ELEMENTS OF MACHINE CONSTRUCTION AND DRAWING: OR, MACHINE DRAWING, WITH SOME ELEMENTS OF DESCRIPTIVE AND RATIONAL\ CINEMATICS A Text-book for Schools of Oivil and Mechanical Engineering, and for the use of Mechanical Establishments, Artisans, and Inventors. CONTAINING THE PRINCIPLES OF GEARING; SCREW PROPELLERS; VALVE MOTIONS AND GOVERNORS; AND MANY STANDARD AND NOVEL EXAMPLES, MOWLTY PROM PRESENT AMERICAN PRACTICE. BY S. EDWARD WARREN, C.E., PROIsOR 1N THE RENSSELAER POLYTECHNIC INSTITUTE, AND AUTHOR OF A SERIES OF WORKS ON DESCRIPTIVE GEOMETRY AND STEREOTOMY. NEW YORK: JOHN WILEY & SON, PUBLISHEPRS 15 ASTOR PLACE. 1872. Entered according to Act of Congress, in the year 1870, by S. EDWARD WARREN, C.E., in the Office of the Librarian of Congress at Washington. CONTENTS. PAGE PREFACE................. SoURCES OF MATERIALS, ETC....................................... xvi ELEMENTS OF MACHINE CONSTRUCTION AND DRAWIN G............................................................. 1 BOOK I. SIMPLE, OR SINGLE ELEMENTS OF MACHINES. PART I. Introduction. General principles............................................. 1 Scales....................................................... 2 Elements of projections....................................... 4 Constructions of the ellipse.................................... 10 Special definitions.............................................. 12 Classification of Machines.................................... 13 Functional classification of mechanical organs.................... 15 Geometrical classification of mechanical organs.................... 19 Reduction of scales............................................ 20 PART II. Theorems, Problems, and Examples on Elements of Machines. CLASS I.-SUPPORTERS. SECTION I. -LOCAL SUPPORTERS A-Point Supporters. Pillow Blocks. EXAMPLE I.-A heavy pillow block.................................. 22 EXAMPLE II. —A Putnam pillow block:.............................. 24 47. Heavy eins. EXAMPLE III.-A French pillow block............................... 25 iv CONTENTS. PAGE EXAMPLE IV.-A locomotive main axle box.......................... 26 48. Shaft hangers. EXAMPLE V. -A bracket hanger..................................... 28 EXAMPLE VI. —A self-oiling drop hanger............................ 29 EXAMPLE VII.-Turbine and spindle footsteps....................... 30 49. Cold rolled shafting. B —Line Supporters. EXAMPLE VIII. —Locomotive guide bars and cross head................ 81 50 —Progressive forms of cross heads. C-Surface Supporters. a-.Plane Supporters. b-Developable Supporters. EXAMPLE IX.-A local bed plate.................................... 84 D —Volume Supporters. EXAMPLE X. —A locomotive cylinder................................ 84 EXAMPLE XI. —A jet-condenser..................................... 36 EXAMPLE XII.-A surface-condenser................................ 88 SECTION II.-GENERAL SUPPORTERS. A —Point Supporters. B-Line Supporters. 53. Standards. EXAMPLE XIII. —The standard of a power hammer................... 89 54. Comaparative examples. C-Surface Supporters. a-Plane Supporters. 55. Prames. EXAMPLE XIV.-Locomotive frames................................ 45 b-Developable Supporters. 56. Beds. EXAMPLE XV. -A prismatic beam-bed and pedestal................. 48 D-Volume Supporters. EXAMPLE XVI.-.-A tank bed-plate................ 49 ErXAMPLE XVII. —Housing, or chambered frame, for a reversible rolling mill engine.................................................... 51 CONTENTS. V PAGE EXAMPLE XVIII. —Housing for a rolling mill........................ 54 EXAMPLE XIX.-A passenger car truck. Practical remarks........... 56 CLASS II.-RECEIVERS. A-Point Receivers. B-Line Receivers. C-Surface Receivers. a-Plane Receivers. EXAMPLE XX.-Locomotive piston; with Roth's steam piston packing.. 63 EXAMPLE XXI.-Thirty-sir, and fifty-four inch pistons................ 67 b-Developable Receivers. EXAMPLE XXII. —A Fourneyron wheel plan.......................... 68 c-Warped Receivers. EXAMPLE XXIII.-A Jonval turbine wheel and bucket................. 71 d-Double Curved Receivers. EXAMPLE XXIV.-The Swain central discharge wheel................ 75 CLASS III.-COMMUNICATORS. A-Point Communicators. EXAMPLE XXV.-Collins' shaft coupling............................ 79 B-Line Communicators. Band, Card, and Chain, Wheels. THEOREM I.-A rotary motion of two parallel axes may be maintained indefinitely, and in one and the same direction for both, by a band passing directly around cylindrical pulleys in the same plane, on those axes; but, if the band be crossed, the rotations will be in opposite directions; but, in both cases, the ratio of the velocities will be constant........................................... 81 THEOREM II. —A band should be crossed by giving it a half twist, in a plane perpendicular to that of the wheels which hold it: it should be shifted, laterally, by operating on its advancing side, and if applied to a cone-wheel, will work itself towards the larger end of the cone............................................... 82 PROBLEM I.-To connect wheels lying in different planes, by a band; when the intersection of their planes is also a common tangent to the two wheels................................... 84 PROBLEM II.-To connect band wheels, in different planes, when the intersection of those planes is not a common tangent to the wheels................................................. 86 VI CONTENTS. PAGE Notes on band wheels............................................... 87 Transmission by ropes, cords, etc.................................... 91 Chains................................................:......... 92 I'fiexible linear communicators..................................... 93 EXAMPLE XXVI. —A locomotive parallel, and main connection......... 93 THEOREM III. —The effective power of a locomotive, taken at the axle, is the same, whether the crank pin is above or below the axle.... 95 THEOREM IV. —The pressure between the axle and the front axle box, of an engine going forward, is double that between the axle and the back axle box........................................... 96 THEOREM V. —The piston, etc., move in space, faster than the engine, going forward, in the forward stroke; and slower. in the backward stroke...................................................... 97 THEOREM VI. — The crank-pin has an accelerated motion in space, from its lowest to its highest point, and a retarded one from its highest to its lowest point, etc................................... 98 THEOREM VII. —The wear of the two crank-pini-boxes is equal..... 99 THEOREM VIII.-With a given boiler capacity, and size of cylinder, the larger the driving wheel, the greater the adaptation to a light load at a high speed; and, conversely, etc........................... 100 EXAMPLE XXVII.-A working beam................................. 104 EXAMPLE XXVIII. -A Stephenson link............................... 105 C-Surface Communicators. a-Plane Communicators. EXAMPLE XXIX. —A circular eccentric, strap and rod................. 107 EXAMPLE XXX. —A heart cam, or eccentric........................... 109 b-Developable Communicators. Gearing. Two classifications.......................110....... 110 THEOREM IX.-The number of revolutions in a given time, and the, angular velocities of toothed wheels, are inversely as their radii... 112 Definitions and Principles (86-93)................................ 113 PROBLEM III.-To construct a cycloid. Two constructions......... 116 PROBLEM IV. —To construct an exterior epicycloid........ 1........ 117 PROBLEM V.-To construct an interior epicycloid.................. 117 PROBLEM VI.-To construct any hypocycloid...................... 118 PROBLEM VII.-To construct the involute of a circle............... 118 THEOREM X.-In all the curves just described; the tangent. at any point is perpendicular to the line from that point to the corresponding point of contact of the rolling and fixed lines.................. 119 THEOREM XI. —The relative position of two circles is the same, whether one rolls over a certain arc of the other, which is fixed; or both revolve on fixed centres till they have had the same amount of contact as before............................................ 119 THEOREM XII.-The relative position of three circles, which maintain a common point of contact, is the same, whether one of them is fixed, or all revolve on their centres........................ 120 CONTENTS. Vii PAGE THEOREM XIII. —In the rolling of three circles, with a common point of contact, any point of the inner circle will describe an epicycloid upon the circle on which it rolls, and a hypocycloid within the remaining circle. These curves will be the proper curves for teeth acting tangent to each other, to give a rolling motion to the circles to which they belong.................................. 121 THEOREM XIV. —When any circle, less than either of two given pitch circles rolls on the exterior of both, and on the interior of both, it will, in the former case, generate the faces of the teeth of both wheels, and in the latter their flanks.......................... 121 THEOREM XV. —Involutes are proper curves for the teeth of wheels.... 122 TIIEOREiM XVI. —The teeth act by sliding contact, and their point of contact is on the- generating circle............................. 123 THEOREM XVII. —Teeth formed by the preceding methods, give a constan#ngular velocity ratio to the wheels which carry them...... 123 THEOREM XVIII. -Within certain limits, the face of a driver acts best on the flank of a follower, and during the arc of recession; but, for either of a pair of wheels to be a driver, the teeth of each must have both faces and flanks.............................. 124 T. —Gener a Solution................................................. 127 1.-Common spur wheels..................................... 127 2. —Spur and annular wheels................................... 127 3.-Spur wheel and rack....................................... 127 II.-The describing circle equal to half the pitch circle.................... 127 1. —Spur wheels.............................................. 127 THEOREM XIX. —Any two wheels of the same pitch, formed by the general solution, with a constant describing circle, will work together; but one made by the second solution will work perfectly only with those of one other number of teeth, and the same pitch......... 128 2.-Spur wheel and rack....................................... 128 III.-The describing circle equal to one of the pitch circles............ 129 1. —Pin wheel and spur wheel................................. 129 2. —Pin wheel and rack.................................. 130 3. —Annular pin wheels....................................... 130 IV.-Solution by involute teeth....................................... 131 1.-Spur wheels.............................................. 131 2. —Spur wheel and rack............................ 131 EXAMPLE XXXI.-To construct the projections of a spur wheel, seen first perpendicularly and then obliquely.............................. 132 EIXAMPLE XXXII. —To construct the projections of a bevel wheel, whose axis is perpendicular to the vertical plane........................ 138 EXAMPLE XXXIII.-To construct the projections of a bevel wheel, seen obliquely relative to the vertical plane.......................... 140 Use of only three projections (101.................................... 142 Practical forms of the teeth of wheels.................................... 143 THEOREM XX.-Circular tooth curves, with centres on a line through the point of contact of the pitch circles, will give a sensibly constant velocity ratio to those circles........................... 145 Vlll CONTFNTS. PAGR PROBI,EM VIII. -To find the radii of the tooth curves.............. 147 PROBLEM IX. —To find centres for approximate involute teeth...... 148 EXAMPLE XXXIV.-To construct teeth having separate faces and flanks, by the odontograph............................................ 149 EXAMPLE XXXV. —To construct approximate involute teeth by the odontograph................1.............................. 152 EXAMPLE XXXVI.-Projections of bevel gearing.................. 153 c-Warped Communicators. EXAMPLE XXXVII. — The complete projections of a screw and nut...... 154 EXAMPLE XXXVIII. —The abridged drawing of screws................ 157 Uni/form System of Screws.......................................... 1t58 EXAMPLE XXXIX. —Endless screws and spiral gear................... 160 EXAMPLE XL.-Detailed construction of a tooth in spiral geatng...... 163 113. Mranufaceture of worm wheels......................... 164 CLASS IV.-REGULATORS. A-Point Regulators. B-Line Regulators. EXAMPLE XLI. —A fly wheel....................................... 165 C-Surface Regulators. Plane throttle valves. Single poppet valves. Cage valves. Cylindrical throttle valves. Ball valves.................................. 167 D-Volume Regulators. Cocks. Globe valves. Water gates.................................. 167 EXAMPLE XLII. —Chambered, or D locomotive slide valves; plain, and anti-friction................................................ 169 EXAMPLE XLIII. —Tremain's balanced piston valve.................... 171 EXAMPLE XLIV. —Balanced poppet valves. Data from racticee........ 175 123. Exa(mples of engine actimn................... 179 EXAMPLE XLV. —Richardson's locomotive and lock-up safety valve...... 180 EXAMPLE XLVI. -A double beat pump valve......................... 183 EXAMPLE XLVII.-A Cornish equilibrium valve....................... 185 EXAMPLE XLVIII. —Giffard's injector............................... 185 CLASS V.-MODULATORS. A-Point Modulators. Idler pulleys.................................. 190 B-Line Modulators. Escapements. Band shifters. Clutches. Etc.......................... 190 CONTENTS. ix CO-Surface Modulators. a-Plane Modulators. PAuE Variable crank..................................................... 191 b-Developable Modulators. Speed pulleys...................................................... 192 THEOREM XXI.-If the band be crossed, it will be equally tight on every pair of opposite pulleys................................. 192 PROBLEM X.-To form a set of speed pulleys, to give a series of velocity ratios in geometrical progression.......................... 193 Cone pulleys. Dead Pulleys. Sectoral motions. Elliptic gears.......... 195 c —Warped Modulators. The helicoidal clutch................................................ 197 d-Double-curved Modulators. Double-curved speed pulleys....................................... 198 CLASS VI.-OPERATORS. A-Point Operators. EXAIIP~LE XLIX.-Movable saw teeth................................ 199 EXAMPLE L. —Lyall's positive motion shuttle......................... 203 B —Line Operators. Cutters (143)........................................................ 209 0-Surface Operators. a-Plane Operators. EXAMPLE LI. —Air pump bucket of a marine engine................. 209 b-Developable Operators. c-Warped Operators. THE SCREW PROPELLER. Preliminary remarks................................................... 21 Introductory geometrical principles........ 211: The helix and helicoid..212 The helix d helicoid................................................ 212 Slip..... 217 Lateral slip.......................................................... 217 Negative slip....................................... 218 Irregular screws..................................................... 220 PROBLEM XI.-To construct a helix of uniform pitch and radius.... 221 PROBLEMl XII.-To construct the projections of the common; right helicoid, generated by the radius of a vertical cylinder..........223 gX.(CONTENTS. PAGE PROBLEM XIII. — To construct the projections of a common right helicoid, which is generated by the diameter of a vertical cylinder....................................................... 223 PROBLEM. XIV.-Having given either projection of any element of a helicoid, to find its other projection.......................... 224 PROBLEM XV.-To represent a common right helicoid by its helical lines..........2......................................... 224 PROBLEM XVI. —To construct the lines of a helicoid, made by its intersection with any plane parallel to its axis..........2........ 225 PROBLEM XVII. —Having given either projection of any point upon a helicoid, to find its other projection................... 226 PROBLEM XVIII.-To develope one or more given helices........... 226 PROBLEM XIX. —From the circular projection and development of a helix, to construct its spiral projection..................... 227 PROBLEM XX.-To construct a helicoid of axial expanding pitch, by means of its helical lines................................ 228 PROBLEM XXI. -To develope the four helices last drawn..... 228 PROBLEM XXII.-To make the projections of a helicoid of radially expanding pitch..................................... 229 PROBLEM XXIII. —To develope the helices shown in the last problem. 230 PROBLEM XXIV. —To construct the projections of the acting faces of a four-bladed common screw propeller................... 231 EXAMPLE LII.-To represent variously limited propeller blades, with their concentric and radial sections............................... 232 Ideas expressed in modified forms of Screws....................... 235 Historical note................................................ 236 EXAMPLE LIII. —The screw of the " Dunderberg.".................. 238 D-Volume Operators. EXAMPLE LIV.-Andrews' centrifugal pump.....p................... 240 BOOK II. COMPOUND ELEMENTS, OR SUB-MACHINES. SUPPORTERS. EXAMPLE LV. —A compound chuck................................. 245 COMMUNICATORS. EXAMPLE LVI. —A beam-engine main movement................... 249 EXAMPLE LVII. —Wheeler's tumbling-beam engine.................... 250 EXAMPLE LVIII.-An eight day clock train......................... 252 Other trains...................................................... 252 Change wheels........................................, 25( THfE SLIDE VALVE AND ITS CONNECTIONS. General description of parts........................................ 257 General action.................................................. 259 Modifications and adjustments.................................... 260 -CONTENTS. xi PAGE Definitions....................................................... 261 THEOREM XXII.-In either mode of connection the velocity of the crank pin is uniform, and that of the piston is variable.......... 263 THEOREM XXIII. — The piston positions, corresponding to crank pin positions, which are equidistant from the same dead-point are identical, for each connection separately............................ 264 THEOREM XXIV.-The segments of the double stroke are equal, in the direct connection, and the front one is the greater in the indirect connection. Conversely, etc.......................... 264 Natural zero points of the piston and crank-pin motions, and segments of the double stroke........................................... 265 THEOREon XXV.-The crank piston is ahead of the yoke piston during the stroke toward the shaft, and behind it during the opposite stroke...................................................... 266 Cut- Off...................................................... 267 THEOREM XXVI -The effect of a given angular advance of the eccentric will be to afford " admission " for a new stroke, " cut-off," "exhaust closure," and release, all at an equal number of degrees before reaching a dead-point................................. 267 THEOREM XXVII. —The effect of a given lap, alone, corresponding to an equal number of degrees from the zero diameter, is, to postpone admission for an equal number of degrees beyond the deadpoint; to produce cut-off at the same number of degrees beyond the dead-point; with release and exhaust closure at the dead-point. 268 PROBLEM XXV. —To produce a cut-off at a given crank-pin position, without preventing proper admission, etc.................. 269 PROBLEM XXVI. —To determine the exhaust closure and release, for the adjusted cut-off and admission.2...................... 270 THEOREM XXVIII. —The travel of a valve, with lap, is the sum of twice the lap, added to twice the steam port opening. 270 THEOREM XXIX.-Inside lap prolongs expansion, and hastens compression; while inside clearance hastens the release, and postpones the beginning of compr ession.............271 PROBLEM XXVII. —To determine the effect of the eccentric upon the valve motion and to counteract it in part..................... 272 Distribution of power............................ 273 Lead........................................................... 273 PROBLEM XXVIII. —To provide a certain lead angle, without disturbance of the cut-off........................................ 274 PROBLEM XXIX. -To determine the effect of lead on exhaust closure, release and travel......................................... 274 THEOREM XXX.-The angular advance, estimated from the zero radius hitherto taken, is equal to the sum of the lap and lead angles, estimated from the same point............................... 275 THEOREM XXXI.-When the steam port is open by the amount of the lead, the exhaust opposite port is open for exhaust by the amount of the lap and lead.......................................... 275 Port opening................................................ 276 Summary of elements........................................ 278 ~Xl~~i ~CONTENTS. PAGB PROBLEM XXX.-To reverse the motion of an engine. Drop hook... 280 EXAMPLE LIX.-A. Stephenson link motion.................283...... 283 To find one position of the link.................................. 285 Data for finding any position of the link.........................( To adjust the model............................ 287 Remarks and results.......................................... 248 EXAMPLE LX. -Data of valve motions............................... 290 I.-Of a 15" x 22" cylinder..290............................ 290 II.-Of a 16" x 24" cylinder.......................... 291 III. —Of an 18" x 22" cylinder................................. 291 Experimental determinations................................... 292 Setting the valve motion of a locomotive.......................... 294 REGULATORS. Governors. Elementa/ry Principles....................................... 297 EXAMPLE LXI. —Chubbuck's fan throttle governor.................... 300 EXAMPLE LXII. —The Huntoon oil throttle governor.................. 302 EXAMPLE LXIII.-Wright's variable cut-off by the governor........... 304 EXAMPLE LXIV.-Babcock and Wilcox governor and variable cut-off... 307 Indicator diagrams........................................ 314 EXAMPLE LXV.-The Putnam Machine Co.'s variable cut-off........... 320 EXAMPLE LXVI.-The Rider cut-off................................. 321 EXAMPLE LXVII.-Sibley and Walsh's water-wheel governor.......... 322 MODULATORS. EXAMPLE LXVIII. —Compound speed and feed motions........... 325 EXAMPLE LXIX. —Whitworth's quick-return motion.................. 327 EXAMPLE LXX.-Mason's friction pulleys and couplings, or clutches.... 328 EXAMPLE LXXI. —Reversing gear for the compound Rolling-Mill Engine........................................ 332 EXAMPLE LXXII. —Bond's escapement, No. 2....................... 334 EXAMPLE LXXIII.-Bond's auxiliary pendulum gravity escapement.... 837 PREFACE. THIS book may be compared to an excursion train. Everything mechanical has called, or striven for a place in it, if only to cling to the platform of briefest mention. Yet not a tithe even of the beauties of mechanism have been admitted, for want of room. Indeed, one of the most arduous labors connected with the composition of this work has been to keep out the nearly irrepressible crowd of topics and examples that were pressing into it. Those that have been admitted have been selected with great care, after personal inspection in machine shops, and from valuable circulars, correspondence, and published authorities. Other examples have been partially represented by woodcuts and brief notices, or have necessarily been excluded, and remanded back to the great world of mechanism. In either case the governing idea has been, to develope the comprehensive scheme of the General Table proportionately, though briefly, with examples that should be American (mostly), new, and good. There are, at least six topics in this work, about which the troublesome problem has been to touch them at all, unless superficially, without devoting a volume to each. These are, Tu'rbines, Gearing, Propellers, Valve moti6ns; Governors, with or without Variable Cutoffs; and Trains of gearing, as clock trains. Of turbines, I have only taken one of each of the three essentially different kinds, with a simple description of its construction and action. On gearing, I have been reasonably full, giving the substance of what need be known in behalf of proper practice, and with simple explanations. As to propellers, their theory is so intricate, owing to the variety and indefiniteness of the data, for calculations concerning them; and Xiv PREFACE. numberless experimental results are so full and accessible in Bourne and in similar works, that I have mostly interested myself in giving exact instruction, nowhere else accessible so far as I know, about making their projections; so as indirectly to correct grievous errors, and supply deficiencies, which have been found in print on this subject. With the ample geometrical treatment of valve motions by Mr. Auchincloss,* and the masterly analytical work of Prof. Zeuner, to supplement the little that I have found room for on the same topic, I have had a narrowly limited and definite object in view in what I have had to say on that subject. The treatment of valve motions in the encyclopmedias and the extended serial works, like Colburn's Locomotive Engineering, is generally unavailable. The works of Auchincloss and Zeuner, presuppose, expressly or impliedly, a good deal of familiarity with the subject. But many persons are wholly unfamiliar with it, and, unless apt to conceive readily of combined motions, find it a puzzling subject. My work has therefore been little more than to begin at the very beginning, and virtually to prepare for the beginner an introduction to those works. More, indeed, could not be attempted in a volume in which so many other topics have been introduced. Governors, a plaything of American ingenuity, have been summarily, but with the most instructive variety attainable within small limits, passed over with the selection of the most marked varieties of governor and valve. Trains of gearing, though very briefly noticed, have, it is hoped, been so treated as to afford some clear and accurate ideas on that subject, as a foundation for further study. The classified table of machines has been prepared with great care, and compared witlf that in the Encyclopaedia Britannica by Prof. Rankine, which is quite different, without material alteration in the result. I have endeavored, in the paragraphs immediately preceding the table, so to distinguish machines from instruments, as to rationally exclude engineering, astronomical and musical instruments, from the province of machines, in which he includes them; contrary to those common usages of speech, which I believe will be found upon analysis to be grounded on real differences. Graduate of the R. P. I., 1862. PREFACE. XV Still, the number and uses of machines are so endless, that I cannot profess to have found a strictly scientific, and therefore exhaustive classification of them. A word now as to the intended use of this book in the class-room may be considered seasonable. Previous to its appearance, the subject of it was taught orally, and with no small labor, to classes, which, in their turn, could progress neither so rapidly nor pleasantly as if provided with a text-book. The present volume is naturally much fuller than an oral course could well be, and is intended as a text-book upon which daily interrogations and black-board exercises are to be held, as well as a manual, to be constantly open before the student for a guide in the preparation of his drawings. With the plates of uniform size for each student, so that they can be agreeably bound together, but with a choice as to that size, from quarter super-royal to semi-super-royal, until the best size can be experinlentally determined, it may be well, wherever practicable, to require that one of them should be constructed from actual measurements, made by the student, and accompanied by a plate containing an inked copy of the sketches and measurements. Some of the plates should have the measurements recorded substantially in the style of office practice, and they should generally be titled, in addition to the general title-page of the collection. Or there should be a separate plate containing the several titles. The "heavy lines" are omitted, or displaced in some plates, as an exercise for the student in supplying or correcting them. Finally, the following lists will show to what helping friends I am indebted, and what sources of information I have diligently consulted. Also the signatures of student draftsmen, of the classes of'70 and'71, R. P. I., on many of the plates will always happily remind me how kindly my labors in that direction were lightened. TROY: November, 1870. Xvi PREFACE. ESTABLISHMENTS VISITED OR DRAWN UPON FOR MATERIALS USED IN THIS WORK. American Saw Co........................ Trenton, N. J. Andrews Bros....................... New York. Atlantic Works........................... Boston. Babcock and Wilcox Eng. Works............. New York. Bement and Dougherty.................. O... Philadelphia. Bessemer Steel Works....................... Troy, N. Y. Boston and Albany R. R. Shops.............. Boston. Boston, Hartford and Erie R. R............ Bond's Chronometer Rooms.................. Brooklyn Water Works..................... Brooklyn. Brown's Machine Works................... Troy, N. Y. Bullard and Parsons......................... Hartford, Conn. Cambridge Machine Works........... Cambridge, N. Y. Chubbuck & Sons.......................... Boston. Collins' Turbine Works...................... Norwich, Conn. Delamater Iron Works................... New York. Gurley W. & L. E.......................... Troy, N. Y. Harmony Mill............................. Cohoes, N. Y. Hopedale Mach. and Furnace Co.............. Hopedale, Mass. Hinckley & Williams' Loc. Works Boston. Horton E. Machine Works.................. Hartford, Conn. Hotchkiss, Power Hammers.................. New York. Huntoon Governor Co....................... Boston. Jones and Laughlin......................... Pittsburgh, Pa. Judson Governor Works.................R.... Rochester, N. Y. Lowell Machine Shop....................... Lowell, Mass. Ludlow Valve Co........................... Troy, N. Y. Lyall Positive Motion Loom Co............... New York. Mason V. W. Friction Clutches, etc............ Providence, R. I. McMurtrie & Co., Machine Agency........... Boston. Milwaukee & St. Paul R. R., E. M. Hall, Supt. Power................................ Milwaukee, Wis. Morgan Iron Works................New York. PREFACE. XVii New York Central R. R. Shops Albany, N. Y. Novelty Iron Works......................... New York. Pennsylvania R. R. Car Shops................ Altoona, Pa. Putnam Machine Works..................... Fitchburg, Mass. Rensselaer Iron Works...................... Troy, N. Y. Ruggles' Machine Works.................... Poultney, Vt. Sault M. & T. Co........................... New Haven, Conn. Sellers, Wmin. & Co.......................... Philadelphia. Shaw & Justice, Hammers................... " Starbuck Bros., Engineers................... Troy, N. Y. Steere, E. N., Cotton Machinery.............. Providence, R. I. Swain Turbine Co........................... Chelmsford, Mass. Tremain, Balance Valves................... Chicago, Ill. Troy & Boston R. R. Machine Shop........... Troy, N. Y. U. S. Navy Dep't........................... Washington, D. C. Washington Iron Works..................... Newburgh, N. Y. Wheeler, N. W., Eng. Office................ New York. PERIODICALS AND WORKS OF REFERENCE CONSULTED. American Artisan. Auchincloss, Link and Valve Motions. Belanger, Cinematique. Borgnis, Composition of Machines, 1818. Bourne on the Screw Propeller. " Catechism of the Steam Engine. Brown, H. T., 507 Mech'l Movements. Burn, R. S., Mech. and Mechanism. Colburn, Locomotive Engineering. Engineer, The. Engineering. Engineer and Mach. Drawing Book. Fairbairn, Mach. of Transmission. Francis, Hydraulic Experiments. Hughes on Water Works, Weale's Series xviii PREFACE. Imperial Cyclopaedia of Machinery. Jour. of Franklin Inst., 1860, 1864, 1867-70. Joynson, Gearing. King, W. H., Notes on Steam. Leroy, Geometrie Descriptive, Applications. Long & Buel, Cadet Engineer. Olivier, G6ometrie Descriptive, Applications. R. P. I. Collections of Mechanical Lithographs. Roebling, Wire Rope Transmission. Scientific American. Sellers on a System of Screw Threads. Weisbach, Mechanics. Weissenborn, Amer. Eng'g illustrated. Willis' Principles of Mechanism. Zeuner, on Valve Gears. GENERAL TABLE. FUNCTIONAL CLASSIFICATION. CO0MPOUND ELEMENTS OR SUBMACHINES. M Local. General. RECEIVERS. COMMUNICATORS. REGULATORS. MODULATORS. OPERATORS. SUPPORTERS. Pillow Blocks.* Chain hooks. Crank Pins. * Saw teeth. Compound Chucks. POINT Axle Box. * Block Cross Heads.* Governor balls.* Idler pulley. Slide Rests. ELEMENTS. Hangers. * Fixed Couplings. * Shuttles. Tool HcXders. Footsteps. * Escape wheel. Winches. Bands.* COMT Guides. Straight. * Drill posts. Treadles. Flexible Cords. Fly wheels. * Band shifter arms. Drills. Curved. Standards.* Engine hand levers. Chains. Cutters, Helical as in Levers of Horse powers. Hay cutters and Beam Engine Connections. Beam Engine C onnets.* LINE Arms. Solid beam-frames or r Cranks. Pin clutches. Ruggles' Slate trim- TumblingBe ELEMENTS. beds.* Rockers. mers. Clock Trains. Connecting I rods. Simple slide rests. Rigid. q Links. * Excentric rods. Working L beams,etc.* REGULATORS. Oi I I Planer tables. Flat pistons. * Throttle valves. Face plates. Plane bed 4lates. Endless platforms in Excentrics. * Puppet valves. Variable orank. Air pump buckets.* PLANE. Flat brackets. Flat Frames. Web.* Horse powers. Flat slide valves. Printing press platens. Valve Motions.5 Open Flat oscillating valves. Governors. BallS Throttle beam.* (Corliss.) I Valve c/2 _ 1- — l | | — | | Fan* l for ~ or F Cylindrical brackets. Prismatic beds.* Cylindrical water wheel Band wheels. * Cage valves. Cone pulleys. Oil *J Valve. Local prismatic beds.* Corliss Vert. Eng. floats. Spur wheels. * Cylindrical throttle Speed " Bending, polishing, Gauges. a Stuffing boxes. * Frames. Fourneyron Turbine Bevil wheels. * valves. Dead " and shaping rolls. - z1 | t | ||Buckets.* Sectoral motions. Z!~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~Elliptic gears. O P N JonvalTurbine buckets.* Hyperboloidal wheels. HIelicoidal clutch. Screw propellers.* MODULATORS. tI < a Turbine guides. Jonval. Wind mill vanes. Screws. * S n; Worm wheels.* |Spiral gear.* 0 Feed Mlotions. cc~~~~~R~~~~~~ ( I I II I I I I ) ~~~~~~~~~~~~~~~~~~~~~~~~Band Shifters. Quick Returns.* Friction Clutches. Bell-shaped pedestals. Segmental spherical pis- Ball valves. Paraboloidal pulleys. Clock bells. Componnd reversing gear.* tons. * Escapements. p Swain turbine buckI ets.* Slide rests. Cocks, and water gates. Steam Hammers. Valve Chests. * Bed Plates-Tank. * Overshot water wheel Chambered slide valves.* Pile driving HamVOLUME Steam Cylinders. * buckets. Balanced poppet valves.* mers. BLEMENTS. " Condensers.* Chambered Frames.* Lock up valve, safety. * Pump plungers. Pump Barrels. Double beat pump Etc. Housings. * valves. * Trucks. * Giffard's injector.* * Illustrated in the plates. Most of the others, by woodcuts. ELEMENTS OF MACHINE CONSTRUCTION AND DRAWING. BOOK FIRST; SIMPLE OR SINGLE ELEMENTS OF MACHINES, PART I. INTRODUCTION. GENERAL PRINCIPLES. 1. Bodies, in addressing the eye, exhibit not only the attributes of color, transparency, or opacity; polish, or roughness; but the two fundamental geometrical attributes of Form and Size. 2. FORM is a determinate arrangement of an assemblage of points, according to some law. It depends upon the relative lengths and directions of the bounding lines of a body. 3. SIZE is the amount of space occupied by a body, and is due to the extent of its bounding lines, as compared with a unit of measure. 4. Drawings may represent objects, in respect to their 8ize, as larger, or smaller than they really are; or, in their real size. 5. Drawings which represent the apparent forms of bodies as presented to the eye, are called perspective drawings, orpictures; and are intended chiefly for ornament, or for popular illustration. 6. Drawings which represent the realforms of objects, as determined by the sense of touch, in taking measurements, are called projections. Since such drawings show the real propor 2 ELEMENTS OF tions of objects, they constitute a graphic language, by which the thoughts of a designer can be most clearly conveyed to a workman, who can thence construct the objects represented. Hence projections are often called workin,g drawings. 7. While working drawings represent the real forms of objects; they represent them, in a majority of cases, in less than their real size. But to preserve the true proportions in the drawing, all the parts of the object must be similarly reduced in the drawings. This is what is called drawing by scale. That is, each distance, as a foot, on the object, is represented by some less distance, as an inch, or a quarter inch, etc., on the drawing. Scales. 8. The only scales necessary to be understood by students of the present work, are the linear scale, and the diagonal scale of egual parts, which we will now explain.!ll. 1. -I! YI I 12 0 I 2 3 4" 5 FIG. 1. Fig. 1 represents a plain linear scale of three feet to the inch,,ach of its units as from 0 to 1 being one-third of an inch. The equal space from 0 to 12, is divided into twelve equal parts, representing inches. Thus, from 4 to the fifth mark to the left of 0, represents four feet and five inches, and is therefore called four feet and five inches. For brevity this is written 4':5". 9. Other linear scales of equal parts, being similarly constructed, can readily be understood from this example. The ivory scales used by draftsmen, contain a variety of such single scales, with the left hand unit divided both into tenths and twelfths. It also contains others, expressed in inches to the foot, as one inch to one foot, three-fourths of an inch to a foot, etc., and numbered accordingly IN = inch; i; A; J; etc. s' do 1 2 3 4 5 111/,1 II' C ba e ae. 2d. 10. Fig. 2 represents a simple diagonal scale of units, ttli'; MACHINE CONSTRUCTION AND DRAWING. 3 and 4ths of the 5ths. The space 0-5', which is equal to 0-1, etc., is then divided into five equal parts; and so is ac. The five equidistant horizontal lines afford four equal spaces. We then reason thus: If in coming down four spaces on Ob, to b, we depart from the vertical Oa by the space ab, which is onefifth of the unit 01, in coming down one space, we should depart one-fourth of ab, which equals one-twentieth of 01. We thus have the rule for reading the scale: proceed to the left of 0 as many spaces as there are 5ths required, and then down on the diagonal thus reached as many spaces as there are 4ths of 5ths required. Thus the distance between the stars is 3 units, 3-fifths and 2-fourths of a fifth, or 3 + + Z of I = 314 = 3-. All other diagonal scales, including the more familiar decimal diagonal scale, are made and used in a similar way; so that if any one of them be rationally and fully comprehended, all others may easily be understood. If, as is sometimes done, the diagonals were drawn in the direction ad, the numbers 0, 1, 2, etc., would be on the lower line ac. 11. In regard to the manual operations of machine drawing, the proper standard of precision should be carefully observed. That is, the student should always imagine himself in a drafting office, working as if his compensation, or position,. depended upon the accuracy of his work. And the latter should be the same as if his plates were to form working drawings for the actual construction of finished machinery. To this end, all 0points should be accurately located, and finely marked; and all lines should be finely drawn with none but the hardestpencils, and exactly throuzgh the proper points. In much of machine drawing, the distances to be laid off are numerous, and quite small, hence the fine spacing dividers, pens, and pencils, are of especial use, as well as the most accurate scales. 12. Tike instruments called scales, are simply pieces of metal, ivory, wood, or paper containing a variety of linear and other scales. The leading forms of scales are edge scales and surface secales. An edge scale is a scale whose graduations are on the edge of the substance containing it. This form of scale is 4 ELEMENTS OF most convenient, because a distance can be transferred from it to the paper, directly, by laying the scale on the paper and pricking off, with a needle-point, the extremities of the given distance. The best form of edge scale is the triangular scale, which contains six linear edge scales. The other form, or Xgatedge scale, having its edges chamfered on one side to ensure greater accuracy in its use, can conveniently carry but two edge scales, except as two or more, each of which is just double the other, may lie against the same edge. 13. Surface scales are flat pieces of some hard material, usually ivory, containing a set of various linear scales, side by side, and, all together, covering the surface of the instrument. These give more scales on a single instrument than edge scales; but to transfer a distance from them to paper, we must proceed indirectly by taking up this distance in a pair of dividers, and then laying it down on the paper. Elements of Projections. 14.'The following brief rehearsal of the elements of projections may assist many, or all who make use of this volume. A solid has three dimensions, at right angles to each other. Therefore if a horizontal plane, as RQ, Fig. 3, be placed parallel to two of....... I c the dimensions, as AB and BC, of a solid, and if the d /i ----- Xlatter be then viewed in a a-!...- 1 direction, Aa, perpendicui lar to the plane RQ, those ------ /. G. V,#""37rdimensions can be seen, cor-.. rectly represented, in length and direction, upon that...... a............. plane, as at ab and bc. R The figure abed is thus FIG. 8. equal to the visible top of the given body, and is called its horizontal projection; or, in the language of practice, its plan. 15. In like manner, if a vertical plane, RS, be taken parallel to the diniensions, AB and AG, and if it be viewed MACHINE CONSTRUCTION AND DRAWIN. 5 perpendicularly, as in the direction Dd', these dimensions can be correctly shown on that plane. The figure d'c'ef, equal to DCEF, is called the vertical projection, or the elevation of the given body. Thus we see that the two projections of a body, taken together, show its three dimensions, when the latter are parallel to the planes RQ and RS, which are called the planes of projeetion. 16. Observe now that d'g, the height of the vertical projection of D above the ground line, is equal to Dd, the height of D, itself, above the horizontal plane. In like manner, cn, the distance of the horizontal projection of C from the ground line, is equal to the perpendicular distance of C, itself, from the vertical plane. That is: The perpendicular distance of a point in space from either plane of projection, is equal to the distance from the ground line to its projection on the other plane. 17. Analyzing the figure a little, we see that when a line, as AB, is parallel to a plane of projection, its projection ab, or c'd', upon such plane, is equal andparallel to itself. Also if a line, as DF, perpendicular to the horizontal plane, or DA, perpendicular to the vertical plane, is perpendicular to a plane, its projection on that plane, as d or d' respectively, is a point; and on the other plane, as at df' and da, respectively, is perpendicular to the ground line, and parallel to the line in space. With this suggestion, the reader can make out the projections of lines in other positions, as the diagonals, AC, DE, AF and AE, not shown. 18. Summary of definitions. A plane of projection is one on which an object is represented. A projecting line is a line from any point of an object, perpendicular to a plane of projection; and it represents the direction in which the object is looked at. The projection of any point, is the intersection of the projecting line of that point with a plane of projection. The projection qof any object is the figure formed by joining the projections of the bounding points of that object. The intersection, Rn, of the two planes, is called the ground line. So simple an apparatus as a folding slate or sheet of stiff 6 ELEMENTS OF paper, with the leaves placed horizontally and vertically, and a few straws, will serve to illustrate the principles here stated. 19. The planes of projection, which are at right angles to each other in space, coincide upon paper. This is accomplished by supposing the vertical plane, RS, Fig. 3, to revolve backward about the ground line, until it co~d 4C incides with the horizontal plane produced backwards.. e' Supposing now the planes to be of indefinite extent, Fig. 3 would be thus transformed into Fig. 4, which shows the projections abed and d'c'ef', as they really are, instead of pictorially, as in Fig. 3. GL, the ground a b ~ —-----— iline, represents Rn in Fig. 3. Th7e two projections, as d and ~d C C d', of the same point, are thus FIG.4. in the same perpendicular to the ground line. This should be carefully remembered. 20. A point is named by naming its projections. Thus the point dd', Fig. 4, means the point itself, D, Fig. 3, whose projections are d and d'. The like is true of lines. Thus the line dc —d'c', Fig. 4, means the line DC, Fig. 3, whose projec-,,,, tions are de and d'c'. 21. Resuming, Iow, the conclusion of (15) if the dimensions ______ B' of a body are not parallel to the planes of projection, they may be made so either by turning the body, or by taking a new plane m.! tA ofprojection. In turning a body, it is sufficient to study the motion of one of its points. This understood, the followFIGe. a ing principles pertain to the revolutions of points a —If a point as mm', Fig. 5, revolve about a Mvertical axis, MACHINE CONSTRUCTION' AND DRAWING. as A-A'B' (see DF, Fig. 3), it will describe a horizontal arc, as mm/ —n'm"', whose horizontal projection, mm", will be an equal arc, with centre at A, and whose vertical projection, mn'm"', will be a straight line parallel to the ground line. b-Similarly, if a point mm', Fig. 6, revolve about an axis, AB-A', which is perpendicular to the vertical plane (See DA, Fig. 3), it will describe an arc parallel to the vertical plane; 0;t oIl l. i, oii,_ FIp l 6. tFIG 7. whose vertical projection, m'o'm"', will be an equal arc, with centre at A', and its horizontal projection, mom", a straight line, parallel to the ground line..n c-If a point mm', Fig. 7, which is vertically over a horizontal axis, A B, revolves 900, it will appear, as at in", on a perpendicular, mm/, to AB, and equal to its height 6 L m'n above the axis. (16.) For the arc of its revolution is in a vertical plane, perpen- dicular to AB, and its horizontal projection is therefore. straight, and perpendicular to AB. Here the axis is in the horizontal plane. If it hadt been merely parallel to that FIG. 8. plane, m'n would have been estimated from its vertical projection, which would have been parallel to the ground line. 8 ELEMENTS OF Like results would be true for a revolution about an axis in, or parallel to the vertical plane. The student should construct figures to represent these cases. d-JIf a point mm' not vertically over an axis, AB, in the horizontal plane, be revolved about that axis, into that plane, Fig. 8, it will appear at a perpendicular distance, kin", from that axis, equal to its true perpendicular distance, in space, from AB. This distance, as may be made evident by the simplest model, will be the hypothenuse of a right-angled triangle, whose base equals ink, and whose /%Xt altitude equals m'n, which % nia'!m~last is the true height of the "-_..'6:ELEMENTS OF bolted together in the usual manner, as shown. When these two parts are bolted together, they form an annular space between their flanges for the reception of the central ring C, which is supported by the ends of radial arms of the portion A, so as to lap over the joint of the two parts AB, as shown in Fig. 4, and thus prevent the entrance of steam in the body of the piston. This ring, C, is fitted so as to be steam tight and immovable, when screwed between the spider A, and follower B. The ring, C, has a central rib which leaves on each side of it an annular space for receiving the expansible packing rings. This centre ring, C, forms one side for each expansible ring (the inner side), the other side being formed by the circular flanges of the piston as shown in Fig. 4. The inner corners of the circular flanges of the piston are bevelled as shown at aa for receiving the corresponding bevelled surface of the packing rings bb66, portions of which are closely fitted within the annular chambers above described, so that their circumferences project a short distance beyond the circumference of the piston. The packing rings bb are made up of segments or sections, and are held out, so that their outer surfaces press gently against the inner surface of the steam cylinder, by means of springs cc, the ends of which are bent outward, as shown in Fig. 5, so as to act upon the ends of the segments bb, and thus to keep the ends of all the segments snugly together, except those segments between which the springs are bent outward. These springs, cc, not only act to expand the sectional packing rings, but they also operate to keep the ends of the sections together so as to form tight joints. To prevent the entrance of steam within the chambers between the sections, b, and the flanges of central ring C, there are recesses in the ends of those segments which receive the ends of the springs, cc, and, inserted into said recesses, short pieces, G, which break joints with the joints of said segments. The segments G may be screwed, or riveted to the segments b on one side of the joint. For horizontal working pistons, where the weight of the piston is supported upon the lower inner surface of the cylinder, there is a segment, g, which is inserted into a recess formed in the ridge of ring C, and acted upon by a spring or set screws, m, or both combined, which supports, or nearly supports, the weight of the piston upon said piece g. This piece, g, may be made of hard brass or any other suitable metal, land as its outer surface wears away, the spring will MACHINE CONSTRUCTION AND DRAWING. 67 force it outward, or, from time to time, it can be set out in proper place by set screws, m, so that the axis of the piston, and axis of the cylinder, within which the piston works, will always coincide. This will prevent the piston, stuffing box, and its rod, from wearing untrue. Where the weight of the piston is supported by the piston rod, as in upright cylinders, the piece g, with springs and set screws m, can, and should be dispensed with. It will be seen by reference to Fig. 4, that the circumference of the packing rings projects sufficiently beyond the circumference of the piston flanges, to allow steam to pass those flanges, and act upon the projecting bevelled surfaces of the packing rings, and expand them against the surface of the cylinder with a pressure commensurate with the force of steam. The springs, cc, are designed merely to keep the packing rings expanded and in position to be acted upon by steam, and forcibly expanded thereby. This invention is not confined to steam engine pistons, as it is applicable to pistons for air and water engines. The segments b may be made rectangular, or of other suitable shape, in cross section. Construction. —16 ins. may be assumed as the diameter of this piston, and it may then be drawn on a scale of one-sixth, showing a little more than half of Figs. 5 and 7. EXAMPLE XXI. Thirty-six and Fifty-four-inch Pistons. Description.-In P1. XI., Fig. 1, R represents a vertical section of a piston, with partly spherical upper and under surfaces. All of its horizontal sections being circles, a plan view of it could easily be added, to make a separate example. P1. IV., Fig. 3, represents a fifty-four-inch propeller engine piston; hAb is half of the part called the "sider," and H'A'K' a vertical section of it on KIH, so that all below A'B', except E'dD' and F'G', should be filled with shading lines. The correspondence of the letters will show the relative heights of the different points above A'e". Thus, the tops of the arms and rim, b, ae, A, are in the plane, D'I'. LM is the cover which closes on to the spider, its under face resting on the plane, D'I'. The bore for the piston rod is slightly conical. 68 ELEMIENTS OF In the annular space, tI'G'I', between the spider bottom and the cover, is inserted the packing. Formerly this consisted of two or more rings cut once or more, so as to be adjustable to the inner surface of the cylinder, and breaking joints, so that the cuts not lying together, steam could not pass through them froim one side of the piston to the other. These rings were then set out by springs, bearing on the inside of the rings, and pressed against the rings by screws bearing at their inner ends against some of the unyielding parts of the piston. At present steacm-acking is very generally used. In this example, 00' is a part of the skeleton ring, resting against the outer ends, IJ, of the spider arms, n'n'; n,.N shows the packing itself, consisting of two rings, of which the inner one, i" square in section, fits within the other, as at n', n". Shallow cuts filed away, as atf, admit the steam over the edges of the piston body, which does not quite fit the cylinder, into the space between the skeleton ring, O', and the packing. Steam can also be admitted through small holes through the piston bottom and cover, near their edges. Construction.-A larger section of the packing, on a scale of. or i, might be made, including the adjacent parts of the piston body. b-Developable Receivers. EXAMPLE XXII. A Fourneyron Wheel Plan. Description.-This example, P1. IV., Fig. 5, does not include the finished wheel, but only what is most essential, viz.: the laying out of the bucket and guide curves, as seen in plan, where their true curvature is shown. 0, near the bottom of the plate, is the centre of the wheel, whose extreme radius is 49f ins.; radius to outer ends of buckets, 49 ins., and to their inner ends, 40 ins. There are 44 buckets, 8j ins. apart at the outer edges, and 9 of an inch thick, made of polished Russia iron. The water enters the wheel from above, through a trunk of nearly the same diameter as the inner radius of the wheel, filli ng the guide channels, and issuing thence against the buckets, and producing rotation in the direction of the arrow. As the wheel MACHINE CONSTRUCTION AND DRAWING. 69 gives way by its revolution from before the water, the latter does not bend round and run out as if in a fixed channel, AGBE, but goes directly on, as indicated, in a general way, by the lines IKM and hN, carrying the bucket with it, and hence the wheel. Still, as a particle of water, relatively to the bucket, follows its curve, AFB3, the form of the bucket is not a matter of indifference. The guides are fixed, and their number somewhat arbitrary, but usually taken at from half to three-fourths the number of buckets. To avoid the injurious pulsation which might follow if many guide edges should coincide at once with bucket edges, it is doubtless best to have the bucket and guide numbers prime to each other, that is, with no common divisor except 1. HIence we have proposed 31 guides to 44 buckets in the present example. The regulating gate is a vertical thin cylinder, which shuts vertically downward in the annular space, HG, all around the wheel, between it and the guides. Water, therefore, enters the guide passages throughout the entire circumference of the guide case, deducting only the thickness of the guides themselves; while in the Jonval wheel the guide openings, as they would be, without a gate, are half closed by the gate. This, however, is not an essential point, for in both cases water issues against the bucket from the entire outer circumference of the guide case, with the above deduction, and the whole structure can be designed with such dimensions as to give any desired area of guide opening. Consttruction. —With a scale of one-tenth, describe the principal circumferences with the dimensio(ns given. The circumference containing the bucket ends, divided by the number of buckets, will give the distance BE. The shortest distance, EF, between the buckets being fixed by the designer, the following form and construction has been proposed: Put EF=a, and the thickness of the bucket=b. Make BC=5a, and draw the radius, CO, to determine A, the inner end of the bucket. Draw AD, tangent to the circle OA, and the arc E-F, with a radius equal to a+b. Then the direction of Ba must be found by trial, so that AD being marked on the edge of a slip of paper, shall be applied with D always on Ba, and the segments te, or erF, etc., constant