, in which S ^ 0, and we may have a '^ r, a ^ r, or a <^ r. The inclined globoids of this class have forms, the limits of which are found in those of the secoud class. Fig. 635. If (5 = 90° we have again the plane cone or plane ring.'- The practical applications of the globoid spiral gears are va- ried, and are found mainly in right globoids of classes III and * Two right globoid rings may unite to form a pair of machine elements when the thickness of one is made equal to the hole in the other, as in Fig. a. The two parts then bear the relation to each other of journal and bear- ing, and are similar to a balljoiut. Each of the two elements describes by the relative motion of any point a corresponding path en the other member. * More properly a spherical cycloid, see §216; its kinematic axoids are normal cones. These conditions are approximately found in a pair of chain links. Such a pair may also be considered as a contracted form of universal joint, A B C, Fig. b, the same relative motion existing between A and C. The same thing is shown in a fractional form in Fig. c, when some method of holding the parts together, such as bands, etc., must be used. This latter resembles closely the ball and socket joints of the human skeleton. ^ THE CONSTRUCTOR. 143 IV. In the valve geai" of Stephenson's locomotives, Fig. 640, is found a globoid worm of class III, using the middle part of the globoid apple, Fig. 637 b, (a <^ r). In this case the reversing lever B is really a part of an internal gear with a radius A\ = the radius r of the describing circle.* In this case the internal gear has but a single tooth, although more might be used. Fig. 637. It will be seen that the globoid forms can be used as internal gears. This is shown in Fig. 641, which represents a worm formed as a globoid screw. Its form is practically the same as that of the hole in the right globoid ring, Fig. 637 a. The sec- tion shown in the figure is of such length that it includes cue- fourth of the entire circumference of the worm wheel B, al- though it could be extended so as to include almost one-half. The most important point to Tae considered is the formation of the teeth. i?j is again made equal to r. Since the globoid is used in the internal form, the two tooth profiles, qn r and 7?, , fall together. The sliding is in the plane of a normal section through B and A A^ and not endlong, and hence the shape of the teeth is absolute. Fig. 640. (Internal gear tooth, with i? = 7?, ). The teeth can be made of straight profile in the worm wheel as well as in the worm.f The production of the globoid worm in the lathe is not diffi- cult. This form has been frequently used in recent work. The advantages appear to be in the simple form of tooth and in the completeness of the engagement. An interesting modification is that of Hawkins, Fig. 642*. In this case the wheel B is composed of friction rollers of quite large size and the friction is thereby greatly reduced. Instead of there being only four teeth, as would at first appear, there is in reality an ideal immber of teeth, a condition refeired to in Fig. 641. the fundamental discussion in ? 200. If for every revolution of the globoid screw, one tooth of the wheel engages, there must for each space formed between the rollers be 10 teeth to a quarter revolution, so that instead of 4 teeth in B, there are 4 (i -|- 10) = 44 teeth. Fig. 642. The gearing used in Jensen's Winch, Fig. 643, belongs to the globoid class IV, of the form shown in Fig. 639. Usually in this form a = r, although sometimes a < r, as in Fig. 639 c. 7?j is again made = ;', and the internal globoid form used. The ratio is so chosen that a slow motion can be converted into a I * The worm and internal worm-wheel, Fig. 621, is another example of the preceding case. ■j- This form is described by Snieaton as used in a dividing engine by Hind- ley, see also Willis. Principles of Mechanism, ist edition, 1S51J p. 163. fast one, as may also be done ^-ith fhe form shown in Fig. 641 if the pitch of the worm is made sufficiently great The use of rollers instead of teeth makes a very satisfactory construction.t * Hawkins' Worm Gearing. Sci. Am. Supplement, No. T04, p. 1643. f See Uhland's Prakt. Masch. IConstrukteur, also Engineer, Vol. 24, p. 493, 144 THE CONSTRUCTOR. If in the first two classes of globoids the supplementarj- axis is removed an indefinite distance, the globoids become plane snrfaces, and the globoid screws thereby reach the limit. The limiting case of Class III is the ordinary worm and worm wheel, and another form is Long's spiral gearing, which also belongs to Class III ; a is chosen at will, c-=- o, 6^0. The globoid be- comes a plane cone and the globoid screw becomes an Archi- median spiral. If J? becomes indefinitely great we obtain a disk with a spiral groove engaging with a rack, the middle sec- tion having full tooth contact from top to bottom.* When this is brought into Class IV, we obtain the Archimedian spiral in its most general form, i.e., the evolute of a circle. whence : £. C.iLCULATION OF PITCH AND FACE OF GEARING. I 225. Pitch os Ge.a.r Wheels. Tooth Section. bt The dimensions of gear wheels must, for the same pressure on the teeth, be increased to meet shock in proportion to the increase in initial velocitj'. For slow running gears this action can be neglected. We may in this respect, therefore, divide gears into two classes, viz. : Hoisting Gears and Transmission Gears ; and includes under the term hoisting gears all those having a linear velocity at the pitch circle of not more than 100 feet per minute, and under transmission gears all those running at a higher velocity. For a pitch /, face b, length of teeth /, and base thickness of tooth //, we have for a tooth pressure /'corresponding to a stress S, the general formula; -.'4(I)(4J (-) and for the proportions of length and thickness already adopted we have : b t = 16.8 :^ (213) This assumes that the resistance of the teeth is proportional to their cross section, which is also equally true for those which have the same ratio o£ b to / to each other, a condition which is often of much service in practice. ? 226. Pitch and Face oe Hoisting Gears. For a hoisting gear of cast iron let : (PR) = the statical moment of the driving force, .^= the number of teeth, R = its previously determined pitch radius, in inches, t = the pitch, we have for the given dimensions : t=o.2^ >/J_-_J_, -- = 0.073 V— ^— • . . . (214) / = o.045 \-!-— ^, -^ = 0.0145 V -!___!_. . .(215) the face b being made R b = 2t (216) These are intended to give a fibre stress .S" of about 4200 pounds. The actual stress is properly somewhat less, because the thickness of the tooth at the base is usually more than ^ t, as assumed in (213). P R Since the value of — y;— is the same as the pressure P, we can use (215) in cases in which Ponly is given, as for rack teeth. In discussing the preceding formulEe, consideration must be given to the elements which are usually given or selected in practice. Let t' and t be the pitch for two cases respectively, and Zand Z' the number of teeth. Also let S and S' be the strefs at the base of the teeth, and let the constant, 6 f — J \ ~r\ , which in (213) is made equal to i5.8, be called Cor C ; we then have, according to (214) : i= \ 2 71 C{PR) SZ t ^ C S'' Z' ^' '7) and for the radii R and R' R^ R ~ Zt ~ ^ C S' \zl (218) The value of C depends upon the ratio of the teeth, and upon the value of S for the material used. If we assume the latter to be the same for both cases, the number of the teeth alone re- mains to be considered. A reduction in the number of teeth increases the pitch, according to (217) ; and according to (218) reduces the radius. . 1. 16; so that the 7 toothed gear will be about 5^ as large as the ii toothed gear, or a 42 toothed gear for the same case would be about J+ as large as a 66 toothed gear, and with 1.16 times greater width of face. ^.xainple i. Z = II, Z' = 7, hence f=NfT=N' i' = 2i'=I.l6^. -§=Nf^=N/ • • Fig. 644. The constant C, for a given series of gears, should be invari- able, and for ordinary spur gears may be taken equal to 16.S, as in (213). For the so-called ' ' thumb teeth, " (2 2 1 2), the constant may be much smaller, and hence permit an important reduction in dimensions. The value of --- for wheels of more than ten teeth is not less than 0.7, and introducing this value we get C ^ S.4, that is o.s C; hence "thumb shaped " profiles are capable of sustaining twice as great a load as the ordinary form. Example 2. If, for a given moment {^P R) the thumb profile is substituted for the ordinary form, without reducing the number of teeth, the pitch may be reduced in the proportion i' =t -$/~^i = 79 t, or about o S times, with a proportional reduction in diameter and face. If. liowever, the teeth are taken in the above ratio of it : 7, we would have for the pitch. and the radius R' 0,202 = 0.5S R. * See Civil Engineer and Arch. Journal, July, 1852, also Dingler*s Journal, Vol. 125. Weisbach, III, ist Ed., p. 449, »d Ed , III, 2, p. 87. The influence of the stress Sis always important, and it should not be increased above the normal value for the given material, ■which latter is usually cast iron. An increase of one-fourth in the permissible stress would reduce the pitch and diameter only 7 per cent., but on the other hand it must be remembered that too low a value of 6' causes an unnecessary increase in the size and weight, not only of the gears butalso of the bearings, frame work and other parts of the machine. The value of .5' used above, viz. : 4200 pounds, has been show in practice to give sat- isfactory results, and there appears to be no good reason for any great variation from it. When the gears are made of wrought iron, as is sometimes the case, 5 may be made much higher, and may indeed be taken double, say 8400 pounds. This gives a reduction in t' in the proportion of /^^o.5 = 0.79 /. THE CONSTRUCTOR. 145 Example 3. For comparison between a wrought iron gear of 7 teeth of thumb shaped outline, with a cast iifcu gear of 11 teeth of ordinary shape, we have : R' = 0.5 :■: 0.5 f:.)=-^ \^ o. loi = 0.47 R^ ■ > 4 7 ' t \K 0.393 = 0-7 t. amdy = 0.7 b. In Fig. 644 the five cases given in the last three examples are shown on the same scale, side by side. In order to indicate the fact that the moment {P R) is the same in all cases, the shaft diameter has been shown. It will be apparent that there is no definite relation between the diameter of the shaft and the ra- dius of a gear. The invariability of the moment, which has been maintained in the preceding examples, does not exist of the tooth pressure Pupon the driven gear is again transmitted through a second so-called compound gear. If the pinion of a radius R, driving a gear R', compounds by a pinion R.^ on the same shaft into a rack -R'/i f°^ example, with a given pressure P, we have from (2«4) ^ = Const. V-^, '-^f whence This gives R'^R 4J£ ?Jl ^ S' Z ' ^ ^ R, C S' Z' 4 (219) R., a ■ c Z,' t,' s_ zi But R.j. = Z-^i and R^' = Z./ //, and from formula (215) ; t'-t J^~I- C V Hence we get : t l-slf S_ Z/ ^ S'' Z' Z'' By selecting the number of teeth we may make Z,' Z' z.,. and then obtain ; ^ = v/^' c s_ S' and for the radii : R^ R Z ^ c S' (220) (221) Example^. A rack with a tooth pressure /", gearing with an 11 toothed pinion, is driven by a larger gear which again engages with an ir toothed pinion, Fig. 645, the teeth being of the usual sliape, and the material cast iron. This is to be replaced by making all parts of wrought iron, and reducing the number of teeth in the rack pinion to 4, as shown in g 212, all teeth being also altered to the thumb-shaped form. We then ha ve C'= 0.5 C, y = 2 5", and hence : t' = x/lT = J^ '. and R' = R -t^ \/ y^ = -^i R. It will be noticed that in this case the ratio between the larger gear and the pinion on the same shaft is such that in (217) and (218) both are determined for the same moment {PR.) Example 5. If, in order still further to reduce the dimensions, steel is used instead of wrought iron, thits parinitting a stress of 14,000 pounds, we have i' = jl v/ 0.5 X 0-3 = °-387 t, and R' = 0.4. jij R = 0.145 R, or about J R. The proportion of the results of the last two examples is shown in Fig. 645. The force P on the teeth of the rack is the same in all three cases. The statical moment on the main shaft is, however, reduced with the reduction in R', as is comsequently that of the inter- mediate shaft. The advantages of steel as a material for gear wheels have al- ready been referred to in ? 222. Its greater strength enables much lighter wheels to be used for the same service, than with cast iron. For a gear of cast iron and of steel, to act against the same moment, all other things being equal, we have, taking (' j^' ,. — 2 5'== 14,000 and S:=42oo — , and also — '^0.3 = about — in t R 3 favor of the steel. This gives for the ratio of weight [fiY, that is 0.3, the same as the ratio of S to S' , or say three to one. This advantage also exists for transmission gearing, although not to the same extent. If the velocity ratio in a compound train is comparatively great, it is interesting to note that the most advantageous ratio between gears lies between i : 9 and 1 : 10, this giving a miav- mum of shafts and of teeth.* k ^^K. SiV?i i^K -m-i Fig. 645. 'i 227. Table of Cast Iron Hoisting Ge.a.rs. t p (^^) R PR Z t IT R (PR) Z % 127 10 o.is 107 8.67 H 200 20 0.20 190 20.56 ¥r 2S7 35 0.25 297 40.16 n 391 55 0.30 428 69.40 I 511 82 0.35 583 110.20 iX 798 160 0.40 761 164.50 ^'A 1 150 277 0.45 963 186.00 ^H 1564 440 0.50 1020 320.50 2 2044 65 8 0.60 1712 555-20 2A 3200 1284 0.70 2330 S81.70 Example i. A force of loo pounds is exerted on a hand crank of 15 inches- radius ; what should be the pitch and face of a 10 toothed pinion for the fur- ther transmission? Here we have -^r- = ~ = 150, and the nearest value in the table Z 10 in the third coUimn, is 160, which corresponds to a pitch of ij^ inches. The face is = 2 rf = 2j^ inches. Example 2. A rack is to work with a pressure of 2000 pounds on the teeth. This would give a pitch of about 2 inches, or as given in the 4th and 5th. columns, a pitch t : 0.65 tt, which is practically the same, and the width of face = 2 I. If the rack is made of wrought iron, we have t = 0.707 X 2 = 1,414", and the face = 2.S''. ^6 22S. Pitch and Face of Gearing for Transmission. The fibre stress S, which is exerted upon the teeth by the ac- tion of a given force P, should be taken smaller for transmis- sion gears as the circumferential velocity Z' increases, since the *If (i> be the total ratio, and /c the number of pairs of gears, and the ratio Z k between each pair be .ar = — we have qb = jr . The total number of teeth :u the train, _j. = ^ (Z + Z') = A Z' (i + x). Now k = -. , and the product of the number of teeth and the number of pairs gives [Inx]-^ I w . yk = - Diffenentiating and making the differential coefficient equal to zero we- get I n X = • — ^ which equation is satisfied by x = 9. 19- For example fj) = 600, and the number of teeth in smallest pinion = 7. We have the fol- lowing combinations: (a) ^ = 20, 30, gives jj/ = 7 (2 + 20 + 30) = 364, jy k = 728. (d) ^ = 4.5.5.6, gives jf =7 (4 +4+5+5+ 6) = 168, j'k = 672. {c) (/) = 6.10. 10, gives^ = 7 (3 + 6 + 10 + ic?) = 2Qg,_y k = 609. The last solution is the best, for although it requires m-ore teeth than (5>- it has one less pair of gears, and for solution {a) the number of teeth, viz.s 210 is inconveniently great. 146 THE CONSTRUCTOR. dynamic action of shock and vibration also increases. For cast iron we may take : 9,600,000 V + 2164 in which v is the lineal velocity in feet per minute. For steel ^may be taken 3'/^ times, and for wood j'^j times the value thus obtained. For cast iron we obtain, for : S = (222) v^ 100 I 200 I 400 I 600 I Soo I 1000 I 1500 I 2000 I 2500 5=4240 I 4060 I 3744 I 3473 I 3238 I 3034 I 2620 I 2302 I 206S For Steel : The velocity v may be obtained when // and R (the latter in inches) are given, by the following formula : v = = 0.5236 jV « {223) The selection of a proper value for v will be discussed below. It is also found that the breadth of face (5 should increase with the increase of P. Tredgold states that the pressure per inch P of face, that is — should not exceed 400 pounds. This, how- ever, is not to be followed implicitly, since pressures as high as 1400 pounds have been successfully used in practice. It is bet- ter, however, to consider the question of wear from the product P . of -^ into «, which should not exceed a predetermined maxi- . P inuni. It is found that if-;- X « exceeds 67,000 the wear be- b comes excessive. In a pair of wheels where the teeth of both are made of iron, the greatest wear comes upon the teeth of the smaller wheel. In this case we may make — , — = not more than 28,000 b (224) and if possible it should be taken at less than this value. For smaller forces this constant, which we may call the co-efEcient of wear aud designate as A, may readil}' be made as low as 12,000, and even 6,000, without obtaining inconvenient dimen- sions. When the teeth are of wood and iron the wear upon the iron may be neglected, as the wear comes almost entirely upon the wooden teeth For wooden teeth the value of ^ should not exceed 28,000, and is better made about 15,000 to 20,000.* It is impossible to give exact values in such constructions, and it must be left to the judgment of the designer as to how far it may be advisable to depart from the values obtained from exist- ing examples. It must be remembered that the different values of A do not appreciably affect the strength, but rather control the rapidity of wear. When sufficient space is available and a low value can be given to the co-efficient of wear, it is advisable to do so ; if this cannot be done, the co efficient which is selected will give an indication of the proportional amount of wear which may be expected. In cases where a number of wheels gear into one other wheel, it is better to take, instead of the number of revolutions of the common wheel, the number of tooth contacts, that is the pro- duct of the revolutions and number of wheels in the group. If 7? is given, as is often the case with water-wheels, fly-wheels, &c.. Pis also known, and since A can be chosen we have, tak- ing jVto be the horse power transmitted : P^n_ A 63,000 N^ ^ 16.8 /> *^ S'b~ T (225) hence from (213) for ordinary teeth, JA.Z_A_ S Ji and for thumb shaped teeth, _S.4_/^ ^ __8.4_^ S b S 11 If, however, as occurs in many cases, R is not j^reviously de- (226) termined, the choice of the number of teeth Z is unrestricted. In such cases we have for the width of face b : 396,000 N If we give to A the successive values 30,000, 25,000, 20,000, 15,000, 10,000 and 5,000, we get the following numerical rela- tions : Commo7i and TJiitmb Teeth. Common Teeth. Thumb Teeth. Pii _^ iV__ 30,000 "" R i = S^ 14, 1 1 2, 1 3, 520112,467 1 1 1,565] 10,782; 10, 103 18725 [7665! 6886 And /or Wood: 5 = 2544 I 2436 I 2246 I 20S3 I 1943 I 1820 I 1572 I I3S1 I 1240 b: b = _Pn^ 25,000 20,000 _Pn^_ 15,000 Pn 3-iS = 4.2 1\_ r'' N_ ~r'' N R N_ '"z'r ...Sg;. --26.4 j-y, t. 504,000 ~r5^ 252,000 = ~75~ 420,000 210,000 n S ' 1! S •J ■! 6,000 i5S,oooj ^-' . f/ —- I ' n S ' n S 252,000 126,000 , ?/ 5 ' n S ) (227) N A'' b = = 6-3 -pj = 39.6 1=-. ; / = 10,000 R Zt 1 68, 000 11 S ■\i'-- b-- _Pjt.^ 5,000 N N ^13.6- = 79.2^^; 84,00c ;7T~ ^' = 84,000 42,000 For transmission gears the minimum number of teeth should not be fewer than 20, in order that the unavoidable errors of construction shall not cause excessive wear ; for quick-running gears it is desirable to have still more teeth. The gear wheels on high speed turbines seldom have fewer than 40, aud often as many as So teeth. When wood and iron teeth are used, the least wear is produced when the wooden teeth are on the driver, because the action begins at the base of the tooth and passes toward the point, while on the driven gear the action is reversed. If desired a number of teeth Z can be calculated which will give a desired ratio b : L If we combiiie formuhe (225) and (226) we obtain the useful relation : 396,000 Y6.S^A~^ n'- S' N (0 (22S) This shows the important influence of A upon Z, and the ef- fect of the number of teeth upon the wear ; also the important relation of the tooth profile, since the constant 16. 8 (or for thumb teeth S.4) appears in the second power. It is also seen that Z is dependent on the square of ii, and the square of S, other things being'constant. These points indicate the methods of obtaining the least stress. The value of — is sometimes made as great as 5. For wider faces and sometimes for narrower, the rim of the gear is made of two adjoining parts. Example i. — A water wheel of 60 horse power, 26 feet, 3 inches 111 diameter, moving with a velocity at the circnmference of 256 feet per minute, is to be provided with an intental gear wheel, the pitch circle bein^ 16 inches less radius than that of the water wheel, and gearing into a pinion which is to make 40 revolutions per minute. 256 AVe have : « = ^^-^ — , _^ __ = 3- 1 and "1 40 also n/ 3.14 + 26.25 256(157-5— i6_ 230 ft perininute. P- 33000 X 60 3-1 157-5 \ 230 = 860S lbs. This gives a permissible stress S = ^loo lbs. nearly. We tfiII choose for the smaller wheel — ^ = 25,000, which gives —r- = — — = o n-y 2SOOO , P 860S ~ = 625, hence = r— = - — = 13^+ ■ "^^'e then have from (227) / = 40 02.5 62.5 ^ ''' 420,000 ,,, „, ., , „ ^ TT R 2JTI4I.5 -i-^ = 2.56". We then have Z= —— = — -^ = 347. if we make 40 X 4100 t 2 56 34S teeth the wheel ma}' be divided into 12 segments of 29 teeth each. For the driven wheel we have Zi = "I ^=— X348 = 40 27, whence R^ =. 27 X 2.56 2 IT = II". E-xaviph 2. — A turbine water wheel of 100 horse power has a vertical shaft making q6 revolutions per minute, and it is required to drive a horizontal shaft at 144 revolutions, hence a pair of bevel gears are required. We will select wooden and iron teeth, and let the wooden teeth be on the driver. We will assume v to be between 1200 and 1400 feet per minute, which gives J S = 1600, and make A = 25,000, also — p = 3- We then have from (22S) Z = ■506,000 962 X i6oo2 X ICO ^ J,\, TT ■ ■ = 70- we tnen have ifj = ^- .70 = 47 • 16. S2 X 250003 3 ' 144 ' ^" * See case 10, in g 229 seg. also i 420,000 96 X 1600 We then have Zi -- 2.73" sa3' '2}'^", 5 = 3/--= ^li", V = 1536 feet per minute. THE CONSTRUCTOR. H7 F.xample 3. — In a giveu train of gearing, Fig'. 646, in which the correspond- ing wheels of both pairs are of the same size, the force transmitted in each case i? inversely as the number of revolutio'ns. In order to have the Pit co-efficient of wear x" alike in both cases it is only necessary to make all the gears of the same face. An ex- ample of this kind maj' be found iu the back gearing of many lathes. Example 4. -Let it be required to construct a pair of durable gears of wooden and iron teeth under the fol- lowing conditions: iV= 5, ?i = n-^ = 60, and — = 2. We may make v = 500, which gives, from {222), ^ = 2160, and as great durability is required Fig. 646. ■we will take A as low as 10,000. These values iu (228) give 396,000 6o2 X 2160- X 5 16.8- X g,ooo'* which we may call So teeth. We have from (227) ■■ 1. 167" and or 2 /, as intended. Example 5. — Let N = 40, n = of common form, and let -— = take A = 25,000. This gives for Ihe driver gear : — = 2.33 gooo 80 X 1.167 30, n-i = 50 for a pair of iron gears with teeth If we make v = 300, S = 3400 and we driver gear : 396,000 5o2 X ^4oo2 X 40 = 41.5 70> i6.S2 X 25,000= 2.5 say 42 teeth, and Z-^ Z = we have t = 420000 50 X 3400 and ^ = 2.5 z! = 6.175". If we choose the thumb-shaped teeth, and make — - = 3.5 we get : 120 Z ai 396,000 55L X 3400= 3-5 210,000 X 40 say S.4 \&Zx - X 25,000= = 200. f 1.2 _._ .1 = 4.32. 50 X 3400 This gives smaller teeth, but larger radii than when the common form is used. When steel is used for gear wheels, special proportions are obtained. It is not too much to say that the value of the co-efftcient of wear A should be taken twice as great as for cast iron. The stress S, however, may be taken 3^ times that permissible for cast iron. Taking these points into considera- tion in formula (228) we see that A would reduce the number of teeth by J-g, and S would increase it by ( 1 , that is, about n times, so that the net in- crease would be -Q— , if the above values are accepted. It may therefore be laid down as a rule that steel gears should have more teeth for the same ser- vice than cast iron gears. The ratio of face to pitch may be made quite large, and iu the case of double spiral gears (as Fig. 637) the ratio ■ — is some- T times made as great as 7 or 8. If the formula for thumb teeth be used, in- stead of the usual shape, the constant 16.S will give satisfactory results. The value obtained for the pitch is that for the normal pitch t = t sin -y, but the width of face is the actual width, as /', in Fig. 657, 2 t>' in Fig. 62S. Example 6. — Suppose the wheels given in Example 5 to be made with ■double spiral teeth of steel. We take A = 56,000, and — =6, also 5= 12,800. 7" We then get : 396,c 5o2 X 12,800- X 40 3.42 X 56000- ' 6 8.4 X 56,000 = 87 We have t = also b - i2,Soo X 50 = 4.4" 56,000 87 X 0-74 If we take Z\ = 84, we get Z = 140 and d = 4%"- If ^ = ^o" we have Eia6o 0.866 ^^ We may take / = 0.375", which gives t = 0.866 X 0.S75 = 0.757" and ^ _ 4.3 ^ T 0.757 We have then finally J?i -= 11. 6", I? = ig.47" = 5.93, or nearly 6. i 229- Examples and Comments. The following examples taken from actual practice will be of interest : (see Table on following page). No. I. Prom tlie driving gear of the main steam engine of Fleming's Spinning and Weaving Mill in Bombay. The toothed fly-wheel is the driver, and the teeth are shrouded, as shown in Fig. 651. The coefScient of wear for the driven gear seems high, and does not indicate long endurance. No. 2. A toothed fly wheel engaging with a pair of equal spur gears ; 300 horse-power transmitted by each gear, making a P >i total of 600 horse-power. The value for — — must therefore be multiplied by 2 ; see last column of the table. No. 3. This is from the air compressor for the atmospheric P71 railway of St. Germain (now abandoned.) — - — is evidently too high, as would probably have become apparent had the gears continued in operation. P . No. 4. — - IS very high, but the small number of revolutions P u keeps the value of — — within reasonable limits. N0S.5 and 6. These are from the great water wheel at Greenock. The pressure at the rim is great, but the teeth have worn well in practice, as might have been predicted from the moderate Pn values of — : — The value of the latter is almost the same for o No. 6 as for No. 5, hence the wear should be about the same for both gears. No. 7. The teeth iu the smaller gear are thinner than those of the large fly-wheel, hence the two values for 5'. Probably the larger wheel was originally made with wooden teeth. Pn No. 9. Notwithstanding the high pressure the value of ;/ is reasonably small. The stress upon the teeth is quite high, as is also the case with No. 4, and lower stresses are to be recom- mended. No. 10. This is one of the most noteworthy examples of the whole collection, on account of the very slight wear exhibited. The wooden teeth on the large wheel, (the fly-wheel of the steam engiue of the Kelvindale Paper Mill at Glasgow) ran for 26}i years, for 20 hours per daj", with a wear upon the teeth, measured at the pitch circle, of only about ji inch. For the first half of this time the engine indicated 84 horse-power, at 38 revolutions. The teeth were lubricated twice a week with talc and graphite. The long endurance is doubtless partiallj' due to the great care which the teeth received , they having been cut upOn the wheel in place, but also to the moderate.co-efficientof wear. No. II. The teeth were found too small in practice, as is indi- cated by the stress of 3000 pounds ; from formula (222) we ob- tain S= 1734 pounds. No. 12. Two gears with wooden teeth engage with a single pinion on the screw propeller shaft. The teeth are in two sets of 4:14" width of face each. No. 13. Very high pressure, Which must appear in the wear upon the teeth ; apparently it should be difficult to keep them P in good condition, owing to the high value of ---. No. 15. These teeth appear weak, as has been shown by re- peated breakages. The wear must be rapid, as indicated by the Pn high value of — , — . ° d No. 17. These gears, (designed by Fairbairn) w^ere intended ultimately to transmit double the power at first given, iu which case the stress would reach over 4000 pouuds, which is admissible P/i but the value of — : — would then become rather too high to in- dicate very great endurance. teeth ; it is almost too great also for the iron teeth, and it must be remembered that with wooden and iron teeth, the wear comes almost entirelj' upon the wooden teeth. No. 22. These gears are from an establishment which has used hyperboloidal gears with much success for power transmission. The angle of the axes is 90°. The use of wooden teeth upon the driver is to be criticised, as tending to increase the liability to wear. No. 20. The value of ■ seems too high for the wooden r. THE DIMENSIONS OF GEAR WHEELS. I 230- The Rim. The ring of metal upon which the teeth of a gear wheel are placed is called the rim. For cast iron spur gears, the thickness of the rim is given by the formula (f = 0.4 / -f 0.125" (229) 148 777^ CONSTRUCTOR. EXAMPLES OF TRANSMISSION GEARING. No. N n R Z t 6 V P S P b Pn b REMARKS. 1 1000 36.67 1 14.8 3S.25 144 46 5-25 24 2300 14,000 1877 5S3 fi'39? 66,970 Iron and Iron. Steam Engine. 3 300 100 I46.S 37 230 58 4.00 14 1900 5,100 1614 364 2x9107 36,400 Iron and Iron. 3 270 60 12 19.6 ~9S^ _i9_ 95 6.25 20.6 616 14,300 1S4S 694 41,650 S,330 Iron and Iron. 4 240 44 no 208 68 3-125 16 766 10,200 3270 639 _8,498 28, 1 10 Iron and Iron. Transmission for No. 8. 5 192 15-14 400 3S"-25 704 62 3-6 15 280 22,240 7252 1483 1,972 22,450 Iron and Iron. Water Wheel. 6 192 50 106 208 ~6r 3-i8 15 S40 7,425 2275 495 7,494 24,750 Iron and Iron. Transmission for No. 5. 7 140 30 55 58^ 32 _L32_ 72 2.8 8.6 900 5,000 4266 48:^5 581 17,440 31,970 Iron and Iron. Steam Engine. 8 140 54-5 66.5 35-75 133 76 3 13 1045 4,350 3700 335 10,040 18,230 Iron and Iron. Steam Engine. 9 120 1-51 13-3 _29I 33 560 80 3-125 15 240 16,230 5688 1082 1,634 14,390 Iron and Iron. Water Wheel. 10 100 45 158,8 24 J76_ 50 3 10 5-9 2000 1,635 924 163 7,357 8,175 Wood and Iv-on. Steam Engine. 11 90 26 80 S5-4 27-75 228 74 2-375 1 163 2,500 3000 424 11,010 33,900 Wood and Iron. Steam Engine- 12 82.5 54 83 55- 1 35-S 114 74 3.1 2X4.7S 11.75 1558 3,440 1848 362 19,540 Wood and Iron. Screw Steamship. 2x30,040 13 50 4.o_ 7"32 50-4 27.5 96 52 3-25 10.6 104 15,500 7536 1463 5,849 10,700 Iron and Iron. Water Wheel. 14 20 _Z:Zi 40 85.4 16.5 24S 48~ 2.2 6-3 328 1,980 2420 314 _2,433 12,570 Iron and Iron. Water Wheel. BEVEL, GEARS. 15 300 50 24-37 45-7 50 93 3-1 13 11S7 8200 3270 3697 630 58,660 31,540 Iron and Iron. Turbine. 16 300 100 29.7 26.7 _55_ 49 2.7 10 1576 6170 3840 617 61,700 68,980 Iron and Iron. Transmission for No. i. iii.S 17 240 44 44 42 75 3-5 iS 96S S050 2133 2000 447 19,670 Iron and Iron. Transmission for No. 3. 18 200 41 80 30.1 98 50 3-8 II. 8 1260 5157 437 17,920 34,960 Wood and Iron. Turbine. 19 130 124 31-3 24.S So 60 2.4 8 1523 2772 2276 2417 346 32,220 42,970 Wood and Iron. Turbine. 20 100 93 144.7 *3-4 15 _7^ 45 2.1 6.3 1 140 2860 2985 3840 454 42,220 65,690 Wood and Iron. Turbine. 21 50 93 218 25.6 10.8 _J5_ 32 2.1 6-3 1236 1313 1564 1 848 20S £9,380 45,430 Wood and Iron. Turbine. HJTPERBOLOIDAIi GEARS. 22 16 72 81.6 21.6 19 68 60 1.996 1-993 5-9 S12 640 2il loS 1250 7,Sio Iron and Wood. "8^851 Transmission. THE CONSTRUCTOR. 149 See Fig. 647. The rim is thickened in the middle or at one edge to — ^, and also stifiened by a rib, and for gears of fine l,5rfi Fig. 647. pitch the section of the rim is curved, which harmonizes well with arms of oval section. According to (229) a pitch of \" would give a rim thickness .;" pitch and 2)4" face, we have bi- = 3.90625, which multiplied by 14.74 gives 57.62 pounds. For bevel gears or for gears with wooden teeth and lighter arms (as given at the end of l 232) the weights will run slightly less than given by the table. CHAPTEK. .XVIII. RATCHET GEARING. i 235. Classification of Ratchet Gearing. Ratchet gearing may be considered as a modification or ex- tension of wheel gearing. The object of ratchets is to check the action of certain portions of a machine or train of mechanism and so modify au otherwise continuous motion into some inter- mittent form. Ratchet gearing may be divided into two main divisions according to the nature of the checking action. When the movement of the checked member is impeded in only one direction we have what may be called a Running Ratchet ; and when the movement is checked in both directions, a Stationary Ratchet. The distinction -will be understood b}' reference to the accom- panying illustrations, in which Fig. 653 shows a ratchet wheel aud pawl a b c, the shape of teeth and pawl permitting motion of the wheel in one direction, and hence forming a Running Fig. 653. Fig. 654. Ratchet Gearing, while in Fig. 654 the rectangular notches and pawl for a Stationary Ratchet Gearing. The lifting of the pawl is called the release, and the falling into gear is called the en- gagement of the ratchet gearing. If the two members b and c are held, a becomes the intermit- tent mover, while if a be held, the parts b and c possess the intermittent action ; as for example, the sustaining pawl and ratchet wheel of a common hoisting winch in the first case, and the reverse lever aud quadrant of a locomotive in the second case. Ratchet gearing is a portion of constructive mechanism which will repay close investigation. For this purpose the following six groups may be considered : 1. Ratchets pure and simple, such as a ratchet wheel and pawl for the mere prevention of rotation. Examples: the ratchets of a windlass, or of the beam of a loom. 2. Releasing Ratchets ; those which act to release members which are under stress, and which by such release are permitted to perform and deternd late work. Examples: the pawls which release the drop of a pile driver, the trigger of a gun, or the trip valve gear of some steam engines. 3. Checking Ratchets ; those which arrest parts which are already in continuous motion. Example; the safety check .ratchets upon elevators, and upon mine hoists. 4. Continuous Ratchets ; those in which a combination of pawls acts to drive a member in a given direction with practi- cally a continuous motion. Examples: a ratchet-driven wind- lass ; sotne forms of counters. 5. Locking Ratchets ; those which act to detain certain mem- bers y- a fixed relation against the action of external forces until released. Examples: some forms of car couplings aud of releasing shaft couplings, also the mechanism of locks. 6. Escapements ; those forms which permit a member under the action of an impelling force to make a regularly intermit- tent motion in one direction. Example: the various forms of clock and watch escapements. By following this classification, the various principal funda- mental forms may ^e briefly examined. I 236. Toothed Running Ratchet Gears. In running ratchets, the direction of motion which is not checked by the pawd is called the forward motion, and the re- verse, the backward motion. The teeth on the ratchet wheel must 'therefore be so shaped that when the pawd is in engage- ment the backward motion ouly must be impeded. It is also important that the form should be so chosen that the first ten- dency toward a backward movement should act to produce an engagement of the pawl wilh the teeth. In determining the fornj of teeth. Fig. 655, we observe that the most effective point upon the circumference of the wheel for the action of the pawl is that at which the joining line 1.2 of the centre of the wheel i, with the point of the pawl 2, is at right angles with the pawl radius 3.2. If we describe a circle upon the diameter 1.3, or the distance between centres of wheel and pawl, the intersections 2 and 2' with the pitch circle of the ratchet wheel will give the tv>-o most advantageous points of application. If the point 2 be selected, the attempted reverse movement of the wheel will subject the pawl to compression, while if 2' be chosen the pawl must be of the hook shape shown, and will be subject to tension. If the teeth of the wheel are to be of straight outline, the flanks should be radial. If a point of THE CONSTRUCTOR. 151 action 2, or 2,, in front or behind 2 or be chosen, the mechanism will be operative, but less advautageousl)' than when constructed as above, for the lever arm of the force-couple act- FlG. 55s. ing upon the wheel will be less, and hence the pressure greater. The angle of the flank, which will cause the direction of the force upon the pawl to pass through the axis 3, is found by erecting a perpendicular from 2^ or 2^ upon 2^ . 3 or 2^ . 3. Fig. 656. It is not necessary to bevel the end of the pawl so that it shall bear in but one point of the tooth, as it is not ditScult to shape the tooth profile so that the force /-"shall jjass through the axis 3, when the pawl engages with the tooth. This is accomplished by making the profile of the flank of the tooth a circular arc struck from 3 as a ceutre, as in Fig. 656 a. Fig. 657. Fig. 658. The same result will be attained by giving this curve to the end of the pawl, and making the point of the tooth the bearing, as at b, or both pawl and tooth may be formed to the curve, as Fig. 659. at c. Since the force which acts upon the pawl has no tendency to cause it to lift out of gear, when constructed as thus described we may call this form of tooth the "dead" ratchet tooth. Other forms of teeth will be considered hereafter. Internally-toothed ratchet wheels may also be made with the pawls adapted to act either in tension or compression, as at 2 and 2', Fig. 657. The axis 3 may be within the wheel, Fig. 65S, in which case the above given conditions for the best position of the point of action cannot be fulfilled. If the radius of the ratchet wheel be made infinitely great we have a ratchet rack, Fig. 659, in which a is a pawl acting in compression, and b a form acting in tension. An important application of the ratchet rack is shown in Fig. 660, which is the upper portion of the lifting frame for a screw- propeller.* Fig. 660. The two ratchet racks a, which support the frame as it is grad- ually lifted, are in the middle plane of the ship, being fast to the walls of the propeller well. In order to insure the engagement of the pawls b b, they are held in geai by the loop springs of rubber. The frame is raised and lowered by a rope tackle, the sheaves of which are shown, the so-called " c/icvse-conpling''' (see I 156), permitting the propeller to be lifted, when its tongue and groove are in the proper vertical position. The pawls are held out of gear by means of lines, during the operation of lowering. The frame and ratchet racks are both made of bronze. The bent lever is another pawl which engages in a notch m a blade of the propeller, and prevents it from revolving during the operation of raising or lowering. There are two wooden struts, the bronze shod ends of which can be seen on each side just above the pawls b, their function being to hold the frame firmly in its lowest position, when the propeller is revolving. Fig. 661. Fig. 662. Ratchet racks are also used extensively in connection with the hoisting machinery in shafts of mines, etc. * See Fig, 323, § 117, -where one of the bearings for the same propeller is shown. S52 THE CONSTRUCTOR. Instead of giving the ratchet wheel an infiuitel}- great radius, the arm 2.3 of the pawl may be made iufinitely long. This simply means that the motion of the pawl is guided in a straight line, in some form of slide. In Fig. 661 such an arrangement is shown for a ratchet wheel, and in Fig. 662 for a ratchet rack, such forms being not uncommon. I 237. The Thrust upon the Pawi<. The condition that the thrust upon the pawl, in a ratchet gear- ing, shall pass through the axis of the pawl, is not always ful- filled, and in some cases it is impracticable to attain such a relation of the parts. The mutual action of the pawl and ratchet -wheel upon each other must therefore alwa3'S be considered. If the flank of the tooth of a spur ratchet wheel (or a tangent "to the flank of the outline is curved) does not form a right angle ■with the plane 2.3 of the pawl, there may exist, under some cir- cumstances, a tendency to force the pawl into the tooth, or in other cases to throw it out of gear. T,/ / .•■•■ .Ti Tj, ••• Ni B-; Fig. 663. In Fig. 663 the various cases are examined. If at the point of contact 2 a normal iViVjto the plane of the tooth flank be drawn, this normal may bear one of three relations to the tri- angle 1.2.3. The "thrust-normal" N jVj may fall without the triangle, or within the triangle, or it may fall upon one of the sides of the triangle. If it falls upon 2 . 3, the thrust is neutral ; if it falls upon 2 . i, the thrust is zero ; that is, there will be no action of the pawl ■upon the wheel, or vice versa,, barring the action of friction. The angle S between the line 2. 3 of the pawl and the tangent at 2, which is equal to the angle between the normal to 2 . 3, and the "thrust-normal," is called the angle of thrust. By considering this in connection with the angle of friction (p vari- ous relations are obtained. On the one part, the force applied will act to alter the posi- tion of the pawl, either to or from the centre of the ratchet •wheel : on the other part, it will also act to move the ratchet •wheel forward and backward. These relations are classified for various conditions in the fol- lowing table, in which a ^orce which acts to force the point 2 from I is called an " outward " action, and the reverse, an " in- ward" action. For the so-called " dead " ratchet tooth a = 90°, case i, hence there is tendency neither to inward or outward movement. The variations above given are, however, more or less used in practice, and the table will be of service in considering the action in such cases. Some examples will be given here, and numerous others may be found in subsequent illustrations. In many cases it is desirable that the pawl should be held in engagement with the tooth by the action of the impelling force, as in Fig. 664, this falling under the fourth or sixth case. This Fig. 664. form of tooth insures the retention of the pawl in place after it has once entered the tooth, and is sometimes used in hoisting machinery when heavy loads are to be sustained ;_ au applica- tion is also found in Pouyer's Coupling, Fig. 453, in which the secure engagement of the pawls is an important point. Another secure form of pawl is shown in Fig. 655. Fig. 665. In this case the wheel is made with pin-teeth. The pawl has a forked end, the inner flank tending to produce an inward movement, the outer flank, outward movement. In this case, as in the preceding, the wheel must be turned through a small angle before the pawl can be released. ANGLE OF THRUST c = 90°. The Thrust Action is : The In^pelling Force : Outward Movement: Inward Movement: i) neutral. is without effect. is without effect. is without effect. ANGLE OF THRUST c 90° ^ 3. 1 73. etc-, of the pitch, that is, through ]< the pitch and any multiples of the same. This is sometimes used in saw mill feed motion, where a fine feed is required with a coarse pitch ratchet. Fig. 6S0. A double ratchet is used in Weston's Ratchet Brace, Fig. 6S0. The pawls b^ and b., are placed one above and one below the arm c, and act on the two parts of the double ratchet wheel (7i, (7,. Another ratchet drill, also by Weston, with four pawls is shown in Fig. 5Si. This has an internal ratchet wheel with THE CONSTRUCTOR. 155 five teeth. Double ratchets are also found in Uhlhorn's coup- ling, Fig. 454, and Pouyer's coupling, Fig. 453. If it is desired, the pitch may be halved, or divided into any two chosen portions, in which case the pawls ma}' be made in one piece, Figs. 6S2, 6S3. Fig. 682. In each of these there is one pushing and one pulling pawl upon the axis 3, the pitch being halved and the pawls acting alternately. One form shows a spur wheel, the other an in- ternal wheel. The form of the double pawl has caused this to be called an " anchor" ratchet. If the wheel is a so-called " face " gear, that is, with the teeth projecting from the face of a disc, two similar pawls may be used, both pushing or both pulling, and forming the same anchor, Figs. 6S4., 685. Fig. 6S4. Fig. 6S5. Fig. 686. If the teeth are set alternately in two concentric rings, the two pawls may be merged into one, as in Fig. 686. This latter form appears to be new. I 243. Step Ratchets. A very instructive form of multiple ratchet gearing is obtained by combining more than two pawls into one piece, and arranging two such pawls to work together, and this form is capable of Fig. 687. very extended application. In the ratchet combination a b c. Fig. 687, we have such a combination of two multiple pawls, with "dead" engagement, released by lifting the pawl b. The part a, which is impelled in the direction of the arrow is thus released, but is arrested again by the shoulder 2'. If the flank a 2' is formed in the arc of a circle from the center 3, a farther lifting of b will cause, without resistance, a fresh release of «, again arrested at /5 2", and a similar action again for the flank y 1'" ; the points 2, a, /3, y all lying ou a circle struck from the centre i. Thus a continuous lifting of b will produce three suc- cessive advances of a. The angle of each advance of a may be called the angle of advauce, and the corresponding angle of lift of b the angle of release. In this case the angles of advance are all made equal to each other, as are also the angles of re- lease. When the position in which 2 is arrested by the flank y 2'" is reached, the angle of thrust g becomes so small that further travel cannot well be obtained. If it is required to pro- vide for still further movement it can be done by making addi- tional teeth behind 2, as II, II', III", etc., which will engage successively with i at 2'". The construction of "dead" form of teeth is clearly shown in the diagram. As before, the angles of advance and release are made uniform. The mechanism as constructed will give nine successive engagements. The ratchet surfaces ou b are struck from 2, and the sliding surfaces on a from i ; the flanks on a with a radius 3.2'" ^ 3 }■, the flanks on b with a radius 1.2. It is to be noted that the two parts a and b are interchange- able in their functions, so that when the extreme notch IT'' of a has been reached, a ma}' be reversed in movement and b follow step by step to its former position. Such step-ratchets are seldom used in practice, but many use- ful applications are possible. In Fig. 68S is given a form of step ratchet arranged to give a uniform angle of advance together with uniform drop of the pawl. The pawl a is acted upon by the force indicated by the arrow, and teeth are upon a cam-shaped disc. Fig. 688. An arc with radius 1.2 passes through 3, the angles of release on b are 30°, and the successive angles of drop of (Z are 5°. This form of ratchet is used in the striking mechanism of repeating watches, and is known as a "snail" movement. The arm a in this case is frequently made ot the form shown in dotted lines zX A. The construction of the snail is interesting. In order to fulfill the given conditions the points 2.2'', 2" must lie on an abridged pericj'cloid ; in the given case, where 1.2 = 1.3 it is the form known as a homccentric pericj'cloid.* The points of the re-entering angles lie on a similar curve. The circles rolling together to describe these curves are shown in the figure T a rolling about i, and Tb about 3 ; their radii are inversel)' as the angles of drop and advance. If the parts b and a move con- tinuously, these circles roll on each other, for the actual move- ments which take place, the drops of the pawl occur as the suc- cessive ringed points coincide. * See Reuleaux's Theoretical Kinematics, § 24. 156 THE CONSTRUCTOR. In the preceding step ratchet (Fig. 6S7) the angle of drop and of release were given the ratio 1:2. In this case the points of the teeth were on cycloids, those ou a being on a pericycloid, those on 6 on a hypocycloid. The contact point of the gener- ating circle falls without the figure ou 3.1 prolonged. Since the radii of the circles are as i : 2 with internal contact the hypo- cycloid becomes an ellipse. A portion of the curve is given in the figure ; 3 X , and 3 Y are the semi diameters. The sim- plest form for the line of the teeth will be obtained by making 1.2 = 1.3, since for this case the ellipse for one diameter of the base circle on b becomes the straight line 3 X. Fig. 689. If it is desired to combine in the same piece two step pawls, Fig. 689, of which one set shall be in tension and the other in compression, an anchor ratchet may be used. In this case a back and forth motion of the anchor permits an intermittent forward motion of the wheel. The anchor has ten steps and the wheel four teeth. This may be considered the general case of which Figs. 6S2 to 686 were special examples. Numerous interesting problems may be solved by such de- vices, such as the conversion of continuous rotation of one piece into intermittent rotation of the second. Applications are found in clock and watch-making. The various modifications which may be made in the relative ' positions of the axes 2.1 and 2.3 permit a very great variety of Step ratchets to be made. ? 244. Stationary R.atchets. Fig. 690. Fig. 691. A stationary ratchet may be considered as a combination of a pair of running ratchets with the teeth facing in opposite direc- tions. The scheme of such a combination is shown in Fig. 690. From the four possible positions of the parts 2.2', II and 11' we may make the following double combinations : 2 with II, 2 with ir, 2' with II', 2' with II. The first two combinations are practically identical with the stationary ratchet. Fig. 691. The flanks of the two wheels give a notch for the space, while the teeth assume a dove-tail shape, and this form of stationar}' ratchet may be called a notched ratchet. The wheel will be firmly held by the so-called " dead " tooth, or when (90° — ")<;«, J 237. Many forms of this kind are used in practice. Fig. 692. Fig. 693 Figs. 692 and 693 show two modifications of the notched ratchet. The distinction between tension and compression pawls disappears, since the pawl is the same for either action. If the distance between the axes I and 3 is made infinitely great, the pawl becomes a sliding bolt. Such a form is shown in Fig. 694, which is for non-inter- secting axes. The wheel is a crown wheel, and the pawl may have more than one notch.* Another form of notched ratchet with axes i and 3 infi- nitely separated is shown in Fig. 694. Fig. 695, and is in- ■ tended to hold a shaft from longitudinal motion, being used Fig. 695. in connection with the disengaging gear of hoisting machinerjf, lathes and other similar machines. In this case the radius a is infinitely great ; the wheel a be- comes a shaft. The combination 2 with II' and 2' with II of Fig. 690, if we make 3.2^ III . II, gives a stationary ratchet of the form shown in Fig. 696. Fig. 697. The pawl becomes a segment of a cylinder and works always in compression, or in the modification given in Fig. 697, always in tension. This form may be called a cylinder ratchet. The form of Fig. 696 has many applications, as, for example, the Thomas' Calculating Machine and similar work. * This form of ratchet will be recognized as similar to the common jaw coupling. The shaft A carries the crown wheel a, the bolt corresponds to the other half of the coupling d. The shaft H carries the part d. the latter sliding upon a feather. THE CONSTRUCTOR. 157 The cylinder b may be entirely cut through as in Fig. 69S, so that the segment shall fall entirel3' within the surrounding circle. When it is placed op- posite the teeth the wheel may be revolved in either direction as far as desired. If this move- ment is to be limited, as, for example, to a given pitch, it can be accomplished by cut- ting a corresponding space in the cjdinder, such as is shown in Fig. 699 a. It is not necessary that the spaces in the wheel a should conform to the circular profile of the cylinder b (see \ 237) ; the thrust is at two points on the right and left of i . 3, and it may be formed as at b, or pin teeth used as at c. This last figure shows the modifi- cation made in the notch of Fig. 698 to reduce the back- lash of the wheel a. In Fig. 699 a the pitch circle of the pin gear a passes through the axis 3, and the gap in the cylinder is increased proportionally. When the wheel is impelled in the direction of the arrow, the pin 2 slips into the space in the Fig. 698. Fig. 699. cylinder as soon as the opening is turned towards it far enough, but cannot pass out until the cjdinder has turned back the same distance in the opposite direction, thus forming an intermittent pitch movement. This idea is more fully carried out in Fig. 699 e. In this case the inner profile of the space is concentric with the outside of the cylinder, as was also the case with the form shown in Fig. 697. In this case the tension and compression pawls are practi- cally combined in one. When the opening moves into the proper position, the pin 2 moves to the point 2', and completes the remainder of the pitch movement when the C3'liuder moves to the left again. This form may be made free from backlash by making the outside of the cylinder fill the space between two teeth, as in Fig. 700. If it is required that the intermittent movement should divide the pitch into two equal parts, the arc of the pitch circle of <7, which is the measure of the thickness of the teeth, must be equal to the arc cut off by the space in the cylinder. If backlash is permissible, the thickness of tooth may be reduced.* Fig. 700. If we compare the various forms of cylinder ratchets mth the notched ratchets, as, for example, in Fig. 692, it will be seen how the one may be derived from the other. If the pawl of Fig. 692 is given a row of teeth similar to the tooth 2, placed in a circle about a centre 3, and a space cut in a of the circular profile indicated, we obtain the same general and important form as is shown in Fig. 698. In a .similar manner tie notched ratchet can be derived from the cylinder ratchet, and also inverted by transposing the parts, * If the preceding forms are compared with Fig 6S2, a similarity will be noticed. The " dead" ratchet with pawls of circular profile, of Fig. 6S2, are here, in Fig. 699, replaced by a gap of small angle ; the compression pawl is at 2, the tension pawl at 2', the arc 2 — 2' is made very small, and the re>a- tive diameters very dififerent. and all the modified forms obtained. The interchangeability of the two parts gives the midway form shown in Fig. 701, in which both pieces are the same, each being wheel and pawl for the other.* Fig. 701. For the varied positions which may be given to the axes, a wide variety of cylinder ratchets can be made, many of these possessing useful applications. If the axes are at right angles, the cylinder may become a disc, as iu Fig. 702 ; this form being used in Thomas' Calculating Machine, in which case the wheel a is made with but a single tooth. Fig. 702. Fig. 703. Fig. 704. The form shown in Fig. 703 is derived from the globoid gear- ing of Class III, \ 224, the ratchet being a cylindrical notched ring. Fig. 704 shows how a pitch ratchet can be made on this principle. An examination of the preceding forms of stationary ratchets, in which the pawl consists of a revolving member with a gap cut in it, will show one common property in all of them. This is the fact that an intermittent motion produced by successive release and engagement may be made either by a continuous rotation of the cylinder or by an oscillating movement. If, therefore, we have a continuously revolving shaft to deal with, or a vibrating member, the desired release or intermittent ac- tion of the part to be acted upon may in either case be ob- tained. Both forms are found successfully applied in actual practice. ?245. R.^TCHETS OF Precision. If we imagine the running ratchet of Fig. 682 so modified that upon the release of the pawl 2 that at 2' shall enter at a point nearer the tooth than the middle of the pitch, as there shown, the principle will not be changed. If this modification is made to such an extent that the angle & in both cases be- comes zero, ;'. to revolve in the direction \ of the arrow, that the pawl \ link c is crowded against the axis 4. The radial com- ponent O, in the direction 4 . 3, exerts a pressure upon the brake block b. We also have the tangential component S, which we may consider as composed of two forces S-^ and S,, act- ing in the same direction, which hold the friction at I and 2 in equilibrium. At 3 we have two opposite forces S^ and S^ which are capable of resisting the friction at 3 and 4 res- pectivel}'. The moment 31, of the four friction forces is : y!/= [S-^ + S^ — Sj — 5,) (a-\- b). If we give the angles the symbols shown in the illustration, and make i .2 = (r, 2.3^15, 3.4 = c, 4. i =rrf, and call the radii of the several journals «j, b^, and c^, we have : Fig. 709. 5,: ^Qfa, a + b' Qf , 5,= Q_f_a (7 4- 4' S.= -^f-^^ and S, 5, = --!-(«'« + rfy) dy But we also have (a -j- b) sin a = c sin y. From this we get c cos a cos (7 c cos a (a + b) cos a [a -\- b) cos a This gives for 31 : A_rf s., = - M=Qf ^) {a + b) a -\r a-i i b-^d a-\- b cos- a c{a -\- b) cos a The force P which acts at 2, to revolve the wheel in the direc- tion of the arrow, may be considered as a couple. We then have for 3/^ Pa : Pa ^ , r'? + "1 a + b = Qf [ a + b DS*' C \C __'V'_ L f: [a + b)coza c m * This term only partially expresses the general scope of the German word *' Bremswerke," \ox which there is no exact equivalent in, English. — Trans. THE CONSTRUCTOR. IS9 Pa But (9 is a function of/', aud iu fact we have — ,— 7 ^(J tau a.* ~ a -\- This gives : sin a cos n — y sin a \.\ b^d b) cos a + ^ ?)] c [a ■ and since the angles cr and a are small, and become smaller under the action of the pressure, a sufficiently close approxima- tion will be obtained by putting : s/K^-G ^)] + b \^c(a + b) The following conditions must be noted. If an independent force outside of Q exerts a normal pressure IV upon the circum- ference of the wheel, the friction iV/will diminish the force acting to turn the wheel backward. If this is to enter into the resistance which is produced by Q, the magnitude of a as given by equation (233) must be modified. If jV becomes sufficiently great, Q maj' become zero ; in such a case we obtain a stationary instead of a running ratchet. The pressure 7? on the pawl may become very great. We have J? = — ^ — which may be made approximately : J? = P_a {a -+- 6) sin o- (234) Mjcantple. — I,et a -- /"at all four points = T4.2", a-i = 1.6", d = 2", ^i = 0.6", c = ir.8", ^1 = 6", and o.io,-f we have d =^ a -i- ^ -i- c = 2S" approxiinatel}', and 15 s 0.6 -I- 28 0-6 ^ sin 5 ^ o.io ( — — \x6. - + ^8 0-6 \ whence sin a- < 0.0834, which gives a- = 4° 47' make a- = 5%°, or sin o- = 0.07S7, and then get R To be on the safe side xve will - = II. 17/*. 16.2 -j- D.0787 The exact length of d will be very slightly' less than a + i + c. As will be seen the ratio comes out unfavorably. The method of remed}'ing this will be discussed hereafter. The pawl c may also extend within the circle of the wheel, as in Fig. 710, in which a is an obtuse angle. The axis of the pawl Fig. 710. Fig. 711. may be either at 4 or 4', on 3 . 4 prolonged ; the pawl is in this case a tension pawl. If a is made an internal wheel, we have the arrangement shown in Fig. 711, the pawl being under com- pression. Especially noteworthy are those cases in which one or more of the axes are infinitely distant. In Fig. 7 [ 2 is shown the case Fig. 712. Fig. 713. in which the length of pawl and also d and ^, are of infinite length. We have for the angle of thrust, from (233) P ^- The moment of the frictions produced at 2 and i b}' the force Pis {a + ai) — Pill = P^- t If various coefficients of friction are to be used we have for 5i, S., Ss and S4, corresponding values yi, /n, J3 and 74. c- Ho-] (a + b) [a + , When rt, is very small, release is difficult, and the arrange- ment does not appear to be verj- practical. If the arm a is made infinitely long, so must also a^, aud we get the case of Fig. 713. The wheel becomes a sliding bar. The relations o- coefficient/ we have the value If the wedge angle 6 = 60° this gives 2/; for 9 =: 30°, nearly 4/. By combining this prin- ciple with the preceding forms, some very useful devices may be made. I ._, It is desirable to arrange the application of ! ' ; the force R so as to exert as small a distorting '; i i action upon the parts as possible. This may i 0j' sometimes be done by arranging two or more \\ ; friction pawls of similar kind to act upon one i ' wheel. Some examples of such devices will ■!■ be found in the following section. It must ';■; not be forgotten that the conditions for 6 are s not changed by the repetition of parts, since 1 the numerical value of /'does not enter into Fig. 718. its determination. There is yet another form of friction ratchet which is capsble of being made very useful. By an examina- tion of formula (223) it will be seen that the influence of the di- mension (2, is almost as great as that of a itself. If we inlrrease «! to nearly the same magnitude as a. Fig. 719, we may approach Fig. 719. Fig. 720. closely the minimum value of a. This carries with it the dis- advantage that the frictioual resistance to the backward and forward movement at I, is greatly increased, but this effect may be avoided by making a special bearing for the friction block ^ axis and rearranging the parts somewhat as shown in Fig. 720. The attempt of a to move backward causes the pieces b and d to press upon the rim of a from without and with- in and grasp it firml5'. The angle c may now be made twice as great as in the previous forms with- out danger, all other things remaining the same. A practical form of this device is shown in Fig. 72 1 , as applied to saw mill feed motion. Here the screw motion F G \s intended to permit of a suitable degree of play for the lever <:.* If we make Sj > n, we have the form shown in Fig. 722, which seems qviite practical, and when applied to a friction rack we obtain the form in Fig. 724. We shall return to the consideration of these double friction ratchets hereafter. It must be remembered that these forms of friction ratchets are also applicable to other positions of axes and scune resulting devices are in practical use. Fig. 721. Fig. 722. Fig. 723. ? 249. Running Friction Ratchets. If the force to be transmitted is not very great, the intermed- iate friction block may be dispensed with and the curved con- tact surface be made directly upon the pawl. This reduces the mechanism to three parts : the wheel «, pawl b, and arm or con- necting bar c. Fig. 724. Fig. 724. This form may be called a clamp ratchet, or since the pawl resembles the thumb-shaped teeth already described, the term "thumb-ratchet " may be used. The determination of the angle B may readily be determined by what has preceded, and the fol- lowing relations established : (235) A suitable profile for the thumb pawl may be obtained as in Fig- 717) tiy using the evolute upon a base circle of radius c sin a, f bout 3 as a centre. This may be approximated by a circular arc struck from M, in which 3 M and i M are at right angles to each other. If a and c are made infinitely great we have a form similar to Fig- 7I3> the straight profile 2 being an evolute of infinitely long radius, and the profile 3 a portion of the circumference of an infinitely great cylinder. If the wheel be made a wedge friction wheel we have the form shown in Fig. 725. The wheel may be made with several grooves, by which means the pressurs on each surface can be reduced (see \ 196). Fig. 725. Fig. 726. A variety of modifications can be made in the arrangement of the pawls. A clamp ratchet in which a repetition of pawls is used to distribute the pressure, is Dobo's ratchet, Fig. 726, which is very effectively used by A. Clair in his indicator. f If we adapt the idea of Fig. 717 to a revolving journal using the "thumb " pawl at 3, we obtain a very useful modification of the clamp ratchet. The curve which is applied between 2 and 3 may be variously arranged. A very simple form is obtained ♦See Goodeve. Elements of MecUauism, London, 1S60, p. 49. fSee Morin, Notions geometriques sur les mouvements, Paris, 1861, p. 200. THE CONSTRUCTOR. i6i by making the curves at 2 and 3 portions of the same circle, and the corresponding curve at 3 so found as to produce the required clamping action. The clamping piece b becomes a cylin- der, Fig. 727. If we make the angle O 2 3 = rf, prolong the radius 3 O to N, then -will 3 O yVbe the normal to the curve at the point of contact with b at 3, since the angle 3 . 3 (9 = S. The curve for c is an arc of a circle struck from a centre DI, on 3 (9 N, found by making i jl/per- pendicular to 3 O N. This curve is practically correct for a smaller clamping C3'linder as at O' 3', since the angle of thrust is very nearly the same as at (9 i . 2, or in other words the ef- fectiveness of the clamping action is not impaired as the cylin- der is reduced by wear. T The pressures at 2 and 3, Fig. 72S, are T = , R cos cr 7—, whence O - cos- IT P (■ (a + a,)f Fig. 728. A practical application of the preceding form is shown in the checking device for sewing machines. Fig. 729. In this case a ball of rubber is substituted for the cylinder. Another similar device is the ratchet check used on the old LangenGas Engine, Fig- 73°- III this case a number of roller checks are used in order to distribute around the wheel a. The whole forms a sort of continuous ratchet gearing in which the backward and forward movement of c imparts a continuous Fig. 730. Fig. 731. forward movement to the wheel a. AVhen c moves in the direc- tion of the arrow II, a is clamped and driven, while the parts are released when the motion is reversed in the direction I. The action of the centrifugal force tends to keep the checking cylin- ders in contact with the outer ring, and so insure prompt action upon the reversal of motion. The piessure upon these roller checks in the Langen Gas Engine was very great ; wrought iron rollers wore out rapidly and phosphor bronze was substituted, although even these gradually altered their form under the pressure. Another ratchet check used by Langen for the same purpose, is shown in Fig. 731. Here again we have a repetition of the parts, and also a return to the friction block, the rollers occu- pying the place of pawls. Comparing this with Fig. 709, the curved bearing surfaces correspond to the journals 3 and 4, and the action is similar to Fig. 727. The block b is arranged so that full clamping is obtained in a quarter turn. Friction ratch- ets with double clamps are also used as in Fig. 721 and the same principle appears in Fig. 732, which shows Saladin's "friction pawl."* A similar de- vice is shown in Fi.g- 733. as ap- plied to a rod movement, and upon inspection the resemblance to the action of the "thumb" pawl will be seen.f As long ago as 1798 Hornblower applied this idea to a rotary engine as shown, in Fig. 734.; Fig. 733- Fig. 734. 250. The ReiP«,or: If Q is less than the right hand expres- sion, /^ will only be par- tially opposed, there will be motion from a toward d, with slip- ping at 2, or in other words, we have a brake, see \ 248. This construction is frequently applied, although it requires a relativelj' large force at Q' , acting through the lever c c', giving increased pressure on the axle and much wear on the block. Various forms oi lever connection are used to modify the ratio O' : O- By clearing the angle which the axes l and 4 make with each other, variotis con- venient modifications may be made. The general scheme of such constructions is indi- cated in Fig. 73S, in which the toggle connection gives a high ratio of Q' to Q; the block being guided in slides. By making a an internal wheel, a very practical arrangement is obtained as shown in Fos- sey's coupling, Fig. 450. Koechlin's coupling. Fig. 449, is also another form of fric- tion ratchet gearing, the pres- sure in this case being applied by the medium of a right and left hand screw. The same is true of other forms of friction coupling, and the various me- thods of applying the pressure ■ and reducing the wear, given in ? 248, may also be applied in the design of mechanism for the purpose. Fig. 73S. ^252. REI,E--^SING R.4.TCHETS. Following the classification given in J 235, we have first dis- cussed the various forms of ratchets for the general meaning of the term, and the five special classes remain to be considered, the next being the so-called Releasing Ratchets. Such ratchets must be considered primarily with regard to the question of re- lease. When the release is to be effected by hand, various forms of handles or other connections to the pawls are readily devised. In most cases, however, the release is automatically effected, in which event, some mechanical tripping device is required. The resisting force in such gearing is practically the same as the force required for release. It is applied usually bj' weights, springs, steam or air pressure, etc., and is variously intended to cause the released member to act with a predetermined velocity, either slow or rapid, as may be required. Many millions of releas- ing ratchetshave been made for gun locks, and the various forms of releasing valve gears for steam engines, introduced by Corliss, but first invented by Sickles,* are of this class. In designing releasing valve gears, it is important that the valves should be closed quickly yet without sudden shock, and hence some form of buffer is essential. It is in the various devices for applying the force, for releasing, and for cushioning the released force that the many gears differ from each other. The original form of Corliss valve gear, and the modified form of Spencer & Inglis, are but little used on the continent, but these are well known, and hence examples will be given of some of the numerous modified trip valve gears which have been put into practical use. Exa}iiple I. — Valve gear by Cail & Co., Paris, Fig. 739. a is the driven piece, a sector with one tooth, fast to the valve stem : b is the pawl ; c the arm, loose on the hub of a ; b' is the pawl spring; d the releasing cam. The Fig. 739. exerted by a spiral spring acting on the rod/, and the valve is opened by the rod connecting the arm c^ with the engine motion. The cushion is effected by an air dash pot. also acting through the rod /, and the instant of release is determined by the governor. Example p.— Valve Gear by Wannich, of Briiun. In this case there are two vry * F. E- Sickles, of Providence, R. I., took out his first patent for a " trip cut-off" valve gear in 1842. Fig. 740. flat slide valves to be operated by the reciprocating movement of the piece c. It will be seen that this is a form of ratchet rack ^rearing. The valves are THE CONSTRUCTOR. 163 closed by steam pressure acting upon small auxiliary steam cylinders on the rods a, tJie cushion being provided by air buffers as in the preceding exam- ple. There are double pawls (^, i^, with "dead"' tooth profiles, faced with steel ; c is the pawl carrier, moved back and forth by the rod c' ; b' b' are the triggers, and ^rfthe releasing stops, the latter shown in three successive positions ; e is a guide rod. The rod c receives motion from an eccentric ou the engine shaft. Example 3. — Valve Gear by Powel, of Rouen, Fig. 741.* This is a form of rod ratchet gearing with bolt pawl. Here b is the driven piece in which the Fig. 741. bolt h and its spring are carried. The rod a is moved up and down by an eccentric. The piece c is guided at Cn. The trigger d acts sooner or later, as the governor changes the position of the trip £. The force to close the valve is steam pressure acting on the upper part of the rod Ci, which also carries an air buffer. The use of releasing ratchets in valve gear of steam engines is very old, being found in the old Newcomen pumping engines, and in the Cornish engine a similar gear is used to-day, while in recent times trip valve gearing of various designs have come into extended use, and some of the forms are shown in Figs. 670 and 671, not only for closing the valves, but also for open- ing them. These latter valve gears are intended to be operated by direct connection with the piston movement, while those in the preceding examples are operated from revolving crank shafts. Releasing gears which are to be operated by reciprocating members, are sometimes constructed on quite a different prin- ciple, viz. : that of a weighted lever in nearly unstable equi- librium, so that it can be caused to fall to the right or left by means of a slight thrust, and so operate a releasing member. A form which was formerly much used, in which the lever is carried on a horizontal axis, is shown in Fig. 742. Fig. 742. Fig. 743- When the weight G is in the vertical position i . 2, the pres- sure acts directly downward upon the axis, the journal friction acting as a ratchet. The form is sometimes used on planing machines, screw-cutting machines, etc. Another form is shown in Fig. 743. Here the pressure is due to a spring, acting through a link 3-2 upon 2 . i. A third form is that used in Shanks' planing machine, Fig. 744. In this case the lever, with its axis a, is at right angles to b, and the latter is pro- vided with a roller. The limit of measurement of « is between 2' and 2". The forms of tumbling ratchets described in \ 239, may be adapted as releas- ing gears, but it must not be forgotten that in such me- chanisms provision must be made for the middle position of the ratchet. A fourth form of tumbling gear, of which, indeed, there are many varieties, is the so-called "loop" of Hofmann's valve gear. Fig. 745. The loop a is made in the arc of a circle from a c' ^ a ■J .1 1 "^ ^k. ^ -(^ Fig. 745. Fig. 746. centre at 2, 5 is a heavy roller, with additional weight suspended at d' . When the loop or curved link is in either of the positions, 3o or 3', the weight acts to continue the motion in the direction in which it started until the limit of travel is reached. A swinging arm b may be substituted for the slot and roller, Fig. 746, and it will be seen that during the movement from the position 2;j 3g to 2' 3' the tumbling action will take place and the arm « be carried over. The similarity to the previous tumbling gear will be apparent. If 2. 3 bemadeinfinitely long, the loop will become straight and the two forms will coincide. Hofmanu has made the analogy to a ratchet train more complete by placing a ratchet so as to engage with the point 3 in the positions 3,, and 3', the release beingmade at the proper time by means of a cataract.* In some cases it is desirable to make a gearing which shall be released by the action of a very small force. For this purpose a second releasing gear may be introduced, itself being readily released, and by its action permitting a blow to fall upon the trigger of the main gear. Such a device forms a releasing gear of the second order. Such an example is shown in the hair- trigger of a rifle. t Releasing ratchet gearings of higher orders are also found in textile machinery, as in the Jacquard loom, also in the striking gear of tower clocks and of repeating watches. Another example is found in the relay of the Morse telegraph, be- sides many other applications which will be considered hereafter. ? 253. Checking Ratchets. Checking ratchets are used in a great variety of machines, but their principal applications are found in machinery for hoisting and lower- ing heavy loads, as in mine lifts, ele- vators, and the like, to guard against accidents in case of the breakage of the 2 ropes. In the opinion of the writer these devices have not been as yet regarded as they should be, merely special cases of ratchet con- struction, and as such capable of util- izing all the various principles here- tofore considered. When examined in this light their study will be greatly facilitated. As a scheme of a general system for checking ratchets a rod friction ratchet may serve. Fig. 747, in which the rod a is held stationary, the loaded mem- ber d carries the ratchet, and the pawl c and friction block b are held out of * Further details of this and the preceding gear will be found in the Aus- trian Report on the E-xposition of 187S. Section ou Steam Engines by A. Riedler, Vienna, 1S79. * See Zeitschrift des Vereins deutscher Ingenienre, i860, Vol. IV., p. 209. t Such hair-triggers were ingeniously applied in former times upon cross- bows. 164 THE CONSTRUCTOR. engagement by the releasing lever f, and rod e as long as the hoisting connections^- and /;, are under stress. If the tension is released, the ratchet is thrown into gear and the parts clamped. If a toothed ratchet is used instead of a friction device, the block b is omitted. According to the manner in which the various constructive details from a to h are arranged, we obtain the various systems of checking ratchets which have found practical application. A collection of such devices was exhibited by the Industrial Association (Verein fiir Gewerbfleiss) in 1879* More than So designs were shown, of which only a few can be described. Many of the device? were rather designs for improved construc- ti ^n as regards strength and rigidity, rather than examples of im. chanical ingenuity. 1 1 most cases the clamping action takes place upon the up- rignt timbers of the shaft ; sometimes guide ropes are used. The greater number of designs shown used friction clamps, those of the type of Fig. 724 being shown, the thumb pawl being roughened, however, or finelj' toothed. The one which showed the most evidence of careful constructive design in accordance with the principles previously laid down in 'i 24S, was that of Hoppe, shown, as attached to each side of the hoisting car, in Fig. 74S. Fig. 748. The form of friction pawl used is similar to that shown in Fig. 713, there being four pawls on each side of the car, or eight in all. The clamping action takes place upon the guide bars (7, made of T iron, as shown. At i are the guide rods between b and a ; at 2, the double clamp blocks of hardened steel, which are connected at 5 to the coupling rods e, e. The actuating springy is a torsion spring (see Fig. VII, p. 19 ; also Fig. VIII, p. 19), secured to the roof of the car at^, g, and operated by the releasing gear^ at 8, and transmitting action from 6 to 5 by the rods e, e, the connection being made by the links 9 to the double chain in such a manner that the army" cannot be drawn too far out of position. The proper adjustment of the pawl arms is obtained by the keys on the rods c, c. Hoppe has taken into consideration the fact that the angle j, see (233), must not pass beyond certain limits, or too great pressure would be ex- erted on the frame d, -d, and hence has provided stops in the frame for the tiavel of the pawls c, c. The parts are so propor- tioned that a load of double that ever placed upon the car would be supported by the friction clamps before there would be an appreciable elastic yielding of the frame. The adjustments of the rods c, c provide for the change of relations due to wear. This apparatus does not bring the lowering car to a sudden standstill in case of breakage of the hoisting gear, but the shock is avoided by the gradual action of the friction brakes. By using the author's device, shown in Fig. 717, at 3, the value of o might be maintained constant, or by proper construc- tion of the guides the wedge friction pawls, similar to Fig. 718, may be used ; the blocks acting on both sides of the guide. This would reduce the stress upon the frame very materially. The system of brakes used upon railway trains are really forms of friction checking ratchets. The shocks due to suddeE stoppage are also to be avoided, and if the wheels are braked too firmly the sliding action is simply transferred to the rails. \ 254- Continuous Running Ratchets. Continuous ratchets (^ 235, No. 4) consist of such combina- tions of pawl mechanism as act to drive a member in a given direction with practically a continuous determinate motion. This may be effected by combining two single running ratchets in such a manner that they both act upon the same wheel, one pawl attached to the arm c, which is stationary, the other swing- ing about the axis i, Fig. 749, this being a very common form. a, b. \ ^ ■^^L ._n 7r Ol \ T r^ ■ Fig. 749- In this case 3 . 2 is the checking pawl, and 3' . 2' the driving pawl. A movement of the driving pawl, if a little more than one tooth space, moves the wheel one tooth ; a little more movement than two spaces moves it two teeth ; and a regular back and forth motion gives a forward movement at intervals of a single pitch space. If this device is made with step ratchets, as in ? 243, the pitch, may be subdivided into 2, 3, 4 or more parts, and for some pur- poses, such as saw-mill feed motions, this is very desirable. If the arm which carries the feed pawl swings about an axis- 4, removed from i, Fig. 749 b, there will be a movement be- tween the pawl and the poiut of application 2 on the wheel ; while in the arrangement shown at a the motion of the two is identical, and hence no wear occurs. The two pawls may be connected so that both of them be- come drivers. If they are arranged so that their movement is alternate, as in Fig. 750 a, the wheel will be moved forward for a. b. Fig. 750. the movement of the lever in each direction, giving a double- acting ratchet motion, the so-called Lagarousse Ratchet.* This may be also accomplished in various ways, as in Fig. 750 b. For any movement of the arm which is less than i and more than ;-2 the pitch, the wheel will be moved i pitch for each vibration, and hence for a half vibration a feed of a half tooth may be obtained. Step pawls may also be used with these de- vices to obtain further subdivisions. If, in Fig. 750 a, we hold the lever c-^ c^ rigidly, and instead permit the arm d to vibrate with the same angle about the axis 4, the wheel moving with it, we obtain the same relative feed motion. t This has been used by Thomson in a telegraph, apparatus. * Berliner Verhandlungen, 187^, p. 345. Prize essay by Dr. F. Nitzsch, on Safety Checking Devices for Mining Apparatus. See aiso Mairs, Berg und Huttenmann, Z., 1879, p. 361. * Named from the inventor, M. de la Garousse, and used in 1737. Belidoi^. Arch. Hydraulique. t This' is the ordinary kinematic inversion. THE CONSTRUCTOR. 165 A continuous ratcbet gearing may be so arranged that back- "ward movement of the wheel is utilized to compel a uniform division of motion. This is the case with the feed motion used by Gebriider Mauser, of Oberndorf, in their revolvers, Fig. 751. In this case Fig. 751. Fig. 752. a crown wheel is used (see Figs. 677 and 67S). The wheel is at ■a ; b'xs the feed pawl, jointed at 3 to the slide r, the whole being carried in the frame d. The zig-zag profile is formed in the rim of the crown wheel, one portion being parallel to the axis, the other spirally inclined, so that the angle of thrust is ct < 90° — <^ and > (? 237, cases 4 and 5). The movement of the pawl produces a backward movement of the wheel. It should be noted that at 1' and 2" steps are made in ends of the tooth pro- files in order to guide the pawl into the proper path and keep it from reversing. The anchor ratchet of Fig. 6S2 may be used for a feed motion, as in Fig. 752, in which there is also the reverse action of the wheel, in accordance with the notation of ? 237. Here the wheel is at a and the anchor at b' b" . When the latter is moved into the position shown by the dotted lines, the wheel is moved backward yi pitch, and the return vibration completes the pitch movement. In order that the anchor shall enter the teeth pro- perly, the movement should be quick, especially at the entrance of the pawl into the space. This is well obtained by electro- magnetic action. Fig. 753. ?255. Continuous Ratchets with Locking Teeth. If it is desired to use ratchets according to the method given in Fig. 749, additional parts must be devised to move the pawl in and out of gear. A simple method of accomplishing this re- sult is to use a single tooth wheel for the driver, and operate the pawl in the same mauner as in Fig. 753. Before the single tooth 5 begins to drive the wheel a, the arm 6 lifts the pawl b and lowers it into the next space just as the tooth ceases to drive. In this case the usual gear tooth profiles may be used. Still better is the "dead" tooth profile of Fig. 754, in which the entrance and withdrawal of the pin tooth both lock the wheel while the pawl is being lowered. This form may also be used for rack feed movement. Fig. 755. In this case the profile of the pin tooth is formed in several arcs ; 2' 2'" being struck from 3, and 2" 2.'" and 2' 2'^ being the paths of the corners of the space (see \ 203). By using the cylinder ratchet, as shown in Fig. 696, the num- ber of parts can be reduced, since the driving gear and check- ing pawl may be combined in the same member. The resulting forms, Figs. 756 to 75S, are variousl}- called : Maltese Cross ; Fig. 756. Fig. 757- Fig. 758. Geneva Stop, used in Swiss watches, in which case one of the tooth sections is filled out ; or after Redtenbacher we may call them single tooth gears, although this is hardly correct, for the general form of Fig. 75S may have several teeth, and a second tooth is dotted in Fig. 756. A great number of variations may be made of these cylinder ratchet motions. An interesting form is the intermittent gear- ing of Brauer, Fig. 759.* Fig. 759- Fig. 760. The pinion a is the driver, and the wheel b is driven, and between the passage of each tooth of the pinion the driven gear remains stationar3' for a short space, about i of the pitch. The points of the teeth of the driven wheel here act as ratchet teeth, in a similar manner to the arc of repose of the single ratchet gearing of Fig. 756. The cylinder ratchet gearing of Fig. 760 is similar to that shown in Fig. 700, and is used in the counting mechanism of English gas meters. In Fig. 761 is a modified spiral ratchet of Fig. 761. Fig. 762. the same general type as Fig. 702, with only a portion of the path of (5 in a spiral, and a similar variation of Fig. 704 is shown in Fig. 762. Fig. 754- Fig. 755- * Royal Germau Patent, No. 5583, 1S7S. i66 THE CONSTRUCTOR. 1 256. Locking Ratchets; Locking ratchets include all the numerous devices by which the parts of a mechanism are firmly held against the action of external forces, and yet readily and definitely released when desired (see i 235, No- 5) ; thus the various clutch couplings are included, also car-couplers and similar devices. Locking ratchets occur frequently in the mechanism of fire- arms, especially to prevent the danger of premature discharge, etc. The great refiuements which have been introduced in such ■weapons during the last ten years include especially the appli- cation of various forms of ratchets. The following single in- stance will serve to illustrate : The mechanism of the well-known Mauser revolver may be divided into two series; one to effect the discharge and the other to unload or remove the empty shell from the chamber. The first may be called the discharging mechanism, the second the unloading mechanism. We then have the following details : A. Discharging Mecha]iisin. This includes the revolving chamber, barrel, hammer, spring and accompanying smaller parts, giving as combinations : 1. Hammer, spring-rod and trigger = ratchet rack, as Fig. 659. 2. Spring-rod and trigger, acting as locking ratchet for the above, as Fig. 664. 3. Spring-rod, pawl and revolving chamber = continuous ratchet with crown wheel and bolt pawl, as Fig. 751. 4. Securing pawl and revolving chamber = locking ratchet, as Fig. 677. 5. Revolving chamber and pawl, forming a ratchet gearing with limited travel. 5. Tumbling ratchet and securing pawl = ratchet gearing for three positions. Fig. 669. 7. Catch on the axis of hammer ^ locking ratchet, as Fig. 695. 8. Trigger guard and pin = locking ratchet and stationary pawl. 9. Checking-plug and trigger ^ locking ratchet with sta- tionary pawl. 10. Rifled barrel and bullet = screw and nut. B. Unloading Mechanism. This includes au axial slide which catches under the rim of the empty cartridge shell to withdraw it, actuated by a toothed sector and revolving clamp and axis called the ring clamp. These include the following combinations : 11. Unloading slide and sector ^ slide with rack and pin- ion. Fig. 381. ^ 12. Axis of revolving chamber, with pawl to prevent end- long motion, ^ locking ratchet gear, as Fig 695. 13. Ring clamp, barrel and chamber bearing = locking rat- chet gear with stationary pawl, as Fig. 654. 14. Ring clamp axis and axis of securing pawl = locking ratchet, as Fig. 701, forming with (13) a locking ratchet gear of the second order. 15. Ring clamp axis upon the reverse motion of the ring clamp forms, with the axis of the securing pawl, a locking ratchet .gear, which combines with (4) to form a similar gear of the second order. 16. Securing pawl acts as a catch for the axis of the ring clamp in the axial direction to form a locking ratchet gear, as Fig. 695, forming also with (4) a similar gear of the second order. 17. Ring clamp hub and axis of securing pawl = locking ratchet, as Fig. 695, and with (4) gives one of the second order. This analysis shows that in the Mauser revolver there are 17 mechanical combinations ; these are composed of 26 pieces. Classified, these are as follows : i releasing ratchet, i continuous ratchet, 2 driving ratchets, 11, locking ratchets, of which four are of the second order, i screw motion and i slide motion. A very important application of locking ratchet mechanism is found in the signal apparatus of Saxby & Farmer for use on railways, and made in Germany by Henuing, Busing and others. This includes many ratchets of higher orders, reaching to the tenth, twelfth, or even higher. When this is used in combina- tion with the electric systems of Siemens & Halske, as in the block system, we have the further combination of two systems of the higher order with each other. A branch of locking ratchets which exhibits a great variety of applications is found in the different kinds of locks, such as are used for securing doors, gates, chests, etc. These extend from the most primitive forms, made of wood, to the most re- fined productions of exact mechanism, and their study possesses an historic and ethnographic interest in addition to their me- chanical value. A door forms itself a ratchet combination ; the door being the part b, the strike the part c, and the bolt or other piece which keeps it from being opened is the part a ; doors with latch bolts being running ratchets, and doors with dead bolts being stationary ratch- ets. A simple lift latch and door, as the furnace door shown in Fig. 763, is really a section of a crown ratchet wheel with running ratchet gearing. A door with sliding dead bolt, as used on common room doors, is a similar section of rat- chet gear with station- ary ratchet. ■ In key locks, the key is the releasing member of the ratchet train, and also serves to actviate the bolt after it is released. The key and ratchet mechanism are arranged in most ingenious manners, so that numerous permutations can be made to effect the release- Some of the most important systems of lock construction are given as examples : Example I. — The common so-called French lock. Fig. 764, is similar to the ratchet of Fig. 753. The bolt is a sliding rack, the " tumbler " b being often. Fig. 764- FiG- 765. as in this case, made in one piece with its spring. The case of the lock cor- responds to the frame for the ratchet mechanism, and the key acts as the releasing and actuating member. Example 2.— T)\& Chubb lock. Fig. 765, which is always made with a dead bolt, forms with the door and door frame a ratchet gearing similar to Fig. 69E. Tne bolt is secured by means of several ratchets of precision, as in Fig, 706, and is moved by a ratchet as Fig- 755- The key, the axis 4, and the vari- ous bittings of the key form a system of pawls. The whole is a ratchet sys- tem of the second order with precision gear. Example J. — The Bramah lock. Fig. 766^ and Fig. 7666, is differently con- structed- In this case the dead Jcg bolt is actu.ited through the medium of a cylindrical driving ratchet , gear, which does not contain the mechanism of se- curity, the latter being in a distinct portion of the lock. Fig. 766 A. This consists of a number of sliding precision pawls, as Fig- 707, the number being 6 to S (in the illustration 5)- '1 he member a of Fig. 707 is here made in the form of a ring with internal teeth, se- cured to the escutcheon a by screws- The key is a prismatic adjuster of the slides, and the whole is a locking mechanism of the third order with ratchets of precision. The spiral spring around the pin restores the slides to their extreme position when the key is withdrawn. lo I lo TT i "0 i -■0 & ^ 5^ )A Fig. 766 a. Fig. 766 d. £xam//e 4.— The Yale lock, Fig. 767 a and d. is also a system in which the mechanism of security is separated from the bolt mechanism. This is again a system of the third order, with ratchets of precision. The key is a flat prism, (corrugated in recent locks) and serves to place precision bolts, or THE CONSTRUCTOR. 167 piu tumblers in proper line, and also operate the bolt. The figure shows the method of connecting the cam b^ to the plug a. The so called combination locks are locking ratchets with precision pawls, operated without a key; by being placed successively in the positions for release in accordance with a previously selected series of numbers and dial marks. h ; 1 1 lo _^r '?V'-" ^ -i — . . Fig. 767. The numereus systems of Arnheim, Ade, Wertheim, Kleinert, Polysius, Kromer, and others are mostly locking ratchet systems of the fourth order, or combinations tliereof The American manufacturers, especially the Yale and Towne Manufacturing Company of Stamford, Connecticut, have shown great ingenuity in this industry.* 1 2:7. Escapements — Their Varieties. Escapements may fairly be considered as among the most im- portant mechanical devices, since it is by their means that the elementary forces are used to regulate mechauical work. For this purpose they are used in the greatest variety, all forming ratchet devices in which the driven member is alternately re- leased and checked. The arc, angle or path through which the driven member passes between the interval of release and check is called the "range" of the escapement. During the passage over this range there elapses a definite amount of time, which maybe called the "period " of movement of the escapement. This is followed by an amount of time when the driven member is stationary, called the period of rest. The sum of the two forms the "time of oscillation." The range and the period of oscillation may be [a) constant, {b) periodically variable, or (..-) variable at will. We therefore have Uniform escapements. Periodical " Variable " and these will be briefly considered. I 25S. Uniform Escapements. If, in ordinary running ratchet, Fig. 76S, we have the wheel a, impelled by a weight or other force, and suppose the pawl b, lifted and dropped quickly, as by the arm b-i, the wheel will move one space, and an escapement will have occurred. In this case the range will be one pitch. If, after a definite time, this operation is again and again repeated, we shall have a uniform escapement. In mechanism the releasing and check- ing action is produced mechanically and not by hand, the im- pulse being obtained from the movement of the wheel. * The ancient and modern Ej^yptian locks, also those of ancient Greece, Rome, India and China, contain the principle of running ratchets with flat pawls, actuated by a key pushed directly into the lock. The Egyptian lock, with pin precision pawls, is quite similar to the Yale lock in principle, al- though very ditferent in construction. Ancient Roman locks, found in Pompeii, are similar in principle. Wooden locks are still in use in China, Persia, Bulgaria, Russia and Southern Italy, also in the Faroe Islands and Iceland. At the suggestion of the author. Professor Wagner, of Tokio, suc- ceeded in inducing some Japanese lockmakers to make a very complete and intelli^bie collection of native locks for the kinematic cabinet of the Royal Technical High School at Berlin. The most general examples of uniform escapement are found in watches. In these impulses are isochronous, and obtained from the inertia of a vibrating body. The wheel a is called the escape wheel. The vibrating member, or balance wheel, makes its oscillations in nearly equal times for great or small vibra- tions. If, therefore, in a watch escapement, the time of the fall of the pawl is less than the time of oscillation, the most impor- tant requirement is fulfilled, namely, that for uniform periods of time the same number of teeth of the escape wheel shall pass, and the corresponding angle may then be used as a measure of time. A given amount of work may ajso be abstracted from the ^ motive power and used to produce the impulse. These impor- tant points have been fulfilled in the design of escapements, and it has been made possible to measure time with a great degree of accuracy. When the highest accuracy is demanded the greatest care must be given to the construction and execu- tion, and to the reduction of friction and compensation of the balance. In the case of watches the duty of the impelling force is simply that of overcoming the resistance of the mechanism, the function of the escapement being to provide against any acceleration of the rate motion, and the impulse which is re- quired to operate the escapement may be considered as a por- tion of the resistance of the mechanism. A systematic discrimination between the various kinds of watch escapements will show that they vary as to the checking ratchet device, the impelling device, the release and the accel- erating device. We may have Simple or Compound escape- ments of the lower or higher orders. Some examples are here given. A. Simple Escapeiiienti. Fig. 769. Example /.—The Free Chronometer Escapement (Jullien le Roy, Earn- shaw, Arnold, Jiirgensen), Fig. 769. The running ratchet gearing a, b, c, is similar to Fig. 76S. The pawl b is provided with a flat spring 3. The im- pelling device is the balance wheel rf, which acts as a pendulum. The re- leasing device is at 4 . 5, and is attached to d, and when it swings to the left, impelled by the movement of the watch, it releases the pawl by means of a second running ratchet at 5. .^t c' is a stop for the pawl b. At 5' is the ac- celerator which, for each tooth of the escape wheel «, swings from 5' to 5". As it returns, the pawl I* engages with the tooth which has just left the point 5". The spring b' permits the releasing tooth 5 to pass back dnring the return oscillation. Tlie balance wheel can swing freely beyond 5" and back without engaging with the escape wheel, hence the name " free " escape- ment. ■'■ Example 2,— The Duplex escapement, Fig. 770, is derived from the ratchet of Fig. 699. The escape wheel is upon the same axis as the checking pawl Fig. 770. Fig. 771. * This beautiful movement is apparently the first form which was applied as a pendulum escapement, having been used by Galileo in 1641. i68 THE CONSTRUCTOR. I> \ the accelerator is at 4, acting upon the impelling pawl at every vibration "between 4 . 4'. The so-called "verge " escapement is similar in construction, except that the arm b' is longer and curved. The simplicity of this form as compared Tvitli the preceding is due to the fact that the impelling and checking pawls are made in one member It will be noticed that the entrance of the tooth ■of the escape wheel into the space, causes a slight reverse movement at a, due to the fact that b is really a tumbling ratchet gear. This escapement has been called duplex: by its Knglish inventor, although some contend that it is properlv a double wheel escapement, although the two wheels are combined in one. Example 3.— KwoW^^x method by which the checking and impelling pawls may be combined is shown in the Hipp escapement, Fig. 771. This consists ■of a simple running ratchet a, /•, c. The pawl 6 is a plate spring, which is lifted and dropped by the passage of the teeth. The acceleration is given by the deflection of the spring. If the impelling force upon the wheel a is great, two teeth will pass, but this can be detected by the note emitted by the spring, which will then be one octave higher than before. B. Compound Escapeinenis. Example 4. — Lamb's escapement. Those escapements which have two escape wheels are properly classed as compound, and to this class belongs I^amb's escapement. This consists of a running ratchet gear, similar to Kxauiple I, and the same form of impelling device, but between these is an internal wheel with pitch ratchet gearing, similar to Fig. 6S6, which is im- pelled with each direction of vibration. Another double-wheel escapement 15 Enderlein's, based on Fig. 70-, also one devised by the author, like Fig. 6S6. Example 5.— ls\\x^%^' & Escapement (also invented by Tiede), Fig. 772. This is a double ratchet gear system, with one pawl in compression and one in. Fig. 773. Fig. 774. are moved alternately by the pendulum ;' for example, the arm b\, being moved into the dotted position, lifts the pawl out of gear, and the weight of the pawl and arm (sometimes assisted by a spring), gives an impulse to the return vibration of the pendulum, the acceleration being provided by the escape wheel acting on the portion //'. A similar action takes place on the other side. Example 6. — Bloxam's or Dennison's so called "gravity'' escapement, Fig. 773. The escapement is controlled by a pendulum suspended by a spring at 4. The escape wheel is made in two parts, as Fig. 686. The accelerating surfaces //' and //" are much better arranged than in the preceding exam- ple, the friction being reduced. A fan is used also, as shown at e, for the purpose of preventing great acceleration of the escape wheel, which might otherwise occur in the large angle (60°) of escape. The fan is not fast to the axis of the escape wheel, but connected by running ratchet so that its mo* mentum. is not checked as the escape wheel is stopped. Example y.—Fvee Anchor Escapement, Fig. 774. The two pawls are com- bined into one anchor, as in Fig. 682, and the action is much the same as Fig. 772. The escape is controlled by a balance wheel at tf. The pawls 2' and 2" are operated through the arm i% and at the same time the impulses are given by the action of the escape wheel upon the inclined surfaces//' and //". The pawls are technically known as pallets. The tooth action at 5 is a continuous ratchet gear similar to Fig. 754. The arm 1^3 is limited in travel by pins at 3' and 3", or in some forms by a fork at 4 Since there is a ratchet at 5 and also at 2, this forms a system of the second order.* (^^=f^ Fig. 776. tension, ^1 and i^- At 2' and 2" is a "dead'' pawl action for checking, and .at //' and //'' a running pawl action for impelling. (See Cases 5 and 7, § 237). The pawls are lifted by the pendulum d. The releasing arms 3' . 5' and 3". 5" * A watch escapement of the third order has recently been designed by A. E. Miiller, of Passau. This is made with a cylindei ratchet, as Fig. 699^, between the arm and the escape wheel. THE CONSTRUCTOR. 169 Example 5.— Graham's Escapement, Fig. 775. The construction is very similar to the preceding. The connection 5 between the anchor-arm A) and pendulum d, is different, and the arm b-^ does not come to rest, but both it and the pallets 2' and 2" slide upon the teeth while the escape wheel is stopped. An earlier form of pallets for this escapement is shown at b'\ and b\ (called Clement's Anchor, frona Clement, 16S0 ; but described by Dr. Hooke in 1666). This form produces a brief reverse movement to the escape wheel at each oscillatiou. Example g.—Th^ form of ratchet of Fig. 6S4 is used in Lepaute's escape- ment, which was really invented b}- the watchmaker Caron, afterwards Marquis Beaumarchais. Example jo. — Cylinder Escapement, Fig. 776. This is made from the cylinder ratchet of Fig. 700, the impelling surfaces being divided between the anchor and the teeth of the escape wheel. The cylinder b is attached to the axis of the balance wheel, and the wide spacing of the teetli of the escape wheel permits a correspondingh' wide amplitude of oscillation. If we im- agine the pallets of Graham's anchor to be formed between two concentric circles (as, indeed, most watch Tiakers construct them), the " cylinder'' will be seen to be a similar anchor. Example II. — Crown Wheel Escapement, Fig. 777. Escapements con structed with crown ratchet wheels (5 z\\) are the oldest forms used iu Fig. 777- ratchets.* The form of the pallets causes a reverse movement, and in the old watches using a balance with its centre of gravity in the axis of oscilla- tion, without any assisting spring action, this reverse movement was a necessity, which accounts for the long and extended use of this form of escapement. Toward the end of the fifteenth centur}' the hair spring was introduced by Hele. in the form of a hog's bristle, and in 1665 Hayghens made the steel hair spring, which made the construction of the modern chronometer possible. The crown escapement is easily modified so as to remove the reverse action, as was done by the author in 1864. We then have a "dead" tooth action, as Fig. 699. The modified escapement is shown in Fig. 778 ;, the pawls are practically hyperboloidal in form f C. Poiuer Escapements. Ill the case of watch escapements the impelling force is only used to overcome the resistance of the watch mechanism. Escapements can also be used to regulate greater forces, such as are intended to perform useful work, and these may be Example /r.— Power Escapement for a Reciprocating Movement, Fig. 779. At a by Cx and a b.> Co are ordinary running ratchets, the pawls b\ and b^ of which can be released and engaged by suitable auxiliary mechanism. This mechanism is either a substitute for or identical with the legulating device (balance wheel, pendulun, etc ) of a watch escapement. The escapement is intended to control the motion of the swinging arm Cby means ot the lever ^i and tlie descending arm -^i- This is accomplished by a double acting ratchet system di dq 5 (as Fig. 671), by means of the slide e, "driven from 8 by the arm cj. The action is as follows: When the parts are in the position shown in the figure, the motion of the wheel a to the right moves the arm ci by means of the pawl bi until the trigger 10" trips the pawl d.^ and shifts the engagement at 5 into the position 5' (in the small figure to the left). This action, by means of the trigg-er at 6", throws in the pawl Aj and stops the wheel a. At the same time bi is thrown out of gear by the connections d^, 6' and 7', and the counterweight Q returns the arm ci to its original position. This brings the trigger 10' against the lever di, and again shifts the engagement at 5. The pawl b] falls into gear, and the pawl b.^ is disengaged, leaving the wheel a free for another forward movement. The preceding escapement can be readily converted into a double acting one by introducing a second ratchet wheel toothed iu the opposite direction, with proper pawl on c^ and trigger connections to d^ ; the other portions would remain the same. This escapement appears to be new, and many important appli- § 258. Periodicai, Escapements. A great variety of periodical escapements are to be found in the striking mechanism of clocks and repeating watches. The entire period is the revolution of the hour hand, and if the half hours are struck the order will be 4, making iu all 90 strokes iu the twelve hours. A fan regulator is used to cause the strokes to follow each other uniformly. There are two systems of escapement in use for this purpose, the German and the English, the latter also used for repeaters. An essential piece of the latter, the so-called "snail," has been shown in Fig. 688 ; its function is to control the number of strokes. Further subdivisions cannot be here discussed, but it must be remembered that the striking arm is itself a ratchet mechanism.* Important applications of periodical escapements are found in the self-acting spinning mule, and both these and the clock striking mechanism are examples of power escapements. The mechanism in Piatt's mule is here briefly shown. Fig. 780, a aud b. The shaft l is required to make rapid turns Fig. 779. called power escapements. Alarm and striking clocks are of this class, and there are numerous other forms. The following example will serve to illustrate : * This has been used since the tenth century, having been invented by Bishop Gerbert, afterwards Pope Sylvester II, about 990 ; also by Hcinrich von Wyck about 1370, and applied to a pendulum hv Huyghens. The oldest tower clock in Nuremberg, built about 1400, has such an escapement- t In the Kinematic cabinet of the Royal Technical High School there is a schematic series of models of clock and watch escapements. through 90° at intervals of different lengths of time. The wheel a is an escape %vheel with teeth in four concentric riugs, I, II, in, IV (compare Fig. 686), each ring having one tooth. The other side of the wheel a is shown in Fig. d, where is the rat- chet chain a d e. When a is released, the pressure of d at 5' moves it slightly and brings the running friction wheel e into contact, thus driving a through a quarter revolution, toward the close of which the pawl (/ again enters iuto engagement. ► See Ruhlmann RMtenbacher. Denison. 17 o THE CONSTRUCTOR. The recesses in a permit the friction wheel to run free when a is at rest. This is evidently a form of ratchet gearing in it- self. The order of escapements at 2 is as follows : I II, II III, III IV, IV I. This is controlled by a second escapement, shown in Fig. 781. We have already intimated that the various forms of coup- lings may be considered as varieties of ratchet gearing- The Fig. 7S1. The pawl b of Fig. 7S0 is connected by the rod/ to the beam a, as shown. This mechanism is a step ratchet of four steps. The steps are the pawls ip b.^, b,, and the stop on the frame c ; giving the positions 21, 211, 2m, 2iv. The action takes place in the four following periods : 1. Drawing and spinning — a checked at 21 2. Stretching and twisting " " 211 3. Holding aud spun thread " " 2111 4. Winding and returning " " 2iv The succession of movements is as follows : At the termina- tion of the first period a projection on the carriage strikes the pawl b-i at 5'. The step lever, which is heavier on the right end than on the left, moves from position I to position II, in which it is held by the pawl b.,\ this, bj' means of the rod /, places the pawl b of Fig. 780 in the position 3 II, thus starting the second period. At the close of the second period the pawl b.^ is released, the lever falls to the position III, shifting the pawl 5 to 3 III, and is held by the pawl 63 at 2'". The third period, which is very brief, is terminated by the winder striking 5'", releasing the pawl b-^, and the lever as- sumes the position IV, and the rod f moves the pawl b into the position 3 IV, and the fourth period begins. During this period the carriage returns, and just before the close of its motion a roller acts upon the portion 5°, bringing the lever back into the first position. This returns the pawl b to its original position 3 I, and the succession is repeated. The entire mechanism forms a periodical escapement of the. second order, or, when the connections are included, the third order, and when taken together with the ratchet gearing, of the fifth order ; while a sixth ratchet mechanism is used for the primary control. I 259. Adjustabi^e Escapements. An escapement can be so arranged that the checked member, after the release, will again be checked by the impulse of its fresh start, thus forming what may be called a self-acting escapement. In a mechanism of this kind, the amplitude of the escapement is dependent upon the amount of displacement which is permitted to the releasing member. This may be made greater or less, and hence such devices maj- be called adjustable escapements. These devices are likely to play an important part in modern machine design. A simple form of adjustable escapement is shown in Fig. 782. This apparatus, designed by the author, is based upon that of Fig. 674. The ratchet wheel a is stationary, being fastened to the frame a' ; the pav*! is at b, and the link is in the form of a disc f, driven by a force C, and checked by the escapement. At 3 . 5 is the guide for the pawl. This can be adjusted b}' the wheel d, by turning the latter more or less in the direction in which c is impelled. If d is turned so far that the pawl b is lifted out of gear, the force at Cwill set the disc c in motion. This latter carries with it the axis 3 of the pawl, which, by the action of the guide 5, draws the pawl into engagement again, entering the space 2 and checking the disc. In order to avoid an uncertain or irregular action, a brake may be used as at a". If the wheel d be moved forward regularly through two, three, or four arcs, the disc iTwill be released and checked successively in similar manner. 'It will be evident from the foregoing that the ratchet gearings which form the foundation of the various kinds of adjustable escapements are so varied that the different constructions which ma)' be vised are very numerous. Among them may be men- tioned those in which friction ratchets are used, these posses- sing the advantage that the arc of motion of the escapement may be varied from the smallest to the greatest without being dependent upon any especial pitch. Fig. 782. same is true of the present subject. If it is desired to use this adjustable escapement as a disconnecting coupling, the follow- ing arrangement may be adopted : The disc c can be attached to the shaft which is to be set in motion, and the wheel a to the driving shaft, which is supposed to be in continuous revolution and is to be coupled to c. The teeth are then to be so arranged that by the revolution of a the pawl b, disc c and wheel d will be carried around together. When the disconnection is to be made, it is only necessary to hold the wheel d from revolving. The pawl-axis 3 will then move on and cause disengagement of the pawl at 2, and the disc c will come to rest. If the wheel d is then turned a short distance in the direction of rotation the pawd will .again, be thrown into gear and the parts once more connected. A coup- ling thus formed from an adjustable escapement may be called an adjustable coupling. The suitability of the application of toothed ratchet gearing for this purpose is open to question, and indeed toothed gearing, is only to be recommended for the lightest service of this kind. In most cases, if indeed not all, friction couplings are much better. An adjustable friction coupling is to be seen b)' refer- ence to Fig. 44S, in which A is the friction wheel, B is the pawl, disguised in the form of a cone, and b is the adjusting member. If a combination is made of an adjustable friction coupling with some form of transmission to a machine, such as a rope or belt gearing, so that it is thrown into action when any re- verse motion is attempted, we have what may be termed an automatic friction brake.* Fig. 783. *See German Patent, E. Langen, No. 21,922. THE CONSTRUCTOR. 171 Example. — Figr. 783 shows such an automatic, brake device as applied to the pontoon bridge at Cologne. At a is a friction cone combined with a spur gear a', driven by the shaft and pinion rt" in the direction to wind up the cord on the drum c'. The drum is fast to the chaft c, but the cone a is loose on the shaft. The wheel a is connected firmly to the shaft t:, when the cone b, which slides on a leather, is forced into engagement witii it, and this en- gagement is effected by the differential screw d and hand wheel d' . The use of the differential screw enables the equisite pressure to be obtained, and also causes tiie motion of d' to be in the same direction as c' when lifting. The friction of tlie cones binds the parts firmly together, so that a is practi- callj' secured to the shaft until d' is revolved backwards, w^hen c' follows by the action of the weight C, the cones slipping upon eacli other and the pressure being automatically regulated, and the motion at once checked when d' is stopped. Other and most important applications of adjustable escape- ments will be given hereafter. It maj', however, be here noted that by means of such mechanism the most powerful combina- tions may be controlled with the exercise of a minimum effort. \ 260. Generai, Remarks upon Ratchet Mechanism. Ratchet mechanism, as already discussed, is applicable to a most extensive range of uses ; in this respect far excelling every other form of mechanism. This is plainly due to the fact that ratchets are suited either to produce the effect of relative motion and relative rest. Considered in this light the six preceding classes may be grouped as follows : Common ratchets, checking ratchets, and locking ratchets are those which act to hinder motion, while releasing and continuous ratchets, as well as escapements, act to produce definite motion. The motion pro- duced by ratchets is intermittent while that produced by the forms of mechanism previously considered, such as cranks, friction, or toothed gearing, etc., is continuous. Mechanism for continuous motion may be called "running gearing,"* and practically merges into ratchet gearing. The general province of ratchet gearing has only been partially covered in the pre- ceding pages, where such forms as may strictly be considered machine elements have been included. An exception might be made as to the allied forms of springs, some of which, indeed, were referred to. There is, however, a large number of machine elements of a different kind, which usually involve the continu- ous action of the operative forces in one direction ; these in- clude tension organs, such as ropes, belts, chains, etc., compres- sion organs, fluid connections, and many others, all of which are considered in the following chapters. It will be seen that these may all be so arranged as to be fairly considered ratchet devices also ; as belts or chains may become friction or toothed ratchet gears, and even the valves of fluid connections are really pawls. t The pawl mechanism must also be extended to include these classes of machine elements, and their limits thus greatl}' widened, especially in the case of pressure organs. Examples of this will be found in the pistons and valves of pumps, both for liquids and gases, which may act as checking or locking ratchets, or in hj'draulic motors and steam engines as escape- ments, and in gas engines, as escapements and continuous ratchets combined. Similar comparisons may be made of the ratchet principle in the use of accumulators for hj'draulic cranes, presses, riveting machines, and the like, and in the cataract for single acting steam engines we find a complete analogy to the ratchet. In these cases we have ratchet systems of the higher orders. The history of the development of these machines is really that of their pawl membeis. A very interesting example is that of Fig. 779, in which, if we substitute a flow of steam for the ratchet wheel, we have the arrangement of the single acting high pressure steam engine with Farey 's valve gear. The numerous modifications of escape- ment gear, which are included in the steam engine, have occu- pied the activity of designers down to the present time. A number of the more recent valve gears have been shown in \. 252, and similar devices are used on engines for steam steering gear, called by the French " moteurs asservis," and such gear also plays an important part in them;ihanism of some of the so- called ''fish" torpedoes. In this manner the applications of pawl ratchets may be ex- tended before our eyes and yet the limitations are not reached, and the further researches are carried the broader and more general does the scope of this division of mechanism become. Not only does it include fluid pressure organs, both liquid and gaseous in a strictly mechanical sense, as in the case of pumps, etc., but also when these are considered in a physical sense with regard to their internal stresses. This gives a branch which may be called "ph^'sical" ratchet trains, of which the steam boiler is the most important example. In this, when taken in connection with a pipe full of steam, and suitable valves for opening and closing, forming what has been termed a steam column,* we have undoubtedly a physical ratchet train in which the particles of vapor are considered as a physical aggregate, which from the higher temperature, are under higher stress. Another example of a physical ratchet train is the apparatus for operation by liquid carbonic acid which has been recently used. Electrical accumulators are also instances of physical ratchet trains, as well as some applications of galvanic batteries, the action taking place by make and break of electrical contact. The dynamo-electric machine also becomes a physical running ratchet and the electric motor a physicabescapement, the whole forming a phj'sical running gear train. Again we may consider a "chemical" ratchet train, such as coal or any fuel, which, during combustion, releases the energy which is stored in it. This may be utilized in numerous ways, but for our present considerations, mainly iti the production of motion. Chemical action is also included in hot-air engines, and in the operation of telegraph apparatus in a similar sense. We maj' consider the principal factors in a steam motor plant as portions of a ratchet chain, somewhat as follows : Chemical ratchet ^ combustion of fuel, Physical " = steam generator, etc., Mechanical escapement = steam cylinder and attachments. Mechanical running gear = crank shaft and wheel, these four uniting to convert the released energy into mechani- cal motion. If we consider a locomotive engine, we have added to this another running gear in the shape of the driving wheels and rails, while the train and wheels and journal bearings unite to form a combination of the sixth order. Another chemical train may be formed by the use of explo- sives, which are released either mechanically, as by percussion or friction, or chemically, by combustion of some auxiliary material. Again, we may have releasing gear of the first, second, or higher orders. In the case of most firearms the release is of the second onler, since the mechanism of the lock acts upon a fulminate by per- cussion, and the heat of the latter releases the pDwder. If we examine and classify all mechanism of transmission in the above manner, it will be apparent that all forms are included in one or the other of the following classes, viz.: mechanical, physical, or chemical ; these also entering into combinations of the higher orders with each other. The steam engine itself, as we have already seen, consists of a driving train of the fourth order. Trains of still higher orders are of frequent occurrence. In the recording telegraph, with relay, we have a ph5'sical ratchet train of the second order, releasing a mechanical run- ning train and operating a recording train, both physical trains actuated by chemical trains, the whole forming a combination of the fifth order. The ordinary si.gnal mechanism of a railway station, when mechanically operated, is a system of the fourth order. The Westinghouse air brake, not considering the boiler, is a train of the fifth order, consisting of an escapement (steam C5'linder), driving ratchet (air cylinder), intermittent ratchet (air vessel), escapement (piston and valve connections), friction checking ratchet (brake gear). If we include furnace and boiler, this becomes a train of the seventh order, and may be still further extended. A still more noteworthy example is found in the application of compressed air for the purpose of operating pumping ma- chinery at the bottom of deep mine shafts. In this case we have : 1. Furnace = chemical ratchet train. 2. Boiler • = physical " " 3. Steam engine ^ mechanical escapement train. 4. Shafting and transmission to " running 5. Air compressor, " driving ratchet. 6. Air chamber, " intermittent "^ 7. Air cylinder in mine, " escapement train. fS. Water cylinder in mine, " driv'g ratchet " The preceding discussion and ilhistrations of the relationship existing between mechanical, physical and chemical trains shows the necessity of combining mechanical and technical research, and a complete mechanical training therefore in eludes these three branches, and also 'the later science of electro-mechanics. Modern methods of invention require research into all of these lines of science, and the constantl}' widening field of mechani- cal engineering is thvis extending its work, while at the same time gathering into systematic form the many branches of applied mechanical science. * See the author's Theoretical Kinematics, p. tion was originall.y made. tSee Theoretical Kinematics, p. 458 et seq. vhich this classifica- * See Theoretical Kinematics, p. 4Q3 t The system of clocks operated by pneumatic pressure from a central station, designed by Mayrhofer, at Vienna, forms a combination of 33 dis- tinct systems. 172 THE CONSTRUCTOR. CHAPTER XIX. TENSION ORGANS CONSIDERED AS MACHINE ELEMENTS. I 261 Various Kinds of Tension Organs. The various forms of ciacbine elements which have alreadj' been discussed, have been those which offered resistance to forces acting in any given direction, forming more or less rigid constructions. We now have a series of elements which are only adapted to resist tension, and which are very yielding under the action of bending, twisting or thrusting forces. These include a great variety of rope, belt wire, chain laelt and similar transmission devices, all of which may be included under the general term of Tension Organs. Their usefulness is limited .by reason of the fact that they have only the single method of resisting force, but at the same time the element of flexibility permits the use of one and the same organ to transmit power in changing directions, and hence gives rise to many useful com- binations. An especially valuable feature of tension organs in practice lies in the fact that many materials are excellently adapted for such use, and can be more economically applied. Fig. 262. Methods of Application. A distinction is to be made between " standing and running" tension organs. The first are those used to suspend weights' support bridges, also in the construction of many machine de tails. E.xaniples of such use are found in suspension bridges' pontoon bridges, hawsers, guj' ropes, standing tackle, etc Running tension organs are used in machine design in connec- tion with other machine elements principally for the transmis- sion of motion. Running tension organs may again be divided into three classes according to their action iu connection with other machine elements. According as they are used : 1. For guiding. 2. For winding (hoisting or lowering). 3. For driving, this also being possible by winding and un- winding. Combinations of these applications may be made, either with or %vithout the use of standing tension organs. In order to understand the various applications it is desirable to consider some of the most important combinations, henre these will be briefly examined, abed e arc of contact. This action, which here opposes the motion of the cord, is in other instances made of great utility. Cord- ^H^ ?f ^/^ :^^ { i^ (0L l5) f /«\ IT li ;: y \s Fig. 784. I. Guiding. — Fig. 7S4 shows several combinations, adapted solely for guiding. At a is the so-called stationarj' pulley, in which a cord, led off at any angle, is rised to raise and lower a load 0. The dotted lines show the position of guides, or in the absence of these the direction of motion is governed by the action of gravity. At b we have the so-called movable pulley, the pulley being combined with the moving piece ; the weight Q is here supported on two parts of rope. Form c is a combina- tion of a and h, and is the well known tackle block. Form d consists of four sets of form a, and the action of the cords com- pels the piece Q to maintain a parallel motion. This is practi- cally applied in Bergner's drawing board. In like manner four pulleys of form b may be combined as in form e. This is the old parallel motion for spinning mules, also used as a squarin.g device for traveling cranes.* The use of pulleys and bearings is to reduce friction at the point of bending, and roller bearings, as Fig. 566, are also used, but when the bending surface is well rounded the pulleys may be dispensed with. Fig. 7S5, at a, b, c, shows such arrangements, the action being the same as before, but with greater friction. The arrangement at rf is a six-fold cord, aod in sail making eye- lets are often used in similar manner, as at e. The friction is great in all such devices, because the cord presses hard upon the point of curvature ; its magnitude increases rapidly with the I1 Fig. 7S5. friction, which is to be considered as a particular case of sliding friction, plays a very important part in constructions, involving tension organs, and will be more fully considered hereafter. In Fig. 786 is shown Riggenbach's rope haulage system for use on inclined trackways, or so-called "ramps." In this arrangement, the descending car is loaded at the top of the ramp with sufficient water to enable it to draw up the ascending car by the power of its descent. The speed can be controlled by the descending weight, and also a weight acting upon wheels gearing into a rack ir.f 2. ll'iiiding. — The most important forms of winding gear are Fig. 787. shown in Fig. 7S7. At a is the common windlass, also known as a winding barrel or drum, extensively used in many forms of hoisting machinery ; i is a drum for spiral winding of a flat belt, the belt being wound upon itself, and side discs being provided as guides for the belt ; c is a spirally grooved drum for winding chain ; (/ is a conical drum, with spiral groove, used in clocks (there called a fusee), also for hoisting machinery with heavy rope ; and T^ Fig. 793. rope. The travel on the drum causes the angle of the rope be- tween W and L to vary, and to prevent this the device shown in Fig. 793 has been used by Riggenbach on the cable incline at Lucerne ; two forms being given. The guide sheaves are trav- ersed by screw motion, the rope being led off iu a plane parallel to the axis of the drum, and in the second form two guide sheaves are used for a double cable. 3. Driving. — This application of tension organs is most ex- tensive. The principal forms are given in Fig. 794. The cap- Fig. 794. Stan a consists of a hollowed drum, the surface of which is composed of numerous ribs and the rope is given several turns about it. The axial travel produced by the spiral path causes the rope to climb upon the larger diameter, from which it is easily forced back to the middle from time to time by hand. At 4 is a sprocket wheel with Y-shaped sprockets, much used in many modifications ; c is Fowler's drum, a form of grip drum which grasps the rope automatical]}', and which is discussed more fully hereafter. At d is a simple rope pulley, partly en- circled by a tension organ under such load as will produce suf- ficient friction to prevent slippage ; t' is a chain wheel with teeth to prevent the slipping of the links. In all iive cases the wheel may drive or be driven by the tension organ. By combination of driving and guiding devices many useful transmissions are made. Several forms are given in Fig. 795 : a is David's Capstan, with conical windlass, with a ring-shaped guide roller which constantly leads the rope from its travel toward the base of the cone. At 5 is a counter-sheave device, the main sheave 7" being made with two grooves and the counter-sheave set at a corre- sponding angle. This gives increased rope contact, which may be multiplied still more by increasing the number of grooves. The counter-sheave may also form tlie second pullej' of the combination, as at c\ this is used in rope transmission devices. Driving tension devices are often capable of being used to 174 THE CONSTRUCTOR. greater advantage than winding devices, since the direction of motion need not be changed and is not limited. For these reasons driving combinations are frequently used instead of drums, as in hoisting machinery. Chain sheaves with pockets to receive the ordinary oval link chain are here applied (see I -75). or with flat link chain the sheave engages with the pins of the chain. Fig. 796. Other driving sj'stems are shown in Fig. 796. At a is a double lift with water counter-weight. T'is a pulley for round or flat belt; the weights O, and Q., are nearly equal, so that a semi- circle of contact is .sufficient to j^revent slipping at Zj and the friction of contact is sufBcient. A reference to the Riggenbach cable road gear, Fig. 7S6, will show a similarity to this device, but in Fig. 7S6 a braking de- vice is provided at Oj and 0,_ to protect from accident in case of breakage of the cable. A similar device, using strains at T, has been applied by Green for operating the sluices of the Great Western Canal. A*" b js shown the grip-wheel, which has also been used for cab'^e driving. In this form the loads may be quite unequal without apprehension of a deep groove cutting in the drum. Koppen's system is shown at c\ this uses a round or flat belt with tightening pulleys L, L, so that sufficient fric- tion can be obtained for any given difference of loads ; this avoids the unequal action upon the heavily-loeded side of the belt, by producing tension upon the otherwise slack side, and might be applied with advantage to the driving system of Fig. 795 r, requiiing but a single tightening pulley, and subjecting the rope to only one kind of bending. At d is shown a bucket gear, which combines driving and guiding, and is much used for conveying in mills, grain eleva- tors, etc. If the difference in weight between the sides is slight, the tension organ may be a leather belt, but for heavy service a chain is used. This device has been in use from a very early period for well buckets, and in modern times in mud dredging machines. At e is the Weston differential pulley block, a modi- fication of the Chinese windlass. Fig. Jgoa. T^ and Tjare chain sheaves fast to each other, producing a differential action due to their difference in diameter, the whole forming a substitute for the older tackle block gear. Fig. 7S4 ,:. The form shown at Fig. 796 d demands further consideration, as it can be given a series of most important applications. ,If the tension organ is made a band and placed in a horizon- tal or nearly horizontal position, it can be used to convey finely divided material simply poured upon its upper surface. Exam- ples of this are found in the transportation of grain, also in the movement of paper pulp, and many other such purposes ; also for convej'ing straw upon chain lattice conveyors, etc. In all of these cases the material is kept on the conveyor simply by gravity. This condition may be avoided and the capacity ex- tended by using a pair of belts, the material to be conveyed being carried between them. .\ very important application of this principle is found in power printing presses, the delivery of the sheets being effected by S3'stems of tapes and bands with great speed and accuracy. Band conveyors are also used in needle machinery and in match making machines, and many similar situations. An important application of driving gear is found in the con- struction of inclined haulage sj'stems for mine ramps. Fig. 797. In Fig. 797 is shown the inclined cable system of the Rhenish Railway. The driving wind T L, operated by a steam engine, works the descending cable on one track and the ascending cable on the other. At L' is a tension pulley to take up the slack cable and maintain a proper tension. The trains Q^ and ft, are connected to the brake cars i?, and i?,, which are extra heavy and control the rate of descent by proper brakes. In the anthracite coal region of Pennsylvania haulage systems are in extensive use for the transportation of coal, some being constructed with iron bands, but most of them using ropes. The arrangement will be understood from the diagrams in Fig. 798 and 799, which, with the accompanying data, have been ob- tained by the author from their engineer and constructor, the late Mr. W. Lorenz. The car in which the coal is hauled is not attached directly to the cable, but is driven by a dummy D, which is permanently connected to the cable. This dummy runs on a narrow gauge track, and at the foot of the incline the narrow track continues on, so that the dummy /? can go below the main track, as shown in Fig. 799, and ou the ascent it can thus be drawn up behind Fig. 799- the cars which have been placed by the shifting locomotive The steam engine and drawing gear is placed at the head of the incline, as shown in Fig. 798, and the cable is led, as shown by the arrows, that it passes twice over the driving wheel T, each time covering about J4 of its circumference. The dummy cars /), and D., are connected by a secondary cable passing over the tension sheave L' ; this secondary cable maintains the proper tension on the main cable, whether the load is at the head or foot of the incline, or on the horizontal. The tension car. is given a play of 75 feet to provide for the necessary variation. A different form of cable haulage is found in the .system in use between Liittich and Ans, and sketched in Fig. 800.* Fig. Soo. In this case the incline is divided into two sections, which make an angle with each other as shown ou the plan, and be- tween which is a short level space. On this space is placed the steam engine and driving wheels Ti, 7",, T^, T^, each wheel having its own engine, two engines always diiving and two being at rest ; L' are the tension sheaves. In this, as in the preceding case, it wiil be noticed that the cable runs continuously in the same direction, differing in this respect froin the previously described winding and reversing system. The cable is brought to rest in transferring the cars from one plane to the other in order that this may be readily and convenientl}' done, but should this be avoided by running them over the connection, by momentum or otherwise, the ad- vantage and usefulness of the system would be greatl)' increased. This has been done in the cable tramways of Halliday and Eppelsheimer, first used in San Francisco, .and shown in dia- ^ See Weber'.s " Poitfolio John CockeriU." THE CONSTRUCTOR. 175 gram in Fig. Soi. This is most effectively applied on the trolley streets of the city, for which it is admirablj' adapted. Fig. Soi. The endless cable runs in an iron way between and beneath the tracks, the power being at T and guide sheaves at L, L, with suitable driving and tension mechanism. The cars grasp the cable by a gripping device through a narrow slot in the trackway. The guide sheaves at the bases of the inclines and sides of the curves permit the grip to pass, and when the foot of the hill at the end of the road is reached, the grip is released and the car transferred to the other track as at M\, and in sim- ilar manner shifted at the other end, IV^_. The weight of the cars on the down grades counterbalances those on the up grades, and so the motive power has only to overcome the fric- tional resistance. The cable system of tramways has been ex- tended to Chicago and manj' other American cities ; also in London, and a cable system of canal towage has been projected by Schmick for the proposed Strasburg-Germersheim Canal. When it is practicable to propel the cars by a .suspended cable from overhead a different arrangement maj' be adopted. Fig. S02. Fig. S02 is a diagram of a system operated by a suspended chain. The descending cars Q^ are loaded and the ascending ones C?., are empt)^ and the speed is controlled by a brake at B. If the action is in the reverse direction, a driving engine must be applied at T. A similar arrangement is much used in coal mines which are entered by inclines. The chain is attached to a fork on the cars. The system of overhead cable tramway, which has been brought to a high state of efficiency by Bleichert, is based on the same principle as the preceding, but for much lighter loads. The sj'stem consists of a cable tramway in which a stationary cable is substituted for the trackway. The running cable is commonly called the pulling rope, and runs underneath the stationary rope. The cars consist of a combination of grooved sheaves, from which the bucket or other receptacle is suspended by curved arms. The stationary cable is supported upon round poles, and the arrangement of the stations is shown in the dia- grams of Figs. So3« and 803^. Fig. 80317. The stationary cable connects with the suspended tramway at Sr S// and Sifll S'V. At So is the anchor of the stationary counter-sheave, as in Fig. 795^, to obtain increased tractive power. Fig. 804 shows a plan view of a double system. pa fj SL. C/^// i,/,/jJu//u/^,„J//J^-^. Fig. %oib cable, with a tension weight at Z,. The driving sheave is at TJ driven by connections to the engine at A', and at L' is the ten- sion device for the pulling cable. If the service is heavy the cable is carried twice around the driving sheave 7, using a Fig. S04. At A'l is the motive power for systems / and //, and at K^ the motor for system ///. The driving sheaves are at 7, the coun- ter-sheaves at G, and the tension sheaves at L' . The supporting columns for the stationary cable must be stiff, and often quite high. a. b. Fig. 805 shows the forms used b}' Bleichert, a being used up to 24 feet high, b for heights between 24 and So feet.* In Fig. 806 is shown a combination of driving and guiding systems in which the guiding and driving sheaves are combined upon the car Q, and the tension organ is fastened at two points So So on the path of the car Q. SflSo Fig. S06. The motive power is on the car and operates the shea-^ e 7. In the form shown at a, a Fowler grip sheave is used at T, this form being suitable for a rope system, while the form shown ai b is better adapted to be used with chain. The system shown in Fig. 806 5 is also adapted for hauling boats, and has been used by Harturch for operating the railway ferry across the Rhine at Rhinehausen. The ferry baat in this case is guided by a stationary cable securely anchored, as in Fig. S07, the anchorage being up the stream, and the force of T ^ T Fig. 807. the current keeping the cables taut. The equilibrium of these forces enables this to act in the same manner as the stationary cable of the Bleichert .sy.stem, the difference only being that the load, instead of being suspended from the cable, exerts a lateral stress. The driving cable is similar to Fig. S06 b, and is beneath the surface of the water. If we imagine, in the combination of Fig. 806, that the traveling vehicle O may be longer than the distance S^ S^y which is the full length of the tension organ, the principle w'ill not be altered, but the action will be modified, since the rela- tions of the traveling vehicle and the tension organ are now inverted. The ends of the tension , organ can now be joined * On the tramway at Liker-Vashegy, poles of 140 feet high are used. 1/6 777^ CONSTRUCTOR. together, or in other words it can be made endless, and if heavy enough, its weight can be caused to produce enough friction on the bed of the stream to furnish the necessary resistance. This is the construction of Heuberger's chain propeller, Pig. So8, as improved by Zede L L ^^. L L r L ^— o^^-o — o — o — ^ Fig. 808. T is the driving sheave for the chain, L, L, L are guide sheaves, Zj is a movable sheave to take up a portion of the slack chain when passing into shallow water. The system is made double, being placed on each side of the boat, and each side is driven independently, so that sharp curves can be turned.* If, in the case of a tension organ driven by a revolving pulley, there is not sufficient tension given, the friction becomes insutS- cient to overcome the resistance of the load ; if the necessary tension is externally supplied and removed periodically, a con- tinuousl}' revolving piilley can be caused to produce a lifting and dropping action of a given load. This plan has been a'^opted in some forms of drop-hammers, of which P'ig. S09 is the arrangement. 7" is a pul- ley running continuously in the direction of the arrow, O is the drop weight, //a handle by which the operator applies and releases the tension which causes the pulley to drive or slip. The applications of running tension organs which have been thus far considered, are +nose in^which the device has been used either to lift weights or to transport the same from place to place. One of the most important applications, how- ever, is that of transmitting rotative motion from pulley to pulley, an operation which can be almost indefinitely repeated. This combination includes all numerous forms of belt, rope and chain transmission. Fig. Sio. The necessary tension for this purpose is sustained by the journals and bearings of the pulleys, also being modified by supporting or by tightening pulleys. The two portions of the tension organ are distinguished as the tight and slack sides respectively," and many modifications of this form of transmission are discussed more fully hereafter, (see Chap. XX to XXII ). There is one application, however, which is appropriately discussed in this place, namely, that in which rotative trans- mission between pulleys upon stationary axes is combined with pulleys upon a movable member, thvis enabling motion to be transmitted from a stationary source to a moving body, Fig. Sli. Fig. S09. Fig. Sio. Fig. 811. In case a, one of the driven pulleys is mounted upon a car- riage, saddle, trolley, or the like, and may be shifted in posi- tion upon its wa3's or track ; the tension is sustained by the three guide sheaves. Applications of this form, using belting, are used upon planing machines by Sellers, Ducommun & Du- bied and others. With rope driving gear it is used to operate the spindles upon the carriage of the self-acting mule, also for operating traveling cranes by Ramsbottom, by Tangye, and by Towne ; being combined by the latter with the squaring device * The following data of performance are given by Zede : Capacity, 500 tons ; length over all, 230 ft. ; breadth, 21^$ ft. ; depth. 6^ ft ; midship draught, 3iJ4 in. The chains were of cast iron, weighing 275 pounds per yard" two en- gines cf 150 I. H. P. gave a speed of 3.72 miles (!) per hour. as shown in Fig. 784 f, and effecting all the functiotis of the crane, including bridge and trolley travel, as well as the hoist- ing and lowering of the load. The form of Fig. '&\\b diifers from a in that both sides of the belt or rope are used to transmit power. .The stationary pulleys 7"j and T^ here drive the movable pulleys T,_ and T^. These driven axes can be utilized in various manners, as, for example, to operate a windlass device for the propulsion of the carriage O ; an example of which is found in Agudio's cable locomotive, f In this device the pulle}'S T^ and T^ drove a friction train which operated a drum connected with a stationary cable as in Fig. 806. A more recent device is shown in a modification of Fig. 811 a, as shown in Fig. 812.J W^vLj/L^ ii ^»ff^^M( 4h if^M,^ f. // - ^-.^ Fig. 812. This construction, which is in use at the Soperga-Rampe at Turin, consists of a double rack, placed between the rails as shown at b, which also shows the gearing by which car is driven. The motive power is placed at the foot of the incline at T, G, the 500-horse power engine running continuously in one direction. The cable is carried upon the overhead guide sheaves Z, and passes around the pulley Z,, and through the sheave system T T' of the locomotive, and is supported also on guide sheaves under the track, a tension pulley being placed at L'. The velocity of the driving cable is four times that of the cars, and the descent is effected by gravity alone under control of a brake. During the descent the bevel gears on the shaft of the driving pulley are released by friction clutches at K, thus rendering the car independent of the cable. The foregoing condensed description is nevertheless fully sufficient to indicate the extreme service of which tension organs arc capable in machine design. No less than seven sj'Stems have been shown for railway use, and four for boats. This is the more significant since it will be remembered that cable pro- pulsion had been abandoned for railway use, but yet appears to now be revived with increasing success. Our division into Guiding, Winding, and Driving systems enables different devices to be placed in corresponding classes. There yet remains to be considered the co-existing action of many of the devices, such as pulleys, windlasses, cranes, etc., in which a negative motion may be given to the tension organ b}^ the descent of the load O under the action of gravity. \ This action can be fully determined by reversing the previously considered movement for the backward motion. In the com- mon belt transmission. Fig. Sio, the action is reversible, as is also the case with the simple pulley. Fig. 794 fi'- The case is different, however, with the rope tackle Fig. 784c and the differential block Fig. 796(', which are therefore here considered in the more general form of Fig. S13. If in these forms the cord Z is pulled in either direction the lower sheave will be also moved up or down pro- portionally. At the pre- sent time systems using endless cords are under ? I [,1 f i f I consideration, but fre- VJ.V 2 \,^ „ VJ^^ 2 qusutly choice is to be /^l"^ /l^ made as to which por- i I X ( J, « tion is best used. It will be seen that the system of Fig. 806, which is made with both ends of the cable secured, can also be considered as a portion of an endless system similar to Fig. 80S, and other endless systems are found in Fig. 7S4 d and e ; also Fig. 813 b, which differs from a only in the run- ning of the rope, the united ends being marked by a cross. If L Fig I See Thomas Agudio. Memoire sur la Locomotive funiculaire, Turin, 1S63. tSee Bulletin de la Soc. d'Kucouragenient, Vol. XVI., 1869, p. 48. I Kinematic force closure. First discussed in the Author's Theoretical Kinematics, p. 575. THE CONSTRUCTOR. 177 If we bring the applications of Figs. S06 and Si i into a general form in which the path of travel shall return upon itself, we have Fig. S14 a. If the guide sheaves are removed and the 1 1 Fig. S14. cord crossed, the simpler form of Fig. Si4 b is obtained. The rotation of the pulley 7", causes travel around the stationary pulley T^. The old form of Agudio's cable locomotive may be represented by a similar diagram, Fig. S14 c. The shaded pulley T.^ is held stationary, while the concentric pulley Zj, is assumed to revolve ; this causes the system to revolve in a circular path, the whole forming a differential or epicyclic system. Fiualh' it may be remarked that in electric transmission systems a similar analogy exists to the above combinations of tension organs of wire and cable in various forms. I 263. Technologicai, Applications of Tension Organs. In addition to the preceding applications of tension organs, they are also used in numerous forms of machine tools, ;'. e., as organs for the alteration of the form of bodies. A straight blade of steel furnished with teeth forms the well- known frame or gang saw used in numerous wood-working machines. When made without teeth, and used with sand and water, it becomes a stone-cutting saw, or in the form of a wire charged with oil and emery or diamond dust, a saw for the hardest materials, in which case a high tension must be given to the wire to prevent lateral displacement. The saw blade may be given a vibrating motion in a device such as Fig. 7S9 b for use as a scroll saw. In all these cases a reciprocating motion is used. Tension organs are also used as running members for sawing, the form of Fig. Sio becoming the well-known band saw. Very fine band saws have been made, and also saws of wire, these having been used as long ago as 1S77 by the writer, suggested by the saws used for precious stones. An ingenious form of wire saw has been made by Zervas for cutting blocks of lava or stone from the original bed, as shown in the diagram Fig. 815. Two small shafts are sunk in the stone, and the guide pulleys inserted as shown, the endless wire being fed down by the screws. The cutting is effected by using water and sand, and the cord is formed of three twisted wires, although more re- cently a single smooth wire, with twisted one wound above it, has been used, the outside diameters being %" to -f/'- A patent was taken out in Germany b}' Paulin Gay in 18S2 for an apparatus for cutting a block of stone into slabs by the use of a number of wire saws. Polishing belts are another example of tension organs used as tools, the flat side of the belt being used, impregnated with polishing material. Such belts, used in the nickel-plating establishment of Neumann, Schwartz & Weil, at Freiberg in Breisgau, are operated at a speed of over 6500 feet per minute. Tension organs are of frequent use in many details of spin- ning machinery, acting both for guiding and winding ; also in numerous other forms of textile machinery. Chains are especially useful for dredging machinery, working in wet or dry material, also for handling coal. In musical instruments we find tension organs of definite dimension and stress, as sound producing machines. §264. Cord Friction. When a tension organ which is loaded at both ends is passed over a curved surface, there is produced between the tension organ and the surface a very considerable sliding friction. Since this friction will first be mathematically considered in connection with the subject of cords, it will be given the gen- eral name of cord friction. The curved surface over which the cord is passed is the pulley, and the motioii of the cord takes place in the plane of the pulley. If the 'ension T on the driving side of the cord is to overcome the cord friction F, as well as the tension i of the driven side, we have for the value of the friction, F=s T— i. It is dependent upon the magni- tude of the angle of contact a and upon the coefficient of fric- tion/, but is independent of the radius 7? of the pulley ; it is also dependent upon the influence of centrifugal force. For these conditions we have : r=;?tf/»(i-z) (237) (23S) In these e is the base of the natural system of logarithms := 2.71S2S, and .:: = 12 gS V being the velocity of the tension organ in feet per second, S the stress in its cross section, y the weight of a cubic inch of the material, and g the acceleration of gravity = 32.2. Example. -In the capstan shown in Fig. 794rt,let/= 0.21, a = 6 tt- = 3 con- volutions, z = o. We then have / a = 0.21 X 6 X 3. 14 = 3-958, say 4, and F = t {2.718^ — A = t {54-6 — = 53 6 t. This shows the friction npon ti;e capstan drum to be nearly 54 times the puU upon the free end of the cord. The influence of centrifugal force becomes important at high speeds, and when the tension organ is under small stress. For hemp or cotton rope, or for leather belting, we may take y = 0.035, and for wire rope about nine times so great. The value of 5 in the formula ; is properly con- sidered a function of ; stant value for the arc 1 for the values of i — z and we may therefore assume a con- and thus calculate the following table TABLE. 5-. Value of Coetlicient i — z for Centrifugal Force. s. I'elacity of Rope in Feet per Second, Rope. 20 40 60 80 100 Rope. 400 lbs. 0.987 0.948 0.882 0.791 0.674 3,600 lbs 600 " 0.991 0.965 0.922 0.861 0-783 5,400 " 800 " 0.993 0.974 0.941 0.896 0.837 7,200 " 1000 " 0.995 0.980 0-9.S3 0.916 0.870 9,000 " 1200 " 0.996 0.982 0.961 0.930 0.S92 10,800 " 1400 " 0.996 O.9S5 0.966 0.940 0.907 12,600 " This table serves both for hemp and for wire rope by taking the ninefold value of.? in the right hand column for wire rope. It should be observed that the velocities are in feet per second. It will be seen that for high speeds a high stress in the tension organ is necessary in order to oppose the action of the centri- fugal force. In order to simplify practical calculations we may substitute for the e-xponent/a (1 — z^ in each case the form/'' a ; that is, instead of using the actual coefficient of friction _/, taking an- other oney, which is equal to (i — z)f. If it is a transmission system, as Fig. Sio, which is under consideration, the friction of the cord, belt, chain, etc., must at least equal the transmitted force P, hence also must the stress be that of a cord friction ^ /", which gives for a minimum value of T: T efo- p — i (239) whence ^ ^ p = e/'a Both or' these values are absolute numbers. The ratio (240} 7 "P indicates the amount of stress which must be given to the ten- 178 THE CONSTRUCTOR. sion organ, and hence may be called the stress modulus, and is T designated as t. The ratio — , we may, in like manner, call the modulus of cord friction, and indicate as /; for both are given in the following table. A series of values Moduli for Coid Friction and Stress. Exaiiipie — Krc o^ contact = tt ; coefficient of friction^= oi6, velocity z' = 80 feet. The teiisi on orgran is a leatlier belt under stress of 400 lbs. per square iuch. We have from the first table i — r = 0.791, hencey' a = 0.791 X 0.16 tt = 3 976, or nearly o 4. From the second table this gives p = 1.49 and t = 3.03, that is' over three tin-es the above stress on the belt would be required to overcome the frictional resistance. If z/ = 20 ft., the value i — 2- = 0.987, and /' a = 0.496 or about o 5, and the modulus of stress t = 2.54. In order to make these relations more apparent, they are shown graphically in the diagram, Fig. 816, in which the 'scale upon the upper horizontal linegives the values for both moduli, wliile the vertical scale on the left gives corresponding values 01 the product y"'' n. The superficial pressure/ of the tension organ upon the cir- cumference of the pulley increases as the belt or cord passes from the slack to the tight side. It is equal to ~-^^, in which b' is the breadth of the surface of contact of the belt. Now for any cross section q, the force Q = q S. Hence we have : / Q S b' 1< (^41) from which it will be seen that the pressure p can easily be kept with moderate limits. Special applications of this formula, aud of the diagram, Fig. S16, will be given hereafter. 'i 265. RoPE.s OF Organic Fibres. Hemp Rope. — The form in most general use is a round hemp rope twisted of three strands. This is twisted "loose" or "tight," according as it is desired to be more or less flexible. The cross section of a three-strand round rope, in which rf is the diameter of a single strand, is 3 — ('', whence ' Testing Machine at the Watertown ArseTial. Trans. Am. Soc. Mech. Eng'rs, Vol. V. t The Prussian rule requires S = -^ K, which gives about 28,000, and R — 37s 5, which gives i" — 38,000, hence the security is only about 2^, or less than given above. Prschibram has used with best results, S = 23,000, 5=27,000; also 5" = 22,750, s = 36,000, but finds that a value i 27.000 to 28,000 lbs. is better for the preservation of the rope. (See j; 26S.I In considering the question of pullev diameter, the ratio to the diameter 5 of the wire should be taken, not tha't to the diameter d oi the rope. R t If -J- is made so small that S + J is greater than the elastic limit, the rope will receive a permanent set. This, however, is uot always dan- gerous. In Fig. 822 the curvature i . i may produce a stress upon the concave side of the wires which, ■when added to 5, may not exceed the elastic limit. If. however, a reverse cnr\'ature be given, as at 3.3, there may result a set, as 3' . 3', and too frequent repetition of this reversal may become dan- gerous. This is shown in the case of hoisting drums, such as Fig. 792 c, in which the rope ll\ L«, ■which is subjected to reverse bend- ing, has been found to last only about ^^ as long as the rope U\ Zj. fi .'^mong important suspension ■bridges are those built by Roebling in America, notably the Niagara, snd llie.East River bridges. I 2 3' 3 Fig. 822. ? 267. Weight of Wire Rope axd its Infeuence. A rope of parallel iron or steel wires, exclusive of any bands, will weigh, per foot, 0.28 ( 12 — i S'' ) , in which i is the num- ■•)• ber of wires and i the diameter of each wire. For twisteri rope, the twist and the hemp core increases this value from ijs to 1% as much, or an average of !](, times. This gives for the running weight per foot Go 1.92 — z (!-' = 3.07 i 6- (247) This is also true for flat ropes, the value of the coefficient for cable ropes being increased as above from i Vs to IJ4, usually about lib times. For deep mine hoists the weight Go exercises a marked influence upon the section of the rope. If Z /t is the length in feet of the vertical hanging rope carrj-ing a load P sX. Its end we have : P-{- L Go =^ S — / c!-, whence for ordinary- round wire rope : P=S- /,S^(.-3.92^) (248) Example i. — Let the depth of shaft L = 1640 ft. Wire rope of steel, A' = 170,000, i-= 28,000, P= 44oi lbs., and i = 36. — X 440= ^ 0.0056 which gives I 28,000 X 36 (i — 0.229) : 0.075. \i L ^ O we get 6- =^ 0.0034, and 5 = 0.058. The above discussion enables us to determine the length Lt of rope which ■would produce by its own weight the stress S in the uppermost cross section : ■■ 0.25 5 (249) This may be called the load-length for the stress 5. Should the shaft reach a depth equal to the load-length, no weight could be suspended to the rope without exceeding the permissible stress ir. If S is equal to the modulus of rupture, the rope would be broken by its own weight. This rupture-length may be designated bv Lz, and is = 0.25 K (250 Example 2. — For round wire rope of uniform cross section the rupture- length L= is as below for the corresponding strength : K 56,000 80,000 85,000 142,000 I.Z 14,000 20,000 21,250 35,5°o 1 70,000 42,500 213,000 23,250 256 000 64,000 For very deep shafts it has been found advantageous to make the rope a body of uniform resistance, which would make both load-length and rupture-length unlimited. The formnlse for this purpose have been already given in § 4. The taper to the rope may be given in two different ways. Either a constant diameter (J of ■wire, and varying number i, may be used ; or a constant number i, and variable diameter 6. If the smaller diameter of wire = t'o. or the minimum number of wires = io, we have for any depth .i- : log T- or log --- = 0.4342945 ;' --;. lo Oo .0 In this )' is the coeflicient of weight which, for round rope, we have found to be = 3.92. Substituting this value we get : log ^— or log --^ =^ 1.68 -- lo Oo ' o (2SI) £.xa>nj>!c 3.— If the value of ^ be takeu as 28,000, we have for the following depths : X = 1000 1500 2000 2500 3000 3600 — 0.036 0.054 0.0714 0.0S9 0.107 0.121 -7— 1. 115 1.123 i-3^S 1.4.11 1.512 1.597 lo r— 1.072 I.IIO 1.148 1. 187 1.230 I-263 60 These values will serve to approximate the intermediate cases Q [1 In the Prschibram mines taper ropes are in practical use. The rope ia the Adalbert shaft is as follows; j°= 3850 lbs., 01 which 2200 is useful load, J^ = 74.S", and the rope is madf in 7 sections of six part strands and eight hemp strands. THE CONSTRUCTOR. i8i The great weight of the twisting rope has led to the use of a double lift, each half of the rope assisting to counterbalance the other half, or another plan is to use a conical drum, to equalize the power.* The spiral winding of flat ropes also serves to equalize the leverage of the drum, and by a judicious selection of drum diameter, this may be very successfully done. Flat ropes are little used in France, but are common in Bel- gium, and their use is iucreasiug in England and America.f Ropes of copper wire are used for lightning conductors, and these are also made of iron wire rope with a core of copper. I26S. Stiffness of Ropes. The resistance of stiffness of ropes must be considered both in hoisting and in driving ropes. The measure of this resistance is the force required to move a rope hanging over a very easy running pulley, both ends of the rope bearing the given load Q. It will be observed that the winding-up side of the rope does not hangas closely to the pulley as does the other side, and that the lever arm of the two sides is constantly changing. Ej'tel- wein's formula gives for the stiffness ^ of a hemp rope of diam- ter f/: S = .5^G (252) in which, when R and d are given in inches, (5 = 0.463. Cou- lomb gives the very inconvenient formula .S - — ^^ ■ Weisbach gives, from ver\- limited data, for wire rope : 6 S= 1.07S + 0.093 ~j^ (253) Exaiitple I. — Given a hemp rope 1" diameter, wilh a load of CGo lbs., bent ■over a pulley 4" radius, from Eytelwein's formula we have : S80 S = 0.463 = 101.8 lbs. 4 ^'hich seems very high. Coulomb's formula gives 66 lbs. Ej:ajnple 2. — \ wire rope, composed of 36 wires, each 0.039" diameter, with Q. load of 550 lbs., is bent over .1 pulley 44 inches diameter. From Weisbach's formula we get : S = 1.078 -I- 0.093 = 3.403 lbs. 22 The utility of these formulas is doubtful, and a fuller investi- gation of the subject is much to be desired. It will be seen from formula (253) that for wire rope the value of R should be taken still greater than already considered for bending stresses (formula 246) ; this subject is also discussed in Chapter XXI. The above rules are deficient in that they do not take into account the kind of mechanical work absorbed by the stiffness of ropes. The angle embraced by the rope is, in the investiga- tions of Amontons, Navier, Poncelet and Morris, assumed to be constant, while in practice it is constantly changing, and exerts a. very material influence upon the result. The author's consideration of the subject is here given : Referring to Fig. S23, it will be seen that the fibres or wires on the concave side of the rope which passes over a pullej^ i?, are compressed, produc- ing a reduction in the form of the convex side, the compression originating with the load Q, being trans- mitted along the whole length of the twisted strands. The bent position of the rope can no longer retain its original sec- tion, of diameter d, but its volume must be the same as that Fig. 023. gf ^ corresponding length of the straight portion. The alteration in cross-section The details are as follows : Depth jr. Dia. ofwirea. Weight Go, S. 3936 0.1043 ^-52 23,2'o 32S0 0.09S4 1.36 22,gio 2624 0.0925 1. 19 22,820 1968 0.0S66 i.oc6 22,860 1312 0.0807 0.912 23,130 656 0.074S 0.785 23.690 The twisting of the rope was commenced at the small end, and the diam- •eter of wires increased every 5 meters (16.4 ft) after ihe first 200 metres (656 it). These ropes are very satisfactory, and last 3 to 4 years. * Conical drums are used in the American anthracite coal mines. + See Dwelshauvers-Dery in Cuyper's Revue des Mines, 1S74 ; also F. Krane in Zeitschrift der Berg u. Hutteuwesen, 1864. s. s-)-i. 27,260 50,470 25,710 48,620 24,170 46,990 22,640 45,500 21,090 44,220 19.550 43,240 may be of two kinds ; first ; uniform compression ; second, when this has reached its limit, a flattening of cross section. Both deformations are observed in practice. Ropes which are very flexible are loosely twisted, and therefore readily com- pressed as they pass over puUej'S. The general compression due to the tension r>f the load in the straight portion causes the twisted strams to pre -s firmly together towards the axis, so that a heavily loaded rope is very harnd breadth 3,2 d. The Magde- burg-Bodenbacher chain is very strono d being given - to 16 I's'', the links being proportioned as at b. Flat link chains have been used by Neustadt, made of multi- ple plates (see \ 94). The plates are made of the best quality and the pins made to project a little, and riveted over cold. Chains of this sort are also used for driving where heavy resis- tances are overcome, as in wire drawing, and in some fpinuiug machinery. I 272. C.-^LCUI^-iTIOXS FOR Ch.^INS. The chains which are made at the best establishments are always thoroughly tested, ever}- link being subjected to a stress closely within the limit of elasticity, or in some cases, slightly exceeding the elastic limit. A few links, usually three, are taken at frecjuent intervals every few weeks, and broken in the testing machine. The usual proof-load is such as to give the following stresses : 5= 20,000 lbs. per sq. in. for open link chain. 5'^ 25,000 " '• " " stay " The tests of chain for the German navy give for S: 17,000 lbs. test of elasticit}-, 1 r ,■ , ,„' „„ .. 1,- t J- < t r tor open hnks ; 19,000 highest test, J ^ ' 25,600 " proof load, | 38,400 " breaking load for - for staj'ed links. three links, J For hoisting chain the elongation should be considered, and the metal should show an elongation before rupture of upwards of 20 per cent.j The permissible working stress per square inch section in Germany \ is ; 9,000 lbs. I ^,000 lbs. 11 For open link chain : For slay link chain From these we get for the proper total load P: For open links, P^ 14,000 d' \ For stay links, /"= 21,000 (/- ] (254) Flat link chains are subjected to the heaviest stress at the por- tion which is in engagement with the toothed chain-wheel. (See Fig. S37.) For this reason there should be not less than five link pins in gear with the Vv'heel at any time. If we assume that the tooth pressure is in arithmetic progression as i : 2 : 3 : 4 : 5 the pressure on the body of the last pin will be \A, Pi and on each journal also ' ;' P, they being impelled forward by y'z P. If we put as a maximum stress in the bolts of 17,400 pounds,* we have for the thickness of plates <', pin diameter d, and num- ber of plate i, for a given load P, the following values ; * The length of fiat links in Fig. S30 a is given as 5 -{- 2.S a, and the projec- tion of the ends as 2 4- 1.4 d. These are in millimetres, and for inches the values o 1S75 + 2.3 rf, and o.oS -I- 1.4 rf should be used. I Excellent pitch chain is made at the Guttehoffi^ungshiitte at Oberhausen ; also by Scihlieper at Iserlohn, and by Dori^mieux at St. .\rnaud, and Plinchon Havez at Guerigny, and by Hawkes Crawshay at Gateshead on T3'ne. \ .-it the Guerigny Works the required elongation is : For rods I '2" to 1" iS per cent. For rods 1" to J.2" 16 per cent. For rods -;s" 14 per cent. For rods yV 12 per cent. For rods V-i" 10 per cent. I At the Gutenhoffnunghiitte. II Henry R. Towne, Treatise on Cranes, Stamford, Conn., gives a permissi- ble stress of g.ooo to 10,000 pounds. (S = 0.0107 ^P / -1- 2 ., — d = 0.0063 —— ^ P =. 0,5s (/ -f 2) (25s) The thickness R is made the nearest convenient value, and x must be a whole even number. For the latter we may take the nearest whole number to the value given by the relation : :0.26 X_f_ / 6.29 X >' X y beiug the -weight of cubic inch of wrought irou =^ 0.27 lb., and heuce : 1[ The pitch for stay link chain in the German navy ■ but has recently been made 4 d. ;as fonnerU' 3 tf; 1 84 THE CONSTRUCTOR. Lt open Links. 4672 8672 Close Links. 4377 S127 Stay Links. 545S 8567 Chains must also be provided with hooks for attachments to the load to be raised. Ch.\in Couplings. Chains which are used for transmission of motion (so called "endless " chains) require devices for coupling, as do al.=o those constructions with which chains are to be connected, and hence we have a variety of eyes, rings, coupling links, swivels, and the like. Fig. 832. A piece which is sometimes used with anchor chains is the so-called "twin" link, Fig. 831. This may be made of cast steel, and because of limited space is formed with circular open- '.ngs. The 9rdinary coupling link is shown in Fig. 832 a. The link isof wrought iron, the bolt and pin of steel, both galvanized. The pin is shorter than the diameter of the eye, and is secured on both sides by a plug of lead. The next link is made some- what longer than the other links of the chain, so that the coupling link may be more readily introduced. This form is used for joining pieces of chain to form greater lengths. The German Admiralty anchor chain is made with stay links, in seven lengths of 25 metres (82 feet) each, joined with coupling links, two of which are swivels. A bow anchor chain is given two more lengths of chain and made of iron 3inm. (o.iiS") thicker.* The chains for the system of boat propulsion are fitted with a coupling Mnk with rounded edges, and two are used together, as in Fig. S32 b^ which shows the chain u.sed on the Elbe. This coupling might also be suitable for power transmission chain. The swivel is used to permit the chain to have a rotation about its axis of length without twisting the links together. Fig. 834. A single hook is given in Fig. 834 «, and a double hook at Fig. 834 b. The construction of such hooks demands the great- est care, and according to Glynn, more lives have been lost and damage incurred by the breakage of hooks than by any other part of a crane. The case is one of combined resistance and leads to unexpectedly great dimensions. The diameter d^ of the shank of the hook may be obtained from formula (72), so that we have for a load P: (/j = 0.02 •■y P (257) This is based upon a stress of 3500 pounds, but an angular pull may increase this five-fold. Taking d^ as the unit, we may obtain the proportions given iu the illustrations in the following manner. Let lu be the width of the opening of the hook, and h the width of the body of the hook, the thickness at the same point is made -3 //, and for a stress of 12,800 lbs. upon the metal of the hook we have : — - = 1.30 \ -^-\-^^ or = 0.026 o'l -^ /i ' 4 — VT V^ -+- (25S) The thickness at the point of the hook is made — , and hence the outside of the hook is a circle of diameter D = w -\- 1.5 h. We then have for : C.6 0.7 o.S o.g Fig. 833. The form of swivel used in the German Navy is shown iu Fig. 833 a, and at Fig. 833 b is shown the English swivel. = 1.77 1.82 1.S6 1. 91 1.95 1.99 2.03 2.0S 2.12 2.16 — :^2 = 0.035 0.036 0.037 0.038 0.039 0.040 0.041 0.042 0.042 0.043 ■^ P 1.06 1.27 1.49 1.72 1.95 2.19 2.44 2.70 2.97 3.24 4 = '3.72 4.00 4.28 4.59 4.88 5.18 5.48 5.S2 6.13 6.48 The most useful ratio is I. In wharf cranes a weight is often combined with the hook in order to facilitate the lowering of the empty chain. This is shown in the dotted lines in Fig. 834^. In the case of a double hook each portion is cal- culated for its component j°, of the entire load P. From this a special unit d^' is obtained only for the dimensions w, /i, and D. Exaviplc. — Let the load upon a hook be .1400 lbs. We have from (257) /fi = 0.02 \/ /*= 0.02 \/ 4400 = 1.326". If v/e take lu = /: we get from the above h = 1.99 X I 326 = 2.63S'', and w is the same ; wliile D = 2.63S -f- 3 957 — tA", In the case of a double hook the angle between the components is 60° ; we ^, , n 0.5 P 2200 then have P\ — cos 30^ If we make -7- = 0.866 - o.g, h = = 2540, whence d\ = 1.91 and 2v = 1.72. D >.02 \' 2540 = I.OoS = 1.92 + 2.86 = 4.5S. * The lengihs in the English Navy are 12;^ fathoms. For the upper portion we have as above twist taken out. - This twisting may be prevented by using the drum arrangement shown in Fig. 839. This "" consists of simple drums all lying in one plane driven by gear- ing so that the proper relative motion is com- pelled. Fig. 539. ? 276. Ratchet Tension Orc^ns. Tension organs may be combined with pawls, which in the case of cords are friction pawls, {\ 24S, 249), and for chains are toothed pawls, acting upon the links iti the same manner as upon ratchet wheels and ratchet racks. The establishment of Felten & GuiUeaume, at Miilheim a. Rhein, have devised a grip pawl for boat-cable driving, in which the rope is clamped to and released from a driving drum by an evolute shaped thumb clamp, the shock being reduced by a spring buff"er. Pawls for chains may be found used in connection with the heavy bow anchors of large vessels ; Bernier, of Paris, has also used such devices upon chain hoisting machinery. 1 86 THE CONSTRUCTOR. CHAPTER XX. BELTING. ?276. Self-Guiding Belting. Belt p^ulleys are indirect acting friction wheels (? 191) and the belt itself is a tension organ combining the functions of driving and guiding (? 261J. Those belts which act without requiring, the use of special guiding devices may be called self-guiding belts. This action is attained by the use of cylindrical pulleys when the edge of the prismatic belt runs in a plane at right angles to the axis of the puUej- ; or in other words, when the middle line of the advancing side of the belt lies in the plane of the middle of its pulleys. When a belt runs upon a conical pulley in a direction normal to its axis, its tendency will be to describe a conical spiral path upon the pullej', as will readily be seen upon the examination of the development of the surface of the cone, Fig. 840. The leading off angle may be made as much as 25°, which occurs when the distance between the axes is equal to twice the Fig. S41. If the pulley is made with a double cone face or a rounded face, Fig. S41, the tendency will be for the belt to run at the middle of the face even when the direction of the belt is not . exactly correct. For leather belting, with a height of the crowning or curva- ture of the face s = J^ of the width of face, the belt may devi- ate from the plane of the pulley hy iY2° (tan = four per cent ), while for cotton belting, on account of the lesser elasticity of the material, the crowning s should not exceed -fij of the face, thus reducing very materially the permissible deviation. In ordinary circumstances at least one of a pair of pulleys should be made with rounded force. ( E!..-h'i3 Fig. S43. diameter of the largest pullej'. Another rule for the minimum distance between shafts for quarter-twist belts is to make the distance never less than ^ b'jD. Fig. S4?. The simplest arrangement of self guiding belting is that for parallel axes, Fig. S42 a and b, a being for open belt and b for crossed belt, eHher arrangement being suitable to run in either direction. For inclined and intersecting axes self-guiding belts are not suitable, except in the case of inclined axes in which the trace .S 5, Fig. S43, of the intersection of the planes of the two pullej-s passes through the points at wliich the belt leaves the pulleys. The leading line then falls in the middle plane of each pulley, but the following side of the belt does not, hence such systems can only be run in one direction. The leaving points'in the figures are at a and b^. The arrangement gives an open belt when the angle ft between the planes of the pulleys = 0°, and a crossed belt when /? = 183°. In the intermediate po.sitious a partial crossing of the belt is produced. If /J = 90°," the belt is half crossed (or as commonly called, quarter twist) ; if /i = 45°, it is quarter crossed.* i 277- ' Guide Pulleys for Belting. When a belt transmission is arranged with guide pulle3-s, the proper guiding action is obtained when each guide pulley is placed at the point of departure of its plane with that of the next following puUey.f Fig. S44. In Fig. S44 examples are given of guide pulleys for parallel axes', all three pulleys l5"iug in the same plane. At a is shown a belt transmission with tightening pulley, b is a device for transmitting motion when great difference of speed is desired. In this case the guide pulley C\s as large as the driver A, and if desired may also be arranged to act as a tight- ener..!: At c is Weaver's device for similar uses.? In this case two belts are used, and the device has been used for driving circular saws. The pulleys should be fitted to run very smoothly in such devices. The cases in Fig. S45-846 have parallel axes with two guide pul]e3's. In the first case the guide pulleys are placed in planes tangent to both operating pulleys, and hence driving may occur in either direction. Usually, however, it is required to provide * The above geometrical construction is only approximate ; for an exact solution see a paper by Prof. J. B. Webb, Trans. Am. Soc. Mech. Eng'rs, Vol. IV., 1SS3, p. 165. ° ' t See also the paper of Prof. "Webb, referred to in the preceding note. i Eckert's patent (German) for driving the drum of a threshing machine g See Cooper's Use of Belting, Pliila., 1S7S, p. 171. THE CONSTRUCTOR. iS; ^^^^.. Fig. S45. Fig. 846. for motion in but one direction, in which case the second form is used as being simjjler of installation. The pulley B may be used as one of the guide pulleys, in which case it may be placed loose upon the same shaft as A, and C or D be made drivers or driven. By placing the guide pulleys between the axes of ^ and B, instead of beyond them, they will revolve in the same direc- tion, and may be made fast upon one shaft, as in Fig. S47 ; this arrangement admitting of motion in only one direction. In Fig. 84S is an arrangement for inclined axes, which is a modification of Fig. S46, as will be seen by the dotted lines. The guide pulleys run in oppo- site directions, but ma)' con- veniently be placed upon the same shaft. In Fig. S49 is shown an arrange- p o ment of quarter-twist belts with ' '' guide pulleys. One side of the belt is placed in the intersec- tion S S o'i the planes of the xvvo pulleys. From any point e Fig. S48. on 5' 5, the tangents c a and e b are drawn, and in the plane of these the guide pulley C is placed. This arrangement permits of rotation in either direction. Another arrangement for the same purpose is shown in Fig, S50. The side of the belt leading off from A is inclined towards B, the other side passing over the guide pulley C, which is in the same plane as A and 5 S. This arrangement is well adapted for driving a number of vertical spindles from one horizontal shaft.* Fig. S40. Fig. S50. Fig. S51 shows the general case for inclined axes. Two points c and r, are chosen on the line of intersection 6' ,9 of the planes of the two pulleys, and the tangents c a, c b, c^ a^, c^ b^ drawn, and in the planes of these tangents the guide pulleys Cand Ci are placed. Under these conditions the rotation may be in either direction. The arrangement shown in Fig. 85 2 occurs when the line 5 S passes through the middle of one of the puUej-s. Fig. S53. A simplification of the general case occurs when, as in Fiji. 853, the guide pulleys fall upon one and the same geometrical axis which is parallel to the axes of both transmitting pulleys. In this case the only inclination of the belt is that given to it by the guide pulleys. The rotation can be in but one direction, viz. : that shown by the arrows ; if the reverse is desired, the guide pulleys must be placed as shown in the dotted lines. If the inclination of the shafts is too great the belt will be liable to drop off when the pulleys come to rest. The use of guide pulleys involves special hangers, a practical form for which is shown in Fig. 854.I * An example is Jacob's grinding mill with 40 sets of stones : see Uhland's Praks. Masch. Koustr., 1S68, p. 83, 1869, p. 242. t Patented in Germany by the Berlin-Anhaltischen Maschinenbau-Aktien- Gesellschaft. THE CONSTRUCTOR. The vertical axis is provided with an oil hole, and is fitted "by a ball and socket bearing to the bracket D. The flange ou the lower edge of the pnlley keeps the belt from falling off the Fig. 854. Fig. 855. pulley when at rest. The form in Fig. 855 was designed by the author for the arrangement of Fig. 848, both pulleys being loose upon the wrought iron shaft. If the position of the shafts can be so chosen that the line ^ iT touches at least one of the pulleys, the very practical arrangement showu in Fig. S56 can be applied. If the distance Fig. 856. Fig. 857. ^ Cis great in comparison with the width of bel':, the pulleys Cand C[ can be placed side by side instead of over each other, Fig. S57, in which case round face pulleys should be used. C2- Wh^. 'SrSifiSSiiSUMB Fig. 85S. By the use of a fifth pulley the preceding arrangement may "be so modified that two pulleys, i?i and i?,, can be^driven from one driver, A. This is shown in Fig. 858 as applied in a spin- ning mill, in which the pulleys i?, and B.^a.re on different floors of the building, and are also provided with loose pulleys.* Fig. 859. lu the arrangement of Fig. S59 the pulley A drives two parallel shafts, one of which intersects its axis at right angles, t"he other passing beneath. Fig. 860. Another arrangement, devised by the author, is given in Fig. 860. In this case the following side of the belt is passed over an idler pulley, Cj or C.,, and a second time arouud the driver (see also Fig. 795) by which the angle of contact a is doubled, and the modulus of friction f/« (^ 264) increased. This may be called a double-acting transmission. The cross section of belt may be made j% of a single acting transmission, so that in spite of the increase of length an economy of belting is obtained. One of the guide pulleys may also be used for a tightener. These devices will also be considered in connection with rope transmission (Chapter XXI.) to which they are especially appli- cable. ? 278. Fast and Loose Pui,i,eys. Fast and loose, or tight and loose pulleys, as they are some- times called, are generally used in connection with another belt transmission in order to throw the latter in and out of action, the belt being guided by a belt shifter, which by the means of forks or finger-bars, enables the moving belt to be shifted. These shifting devices may properly be regarded as guide pulleys, and are sometimes fitted with rollers, as shown dotted in Fig. S61, at c and r^.f Fig. 861. Fig. S62. It is preferable to have the loose pulley upon the driven shaft, since the belt then can be shifted with a gradual spiral action by the shifter /^ Fig. 85i. It is best for the driving pulley to be made straight face, or if two fast pulleys are used side by side on the driving shaft, these should have very slightly rounded faces, if the belt is to be shifted promptly and readily, and for the same object the shifter should be placed as close to the driven pulleys as possible. The loose pulley should be kept thoroughlj' lubricated, and for this purpose numerous oiling devices have been made. The friction between the hub and shaft acts as a driving force upon the loose pulley, and this has been a Source of numerous accidents. This action is avoided in the arrangement in Fig. 862, in which the loose pulley is carried on a consecutive and stationary sleeve D.t A variety of mechanical belt shifting devices have been made, J the desire being to prevent the action of the belt from moving the shifter. A useful form is Zimmermann's Shilter, Fig. '863. •'^ See Fairba ., Mills and Millwork, II., London, 1863, p. 103. For the theoretical discussion of these various arrangements, see § 301. t Such rollers as especially necessary for shifting cotton belts, which are liable to catch on the shifter lingers, and even larger rollers are best in such cases. X See Berliner Verhandlun^en, iSfjg, p. 127. This has been used by the Society tor Prevention of Accidents, of JNIiilhonse. g See Berliner Verhandluugen, 1S68, p. 171, Rittershaus, Belt Shifters. THE CONSTRUCTOR. 189 The shifter bar F, to which the fork G can be clamped at any desired point, is operated by the lever H, which turns upon an and also a sin ji --= R — A*,, which gives : axis at/, forming a "dead" ratchet mechanism. The similarity to the ratchet devices of Figs. 754 and 755 will be observed. The movement of the bar is effected by connection at K or A'j. Fig. S64. Fig. 864 shows a shifter for quarter-twist belt. In this form, devised by the author, the guide pulley, which is required to support the belt, also serves as a shifter to move the belt to and from the belt pulley B, and loose pulley B^. If these pulleys are given greater width than that of the belt, as shown on the right, a vertical adjustment can be given to the upright shaft ; a condition sometimes required in grinding mills and similcr machines. I 279- Cone Poi,i,eys. When a number of pulleys are placed side by side in order to enable varied speeds to be obtained with belt transmission, and are united together in one member, we obtain what is called a cone pulley, such pulley being used in pairs. This construction involves the problem of determining the proper radii for the various pulleys, so that the same belt shall serve for all the changes, i. e., so that the length of the belt shall be the same for each pair of pulleys in the set. The problem may be solved as follows : !■#..' a. Crossed Belts, Fig. S65. The belt makes the angle ji with the centre line of the pulleys R and R^ ; and the half length of the belt, 1= R (-^+!^) + -^i (—+ P) + « cos /3, (! being the distance from centre to centre of the pulley. We then have : '={R + R,) (y + "^" ' (R+R,Y . . (260) This value is constant when R -]- R^ is constant ; that is, when the increase to the radius of one pulley is equal to the decrease in the radius of the other. Crossed belts are seldom used for this service, however, because of the injurious friction between the rubbing parts of the belt. d. Open Belts, Fig. 866. In this case we have : l=:(R + R^)^+{R-R^)fl + a cos ft Rr- {0 sin (3 + cos /3) + ^sin/^l — {j3 sin (3 -f cos /3) sin /? I ^ 2 . • (261) This function is transcendental, but may be graphically repre- sented in the following manner. Fig. S67. In the rectangle ABB' A', with a radius A B = a, strike the quadrant B M C about the centre A. Within this arc will fall all the values of /? which can occur. For any value oT (i = C A M, draw M N perpendicular to MA and make iI/iV = the arc M C =^ a ft Drop the perpendicular M P io A C, and draw A'' O perpendicu- lar to MP. NO will then = a p sin /3. Through A^ draw Q N K parallel to A B, and we have A Q = P O + A P=a (/3 sin [i -\- cos /?). By taking successively all the values of j3 between 0° and 90° in this manner, we can determine the path of the point TV, which will be the evolute of a circle, C N D B D being equal to the length of the arc B M C ^ — a. If we 2 now draw D E parallel to B A, and take its middle point P, we a have Z? F ^ E F- and hence the proportion : £1 F : D B = — : — a = a : i^, and by similar triangles : T JiT^ — Q A = — (13 sin 13 + cos /3). This value is dependent upon — . If we prolong B F until it intersects A C prolonged, the resulting length A A' = B B' will bear to A' B' the ratio — . By then working B G = I, and drawing G H parallel to A' B', we have G PI ^ — . This TT I a length being transferred to /A' gives I T= — — — {j3 sin p TT TT -\- cos j3). We then have onl^- to use zh — sin /3 to solve the problem. Make A R ^ — , and we have the perpendicular R S= — sin ft By laj-ing this length off above and below T on G K, we obtain the points U and V^ and this finally gives / U for the radius R of the larger cone pulley and I V ^= Rj, the radius of the corresponding smaller cone pulley. By solutions for successive values of /?, we obtain the curve DUX T E, which can be used for the determination of the radii of any desired pair of pulleys, each pair of ordiuates measured from Pf I belonging to corresponding pulley on each cone. In practice it is usual to find one of the cone pulleys given and the dimensions of the other required. In this case V U may be taken as the difference R — A*,, between the radii, were the steps uniform. By taking this difference A" — A", in the dividers, and finding the equivalent ordinate U V on the curve, and then adding J' / = R^, the axis U I is found. In order to use the curve conveniently, it may also be laid off left-handed, as shown in the dotted lines D' X E'. The use of the diagram will be rendered still more convenient if we omit the unnecessary value /. This enables us to distort the curve in the direction of the abscissas to any desired extent. iqo THE CONSTRUCTOR. off toward C, the corresponding radius Xd and prolong the axial line d d' to its intersection d' with B E. Then Iiy off the given geometric ratio on C X, considering X d as i (shown in the diagram by the small circles for the ratios \, \, J, J, f), and draw rays from d' through the points of division, and these rays will intersect the curve at the correspouding points for the pulley radii Ry We then have for the radii : a I and a \' for the ratio i : 4 b 2 " b 2' " "2:4 c ?, " c z' " "3:4 dX" dX" " 4:4 e s " c i' " "5:4 e " c 0' " "6:4 Cone pullej s may also be made continuous, thus becoming conoids upon which the belt can be shifted to any point by au adjustable guide or shifter. Such conoids are used for driving the rollers in spinning machinery. Such a pair of conoids are shown in Fig. 869, the proportions having been determined by the graphical scale. The angular velocity varies in au arithmetical ratio as shown. The curve E i' A in the scale shows the limit to which the axial line may approach A E ; this dis- tance must not be less than R -\- R^=- a, from which V Y=\{AB— V U). ? 2S0. Cross Section and Capacity of Belts. A belt of rectangular cross section of width b, and thickness (5, will be subjected to a tension 7" on the tight side (see | 264), which it must be proportioned to sustain. If^'isthe permissible stress for the unit of cross section, we have, therefore. T ^ b v\;;-S:^\ .sV '■**«¥l^'^-i\':.';^^ ^N-\-s^\S*Ki»Nl^l ^<^' ^■.///^i'A'.y /■v//M/^////i,//////^/,'/'./y ri \\ \ i // * Pulleys with wrought iron arms are made in Germany bv Starck & Co., Mainz ; in Eng^land by Hudswell, Clark & Co., Leeds, these latter with arms of round bar iron. tSee an article by the writer, " Ueber das Zentrifugal-Moment," in Ber- liner Verhandlung, 1S76. p. 50. X See Am. Machinist, May 23, 1885, p. 7, Efficiency of Bei^ting. Three causes of loss exist in belt transmissions, viz. : journal friction, belt stiffness, and belt creeping. For horizontal belt- ing we have, according to formula (99) for the journal friction, expressed at the circumference of the pulley a loss E^ when 7 =2.5 P, t = 1.5 P: ^' -F - + P t 4 At^- ^1^ - -^yi^i :). (26S) in which rfandrfj are the journal diameters, and/" the coefficient of journal friction. This loss is doubtless the greatest of the three. For lack of better researches the loss of belt stiffness may be deduced from Eytelwein's formula for ropes. For the coefficient of stiffness s, for force S', which includes both pul- leys ; -^ = E' =s T-\- t P V7? + R ' y?i -.\s fi2 &"• ) (269) in which i =: 0.009 — = 0.012. The loss from creep is due to the fact that the greater stress on the driving pulley over that on the driven requires for a given volume of belt, a longer arc of contact ; for the expendi- ture of force G' for creep on both pulleys, we have for a stress 5i on the leading side of the belt : ?--■ = J ! + ■ E E^S^ (270) S, In this E is the modulus of elasticit}' of the belt, which for leather is 20,000 to 30,000 pounds. The losses from stiffness and creep are .small. Example. — l^et d and dx ^ a" \ R = R = 20", S = 0.2, _/ = 0.08, 5= 0.012^ ii = 28,440, Si = 425, we have F^ = P ; X o.oS ■ X 0.4 — 0.08 P; also S- = P (0.048 X 2) ■ = 0.004S /; 20 and Gi = /> "-' ^ ^-^ ■ = 0.0059 ^• 28,440 -t- 425 The total loss is therefore : o.oS -f- 0.0048 -j- 0.0059 = 9.1 per cent CHAPTER XXI. ROP£ TRANSMISSION. § 285. Various Kinds of Rope Tr.4.nsmission. If in the tension driving gear, shown in Fig. Sio, the rope be used only for the transmission of power we have what is called a Rope Transmission. Since the details of construction must vary, according as fibrous or wire rope is used, we may distin- guish between three kinds of rope transmission, viz. : those for Hemp, Cotton or Wire Rope, and these will be considered in this order. The oldest of all these is hemp rope transmission, but this was gradually being superseded by belting until Combes, of Belfast, revived it, about 1S60, since which time it has been extensively used for heavy transmissions. The char- acter of the material permits a wide variety of applications. The same is true of cotton rope, which is extensively used for driving spinning frames, travelling cranes and many other ma- chines, the softness and flexibility of the material giving it ad- vantages, but within limits. Wire rope transmissions, since its introduction by the brothers Hirn, at Eogelbach, in 1S50, have developed a high degree of efficiency and utility for long dis- tance transmission. As will be seen hereafter, the applications of rope transmission appear to be capable of still further ex- tension. THE CONSTRUCTOR. I9S A. HEMP ROFE TRANSMISSION. I 2S6. Specific Capacity. Cross Section of Rope. It is importaut first to determine the specific capacity for ieinp rope (§ 2S0). This is obtained from the general state- ment according to (262) : N« = A ^, 3 ^ in which 5'i is the stress on the tight side of the rope, and r the modulus of stress. The value for the co-eiScient of friction /, depends upon the form of the groove or channel in the sheave over which the rope runs. Fig. S79. If the groove is semicircular, as at b, Fig. S70, the friction is but little greater than it is upon an ordinary cylindrical pulley, as at £7 / if, however, the groove is made wedge-shaped, as at c (see wedge friction wheels | 196), the driving power is increased although the surface of contact is reduced. In determining the value of r, from formula {239) the iuflueuce of the shape of the groove can be included by using a correspoudiug co-efficient of friction /K According to the recent investigations of Leloutre and others, the value of^^for cylindrical pulleys and new hemp rope is 0.075, 'o'^ semicircular grooves, o.oSS, and for wedge grooves with an angle of 60°,/'= 0.15, which accords well with the action of the wedge, doubling the pressure, see (1S5). For y = o.oSS and a contact of a half circumference, we have y a = 0.3, and hence r = 3.S6 ; with y = 0.15, y ■ a = 0.47, and r ^ 2.67. The latter value, which is even' reduced in actual practice, may be adopted, since wedge grooves in general use. The stress is usually taken while low, and may be put at 5 ^ I 350 350 lbs., which, taking t = 2.67, gives N„ = . — - — = 0.0039 ; see (262). In practice No is found even one-half this value, and we may take as a practical rule in hemp rope trans- mission for the specific capacit}', ;. e, the horse power trans- mitted per square inch of cross section, for e^ich foot of linear velocity per minute ; No ■ 0.004 to 0.002 (271) the cross section being taken as in ^ 265, as that due to the full outside diameter of the rope. When great power is to be transmitted a number of ropes are used side by side, the pulleys being made with a corresponding number of grooves. For machine shop transmission such Topes are conveniently made about two inches in diameter, although they are inches. used as small as i}{, and as thick as 2i{ Example i. A steam engine of 60 H. P. has its power transmitted through "five ropes of 2 inches, the pulley being 11.28 feet diameter, making 45 revo- lutions perminute. This gives v = 1592 feet per minute. The cross secLiou of the rope 3.14 so. inches. Hence A'o = . 0.0024. This 5 1592 X 3-14 is taken from an existing installation.* Example 2. In the jute mills at Gera the fly wheel of the engine is grooved for 30 ropes, ol 2.36" diameter, each rope transmitting 25 H. P. ; the velocity being 3000 feet per minute. This gives a specific capacity of N^ = 3000 X 4-375 : o.ooig. Example 3. The Berlin-Anhalt ZMachine Works has design rope transmis" sions in which ropes of i 18", 1.57", 1,97" diameter transmit forces, respec- tively, of 92.4, 165 and 264 pounds. The respective cross sections of the ropes N P are 1.09, 1.^3 and 3.04 square inches. Since = we have Nr. = V 33000 P which gives in each of the three cases A'o = 0.0026. The cross section of the rim of a pulley for five ropes is shown in Fig. 880. For large steam engines the grooves are sometimes made on the fly wheel, such con- structions sometimes being very large and heavy. I The application of rope transmission in manufacturing estab- lishments simplifies the mechanism very ma- terially, since it enables the jack shaft and gear- FiG. 8S0. ing to be dispensed with. Such an arrangement is shown Fig. 88 1, in which five different lines of shafting are driven from one horizontal steam engine, sixteen hemp ropes beiug used in all. ? 287. Sources of Loss in Hemp Rope Transmission. The use of hemp rope transmission reduces many losses which e.Niist in other methods and which materially reduce the efficiency ; the principal ones which need to be considered are the resistances due to journal friction, stiffness of ropes, and creep of ropes. a. Journal Friction. — In rope transmissions from steam en- gines the journal friction is usually great, because the large fly wheel requires journals of large diameter. The usual calcula- tions can only be given by indeterminate results, because the tension of the ropes sometimes acts with the weight of the other parts, and sometimes against it. If we consider the rope tensions T and t by themselves, as acting horizontally, we have from formula (100) the friction F = — f (,7" 4- {), which reduced to its corresponding resistance IK 2 to theroje, taking r ^2 ^^, gives a loss due to one shaft — -/ + 1- we have : Ez -) \ — jy ) . If we take f =- 0.09 t and "ts, c double the result for both shafts, calling this combined loss Fz 8 X 0.09 X 4-; I which reduces to ; F,: 2.R (272) Example I. In the first of the preceding examples we have also d = 6.3 inches, and 2K = 135!^ inches, hence — — = 0.046 or a little over 4 per 2/C cent. *See Zeitsclirift d. Verein deutscher Ingenieure, Vol. XXVIII, 18S4, p. 640. -{■See Engineer, Jan., 1S84, p. 38, for such a fly wheel 15 ft. fac^, 30 ft. dia.g weighing 140 tons, to transmit 4000 H. P. by 60 ropes. J See g 300. 196 THE CONSTRUCTOR. b. Stiffness of Ropes. — If we apply Eytelwein's formula (252) we have t? = j^ ( T-\- 1) taking both pulleys into consideration, and taking r ^ 273 and introducing T -\- t, gives Q =■ A\ P- It must be considered that the ropes are usually' quite slack, and that the co-efficient stiffness 6', may be taken somewhat less than Eytelwein's value. If we take -/^ as a fair approxi- mation, the ratio of loss is 5 2 ^ \r- I = X 0.463- X4_ and calling this loss Es , we get : Es -= •33 R (273) in which d is the diameter of the rope. Example 1. In the case of the preceding example, d = 2", R = «7.75". This gives Ee = 1.33 67-75 = 0.07S or 7.S per cent. c. Creep of Ropes. — The loss through creep is more important in rope transmission than with belting (see I 2S4) and should not be neglected, although it cannot be so readily determined, owing to the division of the power among a number of ropes. It is practically impossible to insure a uniform tension upon a number of adjacent ropes, or to have them of exactly the same diameter, besides which the "working " diameters of the vari- ■ous grooves differ slightly, so that additional slippage must oc- cur.* The resulting frictional loss is estimated by some at as much as 10 per cent., when the number of ropes is 20 to 30, and it is at all times important enough to be given considera- tion. The losses from stiffness and creep should be investi gated whenever practicable, as the resulting information would be of much technical value. Assuming the loss from creep in the case previously consid- ered to be 5 per cent., we have a total resistance of 4 4- 7-8 + 5 = 16.8 per cent; which, since small values were taken in all cases, is not to be considered higher than the actual loss. This explains the numerous objections which have been raised (as in England) against the use of hemp rope transmission for very large powers (see J 301). ?2S8. Pressure and We.-vr on Hemp Rope. As already seen, the surface of contact of the rope and pulley may be one of three kinds : upon a cylindrical pulley, in a semicircular groove, or in a wedge-shaped groove (Fig. S79), and to these f'ormula (241) can be applied. In case a, we can approximate b' as equal to ^ the circumference of the rope. This gives for the superficial pressure -. whence : '-'-^dR S R For case b, we have 4' = — d, whence 2 5 i_d 2 R (274) (275) In case c, the radial pressure Q, of the rope is divided into two forces Q' acting normal to the wedge surfaces and equal i Q to -^ — n— - in which d is the angle of the groove, and taking the sin if & £. I o contact surface on each side as \ the circumference of the rope, we have S sin a d_ ~R which, for 8 = 30°, gives approximately ; P^ _ d^ S ~^ R (276) ■^ The variation in adjacent ropes :nay be shown by pvitting coloring matter on the ropes and watching its distribntion . little Even under these unfavorable conditions the superficial pres- sure is not important, on account of the small value of S ; which, as already seen, is about 350 pounds. Example. — If .S = 350 pounds, and we have for a cylindri- cal pulley / = 350 X 2 X ~ = 2S lbs. for semicircular grooves, p = lb."? and for wedge grooves, when 6 ~ 30°, / = 56 lbs. per square inch. These low pressures cause but little wear upon the rope, hence the great durability of hemp transmission ropes, some- times extending to two or three vears ot use. £. ^289. COTTON ROPE TRANSMISSION. Cotton rope is not so extensively used for purposes of trans- mission as hemp rope, although it possesses the advantages of great strength and flexibility ; the impediment to its use being its higher price. The application of cotton rope for driving spinning mule spindles, referred to [in l 265, is shown in Fig. 8S2, in which T^ is the driving pulley and Z'j the driven pulley Fig. SS2. ou the carriage. This latter pulley is on the axis of a drum 7j from which light cords drive the spindles 7 ^. At L, L, are guide pulleys. The usual diameter of rope for 7j 7T, is 0.S6'', and for large machines with many spindles two such ropes are used, the pulleys being made with double grooves, these always being of semicircular section. On the ring spinning frame cotton rope of 0.4'' diameter is used on cone pulleys of 12 steps, giving changes of speed from 3:1 to 2:3. The proportions of such pulley may be determined as shown in I 279, the grooves being semicircular. As already shown in ^ 265 cotton ropes have been used by Ramsbottom for driving traveling cranes. For this purpose ropes of i to J inch diameter are used, running at speeds of 2500 to 3000 feet per minute, a weighted idler pulley keeping the rope taut. In view of the slow movement of the load, viz. : 20 to 40 feet per minute, it is questionable whether cotton rope transmission involving such a great transformation of speed, is advantageous, f C IP-IRE ROPE TRANSMISSION, i 290. Specific Capacity. Cross Section op Rope. In considering the transmission of power by means of wire rope the points to be determined are the cross section of the rope, and the deflection of the two portions of rope due to its weight. The cross section will first be considered by determin- ing the specific capacity (See ^ 2S0). This we get from (262) N,=- S, 33000 in which S-^ is the stress in driving half of the rope, considered either in connection with the driving or the driven pulley. The modulus of friction p is taken somewhat higher than for belting, since the angle of contact a is greater, and also because the co-efficieut of friction f for pulleys fitted with diagonal leather strips (see below) is very high ; early and recent tests giving_/= 0.22 to 0.25 and higher. The first value gives f/>» = 2 2 (See Fig. Si 6), and also the stress modulus r = -.=--. 2 (See 239). This gives, in (262) if we neglect centrifugal force : I Si _ 5i 33000 ■ 2 iV„=- 66000 (277) t In some instances leather transmission ropes are used, formed of twisted thongs, these being used for light driving, as foot lathes, or light spindles. THE CONSTRUCTOR. 197 This gives high numerical values, which is also borne out in practice, since large powers are successfully transmitted with wire ropes of small diameter. It is good practice to take .S, for iron wire as high as 8500 pounds, and for steel wire up to 20,000 pounds and even higher. This gives for the specific capacity, when : 6'i = 2000, 4000, 6000, 8000, 10,000, 12,000, 14,000, 16,000, 18,000, 20,000. N^ -.— 0.03, 0.06. 0.09, 0.121, 0.151, 0.182, 0.212, o 242, 0.273, °-3°3 or approximatel}' : For Wrought Iron Wire JV^ = 0.03 to 0.121. For Steel Wire .... IV„ = 0.03 to 0.303. The cross section ^ is readily obtained, since N ^ q v N^ hence : Steel Wire. q = 66,000 V Si (27S) We then have, if i is the number of wires in the rope, a diam- eter of work : f! = 2 — S'-. The speed v, of the rope may be as high as 6000 feet, but should not exceed this velocity on ac- count of the great stress upon the rim of the cast iron pulleys. I 291. Influence of Pulley Diameter. The bending of a rope about a pulley of a radius R produces a stress in each wire equal to ■^i which, if we take both for iron and steel wire E = 28,400,000, gives : .J = 14,200,000 J? (279) The driving half of the rope is therefore subject to a tension stress, both at the point of advancing and departing contact equal' to .S, + j in each wire. It is this sum which must be considered in determining the stress upon the material, and it must not be permitted to exceed the proper limits (See ^ 266). A practical upper limit for wrought iron wire is 25,000 pounds, while for steel it may be taken- much higher ; for hard drawn steel wire of good quality as high as 50,000 or even 60,000 pounds. If we take as upper limits 25,000 lbs. for wrought iron and 50,000 lbs. for steel wire, we have for the given values of S, the following values of 5 and of -^: Wrought Iron Wire- J? 5 5 T 7n 24,885 571 1422 24,174 588 2844 22,752 625 4266 21,330 667 5688 19,908 714 7110 18,486 769 8532 17,064 833 9954 15,642 909 11,376 14,220 1000 12,798 12,798 iiii 14,220 ",376 1250 15,642 9954 1429 17,064 8532 1667 18,486 7110 2000 19,908 5688 2500 21,330 4266 3333 22,752 2844 5000 24,174 1422 10,000 5 J 1422 49,770 285 2S44 48,348 294 5688 45,504 313 8532 42,660 334 11,376 39,816 357 14,220 36.972 385 17,064 34,128 417 19,908 31,284 455 22,752 28,440 500 25,596 25,596 551 28,440 22,752 625 31,284 19,798 718 34,128 17,064 834 36,972 14,220 1000 39,816 ",376 1250 42,660 8532 1667 45,504 5688 2500 48,348 2844 5000 If a still greater value of -r- is used for any given value of S^ than in the above table, the durability of the cable will be in- creased. The minimum pulley radius for any given sum of stresses 5i + 5 is obtained when — = .2, which in the tables gives for -r- = 833 and 417 respectively, as indicated by the full-faced figures. Even in this advantageous proportion the stress due to the bending of the wire around the pulley is double that due to the tension of the driving force. Example I.— Let jV= 60 H. P. K = 2052. The material is iron wire, .Si = 8532 and the number of wires i = 36. We then have for the cross section of rope: : 66,000 ' 60 295= X 8532 : 0.16 sq. in -V o.ib 0.076" and the minimum pulley radius KsR = 833 X 0.076 = 63.3" or appoximately 10 feet diameter. In order to obtain a velocity of 2952 feet per minute this requires about 93 revolutions. If we take S\ = s = 12,798 we find : q = 66,000 - 63 2952 X 12,798 = o.ioS" fo.io whence 6 R = IIII X 0.06 = 66.6" and n = 72. The question here arises, to what extent should the effect of centrifugal force be taken into account? If the velocity zi = 100 feet per second, with a stress 6' = 9000 lbs. we have from the first table in ^ 264 the value i — ;2 ^ 0.S7, so that instead of /a we have _/s' = 0.87 /a. Ufa = 0.22 we have/'a =; 0.87 x 0.22 x 'r = 0.70, and if y =, 0.22 we have /■'a = 0.S7 X 0.22 X 'T = 0.60. These give, by reference to the second table, in | 264, for the first value, the modulus of T friction p ^ —- 2.01, and for the second, p = 1.82 and a modu- lus of stress 2 2.22, which makes the specific capacity : — as great as previously obtained. This may be com- pensated for by making the cross section of the rope i.i times that obtained by the previous calculation. For lesser velocities up to 20C0 to 30CO feet per minute the effect of centrifugal force is much less and may safely be neglected, especially in the case of steel cables, in which much greater stresses are per- missible. Example 2. — How many horse power can be transmitted by a cable of 36 wires, each 0.078'^ diameter; the velocity being 6500 feet per minute. We have iVn = —rz = 0.117 ; also q = (0,078) X 36 = 0.172 sq. in. II 66,000 4 198 THE CONSTRUCTOR. "VVe then have N = q v No = o 172 X 6500 X 0.117 = 130 H. P. For R we have r / ^ r' M.Z'^.OOO X 0078 , „ - „ from (275) K = ■ '— = 64.9" say 65 . 17,064 Hxainplc 3. — What would the horse power be if steel wire were used ? (See \ 266). ^1 = 17,000 lbs. and y^ will be twice = 0.234 whence N = 274 H. P. If we ^desire durability of the cable we may make 5 only 28,440 in- stead of 34,128, and thus obtain R = — ' '— — '■ — = 3S.9 say only 40" 28,440 When the resistance P is directly given, which is rarely the case, we have from the relation g S^ ^ r P^ taking - = 2. P 's,- (2S0) The maximum statical moment which may have to be over- come upou the driven shaft is sometimes given, as in the case of pumping machiuer}', etc. Dividing the precediug equation tiy (-79) we have 14,200,000 [5 .S" . PJ^ and since o =: i — (5-, this reduces to : 4 (5 = 0.00564 V 4- V^j/'y? (281) and if we substitute for the moment P K the quotient of effect N from formula (135) P R = 63,020 — we get ; = 0,251 V4- V S, 11 (282) Example 4. — A pressure pump operated by a crauk on a shaft driven by Tvire rope, offers a resistance of 880 pounds, at a crank arm of 14.2 inches. This gives a maximum statical moment of P R . = 14.2 X SSo = 12,496 inch lbs. If we take i = 36 wires, and Si = 8500 pounds., we have from {2S1) : ■- 0.00355 vr^^ 7,000 12,496 = 36 Af S500 This gives from the table Ji = 833 )< 0.05 = 41.65, say 42". § 292. Deflection of Wire Ropes. In order that the desired tensions T and / shall be attained in the two parts of a wire rope transmission, the deflections must be of predetermined values. The centre line of the rope will hang in a curve which lies between the catenarj' and the elastic line and which approximates closely to a parabola.* For the parameter c, of this parabola, we have for a deflection Ji, in a horizontal rope, Fig. SS3, S/i (2S3 in which a is the distance between two points of suspension ; the deflection in the driving half being called //j, in the driven ■^ The equation of the catenary is as follows a .(285) in which the tangential, vertical, and horizontal forces at a point xy may be designated as px^ ps and pc,p, being the weight for a unit of length, and 5" = v x- — c-. For the point of suspension this gives : K=p{h + c), \/=/ \/~h^~2h^, H = pc (286) in which the parameters is yet undetermined. In order to determine the latter let the equation of the curve be developed into the following series : ^(..^ :.2.3.c3 ■ • + I — ■L.2C- Since the curve is always flat in rope transmission, the quotient -'— is a proper fraction, and both series are converging. Stopping at the third mem ber as giving sufi&cient accuracy we have : x=yzC (4^) = c + half, //2, and in the stationary rope ho. This gives for the tan- gential force A' at the point of suspension : K=p 0+£) (284). X'- = X — c = ■ , which is the equation of a parabola. Fig. 883. All dimensions are to be taken in inches. For any cross sec- tion (7, we have A' = q S. and p ^ q ip \i ), in which ) is the ■weight of the rope per cubic inch, and

is not always constant, but may be taken = |. These values give p ^ \ xo.28x<7 = o.3266§', and calling the coefficient 0.3266 := li, we have : From this we get : /,= ! A±/IZC^ ". . . (,88). 2 li T 4 1//2 8 ^ ' I Since — ^ 3.061 we have, taking the negative sign. /, = i.53S-/^i.53^)-|;^ (2S9)- If we neglect the first member in the parenthesis in (287) we have for a close approximation : a'- h = 0.040S -^ (290). Example. Let a^ which we may take as the distance from centre to centre of pulley, be 262 feet, or 3144 inches; also let 5 in the driving side of the rope be 8500 pounds, and on the driven side 4250 pounds. We have from (289) K = 1.53 X S500 — J (i.S3 X S500)! — ?i|^" = 48" /'= = 1-53 X 4250 — J (1.53 X 4250)2 — -~^ = 95". The appro.xiniate formula (290) gives: h^ = 0,040s ^ = 47.45", and 8500 /io = O.O40S ^ = 94.89". 4250 The following method may be used to show the deflection h, graphically. The positive and negative signs before the radical sign in (288) indicate two values for /;, as will be seen in Fig. 8S4. The greater value is not of practical use, as it gives unstable S Uabil) equilibrium. Between the two lies a value h = V,. — , V' which is obtained when the quantitj' under the radical sign r^ (9, i.e. when 5 = — — . This we will call the "mean " deflec- V2 tion and designate by hm. This deflection is important because with it the absolute minimum stress exists in the rope (see note at the end of this section) ; and this stress, which occurs with the deflection hm, will be designated Sm, and is : Sm = li — >— (291). V2 or for the preceding value V = 0.3266, Sm = (292). 4-33 c and since hm = J'a . — tj we have for the mean deflection THE CONSTRUCTOR. 199 hm = ■ Vs Dividing (2S8) by {293) we have, after some reductions : tint K^m (293). (294). From this we obtain the following geometrical construction of Fig. 8S5. With a diameter = ^i''. describe the semi-circle 1.2.3, and join the point 3 of the quadrant 2 . 3 with 2, or i ; a 2 d then 3.2^3.1 = — ■— =: — — = the mean deflection /hu. V 2 v8 Fig. 8S5. Lay ofrthis distance perpendicular to i . 2 at 2 . 4, and on any scale (not too small) lay off from 2 to 5, the stress Sui. deter- mined from (292). From 5 lay off, on the same scale, 5 . 6, equal to the given stress S, and from 6 draw the arc 5 . 7. This gives 2. 7 =6. 5 — 6.2=6.5— n/(6 . 5)^^- (5'. 2)-, which is = 5 — "v .S^ — Sm''. If we now draw 4 . S parallel to 5 . 7 we have 2.8 2.7 A 2 . 4~ 2 . 5 -, and hence 2 . 8 = /;, The value /lo of the stationary rope is that of a parabola of a length midway between those forZ/i and //^ and is equal to : /to — — -- = 0.67/;,, + 0.2S//1 . (29s) _ It may readily be constructed graphically from the first expres- sion. It is not essential that the driving part of the rope should be the upper portion, as the lower part may drive, as in Fig. 886. The Fig. SS6. ropes will not touch, when stationary, if //, — //j -c 2J?. Owing to the fluctuations due to the action of wind, or of sudden changes of load, the minimum distance should not be too small, and is best kept greater than 20 to 24 inches. Note.— We have from (287) : rf 5 = V j d/i + }s\o — -p; ] dh I which gives for the mini- "] mum of J> ; L ^ ~ V ' ^ l^J J ■ "^"* Wi ^^ ^^^ paramet. er, or C, Wj. hence = 1 — -— or Cm = hm = — — and hence from (2S7) : llm v^g , = .,.( hm -t- Vln =.r- L-x/S + —7^ + V' n/8~ a In Fig. SS4 is shown graphically how for each value of//, the parameter c can be found, by constructing the proportion a — = — . In the figure, 2 . 5' =: 2 . 4' == //' ,• also 4' . 6' = — ; ci c 2 2 and 6' . 7' perpendicular to 6' . 5' gives at \' . 7' the parameter c'. In like manner : 2 . 5' 2.4' h'^ ; also A," . 6" = 2 and 6" . 7" perpendicular to 6" . 5" gives the parameter 4" . 7" = c'' To determine the vertex 4" of the lower parabola we have : h' + -^. = h" + -^— whence h' — h" ■- 8 \h" h' J a} h' h" d- ^= -^ — T , , „ ■ This gives A' /;/' = -3- which as shown above = hm-- If, as before, we make 2.8^ hm, and 2.5'^ //' and draw through 8 a normal to 8.5' the normal will intersect 2 . 4" at 4" which is the desired ape.>i. The lines 5 . 6, 5' . 6', 5" . 6'^ intersect each other at the middle of the half-chord of the parabola at 9. This may be used in the construction by draw'iug from 9, the line 9 . 6, 9 . 6', 9 . 6", and the correspond- ing normals give the parameter points 7, "', 7". The directrix of the parabola lies at a point distant )4 c from the vertex. For the mean parabola the directrix is Lm, midwa}' between 4 and 7, and the focus Fm is at the middle of hm, and is also the centre of the circle 5.6.7. i. In the figure is also shown another curve which indicates the values of S. The proportional value of // from formula (2S7) taken from the line 2 . 11, shows that h is in inverse proportion to the hyperbolic line 10' . 10 . 10". The ordiuates of the hyper- bola, taken from the axis of abscissas 2 . 7 gives the values of 6" for the corresponding values of //. The ordinates 4' . 10' and 4" . 10" give the equal stresses S' and S", and 4 . 10 the mini- mum stress Syn The dotted hyperbola on the upper right, gives the corresponding thrusts in a parabolic arch, and the curve in an arch corresponding to the catenary is the line of thrust. In this also we find the mean height the most economical, the lower ones being stable, and the higher in an unstable equilib- rium, dependent upon the thickness of arch ring and distribu- tion of load for their stability. 200 THE CONSTRUCTOR. \ 293- TIGHTENED DRIVING ROPES. The deflection of transmission cables often becomes incon- veniently great. In many cases, however, it is possible to reduce its amount by increasiug the tension to a greater extent than is necessary to prevent slippage. This requires the cable to be made correspondingly stronger in order to resist the in- creased tension. The modification in the preceding discussion of forces and dimensions is here given, the various terms being given the subscript j to distinguish them, {Ts, is, Ss, <5s, instead of T, t, S, i). The tension T, as shown in I 290, should not be c ^ , J. I / > r, 14,200,000 X 0.026 ^, . , 12 feet. According to (279) R = = 32.45, say 32J4 inches, 11,376 which gives Ao — Aj ^ 2 7? and the driving part of the cable must be. above. The above result shows that the centre of the pulleys must be more than R -t- h., or 24ft + 2^' 8^^" above the ground in order to clear. To reduce this height we must tighten the cable. Suppose we made the diameter of the wires = 0.04" instead of 0,024". This gives -r- 1.67, and from the table, col- o umns 4 and 6, line 11, .Si's = 0.89, 5 = 12,650, and hence we have tna = 162 — 144 = iS". 0.0408 - — -J— =162", and his — h 12,650 We also have R = 11,376 X 0.04 == 50". These values give hi. ^^^^sm//,^mm7:^.i^M }m-^m^^m^-7^^:^^M less than iP, and if this is increased by a given factor m, we have is ^ Ts — P, and also : Ts = m T= 2m P, ^ is ={2m—i)P, is 2111 — I Ts 2111 (296). In order that the stress ^i in the driving part shall not be changed we have for the stress in the driving part, instead of S, — ^, the following : 2 2m (297)- The diameter 6 of the wire, if calculated from (280) is modi- fied to /m {298). or if taken from (281) or (282), we take 5., =(i\^'^;77 (299). from which the following table has been calculated. Tightened cables are frequently applicable where moderate powers are to be transmitted. TABLE FOR TIGHTENED CABLES. Ts n ts is S„s is ^s .3/ -f--^1n m y. p t P S., Si Ts & "' 1.6 3-2 2.2 0.69 1.26 1. 17 I.? 3b 2.6 0.72 1-34 1.22 2.0 4.0 3.0 0.75 1.41 1.26 2.2 4-4 3-S 0.77 1.48 1.30 2.4 4-8 3-8 0.79 1-55 1-34 2.6 S-2 4.2 0.81 161 1.38 2.8 5.6 46 0.82 1.67 1.41 3-0 6.0 •5-0 0.S3 1-73 1.44 3-2 b.4 5-4 0.84 1.79 1.47 34 6.8 5-8 0.85 1. 84 ■•50 3-6 7.2 6.2 0.86 1.90 I-S3 3.8 7.b 6.6 . 0.S7 1-95 1.56 4.0 8.0 7.0 0.8S 2.00 1-59 4.2 8.4 74 o.SS 2.05 1.61 4.4 8.S 7.8 •0.S9 2.10 1.64 4.6 Q.2 8.2 0.89 2.14 1.66 4.8 9.6 8.6 0.90 2.19 1.69 50 1 0.0 9.0 0.90 2.24 1. 71 Example. — Given, A'= 5 . 5, ?i = 100, a = 590.4 ft. = 7086 in. It is required to cover this distance with a &ing"le stretch of cable. If we take Si = 14,220 lbs., and j- = 11,376 lbs., we have — ^ — X ■ ".376 .s ■ 5 14,220 100 0.044. If ; = 36 we have from {282) 5 = 0.251 ^ — — /Vf c.044 =0.024 inches. We then get from (290) : /n = 0.040S ~ = 144" = 12 feet, A^ = 24 feet, and 7^2 — Ai = <^ 2j? and we may therefore place the driving; side below without danger of interference. The greatest deflection occurs when the cable is at rest, and from (295) we have /los = i^q inches, and the total height for the pulley cen- tres is 149 + 50 = 199" or 16" 7". This example is shown in Fig. 887, in which the dimensions, however, are in the metric system. ? 294. SHORT SPAN CABLE TRANSMISSIONS. When the distance between pulleys is short the deflection must not be too small if good results are to be expected. To this end a small value should be taken for S^, and heuce the de- flection is first to be chosen and the corresponding value deter- mined from (287) which is readily done. For moderate powers wire rope transmission may be used in this way for short spans very sixccessfully. Example.— t,^t iV= 5 horse power, to be transmitted over a span of 65.6 ft., or 787 4 inches; the number of revolutions to be 150, and the deflection 40 inches. We have from (2S7) 5j=o.3266 (40 + - J = 645 lbs. Taking iron 40 X S , wire, and making 5 + -r = 25,600 lbs. We have .s-= 25,600— 645 = 24,955. If we make the number of wires /= 36 we have from (282) ^36 Ar 645 J50 We then have from {279) R = and V = 14,2000.000 X 00? 3 25,600 150 X 2 TT X 46 = 45-9 say 46", : 3600 feet, all of wliich values are quite practicable. ? 295. TR.iNSMISSION WITH INCLINED CABLE. A transmission at which the pullej-s are placed at different heights is called an inclined transmission, and the curve in such a case is uns}'nimetrical. For a given distance a, between the verticals through the ends of the curve, and for a difference in height //, we have for the deflections /?/^.-r, and /;/' =x,, Fig. SSS, and for the ordinates j'l and y.^ of the two branches of the curve: „2 ^ ^2 //- ~ A-i = A' = — + — — _ — and a H a , H in which the parameter c is yet unknown.* (300) (301) * Deduced as follows ; jj-- = 2 c x\^ y^ = 2 c xn, y^^ y'y=a, x» — xi=H^ whence; y-r — y-i- = 2 c {x2~ xi) = 2c H, i.e. {y.. -\-y\) {yi—yi) = 2<://andhencej'o— j^i =: 2 c , etc. THE CONSTRUCTOR. 201 For the parameter c, we have from (286) I\^p{h-\-c) or Sq = '^ q(Ji -\- c) and if we consider the lower pulley as bearing The deflection is : 39)6° I X 12,962 12,962 X 0.0025 — 98.5=67.1", and /;""= ^2' + 197=264.1, whence y\ = 1968 — 0.05X 12,962- ■■ 1319.9. The stress on the rope, instead of being exactly 8500 and 4250 pounds, will be, according to (304) : ■ 8500 -t- 0.3266 X 197 = ^564 lbs., and 4250 + 0.3266 X 197 = 4314 lbs. respectively. -ioora- f IG. SS8. S' the lighter load we have : S' = ip (c + .r,) whence f = -t ^j. Substituting the value of Xi, from (300) we obtain after reduc- tion. -^2— (302) Fig. S89. The arrangement is shown in Fig. 8S9. the vertical dimensions being three times the scale of the horizontal, and all dimensions being in metres. Example 2. — Suppose the distance a = 3936 inches, and S\ = 8500, and S The plus sign before the radical indicates thatwe have chosen the "stabil" parabola (see Fig. 8S4), and hence obtain the greater of the two values for the parameter. The parameter thus being determined, we have x\ and }\ from (300) and (301). For the upper branch of the curve the stress S" is to be de- termined at the upper pulley. We then have S" = 'Z' (i^ + A'2). Subtracting from this 5' :=i/' {c -\- x-^ we have S" = S' ^--^ {x.^ — x,)=S' -{-^ H (303) and if 1/'^ 0.3266 we get : S" = 5' + 0.3266 H (304) ExampU I —Let a = 32S felt = 3936", S' = 8500 lbs. If //= O, we have from S500 4250, as before, but the vertical distance H = 1968", or - iig Side : J- + 9S4 ^ '' ^ We then have {a) For the Driving- Side 8500 2()6 = 23361, whence \ X 23361 andjj'i = 1968 — 23361 X 0.5 ^ — 971- inches, the minus sign indicating; *>hit the apex of the parabola lies without the space between the pulleys. {b) For the Driven Side : 4250 o 3266 + s ! +0.5= <302) C = 0.3266 - + / ^ 8500 ^ - ■+ 39362 V" o 3266 ^ ^ ^ ^ 3936- ■ + 0.25 X 2 4- 0.5- 12271 = 25,95i(inches. 8 X 12271 andj'i = ig68 — 12271 X 0.5 = and the apes again lies outside. i+o 125) = 708 inches, 4167 inches. 12271, whence Fig. S90. This value in (300) gives xi= x-i- =/,, = - 3036' 8c S X 25,951 For the slack half of the rope we have 5'o = 4250, and 4250 0.3266 ■= 74.62". ■ + /(i The general arrangement is shown in Fig. S90 all d mensions being given in the metric system, and the vertical and horizontal scales being the same. The increase in the .stresses is more marked than in the previous example, on account of the increase in the value of H. We have S\ = 8500 + 0.3266 X 1968 = 9142 lbs., and S{ = 4250 + 0.3266 X 1968 = 4892 lbs. 3266' ) 3936- — y 8 whence h^ 3936" '■ 8X 12,86 We then have : = 150-5 '. Suppose now that H ~ o 05 a - [a) For the Driving^Side 8500 c= + 2 +0.05 - X 2 + 0.05: ■which is slightly greater than when H^O. 39362 _ 26,018 \f f ^6 + ^'■A '_ 3936' ' V 2 + 0.05= y 8(1+0.00125) We have also, from (300) h'l = and /^i" = /ii + 197" = 205.45". The distance .j'l then becomes: .3936 8 X 26,018 + X 0.0025- 8.45" j)'l = -'- — 005X 26,o;S = (i) For the Driven Side : 667.1, .=iSi!^+ 4/(11^+93. y__^,^^ 3-Ho.o5= + ▼ V 3 .,■ 0.052 y 8(1+0.00125) '''°'- Fig. 891. The upper limit for this form of rope transmission is that ia 202 THE CONSTRUCTOR. ■which the parts of the rope are vertical, in which case the par- ameter ^= cxi. In this arrangement the necessary tension must be obtained by the use of weights, spring, or the like. B3' using guide pulleys, a combination of horizoutal and vertical trans- missions may be made, as in Fig. S91, and the tension obtained by the deflection in the horizontal part. §296. Construction of thk Rope Curve. We have considered the curve as an ordinary parabola. Fig. S92. When the apex C, Fig. S92, has been determined, bisect the two parts J?! Cand Z?j C"of the horizontal tangent Bi D^, at Q and C, join B C^ and D C,, and these two lines will give the direction of tangents to the curve at the points of suspension B and D. Then divide C Cj into equal parts C, i, 2, 3 and C, B into the same number of equal parts C, I, II, III , and by joining these points we obtain a number of tan- gents which include the curve. The other portion C C, D, of the curve is constructed in a similar manner. When the apes of the parabola falls beyond the lower pulley, only one portion of the curve is used. ? 297. ■ Arrangement op Pulleys. When the transmission pullej'S are far apart, and not high above ground, supporting pulleys must be used for the rope. In some instances this is only necessary for the driven part of the rope, the driving part being left unsupported, as in Fig. 893. '^'W'lMr^Wt^^.twy^iWKS; /I. I ! i Fig. 893. Each portion of rope between two pulleys may be called a "stretch" of rope, so that in the above instance we have the driving part in one stretch and the driven part in two stretches. If it is necessary to support both parts it is often practicable to use half as many supporting pulleys for the driving part of the rope as for the driven part as in Fig. S94. Fig. S94. These pulleys are called guide pulleys to distinguish them from the main transmitting pulleys and their supporting struc- tures are called supporting stations. Another arrangement has beeu used by Ziegler, as shown in Fig. S95. Fig. 895. This consists of a number of shorter transmissions, using double grooved pulleys, or two single grooved pulleys at each station. In this arrangement it is advisable to make the stretches of equal length so that a single reserve cable will answer to replace any one which may give out. It is always desirable to run a transmission in a straight line, and especial care must be taken to have the successive pulleys all in the same vertical plane. If it is impracticable to run the entire distance in a straight line it is necessary to introduce angle stations. These may be constructed as in Fig. 896 a^ Fig. S96. using vertical and horizontal guide pulleys, but this requires six pulleys, three for each part of the rope. A simpler arrange- ment is shown at Fig. S96 b, two pulleys and a pair of bevel gears being used. In many cases it is desirable to take off a portion of the power at intermediate stations either by shafting or rope transmission, and this may readily be done by a variety of arrangements of gearing and shafting. It is most important that the pulle)-s both for supporting and transmission should be amply large in diameter. Many rope transmissions have worn out rapidly, simply because the diam- eter of the pulleys has been too small. The intermediate pulleys for the driving side ought to be the same size as the main driv- ing pulley in order that the total stress S-)- j (see § 291) shall not be greater in the former case than in the latter. The sup- porting pulley for the driven side may be smaller because the stress S, is smaller generally, being ji, S,, or for tightened transmissions (I 2S9) being equal to [2m — 1) 2111 Sy The smallest permissible pulleys may be determined from formula (279) and the table of ^ 291. Example i.— In an ordinary wire rope transmission let 5i = 8500, Sn = 4250, and the wire of wrought iron, 5 being = 0.06". From the table in g 291 we have for the mininiinu radius of pulley, R = 833 X 0.06 = 50" or 8^t 4" dia. and for the supporting pulleys : R.-^ = 667 X 0.06 = 40 or 6" S" dia. Exajnple^, — Let 6= 0.04. S[= 568S, 5^ = 2S44, and for iron wire we have R = 2S.56 say 30", R.2 = is"- Exajnple 3.— In a'tightened transmission let m = 3, and 6 = 0.06", Sx = S500. Sn =5i ■Jill}. \ = i 5j = 7080. R = 50" as before, and /?» = 769 X 2 X 3 o 0.06 = 46", a difference which is hardly great enough to be of practical im- portance. I 298. The Construction of Rope Pulleys. The low value of the coefficient of friction of iron on iron makes it impracticable to run the wire cables directly upon the bare metal rim of the pulley, and hence various attempts were early made to fit the groove of the pulley Ivith some soft material. After early experiments with wooden rims fitted with leather, or rubber, it was practically shown that turned iron rims fitted with leather filling placed edgewise in the bottom of the groove gave the best results. "^ l0+0,6fl sjfl.sd Fig, In Fig. S97, is shown at a, a rim for a single pulley and at b, for a double one, both being of cast iron. The proportions are given in terms of the diameter d, of the cable, and in the illus- trations the constants in the various proportions are in milli- meters. The sides of the grooves are made at an angle of ■yf with the plane of the pulley in the case of the single groove * See D. H. Ziegler, " Erfahrungs resultate iiber Betrieb und Instandhal- tung des Drahtseiltriebs." Winterthur, 1S71. THE CONSTRUCTOR. 203 pulley, but this gives au excessivel}' heavy middle rib for the double pulley, and hence the inner angles are made 15° as shown. The smallest diameter of rope for practical use is d = 0.04". The superficial pressure />, may be calculated from d (274). If, for example, i = 36, we have from (244) — - =^ S, and if J? 1000 and S = S500, we have ; 8(5 p = 2S , = 136 lbs. per sq. in. lOOOfi a pressure readily borne by the leather filling. The bottom grooves are made with a dovetail bevel in order to keep the filling from being thrown out by centrifugal force. The filling of leather may be made of pieces of old belting placed on edge and forced by driving into the dovetail groove; if new leather is used it should be softened by soaking in train oil. Rope sheaves for hoisting machinery, which are only used for guiding and supporting the rope, were formerly used with- out any filling, the rope resting on the bare metal. It is be- coming more and more the practice to use a filling in the bot- tom of the grooves of such pulleys, vulcanized rubber giving good results. Fig. 89S. The construction of the rim of Fowler's " Clamp Pulley," re- ferred to in Fig. 794 c, is shown in Fig. S9S a, the clamps being pivoted to blocks by means of bolts with anchor-shaped heads. The pressure upon the rope is the same as in the case of a wedge groove of equal angle, and the pulley as made by Fowler, has one clamp ring mounted upon a screw thread cut upon the pulley, thus enabling adjustment to be made for wear upon the clamps and for the reduction in the diameter of the rope. Fig. S9S b shows an American form of clamp pulley, somewhat sim- pler in construction than Fowler's. The clamps are pivoted on half-journals (see § 95) and the angle is not so small as in the preceding form. The arms of rope pulleys are usually made of cast iron as well as the rim, although the intermediate supporting pulleys are sometimes made with wrought iron arms, as in Fig. 901. Large pulleys, when of cast iron, are usually made in halves, for con- venience of transportation. The number of arms A, may be obtained from : 40 d (305) Cast iron arms may be either oval or cruciform in cross sec- tion, and the width of arm //, in the plane of the pulley, if pro- longed to the centre is : : 4^ ■ ~A (3°6) For arms of cruciform section, the thickness of the arms e may be made i //, and the rib thickness e' = 73 e. Arms of oval section may be made of the same proportions as for belt pulleys, the thickness being made one-half/; at all points and the width at the rim being 7-^ /;. Arms of cruciform section are usually made straight as at (7, Fig. 899, but arms of oval section are frequently made curved as at 5. To draw the curved arm make the circle O A of a radius = }2 J? and divide it into spaces for the desired number of arms. Make A E = ^<, A B, and draw O Cnormal to ^ O and (Twill be the centre for half the arm, and the other centre will be at D, the radius D £ being equal to C E. When straight arms are used the hub should be divided as in Fig. 899 a, in order to avoid injurious stresses from shrinkage in casting. The spaces are afterwards filled in with fitted pieces of iron and a ring shrunk on each side to hold all together. The proportions of hubs are the same as in \ 283. Fig. S99. The distance between journals for the intermediate and sup-" porting pulleys varies from \ R io J R. The load upon the bearings consists of the sum of the weight of the pulley and the vertical component of the various forces upon the rope, and this can best be determined graphically as shown in Fig. 900. The weight G, of the pulley is so dependent upon slight variations in the thickness and section of rim and arms that a general formula of practical value cannot be given. The follow- ing examples from practice are given : Example I. — In an executed transmission by Rieter & Co., at Oberursel, near Frankfurt a. M., the pulleys are made with twelve straight arms of oval section and are 12 ft. 3.6 in. diameter. The main driving pulleys at the end ol the transmission, with single groove, each weigh 2525 lbs. and the intermediate supporting pulleys, with double groove, each "weigh 2780 lbs. The rope is made of 36 wires, each being 0.07" diameter. Exauiple 2. — The Berlin-Anhalt Machine Works Company nmkes a line of rope pulleys with wrought iron arms as in Fig. 901, tlie weights being as follows : R = 20" 24" 2S" 32" 36" 40" 50" 60" 70" G = 176 211 24S-30S 2S1-343 316-3S7 316-506 528-570 74S g6S In these instances the weight upon the bearings is not great. The journals for these pulleys should be made long, in order to reduce the superficial pressure, and swivel bearings wit'u cast iron boxes (J/, 116) can be used, which with self-oiling devices will give good service. In many cases the journals are made of hardened steel in order to combine the greatest security with the minimum size. Example 3. — The intermediate pulleys in Example 1, give a total pressure, according to Fig. 900 h, upon the bearings, of 3036 pounds, or 1518 pounds on each journal. If we make /= 1.5 d according to the table in §91 we get for d only ii^ in. In the actual case, however, the journals are 3->4 in. diam- eter, giving a greatly reduced superficial pressure and thus insuring the most complete lubrication. In this case we have the actual length / = 4.7, whence P = . = 86 lbs. per sq. in. If, in order to use formula (80) 3-75 X 4-7 we take — - = // and make 5" = S500 as before we have : a -=Nf^^V-^- = and =4^=8''', This gives ; 1518 P='^ X8 1,9" say 2" = 95 lbs. per sq. in. 204 THE CONSTRUCTOR. which is such a low value that even half boxes, similar to those m Figs. 324- 325 could be used. Bv using hard steel bearings even this small fnctional lesistance could be reduced to J4 the amount due to the above dimensions. The pulleys for rope transmission should be most carefully balanced, as any vibration causes serious oscillation of the rope ? 299. Construction op the Puxley St.vtions. The extraordinarily high specific capacity of wire rope tran.s- mission has, as already said, caused it to be used especially for upon which the pulleys are carried. * The following are exam- ples of well designed and constructed stations. Fig. go I shows a design for an intermediate station of masonrj'. The foundation is of rough stone-work and the superstructure of brick-work. Stations similar to this are used in the transmission at Ober- ursel, referred to in the preceding section, and erected in 185S. This installation is used to trausmit 104 horse power over a dis- FiG. 901. the long-distance transmission of power. It has been found particularly adapted for the transmission of the power of natural falls of water to places where it can be utilized and has thus Fig. 903. tance of 3168 feet (966 meters) divided into eight stretches, giv- ing two terminal and seven intermediate stations. Each stretch = ^'/^ = 396 feet long ; /? = 74 inches, « = 1 14.5, z' = III II IK 11 h yiiiiiiiniiPLj_ Fig. 902. materially advanced the use of natural sources of power. In such transmissions one of the most importaut and difficult por- tions of the work consists in the construction of the stations Fig. 904. 4400 ft , (' — 0.07", i = 36. The difference in level between the two terminal stations in this case is 145 feet. The transmission of the water power from Scbaffhausen, con- * These have been fullv discussed in a work by D. H. Ziegler treating ot the installations made by Joh. Jak. Rieter, Winterthur, 1S76, and printed privately. THE CONSTRUCTOR. 205 structed by J. J. Rieter & Co. and iu operation since 1 866, is used to transmit a total of 760 horse power developed by the Falls of the Rhine. Of this 200 horse power is transmitted direct to the left bank by means of shafting ; 560 horse power is carried across the Rhine in one stretch, the distance a being 3S5 feet, using two similar ropes carrying 530 horse power. (« = i8o, j? = 88,;<' -4636 ft.) and a third single rope carrying 30 horse power (;; = 180, i'? = 35.4"). Of this power there is about 4S0 horse power transmitted over three principal stretches of37S, 332, and455feet. Thenumberof wires in the heavier cables is i =: So, the thickness of wire <5 = 0.074", 'be rope being made in 8 strands of 10 wires each. One of the intermediate pullej' sta- tions is shown iu Fig. 902, and this is an excellent example of good style in construction. In this case there are two pulleys, side by side. There is a guard shown over the pulleys, to pre- vent possible jumping of the cables out of the grooves in the pulleys, but this has been omitted in later instances as un- necessary., Messrs. Rieter & Co. have also installed a system of turbines and rope transmission at Freiburg, for the Societe generale Suisse des eaitx /orHs, of which 300 horse power is in a long- distance transmission. The power is carried in five stretches of 502 feet each, to a saw mill, the difference in level being 26S feet. Cue of the stations with two supporting pulleys is shown in Fig- 903. this one being quite high ; a similar station, No. II, is placed in a tunnel, through which the rope passes. The num- ber of wires i = 90, the diameter of wire d = 0.072", the cable being made in 10 strands of 9 wires each, /? = 88.6", n = 81, '' = 3743 ft. From this point the power is divided by an angle station and one part is delivered to the saw mill and balance transmitted to a number of minor establishments. An angle station is shown is Fig. 904, and this form is also used when a portion of the power is to be taken off. A fourth large installation of turbines and rope transmission has been executed by the firm of Rieter & Co., for the Conipagiiie giiierale de Bellegarde, at the latter place, for the utilization of the well-known Perle dii Rhone. The combined power of the Rhone and the Valserine is exerted upon five turbines of 630 horse power each, giving a total of 3150 horse power which is trans- mitted by cable to the Plateau of Bellegarde.* At Zurich, the city has utilized the power of the Limmat by means of turbines and rope transmission built by the firm of Escher, Wyss & Co. In this case the stations, which for various reasons are quite high, are made of wrought iron, as shown in Fig. 905. The entire installation develops 11 50 horse power, of which 750 horse power is used for the city water works. At St. Petersburg a rope transmission in ten stretches is used to drive the Imperial Powder Works, the power being delivered into the buildings bj' shafting from each of the ten stations. Amodificatiou of Herland's device for putting on belts, has been made by Ziegler for the purpose of putting the wire cables upon the pulleys. As shown in Fig. 906, it consists of a curved piece of angle iron, clamped tempo- rarily to the arm of the pulley in such a manner as to lead the rope into the groove of the pulley. The short radius to which the I Fig. 906. Fig. 907. rope is thus once bent does not appear to have an injurious effect. When a transmission rope is car- ried over a public or private road a guard should be used as a pro- tection in case of breakage of the rope. A simple form used by Rieter & Co. is shown in Fig. 907, and consists of a sheet iron trough about 18 inches deep and ten feet wide, carried by two stationary suspen- sion cables is H H. ? 300- Efficiency of Rope Tr.\xsmissiox. The injurious resistances iu wire rope transmission are mainly those due to journal friction and stiffness of the rope ; the slip and the atmospheric resistance of the pulley arms being in- significant, t a) Journal Friction. — We have from formula (100), F^= — fO, in which Q is the load upon the journal. For a circumferential speed c, at the journal, we have a resistance in foot pounds : Fc = fndO (307) Example I. — In the case of the transmission at Oberursel a number of ex- perimental determinations were made. For a pair of journals Q = 2948 Ihs., d = 3.75" and «= 114.6. For a coefficient of frictiony = 0.09 (experimentally determined) we have ; 37«5S 0.09 X 1 14-6 XjvTSX^JmS 1. 14 orse power. : 37,658 foot lbs. This gives for 8 iations a total loss of 8 X 1.14 = 912 horse power. The maximum power iransmitted is 104 H. P. and the minimum 40.3 H. P., so that this gives a less of about 9 per cent, of the maximum and 22 per cent, of the minimum. This shows the objection to the use of too largcjournals. *See Engineerint^^ Vol. 37, 1874. t See Leloutre. 2o6 THE CONSTRUCTOR. b) Stiffness of Rot>e.—\Jsmg Weisbach's formula (253) given in ?i 268"! 6- 1.07S + 0.093-^- (3°«) we have, calling T' the tension on the rope : & = 0.093 r- (^'1-6+ ^J • • • for the resistance in foot pounds. Example 2.— In the preceding case, v — 440° ft-, -f = 73-S", and T' = % (T'-f /) = 0.5 X 202S = 1014 lbs., whence : Su = 0.093 X 4400 [ 73 » J036S ft. lbs. This resistance comes twice at each station, and for eight stations we have a total of 2 X 3 X 10.36S = '63,888 foot lbs., or nearly 5 horse power. Adding to this the journal resistance we have a total of 9.12 + 5 = 14. 12 H. P. The direct measurements of Ziesrler gave '3-341 H. P , which is a reasonably close verification of the calculations. The total loss of efficiency is therefore : 104 .14-12 403 13.6 per cent, of the maximum. : 35 per cent, of the minimum, the lesser of these being a very excellent result. i 301. ReuIvEux's System op Rope Transmission. In the preceding sections the utility and importance of wire rope transmission has been shown. The various applications of the methods alread}' discussed exhibit much ingenuity and abil- ity on the part of the designers. At the same time there ap- pears to be a possibility of improvement, especially in the case of the transmission of large powers over long distances involv- ing a number of stretches. The Ziegler system of intermediate pulleys has given excel- lent results, but the following points may be enumerated as ob- jections ; a. The great height of the supports usually necessary because of the large size of the pulleys. b. The large base required for the supports, not only for clear- ance for the lower part of the rope, but also to resist the tension of the rope. r. The necessity of making the supports of great strength ' when gearing is to be carried. These three points are all well shown in the Zurich station, FifT- 905 • d. The resistance due to stiffness of the rope. This has usually been considered uniinportant, uutil the recent investi- gations have shown otherwise. (See the preceding section.) e. The loss of power when the rope becomes slack. /. The necessity of giving sufficient tension to the rope to in- sure satisfactory action in warm weather and consequent exces- sive tension in winter. £■. The unsightly soiling of the exterior of buildings caused by the grease from the cable defacing the wall upon which the receiving pulley is placed. /;. The necessity of making the intermediate pulleys strong enough to carry the heavy stress of the cable, thus increasing the weight and consequently the journal friction. It therefore appears advisable to devise a system which should permit the supports to be made low and light, to use a light cable under moderate tension, also to reduce the number of splices, and to place the terminal pulleys inside of the building, the pulleys being made as light as practicable. All these points have been attained to a great extent in the following system. In the lirst p?ace, the cable, whenever possible, is made in one endless length from the driving to the driven pulley, thus making the intermediate pulleys merely supports and permit- ting them to be constructed very light. It is also desirable to arrange the cable so that both parts shall be at the same height from the ground and that this height should be as moderate as possible. In Fig. 90S is shown the arrangement of the power house, the first driving pulley 7", being directly upon thf motor shaft aud lying in a horizontal plane. The driving part of the rope then passes around a sta ionary pulley Zj and is c irried off in the desired direction. The driven part of the rop passes around a pulley /,' mounted on a carriage running on • track parallel to the direction of the line of transmission • ud by means of weights a pull somewhat greater than 2/ is rought upon the carriage. This tightener pulley /-' is placed >o as to bring the driven part of the rope to the same height as the driving part. The whole arrangement may be protected under roof as shown and the rest of the building used for other purposes, but if necessary the track and carriage maj' extend out of doors. The intermediate stations may all be supporting stations meiely, unless power is to be taken off at an intermediate point. If the transmission is a normal one, not using the method of in- creased tension (see g 293) the same deflection will be obtained in both portions of the rope by making the stretches for the driven part half as long as those of the driving part, so that every other station may be provided with a double-grooved pulley, Fig. 909. "X-lji Ln' ^10! 1-9! Fig. 909. If no change in direction is necessary the cable is thus carried to the driven pulley, the two parts being separated by a distance equal to the diameter of tbe driving pulley 7",, and entering the building where the power is to be received the cable passes over guide pulleys Z,.,, L-, and around the driven pulley T.,. When the load is reduced by throwing off machinery in the manufactory, the ti,ghtener carriage is drawn toward the turbine (Fig. goS) by the driving part of the rope, since both parts give a pull of j4 ( T-+ i). A spring buffer is provided to check the motion of the carriage in that direction. A spring dynamometer may be connected with the bearing of the other pulley Zj and the tension thus measured experimentally. When the trans- mission is set in motion from a state of rest the tightener pulley L moves slowly back until the tension in the driven part of the rope becomes equal to t. Should the rope have much stretch, the carriage must have sufficient travel pro- vided, and when neces- sary the rope must be shortened. The stretch of the cable is less ju this arrangement than with intermediate driving pulleys, because it is bent less frequently around the pulleys, and the wear of the rope is much reduced for the same reason. If angle stations are needed the arrangement of Fig. 910 is used ; this requiring only two pulleys to each part of rope, instead of three, as formerly, and the use of gear wheels is avoided. If the first driving pulley is in a vertical instead of a horizontal plane, the arrangement shown in Fig. 911 « is used, this requiring one more guide pulley than before. In this case the track for the tightener carriage is inclined so that its weight is used to produce the required tension. If it is desired to place the tightenerpulley horizontal the arrangement shown in Fig. 91 1 5 is used. In the cable of the Brooklyn bridge the tightener car- riage is provided with a brake in order to check the suddenness of motion due to variations of load. A friction device similar THE CONSTRUCTOR. 207 to o Fig. 709 will serve for this purpose if the angle 6 is made somewhat greater than is given by formula (233). If it is desired to place the driven pulley T^ in the same plane as one of the parts of the main line cable, the other part must Fig. 911. be led over another angle pulley. If power is to be taken off at intermediate stations these maybe constructed as the angle sta- tions of Fig. 910. Various other forms of intermediate power stations may be used without involving the use of gearing, as shown in Fig. 912, Fig. 912. in which a is for a shaft at right angles to the cable, and b and c for inclined shafts for either direction of revolution. The very moderate force which this system brings upon the supporting pulleys permits them to be made very light. This has been difficult of accomplishment with a cast iron rim. A light wheel can be made of wrought iron, using angle iron riveted to a special shaped centre piece, as shown in Fig. 914. Fig. 913. Fig. 914. These rims are bent hy means of special rolls, and a tongue is formed in the sides of the groove to hold the leather filling in place. The arms are made of light flat iron and the hub of cast iron ; the arms either being bolted fast or cast into the hub, the latter being made in halves. Pulleys made in this manner are very light. The construction of the supports is also peculiar, as shown in Fig. 913. The two posts are made of channel iron secured to a block of stone iu the ground by means of lead run in around the holes in the stone. The whole is steadied by guy-rods, and brackets are provided so that the bearings can be reached by a ladder. In many cases these supports of iron are cheaper than those built of stone. Fig. 915. For the intermediate driving pulle5-s of cast iron, the form shown iu Fig. 915 is used. The hub is outside of both bearings, but the plane of the pulley is midway between the journals. The connection between the arms and the hub is made by means of a hemispherical shaped device, somewhat resembling the frame of an umbrella, and hence these have been called " umbrella " pulleys. This construction enables the pulley to be firmly secured and readily removed without disturbing either bearing. In Fig. 91 5 b, a modification of this form of pullej', the umbrella-shaped hub being made separately, and a straight arm pulley fitted upon it. This permits a single pattern to be used for the centres of a number of sizes of pulleys, or wrought iron pulleys may be used on cast iron hubs of this form. Instead of two journals a single longer one may be used, two forms of hangers being shown in dotted lines. The use of the umbrella pulley enables a verj- sim- ple form of support to be used, either for single or double stations. Fig. 916 «, is a single station composed of a wooden post upon which a projecting bearing is bolted, and iu which the journal of the pulley runs. At b, is a double station, the post being made of iron The dotted lines at D indicate a small roof to protect the bearings from the weather. A comparison of these forms with the older style, as for example. Fig. 903, will show that merely the use of the continuous rope and the umbrella pulley will effect a great econ- omy in construction. The umbrella pulley is also well adapted to be used for rope sheaves for hoist- ing machinery and for chain sheaves.* Fig. 916. * Various applications of the umbrella pulley will be shown hereafter. The principle is also applicable to bell pulleys. At a, is a simple counter- 208 THE CONSTRUCTOR. 300 will be a practical A coraparative example with that in illustration. Exa^nple. — The transmission at Obernrsel is made in eight equal stretches and seven stations with two pulleys each, one driving pulley and one driven. This gives 16 semi-circular wraps of the rope about the pulleys, causing a loss of 5.13 H. P. from stiffness. Bj' the adoption of the new system there would be three semicircular wraps at the power house (see Fig. 908), one on the driven pulley and two quarter.wraps ou the guide pulleys L^, /.y (see Fig. 909) There are also 11 short arcs of con tact, about 3V o^a circle each, on the supporting pulleys, which latter would be very light and on supports con- structed as already described. The combined arcs of contact make practically about 5 semi-circular wraps or /g of the resistance of the old arrangement, that isTe- 5. 13 or about 1.6 H. P. This is not too I'avorable an estimate, as we have not included the effect of the excessive tension which often occurs by the con- traction of the cable in cold weather, and which is entirely avoided by the use of the tightener pulley and carriage. The reduction of journal friction is also important, as the weight of the puUevsand the effect of the rope tension are both much less. The total weight of the pulleys will be only about 7^ that of the old system, although more pulleys would be used, and the journal diameter may be reduced to J^ of the previous value. This gives a loss of 3 X % = g of the previous value of 9.36 H. P., which is2.oS H. P. To this we must add a resistance of 0.40 H. P. for the guide pulleys which have been added in the new system, giving a total loss ofi\^2s =1.60 4- 2.0S 4-0.40= 4.08 H. P. The loss in the first instance is with the new system 4 per cent, and in the second 10 per cent., as against 13 9 and 35,9 per cent, for the old system. In this example there are no intermediate power stations, the entire amount of power less only the hurtful resistance. In considering the question of the stress in the driving part of the cable it is important to know whether the entire power is to be transmitted to the etid of the line or if a portion is to be taken off at intermediate stations. If the initial forces at successive intermediate power stations be indicated by /",, f,,, P^,, -f^, etc., the successive tensions in the cable will be reduced, and hence the deflection h should be determined for the stretches preced- ing and following each station, and the tension in the cable will vary according to the power taken off at intermediate points. The sum of all the forces P, will in every case be determined by taking the tension t, in the driven part at the first driven pulley, from the initial tension T, so that we have T — t ^=^ S P. From this equation we can deduce important results. As an illustration we can assume the entire power transmitted to be divided up among a number of intermediate stations, all being operated by one continuous cable, as shown in diagram in Fig. 917. Fig. 917. In this case the rope passes the entire round of stations T^i, 7",, T3, T^ to Tji, returning to the main power house. The rope returns to the power house at any angle with a tension I, giving T^ J, P -\- i. All stresses are regulated automatically for each stretch of the rope, as the forces vary at each station. If the work at any station is reduced or even becomes zero, the tightener carriage responds and alters the deflection so that T — t ^ "2. . P, in which i remains constant. A transmission of this kind, in which the cable makes a complete circuit of a num- ber of stations, maybe called a "ring" system. In Fig. 917, the supporting stations are indicated by small rectangles or tri- angles, according as the line is straight or makes an angle, and shaft, at (5, a simple headstock for a small lathe, and at c, is a head for a boring machine, the loose pulley running on a stationary sleeve, as already shown i;i Fig- 862. the power stations as shown are circles. At 7"g the rope passes off into an auxiliary circuit, which may be called a "ring" transmission of the second order (see § 260). The stations may all be constructed very simply. The supporting stations are made with one pulley when the line is straight, and with two at the angle stations ; the power stations can generally be made with only two pulleys, providing the necessary arc of contact a, is obtained, or three pulleys used if necessary, see Fig. 918. Fig. 91S. In many cases it is desirable to use the system for under- ground transmission, as in Fig. 919.* Fig. 919. In order to determine when an arc of contact a , of the proper magnitude has been obtained, we have, from (239), \{ P is the greatest force to be transmitted by the pulley with a ten- sion T' : P-- We will P T' call the ratio ^, which is the reciprocal of the modu- lus of stress, the modulus of transmission, and let it be repre- sented by 6, whence : ef'<^ if/'a (309) Neglecting the influence of centrifugal force, we have, from g 290, for f the values f ^= 0.22 and 0.25 to consider. Taking these we get the following values for various angles : Modulus OF TRANSMISSION e. 0( = 15° 30° 45° 60° 90° 120° 150° 180° 270° 360° 450° 540° _/= 0.22 = 0,25 0.06 0.07 O.II 0.12 0.16 o.iS 0.21 0.24 0.29 0-33 0.3S 0.41 0.44 0.48 0.50 0-54 0.65 0.69 0-75 0.79 0.86 0.81 0.88 0.87 These values are shown graphically in the following diagram, Fig. 920 : 2. 360 460 5-t a ~ , Ji^ U,b ,.i^ -r^ .2'.* J 1 ( , ^n^'"- ^^^■^ - ' ..^ r^ t!S5» ' K^ ^ \^ > f /' r y / Q. / ^ 1 1 Fig. 920. From this it will be seen that an arc of contact of 30° will per- mit the transmission of jij the power due to the tension T\ and an arc of 90° gives about 5-3. A convenient application of this principle is found in the arrangement of a " ring" transmission when a large arc of con- tact is obtained upon the first or main driving pulley by redu- * This has been done in San Francisco by Boone, using a conduit for the rope similar to a cable railway. THE CONSTRUCTOR. 20Q plication of the rope over a counter pulley, as in Fig. 795, and also shown in the case of the double-acting belt transmission in Fig. S60. B3' using a single-grooved counter pulley and double- grooved driver we get ot i 360°, so that is at least equal to 0.75. In this way the specific capacity of the rope can be materially increased, practically about iji times. If we give r = -— the value -|- in the first equation of \ 290, we have for the specific capacity of a cable transmission with a counter pulley : iV„ = - ■s-i or say N^ -- 33000 25000 24750 s„ (310) The adaptatiou of the mechanism to receive the counter pulley is usually not difficult. The adaptability of the "ring" system of transmission to use in- distributing power in manufacturing establishments is appar- ent, and for this purpose hemp rope is very suitable. This will be shown by the following example : Example r.— The transmission shown in Fig. 8Si, §286, contained 16 hemp ropes 2 inches in diameter, each having- a specific capacity Nq ~ 0.0021 and 11 = 2360 feet per minute. The cross section of each rope is 3.14 sq. ins. Hence iV = Nog v = 0.0021 )< 3.14 X 2360 = i5-57 H. P. for each rope, or 2^9 H. P. for the 16 ropes. Fig. 921. Substitutinpf the arrangement shown in Fig. 921, we take a single wire cable composed of 60 steel wires, and use a stress of 17,000 pounds in the driving side of the cable and increase the speed to 3150 feet per minute. We then have from (278) : : 66,000 - 249 17,000 X 3150 and hence the area of each wire is = 0.307 sq. in. 60 - 0.0051 sq. in., and the diameter 6 = 0.08", In the original hemp rope transmission the main driving pulley had a radius of 71 inches, and as we have increased the speed ^ times, the driving pulley must be proportionally increased, and hence the radius will be 95". This gives a stress due to bending, 12,000 lbs. nearly; see formula (279), ;9s this being not too great to 'give satisfactory results. We have, instead of a wide face puUev made with 16 grooves, a single groove pulley made with leather filling, as in Fig. S79 a, of 15 ft. 10" diameter. An important point to be considered is the stress due to the bending of the rope over the pul- leys Ti, T2, etc. These pulleys were 36" radius for the hemp rope, and hence *. 36 = 48" radius for the wire rope, or 8 feet diameter. We then have from {279) 0.08 "76~ 23,666, which added to the working stress of 17,000 lbs. gives a totaFof 40,666 pounds, which is not too high for steel wire, according to § 266. The idler pulley L, is made the same size as the driven pulleys T'g. T^, etc., and the tightened pulley Z.' can be made a little larger. The loss of efficiency will be somewhat less than in the case of hemp rope, since for wire rope there is a smaller modulus of stress t, (/. e. 2 instead of 25, see 'i 287), and the initial force P, is smaller, because of the increase in velocity and the loss from stiffness will be less. The loss from stoppage and creep should also be considered as not unimportant (see § 287). ^^^^^^^^^^^^a Fig. 92 If it is desired to use a counter pulley with theabove transmission, the ar- rangement in Fig. 922 may be adopted. In this case the counter pulley G, and tightened pulley /.', are both inclined so that the rope shall be properly guided for the double grooves m the main driving pulley. The arc contact a is in this case greater than 360°, and the specific capacity will be ij^ times greater. This will enable the cross section of the rope to be reduced to 3 the previous value, or ^ = ^. 307 ^ 0.204 sq in. If we use wires instead of 60^ we have for the cross section of each wire 0.204 36 ■ = 0.005*^ sq. in., and 6 = 0.084 i^* The diameter of the rope will be from 8 to 9 5 or %" to £", the latter when the rope is new. The conditions of this example are hardly such as to demand! the introduction of the counter pulley, but when large powers, are to be transmitted its use is most advantageous. In some instances the counter pulley may be arranged, as in Fig. 911, so as to sustain a part of the weight of the fly wheel of the en- gine, and hence materially reduce the journal friction. In many instances the power in factories may be arranged so as to use the "ring " system of transmission, and dispense with the use of the spur or bevel gearing, and some examples are here given. In Fig. 923 a is shown the usual arrangement of the trans- mission of power in a weaving establishment. ■-dl- -^ 1 Fig. 923 a. In this instance the two shafts which extend each way from A', drive the line shafting by seven pairs of bevel gears, while in some factories as many as 12 to iS pairs are used. Pig. 923 b. Fig- 9-3 b shows how a ring transmission can be used to drive the same shafting, there being seven guide pulleys and one tightener L', the guide pulleys being of the "umbrella " pat- tern, as in Fig. 915. The tension weight for the tightener is equal to 2 T'. , Fig. 923 c. Another arrangement is shown in Fig. 923 <-, this being used when the alternate shafts are to revolve in opposite directions. This permits the rope to be used double acting, as described irt ? 277 and shown in Fig. S60. Those portions of the rope marked i in Fig. 923 c, are in one plane, and those marked 2 in a second plane, giving clearance to the parts of the rope, and the rope is guided from one plane to "the other by the guide pulley Zj and tightener L'. Five of the seven driven pulleys are double acting, and hence are made double grooved. 2IO 777^ CONSTRUCTOR. Shafts which lie at right angles but in parallel planes, one above the other, are also readily' driven by use of a ring trans- mission system. Fig. 924. Fig. 925. In the preceding cases it is desired to obtain a double wrap of the rope about the driving pulley A', the arrangement in Fig. 924 may be adopted. In this case two idler pulleys G-^ and ; Fig. 932. The sheave shown in Fig. 931 is for a 25 mm. {\" nearly) chain, and is made with inserted teeth of steel, and the form of Fig. 932 is similar, and is for an iS mm. (0.7 in.) chain. In both cases the teeth are radial, and formed to recjive the chain links, being secured by jam nuts in the second case, and by nuts fitted with the Belleville elastic washers, which latter have worked well in practice. Fig. 933- THE CONSTRUCTOR. 213 In Fig. 933 is given an arrangement of chain sheave gearing, including a solid massive form of bearing, as used in many English collieries.* Here the sheave is made with eight semi- circular ridges or ribs, similar to the old form of capstan shown already in Fig. 794 a ; and both parts of the chain are carried on supporting pulleys. In many instances this arrangement is used, by widening the face of the sheave, to receive several wraps of chain, as shown in the upper right corner of Fig. 933. If we may safely assume that the ridges increase the coefficient of friction at least three times, in the preceding formulas (311) and (312), we have for the corresponding modulus of friction p' : which gives for p' = 2.5!' (315) In this case the pitch length / of the links is taken = 3.5 d, and making r = 5 ?, we get ^1 = ( 7"+ o-OS^yi, and if we put for the loss at both sheaves : P = Ek we get : Ek = 0,072 /i P+i (316) Example t. — Taking the coefficient of friction ^^ ~ o-^5 on account of the small bearing surface we have for a chain transmission on smooth sheaves with half-wrap ; p being = 1.37, as in the preceding section : 6' = }i I 2 3 4 1.58 2.50 6.25 15-63 39.06 2.72 1.67 1. 19 1.07 1.03 0-37 o.5o 0.84 0.94 0.97 £■4 = from which the security against slippage and also the specific transmitting capacitj' may be determined for any given case. Within moderate limits chain transmission may be used as a " ring " system, as for instance in driving the rollers of carding machines, also in wood pulp grinding mills a ring chain trans- mission is used for driving the feed rolls. Efficiency of Ch.^in Transmission. The loss of efficiency in a chain transmission is due to jour- nal friction, dependent upon the chain tensions 7"and f ; and upon the friction of the links in entering and leaving the sheaves. The journal friction is determined as already shown in § 300, and for high values off?, it is not great. The loss from chain friction is due to the rotation of each link about its adjoining link as an axis through an angle j3. This gives, with a coefficient of friction /i, a circumferential resisting force /^„ due to chain friction (see formula loo) /^i=/, T+ t © (4-)- 2.37 : 0.072 X 0.15 = 0.0692 or say 7 per cent. Example 2. — If the sheave is made with ridges, as in Fig. 933, we have \.i = 2.5, and hence Rk = 0.072 X 0-15 3-5 1-5 = 0.025 * The illustration is from Newcliurch colliery at Burnley. or only 2)^2 P^r cent. Example 3. — By using carefully :nade pocketed teeth and making n = 8, we have p = 12.41, whence Ek = 0.072 X 0.25 ( — - — 1 = 0.0126 \ii.4i/ or only i^ per cent., this reduction being due to the reduction in the tension on the chain, showing the importance of considering the question of chain tension in this connection. In the preceding e.'^amples the friction of the links upon each other has been considered, but not that of the links upon the sheave. This latter is a very variable quantity, being unimpor- tant with a smooth sheave, as Fig. 930 n, and sometimes becoming excessive, as shown already in Fig. S3S b, \ 275. In every case all possible care should be taken to produce as little rubbing contact as possible. I 304. INTBRMEDIATB STATIONS FOR CHAIN TR.4.NSMISSI0N. The most important applications of chain transmission are in mining work, both above and below ground ; and especially in coal mines. In this branch of work England takes the lead, followed by America, where, however, wire rope is more exten- sively applied, while in Germany the most applications are found in the Siiarbruck district. A very interesting application of endless long distance chain transmission is shown in Fig. 934, which gives two views of the ^fw.ii'/ '///M/////y', Fig. 934. (Dimensions in Metres.) 214 THE CONSTRUCTOR. Ganuow mine at Burnley in Lancashire. The driving pulley is at T, and guide pulleys at L, while at L' is a tightener pulley iung between two idlers, a construction which is frequently used. The rotation is modified in various ways in the English mines, stations similar to those of rope transmission systems being used. Fig. 935. In Fig. 935 is shown an intermediate station at 7", T.,, and also on angle station at Z. In manj' instances combinations of bevel glaring aud shafting are found in connection with chain transmission, but the examples here given are confined to the use of chain alone. Fig. 936. In Fig. 936 an intermediate station is shown at T^ T^, and a change station at T-^ T^. At 7"j, Fig. 936, the chain makes an entire wrap around the sheave, the latter being made with a wide groove, and interference of the two parts of the chain pre- vented by guide sheaves. The simple supporting stations are made with small horizontal guide sheaves, with wide grooves. The velocity of the chain varies from 200 to 500 feet per minute. ? 305. Strap Brakes. If a driven pulley is embraced by a tension organ, either belt, rope, strap or chain, the ends of which are subjected to tensions 7" and /, and also held from moving, the pulley is hindered from moving toward /, so long as the force acting to rotate it does not exceed P=r T — t. The tension organ then forms, with the pulley and stationary frame work, a friction ratchet system in which the tension organ forms the pawl. If the tension 7" be reduced until T — t <^ P, the pulley will slip in the strap, over- coming the frictional resistance due to T — /, and the motion can be made slower, if 7" and t be made great enough, so long as their difference is only slightly smaller than P. The mechan- ism then becomes a form of checking ratchet (? 253) better tnown as a friction brake, or simply as a brake. Such brakes, when made with tension organs, are called strap brakes. Strap brakes are made in various forms to suit the applica- tion. "• b. Fig. 937. («■) Clamping Brakes. — When a strap brake is to be used to act as a complete clamping brake, to check motion entirely, the tensions 7" and must be determined. These are obtained from formulas (239) and (240) or from the graphical diagram of Fig. 816. Such strap brakes are frequently made with straps of iron or steel. It is generally desirable to so arrange the parts that the motion of the pulley acts to draw the strap into closer engagement, which may be done in various wa3'S. Fig. 937 shows several such arrangements. The various parts are indicated as follows : i is the axis of the pulley ; 2, the point of application of the brake ; 3, the attach- ment of the tight side of the strap ; 4, the attachment for the slack side ; 5, the axis for the brake lever. In Fig. 937 fi, 3 and 5 are separate ; in Fig. 937 b they are combined in one, and in F'ig. 937 c both 3 and 5 are separate, but 3 and 5 are made mova- ble, and 3 and 5 are so nearly in line with 7" that a very slight effect is produced on the lever by T. In Fig. 938^ 3 aud 4 are combined, aud at the same time 3 and 5 are nearly in line with T. Fig. 93S b is the so-called. Fig. 938- "differential " brake of Napier, in which 3 and 4 are so placed that perpendiculars to the directions of T'and t are inversely proportional to those tensions, thus reducing the action of the strap upon the brake lever to a small amount. Fig. 93S c shows an arrangement adapted to permit the pulley to revolve in either direction. The angle 3.5.4 can be so chosen that the force upon the lever may be very small. For heavy hoisting machinery, the braking power required Fig. 939. makes the arrangement shown in 939 suitable. In this case the strap is filled with blocks of wood in order to obtain a higher coefficient of friction aud at 6 is shown an application of the globoid worm and worm wheel shown in Fig. 641. Example. — Required a brake for a shaft driven by a force of 2200 pounds at a lever arm of 7.875 inches. The form chosen is that of Fig. 938 a. the arc of contact a of the strap being 0.7 of the circumference. The coefficient of friction 7= o.J> the strap being lubricated. M'e then havey"c/ =0.1 X i 4 w. = 0.14 77 = 0.43. We then have from the second table of § 264, the tension modulus T - T P : 2.SS nearly, and the friction modulus p = — ~ = 1.5 (see t 7 also the diagram. Fig. 816). This gives -75- = ^ . —^ ^ ?i X 2.88 = 1.92. If we make the brake pulley with a radius of 15.75 in., the braking force at the circumference of the pulley must be — — ^ . 2200 = iioo lbs., and t = 1.02 V 15.75 IIOO = 2112 pounds, and 7= 2.88 X 1100 = 3168 pounds. If the brake is to be operated by a hand lever with a force of 44 pounds, the ratio of the length of the hand lever to lever arm 4 . 5 must be .^^ =^ 48. The strap is under a 44 tension of T = 316S pounds. If we assume a permissible stress of 5 = 14,220 lbs. and a thickness of strap 6 = o.oS" the width will be; 316 = =78", which is quite practicable. THE CONSTRUCTOR. 215 we have for the tight end, where The question of the pressure between the braking surfaces is of interest. P _ Q b' R S^ 14,220. o.oS p = 14,220 —^^rZT — 7^ ''^^* According to formula (241) -~ = 15.75 and at the slack end, since — - Fig. 940. ■ ^, > = ^ . 72 =; 48 lbs., both of which such small values that the wear must be very slight. This example shows how, in a properly arrauged construction, a great ratio of force to resistance can be obtained. In large winding engines the brake pulley can readily be cast in one with the rim of the drum gear. The method of securing the ends a b of the metal strap is shown in Fig. 940. The form at a, is secured by countersunk rivets, and that at 5, by an anchor head and a single small rivet to prevent lateral slip- page. (b) Sliding Brakes. — In using clamp brakes operated by hand for lowering heavy loads in hoisting machinery, great care must be taken, since the throwing out of the checking pawls puts the entire resistance on the brake. With this arrangement there is always more or less insecurity, the safety depending upon the handling of the lever, and serious accidents have frequently occurred. This danger can be avoided by the use of automatic sliding brakes, the following form being designed by the author, and shown in two forms in Fig. 941. The brake pulley a, is loose on the shaft, but engages with it by means of a ratchet system a' b' c' ■ The brake is subjected to a tension equal to a. b. c. JL_ 3. A^-'_a Fig. 941. the greatest braking force desired; i.e.s,o that the weight K must be raised in order to permit the load to run down. If the lever is let go, for any reason, the descent is checked. In form a, the pawls are attached to the pulley, and the ratchet wheel a' keyed to the shaft ; in b, the pawl is on a disk c' . When the load is raised the combination forms an ordinary ratchet train. A silent ratchet. Figs. 673, 674 may be used for this device. At c, is shown a pendulum counterweight, which can be adjusted so as to vary the braking power to suit various loads. Another form of sliding brake, also designed by the author, is shown in Fig. 942. In this design the strap b, is given such Fig. 942. tension t, by means 'of the screw e 7, and lever c, as to hold the load from descending ; a rubber spring being introduced at 7. If the load is to be lowered, the clamp e, is loosened, but is again tightened on ceasing. When hoisting, the tension t at 2" is readily overcome. This is in reality a form of running ratchet gear, and as shown it is made with a strap of wedge section, the angle 6 being 45°. The wedge portion is made of wood on iron at least 0.20I increased by -^ which when used to multi- sin — 2 ply the value of _/ 0( , requires a very small force to overcome the tension t. ?3o6. Chain Brakes. Chain may be used as the ten- sion organ in brakp construc- tion, in which case it is generallF lined with blocki of wood, as in Fig. -— 943. The tensions T and t, to be given to the two parts of the cha:::. are readily oV*- tained from for- mula (312). The ratio of chain pitch length I, to the pulley radius r, is increased be- cause of the use of the wooden block. When / = J.^ rand the arc of contact is less than iSo° we have : Fig. 943- •=-r=(-i)' (316) For wood on iron we may take y ^ 0.3 (see section 193). This gives : T p = -7- = I . 1' = 2.35 ; also T_ ~P'' "p — ' 1-35 = 1.74 and -g- =: T — I = 0.74, or / ^ 0.74 P. These proportions should not be strictly followed for heavy brakes such as in Fig. 939, as such should be determined for each case. ?3o7. iNTERNAi, Strap Brakes. Strap brakes maj' be used in internal pulleys, in a manner similar to the internal ratchet gear of Fig. 711, for example. The outside of the strap then acts upon the inner surface of the pulley, the strap being subjected to compression instead of ten- sion,* thus becoming a pressure organ, a subject treated more fully in the following chapter. The pressure of the internal strap brake is of the same mag- nitude as with the external brake, but in the opposite direction, so that the previouslj' determined value of p from the forces T and /, may be used. Fig. 944 shows three forms of such brakes, these being used for friction couplings, and not in hoisting machinery (see Fig. 449). Fig. 944 a, is Schurman's friction coupling, t The brake lever c, acts by means of a wedge 4, upon one end of the strap. The other end of the strap is held by a pin 3, to the member d, whic"h is to be coupled to a by means of the strap b. The lever c, is also pivoted to the mem- ber d. For the forces T and /, we may use formula (^39), and since o< is nearly = 2 -, or say = 6, we have ior f = 0.1 the valuey^ ex = 0.6, which from the table of ? 264 gives p ■^. 1.S2, and T =1 2.22, whence t := 1.22 P. The strap must be released by the action of a spring. Fig. 944 b, is Adyman's coupling, j which is made with a heavy cast iron ring. The ring b, is made in halves, b' and i", fitted with projections 4' and 4'^ which engage with an interme- diate sheave keyed on the shaft. ♦See Theoretical Kinematics, p. 167 ; p. 54S. t Zeitschrift des Vereins Dentscher Ingcnuiere, Vol. V. p. 301, X Made by Bagshaw & Sons, Batley, Yorkshire. 2l6 THE CONSTRUCTOR. The levers c' and c" have a common axis at 5, and when separated by a wedge at 6, they press upon the ends of the ring at 3' and 3". A pin at 7, keeps the levers from sliding in the direction 7 . I, as well as the ring b' b" . The coupling shown in Fig. 944 f , acts both ways, as an inter- nal and external strap brake, and is used on a shaping machine by Prentiss. * The steel strap b, is covered with leather. When the arms c' c" are drawn together it acts as an external strap on the pulley a" , and when they are forced apart it becomes an internal strap iu the pulley a' . The arms c' c" are carried on sleeves and are rotated to or from each other by a screw action. force" of the pressure organ serving to retain it within the desired limits. Canals are merely conduits of larger dimen- a 1) c CHAPTER XXIII. pressure organs considered as machine elements. Various Kinds of Pressure Org.^ns. In distinction from the various kinds of tension organs which have been considered in the four preceding chapters, there exists another group of machine elements of which the sole or principal characteristic is that they are capable onlv of resist- ing forces acting in compression. This group includes fluids, both liquid and gaseous, whether limpid or viscous, such as : Water, oil, air, steam, all pasty substances, clay, molten metals ; also granular materials, all kinds ^i grain, meal, gravel, etc. In all these materials the principal feature lies in the fact that the particles are subdivided to such an extent that they can be separated from each other by a very small force, while on the other hand they are capable of opposing more or less resistance to compression, this resistance in many instances, as, for exam- ple, in the case of water, almost equalling that of metals. These materials may be used as machine elements in a great variety of ways, and in the following discussion they will be included under the general title of Pressure Organs. Like their coun- terparts the tension organs already discussed, they are used largely for the transmission of motion in various manners, but are of still greater importance on account of the great variety of physical conditions in which they appear. \ 329. Methods of Using Pressure Org.-vns. The distinction which has been made between tension .and pressure organs enables various points of contrast and compari- son to be made as regards the methods of ulilizing them, and pressure organs may be divided in the same manner as tension organs (see \ 262) into standing and running organs. These divisions have but little practical application in this instance, but the three following subdivisions in 'i 262, viz. : Guiding, Supporting (/. c , raising or lowering), and Driving are here applicable also. We may therefore distinguish pressure organs, when considered as machine elements, into the following classes : 1. For Guiding. 2. For Supporting (including raising and lowering). 3. For Driving. These various methods of action may he used either separately or in combination, and are found in most varied forms in many machine constructions. The great variety of possible combina- tions makes it desirable for a general view of the entire subject to be taken before discussing details. I 310. Guiding by Pressure Organs. In order to use a pressure organ for guiding, /. e., to compel a more or less determinate succession of motions, it is necessary to use also two other machine elements formed of rigid mate- rials. These latter are for the purpose : (?, Of resisting the internal forces of the pressure organ and keeping it within the desired limits. b. Of connecting the pressure organ with the external forces to be received and opposed. Tubes, Conduits, Canals. — The tube a. Fig. 945, limits the toundary of the particles of the pressure organ, and retains it in the desired form and controls its direction. A bend in a tube corresponds to a pulley around which the pressure organ is bent, and thus has its direction changed. Even when no change of direction is made, the tube is necessary to oppose re- sistance to the particle of the pressure organ, and hence at every section it must offer resistance to tension as well as com- pression. Conduits, or channels, as at b, are tubes with one side left open, the force of gravity or the so-called "living Fig. 945- sions, as at e, and natural streams of water often serve the pur- pose. Driving Organs, Pistons and Cylinders. — The bodies by means of which the pressure organ is connected with the exter- nal forces and resistances with which it is intended to act mechanically may be called generically. Driving Organs, and are very varied in character. Among these are movable recep- tacles, also moving surfaces or moving conduits (as in turbines), and also moving pistons in tubes or cylinders. A piston serves to oppose the stress in the pressure organ in the direction of its motion, while the walls of the tube oppose their resistance at right angles to the direction of motion. The inclosure in which a piston acts is called, in general terms, the cjdiuder, and details of construction will be given hereafter. The principal types will here be considered briefly. A complete working contact between piston and cylinder can only be obtained when both surfaces are alike, and this is only geometrically possible with three forms of bodies ; /. e., prisma- tic bodies, bodies of rotation, and spirally formed bodies. Of these the prismatics are most useful, and among the prismatic bodies the form most extensively used is the cylinder. The fit of a piston in its cylinder, entirel}' free from leakage, is very difficult of attainment, and is rarely attempted in practice. In steam indicators the piston is very accurately fitted directly into the cylinder, but in most cases a practically satisfactory result is obtained by the use of some intermediate packing device. In many cases a soft packing of hemp or leather is used, Fig, 946. At a is shown a piston with external packing, at b an internal packing. In these cases one entire end of the cylinder is open, the piston filling the entire cylinder and acting upon the inclosed pressure organ on one side, this constituting a single-acting position. At c and d are similar double acting pistons. Pistons of the forms shown in a and b are sometimes called plungers, and the shorter inclosed pistons, as c or d, are also called piston-heads. At f is a double-acting piston used in connection with a rod and stuffing box, the rod being connected with external mechanism, and the stuffing bo.x made either with external or internal packing, as indicated at i and \' . In many instances pistons are made with openings which are fitted with valves, and hence may be called "valved" pistons, while those here shown are termed closed or solid pistons. The tightness of the packing is usually produced by the appli- cation of some external force, but in the so-called forms of self- acting packing the necessary pressure is supplied by the con- fined fluid. This is shown in the following illustrations. a b LI *See Mechanics Feb., 18 Fig. 497- Fig- 947 <^ and b, Cup packing for piston or stuffing box ; metal THE CONSTRUCTOR. 217 packing, usually for pistons, but also used in stuffing boxes. The fluid in all three cases enters behind the packing rings and tightens the joint in proportion to the increased pressure. In the class of self-acting packing may also be included the various forms of liquid packing, some of which are given in Fig. 94S. The forms at a and b are practically plungers, while a b X Fig. 948 in many cases an ordinary packing lias its tightness increased by a layer of water or oil upon the piston, as shown at c. Another variety occurs when the connection between cylinder and piston is made by means of a membrane or diaphragm, as in Fig. 949. Fig. 949. These are among the oldest forms of transmission organs, but are practically true pistons in principle and action. At a is a single diaphragm, known as the monk's pump ; b is the so- called 'bellows" form ; r is a series of flexible metal diaphragms, usually of steel, brass or copper, used for pressure gauges or other similar purposes involving but little movement. At d is the so-called " bag" pump, in which the liquid does not come in contact with either cylinder or piston, but is confined within a flexible bag. Fig. 950. Another class of pistons is that in which a tight packing is not attempted, these usually being used only for air. Fig. 950 a shows a deep piston with grooves formed in it, the fluid endea- voring to pass the piston in the opposite direction to the motion of the latter, becomes entrapped in the grooves, and before it can pass, the direction of motion is changed and this actiou reversed.* ht b\& a piston with a brush packing, used for a blowing cylinder at Sydenham. In this class of pistons we may also include floats which rise and fall with the motion of the liquid. vSuch floats are shown at c and d, the former being open and the latter closed. A solid block may also be used for this purpose, if its weight is nearly counterbalanced b\' another weight. Details of piston and cj'linder construction will be given in Chapter XXVI. The corresponding machine elements to pis- tons in tension organs will be found for ropes in Figs. S25-S26, and for chains in Figs. S31 to 834. The change of direction from compression to tension dispenses with the necessity for a cylinder. Guide Mech.\nism for Pressure Organs. The combination of a pressure organ and its accompanying guide mechanism forms a pressure transmission system. Fx- a b c d e Fig. 931. amples of such systems are given in outline in Fig. 951. At a is an arrangement for raising the load Q vertically. The plungers b and d are of the same diameter ; the pressure on b must be the same as Q, neglecting friction. The column of water is the same diameter as the plungers, and the direction is changed an angle of 120°. It is desirable that distinguishing names should be given to the various arrangements. If we compare these with the corresponding parts in tension organs, Fig. 7S4 and Fig. 7S5 a, we may properly call such an angle transmission a hydraulic pulley, or water pulley, but a still bet- ter name is the "hydraulic-lever" or "water- lever," which will be hereafter adopted. At b is shown a free water-lever. The plungers b and d are equal in diameter, the load Q is supported on two columns of water, hence, if friction is neglected, the force on each plunger will be I2 Q, the angle of change of direction is 180°. At c is a combination of case a with case b. The plungers ^1, b,, b^, are of the same diameter, and the load O is supported on these columns. These three cases correspond in principle with the similar cases a b c of Fig. 7S4. Since the three plungers ij, b.,, A), of case c all exert the same force, they may also be made to give the same result when made as shown at d, or if the three plungers are combined in one, form e is obtaiued. The latter form is well known in practice as the hydraulic press. The principle iuvolved in all these devices is the same as appears in the various pulley systems of tension organs. A comparison of Fig. 95 1 a with e shows that the same prin- ciple exists in both, and case a may be considered as a water- lever of equal arms, and case if as a lever of unequal arms. Fig. 952. The water-lever has been used in more or less complete de- vices for balancing the weight of pump rods in deep mine shafts. Fig. 952 shows Oeking's water couuterbalance.f The * See Weisbach, Vol. III., Part 2, ? 410. t Zeitschrift Deutscher Ingenieure, 18S5, p. 545. Oekin§ the device « ^ au accumulator. incorrectly call 2l8 THE CONSTRUCTOR. pump rod is carried on the two plungers d^ d.,. and its weight counterbalanced by the weighted plunger and cylinder a~b. In the Emery scales and testing machines water-levers of unequal arms are used in connection with metallic diaphragms. ■^'■^''^^-^^''^■'^■ ^-'^■'■^^yy'^yCi^^i^^ d Fig. 953- Fig- 953 shows a combination of two hydraulic levers, each of the form of Fig. 931 a. The weight Q travels in a straight line, being kept parallel by the four equal plungers bib.,bo,b^, and crossed pipe connections. This construction is similar to the cord parallel motion of Fig. 7S4 d. In all of the devices described the rigid body is guided by the motion of the pressure-organ. It must be remembered that motion is merely a relative term, and the rigid body may move through the fluid. An example of the latter is the rudder of a vessel, which acts in one plane ; or in the case of the Whitehead torpedo .several rudders are used, guiding the torpedo in any direction. I 312. Reservoirs for Pressure Organs. Reservoirs are used in connection with pressure organs in order to enable a number of applications to be operated collec- tively, and also to enable the pressure to be stored for subse- quent service, and in this respect they correspond to the variovis forms of winding drums used with tension organs, and shown in Fig. 787. The following illustrations will show the use of such reservoirs. Fig. 954 shows a tank for use with petroleum distribution, as Fig. 954. used in the American oil fields, and more recently in the oil district of Baku. The oil wells are at «,, a.^, a^, and the oil is forced to the elevated reservoir at c by pumps. From the reser- voir the oil flows to the point of shipment d, and the supply is gauged by the fltictuations of level in the tank.* The reservoirs used in connection with the water supply of cities are similar in principle. Where the configuration of the land demands it, the pipes are run in inverted siphons connect- ing intermediate reservoirs. An illustration of this arrange- ment is given in Fig. 955, which shows the waterworks system of Frank^urtam-Main designed by Schmick. Fig. 955. The highest spring is at «,, Vogelsberg, and the next at a.,, Spessart. These both deliver into the reservoir r,, c,, at Aspeu- hainerkopf. The next reservoir is at <:.„ Abtshecke, from which the water flows through b^ to the reservoir t:^ and c\, from which the city is supplied. The elevations above sea level are given in metres. The flow between the various reservoirs is controlled by suitable valves. f Small tanks are iu very general use at railway stations ; and the various ponds and mill dams used in connection with water- wheels are other examples. In many cases the water ways are large enough to serve as reservoirs also, as in the case of canals. Natural reservoirs are found iu the case of many mountain lakes, the Swiss lakes affording many numerous instances.f Such basins are also formed artificially by constructing dams- across narrow outlets, and so storing the water for use. Note- worth)' examples found in France, the basin at St. Etienne, formed by damming the river Furens, beuig over 164 feet {5'> metres) deep. J Water may also be stored in accumulators at high pressures from 20, 50, as high as 200 atmospheres, and can then be used for operating hydraulic cranes, sluice gates, drawbridges, etc. These accumulators may be considered as a form of releasing ratchet mechanism (see I 260). To this class of mechanical action also belongs the system, used in the Black Forest, by which the streams are temporarily dammed and then suddenly released in order to float the logs down with the sudden rush of the current. Iu using high pressure water transmission it is sometimes desirable to transform a portion to a lower pressure in order to operate a lower pressure mechanism, or by a reversal of the same principle, to convert a lower to a higher pres.'iure. This can be done by uieans of the apparatus devised by the author,, and shown iu Fig. 956. \\ Q f a rL M Fig. 956. This is a form of hydraulic lever of unequal leverage, but is diff'erent from those shown iu Fig. 95'- Referring to Fig. 956(7, the high pressure water is delivered at a, aud connected with the lower pressure water a^ by means of the plungers b, b„ the latter being iu one piece of two different diameters. The difference in pressure, neglecting friction, will be inversely as the areas of the two plungers, or if they are of circular section, inversely as the squares of their diameters. In this case the lower pressure then acts in the cylinder c upon the plunger d. The action of this arrangement may be considered as if the plungers b and />, were upon the same axis and rigidly con- nected, and the leverage compounded in a manner similar to that of the rope crane of Fig. 792(7; this comparison being more clearly shown by referring to Fig. 956 b, This device may also be used as a supporting hydraulic lever, similar to Fig. 951 e. If a communication is made between the two different water columns, as shown in Fig. 956 r, the pressure wdll be equalized. This gives a differential hydraulic lever similar in principle to the Chinese windlass of Fig. 790 a, or the Weston Differential Block of Fig. 796 e. , * A system of this .sort was built in 1SS7 from Balcii to Batoum on llie Black Sea. The length of line is 1005 kilometres (603 miles), 6 in. diameter, and the reservoirs 3000 feet above sea level. t A large inverted siphon is formed by the new Croton Aqueduct, which passes under the Harlem River at a depth of 150 feet below the surface of ther river, and a tunnel of loV^ feet iu diameter driven through the solid rock. See Mechanics. Nov , 1SS6, p. 24Z. I This is examined in detail in a memorial on the better utilization of water, published at Munich in 1883 by the German Society of Engineers and Architects. g For further discussion of this subject the following reft-rences may be consulted ; Jaubert de Passa, Recherches sur les arrosages chez les peuples anciens, Paris, 1S46; Ditto, Memoire sur les cours d'eau et les cananx d'arrosages des Pyrenees orientates; Nadault de Buffon, Cours d'agriculture et d'hydraulique agricole, Paris, 1853-1S5S; Ditto, Hydraulique agricole, ap plication des canaux (I'irrigation de I'ltalie septentrionale. Paris. 1S61-1S63 ;. Baird-ftniyth, Irrigation in Southern India, London, 1856 ; Dupuit, Traite de la conduite et de la distr. des eaux, Paris, 1865 ; Scott-Moncrieff, Irrigation in Southern Europe, London, 1868; Linant de Bellefonds Bey, Memoire sur les principaux travaux d'utilite publique en Egypte etc., Paris, 1873 ; Krantz, Etude sur les murs de reservoirs. Paris, 1870 ; F. Kahn, TJeber die Thalsperre der Gileppe bei Verviers, Civil ingenieur, 1S79, p. i; also an article by Charles Grad in " la Nature," 1S76, p. 55 ; also a t)rief article by the author '■ LTeber das Wasser," Berlin, 1S76. II See Glaser's Annalen, 18S5, Vol. XVIL, p. 234. THE CONSTRUCTOR. 219 The opposite extreme to a high pressure accumulator is found iu those pools or receptacles of water far below the natural sea level, such as are found iu mines, and iu the polders or drainage pools of Holland, Lombard^', and parts of Northern Germany. Reservoirs are not confined to use with liquids. Examples of Other fluids are found iu the gasometers of gas works, in the receivers for compressed air, so e.xtensively used in mining and tunneling, and iu making the so-called pneumatic foundations. Smaller reservoirs are found iu the air-chambers on pumping machinery, and the like. The sewage system of Berlin, designed by von Hobrect, con- sists often drainage pits, with the water level below the natural level, an-auged on the so called radial system. The sewage is pumped from these pits and delivered by means of pipes to sewage farms at a distance from the city. Negative receivers, so-called, may be used for air, as iu the case of the coining presses of the English mint, where a vacuum chamber is used to receive the air already used for driving the machines, and kept pumped out by steam power. The venti- lating apparatus for mines also often contains such negative reservoirs for air. Reservoirs are also used for granular materials, such being extensively used in connection with grain handling machinery. A steam boiler may be considered as a physically supplied reservoir, as well as a physical ratchet system (see \ 260). A combined physical and chemical reservoir is found in the elec- trical accumulator, which ma^' properly be called a current- reservoir. A combined physically and mechanically operated negative reservoir is found in the various forms of refrigerating machines. A modern application of pressure organs, and one which is rapidly extending in use, is that of the distribution of power in cities. Followiug the impulse given by the introduction of the high pressure water S3'stem of Armstrong, the use of gas in motive power eugines by Otto followed, and many other methods of meeting the problem have been applied. In long distance transmissions of this sort, special reservoirs are often used, in which force may be stored, so to speak, and from thence distributed in a manner similar to the ring trans- mission system for rope (see § 301). In this method the pres- sure organ after use is returned to the reservoir to be compressed and used again, or it may be used as in the line transmission and allowed to escape at the end of the line.''' The following cases are given as applications of pressure organs in long distance transmission : 1. The London Hydraulic Power Company distributes 300 H. P. by means of water at a pressure of 46 atmospheres (675 pounds). A similar and earlier installatiou is iu use at Hull. 2. The General Compressed Air Company distributes power by means of air at a pressure of 3 atmospheres {45 pounds) in Leeds and Birmingham. The system is an open liue, and 1000 H. P. are used in Leeds, and 6000 H. P. in Birmingham. f In Paris the Compagnie Parisienne de I'air comprim^, procedes Victor Popp, distributes power from three stations iu quantities varying from a few foot pounds up to 70 or So H. P., a total of some 3000 H. P. The use of compressed air appears to be destined to a widely extended use for this purpose. 3. The distribution of power in New York by means of steam mains is extensive and well known. 4. The vacuum system is used also iu Paris by the Societe auonyme de distribution de force a domicile. This is au open line transmission, operating in 1SS5, about 200 H. P. 5. Transmission by highly superheated water has been used in Washington, by the National Superheated Water Co., dis- tributing heated water at pressures from 26 to 33 atmospheres (400 to 5oo pounds), the water being converted into steam at the point of utilization. 6. The distribution of power by means of gas holders has already been referred to, and the distribution by electric cur- rents is rapidly being developed. I 313- Motors for Pressure Organs. The methods of applying pressure organs to the development of motive power are even more varied as in the case of tension organs. For this reason a general view of the subject will be taken in order to obtain a classification which will simplify the discussion. The main distinctions are those of tlie character of the motion of the mechanism, and of the method of applying the pressure organ to the motor. The great difference in the character of the motion of the mechanism lies in the fact that it may be either continuous or intermittent, so that the motor may be either : A running mechanism, or A ratchet mechanism (compare \ 260). The ratchet pawls for pressure organs are the various forms of valves (see Chapter XXVI). The various forms may also be classified according to the fol- lowing important distinctions based ou the method of driving. The pressure organ may drive, or It may be driven, or The impelling mechanism may itself be propelled. There is also a third distinction to be made, uamely, whether the pressure organ acts merely by its weight, or whether it acts by its living force of impact. This last distinction cannot be sharply observed in practice, but is especially to be considered in discussing the theory of action of the various machines. In the following pages the various applications will be shown in a manner similar to that employed in '/. 262 for tension organs, following the system of classification outlined above, aud be- ginning with running mechanism as the simpler of the two great divisions. A. RUNNING MECHANISM FOR PRESSURE ORGANS. I 3M- Running Mechanism in which the Pressure Organ Drives by its Weight. With a few unimportant exceptions the motors of this class are operated by liquids, which at moderate velocities practically follow the laws of gravity. Iu Fig. 957, a is an undershot water-wheel, and 5 is a half- Fig. 957. breast water. The water is guided in a curved channel and the buckets are radial, or nearly so. The wheel is so placed that the buckets pass with the least practicable amount of clearance over the curved channel. At c is shown a high-breast wheel, and at d an overshot wheel (compare §47). In these latter wheels the buckets are so shaped that they retain the water in the circular path, being closed at the sides also, while on account of the moderate pressure they are left open above. At e is shown the side-fed wheel of Zuppinger. Fig- 958, a is an endless chain of buckets, and b a similar arrangement, using disks running with slight clearance in a vertical tube. In the wheels shown in Fig. 957 the water acts on the wheel much in the same manner as a rack acts when driving a pinion, and iu this sense a water wheel may be considered as a gear wheel. When the water acts only by gravity these construc- tions are only practical when the wheel can be made larger in diameter than the p „ ._g fall of water, and where small diameters must be used the arrangements of Fig. 958 are available. Very small wheels acting under high pressures may be employed by making use of the so-called " chamber wheel work," X of which some examples are here giveu. * See a paper by the author in Glaser's Annalen, 1885, Vol. XVII., p. 226. t See Lupton and Sturgeon, Compressed .-Vir vs. Hydraulic Pressure, Leeds, 1886; Sturgeon, Compressed Air Power Schemes,' London, 1S86 ; also The Birmingham Compressed Air Company, Birmiugham, 1S86. Fig. 959(1 is the Pappenheim chamber wheel train. In this the tooth contact is continuous, the teeth being so formed that the continuous contact of the teeth at the pitch circle prevents \ See Berliner Yerhandlungen, 1S6S, p. 42. 220 THE CONSTRUCTOR. the water from passing, while the points and sides of the teeth make a close contact with the walls of the chamber. The downward pressure of the water enters into the spaces between the teeth and drives both wheels. The axes of the wheels are also coupled by a pair of spur gear wheels outside the case, thus insuring the smooth running of the inner wheels. This is the oldest form of chamber train mechanism known, and can also be used as a pump, operating equally well in either direc- tion. Fig. 959 b is Payton's Water Meter, with evolute teeth. The flow is intermittent, but one contact begins before the action of the previous one ceases. Fig. 959 c is Eve's chamber gear train. The ratio of teeth is I to 3, and the flow is also intermitteut. The theoretical volume of delivery for all forms of chamber gear trains, whether con- tinuous or intermittent in deliver3-. is practically equal to the volume described by the cross section of a tooth of one of the two wheels for each revolution. Fig. 959 rf is Behren's chamber train. In this case each wheel has but one tooth, as is also the case with Repsold's train (de- scribed hereafter), and the gears belong to the class of disc wheels or so-called "shield gears " (see g 2ii). This arrange- ment possesses the great advantage of offering an extended sur- face of contact at the place between the two wheels where, in the previous forms, there is but a line contact. This permits a sutficietit degree of tightness to be obtained without requiring the parts to press against each other. Behren's chamber gear makes an excellent water motor if the impurities of the water are not sufficient to injure the working parts. The flow of water through chamber gear trains is not uni- form, and the inequality of delivery increases as the number of teeth in the wheels is diminished, hence they should be driven only at moderate velocities when used as motors, in order to avoid the shocks due to the impact of the water. §315- Rdnning Mechanism in which ths Prbssure Organ Drives by Impact. In driving running mechanism by impact, fluid pressure organs, both liquid and gaseous, may be used, as will be seen from the following examples. Fig. 960. Fig. 960 a; is a current wheel, or common paddle wheel. The paddles are straight, and either radial, or slightly inclined toward the current, as in the illustration. The working contact in this case is of a very low order. Fig. 960 b is Poncelet's wheel. The buckets run in a grooved channel, and are so curved that the water drives upwards and then falls downwards, thus giving a much higher order of con- tact. At c is shown an externally driven tangent wheel. The buckets are similar to the Poncelet wheel, but with a sharper curve inward. The discharge of the water is inwards, its living force being expended. At d is an internally driven tangent wheel, similar to the preceding, but with outward discharge. The form shown at e is the so-called Hurdy-Gurdy wheel. The water is delivered through curved spouts, and this form is prac- tically an externally driven tangent wheel of larger diameter and smaller number of buckets. This wheel, from a crude makeshift, has become one of the most efficient of motors.* Wheels with inclined delivery as made in the forms shown in a 1) c Fig. 961. Fig. 961. At a is shown a crude form, used on rapid mountain * This is the Pelton Water Wheel, built in sizes as great as 300 H. P. See Mining- and Scientific Press, 18S4, p. 246. and 1S85, p. 21. This wheel is built in Zurich, by Escher, Wyss & Co., with a casing, and used for driving dynamos. streams as a simple ex-pedient, but of low efficiency ; b is the Borda turbine, consisting of a series of spiral buckets in a bar- rel shaped vessel ; c is tne so-called Danaide, the spiral buckets being in a conical vessel, this form being mostly used in France.! In the wheels which have been shown in the preceding illus- trations from Fig. 95S, the living force of the water acts by direct impact through a single delivery pipe. The following forms difi^er from the preceding, in that the water acts simultaneously through a number of passages around the entire circumference of the wheel. This form gives the so- called hj'draulic reaction in each of the inclosed channels, and hence wheels of this class are commonly called reaction wheels, or reaction turbines.J Fig. 962. Fig. 962 a is Segner's wheel, the water entering the vertical axis and discharging through the curved arms ; b is the screw- turjpine, entirely filled with water ; c is Girard's current turbine, with horizontal axis, and only partially submerged ; of is Cadiat's turbine, with central delivery, and e is Thomson's turbine with circumferential delivery and horizontal axis, the discharge being about axis at both sides. In all five of these examples the column of water is received as a whole, and enters the wheel undivided until it enters the wheel ; in the following forms the flow is divided into a number of separate streams. Fig. 963. Fig- 963 ci is the Fourueyron turbine, acting with central delivery ; the guide vanes are fixed and the discharge of the water is at the circumference of the wheel ; (5 is a modification of the Fourueyron turbine, the water being delivered upwards from below, and sometimes called Nagel's turbine; c is the Jouval or Henschel turbine, the guide vanes c being above the wheel, which is entirely filled by the water column ; d\s Fran- cis' turbine, .with circumferential delivery through the guide vanes c* ; e is the Schiele turbine, a double wheel with circum- ferential delivery and axially directed discharge. In the latter three forms a draft tube may be used below the wheel, to utilize that portion of the fall, as indicated in forms c and d. Fig. 964. For gaseous pressure organs, of which wind is the principal example, some forms are here given. Fig. 964 a is the German windmill, with screw-shaped vanes ; b is the Greek and Anato- lian windmill, with cup-shaped vanes. Both forms are similar in action to the above described pressure %vheels. At c is shown the so-called Polish windmill, with stationary guide vanes ; || (/is Halladay's windmill, made with many small vanes, which place themselves more and more nearly parallel with the axis as the force of the wind increases, the rudder r, keeping the wheel to the direction of the wind. The extreme position of the vanes is shown at e- Anemometers and steam turbines are examples of wheels in which other pressure organs than wind are used. , Sccti on 4, t See Wei.sbach-Hernnian, Mechanics of Engineering, Part II., p. 55S., X This use of the term reaction is hardlv desirable for this construction, nor is the proposed name of " action turbine," and the name "pressure turbines " is to be preferred. § This form is well made by J. M. Voith.of Heidenheim, Wtirtemberg. II Recueil des Machines avantageuses. Vol. I.. No. 31, 1699, also from thence shown in Henning's Sammlunmii/:,:^//, Fig. 969. train A delivers water to a distant one B, driving the latter and receiving the discharge water from B through a return pipe to be used again. The combination forms a transmission system of the second order (see \ 26), and is similar to a belt or chain transmission. The loss in efBcienc}' in this device is not an un- important consideration. An important class of machines consists of those made with tension organs for transporting granular materials. For this purpose belts, chains, etc., are used, and when the transmission is horizontal, or nearly so, grain is successfully transported on wide belts. jl Another application is that of Marolles, using an iron belt, 40 in. wide, 0.06 in. thick, for transporting mud. Twelve such machines were used on the Panama Canal work, the distance being 200 feet, and the speed of the band 12 to 40 feet, according to the nature of the material. Similar apparatus at the Suez Caual handled material at a cost of 7.6 cents per cubic yard. ?3I7- Running Mechanism in which the Pressure Organ is Driven by Transfer of Living Force. The method of driving pressure organs by a transfer of living force is one which admits of numerous applications, as the fol- lowing examples show. Fig. 970 (7 is a centrifugal pump for moving liquids. The driving mechanism consists of the curved blades, which iu g The firm of Klein, Schanzlin & Becker, at Frankenthal, make a line of pumps similar to Fig 967 d, of a capacity of 1.77 to 177 cubic feet per minute, and discharge openings from i.iS to ii.8 ins. diameter. These are driven by belt and used beer-mash oil, acids, paper pulp, syrup, etc., as well as water. II An excellent transmission is in use at Cologne. See also Trans. Am. Soc. Mech. Engrs., Vol. VI., 1884-S5, p. 400. At the Duluth elevator a rubber belt 50 inches wide, running 600 to 800 feet per minute, carries grain from 600 to goo feet horizontallv. A ^6" belt has carried 14,000 bushels per hour. 222 THE CONSTRUCTOR. many instances are made in one piece wilh the wheel itself, this adding to the efficiencs These pumps ha\e been most sue cessfuUy made by Gwjnne, Schiele Neut and Dumont among Fig. 970. others.* Centrifugal pumps have been successfully used as dredging machines for lifting wet sand, gravel and mud, in- stances among others being the North Sea Canal at Amsterdam, and the harbor at Oakland, California. Fig. 970 i is the well known fan blower u?ed everywhere for producing a blast of air, and acting by centrifugal force. When used <)s exhaust fan this is widely used in connection with suitable exhaust pipes for removing foul air, sawdust, and other impurities in workshops, as well as for the ventilation of mines. f At c is shown a form of spiral ventilator, known as Steib's ven- tilator; it is similar to some of the preceding forms, but is of limited application, and is better adapted for lifting water, a service to which it has been applied in the polders of Holland. At d is a centrifugal separator, a device of numerous applica- tions for separating materials of different specific gravity by centrifugal force. A notable example of this machine is the centrifugal separator for removing cream from milk. Another variety of machines for driving pressure organs by a transfer of living force, is that in which another pressure organ, either liquid or gaseous, is used instead of a wheel as the im- pelling mechanism. To this class belong the various jet devices, injectors, etc. Fig. 971. Fig. 971 ci is Giffard's injector in the improved and simplified form made by the Delaware Steam Appliance Co. In this case steam is used to drive a jet of water into a vessel already con- taining water under pressure. The jet of steam rushing through the nozzle b^ draws the water in by the suction tube b.^. and both pass through the mixing tube i,, and are discharged through the outlet tube b^ ; the outflow at b=^ provides for the relief of the discharge at starting, before the jet action is fully estab- lished. The regulation of the flow of steam is effected by a steam valve attached above i5,. At 4 is Gresham's automatic injector, which is so made that should any interruption occur in the supply of water at b.^, the suction action is automatically started, and the entering column of water is lifted again. This is done by the introduction of a movable nozzle b^ between b-^ and b^, wliich adjusts its position with regard to A3 according to the variations in pressure above and below. Fig. 972 is Friedmann's jet pump. The mixing tube i;, is divided into a number of sections, which permits a very free entrance to the water, and gives an excellent action ; b is Nagel's jet pump, used for lifting water from foundations by means of another jet of water. The entrance jet is at b-^, the * A recent installation of magnitude is that of five centrifugal pumps built by Farcot, of Paris, in 1887, for supplying the Katatbeh Canal in Egypt. The wheels are 12 ft. 6" dia , and each deliver 17,660,000 cubic feet in 23 hours, the lift varies from i to 12 feet. t Fans (or these purposes are made in great variety by J. B. Sturtevant. Boston, Mass. Fig. 972. the regulation is suction tube at b.^, and the mixing tube at b-^ effected by a valve at the end of b,,. Steam jets are also used to produce a blast of air, or com- pressed air may be used for the same purpose, as can also water under pressure. A reversal of the last mentioned arrangement occurs in Bunsen's air pump, in which a jet of water'is used to produce a vacuum. Recent devices for utilizing jet action are numerous. Among others, a jet of air has been used to feed petroleum into furnaces as fuel. Dr. W. ,Siemens proposed to carry the petroleum in the hold of a vessel in bulk, and substi- tute sea water, as it was consumed, in order to maintain the ballasting of the ship undisturbed. Granular materials have been handled by means of jet apparatus, usually impelled by compressed air, sometimes by water jets. An especial f^eature of jet pumps, and one which should not be overlooked, is that they act either by guiding the pressure organ stream, or that the driving action of the pressure organ stream itself produces a guiding action, and that the existence either of a reservoir or some external means <^ ^=(1 of driving must be presupposed. The use of a pressure organ in motion for driving mechan- ism, is in this respect similar to the so-called inductive action of an electric current. An example of pure guiding action is found in the "Geyser Pump" of Dr. W. Siemens, Fig. 973. The water is to be raised from a depth H , and the tube b is prolonged downward to a depth H^ below the sump 5'. The prolonged tube b-^ is open at the lower end, and in the bottom opening 7" an air tube c is introduced, and air is admitted at a pres- sure slightly under that of a column of water of height equal to H^. The air mingles with the water and forms a mixture in a^ which is lighter than water, and the air pressure is then capable of forcing the light mixture up to the surface. The lifting action is assisted by the expansion of the ascending air. Siemens found that it was possible to produce this action when //was equal to //,, that is, the specific gravity of the mixture of air and water =^ J4. -i-P' Fig. 973. ? 31S. Running Mechanism IN which the Propelled. Motor itself is The third division, in which the motor itself is propelled in the liquid pressure organ, contains fewer varieties than the pre- ceding ones but is of the greatest importance since to it belongs the entire subject of marine propulsion. Fig. 974- Fig. 974 0' is the so-called " flying bridge," the current flow- ing in the direction of the arrow, causing the boats to swing across the stream, describing an arc about the anchor to wbicE THE CONSTRUCTOR. 223 they are held by a chain ; (!i, is a sail-boat, the sail beiug the driving organ transferring to the boat a portion of the living force of the current of wind. At c, is a steamboat with side pad- dle-wheels, and (/, a stern-wheel boat ; tf, is a screw propeller. A screw driven hy a steam engine pressing the water backward and the reaction of the water impelling the boat. Aty, is a so-called jet propeller, the reaction being produced by jets of water forced through tubes at the side of the boat, the water beiug driven b}' centrifugal pumps.* At^, is shown a current wheel motor. The side paddle wheels are caused to revolve by the action of the current, and by connection with a cable. or chain gearing (See Figs. 787 and 794) the boat is propelled up the stream. Direct acting reaction jets have been used for torpedo boats, using carbonic acid gas, but this method has been superseded by twin screw propellers driven by compressed air. Rockets and rocket shells are examples of direct acting pressure organs. B. RATCHET MECHANISM FOR PRESSURE ORGANS. I 319- Fluid Running Ratchet Trains. The pawls in a fluid ratchet train are the valves. They may be divided'into two great classes,t similar to those existing in ratchets of rigid materials, viz. Running Ratchets, or Lift Valves, and Stationarj' Ratchets, or Slide Valves. In the first class we have flap valves, also conical and spheri- cal valves, and in the second, the various iorms of cocks, cylin- drical and disc valves and flat slide valves. In both kinds of valves there exists an analogy to toothed and to friction rat- chet gearing, since by use of contracted openings the effect of friction is produced, and with full openings it is obviated. This gives a division which does not exist in the case of friction and toothed ratchet gearing. Viewed according to the preceding classification, piston- pumps, and piston machines are properly ratchet trains. | This idea does not seem to offer any practical diffrculties, since it can be made to include all the numerous variations without crea- ting more confusion than the former methods of classification. It is not practicable to distinguish between the devices acting by gravity and those acting by transfer of living force, since both are frequentl}' combined. The oldest devices are those using air, and the oldest piston is the membrane piston, (Fig. 949) in the form of a bag of skin used as a bellows. In this primitive device the earliest valve was the human thumb, and in the larger bellows the heel of the operator, these beiug followed at a later date by valves of leather.? The working part of the bag was next strengthened by a plate, (See Fig. 949 The canal is open on the upper side (see Fig. 945 b and c) ; the valves b^ and b., are of the running ratchet form, and are in reality double gates. Smaller by-pass valves b^' and b,' are used in order to enable the inlet and outlet of the water to be started gradually. The boat c forms the piston, and when the motion is upward, b^ is the escapement valve, and when downward, b.^ is used. The above canal lock device, while extremely useful, pos- sesses a very low efficiency, since it not only uses a volume of water equal to the displacement of the boat plus the necessary clearance, but also discharges the whole lock chamber of water each time it is used. Later devices have been made for the same purpose, involving a less waste of water. If it is arranged for the service to be doubled by making two lifts adjacent to each other, it is evident that the descending boat can counter- balance an ascending one of the same weight, the only require- ment being that there must be some connecting mechanism in- volving the overcoming an additional resistance, and capable of a Fig. 994. Fig- 994 shows a double canal lift constructed by Green tax the Grand Western Canal in England in 1840, the connecting mechanism being tension organs in the form of chains. The boats are carried in tanks c^ c.,, the ends of which are closed by valves or gates 6, and b.,. and similar gates b/ and b./ also close the ends of the canal sections. A small addition to the weight on the descending side is sufficient to raise the other tank f " Fig. 995. The substitution of a pressure organ for the chain was first made by Mr. Edwin Clark on the Mersey Canal in 1875, in the form of a hydraulic lever, as shown in Fig. 995. This shows clearly the equivalence of the cord or chain and pulley and the water lever, already referred to in i 311. The tanks rj and c^ are carried on plungers 3 feet in diameter, and are 75 feet long and I5j4 feet wide. A head of 6" of water is sufficient to over- come the resistance of motion, and a lift of 50 feet is effected in three minutes.J Smaller installations have been made by Clark and by Stanfield, and other large ones at the La Louviere Canal in Belgium, and the Neufosse Canal at Les Fontinettes, in France. The lifts are 43 ft. and 50 ft. respectively, and the plunger diameters 6|< feet. The loss of water with these lifts is only about i^Jj of the quantity used by common locks of the same capacity. ^ The preceding escapement devices are made for open canals, but escapements may also be constructed with closed tube con- nections. This latter type includes numerous hydraulic eleva- tors for lifting burdens of all kinds. An example of a direct-acting hydraulic elevator is given in Fig. 996. The two valves are combined in one cock. The water under pressure enters at //, and the discharge against the atmospheric pressure is at A. The weight of the plunger is counterbalanced by two counterweights G with chains and •In miles. ' the length of the postal pneumatic tubes in Berlin was over 26 t See Weisbach-Hermann, Vol. III., Part 2, p. 633. X See Duer Trans. Inst. C. E., 1876 ; Colyer, Hydraulic Machinery, London, Spon., iSSr, p. 17 ; also Robinson, Hydraulic Machinery, London Griffin & Co., 1887, p. 64. g See Colyer, p. 29 ; Robinson, p. 69 : also Zentralbl. der pr. Bauverwaltuug, 1SS2, p. 395 ; Hensch, Schiftshebung in Frankreich ; also Scheiufil, Kanal and Hafenwerkzeuge in Frankreich und England, Wien, Ceroid, 1S82, p. 15; also Ernst, Hebezeuge, Berlin, Springer, 1S33, p. 630. In Green's lift the loaded boats descended and the empty ones ascended, hence an excess of water was raised, which was permitted to overflow. These lifts enable much greater differences of level to be overcome than do the ordinarj' locks, and make it practical to use long stretches of canal and make an entire lift at one operation. It may be here noted that pneumatic lifts for canals were designed in 1863 by the Swiss engineer. Sevier. 228 THE CONSTRUCTOR. pulleys, and the plunger operates the valve automatically by means of the rod b' , when the highest position is attained. This form of lift has been much used, sometimes of very large dimensions. The great passenger elevator of the Hamilton St. Station of the Mersey Tunnel has a plunger iS" in diameter, with a lift of Syjj feet, the car holding 50 passengers.* A practical objection to direct-acting lifts of this form lies in the heavy counter- weights required, and also in the depth to which the cylinder must be sunk. A different form has therefore been designed in which a piston travel of moderate length is multiplied by use of a tension organ system, such devices being extensively used for passenger elevators, notably by the Otis Elevator Company. Hydraulic cranes are also forms of high pressure escapements, first designed' by Armstrong, and since used by many others, especially in connection with Bessemer Steel plant, in which hydraulic cranes have proved most valuable. Fig- 997 shows the mechanism of a hydraulic crane by Armstrong. The piston is double acting, and there are four valves /^i, b,, b^, b^, of the type shown in Fig. 9S6, the external connections also being neces' sary in order to complete the escapement. The high pressure water enters at //, and passes through the pipe /, and is discharged to the atmosphere at IF. The rod fj is made of half the area of the piston e^ Fig. 997. (compare Fig. 946 e). When bi and b^ are open, as in the illus- tration, the forward stroke is made with one-half the full force ; when 5j and 5^ are open, the forward stroke is made with full force. By opening b., and b.^, the return stroke is made by the pull of the load upon the chain. At b' is a safety valve which comes into action should the load descend too rapidly, by the opening of b^ alone.* ? 323- Hydraoi,ic Tooi^. Hydraulic escapements, similar to those used for lifting loads are also applicable to machine tools. Among these may be noted the devices of Tweddell, for riveting, punching, bending, etc. (see ? 54). Figs. 998 and 999 show the arrangement of Tweddell's rivet- ing machine ; d is the piston, bi, b.^ the valves, one of which connects with the pressure reservoir at M, and the other with the atmosphere at ^. When b^ is opened by the lever e, the hydraulic pressure enters above the piston d, and the stroke is made. The return stroke is effected by means of the auxiliary piston rfj, which is fast to d, and under which the water pres- sure is acting at all times. Clusitig b-^, and opening b^, enables this to act and lift the main piston. This gives practically a hydraulic Jever of unequal arms, the shorter arm always being loaded with H, and the load on the longer arm varying between H and A. The lever mechanism d', d'\ d"', controls the length of stroke of the die, by means of the tappets d" and d'", which are connected with the lever e. This is also used on the lift of Fig. 996, and shows the complete escapement. The arrangement of valves is shown in detail in Fig. 999.5: Fig. 998. Fig. 999. The preceding apparatus resembles the hydraulic press. It is in fact quite different, being a genuine ratchet train, capable of all the modifications of such mechanisms as to speed, distance, and arrangement. On account of these points the applications of pres- sure organ escapements are becoming rapidly more important. § 324- Pressure Escapements for IMoving Liquids- The use of unperiodic pressure escapements for moving liquids in machine construction has been practiced from an early period, and at the present time improved de- vices for this purpose are much used. An almost forgotten device of this kind is Briudley's boiler feeding ap- paratus, Fig. 1000, this being based upon the principles already given in Fig. 991. The necessary opening of the valve b is made by the float c, and the closing by the counterweight c, (compare Fig. 950). This apparatus was first applied to Watt's boilers, the feeding of the boilers of Newcomen's engines being effected by a cock operated by the attendant. Fig. lool is Kirchweger's steam trap for the removal of water of condensa- tion. The escape valve b is opened by the float c, which, in this instance, is open at the top, so that the water flows over the rim until it sinks, and thus opens the valve, This valve motion is in itself a ratchet train, checked and released by the action of the float. When the valve is opened the water in the float is forced out by the pressure of the steam. § The slow moving float device, as in Fig. 1000, has also been advantageously for operating steam traps, by used Fig. 1 000. * See Robinson. t See Weisbach-Herrman, III., 2, p. 240 ; Colyer, p. 11 ; Robinson, p. 52. X For fuller descriptions of Tweddell's machine see : Proc. Inst. C. E. LXXIII., 1SS3, p. 64 ; EngiHeer, July, 1S85, p. 88 ; August, p. iii ; Revue Indus- trielle, 1S84, p. 5: 1885, p. 493; Mechanics, 1885, p. 272; also Rol»inson, as above, and Zeitschr. Deutscher Ing., 1886, p. 452. Tulpin, of Rouen ; Handrick, of Buckau ; Puschel, of Dresden; Dehne, of Halle, and others. Similar escapements have been designed to separate air from steam, or air from water, as in the devices of Andral, Kuhl- mann, Klein and others. || Other examples of escape- ments of this kind are found in the so-called Montejus, used for elevating syrup in sugar refiner- ies, in the return traps of .'-team heating systems, and in various other forms of boiler feeders, such as those of Cohnfeld. Rit- ter & ISIayhew, and others. ^ § This form of trap is made in many varieties, tlte one shown being by IvOSeuhausen, of Dusseldorf. A similar one by MacDongal is much used in Bngland, and a feed pump on this principle is made by Korting in Hanover ; German Patent No. 3'^-, 332. (1 For illustrations of these devices s-ee SchoH's Fiihrer des Maschinisten, 10 Bdit., p. 493- H See SchoU, p. 235. Fig. iooi. THE CONSTRUCTOR. 229 B. PERIODICAL PRESSURE ESCAPEMENTS. k. 325- PoMPiNG Machinery. Periodical fluid escapement trains have a wider application than unperiodical trains, since it is practicable, as already shown, to use a fluid ratchet train to operate the valves in a simple manner. This makes it possible to produce the opening and closing of the valves in a periodical succession mechanically, instead of by the fluid column. In this construction the fluid ■column may therefore drive the piston, instead of being driven by it. This idea seems very simple, and yet pumps had been known for two thousand years, and had occupied the inventive energy of the preceding centuries before the siinplest forms of the modern steam engine were devised. It is therefore all the more important in the study of machine design to investigate the fundamental principles involved. It is impossible, in the limited space which can here be given, to go into this subject iu its entirety ; the arrangement of the valve gear of the Newcomen engine with tumbling bob gear, is an instructive example. In Fig. 1002 is shown Belidor's single acting water pressure engine." in communication with the discharge, and since b.^ is larger than 5,, the pressure between them moves them into the position b^' b./. This puts the main cylinder in communication with the disciiarge, and the piston sinks by the weight of the load upon it. At the close of the stroke the tappet 6 moves the arm c/ into the position c-^ again, and places the auxiliary valve in the first position and a new stroke is made.i This machine constitutes an escapement of the second order, since the small and large escapements alternately release each other ; the lever device 5-6-Ci forms a third mechanism, so that the machine, as a whole, is of the third order. Fig. 1002. In the cylinder a' is a piston ; rt, is the entrance of the water, «2 the discharge outlet. The valves *, and A, are united in a three-way cock (see Fig. 987). This valve is operated from the piston rod <: by a tumbling-bob gear (see Fig. 742). The tum- bling lever E t\ e.„ weighted at £, is connected with the piston rod at c,, and moves about its axis independently of the lever/. When the end of the piston stroke is nearly reached, the lever E passes the middle point, and tips over, when the arm/, strikes the lever/ and carries it to the position /, moving the lever of the three-way cock from b to b'. The arm f, is behind E. The return stroke of the piston moves the arm e.^ of the tumbling gear towards the right, and as the end of the stroke is reached, the tumbling bob is again tripped, and the three-way cock moved again into the position b. A cord secured at the ends to the points ^3 and c^, and fastened to E, limits the travel of the latter. The piston rod is connected directly to the pump to be operated. t It will be observed that this machine is a ratchet train of the second order, the piston and valve forming an escapement, and the valve gear a releasing ratchet train each operating the other. Fig. 1003 is the single acting water pressure en- gine of Reichenbach. In- stead of using a tumbling bob gear to operate the valve, Reichenbach uses a second water escape- ment, operating the valve by a piston, the valve being itself a piston valve. The double piston valve ^3 bi of the secotid escape- ment is operated by the main piston rod, the tap- pets 5 and 6 striking the lever fj as each end of the stroke is reached. The water under pressure en- ters at <7| and is discharged at a.,. The tappet 5 moves the auxiliary valve into the position b/ 5/, which places the space above b^ * Belidor, Architecture hydraulique. Paris, 1739. Vol. II., p. 23S. t The atiove described machine, designed by Belidor in 1737, for the water ■works at the bridge of Notre Dame, does not appear to be altogether ^jracti* cable. It has been here given on account of the valve motion, which is his- torically interesting and doubtless good, and has been re-invented several times since. It was not new in 1737, having been in Newcomen's steam engine, as was already known to Belidor, since it is described by him iu the same volume of his treatise. Fig. 1003. Fig. 1004. Fig. 1004 shows the double acting water pressure engine of Roux. § The double action is obtained by combining the four valves in one, and by communicating the admission and discharge alternately with both sides of the piston. In this case the lever connection Ci is replaced by an escapement. The small pistons b./ b/ are acted on at the outer ends by the pressure water through the small passages k./, k/. This gives an escapement of the third order. The cup-shaped ends c.^, r.,, of the main piston c form the pump plungers. This machine should operate satisfactorily. It is readily apparent that the piston steam engine may also be considered as an escapement. The valve gears differ from the preceding forms only on account of the conditions of ex- pansion and condensation. These are reducible to a limited number of simple cases. Fig. 1005. Fig. 1005 is a single acting high pressure engine. The steam t See Weisbach-Herrnian, II., 2, p. 536; a'™ Ruhlmann, allgem Masch- inen Chre., l.,p 34S . . , ^■ 8 See Revue Industrielle, 1S84, p, 114- Built by Crozet et Cie. 2^0 THE CONSTRUCTOR. enters at <7|, and the discharge to the atmosphere is at a^. The opeuing of the valve b^ permits the steam to enter, forcing the piston c down, and raising the weight G. The valves b^ and b.^ are operated by a ratchet train released by the tappets 5 and 6 on a rod moved by the main piston. The pawls are double- acting, and are of the form shown in Fig. 67 r. When c reaches the bottom of the cylinder the tappet 5 releases the ratchet 7, and closes the valve b^ by means of the connectionsy, e^ The release of 7 opens the valve b., by means of the connections e„f^, and permits the escape of the steam from below the piston. This equalizes the pressure above and below the piston, from which the valve b., is called the equalizing valve. The upward stroke of the piston causes the tappet 6 to reverse the ratchet 7 and operate the levers e-^f^, closing the equalizing valve and its connections. The device differs from the preceding in that the principal escapement a b^b„cd changes in character with the stroke. The two ratchet trains can be seen in principle in the double acting tumbling gear of Fig. 1002. The mechanism, when lift- ing the valve, is of the third order, and when closing, of the second order. The gear as shown is Farey's ; Fig. 779 shows this principle in a rigid escapement train, the corresponding form in single-acting train is the chronometer escapement, Fig. If the engine is a condensing one, a condenser valve dj is added, this being opened by the closing of b.^, as is also a jet valve in the condenser. When the steam is to be expanded, the lever e-^ is so arranged, the closing of (5, is produced earlier (see the smaller diagram) by the position of the tappet 5, and the corresponding counterweight lifted. This only operates the ratchet 7, and f,_ is released by a second train 8, which is effected by the tappet rod or by the so-called cataract K, released by a tappet 9, see \ 260. The condenser is a negative reservoir, and was the principal in- vention of Watt. It involves the use of two fluid ratchet trains ; the air pump, and the cold water pump, and also usually in- cludes a boiler feed pump. The entire engine is composed of a collection of ratchet trains. Steam pumping engines are by no means always made with lift valves, and a great number of more recent designs are made with slide valves (see Fig. 9S7). Ritteuger has applied slide valves successfully to single-acting engines, and they are espe- cially applicable to double-acting non-rotative engines. In the last decade especially' have valve motion for steam pumps with slide val.ves been multiplied, and some illustrations are here given. Fig. 1006 is Tangye's direct-acting steam pump. The steam iill ^ \M. Fig. 1006. entrance is at /, and the exhaust at IV. The slide valve b is the so-called E form, combining the four valves of Fig. 9S6 in one ; b^ and ij are the auxiliary pistons to move the valve, and form part of an escapement ot which the valves b" and b'" are operated by the mam piston c at each end of its stroke. The latter valves communicate with the cylinder posts //and ///. When b"' is lifted by the piston, the space R is in communica- tion with the exhaust, and the pressure in Z throws the valve over, equilibrium being soon after established through the aper- ture k,,. The reverse action occurs on the return stroke. This is a steam escapement of the second order, with an independent starting lever, the whole forming a combination of the third order. This has been much used by Tangye for steam pumps. Fig. 1007 shows the valve motion of the Blake pump, which is very extensively used in the United States. In this case there is a movable seat b^ under the valve b, the opeuing through the seat always being in communication with the posts //, ///, 11^, although ba is moved a short distance at each end of the stroke by tappets on the piston rod. In the position of the parts shown the steam entering at /will pass through ///and move the main piston to *he left, as indicated by the arrow. Just before the end of the stroke is reached the seat b„ is moved as much to the left of the centre as it now stands to the right. In 1 f /f n_ // / / /w/m ^M^ Fig. 1007. the seat b^, as shown in the figure to the right, there are addi- tional valves formed, /?,, ft, ft, which act to operate the auxil- iary pistons 1^2, (Jj, under which latter the small steam passages can be partly seen. When i')^ is moved to the left, a small post is uncovered by ft, and live steam enters the cylinder L behind A^, while at the same time ft connects R with the exhaust. This causes 5,, b, b^, to move to the right and reverses the pump. The reverse action takes place at the other end of the stroke, the whole forming a combination of the third order. ■///////////////A n '/>/ ! Y //>. in , )7///////////////////////////////My////^^^^^^ Fig. 1008. Fig. looS shows the valve gear of Deane's steam pump, which has also been extensively used. The main valve is moved by means of auxiliary pistons, as in the preceding instance. The auxil- iary pistons are controlled by a separate valve b' , which itself is operated by lever connections with the main piston rod. This combination b' , b-, b, forms again a mechanism of the third order.* If the last three devices described are compared with the Reichenbach water-pressure engine, it will be seen that the fundamental principle is the same in all. The constructive arrangements which may be adopted are clearly shown in the precedi'ug examples, which may be modified in a variety of ways. Among other widely used arrangements, that of Knowdes maj' also be mentioned ; in it the action of the auxiliary pistons is controlled bj' a slight twisting motion given to the valve stem. ..^.^^oJm Fig. 1009. Fig. 1009 shows Pickering's steam pump t In this design the main piston c acts also as the valve for the auxiliary pistons b,, ^3, so that the spaces R and L are placed alternately in com- * See .A.m. :\Iachinist, Feb. 17, 1SS3, p. 4 Dow, see ;Mining and Scientific Press, 18 t See Poillou. For an excellent steam pump by i, March, p. i6g, and May, p. 313. THE CONSTRUCTOR. 231 munication with / and IV. The whole forms a steam escape- ment of the second order. Fig. loio shows Harlow's valve gear, also used for pumping machinery.* This is also a steam escapement of the second order, similar to the preceding. The valve action for the auxiliary pistons is formed in a prolongation of the piston rod, the grooves c^ and €.-,_ placing the spaces R and L alternately in communication with IV. 59* Fig. ioio. By comparing the preceding designs with the water pressure engine of Roux, Fig. 100.J, the similarity will be apparent. All the examples given show the fundamental relation existing be- tween these devices and the mechanical escapements of watch movements. The escape wheel is replaced by the fluid column ; the anchor, by the valve ; the vibrating member, whether pen- dulum or balance wheel has. here not a free movement but a determinate one against an external resistance. Similar arrangements include steam hammers, also hammers and rock drills, usually driven bj' compressed air, these latter consisting of mechanism of the second, rather than the third order. An example will serve to illustrate the general arrangement of such devices. Fig. ioii. Fig. loii shows the arrangement of Githen's rock drill. f The curved valve 4, is operated by the action of the curved outline formed in the piston c. The middle position of the valve is a dead point, but this is overcome by the momentum of the heavy piston. The devices of the third order are capable of a very import- ant modification which can be considered by examining for in- stance the Deane gear. Fig. looS, or either of the two preceding it. An inspection will show that it is entirely practicable to use the auxiliary piston to operate a pump cylinder, as indepen- dently of that operated by the main piston d. It is only neces- sary to make it larger in diameter and of proper length of stroke ; and there is nothing to prevent making it of the same diameter and stroke as the main piston. The valve of each cylinder will then be operated by lever mechanism connected to the rod of the other piston. This arrangement involves the replacing of the E valve by the com- mon D valve, which is not important, but is nevertheless an advantage. The two escapements are conveniently placed side by side for constructive reasons, and the double arrangement is known as a "duplex" machine, this term being given to two combined cylinders, of which the valve of each is operated by the piston movement of the other. This type is now frequently met, having been made for small apparatus very early, in France by Mazellire and yet earlier, in 1S59, ii^ the United States by Worthingtou. Fig. 1012. Fig. 1012 shows a duplex pump by Mazelline.j The illustra- tion shows one piston c^, at mid-stroke with its valve ij, at the end of its travel, and connected to the rod of the other cylinder by the lever e„. The work is divided into two portions which is provided for by the doubling of the parts. If the two piston escapements (cylinders, pistons, valves, steam, etc.) are indicated bj- [i] and [3], and the valve movements by [2] and [4] the action will be as shown in the following lines, whence we have and [1] M [3] [3] [4] [I], both being of the second order. * See Engineering and Mining Journal, Oct., 1884, p. 23 t See Eng. and Mining Journal. March, 18S7, p. 107. Also Halsey's rock drill, Trans. A. S. M. E-, 1S84-5, p. 71. Fig. 1013. Fig. 1013 shows a perspective view of Worthington's Duplex Pump, the arrangement of which is apparent from inspection. The duplex steam cylinders are at the right, and the double acting pump cylinders on the left. The advantages obtained by using this form of pumping machine practically outweigh the objections which might be made against the duplication of pa.rts. In double acting pumps of the forms shown in Figs. 1006 to loio, the motion of the water columns is interrupted, at low speeds, at each reversal of tSee Poillon, Plate I-X 232 THE CONSTRUCTOR. the piston, while with the duplex pump the discharge is practi- cally continuous, because each cylinder begins its stroke just before the other comes to rest. An objection to all the other forms of direct acting pumps already described lies in the fact that to obtain uniform pump- ing action it is necessary to carry the initial steam pressure for the entire stroke of the piston, or in other words, the best action of the water end is obtained by means of the least economical action of the steam cylinders. This defect was overcome in the earlier pumping engines, such as the Cornish mine engines, by using the steam to lift heavy weights, pump rods, etc., the living force of the mass permitting an early cut-off and high expansion, and the uni- form descent of the weight being used to force the water. By this method the Cornish engines attained a high degree of economy. This method being single acting, caused the entire column of water to come to rest during the time required for the up stroke of the pump rod, and hence the Cornish type of pumping engine gives a most economical action of the steam at the expense of a defective action of the pumps. In the larger sizes of Worthington pumping engines the ex- pansion of the steam has been for a long time effected by using compound cylinders, and excellent results attained in steam economy. The efficiency, however, was by no means so high as was desired. In iSS6 the so-called Worthington equalizer was introduced with a view of enabling the desired high duty to be attained. Fig. 1014. This device, shown in Fig. 1014, is a ratchet train of the tumb- ling type, similar to that shown in Fig. 743, the springs being replaced by water pressure from a high pressure air cham- ber.* The air chamber forms a periodical storage reservoir. The plungers/', f, are attached to a cross-head connected to the prolonged piston rod, and the cylinders are carried on trunnions 7, 7. During the first half of the stroke the plungers are forced into the cylinders the latter swinging about the cen- tres 7, 7 ; and during the second half they are forced out by the action of the stored energy .t The resistance and assistance which the pistons /give to the steam piston is shown by a curve of the form of Fig. 1014 b. as has also been obtained by the indicator. Fig. 1015. If in Fig. 1015 a, we make P equal the component on each portion of the pressure O on the main piston rod, we have : _ D • ^ 2 Ptan/3 in which This gives tan p ■■ •v^i -t- tan /3' A' T P Q-- ^^(ff * A.n equalizer of this t3'pe was patented in Germany, by the Berlin Anhalt Works in 18S5. tin kinematic notation this action is expressed by CP i-CP-l), as shown l)j' ^. See Theoretical Kinematics, pp. 322, 325. or if we make Q the ordinate,]', of the desired curve : __2jPa-_ and substituting c for 2 Pa- we have ., ^ c s/^i 4. tji (317) which equation is readily expressed graphically. If this curve is drawn upon the rectangle which represents the resistance of the water, as in Fig. 1015 b, we get the actual resistance curve/"^ //, and this resembles closely the expansion line for a high degree of expansion, or in other words, the im- pelling force and the resistance are practically made equal to each other througho"t the stroke. The dotted curve a b c d e, is that of an actual indicator diagram, j This shows that with the Worthington high duty pumping engine the most efi&cient action of the steam is obtained at the same time as the best action of the water end. < Fig. 1016. Fig. iot6 shows a longitudinal section of a Worthington high duty pumping engine. The equalizing cylinders and their air chamber are seen on the right ; the dotted lines e,^ show the rod of the second cylinder, which operates the valve b^. As it has already been seen that many forms of the third or- der can be reduced to the second order, it may be inquired as to the possibility of obtaining a pumping mechanism of the first order. This has already been accomplished by uniting the steam escapement with a water ratchet train. The device is the Hall Pulsometer, shown in diagram in Fig- 1017. The steam enters at a, at b, is the anchor shaped pawl, and d, is the vessel corresponding to the frame- work of a rigid escapement, (compare Fig. 775). If the vessel d is closed as shown by the dotted lines and a volume of water c, included, we obtain an action of the first order. The efficiency is very low ; about % to ^ that of a piston pump, but the simplicity and conven- ience is so great that this may often be neglected. Another escapement of the first order is Montgolfier's hydraulic ram, which is a w-ater checking-ratchet train, the effi- ciency of which is low. A more recent device is the application of a water ratchet train to drive a pneumatic rat- chet train, first used on a large scale by Sommeillier in the construction of the Mont. Ceuis tunnel, and by means of which the efficiency was brought up nearly to 50 per cent.|] Pearsall has re- cently improved the hj'draulic ram and raised its efficiency to nearly So per cent., either for water or for air, but this FiG. 1017. \ See Mair, Experiments on a direct-acting steam pump. Proc. Inst. C. E. London, 18S6. j! The Worthington equalizer accomplishes an end sought by designers or steam pumps for the past 200 j^ears, for since Papins first machines in Cas- sel {[690) the desired aim has been to combine the action of a variable elastic driving medium, and a uniform, non-elastic resistance. II See the autlior's paper, Ueber die Durchbohrung des Wont-Cenis. Schweiz. polyt. Zeitschr, 1S57, p. 147. THE CONSTRUCTOR. ^33 has been done by the introduction of a valve gear, making it a device of the second order.* §326. Fi:,uiD Transmission at Long Distance. When the motive power is intended to operate the piston of a pump situated at a distance, some connecting mechanism must be interposed between the two cjdinders. Formerly this was accomplished by using long rod connections; instead of this a pressure-organ transmission may be employed. When water is used as the medium for transmission, this may be termed a " water rod " connection. This is used in connection with water levers (see \ 311). §327- Rotative Pressure Engines. An effective method of obtaining an advantageous action of the steam is to substitute for the reciprocating mass of the Cornish engine a rotating mass. This is accomplished by using the reciprocating motion of the piston to operate a crank shaft upon which a fly-wheel is placed. Since it is practicable to give the rim of the fly-wheel four to six times the velocit}' of the crank pin, the magnitude of the moving mass can be much smaller, and since the value varies as the square of the mean velocity, the mass is reduced at least 16 times. It is therefore possible by this means to give even small pumps an efficiency equal to that of large pumping engines. J It is not practicable to construct single-acting pumping engines into fly-wheels, because the piston speed v is too varia- ble. If we draw a curve representing v, the ordinates being the positions of the piston, we have for a connecting rod of infinite length a circular curve, as in Fig. 102 1 a. When the Fig. 1018. Fig. 1018 shows three devices for this purpose. At a is shown a closed system with pistons of equal diameter ; i is a similar one with unequal pistons ; and c is a form with combined pistons. Such water-rod connections are adapted for use in mines, and the following example will illustrate. / T^ i ,> V 1 .-es'^^ ? I \ ^)1/^ : Vi XU- :' / f i f\ i fmN ! : \ t : fr^\ I il 2 :i .Jx^ Lu'S f^-ft^ L.:-. L.* ^■--^-■'■ V \ f \ i / \ '■-V 3" V- i Fig. 1019. The arrangement of transmission in the Sulzbach-Altenwald is shown in Fig. 1019, which represents the engine above ground, while Fig. 1020 shows the mechanism in the mine. Fig. 1021. length of the rod is taken into account these curves are modi- fied, as shown in Fig. \02\b, which is drawn for a rod four times the length of the crank. This curve also shows the ratio of the tangential force on the crank pin to the pressure on the piston. \ The variations in the value of v, which often differ widelj- from the mean value Vm^ must necessarily be communicated to the mass of water, and hence great variations occur in the stresses. For this reason the velocity of the column of water must be kept within moderate limits, notwithstanding the use of air vessels. These variations become much less serious when two pumps are connected b}' cranks set at right angles with each other. The corresponding velocity curve is shown in Fig. 1021 c, and many pumping engines are now so made. More recently triple cylinder engines are made with cranks 120° apart. The velocity curves in this case are shown at d. It is evident that both these forms involve complications in con- struction which compare unfavorably with the direct-acting pump with equalizing cj-linders (see ? .125). Instead of uring a revolving fly-wheel, the mass of metal may be arranged to swing in an arc of a circle of large radius. An ingenious application of this principle has been made by Kley, in his water works engine with auxiliary crank motion. The proportion between the steam pressure and the vibrating mass is so arranged that the auxiliary crank comes to rest either a little before, or a little be3-ond the dead point, so that the re- turn stroke in each case can be effected by the action of a cataract. In the first case, the flv-wheel swings backward after Fig. 1020. The arrangement is of the same form as Fig. loiSi. The steam piston c operates the two plungers i, b.,, which in turn operate the plungers Cj c/, and c, c/ in the mine, the pump plungers e^ e.^ being placed in the middle f *See Engineering, Vol. XLI., 18S6, p. 47, also H D. Pearsall, Principle of the hydraulic ram applied to large machinery. London. 18S6. t See Zeitschriftfiir Berg, Hiitten und Salinenwesen, XXII. p. 179 ; XXIII., p. 6; XXIV., p. 35. The depth is 820 feet, the speed from 6 to 16 double strokes per minute, with a pause of one second, giving about 420 feet piston speed per minute. This engine, built by the Bayenthal Machine Works at Cologne in 1858. has operated regularly for 29 years without any interrup- tiou worth mentioning. X The Gaskill pumping engine is a duplex pump with fly-wheel, and cranks at right angles, and has given excellent results. See Porter's " Re- port of the Gaskill Pumping Engine at Saratoga." V g Referring to the designations ni Fig. 1022, we have — = sin u) -t- tan a. cos t,). Since Pdz ^ P' rdui and Pv= P' c, P' the ratio —7^ is also equal to the same ex- pression. Hence the curves above given also show the ratio of the force in the path of the crank-pill to the pressure on the piston. Fig. 1022. In Fig. 1022 ti and S, P and c are represented by 1.2'; in Fig. c, by 2' 2"; in Fig. 'i, by 2' . i' : in c and d, the ratio of connecting rod to crank is again takeii as infinitely great. The curves are adapted lor double-acting pumps. When two single-acting pumps connected to right-angled cranks are used, the second half of the cun-es of Fig. i become the same form as the first. 234 THE CONSTRUCTOR. the pause, and in the second case, forward." The valve motion of this form of engine is considered in the following section, ? 328. Valve Gears for Rotating Engines. Rotative engines are distinguished from pure reciprocating pressure organ escapements in that the}' deliver their effort in the form of rotary motion adapted to be used for driving run- ning machinery. Between the two forms there is also the intermediate kind, with merely auxiliary rotative mechanism, such as have been already referred to. The translation of reciprocating and rotary motion may be accomplished in a variety of ways, but by far the most useful and best known is that by which the rectilinear motion of a piston is transmitted to the shaft by crank connection. The variations in the tangential component of the pressure P' on the crank pin, Fig. 1021, becomes still greater when the pressure P, on the piston also varies by reason of the expan- sion of the steam. For this reason some form of equalizer is required in the form of a fly wheel. This latter becomes a reservoir for the storage of living force. Extreme examples of this action are found in rolling mill work in which within a brief time a looo H. P. engine may be called upon to deliver 2000 H. P., a demonstration of action of the flywheel as a reservoir of power. The valve gearing for rotative engines is an important ami extensive subject. In the preceding sections a series of valve gears have already been described. These have all been based upon the principle of operating the valves by a direct recipro- cating motion, taken either from the piston or piston rod. With rotative engines another method is used, the motion beini; taken from the revolving portion of the machine, and this method may also be adopted for pumps with auxiliary crank action. We may then distinguish between : Reciprocating valve gears, and Rotative valve gears. Rotative valve gears are desirable even for direct acting pumps, but in a still greater degree are they desirable for rota- tive engines. Watt's rotative engine was made with a recipro- cating valve gear.t and this form has one advantage in that it is adapted for rotation in either direction. Hornblower, the inventor of the compound engine, also used a reciprocating valve gear. The slide valve, .invented by Mur- dock, in 1799, led the way to the introduction of the rotative valve gear in iSoo, but the old reciprocating gear still continued to be used, and is even re-invented at the present time. The later direct acting steam pumps with auxiliary rotative mechan- ism are almost always made with rotative valve gear. Kle}''s pumping engine, referred to in the preceding section, is made with reciprocating valve gear, since its motion is both before and behind the dead points of the crank. The use of the slide valve, combining four valves in one mem- ber, enables a very simple valve gear to be made for the ordinary double acting escapement, as the diagram of a plain slide valve engine, Fig. 1023, clearly shows. Fig. 1023. The use of an eccentric r, and rod /, to operate the valve (5. is not the earliest form of gear, the first method being the use of an irregularly shaped cam which brought the valve to rest except at the time of opening or closing.^ A feature of the slide valve which was long overlooked was the fact that the time of closing the steam ports //and /// could be regulated so as to effect the proper expansion of the steam. In order to accomplish this result without impeding the exhaust of the steam, the eccentric r, must be given the so-called angle of advance 2° i . 2' beyond the mid-position. The direction of rota- *For a fuller account of this interestiiio; engine (German Patent No. 2345), of which more than fifty are in operation, see : Eerg-u. Hiittenm. Zeituny Gliickauf, 1S77, No. 18, 1S7Q, No. gS ; Mor.iteur des int. matdriels, 1877, No. 20°; Compt. rend, de St. Etienne, 1S77, June ; Bergg-ei.st. 1S79, No. 85 ; Z. D. Ingen- iure, 1879, p. 304, iSSi, p. 479, 18S3, p. 579. Duigler'. 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 S p 1.07 1-74 1.82 1.91 2.01 2.13 2.25 2.39 2-54 2.72 2.92

-^^ m- I" some instances the pressure 0.60 0.60 may reach as high as 2250 pounds, in which case the stress would reach 3200 X 1-5 = 4900 lbs. A 4 in. pipe at the Frankfurt Railway Station at 1500 lbs. has an outside diameter of 6.4 ins. Example 4 —The Helfeuberger Water Pressure Engine at Hersbrugg near Rheineck, of 40 H. P., has a cast iron feed pipe under a head of 1312 feet giving a pressure of 569 pounds. The pipe is 14,760 feet long, and 4.6 ins. diameter, and the thickness in the lower third of its length, is 0.43 ins. This gives Dq — 5.46 ins., D = 4.6, and from (333) ■ ^=569 5.462 . l62 5.462 _ 4.6'- ' Exajiipie 5. — Using the empirical formula (31 sizes under the assumption of 150 lbs. pressure = 3357 lbs. we have for the following Z>= 4" 5 = 0.36 y 0.S9 — 8.62 P S 1293 6" 0.39 0.88 12" 0.46 092 30" 068 0-95 48" 0915 0,96 7.S6 12.01 '9-5 24.51 1179 iSoo 2925 3176 The above values of S are taken as acting tipou the longitu- ilinal section of the cylinder, which is the case when a pipe is open at both ends. When the ends are closed, there is also to be considered the stress on a section at right angles to the axis, ■which is equal to /.< S. This, combined with the previous value, gives for the inclined resultant, \^S'^ -\- {o.^ S)' = \.\2 S as the minimum. These conditions exist in the case of a cylinder for a hydraulic press. These are usually made of cast iron, and the increased thickness adds greatly to the weight. It is therefore important to use mate- rial capable of withstanding a high stress, and to take great care in construction and in the disposition of the material. Repeated melt- ings of the iron give more homo- geneous castings. Good results are also obtained by adding wrought iron in the cupola. By thus im- proving the quality of the metal, the permissible stress may be in- creased. A stress as high as 10,000 lbs. may be permitted when the casting is assuredly sound. Simi- lar conditions obtain when bronze is used. With good bronze, if no alteration of form is to occur, the stress should not be greater than 5000 lbs. If it is desired to go higher, some harder composition, such as manganese bronze, must be used. A few practical examples will be given : Example 6.— In raising the tubes of the Conwav Bridge, a hj'draulic press of the following dimensions was used:* Diam- eter of ram, A'= iS inches; bore of cj'liu- der, /? = 20 inches; thickness, 5 = S3/j in. The load was 650 tons ^ 1,456,000 pounds. The pressure in the cylinder being 5900 lbs. per square inch, we get from (323) the stress i" = 10,500 lbs. The cylinder is shown in Fig. 1045. Example 7. — In the construction of the Britannia Tubular Bridge several forms of hydraulic presses were used. One of these was a double press with cylinders of the same dimensions as in the preceding example. The load on each ram was only 460.5 tons, and the cylinder pressure 4190 lbs., giving a stress on the metal 5=7460 pounds. Example 8.— The press which sustained the heaviest load on this great work was one which lifted 1144 tons, or 2,562,590 pounds. This was made Fig. 1045. with a single cylinder with a ram 20 inches diameter, cylinder 22 in. bore, and 10 inches thick. The water pressure was 8400, and the stress in the metal, according to (323), was 14,500 lbs. ! When the tube of the bridge had been raised 24 feet, tlie cylinder gave way, and the load dropped upon the safety suppoits, but was seriously damaged. The fracture was not longitu- dinal, but in the cross section near the bottom of the cylinder, as shown in Fig. 1046. The fracture was doubtless due to the sharp angle at the bottom. A new cylinder was made aud successfully used, tlie bottom being altered in shape as indicated by the dotted lines. The first cylinder which was cast for this press was moulded with the bottom up, but was rejected as being porous ; tlie second was cast bottom down, and gave way in use, as above described ; the third, for which the iron was melted twice, was suc- cessfully used to the end, while a fourth, which was maue as a reserve, was not required. Example 1:^. — A press designed for making compressed emery wheels has the following diniensious : B = 28.35 in., Dq = 40.94 in. K = 27.56 in. P= 2,640,000 lbs., from which p = 4425 pounds. We have ih = — -?- = o.6q. V 40.94 whence 5 = 12,134 pounds, wliicli must be considered a high stress. More recently the cylinders for hydraulic presses have been cast of steel, permitting stresses as high as 20,000 to 28,000 pounds. Modifications in the method of construction may also be made to enable cast iron to stand higher pressures. The danger due to casting the bottom in one with the cylinder may be avoided. The method used by Hummel, of Berlin, is to make the cylinder as a ring, aud the bottom as a separate plate (Fig. 1047). Lorenz, of Carlsruhe, makes the bottom separately and screws it in, as shown in Fig. 1048. By increasing the diameter of the ram to exert a given force, the pressure of the water re- quired will be reduced, aud the stress 5 will be less. This is not attended with a proportional in- crease in the amount of metal required, but on the contrary with a reduction. If the cross section of the cylinder be F, we have F^=TT [D -\- 6) 'S. Substituting the value of i from (321), nZ) - 2h we get P = % -^ and introducing K, Fig. 1046. 2.P ^= t; s~--p (335) which, for any choseu value of 6", diminishes as/> is reduced. Exajitple lo. — In a hydraulic press by Hummel, for making rollers of compressed paper, there are two cylinders of the form shown in Fig. 1047, placed side by side. The diameter A' of the ram is 23 inches, and the cylin- der diameter D 24 inches, the thickness 6 being 8J-2 inches. The load P on the ram is 2,200,000 pounds. The water pressure is 5174 pounds, and the stress on the material about 10,000 pounds. If we increase K to 26 inches, we have, since this is % the preceding value, the value of / reduced to {%)- = 0.79 of the previous amount, or 4087 lbs. If we now make the relation be- tween the inside and outside diameters of the cylinder the same as before, we have the same relation between .5" and /, hence >S= 10,000 X o 79= 7,900 K find the relation between the cross sections of the two cylinders will be also as 0.79 to I. Hence this alteration in dimensions which reduces the pres- sure in the cylinder also causes a reduction of about 20 per cent, in the amount of material. I 337- Wrought Iron and Steei- Pipes. Wrought iron pipe is in very extensive use for conveying gas, water, air, petroleum, as well as steam. These pipes are made either by the process of welding during passage between rollers or are riveted while cold. The former method produces either a butt- or lap-welded joint, the seam being parallel to the axis of the pipe, and more recently pipe has been made in America with a spiral lap-welded seam.f After welding the outside of wrought pipe is generally made smooth by passing betv.'een another set of rolls after re-heating, whence it is sometimes called "drawn" pipe. Pipe is also made of mild steel in the same manner as if wrought iron. The Mannes- mann system is also used for rolling tubing from the solid rod of steel, copper, delta metal, etc., the product being without any seam. Welded tubing possesses a great resistance to external pres- sure and to tension, but a less resistance to iuternal pressure. Butt-welded pipe should not be subjected to a greater stress than 5^1 500 lbs. ; but for lap-welded pipe 6" may reach Sooo *See Clark, The Britannia and Conway Tubular Bridges. London, 1850. t See Engineering and Mining Journal, April 7 and 14, 18 tific American, June, 1SS8, p. 377. also Scicn- 244 THE CONSTRUCTOR. to 12,000 pounds. Spiral lap-welded tubing has been tested to pressures corresponding to stresses from 30,000 to 40,000 pounds, according to the quality of the material used ; but in practical service lower values are used. The Mannesmanu tubes have been used without deformation almost to the elastic limit of the material, which, with cast steel and with Siemen's open-hearth steel, reaches 25,000 to 50,000 pounds, and there- fore possess a utility to which welded tubes have not attained. -!K V X-0—i.i^ Fig. 1047. Example i — In the oil pipe line showu in Fig. 954, 6 inch lap-welded pipe is used, I'i; in. thick, at a pressure of about 1000 pounds. We get from (324) „ 6 X 1000 ■J = —r^ = 9600 lbs. From (323) we have more accurately 5 = 1000 (6.625)° -f (6)= (6.625)=- (6)2 98S7 lb.=. Example 2.— It a spiral lap welded pipe had been used for the preceding example, the thickness 5 need only have been ^V in. Example 3.— If a Mannesmaun tube of Siemens steel had been used for the high pressure water service of Example 3, \ 3 j6, and the stress put at the moderate limit of 15,000 pounds, we get from' (324) Z? ^ 4",/ = 1500 lbs., 1500 X 4 ^ ^ ■ = 0.2", and from {323) we get more accurately i5,o°o X 2 .S= 1500 (4.4>' -f (4)' (4.4)=- (4)- ■ The steel pipe would weigh onlv about J that of the corresponding cast iron pipe.l As an example of the efficiency of this construction, Mr. Hamilton Smyth cites an installation over two miles long and under a head of 550 ft., the pipe lying on the surface of the ground and only protected from changes of temperature by a roof of roughly nailed boards, and in which the total loss by leakage was only 3 to 4 cubic feet per minute. As a consequence of the successful use of these pipes for mining purposes, they were next used for more permanent service as for water supply of cities, and with excellent results. Two such pipes were put in for the supply of drinking water for San J'"rancisco, and a third pipe, many miles long, was sub- sequently added. For large diameters in permanent installa- tions the sections should be riveted together, while for smaller diameters the joint may be made with lead, as hereafter de- scribed. The following table will illustrate some important constructions of this kind. Cherokee . . . Virginia City < Texas Brook . Humbug . , . ft. 1870I 12,800 1S7: 1873 37,116 37,116 4,44° 1,194 u s ■p ^ in. ft. 3° 8S7 II 1722 10 1722 17 781 2b 120 Description of Pipe. lb. 17,500 Sheet Iron, double riveted. i4,ooo| " " " " 14,000 Lap welded, screw connections. 17,000 Sheet Iron, double riveted. ii,5oo|^" sheet iron, single riveted. Also may be noted the Kimberley water works in England, 14 inches diameter, % in. thick and eighteen miles long. The superior economy of wrought pipe over that of cast iron is worthy of greater attention. lu order to illustrate the arrangement more fully of an in- stallation of such pipe the inverted siphon in the vallej' of the Texas Brook, constructed by Mr. Hamilton Smyth, is given, Fig. 1049. The difference in the level is 303.6 feet, and the total length 4438.7 ft. The pipe is in lengths of 20 feet and the figures in the diagram indicate the gauge thickness of the sheet iron in the various portions. The average diameter of the pipe is 17 inches and the highest value of the stres .S was calculated as equal to 16,500 pounds ; some of the plates were too thin and the stress in such places reached 18,000 pounds. The inlet is so shapted that the coefficient of contraction reaches 0.92. The pipe is bedded in gravel 12 to iS inches deep, and passes entirely under the bed Fig. 1049. Riveted pipe of wrought iron have been successfully used iu America for conducting over long distances, and valuable in- formation has been furnished by Mr. Hamilton Smyth, Jr., upon this subject.* Wrought iron riveted pipes were first used in California, made steel metal yV in. thick, to take the place of the canvas hose then extensively used in the operations of hydraulic rain- ing. The pipes were made of ordinary sheet iron, there being a single row of rivets, driven cold, and the joints made simply by inserting the end of one section into the next, as in the case of stove pipes. These first attempts succeeded Ijeyond all ex- pectations and were followed by numerous installations, in sizes reaching as high as 22" to 30'' diameter and sections 18 to 25 feet long. ^ A satisfactory protection against rust was obtained by immersing the finished pipes for a few minutes iu a boiling mixture of asphaltum and tar. If the fit of the ends was too loose to make a good joint the smaller pipe was wrapped with tarred cord, leaky places being stopped w'th wedges of wood and the small leaks being checked by sawdust admitted with the water. of the stream. During a large part of the year the siphon is not full of water and hence entraps much air. In order to per- mit this to escape, air valves of the coustructiou shown in Fig. 1050, are attached at suitable points, fourteen iu all being used. * See Engineering and Mining Journal, May and Tune, 1884 : also Journal of the Iron and Steel Institute, 1886, No. I, p. 133. Fig. 1050. These are simply heavy cast iron flap valves with rubber ring packing. When the chamber is filled with air the valve falls open by its weight but is closed by the action of the water THE CONSTRUCTOR. 245 when the air has escaped. In case of a rupture in the lower portion of the pipe, the air valves in the upper portion prevent the collapse of the pipe from atmospheric pressure. I 33S. STEAM Pipes. When steam is to be conducted to considerable distances, the condensation which is due to loss of heat through the walls of the pipes becomes so great that it is necessary to surround the pipes with a non-conducting covering. Materials for covering steam pipes play quite an important part in the science of steam economy and their manufacture constitutes an extensive industry. The importance of this subject has long been appre- ciated, having been considered, among others by the Industrial Society of Mulhouse more than sixty years ago. In these in- vestigations the measure of effect is the amount of water con- densed by a unit of surface, as one square metre per second. The following table will indicate some of the results obtained.* Material. Material of Covoriug. Uncovered Pipe Pimont's Mass. . Straw Graninies condensed per sq metre per second. 2.84 gr. 1.56 " 0.9S " Material of Covering. Clay Pipe . . Cotton Waste Felt GraniTOCs condensed per sq. metre per second. i.i2gr. 1-39 " 1-35 " The so-called Pimont's Mass, consists of loam and cows' hair, 60 mm. (2^ in.) thick. The straw was first laid on longitudi- nally 14 mm. (fj in.) thick, and then wrapped with straw 15 mm. (J's in.) thick. The cotton waste was 25 mm. (i in.) thick covered with canvas. The felt was saturated with rubber. Straw shows the best results, the condensation being only one- third that given by the uncovered pipe. These experiments have not great present value, partly be- cause the comparison by condensation of water is not altogether reliable, and partly because new material for covering pipes have since come into use. The Society of German Eugiiieers ( Verein Deutscher Ingenicin'c) has undertaken a series of ex- periments from which results of value are to be anticipated. In the United States, Prof Ordway, of Boston, has made some very beautiful investigations, the results being in two series, the first by the method of measuring the condensed water, the second by the calorimetric method.! The unsatisfactory char- acter of the method of condensation is apparent, as it was found, for example, that a portion of pipe 2 feet long condensed 32S grammes of water per square foot per hour, while 30 feet of pipe gave only 140 grammes per square foot per hour. It is also to be noted that Prof Ordway's first researches showed much less condensation for the uncovered pipe than appeared in the Mulhouse experiments, so that no definite conclusions could be deduced. The calorimetric method appears to be much more reliable, as the results appear to be more consistent. From the great number of experiments the two following tables have been selected. Table I. Air Space Carded Cotton .... Feathers Wool • • • Calcined Magnesia . . Cork Charcoal, coarse . Calcined Magnesia . . Wool Lampblack Carbonate of Magnesia Fossil Meal Wool Asbestos Zinc, White Fossil Meal Pine Charcoal Carbonate of Magnesia Hair Felt Lampblack ..... . Chalk Graphite Calcined Magnesia . . Zinc, White Pumice Stone Per Cent. Ivilo-Cent. Solid Material. Heat Units. 0.0 1302 I.O 310 2-0 321 2.1 301 2.3 335 31 343 4.9 340 5-6 220 5-6 266 6.0 371 6.0 393 7-9 238 8.1 '329 8.8 466 II. 2 426 11.9 376 15.0 416 1S.5 177 24.4 286 25-3 560 26.1 1922 28.5 1156 323 1164 ,34.2 Hi Plaster of Paris Common Salt . - Anthracite Coal , Fine Sand . . Coarse Sand . Per Cent. Kilo-Cent. Solid Material. Heat Units. 36.S S39 48.0 I9S3 50.6 968 51.4 i6go 52.9 1684 Temperature of steam 155° C. All coverings i inch thick = 25.4 mm. This table gives noteworthy, and in many cases unexpected results. It is important to note that in all cases the trans- mission of heat bears a definite relation to the percentage of solid matter. For instance, calcined magnesia gives off 335 to 1 1 56 heirt units when the percentage of solid matter ranges from 2.3 to 28.5. Asbestos makes an unfavorable showing, and lampblack gives good results but is inconvenient to use ; wool, is excellent. In practice the cost is of course an important consideration. Table IL Temperature of steam 155° C. Glazed Cotton Wadding Wool Wadding , Calcined Magnesia, loose . . " ■" crowded . " " compressed Carbonate of Magnesia, loose " " crowded " " compressed Fossil Meal, loose " " crowded j Cork in Strips 'i Silicated Cork Chips Paste of Fossil Meal and Flair . Carded Cotton Rice Chaff, straw board .... rhickness. Milli- metres. Per Cent. Solid Matter. Kilo-Cent. Heat units. 50 I-OJ I29.I 40 1-3 193-4 3° 1-7 205.5 20 2.5 326.4 15 3-4 424.2 10 5-1 502.4 25 5-6 219.8 25 2.3 335.2 25 4.9 340.1 25 28.5 "55.9 25 6.0 370-9 25 9-4 386.7 25 15 416.5 25 6.0 393-4 25 n.2 435-8 15 ? 87.1 30 ? 59-2 9 1.0 69.4 50 ? >57-7 12 ? 71.9 This table gives a comparison of fibrous and granular ma- terials. In the first cases the same material was successfully compressed, reducing the thickness and increasing the density, showing and increasitig loss of heat. Ordway recommends cork as the best material, especiall}' in the form of cemented chips, which may be formed into semi-cylindrical sections, as has already been done in Germany.]] Ordway does not advise air space under the covering, but rather recommends the filling such space with alight powder. Of all the materials tried he recommends in the order given: Hair Felt, Cork, Fossil Meal, Magnesia, Charcoal and Rice Chaff.f Prof Ordway remarks that " it is useless to make the testing apparatus of cumbrous dimensions, for as in chemical analysis we use a gramme or less of the sample, instead of kilo- grammes, so in physical experiments increase of size does not uecessaril3' enhance the accuracy of the results." In long stretches of steam pipe the expansion from the heat demands the use of some compensatory device or expansion joint.** Fig. 1 05 1. Some of the forms in general use are shown in Fig. 1051. As a, is a packed expansion joint; h, is a bent copper pipe; c, a drum with flexible steel diaphragms. * This table has been kept in the metric system, as it is only available for comparison. — Trans. \ See Trans, Am Soc. Mechanical Engineers, Vol V. p. 73 ; Vol. VI. p. 168. t The cork was put on like barrel staves, with a slight air space beneath. ^ The cork was chopped into small chips and mixed with ^Y^ t;mes ita weight of water glass at 30° Beaume. II See Z. D. Ingenieure, 18S6, p. 38. \ A new, and efficient as well as cheap material made of common flour paste and saw dust, is described in the Revue Industrielle, Sept. 1S8S, p. 346. ** This "compensation " does not neutralize the expansion, as in a pendu- lum, but only renders it harmless. 24^ THE CONSTRUCTOR. Fig. 1052 shows a U joint with packed connections. The forms given in Fig. 105 1 generally require one position of the pipe to be held fast ; that in Fig. 1052 permits both lengths of pipe to remain free. Fig. 1052. In calculations the actual amount of expansion due to any- given temperature we may put the expansion, if t, be the dif- ference of temperature in degrees for : Materia!. Cast Iron Wrought Iron Copper . . . , Brass Centigrade. Fahrenheit t t 90, 1 00 162,180 t / ~^476oo 155,280 t t 58,206 104,760 t t 53,500 92,30 Example. A cast iron pipe 98.4 ft. long, (= iiSi.i inches). At a tempera- ture of 50° F. is filled with steam at 63 pounds pressure, =310° F. The ex- pansion will then be J181.1 X 260 ; — .; = T.Sg.in. 162,180 ^' I 339. Pipes of Copper axd other Metal. Brazed pipes of copper when used as conductors of steam, should not be subjected to higher stresses than 1500 to 2000 pounds, since the brazed joint is not reliable and reduces the strength of the cross section of the metal about one-third. The heat due to the temperature of steam at pressures from 60 to 100 pounds also reduces the strength of the copper from 10 to 12 per ceut.* Seamless pipes made from the solid metal, or rolled by the Mannesmann process, can stand stresses from 8,000 to 10,000 pounds, and when made by forcing in the hydraulic press (see ? 333, b- 5) only a stress of about 700 to Soo pounds. Wooden pipes for water conductors, made water-tight with cement, have been made by Herzog of Logelbach with excel- lent results; the most recent being 71 in. diameter, and 5900 feet long. Pipes made of paper coated with asphalt have been used to a limited extent, but do not stand the heat of the sun. § 340. Resistance to Flow in Pipes. The resistances which oppose the motion of a liquid in a pipe are due either to changes in the direction of motion, to changes in the rate of motion, or to the resistance of friction. We can only here consider a few cases, and those will be limited to the flow through pipes. Frictional Resistance. — When a flow of water takes place in a vessel with flat walls, through a cylindrical tube, Fi.g. 1053, the difference of level between the surface of the water and the * Investigations made after the explosions on the Elbe and the Lahn will be found in Engineering for August, 18S8, pp. 113, 116, 125. These gave for the modulus of rupture jT for tension; for hard brazed pipes A' = 33,400 ; for seamless electrically deposited pipes, K = 50,000. The reduction in strength due to heat is given according to the old but reliable experiment of the Franklin Institute. mouth of the discharge pipe being /;, we have, according to Weisbach : // O+co+4) (326) in which / is the length and d the diameter of the tube in feet, and V the velocity in feet per second. The volume of flow will be : Q-- •d'v . (327) per second. In (326) fg is the coefiicient of friction for the orifice of influx, and C the coefficient of friction for the rest of the tube. The coefficient Cq, when the entrance is a sharp angle, be- comes considerable, having a mean value of 0.505, but when the entrance is carefully rounded it may fall as low as 0.08. In the latter case, for long tubes, Co may be neglected. f I'or the coefficient of friction C in the pipe various deductions have been made. The conditions which affect the flow of water in pipes are numerous and variable. In cylindrical pipes the particles arrange themselves in such a manner that those in the axis move with the greatest velocity, and each successive annu- lar sheet moving slower, while the particles in contact with the walls of the tube remain practically at rest, so that the velocity of each annular film, from the wall to the axis is a function a the distance from the wall to the centre, increasing from zei to the maximum. In the case of gases the velocity of adjacent rings appro*, mates much more closely than with liquids. In both instances the resistance is the sum of the friction of the successive annu- lar layers upon each other. In practice, the variation of the section of the pipe from the circular form must be taken into account, and also the rough- ness of the walls. The mathematical expression of these rela- tions cannot be a simple one. In practice also many disturbing influences exist, such, for instance, as ice, weeds, etc. In all comparisons with calculated resistances it is therefore essential that the walls of the pipe should be ascertained to be smooth and clean. The Societ)' of German Architects and Engineers has in progress modern iuvestigations conducted by several of its members with a view of determining the most useful formula for finding the value of C for water. The value which such a formula would possess is undoubted, but before it can be satis- factorily determined the fundamental principles of the subject must be determined.! We are at present obliged to use for- mulas previously determined. Among these the formulas of Weisbach and of Darcy are most available. If we express the loss of head in the height // by friction //j, in feet, we have for water, according to Weisbach : h,. I 0.01439 -I- / — (32S) all dimensions being expressed in feet, and "• being the accelera- tion of gravity.^ We have for: V = 0.1 C = 0.06S6 0.0527 0-3 0.0457 0.4 0.0415 0.03S7 11 = 0.5 C = 0.0365 0.7 0.0349 o.S 0.0336 0.9 0.032s C = 0.0315 C.0297 0.02S4 2 0.0265 3 0.0243 C = 0.0230 6 0.0214 8 0.0205 12 0.0193 20 0.0182 According to Darcy we have for water : I v'^ f ^ , 0.00166 ^, / il^ — — == 1 0.019S9 -^ — j — — . d 2o- \ -^ d J d 2g which gives somewhat .greater values thau does Weisbach formula for the higher velocities. 11 }u = ^ (329) t If the tube starts from another tube instead of from the side of a reser- voir, the coefficient of resistance becomes much greater and much care must be given to the shape of the entrance. See Hertel, Zeitschr. D. Ingenieure. 1SS5, p. 660, also \V. Roux, Jenaische Zeitschr. fur Naturwisseuschaften, Vol. XII., 187S. X See a Memoir of tlie D. Arch- u lug.-Yereine, edited b^' Otto Iben, pub- hshed by Meissner, Hamburg, iSSo. This must be used with caution oa account of numerous typographical errors. g This and the two following formulas may also be used when in addition to the height /ji a second height h-^' is to be added due to the contraction ot discharge. This is only of importance in the case of high pressure water transmission, and experimental researches are to be dej:'*ed. il Recherches Experimentales, etc. Paris, Mallet-Bachelier, 1857, THE CONSTRUCTOR. 247 The formula of M. de Saint Venant, which gives lower results than either of the above, is : Aj = (0.0321 V ' 2 / v^ ^ d 2g (330) If we insert in equation (327) the value for v from the equa- tion /?! 'D — we get : 2g :=r-)^^..4.f=(^)f.4=-^.^^- 2g By assuming f constant, as proposed by Dupuit, * we may- state the practical formula : 'h Q" Cd^- Dupuit makes C = 0.03025649, whence C becomes 1313, and we have : rfS; (331) L(^\ 'H V 36.237 and hence for an approximate formula : •^-^WS) '»" These formulas cau be applied so that first from the given values of Q and / aud the friction loss of head //p the diameter D may be determined, and then by making Z> somewhat larger, and applying the formula of Weisbach or Darcy, the excess of head over friction determiued. This will be illustrated by a few examples : Example i — The large inverted siphon described in ; O = 17" = 1.4-7 f'-i ' = 4438 ft. and i? = 32 cu. ft. 32 337. Fig- 1049. gives From th.s we get — „ , , — , = 21.2 ft. per second. o. 7854 X (1.417)- These ^ve in Weisbach's formula : 0.017155 = 0.01439 + %/- : 0.0181, and from (328) h-^ = 395.6 ft. The actual difference of level is 303.6 feet, and hence the coefficient as de- termined from Weisbach, is too high. The coefficient determined from the given difference of level is ^ = 0.0155, and as a flow takes place the actual coefficient must be somewhat less. According to Saint Venant's formula (33°) ^'1 = 293 feet, which is slightly under the actual difference of level. Example 2. — In the work of Iben, already referred to, is given a case in Stuttgart in which /= 3614 ft., I) = 0.33 ft , v = 2.063 ft. From these data, Weisbach's formula gives ^ — 0.0263, and thence k-y = 19 ft. The actual value is 23.2 ft., which corresponds to ^ = 0.0332. The difference is probably due to the construction, there being two stop valves and six elbows included in the resistance. Example 3. — Another instance in Stuttgart is as follows : / = 302 ft. ; D = 0843 ft ; !J = 5.897 ft. The friction head as determined by observation for / = 328 ft., was ^2.89 ft. According to Darcy, the friction head would be 76.57, which is quite close to the e.'cperimental resuUs. In other instances, however, Darcy's formula has not agreed so well with experiment. When air is used instead of water, Weisbach gives for the height of a column of water equal to the frictional resistance : /'i = fi — r = o-°25 -7- ■ .... (333) a 2gE a 2g t in which e is the ratio of the density of the air in the pipe to that of the external atmosphere. Since is e always greater than unity when the air in the pipe is under pressure, //j is smaller than is the case for water, especially when the pressure of the air is great. Valuable experiments upon the transmission of compressed air have been made by Engineer Stockalper at the St. Gothard tunnel. f These showed that Darcy's formula (329) served well for air when the results are multiplied by the ratio of the density of the air to water. Professor Uuwin has given some valuable researches upon the friction of air, in which he shows the important influence which D exerts upou C.J Exa7nph 4. — At the construction of the Hoosac Tunnel it was observed that the pressure of compressed air fell from 821 pounds per square inch to 801 pounds in being transmitted a distance of about 118,000 feet. Resistance in Angles and Bends. — The resistance due to an angle, such as Fig. 1054 a is important, and is dependent upou what Weisbach calls the semi-angle of deviation, /3, according to the following formula : 7;J K-=L V' (°-9457 ■S'Z'J' /5 + 2-047 sin^ (3) — . (334) *See Dupuit, Traite theoretique et pratique de la condxute et de la dis- tribution des eaux. Paris, Dunod, init ed. 1854; snie ed. 1865. f Stockalper, Experiences, fates au Tunnel de Saint Gothard, sur I'ecoule- ment de I'air comprime. Geneve, 1879. t The coefficient of friction of air flowing in long pipes. Proc. Inst. C. E). tyOndon, 18S0. from which we get : ^ = 10 20 30 40 45 50 60 70 C2 = 0.046 0.139 0.364 0.740 0.9S5 r.260 1.861 2.431 Example 5.— In a right angle beud (3 = 45°, the loss is practically equal to- FlG. 1054. In the case of bends, Fig. 1054 b, the resistance is not so great, but is too large to be neglected since we have : '■ '90 ■ 2g' The ratio of the radius of the tube to the radius of the curva- ture of the bend affects the coefficient as below : 0.5 Z? (335) -^ = O.I r 0.2 0-3 0.4 0-5 C2 = 0.131 0.138 0.158 0.206 0.294 ^— = 0.6 r 0.7 0.8 °9 I.O fj = 0.440 0.661 0977 1.408 1.97S Exaj/tple 6. — For a right angle bend in which r ^ D we have : ,, 45 »- "3 h^= 0.294 = o- 147 90 -zg ^' 2g or only about f the resistance of a sharp bend with any curvature. Resistances due to Sudden Changes of Cross-Section. — When water which is moving at a velocity v^ suddenly changes to another velocity v, see Fig. 1055 a, it experiences a loss of pressure which, according to Weisbach, is equivalent to a height : v^—v^ _ Z' F^ ^ ^ ^ — r ^ (336) F and F^ being the respective cross sections ; also Fzi^= ^\T\- Doubling the cross section causes a loss of head equal to ^S Fig. 1055. For gate valves. Fig. 1055 b, or cocks. Fig. 1055 c, there is a loss due to the amount of contraction. For gate valves we have from Weisbach : Openings = % j{ 3/i K f s 3/ ^ F ~ = 0.159 0.315 0.466 0.609 0.740 0.836 0.94S F C3 ^ 97. S 17.00 5.52 2.06 oSi 0.26 0.07 and for cocks : 20" 40" 50° 60° 65° 825 10.850 0.692 0.535 0.3S5 0.250 0.137 0.091 Angle ■ ^1 F C = 0.29 1.56 5.47 17.3 52.6 206 4S6 CO From the above tables it will be seen how important an in- fluence is exerted by valve chests, mud traps aud the like upon the flow of water. In all such cases it is important to modify the suddenness of the change of velocity by rounding and curv- ing all angles in the passages, and in this way a large part of the loss may be obviated. For gaseous fluids the resistance is less, but is at the same time sufficiently important to be care- fully considered. For a fuller discussion of the resistances offered to water in canals and streams the reader must be re- ferred to special treatises on the subject. 24 « THE CONSTRUCTOR. ?34i. Methods op Connecting Cast Iron Pipes. One of the most frequently used methods of connecting cast pipes is by means of the common flange joint, Fig. 1056. Fig. 1056. The proportions are given in the illustration. Formerly it ■was customary to raise a small bearing surface inside the boit circle, but this is generally omitted now, and the entire surface of the flauges finished, making a much better joint, although a trifle more expeusive. In many instances a ring of copper wire, let into a groove, is used to make the joint. For pipes which are not subjected to very high pressures the number of bolts A, is determined from the following : ^ =2 + D (337) in which D is the diameter of the pipe inches. This would give for a pipe 4 inches in diameter four bolts, and for one 3 inches diameter 6 bolts. According to (337) an air pump cylinder 60 inches in diameter would have 2 -f- -'i- = 32 bolts. When the pressure is known to be great, or for cylinder lids, etc., the following formula is to be preferred : 2400 © (338) in which d is the diameter of the bolts, D the diameter of the cylinder, and a the pressure in pounds per square inch. This assumes the diameter of the bolt at the bottom of the thread to be o.S d, and the stress in the bolts to be 3500 lbs- as in for- mula (72). Example. — .\ steam cylinder 40 inches in diameter, subjected to a pressure of 60 pounds, would have according to {320) a thickness of 5 = 0.7S7 + f^f, = •L^z in. This gives from Fig. 1056 for the bolts, ^ = ^ X i?c = 1.5S, say \^^ in., and these values in (33S} give for the number of bolts : ."I = 60 2400 \ 1.58 (Compare close of Chapter XXVI). i)" 16 bolts. Fig. 1057. Flanges with ears, as shown in Fig. 1057, are frequently used, the thickness being made 2 to 2.5 &, instead of 1.6 t!, on account of the smaller flanges. On the Prussian State railways flange joints are made with a lenticular shaped ring inserted in the joint, as shown in Fig. 1058. This permits a certain amount of motion and gives good re- sults in practice. The following table of dimensions is based on one used on the Prussian railways : D 2 = K 2^ ^H 3 3H 4 4K s sH 6 r>, 3 3K ■IM ■■?« 4 3">l 5 '.H 6M 6% iM r = '/« ^/4 2H ^'A 3 iH 4 AM 4H A'4 s }4 t-E A H H H Ji Ji iS Fig. 1059 a shows a cast iron bend with flange. The bend should not be too sharp, in order to avoid excessive resistance to the ilow of the water. ^See Example 6, I 340.) Bends of this sort require a separate pattern to be made for every different angle. Brown's joint is more convenient in this respect, Fig. 1059 b. The bolt holes in this form should be drilled in only one of the flanges first, and the other flange marked off in JQ — place. For any flange angle c< the pipes may be connected for any angle between 2 c/ and iSo°. In the illustration c< = 40°, which answers for most practical purposes. Fig. 1059. Bell or socket connections are much used for gas and water pipe. The joint is caulked with lead, which may conveniently be made in ha;lf rings and driven in, or run in in place, a pack- ing of oakum being first driven in. Fig. 1060. The large end of the pipe is called the bell, the other the spigot. The dimensions of the various parts in Fig. 1060 may be taken as follows, the thickness d being determined from for- mula (318), i.e., i = 0.315 80 Thickness of bell, i^ Thickness of bead, k Inside length of bell, /, Length of bell reinforcement, /.^ Outside length of bell, / Space for packing, b Depth of lead ring // Length of bead on spigot. a Thickness of bead, c =•0.375'' + 0.0135 Z?. = 0.7" + 0.0025 D. = 2.625'' + 0.1 1 D. = 2'' 4- 0.09/?. =^ 4.625" + 0.20 Z). = 0.1875" + 0.007 .0. = 1.125" -f 0.07 £>. = 1.2 6. = 6 -\- b — 0.0625". Some makers put a bead around the inside edge of the bell to assist in retaining the lead packing, but others consider this but little use, owing to the softness of the metal. More recently the bead has been altogether omitted from the spigot end, a shoulder being cast on the inside of the bell instead. In Belgium a joint is used in which a gum ring of globoid form (see Fig. 637 a) is used instead of the lead packing, the ring rolling in as the spigot is pushed into the bell. Fig. 1061 is Petit's pipe joint. A gum ring is inserted in the short bell, and one clamp being connected the pipe is used as a lever to compress the gum ring, the second clamp can be secured. This coupling, which was used in the extensive water system of the camp at Chalons, is cheap and can be rapidly THE CONSTRUCTOR. 249 connected, and possesses a certain flexibility which permits it to be used in running a line of pipe over uneven ground. ""Hf'^ Fig. 1061. A form of screw connection for cast iron pipe is shown in Fig. 1062. The screw thread is cast on the pipe and a leaden ^-- •^asket is placed so as to pack the joint outside of the screw connection.* This may be considered as a flange joint with a single central bolt, which latter is made large enough to permit the pipe opening to pass through it (see \ 86). Since the pipe must be revolved in making the connection, it is necessary to provide wrenches of suitable size for the purpose. i,5(} Fig. 1063. Fig. 1063 shows Normandy's pipe joint. The packing con- sists of two rubber rings. This very simple joint is ver}' useful under certain circumstances, where the proper packing is avail- able. It possesses the flexibility of Petit's joint and is easily connected and disconnected. A similar form of joint has been made for water pipes, using packing rings of lead. The sleeve may be considered as a double bell and the pipes are perfectly straight without any bead at either end. The distance from the centre of one joint to that of the next constitutes a "length." With cast iron pipe this is made a minimum of about 4 to 7 feet, being made as long as practicable for extensive lines of pipe. For gas and water pipe with bell and spigot connections the following pro- portions occur in practice : /? =: 4 inches, / = 7 to 10 ft. " /= 10 to 12 ft. " and over, / = 12 ft. A form of joint used by Riedler for high pressure water connections is shown in Fig. 1064. t The flanges are faced in the lathe and bolted together without any packing in the joint. A leather ring is placed in a channel turned ir the pipe and held in place by a spring ring in two parts, or this latter ma)' sometimes be made in one piece. Joints with spherical contact surfaces have been used by Hoppe for cast iron high pressure pipes when they are to be laid in yielding ground, j Three forms of constraction are shown in Fig. 1065. At (Z is a single ball joint. * This joint is used by the Lauchhaminer Works for pipe up to lYz inches diameter. tSee Zeitschr. D. Ing, Vol. XXXH., i38o, p. 481. i German Patents, No. 42,126. Fig. 1064. The bearing ring is held in position by a ring of bronze divided at right angles to the axis ; this form permits a deflection of 5°. At b is shown a double joint constructed in a similar manner Fig. 1065. and permitting a deflection of 10°. The third form, which is the most recent, has no packing ring, and the bolts are made with spherical heads to facilitate motion. ? 342. Cox>fECTioNS FOR Pipes of Wrought Iron and Steel. Riveted pipes are often connected by means of wrought or cast iron flanges, as shown in Fig. 1066 a and b. When no b i'l 4a ^M. ■;r^ Fig. 1066. k-j^d--^ other data are at hand, the diameter and number of bolts ma}' be_ determined by assuming the pipe to be of cast iron, and using the proportions given in the illustration. The actual thickness i of the pipe may then be determined independently according to the material, pressure, and other conditions. Exajnfile.—A. wrought iron pipe 3 ft. 4 in. in diameter, for delivering water to a turbine, is to be fitted with flanges of wrought iron. A cast iron pipe of this diameter would have a thickness, according to (318) — « Q,-" 80 whence from Fig. 1056 c? = J X 0.815 = 1.05", say iji^. The number of bolts, according to {337). will be 2 -|- ^'^ = 22. If the internal pressure is 30 pounds per square in. we have, according to (324), taking .S= 4200; S = 0.5 ^° '''' *° = 0.143". saysV- 4200 LM Fig. 1067. For thin pipes a very practical form is that shown in Fig. 1067 a. The ends of the pipes are flanged over, and the turned- over ends countersunk into the cast flange rings, the bolt heads also being countersunk. A similar form with wrought iron flange rings is shown at b. l For the thin pipes described in ? 337' w-hen subjected to a high internal pressure, the joint shown in Fig. 1067 c is adapted. In this form a short sleeve is riveted into one of the pipe ends and a loose ring slipped over the outside of the joint, forming a space into which lead is run and afterwards caulked. This also serves as a sort of expan- sion joint (compare § 338). Many important constructions are made with wrought iron pipe. The connections are usualh- made bj' screwing the parts together, and for this purpose many special pieces are made, known by the generic term of " pipe fittings." For straight connections the ordinary "socket" is used, while for angles the so-called '"elbows " and " tees " are made. g For description of a flange joint with welded conical rings, by De Naeyer, sec Zeitschr. D. Ing., Vol. XXX, 1866, p. 106. 250 THE CONSTRUCTOR. The American practice of making the thread tapering is much to be recommended, since b}- means of a little cementing mate- rial a tight joint may be made. The American Mechanical En- gineers have given careful attention to the proportions of pipe fittings, and since 1887 the forms proposed by the late Mr. Robert Briggs have been generally adopted.* The system is as follows : The thread is of triangular section with the angle 2 ;3 = 60°, as in Sellers' system. The top and bottom of the thread are flattened Jj of the theoretical depth 4, so that the actual depth t = 0.S4, and hence equal to 0.69 of the pitch J, Fig. 1068 a. v/Vj" Fig. 106S. The end of the pipe is given a taper of jV on each side, the length of the tapered part being 7"= (4.8 -j- o.?,D)s, D being the outside diameter of the pipe and .f the pitch. Beyond the taper portion comes a length 7", = 2 i~, which threads are full at the root but imperfect at the top, beyond which there is a length T^j = 4 i, consisting of imperfect threads blending into the full outside diameter. The thickness & of the pipe is such that the thickness of metal below the thread at the end of the pipe is = 0.0175/) + 0.025". The pitch .s is finer than for bolts of the same diameter, there being only five different pitches used, and the various dimensions are given in the following table : Tabi,b; op Standard Pipe Threads. Diameter of Pipe. Screwed Ends. Thickness of Metal. Nominal Inside. Actual Inside. D. Actual Outside. Do. Threads Per Inch. Length. T. Inches. Inches. Inches. Inch. No. Inch. % 0.270 0.405 O.06S 27 0.19 % 0.364 0.540 0.088 18 0.29 Vi 0.494 0.675 0.091 18 0.30 % 0.623 0.840 0,109 14 0.39 X 0.824 1.050 O.I13 14 0.40 I 1.048 I-315 0.134 11;^ 0.51 iJi I 380 1.660 0.140 ">^ 0-54 ^% I.610 1.900 0.145 II "X 0-55 2 2.067 2-375 0-154 II>^ 0.58 ^y^ 2.468 2.875 0.204 8 0.89 3 3.067 3500 0.217 8 0.95 i% 3.548 4.000 0.226 8 1. 00 4 4.026 4.500 0.237 8 1-05 e/z 4-508 5.000 0.246 8 1. 10 5 5-045 5-563 0.259 8 1. 16 6 6.065 6.625 0280 8 1.26 7 7023 7.625 0.301 8 1.36 8 8.082 8.625 0.322 8 r.46 9 9.000 9.688 0.344 8 1-57 10 10.019 10.750 0.366 S 1.68 Taper of conical portion of tube i in 32 to axis of tube. It will be observed in the table that the thickness d agrees very well with the formula (5 = o.ii i ^D^. This gives for the diameters 0.405, 1.050, 4.000 and 10.750, the thicknesses 0.071, 0.114, 0.222, 0.364, which agree quite closely with the actual values. The shape of the sockets is shown in Fig. 1069, the thread being given a somewhat greater taper than jV, so that the greatest stress will come on the strongest part of the socket. The increasing use of such pipe in Germany makes it most desirable that a standard of dimensions should be adopted. The American system is manifestly unsuited for use with the metric system. The general arrangement of the American system may, however, be followed with some approximations to adapt it to the metric measurements. The angle of thread may be the same as in the American system ; 2 /3 = 60°. The depth of thread may also be abbre- viated y'j top and bottom, making / = 0.8 i^^ = 0.68 .y, and the taper can also be made -}, on a side. The length T of the tapered portion may be niade 7" = (5 + ,ij j9„j j, which is about the metrical equivalent of the former expression, the nearest even value being taken. The lengths T^ — 2S and T-^ ^ AS may be retained. For the thickness of pipe the American formula transformed gives S = 0.555 ^/7?o in millimetres. Finally for the pitch we may take s ^ I 1.4 1,8 2.2 3.2 mm. (0.94) (1.41) (I -81) (2.21) (3.17) in. the values in parentheses being the corresponding equivalents of the American pitches. The following table gives the values from 10 to 325 mm. This system has been submitted by the author to the manufacturers of the Mannesmann tubes in Remschied, Saarbriick and Komotau, and by them adopted. Metric Pipe Thread System. Outside Diameter Thickness Inside Pitch Length of Thread Length Length -Do- D. T. 7i — 2 J. 7; = 4 J. iO 1-75 6-5 I.O 5-5 2.0 4 15 2.00 1 1.0 1-4 7.5 2.8 5-6 , 20 2.50 15-0 1.4 8 2.8 5.6 25 2-75 1 9-5 1.8 11 3-6 7-2 30 3-00 24. 1.8 12 3-6 7-2 35 3-25 28.5 2.2 14 4-4 8.8 40 3-50 33-0 2.2 15 4-4 8.8 50 4.0Q 42.0 2.2 ■ 15 4-4 8.8 60 4-25 51-5 2.2 16 4-4 8.8 70 475 60.5 3-2 25 6.4 12.8 80 5.00 70.0 3-2 26 6-4 12-8 90 5-25 79-5 3-2 28 6.4 12.8 100 5-50 89.0 3.2 29 6.4 12.8 110 5-75 98.5 3-2 30 6.4 12.8 120 6.00 108.0 3-2 31 6.4 12.8 130 6.25 1 17-5 3-2 33 6.4 12.8 140 6.50 127.0 3.2 34 6-4 I2.S 150 6-75 136-5 3-2 36 6.4 12.8 '75 7-25 160.5 3-2 38 64 1 2.8 200 7-75 184.5 3-2 42 6.4 12.8 225 8.25 208.5 3-2 45 6.4 12.8 250 S.75 232.5 3-2 48 6.4 12.8 275 9-25 256.5 3-2 51 64 I2.S 300 9-50 281.0 3-2 54 6.4 12.8 325 10.00 305. 3-2 58 6.4 12.8 In the preceding table the pipe is classified according to its outside diameter /?o, but it is a question whether it would not be better to follow the custom of designating the sizes by the internal diameter D. The former, however, has an important influence upon the dimensions of the fittings, which it is most desirable to reduce to a standard system. It will be seen by reference to the table of American pipe dimensions that the actual internal diameter differs frequently from the nominal size, the latter really being only a convenient name. By adopt- ing a strict gradation for the sucessive sizes of D^ it would be practicable to make the thickness & somewhat less than given in the table, but in some cases it would be greater. When D^ is greater than 325 mm., & may in ordinary cases be made = 10 mm. The production of the screw threads both in pipe and fittings must be carefully considered in order to insure the interchange- ability which is necessary. Powerful and accurate machines have been devised for cutting the threads, as well as devices for producing the taps and dies, and also gauges to insure mainte- nance of standards. This branch of the art has been carried to a high degree of perfection in America. Fittings for Wrought Pipe. The simplest pipe fitting is the socket used for connecting two pipes of equal diameter D^, and is made of wrought iron * See Trans. Am. Soc. Mech. Engineers, Vol. VII, pp. 311 and 414 ; also Vol. VIII, pp. 29 and 347. FlG. 1069. or of steel. It is made of sufficient length to give a thread in each end of length equal to T, as given in the preceding tables, together with a slight clearance between the ends of the pipes, Fig. lodtja. In many cases the socket must be made with right THE CONSTRUCTOR. 251 and left hand threads, as in Fig. 106915, this being necessary to connect two pipes which cannot be turned axially. For other connections a variety of fittings are made, examples of which are shown in Fig. 1070. FlO. 1070. In Fig. 1070, a is a right angle ; b an elbow (abbreviated in practice to " ell ");<:• is a T ; rf a cross ; and e a reducing socket. These fittings are used as connections for all sorts of gaseous pressure organs. They may also be used for liquids, as water, brine, oil, etc., when the velocity of flow is not great. For im- portant installations it is becoming more and more the practice to design the fittings iu such forms as to produce a minimum of resistance. By making the fittings of cast iron, as is done in England and America, where pipe constructions are very ex- tensively used, it is possible to adhere to accurately designed standard forms. The most important fitting is the elbow, for the right angle bend occasions far too much resistance to be used in important cases. In Fig. 1071 three forms are shown, all of which are A i> c D i;>. D,;-T I r — 4-D--' i Fig. 1 07 1. designed to be used with the thread already described. Of these, form b is the most popular, although form a is frequently used because of the smoothness and neatness of external ap- pearance. Form c is here proposed as an additional design. A comparison between the three forms will show a difference in resistance which may be calculated as follows : The resistance, may be divided into two portions ; one due to the curvature, the radius of curvature being made equal to /?„ ,• and one due to the enlargement and consequent contraction of the passage. Example 2. — In the three forms shown in Fig. 1071 let the radius of curva- ture D^ = 1 inch, and let the velocity z> be taken at 6.56 feet per second. We then have from (335) for the resistance due to the curvature, h■^ = ^a — — . -^ — = 0.334 f'>, and for ^^ in the various forms 90 64.4 0.5 D whence i2 a h c 0.66 0.54 0-39 0.573 0.352 0.201 0.191 0.118 0.067 We also have from (336) for the loss due to enlargement and contraction : . 64.4 , /;, = : & ]- 336^3 hence a * F Fx V 0.75 / / 1.0625 \ 0.75 C, = 3-444 0.904 hence /<3 = 4.6 1.207 hence /in + ^'3 = 4 79^ 1-325 o 067 It will thus be seen that form a cannot be recommended, except for steam for which the coefficient of loss is much less than for water ; and that form 6 occasions quite a perceptible loss. Form c is much to be preferred, both because it offers the least resistance, and also because it is ligrhter, the pro- portion of metal in the curved portion of the three forms being as 36 : 30 ; 25. The only dimeusion which is important in connection with a standard system of fittings is the distance Z^q ~H ^' which should be taken from the preceding table. The thickness <5^ is mainly dependent upon matters of casting, and is here made = fj 4" 0.04^^ ((S -|- I mm.) the thickness of the collar being An indispensable condition for any standard system of fittings is the constant length from end to centre for each size of elbow, cross, or T, so that at any time one fitting may be substituted for another without affecting the length of the pipes. This principle can also be observed when the fittings are used to connect pipes of different diameters.f Such fittings are always known by the name of the largest opening, whether T, elbow, or cross, this dimension governing the proportions. '. I \: __Z1H±K ' is made equal to 0.7 L>, thus giving one-half the area, and making the velocity the same as in the entrance pipe ; if the side opening had been kept full the velocity would have been reduced one-half. The side outlet is shaped like an elbow, with a sharp internal partition to direct the flow. According to Roux, these partitions are of much importance, acting as wedges to split the flow of the water. At b is shown another form, in which both discharge openings are reduced, and every precau- tion taken to give a smooth flow to the water. At if is a reduc- ing fitting which will double the velocity of flow, the reduction in diameter being made by gradual curves. I . 1 , I ^ _ :■: i 3 BJ.lL.irS^;^-?]]-.-!?" a TTJI j_ fjS3j .*.... -N — fDo-i Fig. 1073. Fig. 1073 a shows a T with equal outlets, formed on the plan of the elbow shown in Fig. 107 1 b. This is made with a divid- ing wedge, which is much better than the straight form shown by the dotted lines. The latter form causes material loss by the sudden reduction of velocity to one-half The form shown at b is intended still further to reduce this loss. At c is shown a cioss with three equal outlets designed on the same principle. The previously described fittings have been given on the as- sumption that the velocity of flow is to be kept uniform from the point of division both as regards the fittings and in the pipes. In extensive installations, whether iu residences, public buildings or manufacturing establishments, this is not often the case. Very often it is found that one portion of a system is possessed of but little velocity of discharge, while a neighbor- ing pipe has a flow of high velocitj' iu it. The resistances in such systems become quite material, but may be somewhat re- duced by giving care to the shape of the fittings. In adopting standard dimensions for pipe fittings, which may be based either upon form b or r, especial precautions must be taken to insure interchangeabilit}', this being the principal ad- vantage to be obtained. This involves accurate tappiug of the threads both in the sockets and in the right-angle fittings, which is accomplished by special devices which enable all these operations to be performed without releasing the fitting, the accuracy of angles and sizes then being readily controlled by the machine. The sizes of the fittings are cast upon them in distinct figures, so that they may readily be determined. ? 343. CONNECTIOXS FOR PIPES OF LEAD .\ND OTHER METAI,S. Lead pipes may be connected by means of separate flanges of wrought iron which draw the expanded ends of the pipes together. * The small clearance for the screw thread may be neglected. t See Trans. Am. Soc. IVIech. Engrs., IV, p. 273. 252 THE CONSTRUCTOR. A good flange connection for lead pipe is shown in Fig. 1074;* the pipes are expanded and a double cone socket of brass inserted and drawn together by bolts. Fig. 1075 shows J r 1 1 1 \w 1 , ■\ lllli il \^^~ -1 11 ! ^jfc i ^ \ai ■ ii |.57to.i.l) Fig. 1074. Fig. 1075. another design, by Louch ; the pipes are drawn together by means of screw flanges and a collar, the three pieces all being made of cast iron. Fig. 1076. Fig. 1076 a shows a connection for joining lead to cast iron pipe, and Fig. 107615 is for lead to wrought iron pipe ; the loose collars in both forms are made hexagonal or octagonal exter- nally, so as to be operated hy wrenches. I 344. Fi,ExiBi,E Pipes. For many purposes it is desirable to have a pipe which shall be yielding or flexible, so that, for example, it may follow the inequalities of the ground, or may accommodate itself to 3'ield- ing supports. In such cases the flange connections maj' be constructed to permit motion by means of ball and socket bearings, as shown in Fig. 1065, such joints being especially adapted for pipes to be laid under water. An example of such construction is found in the water main built by G. Schmidt, of Carouge, for the water supply of Geneva, laid on the bed of the Lake of Geneva. The pipe is 47 '^: inches (1.2 metre) diameter, and is made in lengths of 29^4 feet of riveted wrought iron, 0.197 in. thick (5 mm.). The connections are ball and socket flanges, riveted to the pipes. Instead of making the pipe rigid and the joints flexible, the joints may be made rigid and the pipe flexible. Familiar ex- amples of flexible pipe are various kinds of hose, made of leather, canvas or rubber. Special forms of couplings are made for fire hose. If the hose is to be subjected to heavy pressure, either internally or externally, special methods of increasing its strength are used. This may be done by means of a spiral of wire, or better by two separate spirals, one to resist internal pressure and one to resist external pressure, as shown in Fig. 1077 a. The wire spirals furnish the strength and the hose the Fig. 1077. tightness. This idea may be still further carried out by making the material which makes the pipe tight, also in the spiral form. This is shown in the flexible metallic tubing of Levasseur, of Paris, shown in Fig. 1077 3. f This is composed of a spiral of copper or similar metal, the section resembling somewhat the figure 5. The spiral is wound upon a mandrel in a special machine, a la3'er of rubber packing being wound in at the same time, as shown in the illustration. This pipe has been found to answer well for gas, water, steam, air, etc., and is adapted to high internal or external pressures, being tested to 180 pounds Flanges and other fittings are screwed on to the spiral and soldered carefully. This pipe is used, among other purposes, for connections for air and vacuum brakes. ?345- Pistons. Next to the various kinds of pipes, as already discussed in ? 310, the most important members in pressure organ mechan- ism are the various forms of pistons, and with these the differ- ent methods of packing will be considered. Pistons, properly so called, are fitted with packing which presses outward against the walls of the cylinder, while in the case of plungers the packing presses inward. Both forms will be given considera- tion. The most important forms of pistons are those used in steam engines. Some of the low-pressure engine pistons are yet made with hemp packing ; but for higher pressures, metallic packing is used, this consisting of metal rings pressed against the walls of the cylinder by springs and b}' the steam pressure. In some instances a combination packing is used, the metal rings having a backing of hemp instead of springs. The unit upon which the dimensions of the following pistons are based is determined from the formula : = 0-36S W /) — o 04 — o. iiS (339) in which D is the piston diameter in inches. The following table will aid by giving a series of values for s and D : s n .s V .J £> 0.4 4 0.65 20 0.90 58 0.45 5-7 0.7 24 0-95 70 °-5 8 0-75 30 1. 00 S5 0-55 II o.S 40 1.05 100 0.6 ■ 14 0.85 48 1. 10 120 0,9 _fni K^ 9^^3 * ^g.. <^^^: •.■:-;•-;■, Fig. 1078 shows a hemp packed piston by Pen?^. This is made of a cored casting with a ring follower secured by bolts, screwing into bronze nuts recessed into the piston. For pistons of large diameter an increased depth is given at the centre ; this increase may be made by making the depth in the middle equal to 6i + xV-''^. the depth at the edge being 7.8s, and the piston being made flat — when the latter value exceeds the former. Example. — Let Z> =^ 24 inches— for a hemp packed piston, as Fig. 107S, we then have s = 0.7. This gives for the thiclcness of the packing 0.7 X i-8 = 1.26, say I'X in- ; the depth of packing = 0.7 X 6= 4.2 In. ; the depth of piston at tile edge = o 7 X 7.S = 5.36 = say 5-3 in. The depth in the middle will be equal to 6 X 0-7 X tS = 6.6, say 6;"s ins. Fig. 1079 shows a good form of piston with metallic packing, by Krauss. The packing consists of two steel rings, each cut at an angle, a ring of white metal being cast on each steel ring. If it is desired to make the cut in each ring tight, some one of * German Patent, No, 11,535. t Made by the Metallic Tubing Co., Ld. Loudon, N. C. Port Pool Eane. Gray's Inn Road. THE CONSTRUCTOR. 2S3 the methods shown in Fig. loSo may be used. In the iirst one the overlap makes a tight joint, while in the others the inserted piece is fitted steam tight. By filling the packing rings with white metal the wear comes mainly upon the softer material instead of on the cylinder, a most desirable feature, since the rings are easily and cheaply renewed. For the same reason A piston for a single acting engine, with combination pack- ing, is shown in Fig. 10S3. The metallic packing rings are backed with hemp, this combination presenting the advantage Fig. 1079. bronze rings are used, while iron or steel are not to be recom- mended, with the exception of soft cast iron, which works well, the cylinder being made quite hard. In Fig. loSi is shown the so-called "Swedish" piston, as used in a large blowing engine by Egestorff. This piston is &. r 1 1 \^/7~ •-r;SrilJ < l« + 16 D — *■ Fig. 1082. of elasticity together with durability. This style of packing is well suited also for marine engines, as its elasticity renders it less likely to be injured by the pitching and rolling of the vessel than an entire metallic packing. made with increased depth in the centre, similar to that in Fig. 107S, and the holes shown in the sectional plan view are for the purpose of removing the core from the casting. The packing rings are made of cast iron, with the joint made as Pistons for pump cylinders may be packed with leather so long as the temperature of the liquid to be pumped does not exceed 88° F. (30° C). Fig. ioSi. shown in Fig. loSo a. The rings are kept in their proper posi- tion b}' small pins. The method of securing the piston to the rod is worthy of notice. The large key is secured and tightened by a smaller ke}', the latter being held by a bolt, thus forming a fastening of the third order. Fig. 10S2 shows a metallic piston in which the packing rings are pressed out by an inner spring ring of steel.* The double cone shape of the inner ring enables the piston to be closely fitted to the cylinder by tightening the bolts when the engine is built. The nuts for the bolts are made of bronze, as in Penn's piston, the thread in this case being carried entirely through the nut and the hole closed by a plug. * E. Webers & Co., Machine Works, Rheiiie, Westphalia, makes a specialty of high class steam engines. This firm A form of packing for this purpose is shown in Fig. 1084, the principle being the same as the forms shown in the following section. The units for the dimensions are the same as already given. ? 346. Plungers and Stuffing Boxes. As already observed, the packing for plungers and rods acts from the circumference inward, and such packings, in connec- tion with the necessary parts, are known as stuffing boxes, Two stuffing boxes for leather cup packing, especially adapted for hydraulic presses and for pumps, are shown in Figs. 1085 and ic86, the former being for small and the latter for large plungers. The double cup in Fig. 10S5 is made with a spring ring of iron between the cups to hold them in position before the water pressure is applied. When the form shown in Fig. 10S6 is used in the horizontal position, a ring of bronze made in several parts is introduced below the packing, as shown in dotted lines. This is intended to support the plunger and pre- vent it from rubbing against the cast iron cylinder. The propor- tions given in the illustrations are all based on the unit s, given by formula (339). 254 THE CONSTRUCTOR. The friction existing between a plunger or piston rod in the ordinary stuffing box in which the packing is tightened by screws, cannot well be calculated, as it depends upon the pres- sure which is put upon the packing. In those forms of stuffing box in which the pressure in the cylinder tightens the packing the friction may be calculated. According to the very elaborate s,6to 5,0 Fig. 10S5. Fig. 10S6. researches of Hick,* the friction of a well-lubricated cup leather packing is independent of the depth of the packing, and is directly proportioned to the water pressure and to the diameter of the plunger. If P is the total pressure, D the diameter of plunger, and i^the fractional resistance, we have : ~P 0.04 (340) For a new leather packing the friction is about ij^ times greater. If instead of the total pressure P we use the pressure p, in pounds per square inch we have : ~P — = 0.0393 D (341) Example. — For a piston rod 0.4 in. diameter, according to {340) the loss by friction would be ^^, or 10 per cent., while for a plunger 24 in. diameter it would be 0.0016, or y^ of i per cent. If, for example, the pressure is 4000 pounds per square inch, the friction according to (341} would be F = 4000 X 0.0393 X 0.7854 X 24 = 2963 pounds. The total pressure on the plunger would be /*= 4000 X 0.7854 X 24- = 1,810,000 pounds. Stuffing boxes for the piston rods of steam engines must be capable of resisting the action of heat. Hemp packing is still much used for this purpose. The following illustrations show two excellent forms of stuffing boxes to be used with hempen packing. Fig. 1087. Fig. 10S8. Fig. 1087 is intended to be used on the top of a cylinder; Fig. 10S8 is for an inverted cylinder. Both gland and box are fitted with bronze rings, in order to reduce the wear upon the rod. The wedge-shaped edge which is given to these rings was introduced by Farcot, and is an improvement on the older style of beveling the edge in one direction only, the latter method often drawing the packing away from the sides of the box and permitting leakage. In some designs the edge is left square, as in Fig. 1090, or slightly rounded, as in Fig. 1089. Fig. ic Fig. 1090. Fig. 10S9 shows a form especially adapted to inverted cylin- ders. The construction will be apparent on examination, and it will be seen that the ordinary arrangement is reversed, and the glaud is cast upon the cylinder and the box containing the packing is made separate. This prevents water from the cylin- der from readily getting into the box. In order to prevent the gland from binding on the rod it is important that care should be taken to tighten both nuts equally. In large marine engines, for example, the nuts are made with worm wheels upon a common shaft. For small stuffing boxes this is accomplished by having the screw thread cut upon the outside of the box, as shown in Fig. 1090. This box is intended to be made entirely of bronze. The nut is made with six or eight notches in its circumference, to enable it to be turned by a spanner wrench. The dimensions of all the preceding figures are based upon the unit 5 given by the empirical formula (339). Exaviph. — For a rod 3 ins. diameter, according to (339) we get j = 0.36. The thickness of packing will then be 0.36 X 1.8 = 0.648, say Y^ in. The height of bo.\ for Fig. 1087 will be 0.36 X 12 ^ 4.32 ins., and for Fig. 1088 0.36 X 21 = 7 56 ins., and so for the other dimensions. In horizontal stuffing boxes the length of the bronze collars should be made not less than 8 to 12 .j, in order to reduce the wear. The dimensions given in the illustrations may some- times be modified in order to conform to the thickness of ad- joining parts, so as to avoid difficulties in casting and shrinkage. In some instances the stuffing boxes for valve rods for steam engines are made in two parts, divided in a plane passing through the axis of the rod. The flange of the steam chest is then made in the same plane, so that with this construction the chest can be opened and valve and rod very conveniently re- moved and replaced. The large plungers for mine pumps are packed with hemp, the stuffing boxes having 4 to 8 bolts. More recently metallic packing has been introduced for stuffing boxes of steam engines. An excellent example is Fig. 1091. Fig. 1092. shown ill Fig. 1091, which is made by Howaldt Brothers, of Kiel.t The rings are made of white metal, in double cone * See Verhandl. des Vcreins F. Gewerbfleiss, i366, p. 159. t German Patent, No. 15,418. Over 9000 such boxes had been made up to 888 : one of these had been running eight years without opening. THE CONSTRUCTOR. 255 pairs as shown, thus causing the pressure to be exerted alter- nately against the rod and the walls of the stuffing box. An elastic washer is placed between the gland and the first ring to equalize the pressure. Fig. 1092 shows the standard metallic packing introduced on the Prussian State Railways by Super- inteudent Neumann. This uses a single ring of white metal made in two parts. The pressure is obtained from a steel spiral spring placed in the bottom of the stuffing box, and acting against a bronze pressure ring. The whole is enclosed in a steel cylinder which, together with its contents, can be drawn out by inserting a hook into a T-shaped recess. The form shown in the illustration is intended for a valve rod, but a similar pattern is used for the piston rod. ? 347- Pistons with Valves. Pistons with valves are used in lift pumps and in steam en- gine air pumps. An example of such a pistol,, with leather packing, intended for a mine pump, is shown in Fig. 1093. _i-_i Fig. 1093. The packing is composed of conical rings of leather and canvas, each three adjoining layers being sewed together. The pressure of the water acts to tighten the packing. The acid mine water often acts injuriously upon the leather packing of the pump pistons, and in such cases metallic packing, with rings of soft cast iron, is used. At F'ahlun, in Sweden, after many experiments the best material for packing was decided to be birch wood. The proportions for Fig. 1093 are based upon the unit i. A valved piston for steam engine air pump is shown in Fig. 984. I 348. Piston Rods. Piston rods for steam engines are usually made of wrought iron or steel, and recently compound rods of wrought iron sur- rounded with hard steel have been used. The rod i.: either sub- jected to tension only, as in single acting engines, or is alter- nately subjected to tension and compression, in which case the length and resistance to buckling must be taken into account. For short rods the same results are obtained for both condi- tions, but in no case should a rod subjected to alternate tension and compression be made lighter than a rod under tension only. a. Dimensions of Piston Rods, Tension only. D = diameter of cylinder in inches. p = pressure in pounds per square inch. The total pressure P on the piston will be Z'; ■ p D''. In order that the stress on the rod should not exceed S500 pounds we have for the diameter d of the piston rod when made of wrought iron, and is subjected to tension only ; d^ ~D = o.oioS \f / (342) • (343) or for a close approximation : d^ ^ 57^±_°l5^ D 1000 Example. — If / = 60 pounds we have from {342), — — - = 00836, and hence for a 20 inch cylinder c? = 20 X 0.0836 = 1.67 in. The approximate formula (343) gives — — T-^ = 0.0S7, which for /) = 20 ffives d = 1.74 in. 1000 Steel rods subjected to tension only may be made 0.8 times the diameter of wrought iron rods. If a piston rod is weakened by having a keyway cut through it, or by a screw thread, the reduction in cross section should be provided for by a proper increase in diameter. For this reason the diameter of the rod is sometimes increased in the cross head, an example of which will be seen in the locomotive cross head. Fig. 539. This construction involves the necessity of making the stuffing box glaud in halves, as it could not be slipped over the enlarged end of the rod. b. Dimensions of Piston Rods for Buckling Stresses. Using the preceding nomenclature and indicating the length of stroke by L, we have : d_ D : 0.0295 (iH' (344) from which the following table has been calculated : L /= 50 = 60 = 70 = 80 = 90 = 100 = 120 = 140 = 160 = 180 1-5 0.0967 0,100 0.104 0.108 O.III O.I 14 O.I 19 0.124 0.129 0.133 z.o O.III 0.II6 O.I2I 0.125 0.128- 0.132 0.I3S 0.143 0.J48 0.153 2.5 0.124 0.130 0.135 0.140 0.144 0.148 0.154 0.I6I O.J66 0.J71 These values will serve both for wrought iron and for steel (compare I 1S2, and table in 'i 2). Example. — For a steam cylinder 16 in. bore, 4 in. stroke, with a pressure of 60 pounds, we have —^ = 2.5, and d = 0.130 X 16 = 2.08, say 2 inches dia- meter, either for steel or wrought iron. The dimensions of steel keys to secure the piston to the rod are so taken as to give shearing stresses from 5600 to 7500 pounds in the key. Care should be taken that the key be not made too narrow, and the consequent superficial pressure be- come too great. Pressures of 6000 to 7000 pounds per square inch are found in stationary engines, and 10,000 to 15,00(7 pounds in locomotive engines. 'i 349- Specific Capacity of Pressure Transmission Systems. Having discussed the subject of conductors for pressure organs, we return to the consideration of the various mechani- cal devices which maj'be operated by pressure organs, although these have already been described in Chapter XXIII. We are now prepared to consider these in connection with the subject of long-distance transmission of power, in a manner similar to that in which tension organs are used in Chapter XXI. For this purpose we may use to advantage the conception of specif c capacity. This method is especially desirable because its sim- plicity and general character enables comparison to be made between widely differing systems. The conception of specific capacity can be extended without difiiculty to motors operated by water, air, steam, etc., since for all these we may put the general equation : A^o=^ qv deduced in § 280. In this equation g represents the cross sec- tion of the pipe or other conductor in square inches ; the mean velocity in feet per minute = i\ and yV being the horse power. If, for example, in a water pressure engine, h is the available head of water. Q the weight of water delivered per minute, and //' the head equivalent to the resistance against which the water leaves the engine, we have for the work delivered : 33000 p,ut Q = 0.0361 X 12 qv^= 0.434 '/ ^', the coefficient 0.0361 being the weight of a cubic inch of water, and the pressure p with which the water acts ^ 0.434 //, whence // -= 2.^p. Substituting these values we get : 0.434^1' X 2.3 (/—/O I — qv{p—p') N=- 33000 and the specific capacity becomes 33000 ■ ip-p') (345) a value of the same form as that previously deduced in ? 280 [see formula (262)]. ■56 THE CONSTRUCTOR. Example. — If the effective pressure/ — /' be 320 pounds, the specific ca- pacity will be .'Vo = 0.0097. If the pipe is 4.75 in. diameter, and the water has a velocity of 236 feet per minute, we have ; N = 4-75 - X X 236 X 0.0397 = 40.56 H. P. This is only the capacity of the pipe. The effective capacity will be considered later. Formula (345) can also be used for air pressure or for vacuum, for steam or gas, by expressing the effective pressure in terms of an equivalent head of water. For steam and air it may be considered as an expression of the following form : vV„ = - (/■-/O^ (346) The coefficient /( is very comprehensive ; it increases with p and with the rate of expansion f, and can be calculated from these data, and also confirmed by observation. For f = 2, it ranges from Ij4 to 1-3, and increases to 3 to 4 for e = 20 to 30, results which conform to the higher pressures and greater effi- ciency of compound engines in which such high expansion ratios are used. With some transformations the equation for specific capacity may also be used to solve another important problem, that is the question of the best material to be used for the conducting pipe. If we assume the diameter of the pipe, the horse-power N will be : 4 33°oo For the thickness of pipe, we have from the material : 21), for a stress S, iu 2&-Y D = D\ --.-^ V 5-/ And since 2 (5 -|- /? is the external diameter D^^ the cross section q^ of the pipe. we have for ?i -((^o . z?M = U' (I^^O 4 -3 —p Substituting the value of — £>- from its equation in the above 4 expression for jV, we have I 000 S — p 1^-rp-p^' 33000 V J '/i "■J whence iV„ Qy"" ~~ 33000 (-0 (347) a form similar to the preceding expressions for A^^- This expression is very instructive. It is applicable to all forms of conducting pipes for power transmission. It shows clearly the importance and value of a high value of .S. A high value of 6" reduces the proportional influence of /, to a degree which practically makes N„ dependent mainly upon ,5. It fol- lows that we may consider that the specific capacity of the pipe in a pipe transmission system, is practically independent of the pressure of the fluid used iu it. In other words, the capacity of a given pipe in horse-power is the same, whether the medium be liquid or gaseous, high or low pressure, provided the stress iu the material of the cross section of the pipe is constant. It is therefore desirable to use pipes of small diameter and fluids at moderately high pressures. The friction in the pipe need not prevent this, as care iu avoiding sharp bends and angles can be taken ; and as already shown in § 340 the friction is independeut of the pressure of the medium, at least so it appears from such experiments as have yet been made. The value of the stress in the material of the pipe cannot be taken very high ; S= 7000 lbs. being about the upper limit, and 5* =: 6500 lbs. appears to be quite high enough. Wrought iron and steel, especially in the Mannesmann rolled tubes, per- mit the use of high stresses; for wrought iron .S= 17,000 lbs. and for steel 35,000 to 40,000 lbs., or even higher, if necessary, may be used. By neglecting the value of/ in formula (,347) we have for : Cast Iron, Wrought Iron Steel 5= 6,500, N^ S = 17,000, iVo •S" = 35,000, iVo : 0.197 : 0.515 : 1.060 This gives an indication of the efficiency of the pipe system of power transmission and enables comparisons to be made with other systems. { 350. The Ring System of Power Distribution with Pipe Conductors. Before proceeding with the further discussion of the preced- ing equatious it is advisable to investigate further the subject of power transmission by means of pipe conductors, as already in- dicated in ^ 312. It was there remarked that pressure organs might be used in connection with pipe conductors so as to form "ring" transmission sj'stems in a manner similar to those already described for rope. Taking into consideration first, hydraulic systems, especially high pressure hydraulic systems, we find two distinct kinds of "ring" systems which may be used. O J! Fig. 1094. In the first method, Fig. 1094, the flow of water under pres- sure starts from the power station T^, with a pressure p^, and proceeds to the first station 7",, where it operates a water pres- sure engine, and passes on with a reduced pressure pi. It has therefore operated at the station 7", with a pressure p^ — p^. With the pressure/; it passes on to the .■■econd, third, fourth — — nth station Tn, each time losing pressure until it returns to the power station with a final pressure p «, where it is again raised to the initial pressure of p^. This is practically a coun- ter part of the rope transmission system of Fig. 917. It is apparent that the water pressure engines (escapements) at Ti, 7";, T^, — ■ — ■ — Tn, should all be of equal size in order to uti- lize the entire flow without excessive resistance. Automatic regulation, such as Helfeuberger's, described in § 328, is also desirable.* E P Pi A Fig. 1095. The second system is shown in diagram in Fig. 1095. It will be seen that at each station there is a branch or shunt tube, leading through the motor (or escapement) T„, and then re- uniting with the main pipe. The main pipe A, forks at the station iuto the two branches .B and C, of which the first diverts any required fraction of the power of the main flow, as j^j, ^, J, as the case may be. At the fork is a swing valve C', operated by a speed governor /?, driven by the motor. This governor requires the assistance of some form of power reinforcement, such, for example, as shown in Fig. 1037. The discharge pipe £> of the motor unites with the by-pass C, to form again the main couductor £. At the entrance in the main pipe A, we have the pressure />[ of the original flow ; the motor 'J\ is now supposed to be stationary, the stop valve at i?' having been closed by hand. The flap valve O which has been disconnected * The London Hydraulic Power Company has installed separate ring sys- tems, each with a single generator and motor. THE CONSTRUCTOR. >-':>7 from the regulator before stopping the motor, is also closed. The flow of water then passes through C to i? with the pres- sure /,. When the motor T.^ is to be started, the valve B' is opened and the flap valve C gradually opened until the motor begins to move, when it is connected to the governor, which regulates it thereafter so as to keep the motor at its normal speed. When a heavy load is thrown on, the valve is opened so that the pres- sure p., in B, becomes a greater fraction of /„ and when the work is less it is reduced. The pressure of discharge/, acts as a back pressure so that the motor works with an effective pres- sure p2 — Pi- The flow of water in the by-pass pipe C, also passes the valve C with a pressure p.^, and unites with the dis- charge at E to be further utilized at subsequent stations until it returns to the power station, where if it has reached the min- imum pressure, it is permitted to flow into a tank, from which it is again drawn by the pressure pumps. If the return water is delivered under pressure it may be allowed to enter the suction pipe of the pressure pumps direct and so form a closed ring system to start anew on the circuit. This system has not yet to the Author's knowledge been put into practical operation.* The ring system of h3'draulic power transmission is to be recommended when the various stations are distributed over a wide area and are readily connected by a continuous line of pipe. The pipe can be kept from freezing in winter by occa sional gas flames, as has alreadj' been demonstrated by exper- ience with Armstrong's hydraulic cranes. The ring system should be carefully distinguished from those forms in which the flow of water passes through the motor and is allowed to flow off at lowest pressure of discharge. A corresponding dis- tinction is to be made with other forms of power transmission. The author distinguishes as "line" transmissions, those forms in which the transmitting medium does not return to itself in a complete circuit, in contradistinction with the " ring " systems. The older form of rope transmission (J 297) is therefore a " line" system, while the system devised by the author and discussed in ^ 301 is a "ring" system. A hydraulic system in which there is a free discharge of water from the motors is in like manner a hydraulic "line'' transmission system. There is, however, an intermediate form possible, namely, that in which water after passiug through a series of motors as in a ring system, is discharged freely from the last motor Tn. A similar arrangement is possible with other systems of trans- mission. We may therefore extend the definition of a "ring " system to include those forms in which the medium of trans- mission returns to the place of starting. The distinction can then be made between "open" and "closed" ring systems, the latter being shown in diagram in Fig. 917. High pressure hydraulic systems are well adapted for large railway stations where numerous elevators as well as winding hoists and other rotative machines are to be operated. For such installations a combination of "ring" and "line" systems is best suited. The hydraulic elevators are more conveniently arranged on a line system than in a ring circuit. An apparent objection to the use of high pressure water to direct acting ele- vators lies in the fact that the diameter of the plunger becomes so small as to be hardly stiff enough to support the load on the platform without buckling. This difficulty is readily overcome by use of the hydraulic lever, as shown in Fig. 956 a, the con- struction of which offers no difficulties, and it is unnecessary to go into details. Up to the present time air has only been used upon line sys- tems, either with direct pressure or with vaacum. Gas engines can only be operated on a line system since the gas is burned in the engine. Steam has been used in a ring system in New York for some time, on a long distance transmission, and short ring systems exist in most cases of compound or triple expan- sion steam engines as used in marine and stationary practice. T- ^ D=ra T, K, K "■ ~l Fig. 1096. Steam at a high initial pressure is expanded successively in one cylinder after anether, and between the last cylinder or station Tn and the first, or boiler T^, is placed the surface condenser 7,n, where the medium reaches the minimum pressure and is converted into water to be returned to the boiler and start anew on the circuit. In order that the velocity of flow shall be uni- form the successive passages for the expanding steam should be made with continually increasing cross section as shown in diagram in Fig. 1096. If a jet condenser is used instead of a surface condenser the circuit becomes an open ring. The high economy which has been attained by the application of the " ring " system with steam in the form of multiple expansion engines, points to the possibility of a similar economy in the application of the ring system to wire rope transmission. Lehmann's hot air engine, which is a true closed circuit, is an example of the ring system confined within the limits of a single machine. §351. Specific Capacity op Transmission by Shafting. The subject of the specific capacity of shafting was not con" sidered in Chapter IX, and it is introduced in this place in order to obtain a basis for comparison with the other systems of transmission. If we have the moment PR and shaft diameter d, we have, if S is the fibre stress at the circumference 16 (see ? 144)- If we make the lever arm i? := J d, we have P = the force at the circumference of the shaft and hence P^=i S -^ d^. Taking V = the velocity at the circumference of the shaft and iV the number of horse-power transmitted, we have : N= Pv_ 33,000 ^S—d^'-v 2 4 33.000 But — d'^ = g, the cross sectional area of the shaft, whence 4 2 33000 (34S> and hence the specific capacity of the shaft is : qv 2 33,000 ' (349)- This expression, which is of the same form as those already ob- tained, does not give values numerically great, because S must- be taken low enough to avoid excessive torsion of the shaft. If we require, as in I 144, that the torsion shall not exceed 0.075° per foot of length we must have S < 630 d which gives for shafting from 2 to 6 inches diameter .S = about 1200 to 370c pounds and the specific capacity A^o= o.oiS to 0.056 (350)- In other words, such a shaft will transmit, at one foot per min- ute circumferential velocity, o.oiS to 0.056 horse-power for each square inch cross section. In the application of shafting to long distance transmission the friction of the journal bearings is a very important consid- eration. The influence of friction may be determined in the form of a general expression in a similar manner to that of the friction of water in a pipe {§ 340). According to formula (100) we have for the force P", exerted at the circumference to over- 4 come the journal friction i^^ — y times the weight of the shaft, that is = ^^f tO 12 /, X 0.2S in which L is the length of the shaft in feet, and 0.2S is the weight of a cubic inch of wrought iron. It follows that the horse power iVj re- quired to overcome the friction will be : Fv X ? X 12 Z X 0.2S ♦See the Author's article in Glaser's Annalen, Vol. XVII (1885), part 12. from his paper to the Verein fur Eiiseubahnkunde, Nov. 10, 1885. 33000 33000 and if we take the coefficient of frictiony"^ o.oS we have i'SS THE CONSTRUCTOR. ' linrtn tt 96,422 j3000 - N, = --■ 96,422 and if we wish the specific frictional resistance, we have : ■qv ' ^Jo qv 96,422 (351) (352) This resistance is by no means inconsiderable. Expressed as a percentage it will be : so that in both instances it is less than one-fourth the resistance of the corresponding solid shafts, as given in (354). Hollow shaft- ing thus greatly extends the capacity of shafting for long dis- tance transmission and also permits an important economy in material. The subject of shafting made of steel tubing was not consid- ered in Chapter IX, and a brief discussion will therefore be given here. Let d^ be the outside diameter, d^ the inside diameter, let the ratio '- ^ V- Making i/" = 0.9 as is usual in practice with such tubing, the diameter for resistance to torsion, (compare formula 1,133) ) will be: pr = - N, qv L I N ~ "96,442" ■ ^o 353) 'A'' ~ 96,42 The value pr, it will be seen, is inversely proportional to the specific capacity. If we apply this to (.350) we have for a 2 inch assumed. If instead of 71, the circumferential velocity v, be shaft "■■■" "■^— "'■"-" '-'*■ P' and for a 6 inch shaft L L o.oiS X 96,422 1735 L _ L ■" 5400 0.056 X 96,422 ^0 = 0.39^^^ = 6.18,^^ (359) This requires that the number of revolutions be known or assumed. If instead of 71, the ci given, we have for the same shaft : do = 7.25 ^^ (360) (354) £', being expressed in feet per minute at the circumference of the shaft. The number of revolutions will be : 3.S2 hence 1735 feet and 5400 feet are the limits of length respec- d^ tively for the two diameters given, at which the frictional resis- tance will equal the total transmitting capacity. Much higher The diameterTor strength (compare (131)) will be efficiency is obtained by using hollow steel shafting such as is now produced by the Mannesmann process of rolling weldless tubing. This furnishes a seamless tube, of sufficient truth as to cylindrical shape, the journals of which may be made either entirely of steel or of so-called "compound steel."" = 1.11^^^= 5.35 120 ^ instead of jv, = . 96,422 241,000 - q V ' 60 or dividing again by N : /..= L N 241,000 N^ (357) With the values for N^ as given in the two preceding instances, we have for the 2 inch shaft : do = 6.184 <* = ■t-7 \f j;^ = 3-95 in. as would be the case for^ a solid ^shaft. The hollow shaft, however, weighs only \^'^^ J \ J 120 X 5= I — — III — o-8i I = 0.33 times the weight of the solid shaft. The circumferential velocity -v = - 3.82 - = 163 feet. If a higher speed be N^ = 0.0326 and for the 6 inch shaft : N^ =-o.ioi and these give in (357) : for the 2 inch shaft — ■ 0.0326 X 241,000 L - 7S56 and for the 6 inch shaft — *.- ^ L o.ioi X 241,000 24,341 chosen, as may readily be done, on account of the small journal diameter d' , we have from (360), making v = 300 ft., for example: do = 7 25 sf, whence d' = 0.4 d^ = i-7 in- The number of revolutions will then be (358) 3.82 X 300 n = ^ — = 270 424 The weight ollshaft^will be (ig)-(— ) = •■■ * The Mannesmann " compound " steel tubing- is made with the interior times that of a solid shaft at 120 revolutions. The loss from friction will be of soft wrought iron and the outside of hardened steel. onlj- 0.26 times that of the solid shaft. THE CONSTRUCTOR. 259 Spfcific Vai X 0-434 A = 2 ?/ X 275 r , ■ , 4 X 118 X 0.434 X 9-83 ^ I.- 1 - r t\ J^- ^ from which n — r^ — ^— ^ = 3.6 which gives for the distauce 2 X 275 from centre to centre of bolts, 3-6 = I.I I in. or about i^^ ins. For the joint half-way between the top and bottom of the tank the pressure would be but half that at the bottom and the bolts may he spaced proportionately wider, say about 2 inches apart. The total contents of the tank will be — 742 cubic feel = 5550 gallons. In using cast iron tanks of this sort care must be taken to avoid filling them with licmids which have an injurious action upon the rubber packing of the joints. i 355- Riveted Tanks. ..^ When tanks of large capacity are required, wrought iron or steel must be used in their construction and these involve the use of riveted joints. With tanks of large diameter construc- tive difficulties arise in connection with the flat bottoms. In the United States, oil tanks are made with flat bottoms, carefully bedded iu cement, and similar tanks are used in Ger- many for water. It is, however, found that greater facility of construction, as well as economy of material, is obtained by making the bottom convex, as will be shown. A very frequent and useful form is that in which the bottom ti :Fig. 109S.", is made iu the shape of a spherical segment. Fig. X098 a, the tank being supported on a flanged ring riveted to its circumfer- ence and the ring standing on a support of masonry. The construction of the supporting ring is shown in Fig. 109S d, from the design of Prof. Intze. The tension in the inclined direction of the bottom of the tank is carried by the lower half of the supporting riug, while the upper portion is siibjected to the pressure of the tank at right angles to the vertical. This latter force is well resisted by a ring of angle iron running entirely around the tank. The calculation of the bottom of spherical segment shape is as follows : If J? is the radius of the sphere of which the segment is a part, we have from ^ 19, Case II. : p_ ■2S, in which d^ is the thickness and ^i the stress therein due to the pressure/. The pressure is the greatest at the lowest point of the bottom where the height iu feet of the column of liquid is THE CONSTRUCTOR. 261 equal to A, so that if ff, is the weight of a cubic inch of the liquid^ = 12 /i a. We then have: J? 1 2/1 a h 2 S^ 5i which for water gives, a = 0.0361 R (368) = 0.217 h 5^1 At each higher point of the bottom the pressure is less, until at the edge of the bottom the height /;, is diminished by the depthy, of the bottom. For simplicit}', however, it is custom- ary to make the entire bottom of the same thickness increases the size and thickness of the bottom increases. An approximate formula by which the minimum amount of material will be required is : D •366^ Q (370) in which O is the volume of the material iu cubic feet to be contained in the tank. For the height /I of the wetted portion of the surface we have : // + — = // 2 2 £> (371) if we assume, as we may with sufficiently close approximation, the segment of the sphere to be practically that of a paraboloid. The same remark about the most economical ratio of depth to diameter applies here as in the uote to ^ 354. Example I. — For Q = 47,000 cubic feet we have from (370); [.366 -^ . 47-36- A carefully calculated tank at Halle, of this capacity (1200 cu. metres) was made 51.88 feet diameter. ir O = 65,60-1 cu. ft. xve have D = i.366^7o,ojo = 56.3 ft., while a jank of the same capacity at Essen is 5S feet in diara,eter. The water towir at Neustassfurt has a capacity Q = 21,160 cu. ft., and is 39.36 ft. diameter ; according to (170) it would he D = 1.366^21,160 = 37,79 ft. All three cases thus agree well with the formula. For the depth yj of the concave bottom, we have for any given radius J^, the expression 2 J?/-/' \ D\ from which we get B ^ \D J 4 (372) It is found convenient, but not essential, to choose such a value for R, that t\ = i\ when 5 =: S^. To accomplish this re- sult, the conditions which obtain for the equations both for (S, and (5 must be fulfilled. These are : R_ D in/ h ' (■hence, -— D R_ (373) The following table gives a series of numerical values for these relations : R ! 0-55 0.60 0.625 0.65 0.70 0-75 0.80 0.85 0.90 0.95 1.0 / D 05 0.32 C.7 0.2s 0.23 0.21 0.19 0.18 0.16 "■--i 0.14 0.134 h D I.O 0.71 0.68 0.67 0.66 0.70 0.76 0.88 1.07 1.52 2.84 00 h_ -o.sf __ D 0.75 0-55 0.54 0.54 0.56 0.59 0.67 0.79 0.99 1-45 2.77 CO i-( ^)- = 0.17 0.07 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.01 O.OI These relations are also shown graphically in Fig. 1099, and the results are interesting. It will be seen that in order to have (!j = i when Sj := .S we must always make R <. D. It also appears that the best ratio of depth to diameter occurs when 3D -— is about equal to 0.60, for then /; — ■ 0.5 /nearly approaches 0.5 D ; this, however, is only approximate. It thus appears that the two conditions of greatest economy of material and equality of value c!j and t', cannot be attained at the same time o.c^'> 0,85 0,79 o,6r. 11,60 o,5.'> ,00 aao 0,89 0,75 == 25.14 ft., and combiuing these again we get: (7=25,14X0.7854(50.28)2 = 49 420 cu. ft., which is a little under the required content, but shows the cor- rectness of the proportions. 262 THE CONSTRUCTOR. If we now make /= o 21 Z* = 0.21 X 50.28 = 10.56 we have from the above table, i? = 0.7 Z> = 0.7 X 50.28 = 35.2 ft. We have from (371) h = 0.5 Z? + 0.5/"= 0.605 D = 30.42 It. The height of the wetted perimeter will be // = /i — /= {0.605 — o-2t) -^ = 0.395 D = 19.86 ft. Taking for the stress in the metal at the lowest part of the walls of the tank wc have from (369) : 5 = 2.604/? H ir- 2.604 X 0.395 —^ = 0.372 m For the bottom we have j.6o4 J? ' 0.7 X 2.604 X 0.605 ■ 7^2 : 0.4 m. and — ^ 1.07 ; that is, the thickness of the bottom is 7 per cent, greater than o that of the lowest row of plates in the walls of the tank. If we make the tank with six rings of 3 ft. width and one of 2 ft. we get for the thicknesses: Depth = 5 Calculated = 6 In practice = 19.86 16.86 13.86 J0.86 7.S6 4.S6 1.86 0.372 0-315 0.260 0.203 0.147 0.091 0.035 %" T5 w 'A." Ji" J4" M"- The latter figures show an excess over the theoretical thickness, but the excess is needed for stiffness and for constructive reasons. The thickness of the bottom, as already calculated is 0.4 in., but in practice would prob- ablv be made y'g". The riveting may be made the same as ordinary boiler riveting ; and from the table in g 59, we find for 5 = ^-s", d= \\" and for single riveting the modulus of efficiency is 0.47. This gives a stress of -^-— = 15,000 pounds, 0.47 which seems rather too high. For this reason the two lower seams at least should be made with double riveting ; which gives a stress of ^ = 11,800 0-59 pounds. The seams of the bottom should always be made double riveted. Example 3. — Let Q again be taken as 53,000 lbs. "We will now proportion the tank so that 61 = 5, and take i? = 50 ft. R In order that l\ shall at least equal 5, we will take D 0.625 whence f f = = 0.25 D = 12.5 ft. We then have /i = 0.67 D = 33.5 ft., and h — (0,67 — 0.125) D = 0.545 D == 27.25 ft. We therefore have Q = 0.7854 X 27-25 X (50)- = 53.500 cu. ft. Tphich agrees quite closely enough with the original assumed capacity, //will be = to /i — /={o.6j — 0-25) D = 0.42 D = -21 ft. We therefore have for ihe lowest cylindrical portion of the tank : 6 = 2.604 X 0.42 and for the bottom : 0,625 X 0.67/72 D^ - 502 —=- = 2.004 X 0.42 = 0.3906'' o 7000 01 ■■ - 2,604 ■ ^0.625 X 0.67 X (50)^^ == 0.3894" thus giving practically 5 = ^1. The tank will be heavier than the preceding proportions give, as might be expected, but the excess weight will be only about i per cent. § 356. T.^NKS WITH Concave Bottoms. The question of the action of the forces upon the bottom of a tank as discussed in the preceding section, was first thoroughly investigated by Prof Intze, whose valuable re- searches have practically revolutionized the construction of riveted tanks.* The following discussion is based on Intze's, but the calculations are simplified and abridged. Fig. 1 100 shows two forms in which the spherical segment may be used, a, with convex or hanging bottom, as already dis- cussed, and b, with concave or reversed bottom. In both forms the pressure of water on the bottom produces a stress at the base of the cylindrical portion of the tank in the direction of the tangent to the curve of the bottom, the stress acting in- wards in case a, and outward in case b. It is desirable to make the construction such that this force is received by the base ring and not by the shell of the tank. In every case, however, an increase is required in the thickness of the bottom of the tank. There is also a force ;•, acting at right angles to the tangent or normal to the curve of the bottom of the tank, and the deter- mination of both of these forces is a matter of importance. If G be the weight of the liquid, and a the angle which the tangents make with the axis we have for case a, for the two * See the article by Dr. Forchheimer : " On the Construction of Iron Tanks, for Water, Oil and Gas, according to the Calculations and System of Prof Intze, of Aachen." Schilling's Journal iiir Gas-beleuchtung, 1S84, p. 705. lateral forces which act, each on one half the circutafe the base ring of the tank : Ai 1 'Vf:' i. \ t! i ..;■ ' \ 1 / \ 1 / Fig. II 00. 2 cos a producing a load Si per running foot : 5 ■s, =■ -£) D'+f Substituting for G, its value E'r „ in which y is the weight of a cubic foot of the liquid, we get : ^ r, /■ . 2 -/) + ^/ •)] '4["-^H^(*)-] In this h is the distance from the level of the surface of the liquid to the crown of the curve of the bottom, and for the case b, we have : The last member in the brackets is always very small in value as will be seen by reference to the table in the preceding sec- tion. It can therefore generally be neglected, when we have for both cases : s — y ■ J? i) (374) The detailed determination of the forces /; and 4, need not be gone into here, we have for both cases : i = yJ?(k^/)-s = y^(A^±/) . . . (375) There is also a third force 71, acting upon the rim of the spherical bottom in the direction of a great circle at right angles to the plane of the drawing, for which we have per run- ning foot : (376) and finally for the crown of the curve, where the force ii in a great circle is : "o = '■ T '^ (377) These formulte will be somewhat simplified if we take the height H, of the wetted portion of the cylinder, whence h^H±/. This gives: u^y^H, u, = y^{H±/) ■ -(378) THE CONSTRUCTOR. 263 These are the necessary formulas for the calculations of spherical bottoms. The following points are to be noted : I. For the convex bottom (Form, a) u„ has the greatest value, that is, the stress must be calculated for the deepest point if (S„ is to remain constant; 2. For the concave bottom (Form. ^) t has the greatest value, and must be used to determine dj ; 3. The supporting rim should be capable of sustaining s, if the shell is to be free from any stress due to the bottom of the tank. The determination of (S, is the same as before. If we divide the values for «<, and ;', by 12, we get the stress per running inch, and by using the weight cr of a cubic inch of the liquid and taking R in inches, we have for the convex bottom : <5, 12 /;/ G R ^ 2 5i = 12 2 5, oncave bottom : (Ij 12 /;' a R 2S, 12 G 2 5, (379) / • (3S0) [Fig. iioi. If the bottom is made conical, projecting either within or without as in Fig. iioi, the height of the cone being/", we have for the weight of the body of liquid • and taking the component as before in the direction of the angle of the cone, we have : G 5=— D s = 2 2 cos a ■whence : 5 = r — Z)2 4 TT D 2 cos r(--i) D 2 2 cos .(--i) 2 But IS equal to the radius R oi a. sphere inscribed within cos a the cone ; whence we have : R s=y C-'i) We also have for i the same value as for u, and R t = u — } —H (381) (382) For the inverted hanging cone bottom, form c, the greatest of the three forces is s, while for form d, in which the cone pro- jects into the tank t = u, is the greatest, and we use in practice for form c; and for form d : R A R 2 5, }2.a_H_ 2 J (383) (384) The conical form of bottom, as will be found upon compari- son, requires about 40 per cent, more material than the spheri- cal, but as will be seen, its use under some circumstances is advisable. Instead of using a complete tone, the bottom may be made a truncated cone, the tank being formed of two concentric cylin- ders connected by a ring-shaped bottom, as in Fig. 1102. V •D6 - ..-D-l-... f i ;<-D ^ .--jt:-.. H^ % i, ,-' i \^ g ^,-=^ ^ H Fig. 1102. These may be made either projecting inward or outward. Following the same line of investigation as in the previous cases we have for case e : 7 - (/?o^ - n^: 4 1/ 4 B) H{D + i^o - D) ) and for casey.- This gives for case e R f r ^i?, '"2 ] I \D and for case,/ D R -I \H- Y D, 3 c ] H ■ (385) D ^■] (38h) in which R is the radius of the sphere inscribed within the truncated cone.* The forces / and u are obtained in a similar manner as before. The subject of truncated conical bottoms will be discussed again. We have for the weight (t, of a cubic inch of various liquids : Water 0.0361 lbs. Petroleum 0.02S9 lbs. Linseed Oil, at 12° C. = 54° F o.o339 lbs. Bisulphide of Carbon, at 0° C. == 32° F. 0.0459 lbs. Glycerine, at 0° C. = 32° F 0.0455 lbs. Beer, at 0° C. = 32° F 0.0372 lbs. Alcohol (absolute), at 20° C. = 68° F. . 0.0286 lbs. In the construction of tanks, it is necessary also to consider the peculiar properties of the various liquids. For alcohol no packing should be used in the joints, the tightness only being secured by caulking the riveted seams * \i Dq be made o, the formulae will become those for complete cones, as indicated in the dotted lines. The formulae for the weight might also be symmetrically expressed: the form used has been selected because it makes H the higher of the two walls, which is more convenient in numerical cal- culation. 264 THE CONSTRUCTOR. I 357- Combination Forms for Tanks. In the forms of tanks already described the force 5 sin a acts either to press the supporting ring inward or outward in a direction radical to the axis, according as the forms a, c, e, or b, d, f, are used. This circumstance lends itself very fortu- nately to Prof. Intze's method of construction, since by com- bining both forms iu one bottom the forces may be made to equilibrate each other and thus relieve the supporting ring from all radial stresses. Fig. 1103. This idea may be carried out in many ways, as by combining forms d and/, Fig. 1 103 a, or forms e and b, Fig. 1 103 b. or using all three forms as iu Fig. 1103 c, the inner vertical walls being, in these combination forms omitted.* The forms shown in the illustration also have the advantage of reducing the diameter of the supporting ring and hence re- quiring less extensive foundation walls. In order that the supporting ring may be free from radial stresses, the condition ; s' sm 0.' — s" sm a' O . .... (3S7) must be satisfied. This simple equation cannot be briefly solved numerically, hence an example is here given of its application. Exa7nple.—0\veii a water tank of the form and dimensions of Fig. 1104, th radius of curvature of the bottom being R". The first member of the equa tion belongs to the outer, and the second to the inner portion of the tank Fig. 1 104. For the first member we .have for s', from (385) ; D^ = 12, D = 4cy H = 6,/ = 2.4, whence tan « = — ^ =1.667 = tan 59°. This gives sin « ' = 0.8572, and 2.4 cos a' ^ 0.5150, and R. „ °-^ = ^_ = 3,8f3 and Sin a o 515- .'Sina= 0.S37. V :iferential Seams. The cross section of the boiler shell, when the head is fast to it, is subjected to a force — £>'/> = S.^ tv D i', in which S^ = 4 _4 ; that is half as great as the stress S, in the longitudinal D. If the cylindrical shell is made with flat heads its content will be — Z)^ Z, 4 Z)3 G). and the spherical vessel will have a content = -— D^ ; hence 5 we must have i3°i = — V^ D )■ For the thickness of metal we have : — Dp — D,p and for the respective surfaces : F-- D L-\ D\ and F^ ■■ D\. Assuming the heads of the cylindrical vessel to be made the same strength as the shell, we have for the material required for each case : E- (5 — n^ ~ 4 -ii Making 5 =:= Si and putting for D^^ its value — /)3 ( 75- ) get : L F,6, 3 D F6 4 J^ , I z^+.- (393) for the ratio between the amount of material required for spherical and cylindrical vessels. We have for : -^ = I iK 2 3 4 5 6 00 F6 ~ 0.50 0.56 0.60 0.64 0.67 0.68 0.70 0.75 showing that the spherical vessel is in all cases the lighter form. The earliest boilers were made in the spherical form, but soon abandoned on account of the demand for increased heat" ing surface and small content. The spherical form is, however, well adapted for units for sectional boilers.* For spherical ends of cylinder boilers, as in Fig. 1 107, and for the heads of domes, and auxiliary drums, we have for the thick- ness, R-^ being the radius of the sphere : 2 5 (394) which gives, when .S, = 5 the same value for the thickness d, as in the shell when R^ = D. This latter condition cannot always be fulfilled since the curvature of the boiler head is usually controlled by the dies with which the press is provided. The head is usually joined to the shell by being flanged or turned over around the edge in the flanging press, thus enab- ling a joint to be made as at a. Fig. 11 17 ; or it may be made with a ring of angle iron, as at b. Here the circumferential force, as considered in § 355, may be taken into consideration, especially the radial component s sin o, since this acts to draw the shell inward. It is, however, hardly necessary to take this into account as the flange of the head reinforces the shell amply at this point. . c. Flat Surfaces. Unstayed flat surfaces can onh' be u.sed in boilers of small dimensions, as already shown in § 19, and should only be used for heads of steam domes, auxiliary heaters, and the like. Where extended flat surfaces are used, it is necessary to adopt some method of staying ; or in other words to subdivide the ex- tended surface into supported portions small enougli to be of ample strength and at the same time of moderate thickness. A number of methods of staying flat surfaces are in practical use, those most generally employed being shown in Fig. iiiS. Fig. iriS. Stay bolts, such as shown in Fig. iiiS u, (see also | 61) are used for parallel surfaces which are near to each other. Those shown at a are made with nuts instead of riveting the heads as is sometimes done. Flat surfaces which are farther apart are secured b}* anchor bolts, as shown at b ; these are practi- * The Harrison Boiler, the pioneer of modern sectional boilers, is coni- posed of spherical units. Trans. THE CONSTRUCTOR. 269 cally long stay bolts. These are shown reinforced by large riveted washers under the nuts. Stay bars, as shown at c, are used for staying crown sheets of fire boxes in marine and locomotive boilers. Stay tubes, such as shown at d, are used to strengthen tube sheets. These are heating tubes about '4 to -j% in. thick reinforced at the ends and screwed into the tube sheets. Gusset plates c, Fig. b, are used to stay flat heads to the shell, and are used Ijoth in land and marine boilers.* I 361. BonER Flues Subjected to External Pressure. The stresses which appear in the case of a boiler flue subject- ed to external pressure are similar to buckling stresses upon columns, rods, etc., since beyond a certain increase in pressure when a slight departure from the true cylindrical form occurs a sudden collapse follows. The smaller sizes of flues used in the ordinary tubular boiler possess ample strength against col- lapsing, but for larger flues such as are used in Cornish and Lancashire boilers the question of strength to resist collapsing must be considered. The experiments of Fairbairn have demon- strated that the length of the flue has an important influence upon the resistance to collapsing, practicallj' being inversely as the length of the flue, or rather as the distance between the points at which the flue is reinforced against external pressure. stituting these in the formula it will be found if the flue is safe against collapsing. Exain/^le.— In a Cornish boiler intended to work at 37J: pounds pressure, the dimensions are / = 25 ft., Z) = 23 ins., S = 0.25 ins., the flue being made with lap joints. From (397) we have : p = 368,000 0.25 V: 0.25 =5" X 23 = 303 lbs. at whicli pressure the flue actually collapsed. It is evident that should the thickness of the flue be only slightly reduced by corrosion, etc., an explo- sion might readily follow. A method of increasing the safety without using a greater thickness of metal in the walls of the flue, is to reinforce it by stiffening rings, thus practically reducing the length /, as noted by Fairbairn. Fig. 1 1 19. Two forms of stiffening rings are shown in Fig. 1119, a being Adamson's and b, Hick's. The first form is the more difficult Fig. II 20. Fairbairn deduced from his experiments for the collapsing pressure of such flues : p' = 806,300 r! 2-19 (395) in which p' is the pressure in pounds per square inch, D and i are in inches, and I is the length of the flue in feet. If the dimensions are given iu millimetres aud/i' is the pres- sure in kilogrammes per square millimetres, this becomes : 100 p' : 367,973 (I ■'■■9 Tn (396) Fairbairn's experiments have been discussed more recently, ■with a view of deducing a formula which should be more con- venient to use.t The results of Dr. Wehage in connection with later experiments, J give the following formula : r- 368,000 490,000 J ^ (397) in which the upper coefficient is to be used for flues made with lap joints, riveted ; and the lower coefficient for flues in which the joints are made with flap plates riveted on. This formula gives results approximating verj' closely to Fairbairn's most important experiments. It is best used by selecting the desired dimensions for D, I and '5 and then by sub- * The question has been raised as to whether it is not best to stay only one boiler head to the shell and then tie the other head to the first by means ot a number of parallel anchor bolts, thus closi \\^ one end of the shell in a manner similar to the cylinder of a hydraulic press, and relieving the shell of any stress due to the pressure on the heads, and permitting the use of packing to make the joint tight. The author recollects such a construction having been used in a portable engine and boiler but without knowledge of any further attempts of the sort. t See Grashof, Zeitschr. D. Ing. 1S59, P. 234. Vol. III. ; also Love, (-ivilin- genieur, 1861, p. 238, Vol. VII,, discussed by tlie author for the HUtte Society in the Berliner Verhandlungen, 1870, p. 115. t See Engineer, Vol. 51, 18S1, p. 426, also Dingler's Journal, Vol. 242, 1881, p. 236. of construction, but possesses the advantage of removing the rivet heads entirely from the action of the fire. This form of joint and stiffening plate is also frequentlj' used in other parts of boilers for the sole purpose of avoiding the action of the fire on the heads of the rivets. The use of corrugated iron for boiler flues enables great strength against collapsing to be obtained. Fig. 11 20 shows a boiler with corrugated flue, the lengths being welded together. This boiler is made by Schulz, Knaudt & Co., of Essen, and is 86.6 inches in diameter (2.2 metre). Notwithstanding the con- structive difficulties the use of the corrugated flues is constantly increasing. In England corrugated flues are made by the in- ventor, Sampson Fox & Co., of Leeds. The depth of corruga- tions is usually about 4 inches. Corrugated fire boxes have been used in locomotive boilers. Fig. 1121, showing Kaselowsky's fire box. In this form the Fig. 1 121. stay bars to support the crown sheet, and the stay bolts at the sides are entirely omitted. The cross section shows the method of supporting the boiler by a cross beam below the grate bars. The corrugated flue is attached to the boiler by a riveted joint, either by flanging as in Fig. 1121, or by the use of angle iron, as Fig. 1120. Tubes of small diameter are treated practically as single hol- low rivets, the ends being inserted into holes in the tube sheets 270 THE CONSTRUCTOR. and expanded by an expanding tool, the ends being riveted over> as shown in Fig. 1 122 a. Fig. T122. In many establishments, as for example, the Esslingen loco- motive works, the tubes are fitted with hard copper ferules which stand the expanding and riveting better than tubes of steel or iron. The form of tube shown in Fig. 1122^, is rein- forced at the ends, and one end made conical, thus enabling old tubes to be more readily removed and replaced. This con- struction is used by Pauksch & Freund_, of Landsberg, in Ger- many, and by various French builders since 1867. I 362. Future Possibilities in Steam Boiler Construction. The discussion of the preceding sections has necessarily been limited to a few constructive details, since a complete treatment of such an extensive subject requires a special treatise. It is proposed here to give only 'a broad general view of the subject of boiler construc"tion in its present and prospective condition. The descriptions in the preceding sections and in the previous chapter on riveting show that the art of boiler construction has made little or no advance during the past twenty or thirty years, although there is reason to believe that there is ample room for improvement, especially in the matter of greater economy of fuel. In the author's opinion there are four points in construc- tion which deserve the closest attention and to which efforts at improvement should be directed, while in other directions also serious wastes of force appear. 1. Expenditure of Material. — As already shown in ? 359, the expenditure of material is considerably greater in the present forms of steam boilers than if the spherical form were more generally used. It is questionable to what extent the spherical form may be made practicable, but the possibilities in this direc- tion have not been exhausted, at least for certain purposes, for example, for boilers used solely for heating purposes. The spherical vacuum pans only serve as reminders that this oldest form of boiler (i. e., that used with Newcomen's engine), is no longer used ; but it may be only a question of the increase in the capacity of the flanging press ; or, in other words, of the increased command over the working of iron and steel, when the spherical form shall again be used. Another point in the question of material, is the subject of riveting. One of the greatest sources of weakness in steam boilers is the reduction in strength due to the presence of riveted seams. Even if the very best material obtainable is used for the rivets, the reduction in strength for single riveting is about 40 per cent, and for double riveting, 25 per cent.* This weaken- ing is unimportant so far as the circumferential seams of cylin- drical shells are considered, but is well worthy of consideration in connection with the longitudinal seams, especially since it concerns the largest and heaviest part of the boiler, i. e., the main shell. It is for this reason that attempts have been made to weld the longitudinal seams. The meagre results which have been obtained for welded shells subjected to internal pressure, as compared with welded flues for external pressure, may be seen from the case shown in Fig. 1116. The welded seam is there reinforced by a riveted flap, thus reducing the strength practically to that of an unwelded seam. Experimental results with welded joints in the testing machine, justify this distrust of welded seams, and do not war- rant the idea that the weld is equal to the full strength of the plate. This leads to the remark that the coming boiler shell must be without longitudinal seams of any kind, either riveted or welded^ Heating flues for external pressure are already made seamless, and the Mannesmann process produces seamless tubes adapted for internal pressure, and of a grade of material far superior to that heretofore used, as experimental researches have demon- strated. If this process can be so extended as to be made avail- able for boiler shells, an economy of at least one-third of the material can be obtained. 2. Combustion. — The subject of economy of combustion of the fuel is even more important than that of material. In the general description given in the preceding sections it will be seen that the present methods of firing are all based upon the principle of exposing portions of the boiler to the direct action of the fire and of conducting the products of combustion into contact with various portions of the boiler, arranged to act as heating surface. This means that in nearly all cases boilers are independently fired. For a long time the advantages of this system have been doubted. It is manifestly impossible for a complete combustion of the gases to be effected when they are almost immediately brovight into contact with surfaces which have a temperature of 1200 to iSoo degrees lower than the flame. The production of smoke and soot, that is, of unconsumed fuel, is the necessary result of these conditions, and hence a great re- duction in efficiency. This subject has been actively worked over, and an almost endless variety of furnaces and systems has been proposed. The true method of solving the problem appears to have been first discovered b}' Frederick Siemens (Dresden), and for a number of years he has been engaged in developing the practical applications of his researches, t The previous methods of firing were based upon the idea of bringing the flame into direct contact with the surface to be heated, but since about 1879 the method of construction, espe- cially in glass furnaces, open hearth steel furnaces, smelting furnaces, etc., has been to utilize the radiant heat from the arched roof of the furnace, and to economize the heat of the escaping gases in the regenerator. An ecouomj' in the use of the heat of as much as So to 90 per cent, has resulted. This has been followed by a still more marked separation between the two principal periods of combustion, and by the application to steam generators where such a high economy cannot be expected, although a saving of about 25 per cent, has been shown in actual practice. I It is therefore strongly recommended to use such furnace con- structions as shall not bring the direct flame of the fire in con- tact with the heating surface of the boiler, but to use radiating surfaces and also to conduct the highly heated but full}- burned gases through the flues, both of which can be accomplished in various ways.? The application of the principle to stationary boilers is not difficult, and experiments have shown that it may also be suc- cessfully applied both to marine and locomotive boilers. In all cases it has lieen demonstrated that the fuel should be burned in a combustion chamber lined with refractory material, and the discharge of the heated gases retarded by a fire brick bridge or screen before coming in contact with the boiler. It will be seen from the preceding, that by using the Siemens' method instead of the older method of burning the fuel directly in the boiler, an economy of about 25 per cent, can be obtained, and this fact should always be kept in mind in future designs. 3. Heating Surface. — The third point concerns not so much a variation in construction, as it does the lack of knowledge of the fundamental principles, this suljject having been much less fully investigated than other portions. Recent investigations show conclusively that the axiom that the heating surface is a magnitude proportional to the desired efficieucy of the boiler, cannot be sustained. It is evident that there must be a very considerable difference in the heating value of portions of the surface which are at greatly different distances from the fire. A very high temperature of the gases at the beginning, and a comparatively low temperature near the end, must mean a rapid formation of steam near the fire and a weak production over *When the rivets are made of no better material than the plates, the re- duction for single riveting is about 53 percent., and for double riveting about 41 per cent. Triple riveting, as shown in Fig. 155, is too expensive to come into general use. fThe following list will serve for those who aesire to refer to the original and fundamental publications upon this subject : — Friedrich Siemens, Heiz- ver fohren niit freier Flamment faltung, Berlin, Springer, 1882; Siemens' Regenerativofen, Dresden, Ramming. 1S54 ; Vortrag von Friedrich Siemens Uber Ofeubetrieb mit ausschliesslicher Benutzung der strahlenden Warme der Flamme, Gesundheitsingenieur, :8S4 ; Vortrag von demselben iiber ein neues Verbrennungs-und Heiz-systeni, Busch, Journ. f. Gasbeleuchtung, etc., 1885; Vortrag von demselben in der Ges. Isis in Dresden iiber die Dis- sociation der Verbrennungsprodukte. Dresden, Blochmann, 1SS6; Vortrag von demselben im Siichs. lug. u. .\rchit Verein iiber die Verhiitung des Schornsteinrauches, Civ. Ing. Bd., 32, Heft 5, 1886; Vortrag vom demselben ini BezVer. D. Ing in Leipzig am 8 Dez. 1886 Uber den Verbrennungsprozess, 2 Aufl , Berlin, Springer, 1S87; Vortrag von demselben, gehalten in Ham- burg im Ver. D. Gas-und Wasser fachmanner iiber Regenerativ — Gasbrenner, etc., Dresden, Ramming, 1887; Ueber die Vortheile der Anwendung hocher- hitzter Luft fiir die Verbrennung, etc, 2 Aufl., Berlin, Springer, 1887, X For example, a test by K. H, Kiihne & Co., of Dresden Lobtau on Feb, 16, 1884, showed a gain of 26 per cent, due to the substitution of a Siemens furnace for one of the usual kind ; the conditions of draft and cleanness of flues being alike in both cases. g Two methods have been described by Dr. Siemens, both of which have been applied by him to flue boilers. In the first, the combustion of the fuel takes place upon a grate in a combustiou chamber which is directly over the grate. A bridge wall of fire brick is placed about half the length of the grate further back, and beyond this are two ring shaped screens of fire brick, which are so placed as to direct the products of combustion toward the axis of the boiler flue ; after passing through the flue the gases return about the outside of the shell and are then sufficiently cooled to be permitted to pass over the portions of the shell unprotected by water on the way to the chim- ney. In the second method the fuel is burned to gas in a gas producer separately constructed from the boiler, and the gas mixed with heated air and thus delivered to the boiler flue, where it follows the same course as in the first case. THE CONSTRUCTOR. 271 distant portions of the surface. It lias been shoT\-n that in some instances the heating surface of one and the same boiler ma}- be reduced one-half without caxising any reduction in the steam production. The usual method of proportioning the heating surface in all kinds of boilers appears to be based upon previous results with similar forms, and hence is often one-sided and unsuited for sj-stematic investigation. A new departure in the discussion of this important subject has been made by the chief director and engineer of the Swedish railway's, Mr. F. Almgren. He has made the subject of the proportioning of heating sur- face the object of a series of experiments extending over a number of j'ears, and has placed the matter upon a much higher plane of investigation than heretofore. The practical results are of much importance, and in advance of the publication of the whole the following general discussion has kindly been placed in the author's hands by Mr. Almgren, and is here given in his own words.* PRACTICAL RESEARCHES UPON LOCOMOTIVE BOJLERS WITH SMALL TUBES. BY F. AIy the following formula, in which D is the clear opening through the valve. j = \/i? + 0.16' (401) For round valves D is the diameter of the opening ; for rec- tangular openings it is taken as the smaller side of the rec- tangle. The blow with which a valve strikes the seat increases in force with the amoinit of lift (compare \ 368), and as the lift depends upon the actual size of the valve, this objectionable feature is reduced bj' using several valves of smaller size instead of a .''i.ngle large one. Fig, 1 1 26, A flap valve with metal seat, which is so constructed as to offer as little obstruction as possible to the flow of liquid, is shown in Fig. 1126*. This is tapped out for the standard pipe thread system described in J 342, the cap gives access to the vafve, the screw plug limits the amount of lift, and a flexible connection between the disk and the hinge enables the former to obtain a fair bearing on its seat. The freedom from shock would be somewhat less if the bottom of the case conformed to the shape indicated by the dotted lines. , are the pressures per unit of area on each side, and the weight of the valve is neglected or counterbalanced. From this we have F ' or of the ratio -~ is put : A (403) The pressure/ — p^ is the unbalanced pressure on the valve, and the ratio -^ P-P, is the ratio of unbalanced pressure. Upon this question of unbalanced pressure much depends, and many calculations have been made for various sorts of valves, the pressure tending to close the valve being much reduced in bell shaped valves, such as shown in Fig. 1134. Experimental researches, made upon pumps of various sizes, however, have shown that only a small excess of pressure is actually required, t At the same time the preceding formula shows that the question of the unbalanced pressure is by no means a subject to be neglected.! As an instance of the effect of unbalanced pressure may be cited a bell shaped valve, i metre clear opening, in the shaft of the Bleyberg mine, of which the seats could not be kept down by their own weight, but would adhere to the valve, rising and falling with it until secured by some other means. Riedler has observed the fact that in arranging valves in a series in a cone as iu Fig. ii36(Z, the uppermost valve which is subjected to the greatest excess of pressure according to (403), lifts first, and is followed by the others, the lowest rising last. It appears that a thin film of water is retained between the bearing faces of valve and seat, which responds rapidly to the pressure of the lower column p^ ,and thus tends to reduce the value given by the above equation. If we first make the assumption that such a film exists and acts in the manner indi- cated, we have for two successive ring valves, arranged for example as iu Fig. 1136a, the following stresses in the liquid. The weight of the valves, beginning from the top, is indicated by G-^ and G.^, and their projected areas by Fy and F.,. P'=p,-\- ?i and /" = /. + -^;^-% (404) Now it appears by examination of the weights and areas that G G txnder the circumstances ^^ is greater than — ', which is then F„ F^ also true for the entire second member of the value of /" |, so thatj*' is the resistance which is overcome first. In the case of the Bleyberg mine F^ is very much greater than F^, and p'^ becomes less than p' which explains tne action of the yalve seat. The actual behavior of the film of liquid between the surfaces of contact may not be so definite as indicated above, but it ap- proaches to it as an approximation. This is shown by the very valuable researches made by Prof. Robinson upon a valve acting under steam pressure. || In two extensive series of ex- periments he investigated the actual weight required to lift a valve under pressure. The results showed that the unbalanced pressure was much less than py — p. ♦German Patent, No. 33,103, t Reference is especia'iy made to the numerous and valuable investiga- tions of Prof. Ried'er. X See the comprehensive papers of Prof. C. Bach, in Zeitschr. D. Ing. for i386. " Versuche zur Klarstellung der Bewegung Selbstthatiger Pumpen- ventile.'' § In the case of the arrangement shown in Fig 1136a, the ratio of weight and area for the three valves, proceeding from above downwards, is 50 : 76 ; SS- II See Trans. Am. Soc. M. E-, Vol. IV, 1832-1883, p. 350. r-a* The experimental valve, shown in Fig. 1137, had an annular seat of 6 in. outside and 2^^ inside diameter, and was subjected to a steam pressure />, above, and to the atmospheric pres- sure/ below. In the follow- ing table p' indicates the pressure per square inch which would give the equiv- alent of the actual pressure _ F required to lift the valve, V- while a is the area and d the diameter of a circle for which — ^ iPi — P) = ^- This circle Robinson calls the circle of equilibrium, and it is always smaller than the upper pro- jection of the valve. The valves under a and d are taken approximately at the nearest values. The un- balanced pressure can readily be determined from the table. Fig. 1137. P- i'x-p /' a d d' Pounds per Pounds per Square Square Inch. Square Inch. Inches. Inches. Inches. 5 8 5-^ 26 2.53 10 17 5-8 2-7 2.85 15 26 6.0 2.8 2.92 20 36 6.2 2.8 3.02 25 46 6.4 2.9 3-09 3° 57 6.6 2.9 3-14 35 69 6.8 2.9 319 40 Si 7.0 30 3.22 45 95 7-3 30 3-25 5° 112 7-8 3-1 3-27 55 129 8.2 3-2 3-29 60 150 8.7 3.3 3-31 65 172 5-; 34 3-33 70 1 98 9.8 3-5 3-34 75 230 IO-5 37 3-35 If />! — / = 45 lbs. we have, since aT = 3 in. = )A 6 in. for the excess pressure, one-fourth /, — p ; for p^ — / = 75 lbs. it is equal to 0.3S (/>, — p). The law of reduction of pressure between the surfaces from p^ to p is not simple. The corres- ponding curve is convex towards the axis of abscissas, as shown in Fig. 1 137. If it is desired to determine the mean pressure pm we have from the table for p^ — p =1 ^ the value pm '■ A— A 4-43 forpi — /■ = 75 it is/„ : A-/_ 2.36 For a rough approximation we may put/>„ = }{ (p^ — /). Prof. Robinson has deduced a theory from these experiments. He assumes that between the surfaces there exists between the pressure A at the outer circumference to the pressure p, at the inner cir- cumference, a gradual increase of pressure from/ to/,. Under the assumption that the fluid under consideration is incom- pressible he obtained by pure analysis the following equation for the value of d : 0" = 2 r ./ R (^-0 (405) in which I? and r are the inner and outer radii of the ring of the seat. The values of d' as obtained from this equation are given in the fifth column of the table. They increase nearly as the experimental determinations of d, but with Robiuson's assumption of an entirely elastic fluid they are 10 to 15 per cent, too great. Probabl}' steam should be considered as mid- way between an elastic and a non-elastic fluid. The deductions from Robiuson's experiments are hardly ap- plicable to pump valves because the lifting of the valve by the action of the lower column is effected by a varying pressure, while in the experiments / was uniform. If we accept Robin- son's theory we arrive in fact to what has been already stated, namely, that when the value of/ increases between the surfaces until it reaches /,, the pressure p., will be balanced, since in equation (405) for / — /, the value of a" r= 2 r, that is, the unbalanced pressure becomes zero. This also agrees with Riedler's indicator tests, since experiments with the indicator failed to show appreciable unbalanced pressure. 278 THE CONSTRUCTOR. These experiments appear to indicate that practically the unbalanced pressure cannot be great, and in most cases for self- acting valves it may be neglected. Prof Robinson's experi- ments and theory may serve to determine with considerable accuracy the pressures at which a safety valve begins to lift. I 369- • C1.0SING Pressure of Self-acting Valves. As already shown, a self-acting valve opens whenever the pressure in the under column exceeds that above the valve. As soon as the direction of pressure is reversed the valve should close quickly. This is especially important, as Riedler has shown in the case of suction valves, since when the closing is delayed appreciably after the reversal of the pump piston, the moving column of water is checked with a sudden shock. For this reason the suction valves are given especial attention, as shown in the example already cited from Creuzot, in which there are 38 suction valves and only 27 discharge valves. In order that the lift shall not be too great and to insure prompt closing, the valve may be loaded with a definite pressure, K, obtained either from the weight of the valve, or by means of a spring, or by both. This ques- tion will here be exam- ined. Referring to Fig. 113S, we have for the lifting pressure due to the under column : P- = A + 7?^ = A + ? (406) in which /> — P\^= q the closing pressure per unit of area. For a height //, and putting 11 = the circumference of the cylindri- cal space inclosing the valve, we have : <£', /: :'. = Fv ■Wi being the velocity of flow at the outer edge of the valve, and t' the velocitj' of flow in the under column, h being in feet. Now if 2V is the velocity at the inner edge of the valve we have that is : But we also have W= ^ 2 gh' = ^2^X2-3? (since the pressure per square inch is equal to ^ — 1 and hence : Substituting, we get : 2 ^ X 2.3 ^ Fv Now it is desirable that -v and zc, should not be too great ; that is, the ratio of h u to i^ should be equal to, or less than, unity. If we put h u — j3 F, we have : /5 4- 2.5" X 2.39 and, putting for g its value ^ 32.2, we get : a iP' a v' or say = * 148.12 ^2 from this formula we get for : 424 g = .006667 a v^ .01185 a z^ .02666 a v' .1066 a v' (407) /3= I in which v is at its maximum value when it equals the velocity of the pump piston. For purposes of numerical calculation we still require the value of n. Taking the width of bearing s, and projection in the case of conical valves s^ from (401) and (402) we have : Dia. D = 2 in. 4 in. 6iii. 8 in. 10 in. 12 in. 16 in. Width of seat s = Projection .ri = . . Cone valve a = . Flat Valve a = . . 0.44 0.28 ■■65 2.17 0.56 0.40 1.44 1.64 0.64 0.48 1.36 1.44 0.72 0.56 I 31 1-39 0.80 0.64 1.27 '■35 0.84 0.68 1.24 1.30 0.96 0.80 1.21 1.25 An example will show how the pressure of closing can be calculated : Example I. — For a conical valve -whose smallest diameter Z* = 4 inches, and the greatest velocity v of the lower column is 6^^ feet per second the area of inlet of valve hu = F^ and /3 = 1, we have a pressure of ^ = .006667 X 1.44 X (6.5)" — 0-4 lbs. per sq. in. For the total pressure we have K— (4 + 2 X o-4°)- X 0.4 = 7.24 lbs. 4 Example 2. — For a fiat valve of the same dimensions we have a = 1.64 — whence K = — - — 7.24 = 8.24 lbs. 1.44 The method of calculation is similar for ring shaped valves and can readily be applied. The formula (407) can only be considered as an approximation as the variations in the jet of water affect the pressure. It is evident, however, that K is often quite large. In the preceding calculation the momentum of the water column has not been taken into account. In some cases this is sufficient to hold the valve open until the piston has made a great portion of its return stroke. This is well shown in the case of the pump at the Bleyberg mine (? 319, note) which ap- parently showed a discharge of 104 per cent. If this action can be made to exist during the entire stroke by giving the water a sufficient velocity by contracting the tube that the discharge valve does not close at all, this valve may be entirely omitted. This is the case with the single valved pump of Edmond Henry,* which has only a suction valve and no discharge valve. An analogy to this form of fluid ratchet is found in Langen's fly wheel ratchet train. Fig. 730 and 731. In this case the momentum of the fly wheel is sufficiently great for it to suffer no perceptible loss of velocity during the return stroke of the pawl. Mechanically Actuated Pump Valves. The numerous investigations of recent years have show. ' that by proper loading of the valves, combined with a reduc- tion of lift, the shock of the water in a pump can be verj' ma- terially reduced and kept within practical limits, even for high piston speeds. The reduction of lift involves a great multipli- cation in the number of the valves and a great increase in dimensions. For this reason another solution of the problem has been attempted, namely, that of abandoning the self-acting feature, and actuating the valves by mechanical means. The best arrangement seems to be that in which the valves are opened by the action of the water, but closed by a positive gear in advance of the shock. The application of this method enables the size of the valves to be reduced, and as it is princi- pally used for large pumping engines the valves can be oper- ated by connection to the flj' wheel shaft. Professor Riedler has recently made very valuable investigations upon this system, t Fig. 1 139. Fig. 1 139 shows the valve gear for the Riedler pumping engine at the Wartinberg mine. The revolving cam d, closes * See Revue Industrielle, p. 342, September, 1SS8, where the complete theory of this form of pump is given. fSee Riedler, Mine Pumps with Positive Valve Gear, Zeitschr. D. Ingen- ieure. 1S88 p. 481. THE CONSTRUCTOR. 279 the valve b, just as the plunger is at the end of the stroke, and permits it to open by the action of the water. The valve is held to its seat by a a spiral spring. Pumps of this construction operate very smooth- ly. Further details of this construction are given in the arti- cles already cited. For blowing engines, and especially for air com- pressors, positive ly actuated 'valve gears are much used. A very simple action for the inlet valves is shown in Fig. 1140. The piston rod c moves the valve b, by means of the fric- tion of the rod in the stuffing box, the ac- tion taking place just at the reversal of the stroke. Examples of this construction are to be found in the air pumps for use in physical laboratories. ? 371- Valves with Spir.^i, Movement. It is not so convenient to construct a valve so that its motion shall be both rotary and rectilinear axially, and this construction is mainly limited to valves which are operated by hand. Fig. 1140. This is the counterpart of the throttle ratchet shown in \ 250, and valves of this sort have been much used with throttling" governors for steam engines. The closing of such valves is im- perfect, as the edge must be rounded near the hub of the valve, thus giving only a line of contact* Fig. 1142. If it is desired to use throttle valves for regulation of water pressure, as the case of turbines, etc., it must not be forgotten that the resistance of the valve will materially affect the effi- ciency. For self-acting valves a variety of throttle valve may be used, in which the area of one wing is only about )-l to Ys, that of the other wing, thus partially balancing the valve. This form, which is old, appears to be again coming into use. t Lift valves which are situated in vessels which are not closed at the top may be balanced in a simple manner by making the valve with a tubular continuation which extends above the sur- FiG. 1141. Fig. 1141a shows a conical valve with spiral motion, as used on the Giffard injector. This arrangement enables a very fine adjustment of the opening to be obtained ; a similar form is also used in the so-called " cataract " for steam engines. The sharp point of the cone has caused valves of this sort to be called "needle" valves, and similar forms, without the spiral action, are found in gas regulators. Stop valves for steam and for water are frequently made with spiral motion. An example is shown in Fig. 1141^. When the valve is not in contact with its seat it has both a vertical and a rotary motion. In the parti- cular form shown the valve has a disk of asbestos which forms the surface of contact with the seat. This general form is known as a "globe " valve on account of the form of the body, and such valves are very extensively used for steam and water. ? 372- Bal.anced V.ai,ves. Valves which are to be operated by other means than by the action of the fluid, are advantageouslj- made so as to be relieved from fluid pressure, and thus offer less resistance to operation. Valves of the wing or flap construction are conveniently bal- anced by combining two valves moving in opposite directions into one valve of the form commonly called "throttle" valve. Fig. 1 142. Fig. 1 143. face of the water. A balanced valve upon this principle, as used for an outlet valve in a canal lock, as at b-^ and b./ , *The form shown at 5 is recommended in Revue Industrielle, p. 205, May 26, 18SS, as insuring a better balance, but from Robinson's experiments, already cited, this form would offer too much resistance to opening. tSee Belidor, Architecture Hydraulique, Paris, 1739, Vol. II. These valves were of brass with metallic packing. iz8o THE CONSTRUCTOR. "Pig- 993> IS shown in Fig. 1143. This valve, designed by Constructor Cramer, is made with a cylindrical shell of sheet iron extendmg to the surface of the water. The diameter of this shell is the same as that of the valve, and the weight of the valve, which is by no means small, is partially counter- balanced, leaving only sufficient to insure proper closing and seating. * _ If it is desired to apply Cramer's construction to valves which are subjected to high pressure, this may be done by using two stuffing boxes, one external and one internal, Fig. 1 144. as shown in Fig. 1144, which, however, adds to the complica- tion. For lift valves which are to act under high pressure a better construction is the so-called "double-beat" valve, which, like the throttle valve, consists of two similar valves in which the pressures oppose and neutralize each other. Three forms are shown in the accompanying illustrations. Fig. 1 145a Fig. 1145a. being a double disk valve, and Fig. 1145,5 a tubular valve. Both of these were invented by Horublower in the latter part of the last century. Fig. ii45(; is a bell or Cornish valve. These Fig. 1 1454. valves each consist of a pair of conical lift valves, the varia- tions appearing in the details of the connections and passages. •See Annales des Fonts et Chaussees, 6me serie, Vol. XII, 1886, 11 Semes- tre, p. 248, also Zcitschrift fur Bauwesn, 1880, p. 155. When the projection of one seat falls within that of the other, as in forms b and r, the unbalanced pressure is that due to the projections of both seats. If so desired, however, these may be Fig. 1145c. made as Fig. 1145(7, with one seat directly over the other, in which case the pressure/; — p need only be calculated for one seat. For the preceding double seated valves we may make : for the width of seat i.= '/,' (0.2 »Ad -\- 0.137 I and for the projection j-j = yi 0.2 ^D''^ (40S) In form a the mean diameter D' of the valve is = o.S times the diameter D of the pipe, while in forms b and c the diame- ters of valve and pipe are the same. For the force required to lift the valve, taking the projection jj into account and assum- ing the pressure between the surfaces to be as in \ 36S, equal to Vi {Pi P), we have, neglecting the weight of the valve : P' IT D' s,' VyiiPi-p) ■- (409) while for a single conical valve of the same diameter D it would be : P = G D'- + =.3' s,^{D + s\) ](A-; • (410) P is proportionally very great, while P' is not always unim- portant. Example.— For D' — 12, we have for form a, Si' = yi, { 0.2 ^/Ta ) =0.346". If uow />! — p = 60 pounds per square inch we have ; -f = IT X 12 X 0.346 X 7^ X 60 = 521 pounds. For a single valve the diameter would be Z* =: — -- = 15 inches, and from '402) Si = 0.2 \/ 15 = 0,77, whence P= -— 15- + -A X 0.77 IT ( 15 -(- 0.77 j 60 = 12,126 lbs. so that F is nearly 24 times P'. It is verj' desirable for double seated valves which are to be used for steam, that both valve and seat be made of the same material, in order to avoid unequal expansion. Fig. 1 146. Double seated valves are also used for water, shows such a valve arranged for a sluice. Fig. I 146 THE CONSTRUCTOR. 281 This valve is made with flat seats, the lower seat being faced with rubber, and the upper one packed with leather secured to the housing whicli is shown over the valve. The valve red runs through this housing and through a tube above the surface of the water. The diameter D is 1400 mm. = 4 ft. 7 in. This is practically a tubular valve, similar to Fig. 11456, except that the direction of flow is reversed ; this arrangement has also been used by Hornblower. The leather packing at 2" is made flexible, since the projections of the valve seats lie one within the other so as to make a slight tendency for the valve to lift, without entirely overcoming the weight of the valve. Balanced valves of the kind described above are also adapted to large steam engines. In some instances a small balanced valve is arranged so that it is lifted first and admits steam under the main valve before the latter is lifted. Another device is that shown in Fig. 1147, known as Ait- ken's automatic steam stop. The main valve b, is closed by being screwed up against its seat by the spindle and hand wheel. Before opening, it is balanced by admitting steam through the by-pass vah-e b' . The valve itself is loose on the spindle, and if through any breakage in the pipe beyond the valve a sudden or rapid flow of steam should take place, it will be automatically closed by the force of the current. plug and one for the spindle. The management of screw cap and jam nut enables a fine adjustment to be obtained.* Fig. 1 147. Lift valves may also be balanced by making a balance piston connected with the valve, the pressure of the steam acting upon the piston in the opposite direction to the action on the valve. This construction has also been applied to reducing valves in the place of weighted levers or springs in various ways, but space cannot here be given to the subject. B.— SLIDING VALVES. I 373- Rotary Valves and Cocks. For rotary valves the bearing surfaces are conveniently made conical, so that a simple endlong pressure on the valve will hold it firmly to its seat. Valves of this construction are known as cocks. Fig. 1 148 shows two forms of such cocks which are in general use. The opening through the plug of the cock increased in height in order to obtain a full area without requiring the diameter of the plug to be too great ; the area of the opening through the plug being made equal to the area of the pipe, i. e., = -D\ 4 According to the experiments of Edwards, a good taper for' the plug is \ on each side. For the thickness i of the metal in the body of the cock formula (319) may be used when the D material is of cast iron, which gives & ^ 0.472'' -| ; for bronze 50 the thickness ma}' be made one-half to two-thirds this value. The design shown in Fig. 114814 has the plug entirely inclosed in the body, and is made with two stufiing boxes, one for the Fig. 1 148. Fig. 1 149 shows two forms with hollow plugs, these being much, used for injection cocks for jet steam condensers. Fig. 1 149. When the angle of the apex of the cone becomes iSo° the plug becomes a flat disk, and this form is often found in the throttle valves of locomotives, and less frequently in the valve gear of engines. True cylindrical plugs, i. e. , those in which the angle of taper is equal to zero, are rarely used, although recommended by some. This form is better made in a portion of a c^'linder, and operated by an oscillating motion, as in the Corliss and similar valves. A starting valve of this type, used as the steam admission valve for a triple expansion engine is shown in Fig. 1150. Fig. 1 1 50. At a is a longitudinal section, b a cross section, and at c is shown the seat looked at from above. In the one seat three passages are controlled at /', I" and /"'. All three are closed when the valve is in the position shown at b, but open at the same time when the valve is moved to the left. Tlie trapezoidal opening in /' admits a small amount of steam to the high pres- sure cylinder at the same time that a little live steam is admit- ted through /" and /" to the intermediate and low pressure cylinders, so that the engine is sure to start. The valve is then thrown all the way over, closing /" and /■"' and throwing /' wide open.f * Mosler's German Patent, No. 33,912. t See Zeitschr. D. lugenieure, p. 509, Marine Engine. Meyer, Triple Fxpansion 282 THE CONSTRUCTOR. i 37-1- Gate Valves for Open akd Ci.osed Conductors. A great variety of valves has been devised for open water con- ductors in the form of gates bj' which the flow can be regulated. Such gates have been preferably made of wood with the excep- tion of the operating mechanism. At the present time iron is be- ginning also to be used for the gates, and as in the case of other branches of work, wood is likely to be less and less used, being limited to a few special cases. For very broad streams the con- struction of such gates is now sometimes made upon the princi- ple of subdivision. In such cases the breadth of the stream is subdivided into a number of smaller streams, each with a sep- arate gate, thus keeping the gates small enough to be m.ovable by hand. A weir which is placed in a stream is both in principle and in construction a valve. When the water in the stream is low the flow is entirely checked ; for the mean flow the stream passes through the reduced opening with a velocity due to the reduc- tion in section, while for high water the entire width of the dam is overflowed. Movable weirs are plainly examples of regulating valves. French engineers have given much atten- tion to moveable dams with excellent results. A new design for a moveable dam by Schmick is shown in Fig. 1151.*. This dam consists of a number of pontoons, each three of which are Fig. 1151. secured together by a yoke and anchored by a chain to a point up the stream. All three pontoons of each set are arranged with variable water ballast in two or more compartments, a/ and a./. An adjustable valve 3, enables comnranicatiou to be made with the upper water level, and the compartment a/, and a similar valve b., connecting the compartment a./ with the lower level, while a third valve b., enables communication to be made between the two compartments. By varying the open- FiG 1152a. ings of the valves the pontoons can be caused to regulate the difference of water level above and below the pontoons, while if all three valves are closed the pontoons will rise and fall with the variations in the level of the stream. Gate valves are much used for water mains, and an example of the many varieties used for the purpose is shown in Fig. 1152a. The gate or disk of the valve is made of bronze, and is wedge shaped, in order that it may be firmly pressed against its seat "when the screw is tightened (this forms a pressure of the second order) while the pressure is immediately relieved at the commencement of opening. The screw is in this case made of ' ' sterro-metal ' ' to avoid rusting. Gate valves are also used for gas mains, and a valve for this service is shown in Fig. 'i\^2b: In this instance the valve is operated by means of a rack and pinion. The motion is made in the horizontal direction so that the valve will remain in any position, the only resistance being that of friction. ? 375- Sl,IDE Vai'= a -]- e -\- s, that is half as much as before. It is evident that the laps — and — must bear the same relation to — as the diagram gives for a : e: i, in the preceding forms. 2 5 16 ' ^a = ^5- we have / = s-S X 0.75 + 3.5 X 0.75 + 3 x 0.375 + 0.1875 + 2 X 0.1875 + 0.3125 = 8 75" 2 )- = 0.75 + 0.75 -I- 0.375 = 1.875. This gives for the work of friction of such a valve as compared with an equivalent plain slide valve : 6.S75 X 375 ^ 8.75 X 1.875 ' '' which is an important gain. 5. Borsig's Gridiron Valve, Fig. 1157. This is the same in principle as the preceding, and differs only in construction, the .-.L-l- Fig. 1 157. exhaust passages being carried on each side of the valve instead of above, as in Penn's construction. 6. Hick's Double Valve, Fig. 115S. This ^—: -— fa- is intended for use with L'3^^_LJi^iLi compound engines with parallel cylinders (Hornblower and Woolf), the ports //' and ///' are for the high pressure cylinder, ^g and //" and ///■" for the low pressure cvlin- der. The width 7 of the valve is : ^=5 a + sai+ 6e + 4 s + e^ + t^ X to. piQ I , .3 Usually ffj is made equal to valves cast together, and the over travel beyond the valve seat gives the admission. We have as before : r ^= a -\- e -\- s, and ^ = i-j-r=a + e+^ + s^ i = e+7-=a-\-2e+s y (413) ao =^ a J 284 THE CONSTRUCTOR. This gives for the width of the valve .- / = 3a+2/5+2i(, + 2i'or: "I which is considerably greater than for au ordinary D slide valve. (414) 16 tffic E^iample 3. — If as in Example 1, we make a = £• — 0.75", 0,375",/ = 0.1S75", we then have / = 13 X o-75 + 0.6875 + 4 X 0.375 + 2 X 0.1S75 = 12.3125" against 6. S75'' for the plain Z? slide valve. It will be evident that the £ valve is only available for small port widths and small laps, as will also be seen in Figs. 1006 and looS. The principal vahie of this valve lies in the use of the outer edge of the valve seat as the edge of opening, which principle also has a valuable application in the following valve. 9. Trick's Valve, Fig. 1 161.* This is a double valve and con- sists of one D valve over another, with a steam passage be- tween. As before, we have r -- a -j- e -{- s, and also make bo = 2 1? — t, i. c. , the inner edge of the outer valve when the valve is in mid-posi- tion, is at a distance -= e from the edge of the valve seat. The consequence is that when the valve is moved a distance equal to e, say to the right, the passage through the valve opens to ad- mit steam at the same instant as does the edge of the valve on the left. This gives a steam admission twice as quickly and an opening twice as great as would otherwise be the case. The following positions from a to /, Fig. 1 162, will show the succes- sive actions, the exhaust ports being omitted for simplicity. Fig. 1161. yvnr--'-'Y--7-~ , - » , „ ,,„,, ■,,.,-,■,.,. .-,,,., ^ , M a. The admission is just about to take place both from the edge of the valve on the left and through the passage in the "■'alve. If we apply Zeuner's diagram (compare Fig. 1025) see Fig. 1163. Fig. 1 163, we must from the point A, which indicates the port opening, double the width given by the Zeuner circle until the entrance to the passage in the valve is wide open, as at b. By thus doubling the opening in the diagram we obtain the curve A i?,. b. From this position on, the opening at the left continues to grow wider, but that through the valve on the right does not, hence on the Zeuner diagram from this point we return to the opening which the regular valve circle gives, to which is added the constant opening c ^= B B^=^ C,C^ indicated by the curve Bi Cy This continues until the inner edge of the opening of the valve passage on the left reaches the edge of the bridge as at <-. c. As the valve continues to move the passage through it is gradually closed, but the steam tort is opened to the same amount, and hence the actual port opening remains constant. This continues until the position d is reached, when the passage through the valve is entirely shut off. This is indicated in the diagram by the arc C, D, struck from the centre at I. (/. The valve continues to move to the right until it is en- tirely upon the bridge, the corresponding portion of the dia- gram being the arc D E of the valve circle. e. The valve from this position moves on the bridge beyond the port until it has traveled a distance equal to s, as shown at f, during which time the port opening remains constant, as in dicated in the diagram by the arc E E' struck from the centre I. From this point the same actions take place successively in the reversed order. It will be seen that Trick's valve gives a much quicker opening and also a much longer duration of the full opening than does the plain slide valve. It remains to be seen how these features can be used to the best advantage. According to Trick's prac- tice this is best done by making the value of j negative, and also^/. This makes the port opening from Ci to (T/ in the diagram constant, as shown in the diagram. In order that the apparent contraction of the ports by the change in the sign of .s shall not occur, the value of a is made greater than would otherwise be the case. Under these condi- tions we have for the exhaust port Oo, the equation : ao -\~ t — e-^ — a — « = a — .y, in which .j is given the magnitude equal to the distance which the edge of the valve is moved beyond the edge of the bridges. (See Fig. 11 62/). We then have : For the exhaust port, «» = 2 j + c, -|- ?' — s — / | For the bridge, b ^ e — <'i + -^ — t \ c^jci For the passage through valve r = c — / — t\ 1 V4 0; For the total valve, l=\a-\-ii,e—i\-\-i — 3^+;" J E-vample 4. — Making s -- other data the same as the plain slide valve of Example i, and we have : a = 0.75 4- 0.1875 = 0.9375"; e = 0.75 ; i ^ 0.6875 fj = 0.078", whence ; f— a + e — s = 0.9375 -I- 0.75 — 0.1875 = 1.5" 2.2655" and - / = 2 X 0.9375 + 0.07S -t- 0.6875 — 0.1875 — 0.1B75 - ei + s — t = 0.7; c = e — t- — 0.078 -f- 0.1875 . ■ ^1 = 0.4845" -o 1875. !=4i + ie — e-i+i — 3S-i-i — 3.ys+3 — 0.078 + 0.6875 —0.5625 4-0.1873 = 6.9835" as against — 6.875" for the plain slide valve, which compares very favorably. * This valve was invented and made by Trick at Esslingen in 1857, and by Allan in England in 1858-1860 ; in the United States it Is correctly known as Trick's valve. Fig. 1 164 shows the application of the author's valve diagram, already shown in Fig. 1024. The action of the inner portion of the valve is the same as with the ordinary slide valve. For comparison the following dimensions of an executed valve by Trick, are given : a = 1.77" (45 mm.), f , = / := 0.07S" (2 mm.), / = 0.216" {5.5 mm.). THE CONSTRUCTOR. 2^5 b = -i." (25.5 mm.), e = 0.846" (21.5 mm.), c = 0.55" (14 mm.). s = — 0.374 (9.5 mm.), ;- = 2.24" (57 mm.), / = 5.27" (134 mm.). bo = 1.45'' (37 mm.), lo = 5-83" (148 mm.), i5 = 30°. Trick's valve is especially well adapted for use on compound' marine engines, and has recently been used in the forms of double and gridiron valves, as in Nos. 4 to 7 preceding.* l 376. Bal-A-Nced Si N whence a P' . The value n P' shows the Revolutions per Minute. Pressure Pouuds. H. P. of Eugiue— A''. H. P. .V for Valve. Ratio of preceding. Ratio 125 175 200 10 30 40 3 9 13-5 2 per ct. r.2 " 1.4 " 48 61 91 Unbalanced Valve. — Cylinder 9" x 12". n = 100. Ratio H. P. by Brake. JV' Ratio a f 4-5 4.5 per cent. 247 7.0 3-5 245 8.25 4.0 330 8.9 6.0 534 II. I 7-3 " Sio Balanced Valve. — Cylinder <)" x '4 Ratio H. P. by Brake. .V Ratio a /» 1 1.4 1.2 per cent. 137 13-5 II 149 14.0 1.0 " 140 15.6 1.0 " ■56 The last column in each of the three tables has been added by the author of this work, and is obtained as follows : If N and N' are the values in horse power of the engine and of the resistance of the valve, v and v' , the corresponding mean velocities of piston and valve, and P and F' the force upon N' each, the experiments give the relation — ^ipor P'v' ^^ Pv. Hencejit follows for the force required to move the valve: 1p P V P =■ Now for a given engine v' bears a constant re- lation to the number of revolutions n, so that we may put * See Zeitschr. D. Jng. 1S88, p. 509, Triple Expansion Engine by'G. L. C. Meyer, of Hamburg. t See Trans. Am. Soc. Mec. Engrs., Vol. VII, p. 631 : C. M. Giddings, Descrip- tion of a Valve Dynamometer for measuring the power required to move a slide valve at dilTerent speeds and pressures. Fig. 1165. the increase in power required to operate the valve. It is evident that /" increases more slowly than the increase in steam pres- sure, but the resistance becomes quite great for unbalanced valves. The present difficulty lies in the limi'ed number of engines upon which experiments have been made. Fig. 1165 shows the character , of diagram made by ' ' Gidding's apparatus, and it will be seen that the greatest re- sistance occurs at the beginning of the stroke, diminishing toward the end to nearly zero. The inequality between the resistance of the back and forward strokes is due to the action of the steam pressure upon area of the valve rod. In considering the pressure upon unbalanced slide valves ihe consideration mentioned already in connection with Robin- son's experiments on lift valves is that there exists a counter pressure between the valve and seat which overcomes an im- portant portion of the pressure on the valve. As a rough ap- proximation we may take this pressure between the surfaces as ■^ (/>! — p). Gidding's experiments show that the coefficient of friction is not constant, but diminishes with increased speed. More extensive experiments are much to be desired. The method of balancing slide valves may be divided into three classes : a. Removal of pressure from the back of the valve. b. Opposing the pressure on the valve by counter-pressure. c. Equalization of pressure on all sides. Typical examples of these three systems will here be given : a. Removal of pressure from back of valve. I. The so-called long Z^-valve, invented by Murray, and used on engines built by Watt, the pressure was relieved from the back by a form of stuffing box, which answered well, but was not adapted for high pressures. b. Fig. 1 1 66. 2. Fig. Il66a shows the balance ring of Eoulton & Watt. The under side of the steam chest lid is finished parallel to the valve face. Against this surface a ring of soft cast iron or bronze is fitted steam tight, this ring being fitted to the valve by an elastic packing and moving back and forth with it. The space within the ring is subject only to the exhaust pressure. This form was used on the Great Eastern. Fig. ii66d shows the balance ring of Kirchweger, much used for locomotive engines. In this form the ring is pressed against the lid by steam pressure instead of spring packing. Both of these devices, as well as the similar ones of Penn, Borsig and others, leave too great a portion of the steam pres- sure unbalanced (at least 30 per cent, being left), and also pre- vent the valve from leaving its seat should water be carried into the cylinder X b. Balancing by counter-pressure. 3. Cave's Valve with Balance Piston. The valve in this form. Fig. 1 167, is connected by a link to a piston, which works in a cylinder formed in the steam chest lid, and is subjected on the outside only to atmospheric pressure. Bourne's method of balancing is similar, except that the other side of the balance pisjon is in communica- tion with the exhaust. 4. Valves with Rolling Support. Fig. 11 68. At a is shown I,indner's valve. The top of the valve itself is formed into a piston, sliding up and down in the valve and supported by two segmental rollers. J An elegant construction for this form of balanced valve is that of Robir/" son. (See Trans. Am. Soc. Mech. Engrs., Vol. IV, p. 375. 286 THE CONSTRUCTOR. The degree of balancing is dependent upon the size of the piston. At b is shown Armstrong's roller supported valve. In this case the valve is closed at top as usual, and the construc- tion is very simple and practical. At c is Bristol's valve, in (j ^ \ 'i' '1 y ^ 1' SjWAV, >^,\ •i\ , iiiiii 1 I 4 1 IIV Fig. 1 167. which the valve is supported on a system of friction rollers. This has been used by the works at Seraing for large marine engines. To this class of methods of balancing belongs also that used by Worthington, in which a cylinder is formed in the top of the valve, as shown in Fig. 1016. Fig. II 68. 5. Cuvelier's Valve, with pressure beneath,* Fig. 1169a. The ordinary slide valve is here combined with another, both being made in one piece, and the combined valve held down to its seat by pressure rollers. Live steam is admitted through the passage / into the space between the tv/o valves. At b is Fitch's valve, also with pressure beneath. In this form the Fig. 1 1 69. pressure rollers are omitted, and the valve is held down by live steam pressure in the steam chest. The steam is admitted through very small holes £ B, and escapes to the exhaust through similar holes B' B', so that the supply is about equal to the loss by condensation. An objection to the use of valves with pressure beneath is the large area of valve seat which is required. Fig. 1770. 6. Double Seated Valves, Fig. 11 70. At a is Brandau's valve, and at b is Schaltenbrand's valve ; the former is analogous to Horublower's lift valve, Fig. Ii45ff, and the latter to the bell valve of Gros, Fig. 1 145^. In neither form is the degree of bal- ancing so complete as is desirable. c. Equalization of Pressure on all Sides. Fig. 1 171. 7. A very complete equalization of pressure is obtained by making the valve in the form of a piston. Fig. 1171 shows a recently designed piston valve with its steam cylinder. The flat valve seat here becomes a cylinder, and the valve a double piston, the flat sides of the valve disappearing. The valve pis- tons are each fitted with a single ring behind which steam is admitted through a small hole, thus rendering springs unneces- sary. The principal defect in piston valves is the question of wear. The best results appear to be obtained by making the piston valve solid, and very accurately turned and polished, and made about yjjj" smaller in diameter than the boreof the valve cylinder, both valve and valve cylinder being made of the same material. Piston valves fitted in this manner last a long time. S. Rotary Valves : For steam hammers, in which valve gear operated by hand has been found preferable to automatic movements, valves with rotary movement, formed like cocks, are used to advantage. These have been well designed by Wilson, the superintendent of Nasmyth's works. a. h. Fig. 1172. Fig. 1172a shows an oscillating valve by Wilson. Opposite to the ports //, ///, II', are false ports or recesses of shallow depth. The steam enters at the end of the valve into the sym- metrical spaces /, /. The unbalanced area of the steam of the valve causes a corresponding endlong pressure which is received by a thrust bearing. If we neglect the slight pressure due to the steam in the false ports when expansion takes place in // and ///, the valve is balanced on all sides. Very large oscil- lating valves of this sort are easily moved by hand.f A modification of this valve enables it to be operated by rota- tion instead of oscillation, as shown in Fig. 1172^. Here the parts are symmetrically arranged, as was not the case with the old four way cock of Fig. 9S7. The exhaust passages IV con- nect with one end of the valve, and the admission /, with the other end. There remains here also an unbalanced end-long pressure which is received by a thrust bearing. With this ex- ception the valve is entirely balanced, and when well made the thrust bearing offers but little resistance. The construction of such valves demands a high degree of accuracj-, and a specialty of this form is made by the establishments of Dingier of Zwei- briicken and of Pfaffin Vienna. The brief examination which we have given to the preceding methods of balancing does not include a method which, while offering great difSculties of construction, appears to be gradu- ally coming into use. This method consists in surrounding the ordinary flat side valve with equalizing pressure plates. Several practical illustrations of this method will be given. * Compare Fig. ir34. tSee Zeitschr. D. Ing., i36S, Vol. II, p. 207. THE CONSTRUCTOR. 287 9. Wilson's Balanced Valve. Fig. 1 173. (First shown at the London Exhibition of 1862.) The valve is symmetrical, and slides between two parallel and similar faces, the lower face having opennigs correspond- ing to the ports, the upper face having similar false ports. The close fitting and accurate parallelism of the surfaces was de pended upon to obtain the balancing. In practice it was found that the balance i)lates would spring under the pressure of the steam unless made very stiff and strong, and that the weight of the valve caused much friction and wear. Both of these difficulties have been met in more recent designs, as will be seen below. a. b Fig. 1 174. 10. Fig. It74n; shows the balancing, of the valves of the Por- ter-Allen engine.* The pressure plate is made very deep and stiff and formed with inclined plane bearings and set screws, by which the pressure can be very closely regulated. Fig. 1 1746 is Sweet's balanced valve. The pressure plate is here also made heavy and stiff, and is supported on longitu- dinal wedge bearings on each side, adjustable at the ends by screws. In both forms the pressure plate is fitted with springs to allow the plate to yield in case of water getting in the cylin- der. These forms of balanced valve have the objection that the ignorant mechanic may render the balancing ineffective by improper adjustment of the screws, permitting the full pressure of the live steam to act upon the valve. ? 377-. Fl,UID V.\LVES. Valves may be formed of fluids, or, more generally speaking, may be constructed of pressure organs. Ratchets adapted to pressure organ?, as fluid valves are properly called, are in ex- tensive use, but have not generally been recognized as valves. a. b. ^^r^aj p^^ ^^^a^ ili'i Pig. 1175. They are all reducible to one of two principal forms, either the direct or inverted siphon. Fig. Ii75 /;, + '''' the fluid valve will be thrown out at the top of the pipe, the arrangement thus form- ing a safety valve against an excess of pressure in a.,. This device was for a long time in use for low pressure boilers, Brindley's feeding device, Fig. 1000, being constructed on this principle. Natural stand-pipes with periodical discharge exist as geysers. Fig. 11771^ is a closed stand pipe for steam boilers. The pipe which has first been filled with steam gradually fills with water as the steam condenses. If the water level in a sinks below the end of the pipe the water runs out and live steam fills the pipe again. This action is utilized in safetj' devices by Black and Warner, and by Schwartzkopf. In the blast furnace the fluid iron with the slag floating upon it forms an inverted siphon which checks the blast. In the Bessemer converter the air pressure is so great that the iron is kept in agitation by the air bubbling through it. Fig. II 76. * See Trans. Am. Soc. Mech. Engrs , Vol. IV, p. 26 anced Valves. C. C. Collins, Bal- \1\\ this statement is included such fluids as do not :ningle by simple contact. In this sense steam and water will not mingle, and it thev are not of the same temperature the warmer will be transferred to the other. Air and water will not mingle because the w-aterhas become saturated with air- According to the researches of Colladon & Sturm (Memoire snr la compres- sion des liquides, 1827, reprinted by Schuchart, Geneva, 1S87), the saturation of water with air appears to partake of the nature of an internal, chemical combination. As m'ght be expected, water which is saturated with air shows a smaller compressibility in the Piezometer than water which is free from air, being 4S.65 millionths to 40.65 millionths. The combination of air with water ceases upon heating to the boiling point. X Natural inverted siphons with branches of varying levels exist in the case of artesian wells. 288 THE CONSTRUCTOR. In gas holders the water in the tank forming the seal is a fluid valve of tne inverted siphon type (compare Fig. 9486), a. b. Fig. 1 1 77. and a similar device is used with sand instead of water in Hoff- man's furnace, Fig. 1178, in which a^ is air, and a, smoke, the the bell-shaped lid being sealed with an annular valve of sand. F;g. ii;8. Fig. 1 179 is Wilson's water gas farnace.* In this a mixture of waste-slack and water forms a fluid valve. The mixture is propelled by an endless screw and discharged at the end. The atmosphere is at a^ and the gas at o^. the latter being kept under pressure bv a stesm jet. Fig. 1179. Hero's Fountain, Fig. 11 80, consists of two inverted siphon valves, in which a^ and «, have air at the atmospheric pressure, a^ is air under pressure, and a is water (often Cologne water). The action continues until the column /;', = h„. A practical application of the principle of Hero's fountain is the water trap of Morrison, Ingram & Co., Fig. iiSi.f In this device there is a periodical action of fluid valves as follows : a stream of water flows into the tank F aX E, gradually filling it, Fig. iiSia. The inner tube C, and fixed bell D, form an in- verted siphon, the shorter branch of which is the space between FiG. II 80. (Tand D. As soon as the level of the water in the tank jP rises above the top of Can overflow begins, filling the cup £, at the foot of the pipe C, and forming there a second siphon and making a seal between ^3 and a^, Fig. iiSiA. The two siphons now form a Hero's fountain, in which the continuing flow at E ^ b. .1' i Fig. 1181. causes an outflow iuto the discharge pipe A. As the level con- tinues to rise in F, the air in a., becomes more and more com- pressed, until finally the pressure column h becomes greater than the difference in level of the lower siphon, causing its dis- charge and cousequent opening of the fluid valve into a^. This relieves the pressure on the air in a.,, thus permitting the upper siphon to act, and causing an immediate and rapid discharge of the contents of F. By adjusting the rate of flow at E this action can be regulated so as to take place periodically at any desired intervals of time. Richard's manometer. Fig. 11 82, consists of alternate direct and inverted siphons ; a is quicksilver, a, steam, a^ water and fij atmospheric air. The spiral pump and the Cagniardelle shown in Fig. 966a and b contain successive fluid valves in the same pipe, alter- nately direct and inverted. lyangen's device for discharging bone furnaces of the hot granular burnt bone, is a ratchet system involving valves con- *See A. Wilson,^Generation of Heating Gas, etc., Journal of Soc. of Chem- ical Industries, Manchester, Nov., 1883. t See Revue Industrielle, June, iS THE CONSTRUCTOR. 289 sisting of a granular pressure organ, Fig. H83. The discharge pipe d of the furnace is closed at the bottom by the sliding plate t: which is given a reciprocating movement (in this in- stance operated by a small hydraulic motor). This plate c is made with a step as shown in the figure at a, which receives a layer of the material, and on the return stroke, as shown at b. Fig. 1 182. this layer is discharged on the plate. This layer forms a suc- tion valve when acting as at a, and a discharge valve, as at b, while the plate c corresponds to a single acting piston, con- sidering the whole as a pump. If the plate c is made with a middle rib, as shown in Fig. iigoi;, it works both ways and b. c. Fig. II 83. becomes a double-acting pump. This is an illustration of the fluid valve in its most general form as applied to a pump. In many instances fluid valves are as good and sometimes even better than valves composed of rigid materials. Especially is this the case when they act continuously in one direction in in a free, open pipe, for which purpose they excel all other forms of valves, as in jet pumps and the like (see Fig. 972). S378. St.^tionary Valves. We have thus far considered valves as ratchets for pressure organs, when they operate so as to check the motion of the fluid at the intervals of time (see \ 365). If we consider this definition in its most ijeneral sense we may take it to include certain kinds of fastenings for closing apertures, and call these also valves. These we may distinguish from ordinary valves by the fact that they are not operated by the motion of the machine, and hence to them may be given the name of "stationary valves."' Stationary lift valves are found in the lids of steam cylinders, these belonging to the class of disk valves. These are required to resist internal pressure, and must therefore be securely bolted in place, the pressure being generally great, and resisted by the bolts. Steam chest covers are generally rectangular, flat, stationary valves, and an example of a stationary flap valve is seen in the valve chest door shown in Fig. 1128, this also being secured by means of bolts. Furnace doors, such as shown in Fig. 763, also belong to this class. The more readily such a valve is opened and closed the more nearly it approaches in construction to the movable valves, and packing is sometimes omitted in order to facilitate opening and closing. The valve chest lids, shown in Fig. 1131, are readily recognized, these being readily slipped into place and held by a yoke, or so- called "gallows screw." Numerous forms of stationary valves are also found in various kinds of bottle stoppers, these being effective substitutes for the older cork stoppers which often were held in place only by friction. Stationary fluid valves are also occasionally still found in use for bottle stoppers in parts of Italy and Greece. Fig. 11S4. In all the cases thus far mentioned the fastening by which the stationary valve is held in place must be at least slightly stronger than the pressure beneath the valve. As a stationary valve in which this is not the case, we have the ordinary manhole plate as used in steam boilers. Fig. 11 84. In _ this the pressure acts to hold the '^ plate to its seat. Other examples are found in the spring valves used in the so-called siphons of soda water, and the particular form of bottle stopper which con- sists of a small ball valve held up to the mouth of the bottle by the pressure within. Stationary slide valves are less frequently used than lift valves, as the con- ditions are less favorable for proper packing, but examples are to be found. It will be seen by the instances already given how far reaching into all buinches of machine design the use of ratchets for pressure organs extends. ? 379- Stationary Machine Elements in General. It is not a peculiarit}' of valves alone to be used conveniently in the " stationary " form in the sense discussed in the preced- ing section. Here, as we have arrived at the close of the book, it is desirable to review the preceding pages in this respect. In the first four chapters of Section III the subjects considered are nearly always used as stationary elements. Rivets do not differ in form from cylindrical journals, but they are generally stationary because of two conditions ; be- cause of the firm binding of the surrounding metal, and because there are generally two or more rivets placed side by side. If only single rivet is used and no impediment to movement in- troduced, the binding of the metal would soon give way to any forces tending to cause rotation. Forced connections resemble journals and their bearings in form. The force by which the external piece grasps the inter- nal one eifectively resists all forces acting to produce rotation. Keyed connections are especially adapted for stationary service. The particular examples shown in Figs. 61S and 619 are in fact stationary keys in form, although really special cases of spiral gear wheels. Screws, in by far the greater number of cases, are used as stationary elements, probably in a greater variety of applications, broadly considered, than anj' other machine ele- ment. In ? 86 a glance is given at the use of the screw as an active machine element. Journals are frequently conveniently used as stationary ele- ments, as in the examples illustrated in Figs. 251, 252, 253, 256, 257 and 258. In \ 90 we have already distinguished between "journals at rest" and "running journals," the former corres- ponding to the definition of stationary elements. Roller bear- ings for bridge truss supports, 'i 19S, are also stationary ele- ments. Crank connections are found in the bottle stoppers already mentioned, and in numerous other applications such connec- tions are properly considered as stationary elements, Here wheels are rarely used as stationary elements, but such applica- tions are frequently found of ratchet wheels. Longitudinal keys used to secure hubs upon their axles are almost invariably stationary elements, practically corresponding to "stationary ratchets," as a comparison between Figs. 1S8 and 654 will show. Ratchets also find numerous applications in stationary mechan- ism for securing bolts, keys and the like. An examination of Figs. 237 to 243 and 246 to 248 will illustrate this point. In the couplings shown in Figs. 423 to 430 we also have a number of stationary ratchets (see also Fig. 67S). In § 309 I have referred to the possibility of using pressure organs as standing or "stationary " elements, but these are as yet unimportant. The pipes used as conductors for pressure organs, however, furnish numerous instances of pressure organs. The above distinctions are by no means merel}^ theoretical, but are of a highly practical nature. Every means which will enable us to obtain a clearer and better comprehension of the use of machine elements should be most welcome. In the preceding arrangement the stationary elements have therefore been grouped together for this end. It follows that those forms which as " stationary " or "passive" elements are extensively used ia building and civil works, as well as in ma- chine design, forming the connecting links between the works of the civil and the mechanical engineer. THE CONSTRUCTOR. 291, SECTION IV. MATHEMATICAL TABLES. ?38o. Tables of Curves, Areas and Volumes. The following tables give in convenient form the most im- portant geometrical and mechanical properties of the more useful curves, areas and volumes. The significance of the let- ters used in the formulae will be found indicated on the dia- grams. The following remarks are also to be noted. By the rectification of a curve is meant the length j of that portion of the curve from the origin to the point x y, corres- ponding to the angle ^ ; and by S is meant the entire length of the curve. In the moment of inertia the mass of the body is assumed = I, in order to reduce the number of letters. In view of the importance of this subject a few points are here given. The mo- ments of inertia for surfaces are both equatorial and polar, each referred an axis of moments. This latter is called an equatorial axis when it lies in the plane of the surface, and a polar axis when it is at right angles to the surface. Each equatorial which passes through the centre of gravity is especially termed an equator-axis, and a polar axis which passes through the cen- tre of gravity is called a pole axis. Every surface, therefore, has but one pole-axis, and an infinite number of equator axes. The moment of inertia is called equatorial or polar, according to the axis to which it is referred. The moment of inertia Jp for any surface referred to the polar axis is found by adding together the two equatorial mo- ments of inertia Jg^ and Jf,,, the axes of which intersect each other at right angles in the polar axis : from the moment of inertia /referred to a parallel axis through S, by the following relation : Jp= Jq\ + Jt: (4.6) J'= J-\-d'- F. (417) The moment of inertia /' of a surface, referred to any axis situated at a distance a, from the centre of gravity 5, is found in which j 'is the area of the surface. This relation also holds good for solids, if the mass of the body is substituted for F. For solids one of the preceding conditions does not hold. For each different shape one of the axes which passes through the centre of gravity, is taken as the pole-axis for all sections normal to it, and the section at right angles to this axis which passes through the centre of gravity is called the Equatorial Section, whence the equatorial and polar moments of inertia are in these cases distinguished according to their position with regard to this equatorial section. In all the examples of solids here given, the actual equatorial and polar axes are meant. For a right prism, of any given base having as the polar mo- ment of inertia ip and the half-height = /, the polar moment of inertia is : Jp = ^-lip {418) and the moment of inertia referred to an equatorial axis : Jp=VifP-\-2lig (419) in which y" is the area of the cross section, and i,/ the equatorial moment of inertia of the cross section referred to the same axis a.^Jg. The centre of gravity and the moment of inertia for a surface of irregular form is often readily obtained by grapho-static methods, with sufBcient numerical accuracy. For this purpose the force and cord polygons are applicable according to the methods already described in Section II. 292 THE CONSTRUCTOR. « M A P < > P o h o to P Pi P O o a; .11 + 1 II + >^ ^ ^ \ 6 t^ ^ s a P^ iu ft a t*. .i^^i 0^ M y II 11 h Co ^ II '^ ^^ ^ ^r ii I .^ ^ 3 z ^ Si « ^1^ I I T II \i J % c ll a H ll + ^, \ \ ^ 1 .) VH 17 \^ 7 A w Oi ^ ^v- // > \\.a f / V \ // / i" \ i+r ^ > Q II <1 a o 6^ THE CONSTRUCTOR. 293 A < W w A H > U O 3 S < ^ g a « a I ho "S 11 A .« .2 c p. Cc; O 13 ii O S ^ o o '0 0. aJ ^ -5 - c; .^ a cti rf ^ o *^ o o r^ -2 ll \1 CJ " •— ' >^ ^ a 8 CD ^0 s a 0) 'tT ,n > OJ 10 >> en u II M OJ 1^ a +-< C:: bf X ^-1 II CO II 1^ ^ p <4-< fTl nd II a; ■^ =b OJ ft? I? il.t 3 I CN Ml ill * ^ o rCrft?: O Q 5 a 3 ° a "mm a a a .2.2-2 a 3£ > > ^ O) OJ OJ ft? . ft, bo a S ft. ^ + > ^ N o pa taOCX2 294 THE CONSTRUCTOR. Moment of Inertia. For polar axis through C: ■''- ? 4 For Pole axis through centre of gravity 5 : >-*/^ I — C05/3' r 1^ .,1 I COS -cosp\ r*[ i—cosf}^ For polar axis through C : '"~ 2 ' ~ ^ ' For polar axis through S : i r I 2 \ For the equatorial axes XX, Y Y : ji f'- 4 ^-=-^' = ^ = T-- For Pole axis through C: '- 1 2 For Equator axis A' A": For polar axis through C : = f('V-'V)=^(4*H+r*3) For polar axis through C : ' 4 4 ^- 12 J •> - 2 siti ft cos^ ji COS /S sin' /3 1 For eauatorial axes A' A' and Y Y Jx y 5 15 35 loi For Equator axis A' A':/ ,= — a b^ 4 For Pole axis C : Jx Jy=, 6 ~ ~i2~'-^-"~'"Ts~~^6" 6 /;2 b h bh^ bli' h, , ., bli («3 4- ifi) /5=— +-^(«'+-.'»)--[8 («-^+Z'-')-3 b--^ THE CONSTRUCTOR. 295 No. Form. XXI. Triamlar Prisi. XXII. Rectangnlar Frisi. XXIII. RloiMc Prism, XXIV. Hexaional Prisi. XXV. Cylinder. XXVI. Hollow CyMer. XXVII. ParaMic Prisi, Sides : F^ = 2l (a-\- b + c] One end : P„ = Q. ^nt .-*-Q Surface. Volume. V= b h I Centre of Gravity. Moment of Inertia. For Equator axis Q Q : 7. Centre of Figure. Sides: f,=4/ {b + /i) One end : F„^ b h v; Sides :/=",= 8/ V/;" + One end : K,= b h For Pole axis P F : Ip = '« |_~i~ + "c: (" + ^) — For Equator axis Q O Jc V=2bhl 4' r=2bhl Sides : /^, = 1 2 / r One end : /^.^ ^-r'y/T,- = 2.sgSr' = "'(y+S) Centre of Figure. For Pole axis /'/': 111 12 // = ^(^'-^ + *') Centre of Figure. = 5.196 / r- = 1 Centre of ) For Eqiiator axis O O : For Pole axis P P\ For Equator axes Q Q and O^ Q^ : /, '^='"(t + - 24 Vertical surface : F, = A^lr One end : F.^= tt i^ Figure. I For Pole axis /'P: r=2^/;'=' Centre of Figure. -// = !; "--^ For Equator axis O Q : 7y = '" For Pole axis P P: Vertical surface : I One end : ^ ^2--^"(''i' — ''■/) =2-rb I For Equator axis O Q ■ Centre of j Figure. ! r /'^ r'^ ^^ L3 2 ^ S ] For Pole axis P P: jp =-- "' ['V-' + '•;-■] = '« Q-'+ -^]] One end : F.^^ — jc y 3 r= — I xy 3 For Equator axis O O : ^-'"[M'] For Pole axis P P: Jp = '« L 5 35 J 296 THE CONSTRUCTOR. No. XXVIII. XXIX. Eectanplar Pyraiid. XXX. RilM CoEe. XXXI. Triicateil Cone, Form. XXXII. SDiere. XXXIII. Sector of spliere, XXXIV. sesmem of Spliere. XXXV. SDtieroil. XXXVI. Paralioloid 01 RevoMion. Surface. F=\^-'-Rr Sides : v; 4 4 Bottom : F„^ a b. luclined surface: i^j ^ 77 JV h- 4- r- = TT >-5 Bottom : F„^=ti r''- Sides ; F,^~ [r, + r.^VlfiI^^^^,^^ = 2 TV }■ S Ends : F^^r'Tz Conical surface ; F^= a~ r=~ r v 2 ;' /; — K' Curved surface : Bottom : F2 = a' ;r, r 2 h Bottom: F^^t^ y' Volume. V^2 n- J?r ^ h a b h 3 3 ~ 3 r=-/;[r,''+r,;-,+r/] 7'= — TT r'^ h 3 Centre OP Gravity Moment of Inertia. Centre of Figure. Jt ^~ 4 For the surface onlj- . h 4\ ri'+ri^+r,- Centre of Fi^rure. -10-4) J'=n/i' I r ■ /^ (3 «» + /'') 4 ir~/i For the surface only . /i V= — TV a be 2 Centre of Figure. For Equator axis Q Q : Jo = "' 2^8 For Pole axis P P 3 . R'-\-- For Equator axis Q Q : Jc :r3./,+ i!-i Lso ^ 20 J ? " I So ■ 20 . For Pole axis P P : 117 ^p- For Equator axis Q Q : Jo = — "' y 20 D-f] For Pole axis P P : Jh = - - "' '^ /^ 10 For Pole axis P P: Jt, 3 i\ — » 2 10 ;■/ — ?v For Equator axis O Q : For Pole axis P P : For Pole axis P P : + 20 J ir — h For Equator axis Q Q, coincident with a : For Equator axis Q Q : /.="' (f +S) For Pole axis P P: in Jp- -r THE CONSTRUCTOR. 297 381 Trigonometrical Table. The following table contains, in convenient form, the sines, cosines, tangents and cotangents for angles from 0° to 90° lor every ten minutes, and also the corresponding arcs to a radius of unity. At the foot of the table arcs are also given for small angles and also for some of the more frequently used angles greater than 90°. ANGLE. " ANGLE. 1 ANGLE. ANGLE. arc. sine, cosine. tan. cot. arc. arc. sine. cosine. tan. cot. arc. deg. inin, deg. min. deg. min. deg. min. 0.0000 0.0000 1. 0000 0.0000 CO 1.5708 90 10 0.1745 0.1736 0.9848 0.1763 5-6713 1-3963 80 10 0.0029 0.0029 1. 0000 0.0029 343.77 1.5679 50 10 0.1774 0.1765 0.9843 0.1793 5-5764 1-3934 50 20 0.005S 0.0058 1. 0000 0.0058 171.89 1.5650 40 20 0.1804 0.179+ 0.9838 0.1823 5-4S45 1-3904 40 30 0.0087 0.0087 1. 0000 0.0087 114.59 1. 5621 30 30 0.1833 0.1822 0.9833 0.1853 5-3955 1-3875 30 40 0.0116 0.0116 ' 0.9999 0.0116 85.940 1.5592 20 40 0.1862 0.1851 0.9827 0.1S83 5-3093 1.3S46 20 50 0.0145 0.0145 1 0.9999 ( 1 0.0145 68.750 1.5563 10 50 0.1891 0.1880 0.9S22 0.1914 5-2257 1.3817 10 I ' 1 0.0175 0-OI75 0.9998 0.0175 57.290 1-5533 89 II 0.1920 0.190S 0.9816 0.1944 5-1446 1.3788 79 10 0.0204 ■ 0.0204 [ 0.9998 0.0204 49.104 1.5504 50 10 0.1949 0.1937 0.9811 0.1974 5-0658 1-3759 50 20 0.0233 ' 0.0233 : 0.9997 0.0233 42.964 1.5475 40 20 0.197S 0.1965 0.9805 0.2004 4.9894 1-3730 40 30 0.0262 0.0262 0.9997 , 0.0262 38.188 1.5446 30 30 0.2007 0.1994 0.9799 0.2035 4.9152 1-3701 30 40 0.0291 0.0291 0.9996 1 0.0291 34.36S I-5417 20 40 0.2036 0.2022 0.9793 0.2065 4.8430 1-3672 20 50 0.0320 0.0320 0.9995 0.0320 31.242 I-538S 10 50 0.2065 0.2051 0.9787 0.2095 4.7729 1.3643 10 2 0.0349 0.0349 0.9994 0.0349 28. 636 1.5359 88 12 0.2094 0.2079 0.97S1 0.2126 4.7946 1.3614 78 10 0.0378 0.0378 1 0.9993 ; 0.0378 26.432 1.5330 50 10 0.2123 0.2108 1 0.9775 0.2156 4.6382 1.3584 50 20 0.0407 0.0407 1 0.9992 1 0.0407 24.542 1. 5301 40 20 0.2153 0.2136 0.9769 0.2186 4.5736 1-3555 40 30 0.0436 0.0436 0.9990 0.0437 22.904 1.5271 30 30 0.2182 0.2164 0.9763 0.2217 4.5107 1.3526 30 40 0.0465 0.0465 0.9989 0.0466 21.470 1.5243 20 40 0.2211 0.2193 0.9757 0.2247 4.4494 1-3497 20 50 0.0495 0.0494 0.9988 0.0495 20.206 1.5213 10 50 0.2240 0.2221 0.9750 0.2278 4.3897 1.3468 10 3 0.0524 0.0523 0.9986 0.0524 19.081 1.51S4 87 13 0.2269 0.2250 0.9744 0.2309 4.3315 1-3439 77 10 0.0553 0.0552 0.9985 1 0.0553 18.075 1.5155 50 10 0.229S 0.2278 0.9737 0.2339 4.2747 1.3410 5° 20 0.0582 0.0581 0.9983 0.0582 17.169 1.5126 40 20 0.2327 0.2306 0.9730 0.2370 4.2193 1-3381 40 30 0.061 1 0.0610 0.9981 0.0612 16.350 1.5097 30 30 0.2356 0.2334 0.9724 0.2401 4.1653 1-3352 30 40 0.0640 0.0640 0.9980 j 0.0641 15.605 1 .5068 20 40 0.2385 0.2363 0.9717 0.2432 4.1126 1-3323 20 50 0.0669 0.0669 0.9978 0.0670 14.924 1.5039 10 50 0.2414 0.2391 0.9710 0.2462 4.0611 1-3294 10 4 0.0698 1 0.069S 0.9976 0.0699 14.301 1.5010 86 H 0.2443 0.2419 0.9703 0.2493 4.0108 1.3264 76 10 0.0727 ' 0.0727 0.9974 ; 0.0729 13.727 1.49S1 50 10 0.2473 0.2447 0.9696 0.2524 3.9617 1-3235 50 20 0.0756 : 0.0756 0.9971 0.075S 13.197 1.4951 40 20 0.2502 0.2476 0.9689 0.255s 3-9136 1.3206 40 30 0.0785 0.07S5 0.9969 0.0787 12.706 1.4923 30 30 0.2531 0.2504 0.9681 ' 0.2586 3.S667 1-3177 30 40 0.0814 1 0.0814 0.9967 0.0S16 12.251 1 .4893 20 40 0.2560 0.2532 0.9674 0.2617 3.8208 1.3148 20 50 0.0844 ' 0.0843 0.9964 0.0846 11.826 1 .4864 10 50 0.2589 0.2560 0.9667 0.264S 3.7760 1-3119 10 5 0.0873 0.0S72 0.9962 0.0S75 1 1 .430 1.4835 85 15 0.2618 0.2588 0.9659 0.2679 3.7321 1.3090 75' 10 0.0902 0.0901 0.99S9 0.0904 11.059 1.4806 50 10 0.2647 0.2616 0.9652 0.2711 3.6891 1.3061 50 20 0.0931 0.0929 ; 0.9957 j 0.0934 10.712 1-4777 40 20 0.2667 0.2644 0.9644 0.2742 3.6470 1.3032 40 30 0.0960 0.0958 0.9954 ! 0.0963 10.385 1.4748 30 30 0.2705 0.2672 0.9636 0.2773 3.6059 1.3003 30 40 0.0989 0.0987 0.9951 1 0.0992 10.078 1.4719 20 40 0.2734 0.2700 0.9628 0.2805 3-5656 1.2974 20 50 0.1018 0.1016 0.9948 0.1022 9.7S82 1 .4690 10 50 0.2763 0.2728 0.9621 0.2836 3-5261 1.2945 10 6 0.1047 0.1045 0.9945 0.1051 9-5144 1. 4661 84 16 0.2793 0.2756 0.9613 \ 0.2867 3-4874 1. 2915 74 10 0.1076 0.1074 0.9942 0.1080 9-2553 1.4632 50 TO 0.2822 ' 0.2784 0.9605 ' 0.2S99 3-4495 1.28S6 50 20 0.1 105 0.1103 0.9939 0.1 110 9.009S 1.4603 40 20 1 O.2S5I O.2S12 0.9596 0.2931 3.4124 1.2857 4(1 30 0.1134 0.1132 0.9936 j 0.1139 8.7769 1.4573 30 30 0.2880 0.2840 0.9588 0.2962 3-3759 1.2828 30 40 j 0.1 164 0.1161 0.9932 0.1 169 S.5555 1-4544 20 40 0.2909 0.2868 0.9580 0.2994 3-3402 1.2799 20 50 0.1193 0.1190 0.9929 1 0.1198 1 8.3450 1-4515 10 50 0.2938 0.2896 0.9572 0.3026 3-3052 1.2770 10 7 0.1222 0.1219 0.9925 1 0.1228 8.1443 1 .4486 83 17 0.2967 0.2924 0.9563 0.3057 3.2709 1.2741 73 10 0.1251 0.1248 0.9922 0.1257 7.9530 1.4457 50 10 0.2996 0.2952 0.9555 0.3089 3-2371 1. 2712 50 20 0.12S0 0.1276 0.991S 0.1287 7.7704 1.4428 40 20 0.3025 0.2979 0.9546 '0.3121 3.2041 1.2683 40 30 0.1309 0.1305 0.9914 0.1317 7-5958 1,4399 30 30 0.3054 0.3007 0.9537 0.3153 3.1716 1.2654 30 40 0.133S 0.1334 0.9911 ' 0.1346 7-4287 1.4370 20 40 0.3083 0.3035 0.9523 0.31S5 3.1397 1.2625 20 50 0.1367 0.1363 0.9937 0.1376 7.2687 I. 434 I 10 50 0.3113 0.3062 0.9520 0.3217 3.1084 1.259s 10 8 1 0.1396 0.1392 ; 0.9903 0.1405 7.1154 1-4312 82 18 0.3142 0.3090 0.95 II 0.3249 3.0777 1.2566 72 10 0.1425 0.1421 0.9899 1 0.1435 6.9682 1.4283 50 10 0.3171 0.3118 0.9502 0.3281 3.0475 1.2537 SO 20 0.1454 0.1449 0.9894 I 0.1465 6.S269 1.4254 40 20 0.3200 0.3145 0.9492 0.3314 3.0178 1.2508 40 30 0.1484 0.147S 1 0.9S90 0.1495 6.6912 1.4224 30 30 0.3229 0.3173 0.9483 0.3346 2.98S7 1.2479 30 4^ 0.1526 0.1507 0.9S86 0.1524 6.5606 1. 4195 20 40 0.3258 0.3201 1 0.9474 0.3378 2.9600 1.2450 20 5^ 0.1542 0.1536 0.9S81 0.1554 6.4348 ' 1. 4166 1 10 50 0.3287 0.3228 0.9465 0.341 1 2.9319 i.2421 10 9 0.1571 0.1564 1 0.9877 0.1584 6.3138 1-4137 81 19 0.3316 0.3256 0.9455 0.3443 2.9042 1.2392 71 10 o.i6oo 0.1593 0.9872 j 0.1614 6.1970 I. 4108 50 10 0.3345 0.3283 0.9446 0.3476 2.8770 1-2363 50 20 0.1629 0.1622 C.986S 0.1644 6.0844 1.4079 40 20 0.3374 0-3311 0.9436 0.350S 2.8502 1-2334 40 33 0.165S 0.1650 0.9863 ; 0.1673 5.9758 1.4050 30 30 0.3403 i 0.333S 0.9426 0.3541 2.S239 1-2305 30 40 0.1687 0.1679 O.985S 0.1703 5.870S 1. 402 1 20 40 0.3432 0.3365 0.9417 0.3574 2.79S0 1.2275 20 50 0.1716 0.1 70S 0.9853 0.1733 5-7694 1.3992 10 SO 0.3462 0.3393 0.9407 0.3607 2.7725 1.2246 10 Angle. arc. cosine. sine. cot. tan. arc. Angle. Angle. 1 arc. cosine. sine. cot. tan. arc. Angle. 298 THE CONSTRUCTOR. ANGLE. ANGLE. ANGLE. ANGLE. arc. sine. cosine. tan. cot. arc. arc. sine. cosine. tan. cot. arc. deg. min. deg. min. Ideg. min deg. 59 rain. 20 0.3491 0.3420 0.9397 U.3640 2.7475 1. 2217 70 31 0.5411 0.5150 0.S572 0.6009 1.6643 1.0297 10 0.3520 0.3448 0.9387 0.3673 2.7228 1.2188 50 10 0.5440 0.5175 0.8557 0.604S 1.6534 1.0268 50 20 0.3549 0.3475 0.9377 0.3706 2.6985 1.2159 40 20 0.5469 0.5200 0.8542 0.6088 1.6426 1.0239 40 30 0.3578 0.3502 0.9367 0.3739 2.6746 1.2130 30 30 0.5498 0.5225 0.8526 0.6128 1.6319 1.0210 30 40 0.3607 0.3529 0.9356 0.3772 2.6511 1.2101 20 40 0.5527 0.5250 0.8511 0.616S 1.6212 1.0181 20 50 0.3636 0.3557 i 0.9346 ! 0.3S05 1 1 2.6279 1.2072 10 50 0.5556 0.5275 ; 0.8496 1 0.6208 1.6107 1.0152 10 21 0.3665 0.3584 0.9336 0.3839 2.6051 1.2043 69 32 0.5585 0.5299 1 0.8480 0.6249 1.6003 1 .0123 S8 10 0.3694 0.3611 0.9325 0.3872 2.5826 I. 2014 50 10 0.5614 0.5324 0.8465 0.6289 1.5900 1 .0094 SO 20 0.3723 0.3638 0.9315 0.3906 2.5605 1.1985 40 20 0.5643 0.5348 0.8450 0.6330 1.5798 1 .0065 40 30 0.3752 0.3665 1 0.9304 0.3939 2.5386 1. 1955 30 30 0.5672 0.5373 ; 0.8434 0.6371 1.5697 1.0036 30 40 0.3782 0.3692 0.9293 0.3973 2.5172 1.1926 20 40 0.5701 0.5398 0.8418 0.6412 1.5597 1.0007 20 50 0.381 1 0.3719 0.9283 0.4006 2.4960 1.1897 10 50 0.5730 0.5422 0.8403 0.6453 1.5497 0.9977 10 22 [ 0.3840 0.3746 0.9272 0.4040 2.4751 1.1868 68 33 0.5760 0.5446 0.8387 0.6494 1.5399 0.994S 57 10 0.3869 0.3773 0.9261 0.4074 2.4545 1.1839 50 10 0.5787 0.5471 0.8371 0.6536 1.5301 0.9919 50 20 0.3S9S ' 0.3S00 0.9250 0.4108 2.4342 1.1810 40 20 0.5818 0.5495 0.8355 0.6577 1.5204 0.9890 40 30 0.3927 0.3S27 0.9239 0.4142 2.4142 1.1781 30 30 0.5847 0.5519 0.8339 0.6619 1. 5108 0.9861 30 40 0.3956 j 0.3854 1 0.922S 0.4176 2.3945 1.1752 20 40 0.58-6 0.5544 0.8323 0.6661 1.5013 0.9832 20 50 0.39S5 0.3S81 0.9216 0.4210 2.3750 1.1723 10 SO 0.5905 0.5568 0.8307 0.6703 I. 4919 0.9803 10 23 0.4014 0.3907 0.9205 0.4245 2.3559 1. 1694 67 34 0.5934 0.5592 0.8290 0.6745 1 .4826 0.9774 56 10 0.4043 0.3934 0.9194 0.4279 2.3369 1.1664 50 10 0.5963 0.5616 0.8274 0.6787 1.4733 0.9745 50 20 0.4072 0.3961 0.9182 0.4314 2.3183 1.1636 40 20 0.5992 0.5640 0.8258 0.6830 1.4641 0.9716 40 30 0.4102 0.3987 0.9171 0.4348 2.2998 1. 1606 30 30 0.6021 0.5664 0.8241 0.6873 1.4550 0.9687 30 40 0.4131 0.4014 0.9159 0.4383 2.2817 1. 1577 20 40 0.6050 0.5688 0.8225 0.6916 1.4460 0.9657 20 50 0.4160 0.4041 0.9147 0.4417 2.2637 1.1548 10 50 0.60S0 0.5712 0.8208 0.6959 1.4370 0.9628 10 24 0.4189 0.4067 0.9135 0.4452 2.2460 1.1519 66 35 0.6109 0.5736 0.8192 0.7002 1.4281 0.9599 55 10 0.4218 0.4094 0.9124 0.4487 2.22S6 1.1490 50 10 0.6138 0.5760 0.8175 0. 7046 I.4I73 0.9570 SO 20 0.4247 0.4120 0.91 12 0.4522 2.2113 1.1461 40 20 0167 0.5783 0.8158 0.7089 1. 4106 0.9541 40 30 0.4:^76 0.4147 0.9100 0.4557 2.1943 1.1432 30 30 0.6 96 0.5807 0.8141 o.7'33 I. 4019 0.9512 30 40 0.4305 0.4173 0.908S 0.4592 2.1775 i.ii03 20 40 0.0225 0.5831 0.8124 0.7177 1.3934 0.9483 20 50 0.4334 0.4200 0.9075 0.4628 2.1609 '■1374 10 50 0.6254 0.5854 0.8107 0.7221 1.3848 0.9455 10 25 0.4363 0.4226 0.9063 0.4663 2.1445 1.1345 65 36 0.6283 0.5878 0.8090 0.7265 1.3764 0.9425 54 10 0.4392 0.4253 0.9051 0.4699 2.1283 1.13ID 50 10 0.6312 0.5901 0.8073 0.7310 1.3680 0.9306 5° 20 0.4421 0.4279 0.9038 0.4734 2.1123 1.1280 40 20 0.6341 0.5925 0.S056 0.7355 1.3597 0.9367 40 30 0.4451 0.4305 0.9026 0.4770 2.0965 1. 1257 30 1 30 0.6370 0.5948 0.S039 0.7400 1.3514 0.9338 30 40 0.4480 0.4331 0.9013 0.4806 2.0S09 I. 1228 20 40 0. 6400 0.5972 0.8021 0. 7445 1.3432 0.9308 20 50 0.4509 0.4358 0.9001 0.4841 2.0655 1.1199 10 50 0.6429 0.5995 0. 8004 0. 7490 1.3351 0.9279 10 26 0.453S 0.4384 0.8988 0.4877 2.0503 I. 1170 64 37 I 0.6458 0.601S 0.7086 0.7536 1.3270 0.9250 53 10 0.4567 0.4410 0.8975 0.4913 2.0353 I. 1141 50 10 0.6487 0.6041 0.7069 0.7581 1.3190 0.9221 50 20 0.4596 0.4436 0.S962 0.4950 2.0204 1.1112 40 20 0.6516 0.6065 0.7951 0.7627 1.3111 0.9192 40 30 0.4625 0.4462 0.8949 0.4986 2.0057 I. 1082 30 30 0.6545 0.60S8 0.7934 0.7673 1.3032 0.9163 30 40 0.465 A 0.4488 0.8936 0.5022 1.9912 I. 1054 20 40 0.6574 0.6111 0.7916 0.7720 1.2954 o.9>34 20 50 0.4683 0.4514 0.8923 0.5059 1.9768 1.1025 10 50 0.6603 0.6134 0.7S98 0.7766 1.2876 0.9105 10 27 0.4712 0.4540 0.8910 0.5095 1.9626 1 .0996 63 38 0.6632 0.6157 0.7S80 0.7813 1.2799 0.9076 52 10 0.4741 0.4566 0.8897 0.5132 1.9486 1 .0966 50 10 0.6661 0.6180 0.7862 0.7S60 1.2723 0.9947 50 20 0.4771 0.4592 0.S884 0.5169 1.9347 1.0937 40 20 0. 6690 0.6202 0.7844 0.7907 1.2647 0.9018 40 30 0.4800 0.4617 0.8870 0.5206 1.9210 1 .0908 30 30 0.6720 0.6225 0.7826 0.7954 1.2572 0.8988 30 40 0.4829 0.4643 0.8857 0.5243 1.9074 1.0879 20 40 0.6749 0. 6248 0.7S08 0.8002 1.2497 0.S959 20 50 0.4858 0.4669 0.8843 0.5280 1 .8940 1 .0850 10 SO 0.6778 0.6271 0.7790 0.8050 1.2423 0.8930 10 28 O.4S87 0.4695 0.S820 0.5317 1.8807 1.0821 62 39 0.6807 0.6293 0.7771 0.S098 '.2349 0.8901 51 10 0.4916 0.4720 0.8816 0.5354 1.8676 1.0792 50 10 0.6836 0.6316 0.7753 0.8146 1.2276 0.8872 50 20 0.4945 0.4746 0.8802 0.5392 1.8546 1.0763 40 20 0.6865 0.6338 0.7735 0.8195 1.2203 0.S843 40 30 0.4974 0.4772 0.8788 0.5430 1.8418 1.0734 30 30 0.6894 0.6361 0.7716 0.8243 1.2131 0.S814 30 40 0.5003 0.4797 0.8774 0.5467 1.8291 1.0705 20 40 0.6923 0.6383 0.7698 0.8292 1.2059 0.8785 20 50 0.5032 0.4823 0.8760 0.5505 1.8165 1.0676 10 50 0.6952 0.6406 0.7679 0.8342 1.1988 0.8756 10 29 0.5061 0.4848 0.8746 0.5S43 1 .8040 1 .0647 61 40 0.6981 0.6428 0.7660 0.8391 1.1918 0.8727 5° 10 0.5091 0.4S74 0.8732 0.5581 1.7917 1.0617 SO 10 0.7010 0.6450 0.7642 0.8441 1.1847 0.869S 50 20 0.5120 0.4899 0.8718 0.5619 1.7796 1 .0588 40 20 0.7039 0.6472 0.7623 0. 849 1 I.177S 0.S66S 40 30 0.5149 0.4924 0.8704 0.5658 1.7675 1.0559 30 30 0.7069 0.6494 0.7604 0.8541 .1.1708 0.8639 30 40 0.5178 0.4950 0.86S9 0.6696 1.7556 1.0530 20 40 0.7098 0.6517 0.7585 0.8591 1. 1640 0.8610 20 50 0.5207 0.4975 0.8675 0.5735 '■7437 1.0501 10 50 0.7127 0.6539 0.7566 0.8642 1.1571 0.85S1 10 30 0.5236 0.5000 0.8660 0.5774 1.7321 1.0472 60 41 0.7156 0.6561 0.7547 0.8693 1.1504 0.8552 49 10 0.5265 0.5025 0.8646 0.5S12 1.7205 1.0443 50 10 0.7185 0.6583 0.7528 0.8744 1.1436 0.8523 50 20 0.5294 0.5050 0.8631 0.5S51 1.7090 1.0414 40 20 0.7214 0.6604 0.7509 0.8796 1.1369 0.8494 40 30 0.5323 0.5075 0.8616 1 0.5890 1.6977 1.0385 30 30 0.7243 0.6626 0.7490 0.8847 1.1303 0.8465 30 40 0.5352 0.5100 0.8601 0.5930 1.6S64 1.0356 20 40 0.7272 0.6648 0.7470 ! 0.8899 1.1237 0.8436 20 50 0.5381 0.5125 0.8587 0.5969 1.6753 1.0326 10 50 0.7301 0.6670 0.7451 0.8952 1.II71 0. 8407 10 Angle. arc. cosine. sine. cot. tan. arc. Angle. Angle. arc. cosine. sine. cot. tan. arc. Angle. THE CONSTRUCTOR. 299 ANGLE. arc. sine. cosine. tan. cot. arc. ANGLE. ANGLE. arc. sine. cosine. tan. cot. arc. ANGLE. deg. min. deg. rain. deg. min. 1 deg. min. 42 43 10 20 30 40 SO 10 20 30 40 50 0.7330 0.7359 0.7389 0.7418 0.7447 0.7476 0.7505 0.7534 0.7563 0.7592 0.7621 0.7650 0.6691 0.6713 0.6734 0.6756 0.6777 0.6799 0.6820 0.6841 0.6S62 0.6884 0.6905 0.6926 0.7431 0.7412 0.7392 0.7373 0.7353 0.7333 o.73'4 0.7294 0.7274 0.7254 0.7234 0.7214 0. 9004 0.9057 0.91 10 0.9163 0.9217 0.9271 0.932s 0.9380 0.9435 0.9490 0.9545 0.9601 1.1106 1.1041 1.0977 1. 0913 1.0S50 1.0786 1.0724 r.o66i 1.0599 1.0538 1.0477 1. 0416 0.8378 0.8348 0.8319 0.8290 0.8261 0.8232 0.8203 0.8174 0.8145 0.8116 0.80S7 0.8058 48 47 50 40 30 20 10 so 40 30 20 10 44 45 lO 20 30 40 50 0.7679 0.7709 0.7738 0.7767 0.779S 0.7824 0.7854 0.6947 0.6967 0.6988 0.7009 0.7030 0.7050 0.7071 0.7193 0.7173 0.7153 0.7133 0.7112 0.7092 0.7071 0.9657 0.9713 0.9770 0.9827 0.9SS4 0.9942 1. 0000 1.035 s 1.0295 1.0235 1. 0176 I.0II7 1.0058 1. 0000 0.S029 0.7999 0.7970 0.7941 0.7912 0.7883 0.7854 46 50 40 30 20 10 Angle. arc. cosine. sine. cot. tan. arc. Angle. ang.=o° \' o°5' 135° iSo° 3.1416 225° 3.9270 270° 4.7124 315° 5.4978 360° Angle. arc. cosine. sine. cot. tan. arc. Angle. arc. = 0. 0003 0.0015 2.3562 6.2 S32 TRIGONOMETRICAL FORMULA. sin (a d= /3) = sin a cos j} ± cos a sin J3 cos (a ± /3) = cos a cos i3 =F sin a sin /? sin 2 a = 2 sin a cos a sin 30^3 sin a — 4 siti a-' = sin a (4 cos a? — i) cos 2 a ^ cos a'^ — sin a' = 2 cos a? — i ^ i — 2 sin a? cos 3 u ^ 4 cos a-'' — 3 cos a = cos a (i — 4 sin a') a -I- /3 a — .3 si?i a -\- sin p ^ 2 sifi cos sin a — sin p ^ a4- /3 . 2 cos sin 2 cos a 4- cos p ^ 2 cos cos 2 a — 13 2 a — j3 II. 12. 13. 14. 15. 16. 17. IS. 19. . a 4- 13 . 13 — a COS a — cos 3 = 2 sin sin 22 sin a' = ^^ (i — cos 2 a) cos a' = yi (i + cos 2 a) sin q' ^ X (3 .f"^ " — -S"' 3 ") cos a^ = % (3 '^"^ "■ + ^os 3 a) iang a ± tang j3 I q= iang a tang /3 colang a cotang /? =F l tang (a ± cotang (a ± /3) - tang 2 a- ^ cotang a + cotang ,■? 2 tang a cotang 2 n = ian^ a — tang a' cotans: a'' - V .^s/S 2 cotang a • cos 2 a cotang tang a zb iang /3 = cos 2 a cos 2 a 1 + 2 cos a sin 2 a I — cos 2 a I — cos 2 a sin (a zt /3) 2 tang Yz "■ 1 — iang ]A, a^ cotang )4 a- — I 2 cotang yi a 22. cotang a ± cotang ft = cos a cos /3 sin (/? ± a) 23. sin a sin J3 sin a 4- sin j3 iang }4 (a + (3) sin a — sin j3 tang j^ (a — /?) 300 THE CONSTRUCTOR. TABL'E OF NUMBERS.-I. » i I n ■,i>- «^ v/,7 1 ^,T 1 ^,~ I ^;7 0.30 0-375 0.60 0.625 0.70 3-333 2.667 1.667 1.600 1.429 0.090 0.141 0.360 0.391 0.490 0.027 0.053 0.216 0.244 0.343 0.548 0.612 0.775 0.791 0.837 1.826 1.633 1.291 1.2(35 1.195 0.669 0.721 0.843 0.855 0.888 1.495 1.387 1.186 1.170 1,126 0.740 0.783 0.880 0.889 0.915 1.351 1.278 1.136 1.125 1.093 0-75 0.875 0.90 1. 10 1.2 1-333 1. 143 I. Ill 0.909 0.833 0.563 0.766 0.810 1. 210 1.440 0.422 0.670 0.729 1.331 1.728 0.866 0.935 0.949 1.049 1.095 1.155 1.069 1.054 0.953 0.9 '3 0.909 0.956 0.965 1.032 1.063 1. 100 1,046 1.036 0.969 0.941 0-931 0.974 0.987 1.024 1.047 1.075 1.024 1.013 0.976 0.955 1.25 1.50 1-75 2.0 2.25 0.800 ' 0.667 0.5-1 0.500 0.444 1.563 2.250 3.063 4.0 5.063 1.953 3.375 S-359 8.0 11.391 1. 118 1.225 1.323 I.4I4 1.500 0.894 0.816 0.756 0.707 0.667 1.077 1.145 1.205 1.260 1.310 0.928 0.874 0.830 0.794 0.763 1.057 1.107 1.150 1.189 1.225 0.946 0.904 0.869 0.841 0.816 2.50 2.75 3.0 3-25 3.50 0.400 0.364 0.333 0.308 0.286 6.250 7.563 9.0 10.563 12.250 15.625 20.797 27.0 34.328 42.875 1.581 1.658 1.732 1.803 1.871 0.632 0.603 0.577 0.555 0.53s 1.357 1.401 1.442 1.481 1.518 0.737 0.714 0.693 0.675 0.659 1.257 1.2S8 1.318 1.342 1.368 0.795 0.777 0.759 0-745 0-731 3-75 4.0 4.5 5.0 5-5 0.267 0.250 0.222 0.200 0.182 14.063 16.0 20.250 25.0 30.250 52.734 64.0 91.125 125.0 166.375 1.936 2.0 2. 121 2.236 2.345 0.516 0.500 0.471 0.447 0.426 1-554 1.587 1. 651 1.710 1.765 0,644 0.630 0.606 0.585 0.567 1.392 1.414 1.457 1.495 1.531 0.719 0.707 0.6S7 0.669 0.653 6.0 6.5 7.0 7-5 8.0 0.167 0.154 0.143 0-133 0.125 36.0 42.25 49.0 56.250 64.0 216.0 274.625 243.0 421.875 512.0 2.449 2.550 2.646 2.739 2.828 0.408 0.392 0.378 0.365 0.354 1.817 1.866 1.913 1.957 2.0 0.550 0.536 0.523 0.510 0.500 1.565 1.597 1.627 1.655 1.682 0.639 0.626 0.615 0.604 0595 8.5 9.0 9-5 IC II 0.1 iS 0.1 1 1 0.105 0.1 00 0.091 72.250 81.0 90.250 1 00.0 121.0 614.125 729.0 857-375 lOOO.O I33I-0 2.915 3.000 3.082 3.162 3.317 0.343 0.333 0.324 0.316 302 2.041 2.080 2.118 2.154 2.224 0.490 0.481 0.472 0.464 0.450 1.707 1.732 1.756 1.778 1.821 0.586 0.577 0.570 0,562 0.549 12 13 14 15 16 0.083 0.077 0.071 0.067 0.063 144 169 196 225 256 1728 2197 2744 3375 4096 3-464 3.606 3.742 3.873 4.000 0.289 0.277 0.267 0.258 0.250 2.289 2.351 2.410 2.466 2.520 0.431 0.425 0.415 0.405 0.397 1.861 1.899 1.934 1.968 2.000 0.537 0.527 0.517 0.508 0,500 17 18 19 20 50 O.OS9 0.056 0.053 0.050 0.020 289 324 361 400 2500 4913 5832 6S59 8000 125000 4.123 4.243 4.359 4.472 7.071 0.243 0.236 0.229 0.224 0.141 2.571 2.621 2.668 2.714 3.684 0.389 0.381 0.375 0.368 0.271 2.031 2.060 2.088 2.115 2.659 0.492 0,485 0.479 0-473 0-376 100 1000 n- = 3.142 2 ff ^ 6.2S3 0.0 10 0.00 1 0.318 0.159 1 0000 I 000000 9.870 39-478 I 000000 I 000000000 31,006 248.050 lO.O 31.623 1.772 2.507 0.10 0.032 0.564 0-399 4.642 1 0.0 1.465 1845 0.215 O.I 00 0.683 0.542 3.162 5.623 1-331 1.583 0.316 0.178 0.751 0.632 -=1.571 0.637 ■ 2.467 3-878 1.253 0.798 1.162 0.860 1.120 0.893 ■K ~ = 1.047 0.955 1.097 1. 148 1.023 0.977 1.016 0.985 1.012 0.989 4 - TT = 4.189 0.239 17.546 73.496 2.047 0.489 1. 612 0.622 1.431 0.699 -=0.785 4 1.274 0.617 0.484 0.886 1.128 0.923 1.084 0.941 1.062 ^ = 0.524 1. 910 0.274 0.144 0.724 1.382 0.806 1. 241 0.851 1. 176 rr"- = 9.S70 O.IOI 97.409 961.390 3.142 0.318 2.145 0.466 1.772 0.564 7r' = 31.006 0.032 961.390 29809.910 5.568 1.796 3-142 0.318 2.360 0.424 = 0.098 32 10.1S6 0.0095 O.OOI 0.313 3.192 0.461 2.1 68 0.560 1.782 3'^ 76= °-5«9 1.698 0.347 0.204 0.768 1.303 0.S38 1.194 0.876 1. 142 g = 32.2 2^ = 64.4 0.031 0.015 1036.84 4147-36 33386.24 267090 5.674 8.025 0.176 0.125 3,181 4,007 0.314 0.249 2.381 2.833 0,419 0-337 THE CONSTRUCTOR. 301 TABLE OF NUMBERS.- -II. n VTi \ n v/17 ^T n x/17 i n v/^ ^17 0.0 1 0.1 0000 0.21544 0.26 0.50990 0.63825 0.51 0.71414 0.79896 0.76 0.87 1 78 0.91258 0.02 0.14132 0.27144 0.27 0.51962 0.64633 0.52 0.72111 0.80415 0.77 0.S7750 0.91657 0.03 0.17321 0.31072 0.28 0.52915 0.65421 0-53 " 0.72801 0.80927 0.7S 0.88318 0.92052 0.04 0.20000 0.34200 0.29 0.53852 0.66191 0.54 0.73485 0.81433 0.79 0.8S882 0.92443 0.05 0.22361 0.36840 0.30 0.54772 0.66943 0-55 0.74162 0.81932 0.80 0.89443 0.92832 0.06 0.24495 0.39149 0.31 0.55678 0.67679 0.56 0.74833 0.82426 o.Si 0.90000 0.93217 0.07 0.26458 0.41213 0.32 0.56569 0.68399 0.57 0.75498 0.82913 0.82 0.90554 0-93599 o.c8 0.2S2S4 0.43089 0-33 0.57446 0.69104 0.58 0.76158 0.83396 0.83 0.91104 0.93978 0.09 0.30000 0.44814 0-34 0.58310 0.69795 0.59 0.768 1 1 0.83872 0.84 0.91652 0.943S4 O.IO 0.31623 0.46416 0.35 0.59161 0.70473 0.60 0.77460 0.84343 0.85 0.92195 0.94727 0.1 1 0.33166 0.47914 0.36 0.60000 0.71138 o.5i 0.78102 0.84809 0.86 0.92736 0.95097 0.12 0.34641 0.49324 0.37 0.60828 0.71791 0.62 0.78740 0.85270 0.87 0.93274 0.95464. 0-13 0.36056 0,5065s 0.38 0.61644 0.72432 0.63 0.79373 0.85726 0.88 0.93808 0.95828 0.14 0.37417 0.51925 0-39 0.62450 0.73061 0.64 0.80000 0.86177 0.89 0.94340 0.96190 0.15 0.3S730 o-53'33 0.40 0.63246 0.73681 0.65 0.80623 0,86624 0.90 0.94868 0.96549 0.16 0.40000 0.542S8 0.41 0.64031 0.74290 0.66 0.81240 0.87066 0.91 0.95394 0.96905 O.I7 0.41 231 0-55397 0.42 0.64807 0.74889 0.67 0.81854 0.S7503 0.92 0.95917 0.97259 0.18 0.42426 0.56462 0.43 0.65574 0-75478 0.68 0.S2462 0.87937 0.93 0.96437 0.97610 0.19 0.43589 0.57489 0.44 0.66332 0.76059 0.69 0.83066 0.88366 0.94 0.97954 0.97959 0.20 0.44721 0.58480 0-45 0.67082 0.76631 0.70 0.83666 0.8S790 0.95 0.97468 0.9S3O5 0.21 0.45826 0.59439 0.46 0.67823 0.77194 0.71 0.84261 0.S9211 0.96 0.97980 0.9864S 0.22 0.46904 0.60368 0.47 0.6S557 0.77750 0.72 0.84853 0.89628 0.97 0.98489 0.98990 0.23 0.47958 0.61269 0.48 0.69282 0.78297 0.73 0.85440 0.90041 0.98 0.98995 0.99329 0.24 0.48990 0.62145 0.49 0.70000 0.78S37 0.74 0.S6023 0.90450 0.99 0.99499 0.99666 0.25 0.50000 0.62996 0.50 0.7071 1 0.79370 0.7S 0.86603 0.90856 1. 00 1. 00000 1. 00000 sin 30 ° = cosi> 3° = K; CO. f 30° = SI « 60° = j4 \/^= 0.8660. sin 75 ° = cos\ 5° = 0.9659 ; ta I So° = c jl 60° = 'Ji ^T= 0.5774 ; cosT^ ° = sin I 5° = 0,258s ; CO 'an 30° = (an 60° = ^T= 1.73 21. log TV = 0.49714 99- log '.§•=1.5 37856. ^LF'H^BETIC^L UsTDEX. ACCUMULATORS 264 Hoppe's 264 " Hydraulic 218 Tweddells' . . . . 265 Action of GearTeeth 129 Adamsou's Stiffening Ring 269 Addition and Subtraction of Forces 26 Addition, Graphical 26 Adjustable Escapements 170 " Gears for Rotative Mot- ors 237 " Hangers 74 " Power Escapements . . . 236 " Pump Gears 236 Admiralty Chain 1S2 Adyman's Coupling 215 Agadio's Cable Locomotive 176 Air Compressors, Riedler's 279 Air Pump, Bunsen's 222 " Voa Gerike's 225 " Watt's 225 Air, Reservoir for Compressed 272 Allan's Link Motion 235 Almgren's Researches on Steam Boilers 271 Althaus' Furnace Hoist 173 Althaiis' Pump 223 American Standard Car Bearing. ... 75 Amos & Smyth's Pump 224 Anchor Escapement, Free 168 Bolts 56 Ratchet I55 Anemometers 239 Angle and T Iron Columns 83 Angle of Torsion, Determination of 93 Angle of Rotation in Torsion 11 Angstrom's Valve Gear 233 Anti-Friction Wheels 123 Anti-parallel 22 Anti-projection 23 Application of Tension Organs 172 Archimedes, Tympanon of 221 Archimedian Screw 221 Area o£ Polygons 23 ' ' Quadrilaterals 24 Triangles 23 Arithmography 22 Arm Sections, Table for Transform- ing 103 Arms of Gear Wheels 149 Armstrong Hydraulic Crane 22S Artificial Draft 272 Atmospheric Railway 227 Attachment of Journals 67 Audemar's Pump 224 Automatic Coupling loi " Friction Brake 170 " Steam Stop 281 Axis, Neutral 3 Axle, American Railway Standard.. 89 Prussian Railway Standard. . . 89 " Simole Crank 106 Axles .'. 85 " for Water Wheels 91 " Graphical Calculation of 86 " Loaded at Two Points 87 Axles 107 " Non-Symmetrical 86 " Proof Diagrams of 87 " Proportions of 86 " Railway 88 " Symmetrical 85 " with Circular Section 85 " with Cruciform Section 90 " with Inclined Loads 90 Axles with Three or more Bearings . 89 ' ' Wooden 92 BAG PUMP 217 Baker's Blower 221 Balanced Valves 279 Balance Wheel 167 Balancing of Pulleys 194 Balanced Slide Valves 285 Balanced Valve, Cramer's 280 Ball Bearings 127 Ball Joints for Pipes 249 Band Saws 177 Base Figures for Hyperboloidal Wheels 136 Bastard Gears 135 Beale's Gas Exhauster 226 Beams 3 ' ' Double Trussed 35 Sections, Table of 5-7 Force Plans for Framed 38 Multiple Trussed 35 Scale Ill " Simple Trussed 35 Triple Trussed 35 Walking no " with Common Load 11 Bearings, Ball 127 Design and Proportion of 68 " Independent Step 75 " Lateral 68 Metaliue 179 Multiple Collar 77 Multiple Supports for Fo Pedestal 71 ' ' Roller 126 Roller for Bridges 126 Simple Supports for 79 Standard American Car. . . 75 " Standard Prussian Car. ... 75 " Special Forms of 74 Step 75 Bearing Supports, General Principles 82 Bearings, Thrust '. . . . 65, 68, 75 " Thrust with Wooden Sur- face 76 ' ' Supports for 79 Wall 68, 71 Wall Step 75 with Three-part Boxes ... . 70 " Yoke 72 Becker's Clutch loi Behren's Chamber Gear Train 220 Belidor's Water Pressure Engine . 229 Bell Crank no Bellegarde, Rope Transmission at , . 205 Belleville Elastic Washers 212 Bellows Pump 217 Bell Pipe Connections 248 Bell Valve '. 276 Belt Connections 191 " Fastening, Bolter's 193 " Fastening, Moxon's 193 Belting 186 ' ' Cement for 193 " Efficiency of 194 Specific Capacity of 190 ' ' Stress on 191 " Various Examples of 1S7 " Table of Examples 192 Belt Lacing 191 Belts, Capacity of 190 " Creep of 194 " Cross Section of igo " Path of 186 Belts, Polishing 1 177 Quarter Twist 186 Belt Shifters ■. 188 Belt Shifter, Zimmerman's iSg Belts, Stiffness of 194 Belt Transmission, Examples of.... 191 Belts, Transporting 221 Bending, Bodies of Uniform Resist- ance to 8 " Load 3 Moment 3 ' ' Resistance to 2 Bergner's Drawing Board 172 Berlin, Sewerage System of 219 Bevel Friction Wheels 124 " Friction Wheels, Minotto's... 125 Gears 135 " " Construction Circles for. 135 " " Spiral 141 '■ " Stepped 141 Beylich's Universal Gears 136 Biquadratic Parabola 10 Blake's Steam Pump 230 Bleichert's Cable Tramway System . . 175 Blower, Baker's 221 Blower, Root's 221 Blowers, Fan 222 Bloxam's Gravity Escapement 168 Boat, Sail 223 Boat, Chain Propulsion of 183 Bodies of Uniform Strength 2 Bogardus Mill 126 Boiler ConstructioB, Economy in Combustion. 270 " " Economy of Material in 270 '■ " Improvements in Heating Surface . . . 270 Boiler Details 266 " Feeder, Brindley's 228 Feeders 228 Flues 269 ' ' Flues, Corrugated 269 Riveting 42 Boilers, Almgren's Researches on.. 271 " Circumferential Seams of.. 268 " Classified 265-266 Flat Surfaces of 268 for Swedish State Railway. 272 Longitudinal Seams of 267 Thickness of 266 " Spherical Details 26S ' ' Steam 265 Boiler Tubes 270 Boiling Water, used for Shrinking. . 47 Bolt Connections, Unloaded 60 " Dead 166 " Gerber's 57 " Heads 54 " Latch 166 Bolts, Anchor 56 " and Nuts, Metric 55 " and Screws 50 " Maudslay's Method of Secur- ing 58 Parsons' 57 Penn's Method of Securing.. 57 " Special Forms of 55 Borda Turbine 220 Bored Guides 122 Botter's Belt Fastening 193 Boxes, Various Forms of Journal ... 6g Brace, Weston's Ratchet 154 Bracket Support for Bearings 7q 304 ALPHABETICAL INDEX. Brackets, Wall 72 Brake, Automatic Friction 170 Brake, Napier's Differential 214 Brakes, Chain 215 Clamping 214 " Internal Strap 215 " Sliding 215 " Strap 211-214 Brake, Toggle Friction 162 Bramah Lock 166 Brasses for Connecting Rod 112 Brauer's Intermittent Gearing 165 Breaking Load i Bridge Bolts 59 Flying 222 " Roller Bearings 126 " St. Louis 60 Briggs' System of Pipe Threads. . . . 250 Briadley's Boiler Feeder 22S Britton's Steering Gear 23S Brown's Valve Gear 235 Brown's Windlass 173 Buckling, Resistance to 13 Buckling Strains, Table of 14 Buffer Coupling? iSi Built up Screw Propellers 57 Bnnsen's Air Pump 222 Butler's Coupling 96 CABLE, Arrangement of Pulleysfor 202 " Drum, Fowler's 173 " Ferry System, Hartwich's. 175 Grip Pawl 185 " Haulage Systems 174 " Incline at Lucerne 173 " Locjrnotivi, Agudio's . . . . 176 " Railways. 173 " Rhenish Railway 174 " Road, Kahlenberg 173 Cables, Table for Tightened 200 Cable System for Canals, Schmick's. 175 " System. Riggenbach's 174 " Tramway, Chicago 175 " Tra-nw-iy, Overhead 175 " Tramways, San Francisco. . . . 174 " Transmission, Ring System 208-211 " Transmissions, Short Span... 200 " TransmissioQ with Inclined. . . 200 Cadiat Tnrbin 3 220 Cagaiardelle 221 Call & Co., Valve Gear 162 Calculating Machine, Thomas'. . .153, 156 Calculation of Springs 18 Calculations for Chains 183 Cambon's Roller Bearing 127 Cam Valve gears 236 Canal Cable System, Schmick's 175 " Lift at Les Fontinettes 227 ■' " Green's 227 " " La Louviere 227 " " Mersey 227 Locks " 227 Cannon, Thickness of 15 Capacity of Belts 190 Capstan, David's 173 Car Bearing, American Standard... 75 Car Bearings, Prussian Standard ... 75 Cardan's Coupling 97 Cardioide 92 Casting 240 Cast Iron Cranks 105 Central Curve of Valve Gear 235 Centre of Gravity Graphically De- termined 33 Centrifugal Force 177 " Force of Wire Rope.... 197 " Pumps 222 Chain, Admiralty 182 " Brakes 215 " Couplings 184 " Drums 185 Flat Link 1S3 " Gemorsch 182 " Madgeburg-Bodenbacher. . . . 183 " Open Link 182 Pawls for 185 Pitch 183 Chain Propeller, Heuberger's 176 Propelling Gear 187 " Proportions of 183 " Propulsion of Boats 183 Chains, Calculations for 183 Chain Sheaves 1S5, 211 Chain, Specific Capacity of 211 Chains, Running 1S2 Stationary 182 ' ' Tests for 183 Chain Strippers 1S5 Swivels 184 " Transmission 211 " Transmission, Efficiency of... 213 in Mines 213 " " Decido Mines. 212 " Weight of 183 Chamber Gear Train, Behren's 220 " " " Eve's 220 " " " Repsold's. . . . 220 " Wheel Trains 2ig Channeled Connecting Rods 117 Checking Ratchets 150, 163 Check Valves 274 Cheese Coupling 99, 151 Chemical Ratchet Trains 171 Cheret's Press, Friction Gear of. . . . 125 Chicago Cable Tramway 175 Chronometer Escapement 167 Chubb Lock 166 Circular Plate, Deflection of 15 Circumference Scale 12S Clamp Coupling 95 Clamping Brakes 214 Clamp Pulley, Fowler's 203 Clamp Ratchet 160 Clark's Canal Lift 227 Clerk, Method of Shrinking Rings. 45 Clocks, Striking Mechanism for, . . . 169 Close Link Chain 182 Clutch, Becker's loi " Cone 99 " Couplings 95,98 " Coupling, Fossey's 100 " Dohmen-Leblanc's 101 " Forks 99 Clutches, Friction 99 Clutch, Garaud's loi " Jackson's loi " Koechlin's Friction 100 " Napier's loi " Reuleaux's Friction 100 " Schurmann's loi Toothed 98 " Weston's Friction loi Coating Operations 241 Cock, Four Way 225 Cocks 281 Coefficients of Resistance i Coefficients of Safety 1 Cold Forcing 17 Forcing, Dimensions for 47 -" Hooping 45 Collar Thrust Bearings 66 Columns, Calculations for Iron 82 Fluted 83 " Forms for Iron 84 " Grouped 84 ' ' Hollow 83 of Angle and T Iron 83 " of Uniform Resistance.... 13 " Strength of Cast Iron 83 Stresses in 82 Combined Levers no Compound Escapements 168 " Link as Thrust Bearing. 67 " Strains, Table of 15 " Stresses 13 Compressed Air for Power Distribu- tion 219 Compression, Resistance to 2 Condenser, Watt's 230 Conditions of Equilibrium 29 Conductors for Pressure Organs 242 Conduits for Pressure Organs 216 Cone Clutch Coupling 99 Cone Coupling, Reuleaux's 96 " Pulleys 189 " " Diagram for igo " " for Crossed Belts iSg " " for Open Belts iSg Conical Gear Wheels 135 Connecting Rod Brasses 112 " End, Cast Iron.... 113 '■ " Krauss' 113 " " Penn's 113 " " Polonceau's. . 114 " Porter- Allen 117 Rods J12 " Channeled 117 " Forms of ij8 " Locomotive 116 " Rectangular Section 117 Rod, Solid End for 113 Rod, Solid End for Lo- motive 113 Rods, Ribbed 117 Rods, Round ij6 Red, Strap End for 112 Rod, Whip Action of . . . . 116 Connections for Belting igi " Cast Iron Pipes.... 248 " Crank Axles 115 " Lead Pipe 251 •' Neck Journals 114 " Wrought Iron Pipes 249 Construction Circles for Bevel Gears 135 Construction of Machine Elem'nts. 39-289 " Pulley Stations 204 " Rope Curve 202 " Rope Pulleys 202 Screw Thread 50 Continuous Ratchets 150 Ratchets with Locking Teeth 165 Running Ratchets 164 Copper Pipes 246 Cord Friction 177 Cord Polygon 26 Corliss Valve 236 Corliss Valve Gear 162 Cornish Valve 280 Cornish Valve Gear 153, 163 Corrugated Boiler Flues 269 Corrugated Fire Box 269 Cotton Rope 179 Cotton Rope Transmission 196 Counterbalance, Oeking's .... 217 Countershaft Hanger, Sellers' 74 Counting Gear for Gas Meter 165 Couples, Force 29 Coupling, Adyman's Z15 " Butler's 96 " Cardan's 97 " Cheese 99, 151 " Clamp 95 " Cresson's 96' " Drag Link 97 ' ' Hooke's 97 " Muff 95 " Oldham's 96 Plate 95 " Pouyer's loi " Prentiss 216 " Ramsbottom's Friction... 99 " Reuleaux's Cone 96 Couplings 95 " Automatic loi " Buffer 181 " Schurman's Friction 215 Clutch.... 98 Coupling, Sellers' 96 Couplings, Flexible 96 " for Chain 184 ' ' for Propeller Shafts 95 " Sharp's 96 " Link gS Coupling, Uhlhorn's loi Cramer's Balanced Valve 2S0 Crane Hook, Proportions for 184 " Pillars 89 ' ' Ramsbottom's 176 ALPHABETICAL IXDEX. Cranes, Cotton Rope Driven 196 Graphical Calculation 27 " Hydraulic 228 Squaring Device for 172 " Varieties of 173 Crane, Tangye's -^ \lb Crane, Towne's 176 Crank Axle, Graphostatic Calcula- tion of 106 " Axles, Connections for 115 " Axle, Simple 106 " Graphostatic Calculation for Return 105 " . Graphostatic Calculation for Single 104 " Pin, Tangential Pressure on. 233 Pins 61 " Pins, Connections for 112 " Return 105 Cranks, Cast Iron 105 Cranks, Classified 104 Crank Shaft, Graphostatic Calcula- tion of 107 Cranks, Hand 109 Crank, Sliding 226 Cranks, Single Wrought Iron 104 Creep of Ropes 196 Creep of Belts 194 Cresson's Coupling 96 Crossed Belts, Cone Pulleys for 189 Cross Heads 118 " for Guides 119 " " for Link Connections... 119 " " for Locomotives 121 " " for Marine Engines.... 120 " Head, Slipper 121 " Head, Superficial Pressure on. 120 '• Keyed Connections 48 " Section of Belts 190 " " Hemp Rope 195 " " Wire Rope 196 Crown of Pulleys 186 Ratchet 154 Wheel Escapement 169 Cup Packing 253 Current Motor 223 Curve, Elastic 3 Curves, Velocity 233 Cycloidal Curves 130 " Curves, Generation of 130 " Sinoide 13 Cycloid, Spherical I35 Cylinder Escapement 169 Ratchet 156 " Ratchet Gearing 165 Cylinders 216 Cylinders for Hydraulic Presses 243 Cylindrical Spiral Gears 138 Cylindrical Vessels 15 DANAIDE 220 Darcy, Formula for Friction of Water 246 David's Capstan 173 Davis & Co., Steering Gear 238 Dead Bolt 166 Dead Ratchet Tooth 152 Deane's Steam Pump 230 Decido Mines, Chain Transmission of 212 Decomposition into Parallel Forces. 31 Deflection 3 " in Bodies of Uniform Re- sistance 8 " of Circular Plate 15 " of Shafting 94 " of Shafting, Torsional. ... 92 " of Wire Ropes 198 Delisle's Screw Thread Systems. ... 53 Dennison's Escapement 168 Design and Proportion of Bearings. 68 Diagram for Cone Pulleys 190 Diametral Pitch 12S Diaphragm Pump 217 Differential Brake, Napier's 214 " Hydraulic Lever 218 Pulley Block, Weston's. 174 Differential Windlass Dimensions of Gear Wheels Disk Friction Wheels " Valves, Flat " Wheels with Pin Teeth Distribution of Weight Division by Lines Division of Gear Wheels, Circumfer- ential . . . Dobo's Ratchet . Dohmen-Leblanc's Clutch Donnadieu's Pump Door Locks Double Acting Pumps " Arm Pulleys " Beat Valve " Friction Ratchets. . . " Spiral Gears Douglas & Coulson's Steering Gear. Downton's Pump Draft, Artificial Draft Keys. Drag Link Coupling Drawing Board, Bergner's Driving by Tension Organs Drop Hammer, Friction Drop Hammer, Merrill's Drums for Chain Dry Gas Meter Ducommun & Dubied's Planing Ma- chine Dunning & Boissiere's Steering Gear Duplex Escapement Pump, Mazelline's Pump, Worthington's 173 147 124 275 133 3 23 128 160 lOI 223 166 224 193 280 160 141 238 224 272 48 97 172 173 176 123 185 240 176 238 167 231 231 ECCENTRICS 109 Eccentric Straps 115 Edge Keys 49 Efliciency of Belting 194 " of Chain Transmission... - 213 " of Rope - Transmission. . . 205 Equalizing Levers 32, iii Equalizer Worthington's 232 Equation of Elastic Curve 3 Equatorial Section Modulus... ■ 5 Equilibrium, Conditions of 29 " of External Forces. ... 27 of Forces 22 " of Internal Forces. ... 28 of Three Parallel For- Elastic Curve Elastic Curve, Equation of Elasticity and Strength of Flexure . Elasticity, Modulus of i, Elastic Limit i " Limit in Beams " Line 92 Washers, Belleville 212 Elbe, Chain Propelling Gear on. . . . 185 Elbow Fittings 251 Elbow Fittings, Friction in 251 Electric Signals, Siemens & Halske. 166 Elements of Graphostatics 22-38 Elevator, Hydraulic 227 Elevator Safety Devices 164 Emery Weighing Machine 173 Enderlein's Escapement 168 Engine, Porter-Allen 236 Engines, Rotative Pressure 233 Engines, Valve Gear for Rotative. . - 234 Enlarged Screws 58 Epicycloidal and Evolute Teeth Compared 135 Erhardt's Flange Joint 47 Escapement, Bloxam's Gravity 168 Chronometer 467 Crown Wheel 169 Cylinder 169 Dennison's 168 Duplex 167 Enderlein's 168 Free Anchor 168 Hipp 168 Graham's 169 3o5 Escapement, Lamb's 168 Lepaute's 169 Mudge's 16S " Reuleaux's 16S Escapements 150, 167 Adjustable 170 " Compound 16S for Measurements of Fluids 239 for Measurements of Volume. . . : 239 " for Moving Liquids, Pressure 228 " for Pressure Organs. . 226 Isochronous 167 " Periodical 169 " Periodical Pressure. . . 229 Period of 167 Power 169 " Power Adjustable. .. . 237 Range of 167 " Simple 167 Time of Oscillation . ... 167 Uniform 167 Escapement, Tiede's 168 Eve's Chamber Gear Train 220 Evolute Rack Teeth 132 Evolute Teeth for Interchangeable Gears 131 Examples of Belting, Table 192 of Belt Transmission 191 of Gearing 147 of Journals 62 Thrust Bearings 78 Expansion Gear, Farcot's 236 '" Gonzenbach's 236 " Meyer's 236 " Joints 245 Valves 236 External Forces, Equilibrium of . . . . 27 Extraction of Roots 26 Eytelwein's Formula for Stiffness of Ropes 181, 196 FABRY'S VENTILATOR 221 Factor of Safety i Fairbairn, Experiments on Boiler Flues 269 Fan Blowers 222 Farcot's Stuffing Box 254 Farcot's Valve Gear 236 Fast and Loose Pulleys 188 Felbinger's Postal Tube 227 Ferules for Boiler Tubes 270 Fink's Link Motion 235 Fire Box, Corrugated 269 Fire Box, Kaselowsky's 269 Fish Torpedo 171 Flange Connections for Lead Pipe. 252 Joints 58 Joint, Erhardt's 47 Joints for Pipes 248 Flanges for Riveted Pipes 249, Flap Riveted Joints 40 Flap Valves 274 Flat Hemp Rope 178 " Link Chain, Neustadt's 183 " Link Chain, Table of 183 ' ' Pivot Bearings 66 " Ropes I Si Flexible Couplings 95, 96 Pipes 252 Rod Connection 114 Flexure, Elasticity and Strength of. 2 Flexure, Strains of 3 Flow of Metals 240 Fluid Escapements for Transporta- tion 227 " Running Ratchet Trains 223 " Valves 287 Fluted Columns 83 Flying Bridge 222 Fly Wheel, Oscillating 233 Fly Wheels 233 Force, Centrifugal 177 Force Couples 29 3o6 ALPHABETICAL INDEX. Forced Connections, Examples of. . . 46 Forced Draft 272 Force Plans for Framed Structures. 34 Plans for Roof Trusses 36 " Polygon 26 Forcing Fil I7 Forces, Addition and Subtraction of 26 Equilibrium of 22 Resultant of Several 26 Forcing 45 Forks, Clutch 99 Forlc Journals 63 Fork Journal, Stub End for 114 Forms for Iron Columns 84 Fossey's Coupling 100 Foundation Bolts, Keying for 48 Fourneyron Turbine 220 Four Way Cock 225 Fowler's Cable Drum i73 Fowler's Clamp Pulley 203 Friction Brake, Automatic 170 Brake, Toggle 162 Clutches 99 Clutch, Koechlin's 100 Ramsbottom's 99 Reuleaux's 100 " Westons loi Cord 177 Coupling, Schurman's 215 " Drop Hammer ■ 176 Feed, Sellers' 126 Gear of Cheret's Press 125 " Gear, Robertson's 125 in Elbow Fittings 251 " in Spur Gearing i34 in Stufang Box 254 of Chain Transmission 213 ' ' of Journals 64 " of Pivot Bearings - ■ 66 " of Screw Thread 50 of Spiral Gear Teeth 140 " of Water in Pipes 246 Friedmann's Jet Pump 222 Friction Pawls i59 Pawl, Saladin's 161 " Pawls, Release of 161 Ratchets 158 Double 160 Rod 163 Running 160 " '■ Stationary 161 " Rollers, Mechwart's 127 " Trains, Special .... 126 Wheels 122 " Bevel 124 " " Construction of 123 " " Disk 124 " '* for Inclined Axes. . . 124 " " for Parallel Axes... 123 " " Material for 123 " Minotto's 125 " " Robertson's 125 " " Sellers' 126 " " Two Applications of 123 " " Wedge 125, 160 Francis' Turbine 220 Framed Beams, Force Plans for. ... 38 Framed Structures, Force Plans for 34 Frankfurt on Main, Water Supply of 218 Free Anchor Escapement 168 Free Cross Heads 119 li'reiburg. Rope Transmission at ... . 205 French Lock 166 Front Bearings 6S Furnace Hoist, Althaus' 173 Furnace, Wilson's Water Gas 288 Future Possibilities of Boiler Con- struction 270 GANNOW MINE 214 Garand's Clutch loi Gases, Reservoirs for 219 Gas Exhauster, Beale's 226 " Holders 272 " Meter, Counting Gear for 165 " " Dry 240 Sanderson's 239 Gas Meter, Wet 239 Gate Valves 282 Gearing, Brauer's Intermittent 165 " Calculation of Pitch and Face 144 " Cylinder Ratchet 165 " Double Pin 132 " E.xamples of 147 " Friction in Spur Tooth. ... 134 " Fundamental Formula for. 12S " Globoid Worm 143 " Hawkin's Worm; 143 " Jensen's Worm 143 Ratchet 1 50 Shield 133 " Gearing, Step 141 Toothed 127 "' Worm. 139 Gears, Bastard 135 Bevel 135 Bevel Spiral 141 Beylich's Universal 136 Cylindrical Spiral 14S Double Spiral 141 Examples of Spiral 140 Globoid Spiral 142 " Hoisting 144 Parallel 133 Precision 139 " Single Tooth 165 " Spiral 13S Stepped Bevel 141 '■ Table of Cast Iron 144 " Teeth for Hyperboloidal 13S Transmission 144 Gear Teeth, Action of 129 Construction of Spur.. 128 Epicycloidal and Evo- lute Compared 135 Evolute Interchange- 131 able 131 Friction of Spiral .... 140 Interchangeable 130 Internal 131 Line of Action of 129 Loss in Determined Geometrically 134 of Circular Arcs. ..... . 131 Pin 132 Pressure on 146 Section of 144 Stress in. . . . 145 Thumb Shaped 134 Wear on 134 " Tooth Outlines, General Solu- tion 129 " Wheel Arms, Table of 149 " Hubs 150 " Plane 136 " Wheels, Arms of 149 " " Circumferential Divis- ion of 128 " Classified 127 " Conical 135 '' Dimensions of 147 " Diametral Pitch of... 128 "' Hyperboloidal 136 " Interchangeable 128 " " Pitch of 144 " Pitch Radius of 12S " Rim of 147 " " Weight of 150 Gemorsch Chain 1S2 General Form of Toothed Ratchets. 158 General Remarks upon Ratchet Me- chanism. 171 Generation of Cycloidal Curves.... 130 Geneva, Sluice Gates at 275 Geneva Stop 165 Gerber's Bolt 57 Geyser Pump, Siemens' 222 Gidding's Slide Valve Experiments. 285 Giffiard's Injector 222 Girard Turbine 220 Githen's Rock Drill 231 Globe Valve 279 Globoids 142 Globoid Spiral Gears 142 Globoid Worm Gearing 143 Gooch's Link Motion 235 Goodwin's Split Pulley 194 Gonzenbach's Expansion Gear 236 Graham's Escapement 169 Graphical Addition 26 ' ' Calculation of Axle 86 " " of Powers ... 24 " " of Shafting. . . 95 " " of Crank Axle 106 " " of Multiple Crank Shaft 107 " " Return Crank 105 " " Single Crank. 104 Graphostatics, Elements of 2 2 -38 Green's Canal Lift 227 Greindl's Pump 221 Gresham's Injector _ 222 Grip Pawl for Cables 185 Grooves for Rope Transmission .... 195 Grooved Fly Wheels 195 Grouped Columns 84 Group Riveting 41 Guide Mechanism for Pressure Or- gans 217 Guide Pulleys for Belting 186 Guides and Guide Bars 121 Bored 122 for Marine Engines 122 Guide Sheaves 185 Guides, Locomotive 122 Guiding by Pressure Organs 216 Guiding, Tension Organs for 172 Gun Lock Mechanism 166 Gun Locks 163 Guns, Hooping of 16 HAIR SPRING i6g Hair Trigger 168 Half Journals 64 Hammer, Merrill's Drop 123 Hand Cranks 109 Hanger Boxes, Sturtevant's 74 Hangers 68 Hangers, Adjustable 74 Hanger, Sellers' 74 Hanger, Sellers' Countershaft 74 Hangers for Rope 181 Post 73 Proportion of 73 Ribbed 73 Harlow's Valve Gear 231 Hartwich's Cable Ferry System 175 Hastie's Steering Gear 236 Hauling System, Riggenbach's 172 Haulage Systems, Pennsylvania. . . . 174 Hawkin's Worm Gearing 143 Helfenberger's Regulator 236 Hemp Rope 178 " Transmission, Specific Capacity of 195 " Wear on 196 " Weight of 178 Hero's Fountain < zSS Heusinger's Link Motion 235 Heuberger's Chain Propeller 176 Hick, Experiments on Stuffing Box Friction . . 254 Hick's Stiffening Ring 269 High Duty Pumping Engine, Worth- ington 232 Hipp Escapement , 168 Hirn'S Experiments on Journals.... 64 Hodgkinson's Experiments 13 Hofmann's Valve Gear 163 Hoist, Althaus' 173 Hoisting Devices 172 Hoisting Gears 144 Hollow Columns 83 Hollow Journals 62 Hooke's Coupling 97 Hooks 1 84 Hooping 45 " by Shrinkage 45 of Guns 16 Hoppe's Accumulator 264 ALPHABETICAL fXDEX. 307 Hose .11 252 Howaldt's Metallic Packing 254 Hubs for Rock Arms 102 Hubs of Gear Wheels 150 Hunting Valve 238 Hurdy Gurdy Wheel 220 Hydraulic Accumulators 218, 264 Cranes 228 Elevator 227 Lever 217 " Lever, Differential 218 " Parallel Motion 2r8 " Power Distribution 219 " Power Distribution, Ring System 256 " Presses, Cylinders for. . . 243 " Press, Thickness of Cyl- inder 16 " Hydraulic Ram, Montgol- fier's 232 " Riveting Machine, Twed- dell's 228 " Steering Gear 237 Tools 228 '• Transformer 218 Hyperboloidal Gear Wheels 136 " Gears, Teeth for 138 " Wheels, Base Figures for,, 136 IDEAL BENDING MOMENT.... 13 Ideal Twisting Moment 13 Impact Water Wheels 220 Inclined Cable, Transmission with - . 200 Independent Step Bearings 76 Inertia, Moment of 5-7 Influence of Pulley Diameter on Wire Rope 197 Influence of Weight of Wire Rope. 180 Injector, Giffard's 222 Injector, Gresham's 222 Interchangeable Gear Teeth 130 Interchangeabe Gear Wheels 128 intermittent Gearing, Brauer's 165 Internal Flow 240 " Forces, Equilibrium of .... - 28 Gear Teeth 131 Ratchet Wheels 151 Strap Brakes 215 Intze's Discussion of Tanks 261 Inverted Siphon 244 Iron Columns, Calculations for 82 Iron, Weight of Round ■ • 55 Isochronous Escapements 167 Isolated Forces in One Plane 26 Isolated Forces, Resultant of 29 JACKSON'S CLUTCH 101 Jacquard Loom 163 James Watt & Co., Thrust Bearing 77 Jam Nut 56 Jaw Clutch gS Jensen's Worm Gearing 143 Jet Action 241 ' ' Mechanism 240 " Propeller 223 " Pump, Friedmann's 222 Joints, Expansion 245 " Flange 58 Strength of Riveted - 40 Joint, Universal 97 Jopling's Water Meter 239 Journal Boxes, Various forms 69 " Friction in Rope Transmis- sion 195. 205 '• "of Chain Transmis- sion 213 Journals, Attachments of 67 " Dimensions - 61 " Examples of 62 Fork... 63 " for Levers loi for Shafting 94 " Friction of 64 Half 64 Hollow,,,,, 62 Journals, Lateral 61 Lubrication of 61 " Multiple 63 " Neck 62 " Overhung 61 " Pressure on 61 " Proportions of 61 ' ' Stress on 61 " — Various Kinds 60 KAHLENBERG CABLE ROAD- . 173 Kaselowsky's Fire Box 269 Kennedy's Water Meter 239 Kernaul's Key 49 Keyed Connections 47 Keying 47 " Foundation Bolts 48 " Peters' Method 102 " Sci-ew Propellers 49 Key, Kernaul's 49 Keys, Concave 48 " Draft 48 " Edge 49 " Flat. . 48 " for Rock Arms. 102 Longitudinal 48 " Methods of Securing 50 ' ' Recessed 48 Stresses on 48 ' ' Taper of 47 ' ' Unloaded 49 Kirchweger's Steam Trap 228 Kirkstall Forge Rolling Mill 126 Kirkstall Forge, The 94 Kley's Pumping Engine 233 Klug's Valve Gear 235 " Knot " in Cord Polygon 27 Koechlin's Friction Clutch 100 Krauss' Connecting Rod End 113 Krauss' Piston 252 Kriiger, Investigation of Rivets ... 39 LACING, BELT 191 Lagarousse Ratchet 164 La Louviere Canal Lift 227 Lamb's Escapement 168 Langen Gas Engine 161 Lap Joints, Riveted 40 Lap of Slide Valve 225 Latch Bolt 166 Lateral Bearings 68 Lateral Journals 61 Lead Pipe Connections 251 Lemielle's Ventilator 82 Lepaute's Escapement 169 Levasseur's Metallic Tubing 252 Lever Arms, Calculation of . . . . .... 103 " Arms of Combined Section. . . 103 " Differential Hydraulic 218 Hydraulic 217 Levers, Combined no Equalizing 32 " Journals for loi " Simple loi Lifting Frame for Screw Propeller. 151 Lift Valves 223, 273 Liquids, Pressure Escapement for Moving 22S Limit of Elasticity i Limit of Elasticity in Beams 8 Line of Action of Gear Teeth 129 Line Shafting 93 Link Couplings gS " Motion, Allan's 235 " " Fink's 235 " " Gooch's 235 " " Heusinger's 235 " " Stephenson's 234 Load, Breaking i Load Length of Wire Rope 180 Lock, Bramah 166 " Chubb 166 •' French 166 Locking Ratchets 1 50 Locomotive Axles 107 " Connecting Rods 116 Locomotive Connecting Rod, Solid End for 113 " Cross Heads 120 Guides 122 " Springs, Screws for. .. . 58 Locks, Canal 227 Door 166 Gun 162 Lock, Yale 166 Logarithmic Spiral 26 Long Distance Fluid Transmission . 233 Long Distance Power Transmission 259 Longitudinal Keys 48 Loom, Jacquard 163 Loss in Gear Teeth Determined Geometrically 135 Loss in Hemp Rope Transmission. . 195 Lubrication of Journals 61 Lucerne, Cable Incline at. , 173 MACHINE ELEMENTS, CON- struction of 39-289 Machine Riveting 39 Mackay & McGeorge, Riveting Ma- chine Ill Magdeburg-Bodenbacher Chain.... 183 Maltese Cross Gear 165 Manholes 289 Mannesmann Tubing 243 Marine Cross Heads 121 " Engine Guides 122 Propulsion 222 Marshall's Valve Gear 235 Materials — Strength of 1-21 Mathematical Tables 291 Maudslay, Method of Securing Bolts 58 Maudslay, Thrust Bearing by 77 Mauser's Revolver 165 Mazelline's Duplex Pump 231 Measurement of Fluids, Escape- ments for 239 Measuring Devices, Running 239 Mechwart's Friction Rollers 127 Medart Pulley 193 Merrill's Drop Hammer 123 Metaline Bearings 179 Metallic Piston Packing 253 Metallic Tubing 252 Metals, Weight of Sheet 43 Meter for Alcohol, Siemen's 239 Methods of Securing Bolts 57 Methods of Securing Pawls 153 Metrical Screw Systems 52 Metric Bolts and Nuts 55 Metric Pipe Thread System 250 Meyer's Valve Gear 236 Mill, Bogardus 126 Minotto's Bevel Friction Wheels... 125 Mixed Tooth Outlines 133 Mines, Chain Transmission in 213 Modulus of Elasticity i, 13 " Resistance.... i Rupture 1,2 Transmission 20S Molinos & Pronnier, Speed of Rivet- ing 39 Moment of Inertia 3, 5, 7 Moment of Inertia, Polar 11 Mont Cenis Air Compressors 232 Montejus 22S Montgolfier's Hydraulic Ram 232 Morin's Experiments on Journals. . . 64 Motors for Pressure Organs 219 Moulding 240 Moxon's Belt Fastening 193 Mudge's Escapement 168 Muff Coupling 95 Mule Post 188 Mule, Spinning 169 Multiple Collar Bearings 77 " Collar Thrust Bearings. .. . 66 " Crankshafts 107 Journals 63 Ratchets 154 Supports for Bearings 80 ' ' Trussed Beams , , , , 35 ;o8 ALPHABETICAL INDEX. Multiple Valves 276 Mviltiplication and Division Com- bined 23 Multiplication by Lines 22 Murdock's Slide Valve 234 Muschenbroeck's Pump 223 NAGEL TURBINE 220 Napier's Clutch 101 Napier's Differential Brake 214 Natural Reservoirs 21S Neck Journals 62 Neck Journals, Connections for. ... 114 Negative Reservoirs 219 Neutral Axis 3 Neutral Plane 10 Neustadt's Chain 1S3 Newcoraen Engine 163 Normandy's Pipe Joint 249 Norton's Pump 225 Nut, Jam ■ 59 Nut Locks 56 Nuts, Washers and Bolt Heads 54 OBELISK, FORCES IN RAISING 2S Oberursel, Rope Transmission at. 203, 205 Oeking's Water Counter-balance... 217 Oil Tanks 21S Oldham's Coupling 96 Open Belts, Cone Pulleys for 189 Open Link Chain 182 Ordvvay, Experiments on Pipe Cov- ering 245 Oscillating Fly Wheel 233 Oscillating Pumps 226 Oscillation of Escapements, Time of 167 Osterkamp's Rope Hanger 181 Otis Elevator 228 Overhead Cable Tramv^ay 175 Overhung Journ als 61 PACKING FOR HYDRAULIC Press 253 " for Pump Pistons 255 " Howaldt's Metallic 254 *' Piston 216 " Standard Prussian Rail- way . 255 Pagel's Elastic Washer 57 Pallets 168 Pappenheim Chamber Wheel Train . 219 Parabola, Biquadratic 10 Parallel Forces — Equilibrium of. .30, 31 Gears 133 " Motion, Hydraulic 218 Rods for Locomotive En- gines 117 Parson's Bolts 57 Pattison's Pump 226 Pawl, Cable Grip 185 Pawl. Saladin's Friction ... 161 Pawls for Chains - 185 " Friction 159 Methods of Securing 153 Pawl, Spring 153 Pawls, Release of Friction 161 Pawl, Thrust upon 152 Pawl, Thumb Shaped 160 Payton's Water Meter. 220 Pedestal Bearings 71 Penn's Connecting Rod End 113 Method of Securing Bolts.. 57 Piston 252 Pennsylvania Haulage Systems.... 174 Periodical Escapements 169 Periodical Pressure Escapements... 229 Peters' Method of Keying 102 Petit's Pipe Joint 248 Pfalz-Saarbruck Screw Thread Sys- tem 53 Phoenix Column 59. 83 Physical Ratchet Tram 171 Pickering's Steam Pump 230 Pillow Blocks 68 Blocks, Adjustable 69, 70 Block, Sellers' 70 Pillow Blocks, Large 69 Blocks, Proportional Scale for 68 Block, Sturtevant's 70 Pin Gearing, Double 132 " Gear Teeth 132 ■' Ratchet Wheel 152 Pins, Crank 61 Pin, Split 56 Pipe Connections, Socket 248 ' ' Coverings 243 ' ' Fittings 249 " Joint, Normandy's 249 " Petit's 248 Riedler's 249 ' ' Riveted 244 Pipes, Ball Joints for 249 Pipes, Connections for Cast Iron . . . 248 " Connections for Wrought Iron 249 Copper 246 Flange Joints for 248 Flexible 252 for High Pressures 242 Resistance of Bends in 247 Resistance to Flow in 246 Pipe Sockets 250 Pipes, Steam 245 Pipe, Steel 243 Pipes, Thickness of Cast Iron 242 Pipe Threads, Briggs' System 250 j" Thread System, Metric 250 " Weight of Cast Iron 242 " Wrought Iron 243 Piston, Krauss' 252 Packing 216, 253 Packing, Metallic 253 Penn's 252 Pumps 223 Rods 255 Pistons 216, 252 Piston, Swedish 253 Pistons with Valves 286 Piston Valves 286 Pitch and Face of Gearmg, Calcula- tion of 144 Hoisting Gears. , 144 Transmission Gears . 144 Chain 183 Circles, Table of 1-28 " of Gear Wheels 144 " Radius of Gear Wheels 128 Pivot Bearings, Flat 66 Pivots, Formula for 65 " Pressure on 65 Proportions of 65 Plain Slide Valve 282 Plain Slide Valve Gear 234 Plane Gear Wheel 136 Planing Machine, Ducommun & Du- bied's 176 " " Sellers 176 " Shanks' 163 Plate Coupling 95 Plungers 216, 253 Plunger Pumps 223 Pneumatic Power Distribution 257 Pneumatic Tube 227 Polar Moment of Inertia 11 Polishing Belts 177 Polonceau's Connecting Rod End.. 114 Polygons, Area of 24 Poncelet's Chain 173 Poncelet's Water Wheel 220 Porter- Allen Connecting Rod 117 Porter- Allen Engine 236 Post Hangers 73 Pouyer's Coupling loi, 152, 153 Powel Valve Gear 163 Power Distribution, Compressed Air 219 " " Hydraulic 219 " " Hydraulic Ring System 256 " " Pneumatic 257 " " Steam 219 " " Systems 219 '■ " Vacuum 219 Power Escapements 169 Powers, Graphical Calculation of , . . 24 Powers of Trigonometrical Func- tions 25 Power Transmission by Superheated Water 219 Practical Resistance. i Precision Gears 139 Precision Ratchets 157 Prentiss' Coupling 216 Pressure Escapements for Moving Liquids 228 on Gear Teeth 146 Journals 61 '■ Lift Valves 277 " Pivots 65 Screw Threads 58 Organs , 216 " Conductors for 242 " " Conduits for 216 " " Escapements for... 226 " " Guidmg by 216 " Guide Mechanism for . 217 " " Motors for 219 " " Ratchet Mechanism for 223 " Reservoirs for. .218, 260 " ■' Running Mechanism for 219 " '■ Technological Appli- cations of 250 " Superficial i Transmission, Specific Ca- pacity of 255 Proof Diagrams of Axles 87 Propeller Bearing, Ravenhill & Hodgson's . 74 Jet 223 Lifting Frame for 151 ' ' Screw 223 " Shafts, Couplings for 95 Propelling Chain : 1S5 Proportions of Axles 85 " Chain. 183 " Flange Joints 59 " Hooks 184 " Journals 61 Pivots 65 " Pulleys 193 Propulsion, Marine 222 Prussian Standard Car Bearing 75 Pulley Block, Weston's Differential. 174 " by Walker Mfg. Co 193 " Diameter, Influence on Wire Rope 197 Fowler's Clamp 203 Medart 193 Pulleys, Balancing of 194 " Cone 189 Construction of Rope 202 '• Crown of Face 186 "' Double Arm 193 Guide 186 " Fast and Loose 188 for Cable, Arrangement of . 202 Proportions of 193 Split 193 Pulley Stations, Construction of . . 204 Pulleys, Tightening 186 Pulley, Sturtevant's 194 Pulleys, Umbrella 207 Pulleys, Vertical Supporting 188 Pulsometer 232 Pump, Althaus' 223 Amos & Smyth's 224 Bag 217 Bellows 217 " Diaphragm 217 Donnadieu's 223 Downton's 224 Friedmann's Jet 222 Gears, Adjustable 236 " Greindl's 221 " Engine, Kley's 223 ALPHABETICAL IXDEX. 309 Pumping Engine, Worthington's High Duty 232 Pumping Machinery 229 Pump, Mazelline's Duplex 231 Muschenbroeck's 233 " Norton's 225 Pattison's 226 Pistons 253 Pistons, Packing for 255 Regulator, Helfenberger's . 236 Repsold's 221 " Rittinger's 223 Pumps, Centrifugal 222 " Considered as Ratchet Trains 223 " Double Acting 224 Pump, Siemen's Geyser 222 Pumps, Oscillating 226 Pump, Spiral ■■■ ■ 221 Pumps, Piston 223 Plunger 223 Rotary 226 Stolz's 224 Pump, Stone's 224 " Valve Gear 225 Valves, Riedler's 278 Vose's 224 Worthington's Duplex 231 QUADRANTS 153 Quadrilateral Figures, Area of 25 Quarter Twist Belts. . ■ • 186 Quarter Twist Belt, Shifter for 189 RACK, RATCHET 151 Rack Teeth, Evolute 132 Railway Axles 88 Ramsbottom's Crane 176 Ramsbottom's Friction Clutch gg Ratchet, Anchor 155 Brace, Weston's 154 Clamp 160 " Crown 154 Cylinder 156 " Dobo's 160 Gearing 150 Cylinder 156 " " Dimensions of Parts of 158 Gears, Toothed Running. . 150 Lagarousse 1 64 Mechanism for Pressure Organs 223 General Re- marks upon 171 '" " Kinematically Discussed.. 171 Rack 151 Rod Friction 163 Ratchets, Checking ....... 150 Continuous 150 " Continuous Running 164 " Double Friction 160 Friction 158 " General Form of Toothed 158 " Locking 150, 166 " Multiple 154 " of Precision 157 ■' Releasing 150, 162 " Running 1 50 " Running Friction 158 " Silent 153 Spring 153 " Stationary 1 50, 156 " Stationary Friction 161 Step 155 Step Anchor 157 Throttle 161 " with Locking Teeth, Con- tinuous 165 " Teeth, Flanks of 153 " Teeth, Form of 150 •' Tension Organs 185 " Tooth, Dead . 152 " Train, Physical 171 Ratchets, Trains, Chemical 171 Wheels, Internal 151 Wheels, Special Forms... 154 '• Wilber's 153 Ravenhill & Hodgson, Propeller Bearing 74 Ravenhill & Hodgson, Thrust Bear- ing by ; 77 Reciprocating Valve Gears 234 Regulator, Guhrauer & Wagner's . . 237 Regulator, Rigg's 236 Reichenbach's Water Pressure En- gine 22g Releasing Ratchets .150, 162 Releasing Valve Gears 162 Release of Friction Pawls 161 Rennie, Experiments on Journals. . . 64 Repeating Watches 169 Repsold's Chamber Gear Train 220 Repsold's Pump 221 Reservoirs for Air and Gas 272 " for Gases 219 " for Pressure Organs. 218, 260 Natural 218 " Negative 219 Resistance, Coefficients of " Modulus of " of Bends in Pipes 24' " of Valves in Pipes 24' Practical Theoretical . . : " to Bending 2 " Buckling 13 Flow in Pipes 246 " Shearing 2 Torsion 11 Resultant of Isolated Forces 29 Load on Water Wheel determ'd Graphically. 34 " Several Forces 26 Return Crank 105 Return Crank, Graphostatic Calcu- lation for 105 Reuleaux's Coupling g6 Escapement 168 " Friction Clutch 100 " System of Rope Trans- mission 206 " Valve Diagram 234 " Winding Drum 173 Reversing Gear, Globoid 143 Revolver, Mauser 165, 166 Rhenish Railway Cable 174 Ribbed Axles 91 Rib Profiles, Construction of gi Richard's Manometer 288 Rider's Valve Gear 236 Riedler's Air Compressors 27g Pipe Joint 249 Valve Gear 278 Riggenbach's Cable System 174 Riggenbach's Hauling System 172 Rigg's Regulator 236 Rigid Couplings 95 Rim of Gear Wheel 149 Ring System of Cable Transmis- sion 208-211 Rittinger's Pump 223 Riveted Joints, Construction of An- gles 44 " " Junction of Plates.. 43 " " Proportional Scale for 41 " ■' Reinforcement of Plates 44 " " Special Forms 43 Strength of 40 Table of 40 Pipe 244 Pipes, Flanges for 249 Rivet Heads, Proportions of 39 Riveting 39 Boiler 42 " Group. 41 Machine 39 " " Mackay & McGeorge in Rivet'ng Machine, Tweddell's Hy- draulic 228 Riveting, Speed of 39 Rivets 39 Robinson's Experiments on Lift Valves. 277 Robertson's Friction Wheels 125 Rock Arms 162 Rock Drill, Githen's 231 Rod Connection, Wiedenbruck's. ... 50 Rod, Friction Ratchet 163 Rods, Connecting 112 Rolled Shafting 94 Roller Bearing, Cambon's 127 Bearings 126 " Bearings for Sheaves 179 Roof Trusses, Force Plans for 36 " Truss, Polygonal 37 " with Simple Principals 36 " with Trussed Principals 36 Root's Blower '.x 221 Roots, E.xtraction of . 26 Rope, Centrifugal Force of Wire. . . . 197 Connections 181 Cotton .^. i7g Cross Section of Wire........ 196 Curve, Construction of 202 Hanger, Osterkamp's 181 Hangers 181 Hemp 178 Influence of Pulley Diameter on 197 Ropes of Organic Fibres 178 Rope Pulleys, Construction of 202 Ropes, Creep of. 202 " Deflection of Wire 198 Flat 178, 181 ' ' Loss from Stiffness 196 Rope, Specific Capacity of Wire. . . . 196 Rope Splice 181 Ropes, Stiffness of 181 " Tightened Driving 200 Ziegler's Experiments on... 181 Rope Transmission 194 at Bellegarde. . . . 205 " at Freiburg 205 " at St. Petersburg 205 " at Schaffhausen. 204 " at Zurich 205 " Cotton 196 Cross Section for Hemp 195 " Efficiency of.... 205 Loss in Hemp. . . 195 " Reuleaux's Sys- tem of 206 " Specific Capacity of Hemp 195 Wire ig6 Weight of Hemp 178 Wire 179 Rotary Pumps 226 Valves 281 Valve, Wilson's 285 Rotative Motors, Adjustable Gears for 237 " Pressure Engines 233 Valve Gears 234 Round Connecting Rods ij6 Round Valves 275 Roux's Water Pressure Engine 229 Rubber Springs 21 Rubber Springs, Werder's Experi- ments 21 Running Chains 182 " Friction Ratchets 158,160 " Mechanism for Lifting Water 221 " Mechanism for Pressure Organs 219 " Ratchet Gears, Toothed. . 150 " Ratchets 150 " Ratchet Trains, Fluid.... 223 " Tension Organs 172 Rupp's Variable Speed Gear 124 Rupture, Modulus of i ;io ALPHABETICAL INDEX. SAFETY, COEFFICIENT OF.. .. i Safety Devices for Elevators 164 Safety, Factor of i Sail Boat 223 St. Petersburg, Rope Transmission at 205 St. Louis Bridge 60 Saint Venant, Friction of Water.... 247 Saladin'.s Friction Pawl 161 Sanderson's Gas Meter 239 San Francisco Cable Tramways. . . 174 Saxby & Farmer, Signal Apparatus. 166 Saw, Zervas' Wire 177 Saws, Band • i77 Scale Beams m SchafEhausen, Rope Transmission at 204 Schiele Turbine 220 Schmick's Canal Cable System 175 Schmid's Water Meter 240 Schmid's Water Pressure Engine . 236 Schurman's Clutch loi Schurman's Friction Coupling 215 Screw, Archimedian 221 " Connections 5S " Propeller 223 Propeller, Lifting Frame for 151 ". Propellers, Built up 57 " Propellers, Method of Keying 49 Screws, Enlarged 58 Screw Thread, Construction of 50 " " Dimensions of V. .. . 50 " " Friction of 51 " " Pressure on 58 " Threads, Special Forms of . .. 57 " Threads, Trapezoidal 58 Section of Gear Teeth 144 Section Modulus 5. 7> n Sections of Uniform Resistance.... 8 Secured Bolts . , 57 Securing Keys, Methods of 5° Segner s Water Wheel 220 Self Guiding Belting 186 Seller's CoupUng 96 " Friction Feed 126 ' ' Hanger 74 Pillow Block 70 '■ Planing Machine 176 Screw Thread System 52 Wall Bearing 71 Sewage System of Berlin 219 Sewing Machine Check 151 Shafting .,..■■ 92 Deflection of • 94 " Dimensions of 92 ■" Examples of Torsion in 94 " Graphical Calculation of - . 94 Journals for 94 Line 93 Rolled 94 " Specific Capacity of 257 " Torsional Deflection of 92 " Wooden 94 " Wrought Iron 93 Shank's Planing Machine 163 Sharp's Coupling 96 Sharp's Strap End 112 Shearing, Resistance to 2, 10 Shearing Strain 2 Sheaves, Chain 185, 211 Sheaves, Roller Bearings for 179 Shield Gearing 133 Shifter for Quarter Twist Belt 189 Shifters, Belt.... • . 188 Shifting Eccentrics 235 Short Span Cable Transmissions. . . . 200 Shrinkage, Hooping by 45 Shrinking Fit 17 Fits, made with Boiling Water 47 " Rings, Clerk's Method. . . 45 " Temperatures 45 Sickles' Adjustable Valve Gear 237 Sickles' Valve Gear 162 Side Wheel Steam Boat 223 Siemens & Halske, Electric Signals. 166 Siemens' Alcohol Meter 239 Siemens' Geyser Pump 222 Signal Apparatus, Saxby & Farmer. 166 Silent Ratchets 153 Simple Crank Axle 106 Simple Escapements 167 Single Acting Steam Engine 229 Single Tooth Gears 165 Sinoide gi Sinoide, Cj'cloidal 13 Siphon, Direct 287 Siphon, Inverted ... .244', 287 Slide Valve, Common 225 " " Gear, Plain 234 " " Lap of 225 -" Murdock's 234 " Valves. . . 223, 273, 2S1, 282 " Valves, Balanced 285 Sliding Brakes 215 Sliding Crank 226 Slipper Cross Head 121 Sluice Gates at Geneva 275 Sluice Valve 2S1 Snail 169 Solid End for Connecting Rod 113 Special Forms of Bearings 74 "of Bolts 55 " " of Ratchet Wheels. - 154 " of Screw Threads. .. 58 Specific Capacity of Belting 190 of Driving Chains. 211 " " of Hemp Rope Transmission.. . 195 " " of Pressure Trans- missions 255 " " of Shafting 257 " " of Wire Rope 196 Speed Gear, Variable 124 Spencer & Inglis Valve Gear 262 Spherical Cycloid 135 " Journal, Connection for. . 115 " Spiral 142 Valves 275 Spiral Bevel Gears 141 Gears 138 " " Double, 141 " " Examples of 140 " Teeth, Friction of 140 Gears, Globoid -. 142 Pump 221 " Spherical 142 ' ' Winding Drums 181 '■ Wire Pipe 252 Spinning Miile 169, 196 Splice for Ropes 181 Split Pin 56 " Pulley, Goodwin's 194 " Pulleys 193 Spring, Dudley's 20 " Pawl 153 " Ratchets 153 Springs, Best Material for 20 " Calculation of 18 Table of 18-19 " Vulcanized Rubber 21 Spur Gear Teeth, Construction of. . 128 Squaring Device for Cranes 172 Square Thread 40 Standing Tension Organs 172 Stand Pipes 2S7 Statical Moment 3 Statical Moment, Graphically Con- sidered 33 Starting Valve 281 Star Pin 153 Stationary Chains 182 " Friction Ratchets 161 " Machine Elements 2S9 " Ratchets 150, 156 " Valves 289 Steam Boat, Side Wheel 223 ' ' Boilers 265 " Distribution of Power 257 Engine, Single Acting Steam 229 " Pipes 245 " Power Distribution 219 " Pump, Blakes 240 " " Deane's 230 .Steam Pump, Pickering's 230 Tangye's 230 Steering Gear 238 Trap, Kirchweger's 228 Steel Pipe 243 Steering Gear 171 " Britton's 238 " " Davis & Co.'s 238 " Douglas & Coulson's. 238 " " Dunning & Bossiere's 238 " Hastie's 238 " " Hydraulic 237 " Steam 238 Steib's Ventilator 222 Step Anchor Ratchet 156 " Bearings '75 " Bearing, Support for , 80 " Bearings, Wall. . . . ^ 75 " Gearing 141 Stephenson's Link Motion 235 Stepped Bevel Gears 141 Step Ratchets 155 Step Valves 276 Stcjvart, Experiments on Springs... 21 Stiffness of Belts 194 of Ropes 181 '* " Eytelwein's For- mula 181 " " Loss from 196 " " Weisbach's For- mula 181 " •' Wire Rope 206 Stolz's Pump 224 Stone's Pump 224 Stop, Geneva 165 Storage Reservoirs, General 273 Strain, Shearing 2 Strains of Flexure ... 3 Strap Brakes 211, 215 Brakes, Internal 215 End for Connecting Rod 112 " End, Sharp's 112 Straps, Eccentric 115 Strength of Cast Iron Columns 83 of Materials 1-2 1 of Wire Rope 179 " Tensile i Stress Curve 87 Stresses, Compound 13 " in Columns , 82 ' ' on Keys 48 Stress on Belting 191 on Gear Teeth 145 ' ' on Journals 61 " S, Value of 8 Striking Mechanism for Clocks 169 Stub End, for Fork Journal 114 Stuffing Boxes 253 Box, Farcot's 254 " Box, Friction in 254 Sturtevant's Hanger Boxes 74 Pillow Block 70 Pulley 194 Superficial Pressure • i Superheated Water Transmission , . . 219 Supporting Pulleys, Vertical 188 Supports for Bearings 79 " " General Prin- ciples 82 " Simple 79 Supporting Power of Beams 5 Swedish Piston .... 253 Swedish Railway, Boilers for 272 Sweet's Valve Gear 235 Swivels 182 Swivels for Chain 184 Symmetrical Simple Axles S3 TABLE OF BEAM SECTIONS. 5, 6, 7 Tables of Curves, Areas and Vol- umes 291-296 Table of Numbers 300-301 Tackle Block 172 Tangential Pressure on Crank Pin . . 233 Tangye's Crane 176 Tangye's Steam Pump 230 ALPHABETICAL INDEX. 311 Tanks, Cast Iron 260 " Combination Forms for 264 " Intze's Discussion of 260-264 " Oil 21S " With Concave Bottoms 262 " Wrought Iron 260 Taper of Keys 47 Technological Applications of Pres- sure Organs 240 Technological Applications of Ten- sion Organs i77 Tenacity i Tensile Strength 1 Tension Organs 172 " for Driving 173 " " for Guiding 172 " " for Winding 172 " Ratchet 185 " " Running 172 " " Technological Appli- cations of 177 ' ' Resistance to 2 Tests for Chain . 1S3 T Fittings 251 Theoretical Resistance i Thickness of Cast Iron Pipes 242 Thick Vessels, Walls of 16 Thomas' Calculating Machine. . . 153, 146 Thometzek's Valve 276 Thomson's Turbine. 220 Three-part Bearings 70 Throttle Valves 161 Throttle Vaves 279 Thrust Bearing by James Watt & Co. 77 " by Maudslay 77 " by Penn 77 " " by Ravenhill & Hodgson 77 " " Compound Link as 67 " Bearings 65, 68, 75 Collar 66 " " Examples of 78 Multiple Collar 66 '■ " with Wooden Sur- face* 76 " upon the Pawl 152 Thumb Shaped Pawl 160 Thumb Shaped Teeth 134 Tiede's Escapement 168 Tightened Cables, Table for 200 Tightened Driving Ropes 200 Tightening Pulleys 186 Tightening Pulley, Weaver's 186 Toggle Friction Brake 162 Tools, Hydraulic 218 Toothed Gearing 127 Tooth Friction in Spur Gearing. . . . 134 " Outlines, General Solution of 129 " " Mixed . 133 of Circular Arcs... 131 Torpedo, Fish 171 Torsional Deflection of Shafting... 92 Torsion, Determination of Angle of 93 Resistance to 11 Table 12 '• Uniform Resistance to. .. . 13 Towne Crane 176 Transformer, Hydraulic 218 Tranforming Arm Sections, Table for 103 Transmission at Long Distance, Fluid 233 Chain 211 Gears 144 " Long Distance Power. 259 " Modulus of 208 Rope 1 94 With Inclined Cable. . . 200 Transportation, Fluid Escapement ment for 227 Transporting Belts 221 Trapezoidal Screw Threads 51, 58 Trap, Water 287 Triangles, Area of 23 Trick's Valvs 2S4 Trigger, Hair 153 Trigonometrical Formulee 299 Trgionometrical Functions, Powers of 25 " Table ... .297-299 Trussed Beams, Double 35 " " Simple 35 Triple 35 Tubing, Levasseur's Metallic 252 Tumbling Gears 163 Turbine, Borda's : . 220 " Cadiat 220 " Fourneyron 220 Francis 220 ' ' Girard 220 " Nagel 220 Schiele : . 220 " Thomson's 220 Tweddell's Accumulator 265 Tweddell's Hydraulic Riveter 228 Twin Link 184 Twisting Moments, Graphically Con- sidered 33 Tympanon of Archimedes 221 UHLHORN'S COUPLING. . . .101, 153 Umbrella Pulley 207 Uniform Escapements 167 Uniformly Distributed Forces 32 Uniform Resistance, Columns of . . . 13 " " Sections of . . . . 8 to Bending. ... 8 " " to Torsion. ... 13 Strength, Bodies of 2 Universal Gears, Beylich's. 136 Universal Joint 97 Unloaded Bolt Connections 60 Unloaded Keys 49 Unperiodic Power Escapements for Pressure organs 227 VALVE, ALLAN'S DOUBLE.... 2S3 Armstrong's Supported.. 286 Bell ■ .. 276 Boulton & Watt's Bal- anced 285 " Brandau's Double Seated. 286 " Cave's Balanced 285 " Corliss 236 " Cornish 280 Cramer's Balanced 2S0 Cuvelier's Underpressure 2S6 " Double Beat 280 " D 283 " " Flap. 274 " Diagram, Reuleaux's .... . 234 " Zeuner's. 234 " Gear, Angstrom's 235 Brown's 235 •■ Call & Co 162 Cam 236 " " Corliss 162 " " Cornish 163 " " for Pumps 225 " Harlow's 231 " " Hofmann's 163 " Klug's 235 " " Marshall's 235 " " Plain Slide 234 Powel 163 " " Rider's 236 " Gears for Rotative En- gines 234 " " Reciprocating 234 " " Releasing 162 Rotative 234 " Gear, Sweet's 235 Gear, Wannich 162 Globe 279 " Gridiron 283 Hick's Double 283 Injector 279 Kirchweger's Balanced .. . 2S5 Lindner's Balanced 285 Plain Slide 225, 282 Porter- Allen 2S7 Rubber Disk 274 Valves 279 Valves, Balanced , 279 Valves, Balanced Slide 285 Valve, Schaltenbrand's Double Seat- ed. Valves, 286 Check 274 Closing Pressure of 278 Conical 275 Considered as Pawls. . . .223, 273 Flap 274 Flat Disk 275 Fluid 287 Gate 2S2 Gidding's Experiments on . . 285 Lift ... 223, 273 Mechanically Actuated 27S Multiple 276 Piston 286 Resistance of 247 Robinson's Experiments on. 277 Rotary 2S1 Round 275 Slide 225, 273, 281 Spherical 275 Valve, Starting 282 Valves, Stationary 289 Step 276 Throttle 279 Unbalanced Pressure on Lift 277 Valve, Sweet's Balanced 287 Valves, Width of Seat 274 Valve, Thometzek's 276 Trick's 274 " Wilson's Balanced 287 " Wilson's Rotary 286 Value of Stress S 8 Vacuum Power Distribution 219 Variable Speed Gear 124 Variable Speed Gear, Rupp's 124 Velocity Curves 233 Ventilator, Fabry's 221 Lemielle's 82 " Steib's 222 Verge Escapement 168 Volume, Escapements for Measure- ment of 239 Von Gerike's Air Pump 225 Vose's Pump 224 V Screw Thread 50 WALKER MFG. CO., PULLEY by Walking Beams Wall Bearings 68, " Bearing, Support for ' ■ Bearing, Sellers' ' ' Brackets Walls of Vessels, Resistance of Wall Step Bearings Wannich Valve Gear Washers Watches, Repeating Water Counterbalance, Oeking's . . . " Meter, Jopling's " " Kennedy's " " Payton's " " Schmid's ■ ' Pressure Engine, Belidor's . . " " " Re ic hen- bach's. . " " " Roux's. . . . " " " Schmid's.. " Reservoir, of Frankfurt on Main Rod Connection ■. . . Running Mechanism for Lift- ing " Trap " Trap, Morrison, Ingram & Co. Wheel, Poncelet's " Wheel, Resultant of Load on Wheels. Axles for Wheel, Segner's Wheels, Gravity " Wheels, Impact. Watt's Condenser Wear on Gear Teeth 194 no 71 79 71 72 15 75 162 54 169 217 239 239 220 240 229 229 229 236 218 233 287 288 220 34 91 220 2ig 220 230 134 312 ALPHABETICAL INDEX. Wear on Hemp Rope i . • 196 Weaver's Tightening Pulley 1S6 Wedge Friction Wheels 125, 160 Weighing Machine, Emery's I73 Weight of Cast Iron Pipe 242 of Chain 183 of Gear Wheels 150 " of Hemp Rope 17S " of Round Iron 55 Sheet Metal 43 of Wire Rope 180 Weir, Camere's 275 Weisbach, Formula for Friction of Water 246 Weisbach's Formula for Stiffness of Ropes ■ 181 Werder, Experiments on Springs. . . 21 Weston's Differential Pulley Block. 173 Friction Clutch loi Ratchet Brace I54 Wet Gas Meter 239 Wheels, Classification of 122 Whip Action of Connecting Rod. . . 116 Whitehead Torpedo 237 Whitworth's Screw System 51 Whitworth's Pipe Thread Scale.... 51 Wiedenbruck's Rod Connection.... 50 Wilber's Ratchet 153 Wilson's Rotary Valve 286 Wilson's Water Gas Furnace 288 Winding Drum, Reuleaux's 173 " Drums, Spiral 181 ■" Tension Organs for 172 Windlass 172 " Brown's 173 Differential 173 Windmills 220 Wind Stresses, Graphically Deter- mined 37 Wire Rope 179 '• Influence of Weight. .. . 180 " Load Length of 180 " " Strength of 1 79 " Transmission 196 " Weight of iSc " Saw, Zervas' 177 Wooden Axles, Proportions of 92 Wooden Shafting • 94 Worm and Worm Wheel 135 Gearing, Globoid 143 Hawkins 1^3 Jensen's 143 Worthington High Duty Pumping Engine 232 Worthington's Duplex Pump 231 Worthington's Equalizer 232 Wrapping Connections 173 Wrenches 56 Wrought Iron Cranks, Single 104 Pipe 243 " " Shafting 93 Walking Beams iii YALE LOCK 167 Yoke Bearings 72 ZERVAS' WIRE SAW 177 Zeuner's Valve Diagram 234 Zimmermann's Belt Shifter . . . 189 Zuppinger's Water Wheel 219 Zurich, Rope Transmission at 205 .\^' 1 ^- (} ,- ,A -^ ^\ ^O' =S; nO°-. -bo ^^/'o , .-^^ ^0 "./>, ° ^,^' /f^^^ .%^"' ,0^ A V s: -^^ ■ «-^^ A" ■\ o '^oo-^ .\^' l.\-' >^ » « I 1 ' 'O, ' , I * jO <, k * ^>-^ .^^'..^ U- V 'OO^ >- v,^' o ,^ N°°<. ,0- z -F ,-.^^ % ^-^ ,^ A <-:i y„. , -i r-,^ ^ '/, . ^ .A a ' ^ >^ ^v\' ,'\'/ * . -o OO \ > -■(•0, '> -1^' -'\ '// ^- V * ' " " ,'- ^^- /--^cv- ,0- "^.Z.*.o.o' v^^ V ■>- ^-^ A- -?=•/_ V QO. \ Oo^ .O' ^^.-i"^ ^-U „> „ N r. . <-, ^ « « ^ ^s^\ .vie- ' ,"• ' ° ', "">°'^^:^'?%*^''"^^ ^^^^^'^%,# : ii'^^ »^ ,0- ^ .c ■iijni-ijo 021 225 270