Class '^d Jc^ OC ^ Book._ CopyrightN^. 1 €i /^/^ Ci)FffiIGHT DEPOSm WORM GEARING McGraw-Hill Dookfompaiiy Puj6tis/iers (^3oo£§/br ElGCtrical World TheEnginGGiin^ and Mining Journal tngiaeering Record Engineering Nows Kailwa/A^e GazettG American Machinist Signal LnginoGr American Engineei* Electric liailway Journal Coal Age Metallurgical and Chem ical Lngineering P o we r s»* -■■:S^'* Frontspiece. A Fifteenth Century Worm Gear from an Etching by Albrecht Diirer. WORM GEAMNG BY HUGH KERR THOMAS M. I. MECH. E., M. A. S. M. E., M. I. A. E. Second Edition Revised and Enlarged McGRAW-HILL BOOK COMPANY, Inc. 239 WEST 39TH STREET. NEW YORK LONDON: HILL PUBLISHING CO., Ltd. 6 & 8 BOUVERIE ST„ E. C. 1916 \ *Si t .» Vi ' .^ C^^ i"* Copyright, 1913, 1916, by the McGraw-Hill Book Company, Inc. m -7 1916 THE MAPIiE PRESS X O R K PA 'CI.A445532 PREFACE TO SECOND EDITION A second edition of this book having been called for, the opportunity is afforded for making corrections of a few typographical errors, which appeared in the first edition. Part of Chapter IX has been rewritten, and three short appendices added to bring the text abreast of the writer^s most recent investigations. It is hoped that the value of the book is thereby considerably enhanced. H. K. T. PREFACE TO FIRST EDITION In the following work an attempt has been made, it is believed for the first time, to deal exhaustively with a little understood branch of applied mechanics. A complete analysis of the principles of the design of worm gearing has been made, and, primarily, this has been treated in its application to the rear axles of automobiles. For a number of years this type of gearing has enjoyed considerable popularity in Great Britain, and its wider use is daily becoming more general. From his experience with his own staff, as well as with many professional acquaintances, the author believes that the book will give information, which, although possibly possessed by a few engineers, has not hitherto been accessible to the designer and draughtsman in anything like a complete form. An important use for worm gearing is rapidly developing in connection with the reduction gear of steam turbines for marine propulsion; while these are, mechanically speaking, simpler problems, by reason of their greater stability, than the gears of an automobile, precisely the same rules can be applied, and the same thing may be said of gears for driving line shaft- ing from electric motors. vii viii PREFACE A brief outline of the work was given in two papers contrib- uted by the author to the "Automobile Engineer'^ for May and June, 1912, and some of the formulae there published, with much amplification, have been incorporated in the text. The subject is a complicated one, demanding lengthy explanations, but by confining the calculations to the use of elementary alge- bra and trigonometry, it is hoped that the solution of the vari- ous problems has been as far as possible simplified. The literature on the subject is extremely meagre; where it has been consulted, reference to the author has been given in the text; with very few exceptions, however, the work is entirely original, and the rules given have been in every case referred to practical experiments for verification. The author wishes to acknowledge many valuable sugges- tions in the progress of the work from his assistants, John Younger, B. Sc, Lewis P. Kalb, M. E., and C. P. Schwarz, D. Sc, to the former of whom is due the method for determin- ing the width of the worm wheel. He is also indebited to A. L. Cox for valuable assistance in reading the proofs and preparing the index. H. Kerb Thomas. London, January, 1913. CONTENTS Page Pkefacb V CHAPTER I Introductory 1 CHAPTER II Choice of Materials and Methods of Manufacture , 5 CHAPTER III Definitions and Symbols 8 CHAPTER IV Preliminary Proportions 13 CHAPTER V Pressure Angle and Form of Thread 22 CHAPTER VI Strength of Worm Wheel Teeth • 34 CHAPTER VII Stresses in Worm Gearing 37 CHAPTER VIII The Width of the Worm-wheel 47 CHAPTER IX The Temperature Coefficient 58 CHAPTER X Efficiency of Worm Gearing 63 CHAPTER XI General Points of Design of Mounting 76 CHAPTER XII Recapitulation of Formula Used 81 IX X CONTENTS APPENDIX A. Alternative Method of Calculating Stresses in Worm Gearing. . 85 B. Efficiency of Worm Gearing 89 C. Reversibility. 91 Index 95 WORM GEARS CHAPTER I INTRODUCTORY. Worm, or more properly screw, gearing is of great antiquity; the word screw, directly derived from the Danish scrue, is defined in the Encyclopaedia Britannica (11th edition) as "si cylindrical or conical piece of wood or metal having a groove running spirally round it." Such a spiral was first studied geometrically by Archimedes (287-212 b. c.) and described in his work Uepl €\tK0)v, which deals in 28 propositions with various mathe- matical problems arising out of the construction of a helix. Un- fortunately but too many inventions of the early engineers have passed out of human knowledge, and the earliest form of worm gearing in which an Archimedian spiral was employed to rotate a toothed wheel is not now on record. We have, however, a series of curious drawings by the German artist, Albrecht Diirer (1471-1528), which were engraved on wood by the Master, to the order of the Emperor Maximillian. It is on record that His Majesty, conceiving for his own aggrandizement a Triumphal Procession, commissioned his favorite painter to prepare designs of the emblematic cars which were intended to figure therein. A number of these designs are still in existence and one is reproduced here in the frontispiece. It will be seen that the vehicle, which is of heroic proportions, is propelled by all four wheels, each actuated by a perfect worm gear. It is thus a "four wheel driven'' car in the most modern sense of the word and is at the same time undoubtedly the earliest worm driven vehicle of which any record exists. Unfortunately, we can hardly believe that so ponderous a machine could have ever been propelled by means of such obviously inefficient gearing operated by only "four man power." It is not known 1 2 WORM GEARS whether this remarkable car was ever made, but it is certain from the existing prints that Dtirer must have seen, somewhere or other, a worm and wheel from which his elaborate design was copied. From the fifteenth to the nineteenth century, almost all gearing was made of wood, and in old treatises on gearing, numerous examples of wooden worm gears are illustrated. Several examples of worm-driven traction engines are to be found in the records of the British Patent Office, and more than one are for the propulsion of tram cars, but as we know it to-day, w^orm gear, which had been used for many years in various forms, was first applied to the purpose of driving the road wheels of automobiles by Mr. F. W. Lanchester, who, employ- ing it in his first cars designed and built before the close of the nineteenth century, has consistently used it ever since. The Brothers Dennis adopted it for 3 1/2-ton commercial vehicles and subsequently for pleasure cars a few years later, and for several years the Lanchester Company and the Dennis Com- pany were the only firms regularly using it. Much opposition to its general adoption existed for years but Lanchester and Dennis holding confidently to their belief in the principle, it began to be realized in England that worm gearing for final drives of automobiles was a serious competitor of bevel gears and could be considered as having passed the experimental stage. Gradually other builders took it up and made it success- ful in wider use until finally the seal of approval was set by the enormous fleet of vehicles built and owned by the London General Omnibus Company, all driven through this type of gear. A few attempts, distinguished by lack of self-confidence, were made to use it in the United States, with little success on the part of manufacturers or enthusiasm shown by the public, until the author, who had experimented with it for two years in England, was privileged to introduce it into the United States, in the year 1911, on an extensive scale, as part of the regular product of The Pierce- Arrow Motor Car Company, and its use is now very general in England, on the Continent of Europe, and is rapidly extending in America. INTRODUCTORY 3 Certain obvious advantages have contributed to its adoption in automobiles of all kinds from the lightest pleasure cars to the heaviest commercial vehicles, and, while there is everything to be said in its favor, its design has been surrounded with so much mystery that many have been deterred from adopting it. It is true that some writers have attempted to ventilate the subject, and some mention of it will be found in most works devoted to gearing, but the author has searched in vain for any book treating the whole subject comprehensively. Literature specially devoted to worm gears for automobiles is, indeed, practically non-existent, and, at the best, consists of occasional contributions to technical magazines. Most of such articles omit altogether the fundamental principles, a careful study of which is a necessity if successful result is to be obtained by other methods than mere chance. Much opposition to the use of worm gearing has been due to a misconception, for which practically all the text-books are responsible, that it is inefficient, and (for automobile work) irreversible; it will presently be shown that it is neither, but mention may be made here of a remarkable kind of gear which appeared in the closing years of the nineteenth century. It was known as the ''Globoid Gear" and the supposed excessive friction of an ordinary worm was sought to be overcome by an arrangement of conical rollers mounted upon radial pins around the periphery of the worm wheel; these rollers were meshed by a worm of very large pitch and the gear certainly worked freely in both directions; beyond that, it was very cumbersome, and after a little use noisy. It was never used in automobiles, and is only referred to here to show how the action of this type of gearing has been misrepresented — even to the extent of inducing inventors to develop complicated mechanisms to overcome supposed faults which do not in reality exist. Since the early days of experiments so much actual experience in daily use with worm gears of all sizes has been possible that it can now be said that something approaching finality has been reached, and the art of making worm gearing has been placed upon a substantial foundation. Such is, in outline, the history of this most interesting me- 4 WORM GEARS chanical device, and with this very brief glance at its early use, we will pass to the consideration of the principles of its design and construction, dealing in as progressive a manner as possible with a subject which is necessarily of a somewhat complex nature. CHAPTER II CHOICE OF MATERIALS AND METHODS OF MANUFACTURE At different times, many combinations of metals have been proposed for making worm gears. Some kind of bronze for the wheel and steel for the worm was adopted as the most natural combination for two sliding sm'faces working together; the first and most obvious improvement being to case-harden the worm to minimize wear. Rise in the cost of copper and tin suggested the use of other metals for the wheel, and cast iron of various grades, mild steel, and even case-hardened steel were tried for the wheels, all in combinations with and without a hardened worm. For many years, bronze has been used for the worm wheels of elevators driven by a steel worm and it is now gen- erally recognized that case-hardened steel for the worm and phosphor bronze for the wheel cannot be surpassed. It is interesting to note here that manganese bronze is unsuitable, presumably on account of its hardness. The bronze must be very homogeneous and close grained, and great care must be taken to ensure uniformity of mixing, melting and pouring, in order to ensure uniform results. The writer recommends a mixture of the following proportions: Copper 89 Phosphorous 1 Tin 10 This will have an elastic resistance to crushing amounting to 22,000 lb. per square inch, and to penetration, under a hardened steel point of .125 sq. in. area, of 5000 lb. On the Shore scleroscope, it will give a hardness reading of 15 to 20. Its tensile strength will amount to 35,000 lb. per square inch. With such a metal, a safe working stress of 7000 lb. per square inch may be permitted 5 6 WORM GEARS For the worm, seeing that its shape conduces to great strength, no unusual strength is required in the material, and almost any low carbon case-hardening steel, which can be heat treated without serious distortion, may be employed. Steel for this purpose should contain 3 to 3.5 per cent, nickel and .16 to .18 carbon. In its untreated state, it will give approximately the following physical tests, on a standard 2-in. specimen. Elastic limit 62,000 lb. per square inch. Maximum strength 82,000 lb. per square inch. Elongation 30 per cent. Reduction 60 per cent. After machining to size with an allowance of say .003 in. for grinding, it can be subjected to the following heat treat- ment. Pack in carbonizing material and maintain for eight hours at a temperature of 1600 to 1650° F., allow to cool in the carbon, reheat to 1350° F. and quench in oil. The above may, of course, be varied at the discretion of the designer, the object being to obtain a thick casing of a hardness which will indicate from 60 to 70 on the scleroscope. After hardening, the worm may be finished to required size by grinding which will, if properly performed in a suitable machine, provide a worm accurate upon all threads and with a perfect and lasting working surface. With regard to the method of manufacture of gears, the cast- ing of the wheel should be rough turned and roughly gashed in a milling machine, after which it is strongly recommended that it be set on one side to season. Being necessarily of consider- able mass, the surface tension set up during cooling, on being released by machining will cause some distortion, and seasoning for a period of some weeks will allow the metal to assume a permanent form which it may be relied upon to maintain. The wheel is then turned to exact size and the teeth cut by means of hobbing. It is safe to say that few makers have, until recently, papreciated the necessity for employing extra rigid machinery for this purpose without which the highest class of work is impossible. By the older method of hobbing, a parallel hob was fed radially into the wheel, but in more modern methods a taper hob is fed at fixed centers tangentially to the wheel it CHOICE OF MATERIALS 7 is cutting; the reason for this is obvious — by such a method alone can the centers be preserved at the required distance and interchangeabihty of similar gears be assured. It is not possible to obtain the same finish on the worm wheel as in the case of the worm, but some makers have been particularly successful in this respect, finishing the worm wheel with a special kind of hob which leaves an extremely accurate surface which immediately beds to the worm. In the case of the worm, it is an advantage to rough turn it and rough cut the thread, and then anneal it, to remove internal strains and subsequent distortion, and while in some steels it is not necessary, it is a practice which may be recommended. The cutting of the worm may be performed in a lathe with a single point tool accurately ground to the required form, and for experimental work, saving the expense of cutters, this has its uses; the better method, however, is of course, to mill the thread in a thread milling machine, and special machines are generally used for repetition work. In the case of the hollow or Hindley worm, as used by Mr. Lanchester and others who have followed his example, a special bobbing machine is used for cutting the worm, to which reference will be made again later. Such a worm cannot be ground, and, after hardening, must be cleaned up on its working surfaces by hand. At the time of writing, the author is not aware of any parallel worms having been hobbed, but it seems likely that there may be developments in this direction in the future. Whatever method is adopted, however, the greatest possible accuracy must be insisted upon in the case of both the worm and the wheel, the high efficiency of modern worm gears being almost entirely due to extreme care in this respect. % CHAPTER III DEFINITIONS AND SYMBOLS A worm gear may be defined as a spur wheel which is rotated by an endless rack, the teeth of which are successively pressed against the teeth of the wheel. By making the rack teeth in the form of a spiral and rotating it upon its axis (sloping the wheel teeth to a corresponding angle), the effect of an infinitely long rack is obtained. Such a rack is called a parallel worm. By revolving the worm wheel, the teeth of the rack may be caused to move along, that is to say, the worm will itself commence to rotate, the relative motions being thus convertible. The worm is then said to be '' reversible, '^ the amount of reversibility depending upon certain fixed principles which will be later discussed. In order to obtain a greater area of surface contact between the wheel and the rack, the latter may be curved to partially embrace the wheel, the form of the worm then partaking of that of an hour glass. This form is usually known as the Hind- ley worm, from its inventor, and has been brought to a high state of perfection for use in automobiles by Mr. Lanchester. As the central portion of a Hindley worm exactly corresponds in its action with that of a parallel worm, the same calculations are applicable to either, with the exception of such formulae as refer to the number of teeth in engagement at one time and consequently the length of the worm; these differences will be treated in their proper place. We will now define clearly the following expressions to which frequent allusion will have to be made. Gear Ratio. — As in any other system of gearing, the ratio of the speeds of revolution which one member will make when meshing with the other is called the gear ratio; thus, with a 3 to 8 DEFINITIONS AND SYMBOLS 9 1 ratio, it is to be understood that the worm will make exactly three revolutions to one revolution of the worm wheel. Unlike spur or bevel gears, however, the ratio is not primarily depend- ent on the number of threads of the worm, but involves the question of lead. Lead. — ^This may be defined as the distance traveled along the axis by one thread in one revolution; it is measured in inches. How this determines the gear ratio may be made clear by the following example. If the worm wheel be 30 in. in circumference and the worm has a lead of 6 in., the gear ratio 30 will be -^ = 5 to 1. This may be further defined as the dimen- sional ratio, obtained by dividing the circumference of the wheel by the lead, so that we may say as a starting-point that wheel circumference ,. ,^. -, — , = gear ratio (1) worm lead By increasing the diameter of the wheel the number of teeth is increased and the gear ratio is reduced, while the converse of this is, of course, true, but by increasing the diameter of the worm the rubbing velocity is increased. By increasing the lead angle, the gear ratio is reduced; the circumferential pitch being first determined and the number of wheel teeth, it follows that a single-threaded worm will give a reduction of 1 to n where n = the number of wheel teeth. A double-threaded worm will have a reduction oi2ton and so on, but this relationship is not primarily due to the number of threads in the worm but to the lead of the thread which may be extended to include at each revolution one, two, or more teeth. Thus, in a worm having five threads, each thread will make a complete engagement with only every fifth tooth in the wheel, and the other four threads will pick up the intervening teeth respectively. Pitch Line of WheeL — ^Properly speaking a worm wheel has no pitch line at all, that is to say, in the sense of the two rolling circles of spur or bevel gears; it is, however, a convenient expres- sion for a circle corresponding to the mean effective diameter of the wheel and must be so understood. 10 WORM GEARS Pitch Line of "Worm. — This also has only an hypothetical existence, but may be defined as the circle described about the axis of the worm which touches, at right angles, the pitch circle of the wheel. All subsequent references to '^Pitch" whether applied to the worm or wheel must be understood to be meas- ured upon these circles. Circular Pitch of Wheel. — The distance in inches, or fractions of an inch, between the centers of the wheel teeth measured along the pitch line. Axial Pitch of Worm. — The distance in inches, or fractions of an inch, between the centers of adjacent worm threads, meas- ured along the pitch line, parallel to the axis of the worm. Note. — In a worm having more than one thread, this is not the same as the lead of the worm. In fact, the lead is the prod- uct of the number of threads or starts multiplied by the axial pitch. The axial pitch of the worm is always equal to the circular pitch of the wheel. Normal Pitch of Worm and Wheel. — This is the distance between the centers of two adjacent threads or teeth measured normally or at right angles to their faces along the pitch line. Owing to the inclination of the threads or teeth, this distance is always less than the circular or the axial pitch. Circimiferential Pitch of Worm. — The distance along the pitch line between two adjacent worm threads, measured cir- cumferentially round the worm. Lead Angle. — The angle formed by the thread of the worm with a line drawn at right angles to the axis of the worm. If the spiral formed by the threads were unwrapped from the worm, it would form an inclined plane having an inclination equal to the lead angle. Addendimi. — The height of wheel tooth or worm thread outside pitch line. Dedendtmi. — The depth of wheel tooth or worm thread inside pitch line. Length of Worm. — ^The length of the cylinder bounded by the pitch line of the worm. If two radii be drawn from the center to the circumference of the pitch line of the wheel, so that each touches the extremity of this cylinder, they will make an angle DEFINITIONS AND SYMBOLS 11 which is subtended by the cyHnder of the worm pitch Kne, and the length of this cyhnder is the chord of this angle. Note. — The length of the worm has nothing to do with the the lead of the worm. Subtended Angle of Worm. — If two radii be drawn from the axis to the circumference of the pitch circle of the worm, they will form an angle which is filled by the teeth of the worm wheel. This is the subtended angle of the worm, and is the only manner in which the effective width of the worm wheel can be measured. The determination of this angle has a very impor- tant bearing upon the performance of the gear. Included Angle of Thread. — In a V-threaded worm, the angle made by the inclination of the faces of the V is the included angle. Pressure Angle.— Half the included angle. Note. — Both the included angle and the pressure angle may be measured normally or in the direction of the axis; unless otherwise specified, however, the axial angles are always to be understood to be implied. The following system of symbols will be employed through- out this work. G =Gear ratio. D =Pitch line diameter of worm wheel in inches. d = pitch line diameter of worm in inches. L =Lead of worm in inches. I = Length of thread per revolution in inches. N = Number of teeth in worm wheel. n = Number of teeth in worm. R = Pitch line radius of worm wheel in inches. r = Pitch line radius of worm in inches. R^ == Radius of road wheel in inches. i^" = Extreme radius of worm wheel on plane of worm axis. V = Rubbing velocity in feet per second. T = Torque in pounds at pitch line of worm. p = Tangential pressure at pitch line in pounds. p' = Normal tooth pressure. P = Circular pitch of worm wheel and axial pitch of worm. 12 WORM GEARS P' = Normal pitch of worm threads and wheel teeth. P'' = Circumferential pitch of worm threads. / = Force separating worm and wheel. w = Weight at rear axle in pounds on the ground. X =Safe load on tooth of worm wheel, in pounds. g = Length of worm in inches. n' = Number of teeth in contact. a =Lead angle. = Included angle of worm thread (axial). W = Normal included angle of thread. o = Stress permissible in wheel teeth in pounds per square inch. ;- = Gliding angle. f = Angle of friction (tan

600 \ \ t \ \ s \ \ V \ X ■V, V *^ --«. -«, — ^ ■-. 10 15 20 25 30 35 Velocities - Ft. per Sec. Fig. 25. 40 45 and consequently H varies as p^V, and if a constant value is demanded for H, it follows that p'F is a constant. In Fig. 25, an hypothetical case, the rubbing velocities are represented as abscissse and the tooth pressure in pounds as ordinates, p' 7 = 50, 600 a constant and the curve is in conse- quence an hyperbola. In the Bach & Roser experiments, it was found that at constant temperatures, a very close approximation to this hyperbola was obtained. Thus, the important deduction from this is, that given the maximum pressure and speed of any gear at which a constant temperature may be maintained, the other pressures and speeds can always be calculated, lower speeds permitting higher pressures and conversely. THE TEMPERATURE COEFFICIENT 61 It is profitable to investigate the effect on the quantity of heat generated by varying the pressure and the velocity. pV 60 H=v 778 Case 1. — Assume p' constant at 3500 lb. then ^^ 3500X.002X60 ,, H = v ^^3 ^..54. TABLE V V f.p.s. B.T.U. per minute 5 2.70 10 5.4 15 8.1 20 10.8 25 13.5 30 16.2 35 18.9 40 21.6 45 24.3 Here the difference = 2.7 B.T.U. per increment of 5 f.p.s. Case 2. — Assume v constant at 45 ft. per second H^2^' .002X45X60 778 .00694 / TABLE VI V' B.T.U. per minute 1500 10.41 1750 12.14 2000 13.88 2250 15.62 2500 17.35 2750 19.09 3000 20.81 3250 22.55 3500 24.30 3750 26.05 4000 27.75 62 WORM GEARS Here the difference = 7.4 B.T.U. per increment of 250 lb. Fig. 26 shows these increments of H plotted against pressures and velocities respectively, and the results show that for a com- 2500 § 2250 ^ 2000 g 1750 o 1500 ^ 10 12 14 16 18 B.T.U. per Minute Fig. 23. paratively small increase in the normal tooth pressure, the , heating effect is increased at a greater rate than with relatively larger increments of the velocity. CHAPTER X EFFICIENCY OF WORM GEARING The efficiency of worm gearing has been discussed by several writers and it will be dealt with next, because closely bound up with it is the coefficient of friction between the worm and wheel, and the determination of the latter is a necessary preliminary. The most complete investigation of mechanical friction that the author is familiar with, is that conducted by Beauchamp Tower, and published by him in various numbers of the Proceed- ings of the Institution of Mechanical Engineers. Taking the materials most nearly corresponding to those employed for worm gears, viz., steel rubbing against bronze in a bath of mineral oil, we find the lowest value he obtained for /i was .0008. No doubt these conditions were ideal, but suppose we take a higher value, .002, as an assumption, and then see how nearly this can be approached and maintained in practice; it will presently be shown that we are fully justified in so doing, see- ing that the conditions favorable to a low coefficient are all present in a worm gear. These are 1. Brief period of contact. 2. Intermittent pressure. 3. Certainty of oil films reaching every part of the pressure surface. 4. No metallic contact. The well-known experiments of Bach & Roser at the Royal Technical High School, Stuttgart, throw some light on the behavior of worm gears, but unfortunately they were carried out upon gears which were by no means as accurately made as is the case in the present day practice; but while their results are of little specific value, they undoubtedly illustrate the laws which govern the whole performance of worm gears. 63 64 WORM GEARS The worm gear upon which the experiments were made •appears to have been of the following proportions: Worm: Number of threads 3 Pitch diameter 3 in. Lead 3 in. Lead angle 17° 40' Wheel: Number of teeth 30 Pitch diameter 9 . 54 in Circumferential pitch 1 in. The tooth pressures were varied from 190 to 2660 lb., and tests were made at six different speeds as follows: V =28.30 f.p.s. 19.34 f.p.s. 9.80 f.p.s. 4.62 f.p.s. 2. 56 f.p.s. 0.85 f.p.s. It will be observed that while the tooth pressures are m some accordance with automobile practice, the velocities are lower than is ordinarily the case. Fig. 27 shows the tooth pressures and coefficients of friction for different velocities obtained in these experiments. It is at once apparent that for velocities between 2.5 and 20 ft. per second, the coefficient is but little affected by the speed but varies curiously with the pressures on the teeth, attaining in all cases a minimum at or near 1000 lb. tangential pressure. Unfortunately these experiments do not state, or give sufficient data to calculate, the specific tooth pressure per square inch, which would have been very valuable. As has, however, been pointed out by Mr. Robert A. Bruce, the deductions from these experiments are only useful when the exact conditions of the original experiments are reproduced, and he further points out that with superior working surfaces, e.g., hardened and ground worms, and efficient cooling arrangements higher pressures might be realized. With a view to proving the comparability of worm gears and the journals investigated by Beauchamp Tower, the author carried out the following experiment. A completely assembled rear axle belonging to a 5-ton com- EFFICIENCY OF WORM GEARING 65 mercial vehicle of the author's design was taken and mounted upon a stand (Fig. 28), using one of the road wheels as a drum round which a weighted cord could be wound. A smaller drum was secured to the worm spindle with another weighted cord wound upon this. The relative diameters of the drum on the axle and the drum on the worm spindle were such that their ratio was the same as that of the worm gear reduction, viz., 7.8 to 1. Equal weights being hung upon the cords, equilibrium 0.07 0.06- g0.05 o m0.04 6 0.03 0.02 0.01 i\t;, ar w — \\V ^V = 19.34 i.p .s. w c ^ \ Y ^=28.3 J / V / e \ !^ to.85^ x^/ \\\ f^ y y -^ y ' - \ i\\ \, -- / V V\ \ ,/ / / Tv^\ V' = 9.8 / / / \ / / / Tx > \^ V= = 4. 62 t I / / / / vS^ S -6- >// A / ^>^ - J^' X kxVJ b^^ y y / ^ r^ .-- ^ 4 = 2.5^ BACH & ROSER Relation between Pressure, Velocity and Coefficient of Friction — 500 1000 1500 2000 Tangential Tooth Pressures - Lbs. Fig. 27. 2500 3000 was established. The mechanism being set in motion by hand, weights were added to the cord on the worm drum until a con- stant speed of rotation could be maintained. On substituting heavier weights, it was found that a greater weight was needed to keep the wheels in motion, and, several observations being thus made, a number of values w^ere obtained for the weight necessary to maintain uniform motion with different degrees of tooth pressure. Plotting these values at ordinates, upon tooth pressures as abscissae, it was found that 66 WORM GEARS 00 ^ EFFICIENCY OF WORM GEARING 67 these fell in a straight line as shown by the line A (Fig. 29) The observed points being TABLE VII Pressures x Load to overcome friction y 2 1b. 4 1b. 8 1b. 3.75 4.75 6.75 It is evident that the loads y are the product of a;X/^ Curves showing Values of jJ- from Observed Friction Readings from Observed Points. V = Xti &. 2y-X=5.5 .'. 2XfJi.-x=5.5 fi approaches V2 as a Limit as X approaches co H .. CO .. .. « ., .. .. Zero. 7 / ' 6 3. 3 / ■?. n / f h 01 -) / A V f •m 4 o / y / s / 1 ^z \ 2X \ I 1 \ "^ ■ — . n 2 4 6 8 10 20 30 Pressure in #iC Fig. 29. 40 50 The equation to a straight line is 2y — x = 5.5 .*. 2xfi — x = 5.5 5.5 + x Fig. 29 shows the curve to this equation giving values of /x for various pressures x, from which we see that 68 WORM GEARS [i approaches . 5 as a limit as x approaches c-o H approaches oo as a Hmit as x approaches o. Turning again to Beauchamp Tower's experiments, the curves in Fig. 30 show the relationship between bearing pres- siu-es and n for three different rubbing velocities, from which it is evident that the characteristic of the curve obtained from the entire axle is identical with these and consequently the rules deduced from Tower's experiments by Unwin for the variations of ji with pressures and speeds will apply equally to worm gears. This is highly important in all that follows. Beauchamp Tower's observed figures for mineral oil bath lubrication. Frictional resistance 2? = /fP = P(7-\ /— Mean value of C for mineral oil .27 0.012 O.QU 0.010 ci 0.009 % 0.008 2 0.007 M 0.006 •♦J .2 0.005 g 0.004 u 0.003 0.002 0.001 0.000 (a) 0.426 0.26 (b^ 0.56 0.27 (C) 0.733 0.28 "\ ^ h X is the angle of repose corresponding to the coefficient of friction. Since it has been assumed that the coefficient of friction is .002, it follows that the angle of which this is the tangent is 7'. The following table shows the values of W for different degrees of the pitch angle. 70 WORM GEARS TABLE VIII + o + o S ^ -e- S T-T « '-* d O -©- + ^ II II "-* 4- ^ Oh 1—1 CI 03 ti ^ ^^ • " i II fin ^ ^1 03 10° lO"?! .1784 178.4 5609. .1763 15° 15 7 .2701 270.1 3700. .2679 20° 20 7 .3662 366.2 2732. .3639 25° 25 7 .4687 468.7 2133. .4663 30° 30 7 .5800 580.0 1724. .5773 35° 35 7 .7032 703.2 1422. .7000 40° 40 7 .8425 824.5 1185. .8390 45° 45 7 1.0040 1004.0 996. 1.0000 50° 50 7 1.1966 1196.6 835. 1.1917 55° 55 7 1 . 4343 1434.3 698. 1.4281 60° 60 7 1 . 7402 1740.2 575. 1 . 7320 The values given in column five would be correct but for the fact that they take no account of the work done in the form of friction, and are only indicative of the gain in power due to the reduction of the worm gearing, they are, however, correct for the angle of 45 degrees, as will presently be shown. In Fig. 32 the parallelogram represents the developed sur- face of the worm, of which A A is the thread. The worm wheel is to be revolved in the direction LM. Since the teeth of the wheel and the thread of the worm are parallel to one another at the point of contact, it follows that the pressure between the two is at right angles to the thread surface. Let aB represent this force, then this may be resolved into two forces La and LB, of which the latter is the useful and the former the useless component. If the pitch of the worm is increased to CC, the pressure applied along the line gB can be resolved into gl and IB, of which LB is the useful component; Ba and Bg are by construction equal, and Bl is less than BL, hence the force which is available for turning the wheel decreases as the pitch is increased and conversely the useless component increases with the increasing angle. EFFICIENCY OF WORM GEARING 71 BLa is a right angle triangle, of which LBa is an angle corresponding to the pitch angle. Ba is the force applied. BL is the usefnl force. The ratio of these two BL Ba is the cosine of the angle LBa. Up to this point, therefore, it would ap- pear that, with a constant worm torque, the force available for turning the wheel would vary inversely as the cosine of the pitch angle. There is, however, another aspect which must be considered, and it is the effect of the relative motion between the worm and wheel. Thus with a approaching zero value the relative motion between the two is very high compared to the useful motion of the worm wheel; hence much work is dissipated in useless friction. Various formulae have been proposed for determining the actual mechanical efficiency of worm gears, and so far as possible, these will be examined next. We will take first, the much quoted formula developed by Professor Barr of Glasgow University: V = tan a (1 —/jl tan a) tan oc-\r2/jL (76) 72 WORM GEARS For various angles of thread, a, the efficiencies have been plotted in column II table IX. Next, Professor Unwin (Elements of Machine Design, Vol. 1, p. 423) gives the following: 1 -u cot a (77) Column III of the table gives these values. Francis W. Davis, M.E., has proposed the following: , = !-.( ^ r-) ('«) Comparing this equation with (77), it will be noted that Unwin's may be written //(cot a + tan a) (79) l+jj. tan a (80) COS a sm a + n sm o:^ very closely resembling (78). The values of /j. in this equation are given in Column IV. It must be observed here that neither of the above formulae takes into account the pressure angle, and this undoubtedly exercises a considerable influence upon the efficiency of the gear inasmuch as it very largely governs the normal tooth pressure, quite irrespective of the torque. See equation 43, Chapter VII. The force of friction in the gears is equal to fiXp^, and this force tends to resist the gliding of the worm over the wheel tooth. It is evident that the path along which the gilding occurs is a helical line wound around a cylinder whose diameter equals the pitch diameter of the worm, and the length of this path is I, the length of thread per revolution measured at the pitch line. See equation 15, Chapter IV. The work lost in friction equals in foot pounds "12" ^^^) EFFICIENCY OF WORM GEARING 73 The work put into the worm is Tndp 12 Hence, the efficiency is and so l=Tzd sec a /jup^nd sec a\ _ /jup^ sec oc" Tnd J ^ \ T , Substituting the value already found for p\ we get ,=1 jJ-\\ (T cot a tan "9" ) + ( ~ — ) X sec a T This may be written W 1 — -. ^ \\ 1 +cos^ a tan^-^ sm a cos a\ 2 (82) (83) (84) (43) (85) (86) In the author's opinion, this is the most accurate formula of the four examples, and the values are given in column V of the table up to the equivalent lead angle of 45 degrees. TABLE IX.— EFFICIENCIES OF WORM GEARING FOR VARIOUS ANGLES OF LEAD I II III IV V a 15° 98.5 99.2 -0 99.2 99.0 20° 98.68 99.39 99.38 99.3 25° 98.8 99.50 99.45 99.4 30° 99.0 99.52 99.54 99.4 35° 99.5 99.59 99.56 99.53 40° 99.5 99.62 99.59 99.56 45° 99.5 99.63 99.60 99.58 50° 99.5 99.5 99.63 99.61 99.59 99.55 55° 60° 99.5 99.58 99.54 65° 99.5 99.50 99.45 70° 99.3 99.40 99.38 74 WORM GEARS It vis clear, therefore, that for all usual lead angles as em- ployed in automobile gears, the efficiency is over 99.5 per cent, and the conclusion is that the whole question of efficiency is one of the simple equivalent of the work put into the worm minus that absorbed by friction, and may, for all practical pur- poses, be expressed thus for all usual angles . = ^^ (87) .998 Numerically, this is ~ — = 99 . 8 per cent. Some apology is due to the reader for introducing so simple an expression as the outcome of so cumbersome an amount of preliminary calculation. The author feels, however, that having regard to the high percentage of efficiency this formula gives, it would hardly be accepted by engineers unless the previous proof were inserted; the demonstration is thus rendered complete. It may be pointed out that if the table be extended in both directions, completing the series of values of a from 0° to 90°, it will be found that at both limits the value of tj is zero, which is so obvious as to need no elaboration further than to observe that if a = 0° or 90°, there will be no movement of the worm wheel. Practical corroboration of these values of rj has been indis- putably furnished by the long series of experiments conducted by Messrs. Brown & Sharpe, and published as a memorandum by Professor Kennerson in the Transactions of the American Society of Mechanical Engineers, 1912, where average values of 7) for the entire worm gear, including the friction of all the bearings, amount to over 97 per cent.,^ and in two instances in the series of experiments, the records of which the author has been privileged to see, readings were obtained o/ 99 . 8 per cent. The efficiencies given in column V of Table IX may, there- fore, be accepted as representing what may with care be ob- tained in practical working, provided the workmanship is ^Also corroborated by Nat. Phys. Lab., London in tests on Daimler worm-gear, Nov., 1912, (seeaute). EFFICIENCY OF WORM GEARING 75 good, and the mounting and lubrication of the gears properly carried out. The late Mr. Briggs, of Philadelphia, says in regard to friction, '^It is established that for ordinary ratio of wheel to worm, say not to exceed 60 or 80 to 1, well-fitted worm gear will transmit motion backward through the worm, exhibiting a lower coefficient of friction than is found in any other description of running machinery.'' The selection of .002 as the value of the coefficient of friction, earlier in this chapter is thus justified. See Appendix B. CHAPTER XI GENERAL POINTS OF DESIGN OF MOUNTING. The design of the worm and wheel have now been fully con- sidered, and the stresses set up in the various bearings in- vestigated to the point where the designer is in possession of all the information to enable him to design the casing in which the gears will work. For a worm gear which is to work stationary machinery the provision of a suitable casing is a comparatively simple matter the only conditions being rigidity and the provi- sion of an oil-tight casing to carry the lubricant. The quality of rigidity must be insisted upon to ensure the maintenance of the correct relative positions of the worm and wheel under the heavy stresses to which they are subjected. Weight in such cases is of secondary importance and metal need not be spared to ensure the desired end. The casing moreover will in such a case be bolted securely to a solid foundation which greatly facilitates matters. Very different are the conditions in an automobile axle where weight must be saved at any cost, and strength and lightness have both to be studied. In the days of the first Lanchester Car, 8 H.P. was all that had to be provided for, and the weight of the car was but a few hundred pounds. Recent worm gears the author designed are capable of transmitting 90 H.P. in a fast pleasure car which weighs over three tons with a load of seven passengers and their luggage; at the time of writing it is believed that these gears are considerably more powerful than any others working in auto- mobiles, although their size is by no means excessive. It may well be imagined that the provision of a rigid case for these is something of a problem; the result has, however, proved entirely satisfactory. / In the first place there is the question of whether the worm shall be placed above or below the wheel. This subject has 76 GENERAL POINTS OF DESIGN OF MOUNTING 77 been discussed by different makers as though it were a matter of great moment; in reality it is of no consequence whatever, and is purely to be governed by the expediency of other factors in the general design. Much more important is the design of the axle casing. Based no doubt upon bevel gear practice, the casing was originally made in two halves, each containing half the bearings required to support the differential, which were bolted together around the center, the portion of the case con- taining the worm was also in halves, being in fact an extension of the main casting. Perfectly satisfactory axles have been built on this plan but the difficulties of assembling are very great as neither the worm nor the wheel can be seen after they are mounted in position. A more modern type is shown in Fig. 33 which is an illustration of a rear axle of a five-ton com- mercial vehicle, here reproduced by permission of the Fierce- Arrow Motor Car Company. It will be seen that the axle proper is a pan-shaped casting, with elongations which carry the steel tubular extensions on which the road wheels are mounted. At the outset therefore a very rigid structure is provided for preserving the true alignment of the driving shafts. The worm and the wheel are mounted in a single casting, provided with strongly ribbed brackets for carrying the bearings of the worm wheel, the whole forming the lid of the axle case proper. By this arrangement the assembling of the gears and their exact adjustment can be carried out on the bench with the worm and wheel in full view, giving every facility to the erector for correct location in their relative positions; such a method also enables an accurate machining operation to be carried out upon the main bearings of the gear, and, when examination is necessary, it can be made without disturbing any adjustments, the entire system of worm wheel, and differential with all their bearings being lifted out in one piece for the purpose. Much latitude is permissible to the designer, who has a wide field of arrangements to select from; it must always be remembered that the stresses in the parts supporting the bear- ings are high, but with the use of the diagram, Fig. 19, they may be determined at a glance, and a selection of suitable ball bearings can bie made. 78 WORM GEARS CO CO 6 GENERAL POINTS OF DESIGN OF MOUNTING 79 It may be observed that in the case of a straight worm, that is, one having a cyHndrical pitch line, a double thrust bearing should be provided at one end (whichever is most convenient) the other end can then be left free to expand or contract with the difference of temperature which occurs when the worm is running. In the case of the hour-glass pattern worm, it is difficult to say what happens when it has to expand, presumably the casing expands too, and in that case a single thrust bearing at either end is the better arrangement. Some designers provide a very heavy thrust bearing to take the forward drive and a relatively light one for the reverse — • it is hard to see any justification for such an arrangement, since the reverse gear is almost invariably lower than the first speed forward, and the torque at the worm pitch line, and all the resultant pressures are in consequence heavier. Provision must be made for oil to reach the thrust bearing at the rear of the worm when the latter is mounted above the wheel. The simplest way, and one which can be confidently recommended, is to bore the worm spindle through the center and drill small holes, say 3/16-in. diameter, radially into the hollow center, these holes being of course drilled between the threads of the worm before it is hardened. In the case of a large worm for high powered pleasure cars, this method has the added advantage of considerably lightening the worm, and the hollow in the spindle carries the oil back to the thrust bearing in a steady stream. Provided precautions are taken to prevent the oil from leaking out through the arms of the axle, it will be found that the same oil may be used almost indefinitely — at any rate from 5000 to 7000 miles, and it may be observed that the oil level should be somewhat below the center of the casing. A small vent pipe may be provided if thought necessary, to allow for expansion of the air when the gears get warm, this will prevent the oil being forced out round the axles by any pres- sure so formed; such a pipe must, however, be closed by a cap drilled with a very small hole, say 1/16-in. diameter and the pipe should be loosely plugged with cotton to prevent the entrance of dust. 80 WORM GEARS The author has intentionally refrained from elaborating to any extent upon the details of the casing, which are best left to the judgment of the designer; sufficient attention has, however, been drawn to the more important features to enable a satisfactory design to be produced if the principles set out in the earlier chapters of this book are closely followed. In con- clusion, let it be pointed out that no workmanship can be too accurate in the manufacture of worm gearing, and unless facilities exist for this, a satisfactory result cannot be looked for. If, on the other hand, proper precautions are taken, a worm gear, correctly designed and mounted, is, as Mr. Briggs has observed, probably the most efficient piece of machinery known; and its efficiency is only equalled by its durability and silence of operation. CHAPTER XII RECAPITULATION OF FORMULiE USED ^ .. ^ Wheel circumference Gear ratio G = , — -^ (1) worm lead L L c-f tf) J Measurement of (a) Axial pitch =P (6) Normal pitch P' = P cos a [ (2) (c) Circumferential pitch P" = P cot a Circumference of worm = 7id = nP cot a (3) Ttd Number of threads on worm = n = -=, — - — - (4) r* cot a ^ Lead = L= -^ (6) G Lead angle = a Length of worm=gf L^Pn (14) tan a = — , (7) ltd ^ tan a = ^ (8) Pn tan a =^ (17) g= 2iS"sm| (9) SI 82 WORM GEARS Number of teeth in contact = n' n' = 2R'' sin ^ P Centers of worm and wheel = c D + d '= 2 Length of thread per revolution of worm = l = 7[d sec a Rubbing velocity of worm = 'i; (11) (12) (13) (15) Tzd sec ap . v = — ^^ — it. per second Uo) Ratio between axial pressure angle d and normal pressure angle ¥ n tan^^- tan -- = (19) 2 cos a we tan— = tan ~ cos a (20) Normal tooth pressure = p' T ^'^sin(a + ^) ^^^^ V'=~~^, (26) cos X rp p'=J(rcot«tan|)=+(^j-^)^ (43) RECAPITULATION OF FORMULA USED 83 Resultant, normal pressure angle = z ¥ tan 2= cos a tan- (40) Tangential tooth pressure (due to tractive resistance) = P=^ (28) Length of wheel tooth =ii Safe load on wheel tooth = x x = 2,2 l^P (34) Worm end thrust = p p = Tcota (49) Separating force between worm and wheel =/ /= T cot a tan - (53) Radial thrust on worm bearings = S S = ^l (7^cotatan|^V-+ r (56) Subtended angle of worm = /? f ^ = 2JJP^ ^''^ 360 g tan a ^ n {d+.63m P) ^ ^ Tooth pressure velocity constant = ^'1; p'v^ 17000 (67) Heat generated in gears = H 778 ^ ^ 84 WORM GEARS Area of axle surface required to dissipate heat = a a = .635 sq. ft. (73) Coeflficient of friction = /i // = . 002 = tan a = tan 7' (75) Efficiency (accurate) = t^ '^ = 1 - sin jWaV^ +(ios'c, tan^ ~ (86) Efficiency (approximate) V- ^ (87) APPENDIX A ALTERNATIVE METHOD OF CALCULATING STRESSES IN WORM GEARING On page 25, a value of 60° is assumed for the axial included thread angle d. The relationship between the axial included angle and the normal angle of the thread is shown in equation 20. In Fig. 19, page 45, the value of this angle is shown in the form of a curve for different lead angles assuming a constant value of 60° for the axial thread angle. A study of Fig. 19 shows that a different milling cutter would be required for every different value of the lead angle in order to produce a constant axial thread angle of 60°. For considerations of manufacturing, it is far easier and more economical to keep the normal thread angle constant Sit 60° by which means all the cutters would be ground to the same angle and the axial thread angle would be allowed to take care of itself. Moreover, although slightly in- creased by this method, the tooth pressures are not sensibly affected and the calculation becomes a much more simple matter. In the following figure (a), let ab be the axis of the worm; and c, d, e, /, the surface of the pitch line cylinder of the worm opened out flat, one thread of the worm being shown. Let Tx represent the magnitude of the torque; px will represent the normal pressure, and for a square thread, the magnitude of this will be expressed by ^^ = .^^ f ^ ■ N (25, Chapter V) sin {a + (p) (p can be omitted as being too small to make any practical difference. Let plane c, d, e, f, be viewed in perspective, as shown in the following figure (h). Draw the perpendicular pq and make the angle pxq 85 86 APPENDIX equal to 30°, that is to say, equal to the pressure angle — • Si Then the triangle pxg can be solved for xq by the following equation xq = "px sec 30 T px = — Therefore, xq = sin a T sec 30 sin a and as sec 30° is a constant, the equation can be written 1.15477^ xq = — ; = V sm a X (a) The following table gives the value of xq, the normal tooth pressure for various lead angles. TABLE X Lead angle Tooth pressure. {T = 1 lb.) 15° 4.46 20° 3.375 25° 2.73 30° 2.31 35° 2.01 40° 1.795 45° 1.630 APPENDIX 87 The other stresses shown in the diagram, Fig. 19, the end thrust of the worm, radial thrust of the worm, side thrust of the worm, and the separating forces are not affected. Tooth pressure for 30° pressure angle may be written T = ^M^ inch lbs. n d = pitch diameter .5d = pitch radius As torque varies inversely with .5d, we get 63,024P _ 126,048P T Sit pitch line = n X .5c^ nd , 1.15477^ and p sm a and substituting , 1 1 ..7 ^ 126,048 X P 145,447P p' = 1.1547 X b-^ " = — J—- nd sin a nd sm a APPENDIX B EFFICIENCY OF WORM GEARING Equation (84) gives the following value for the efficiency -, ( iiv' sec a\ Substituting for p' the value obtained (from Appendix A), we get the following equation / T U X 1.1547 ^ X sec a 1 sm a , = 1 _ ^ _ — This may be written .002 X 1.1547 X ~— X sec a -, , sm a Or more simply 77 = 1 - f .0023094 ^-~^-^) \ sm a/ This can, of course, be expressed empirically for any pressure angle as follows: / fx sec — sec a , = 1 _ I — 4 — \ sm a The efficiencies calculated by this formula are given in the second column of the following table: 89 90 APPENDIX TABLE XI Lead angle Efficiency 15** 99.076 20° 99.28 25" 99.39 30° 99.46 35° 99.508 40° 99.53 45° 99.539 50° 99.53 Basing his reasoning upon the relative velocity of the worm and the worm wheel, P. M. Heldt gives the following formula for efficiency: V = cos ^ — M tan a Jit cos ^ + M cot OL It will be found that this formula gives precisely similar results to those given in the above table, assuming in both cases, at 30° pressure angle. APPENDIX C REVERSIBILITY There is a very erroneous opinion that the reverse efficiency of any worm gear is considerably lower than the forward; that is to say that the efficiency is higher when the worm is driving the wheel than it is when the worm wheel drives the worm. This misconception as to reverse efficiency, frequently referred to by writers on worm gearing, is due to the mistake almost invariably made in over-estimating the value of the coefficient of friction. From various experiments it has been substantiated that this value does not exceed .002 and is sometimes lower. Again the pressure angle is cited by some as having a great effect in reversibility. As a matter of fact it will now be shown' that the reverse efficiency is within six thou- sandths of 1 per cent, of the forward efficiency. We will take the case first of a square thread, ^.e., one which has no pressure angle, and assume a pitch line torque of 1 lb. both forward and reverse (see Fig. 7, page 27). T Tooth pressure p' = ^ and T = 1. sm a -{- (p so p' will be sin a -\- (p neglect (p. * ■ Then for the forward angles we have the following backward angles and other functions as follows*: 91 92 APPENDIX TABLE XII Angles Sin a Sinf Tooth pressures Forward = a Backward Forward 1 Sin a Backward 1 Sinr 15 20 25 3o; 35 40 45 50 75 70 65 60 55 50 45 40 0.2588 0.3420 . 4226 0.5000 0.5735 0.6427 0.7071 0.7660 0.9659 0.9396 0.9063 0.8660 0.8191 0.7660 0.7071 0.6427 3.87 2.92 2.37 2.00 1.745 1.556 1.414 1.305 1.035 1.065 1.104 1.155 1.220 1.323 1.415 1.555 fjL sec ^sec a The efficiency is 77 = 1 — -r^ and since pressure sm a ^ angle = 0, the sec — becomes 1, hence for a square thread 1 - M sec a sin a. TABLE XIII Angles Sin Sec Sec/Sin ^ Efficiency, per cent. 15 0.2588 1.0352 4.00 XO .002=0.00800 99.200 'S 20 0.3420 1.0641 3.11 0.00622 99 . 378 03 25 0.4226 1.1033 2.61 0.00522 99 . 478 30 0.5000 1 . 1547 2.308 0.004616 99.5494 ^ 35 0.5735 1.2207 2.13 0.00426 99.574 40 0.6427 1.3054 2.03 0.00406 99.594 45] 0.7071 1.4142 2.00 0.00400 99 . 600 50 "xi 0.7660 1.5557 2.03 0.00406 99.594 55 03 0.8191 1.7434 2.13 0.00426 99.574 60 ^ r ^ 0.8660 2.0000 2.308 0.004616 99 . 5494 65 03 0.9063 2.3662 2.61 0.00522 99 . 478 70 pq 0.9396 2 . 9238 3.11 0.00622 99 . 378 75 J 0.9659 3 . 8637 4.00 0.00800 99.200 APPENDIX 93 Next take the case of the square thread but including the effect of friction (the value of ^). We have at the following angles the values shown : TABLE XIV a a+ip Sin Sec Sec/Sin Efficiency, per cent. 15 15° 7' 0.2607 1 . 0358 3 97 : X0.002 = =0.00794 99 . 206 30 30° 7' 0.5071 1.1560 2 280 0.00456 99.544 45 45° 7' . 7085 1.4171 2 000 0.00400 99.600 60 60° 7' 0.8670 2 . 0070 2 310 . 00462 99.538 75 75° 7' 0.9664 3.8932 4 03 . 00806 99.194 This may for convenience be written: TABLE XV Efficiency, per cent. Difference, Ol Forward Backward per cent. 15 30 45 99.206 99.544 99.600 99.194 99 . 538 99 . 600 0.008 0.006 . 000 Lastly we will take into account both the friction and the pressure angle, assuming the usual value of 30° for this. The efficiency is expressed as 7/ = 1 —IfJ, sec 30 sec a sm a or 77 = 1 - ( 0.0023094 sec a sin a Taking the values as in column 5, Table XIV, we obtain: 94 APPENDIX TABLE XVI a Sec/Sin Efficiency, per cent. 15° 3.97X0.0023094=0.0091683 99.08317 30° 2.28 0.0052655 99.47345 45° 2.00 0.0046188 99.53812 60° 2.31 0.0053347 99.46653 75° 4.03 0.0093069 99.06931 which may be written: TABLE XVII Efficiency, per cent. Difference, oc Forward Backward per cent. 15 30 45 99.083 99.473 99 . 538 99.069 99.466 99 . 538 0.014 0.007 0.000 INDEX Addendum, 10, 16 Annealing worms, 7 Area of physical contact, 50, 52 Axial pitch of worm, 10 Bach and Roser, 60, 64 ^ value of, 48 ^ variations of, author's experi- ments, 52 Bronze alloy for worm wheel, 5 Brown & Sharpe, efficiency experi- ments, 74 Bruce, Robert A., deductions, 50 Case hardening, worms, 6 Casing, design of, 76 Centers of axes, 19 Circular pitch of wheel, definition of, 10 Circumferential pitch, definition of, 10 Daimler axle, test of, 33, 74 Dedendum, 10, 16 Dennis, early worm gear, 2 Efficiency, Daimler worm axle, 74, foot note, formula for, 72, 73, 74 of worm gearing, 63 et seq., 89 Forms of teeth, 29 Friction, Beauchamp Tower experi- ments, 63 ' coefficient of, 63 in axle, author's experiment, 67 Gear ratio, calculations for, 17 definition of, 8 Gliding angle, 26 Globoid gear, 3 Hardness of materials used, 5, 6^ Heat generated, 58 Hindley worm gear, 7, 8, 29 Hobbing, 6 Included angle, calculations of, 25 of thT-ead, definition of, 11 Interference of wheel teeth, 22 Kennerson, Professor, on Brown & Sharpe experiments, 74 Lanchester, early worm gear, 2, 8 gear, 30 Lead angle, calculations for, 17, 19 definition of, 10 definition of, 9 of worm, 19 Length of teeth, calculation of, 27 of thread, 19 of worm, definition of, 10 determination of, 17 Lost work of worm, 21 Lubricating oil, 57 Manufacturing methods, 6, 7 Materials for worm gears, 5, 6 Mechanical efficiency, 72, 73, 74 Normal pitch, definition of, 10 Oil, film, 56 lubricating, 57 Pierce-Arrow Motor Car Co. worm axles, 2, 77 Pitch, calculation of, 17 in relation to horsepower, 14 line of wheel, definition of, 9 of worm, definition of, 10 selection of, 13 Pressure angle, 22 definition of, 11 95 96 INDEX Proportions of gears. Rack, 8 Radiation of heat from axle, 59 Reversibility, 8, 23, 89 Rubbing velocity, 19 Screw, definition of, 1 Strength of bronze alloy, 5 of steel for worm, 6 of teeth, 34 Stresses in gears, 38-46, 85 Subtended angle, definition of, 11 Symbols, 11, 12 Temperature, coefficient, 58 maximum permissible, 58 of gears, 51 Teeth in contact, number of, 17 Thread, form of, 22 proportions of, 26 Threads, number of, 17 Tooth pressure, effect of varying, 62 pressures, calculations of, 28 Tower, Beauchamp, friction experi- ments, 63, 68 Tractive resistance, 34 Velocity, rubbing, effect of varying, 62 Viscosity of lubricant, 52 Width of worm wheel, 47 Worm gearing, definition of, 8 location of, 77 material for, 6 wheel, material for, 5 size, determination, 16 /