TJ 184 .076 SSi A TREATISE ON Gear Wheels, BY GEORGE B. GRANT. PUBLISHED BY GEORGE B. GRANT, LEXINGTON, MASS. rniLADELPHIA, PA. SIXTH EDITION PRICE, clothi and. gilt, $1.00 post-paid, paper covers, .60 AQENXS WANXKD. Any active and intelligent man can make money selling this book. I allow a liberal dis- count and take back unsold copies. Write for circular of " terms to agents." Con.trit>i_iteci to tti.© A^s^ERICA.N NlACtnNISX. Kor 1890. COPIED from the American Machinist, and published in full by the " English Mechanic and World of Science," and by the " Mechanical World." ADOPTED and used as a text or reference book by Michigan and Cornell Universities, Rose Polytechnic Institute, and by many schools of mechanics and drafting. OopyrigtLt 1S93. By GKORGE; B. ORANT. A Working" Course of Study. It is not necessary that the student, especially if he Is a workman, should learn all that is taught in this book, for it contains much that is not only difficult but also of minor practical importance. The beginner is therefore advised to master only the following sections : I, 2, 7 to 15, 23, 25, 31, 32 of the general theory; 35 to 47 of the spur gear ; 53 to 64 of the involute tooth ; 76, 77, 80 to 83, 89 of the' cycloidal tooth ; 91, 95, 97 of the pin tooth ; 98, 99, III, 113 to 119 of spiral and worm gears; 154, 157, 158, 161 to 169 of the bevel gear. These include not half of the whole matter, but, knowing this much well, the student has a good outline knowledge of the whole, and he can then take the balance at leisure. osm A TREATISE ON GEAR WHEELS 1. THEORY OK TOOTH ACTION. -INTRODUCTORY-. The present object is practical, to reach and interest the man that makes the thing written of ; the machinist or the millwright that makes the gear wheel, or the drafts- man or foreman that directs the work, and to teach him not only how to make it, but what it is that he makes. To most mechanics a gear is a gear. *' A yellow primrose by the shore, A yellow primrose was, to him, And it was nothmg more ; " and, in fact, the gear is often a gear and nothing more, sometimes barely that. But, if the mechanic will look beyond the tips of his fingers, he will find that it can be something more ; that it is one of the most interesting objects in the field of scien- tific research, and not the simplest one ; that it has received the attention of many cele- brated mathematicians and engineers ; and that the study of its features will not only add to his practical knowledge, but also to his entertainment. There is an element in mathematics, and in its near relative, theoreti- cal mechanics, that possesses an educating and disciplining value beyond any capacity for earning present money. The thinking, inquisitive student of the day is the success- ful engineer or manufacturer of the. future. The method will be fitted to the object, and will be as simple and direct as possible. It is not possible to treat all the items in simple every-day fashion, by plain graphical or arith- metical methods, but where there is a choice the path that is the plainest to the average intelligent and educated mechanic will be chosen. A thousand pages could be filled with the subject and not exhaust anything but the reader thereof, but what is written should receive and deserve attention, and must be condensed within such reasonable limits, that it shall not call ^f or more time and labor than its limited application will warrant. Demon- strations and controversies will be avoided, and the matter will be confined as far as is possible to plain statements of facts, with illustrations. The simplest diagram is often a better teacher than a page of description. First, we shall study the odontoid or pure tooth curve as applied to spur gears, then we shall consider the involute, cycloid, and pin tooth, special forms in which it is found in practice ; then the modifications of the spur gear, known as the spiral gear, and the elliptic gear ; then the bevel gear, and lastly the skew bevel gear. Literature. 3, — PARTICULARLY LMPORTANT. Begin at the beginning. The natural tendency is too often to skip first principles, and begin with more ad- vanced and interesting matter, and the result is a trashy knowledge that stands on no foundation and is soon lost. When a fact is learned by rote it may be remembered, but when it follows naturally upon some simple principle it cannot be forgotten. Therefore the student is urged to begin with and pay close attention to the odontoid or pure tooth curve, before going on to its special applications, for the apparently dry and trivial matter relating to it is really the foundation of the whole subject. The usual course is to begin at once with, the cycloidal tooth, to hurry over the in- volute toolh, and then, if there is room, it is stated that such curves are particular forms of some confused and indefinite general curve. Our course will be to study the unde- fined tooth curve first, and then take up its particular cases. 4. — LITERATURE. It is impracticable to acknowledge all the sources from which information has been drawn, but it is in order to briefly mention the principal works devoted to the subject. Professor Herrmann's section of Professor Weisbach's "Mechanics of Engineering and Machinery" is the most important work that can be named in this connection. It treats of much besides the teeth of gears, but its treatment of that branch is particularly valuable. It is not easy reading. Wiley, $5.00. Professor Willis' "Principles of Mechan- ism " is a celebrated book, now many years behind the age, but it is, nevertheless, of the greatest value and interest in this matter. To Willis we are indebted for many of the most important additions to our knowledge of theoretical and practical mechanism. Long- mans, $7.50. Out of print. Professor Rankine's " Machinery and Mill- work" should not be neglected by the student, for, although it is the dryest of books, its value is as great as its reputation. Griffin, $5.00. Professor MacCord's "Kinematics" is a work that abounds in novelties, and is writ- ten in an attractive style. It contains many errors, and some hobbies, and needs a thorough revision, but the student cannot afford to avoid it, or even to slight it. Wiley, $5.00. Mr. Beale's "Practical Treatise on Gear- ing " is really practical. Many of the so-call- ed "practical " books are neither practical or theoretical, but we have in this small book a collection of workable information that should be within the reach of every man who pretends to be a machinist. We have drawn from it, by permission, particularly with regard to spiral and worm gears. Mr. Beale's experimental work, in connection with the spiral gear, has been of great service. The Brown & Sharpe Mfg. Co., cloth fl.OO, paper 75c. Professor Reuleaux's " Konstrukteur " is a justly celebrated work in the German lan- guage. A translation of it is now being published in an American periodical — Me- chanics. Professor Klein, the translator of Herr- mann's work, has lately published the " Ele- ments of Machine Design," a collection of practical examples, with illustrations. J. F. Klein, Bethlehem, Pa., $6.00. "Mill Gearing," by Thomas Box, is a practical work by an engineer, and from it we have drawn much of our matter on the cloudy subject of the strength and horse- power of gearing. Spon, $3.00. "Elementary Mechanism," by Professors Stahl and Woods, is a recent work of general merit. It is well designed as a text book, and treats the subject in a simple and in- teresting manner. Van Nostrand, $2.00. In addition to the above works, reference may be made to numerous articles to be found in periodicals, notably in the ' ' Ameri- can Machinist," the "Scientific American Supplement," the "Journal of the Franklin Institute," " Mechanics," and the " Transac- tions of the American Society of Mechanical Engineers." Ge7ieral Theory This, the science of pure mechanism, re- lates exclusively to the constrained and geometric motions of mechanism, and it has nothing to do with questions of force, weight, velocity, temperature, elasticity, etc. The path of a cannon ball is not with- in the field of kinematics, because it depends upon time and force. A belt and pulley are kinematic agents, because the contact be- tween them can be assumed to be definite, KINEMATICS. and the action is therefore geometric, but the slipping and stretching of the belt is not kinematic. The action of gear teeth upon each other is purely kinematic, but we can- not consider whether the material is wood, or steel, or wax, whether the gears are lifting one pound or a ton, or whether they are run- ning at one revolution per second or one per day. 6. — ODONTICS. The name "odontics' may be selected for that limited but important branch of kinematics that is concerned with the trans- mission of continuous motion from one l)ody to another by means of projecting teeth. Even this restricted corner of the whole subject is too large for the present purpose, for it covers much that cannot be considered within our set limits, and gear wheels must, therefore, be defined as devices for trans- mitting continuous motion from one fixed axis to another by means of engaging teeth. Thus confined, gear wheels may be con- veniently divided into three general classes. Skew bevel gears, transmitting motion be- tween axes not in the same plane. Bevel gears, transmitting motion between intersecting axes. Spur gears, transmitting motion between parallel axes. The last two classes are particular cases of the first; for, if the shafts may be askew at any distance, that distance may be zero, and if they intersect at any point, that point may be at infinity. It would be scientifically more correct to first develop the skew bevel gear, and from that proceed to the bevel and spur gear, but practical clearness and convenience is often more to be admired than strict accuracy, and, as the true path is difficult to follow, we shall enter in the rear, and consider the spur gear first. Odontics does not properly include the consideration of questions of strength, pow- er and friction, but we must admit certain important items in that direction. 7.— PITCH The fixed axes are connected with each other by imaginary surfaces called "axoids," or pitch surfaces, touching each other along a single straight line. "We must imagine that the pitch surfaces roll on each other without slipping, as if adhering by friction. The whole object of odontics is to provide these imaginary surfaces with teeth, by SURFACES. which they can take advantage of the strength of their material and transmit power that is as definite as the geometric motion. The pitch surface of the skew bevel gear is the hyperboloid of revolution, which be- comes a cone when the axes intersect, and a cylinder when the axes are parallel. 8. — NORMAL SURFACES. An important adjunct of the pitch surface is the normal surface, or surface that is everywhere at right angles to both pilch sur- faces of a pair of axes, and upon which the action of the teeth on each other may best be studied. For the skew bevel gear there does not appear to be any normal surface. For the bevel gear the normal surface is a sphere, and for the spur gear the sphere becomes a plane. General Theo ry. 9. — UNCERTATNTIES. The theory of tooth action is not yet fall and definite in all its parts, for there are some disputed points, and some confusion and clashing of rules and systems. This is particularly the case with the theory of spiral and skew bevel teeth, for much of the work that has been done is clearly wrong, and there is little that has been definitely decided. 10.— PITCH Fig. 1. JPitch cylinders Two cylinders, A and B, Fig. 1, that will roll on each other, will transmit rotary mo- tion from one of the fixed parallel axes c and O to the other, if their surfaces are provided with engaging projections. When these projections are so small that they are imperceptible, the motion is said to be transmitted by friction, and it is prac- tically uniform. But when they are of large size, and readily observed, the motion, CYLINDERS. although it is unchanged in nature, is said to be transmitted by direct pressure, and it is irregular unless the acting surfaces of the projections are carefully shaped to produce an even motion. The whole object of odontics is to so shape these large projections or teeth that they shall transmit the same uniform motion be- tween the rotating cylinders, as would be apparently transmitted by friction. These cylinders are imaginary in actual practice, although they are one of the principal elements of the theory, and they are called the axoids, or pitch cylinders of the gears. The normal surface (8) of the spur gear ip a plane, and, as all sections by normal- sur- faces are alike, we can study the action on a plane figure easier than in the solid body of the gear. 11.— THE LAW OF Fig, 2. Tooth action With the above conditions given we can deduce the following law: TOOTH CONTACT. The common normal to the tooth curves must pass through the pitch point. That is, in Fig. 2, if the tooth curves OD and o d are to transmit the same motion between the pitch lines pi and PL as would be transmitted by frictional contact at the pitch point 0, they must be so shaped that their common normal Op at their com- mon point p shall pass through that pitch point. Conversely, if the tooth curves are so shaped that their common normal always passes through the pitch point, they will transmit the required uniform motion. 12. — THE ODONTOID. This universal law enables us to define the "odontoid," or pure tooth curve, for the contact of the pitch lines at the pitch point is continuous and progressive, and, if the tooth curves are to transmit the same motion, their normals must be arranged in a contin- uous and progressive manner. The normals nl, as in Fig. 3, must be arranged without a break or a crossing, not only springing from the odontoid at consecutive points, but intersecting the pitch line at consecutive points. This arrangement may be called. General Theory. Fig Odontoids •'consecutive," and the definition is not a law by itself, but an expression of the given universal law. It is seen that the odontoid is inseparably connected with its pitch line, and that the same curve may be an odontoid with re- spect to one pitch line, and not with respect to some other. The curve Fig. 4 is an odontoid with respect to the pitch \vaQpl, but not with respect to the pitch line pV be- yond the point p at which the normal is tan- gent to that pitch line. The odontoid, so far as defined, is not a definite thing, and, for practical purposes, it must be given some particular shape. It may be involute or cycloidal, or of other form, but must always have normals ar- ranged in consecutive order. 13. — THE LINE OP ACTION. As the tooth curves od and OD, Fig. 5, work together, the point of contact will travel along a line Op W called the "line of action." There is a definite relation between the odontoid and the line of action, so that, if either one is given, the other is fixed. If the odontoid OD is given, with its pitch line PL, the line of action is determined without reference to the pitch line pi or its odontoid; and, conversely, if the pitch line and line of action are given, the odontoid to correspond is determined. 14.— INTERCHANGEABLE ODONTOIDS. This feature leads at once to the broad and useful fact that all odontoids, on pitch lines of all sizes, that are formed from the same line of action, will work together inter- changeably, any one working with any other. Therefore, to produce an interchangeable set of odontoids we can choose any one line of action, and form any desired number of them from it. 15. — INTERNAL CONTACT. The pitch lines of Fig. 5 curve in opposite directions, and the contact is said to be " ex- ternal." But the principles involved are in- dependent of the direction of the pitch lines, and they may curve in the same direction, as in Fig. 6, in " internal" contact. Tooth contact is between lines only, there being no theoretical need of a solid material on either side of the line, so that either side Interndl action Cusps and Terminals. of the tooth may be chosen as the practical j workiBg side. Therefore the internal gear is precisely like the external gear of the same pitch diameter, working on the same lines of action, so far as the odontoids are concerned, as illustrated by Fig. 7. Fig. 7i Intenval and external teeth 16. — THE CUSP AND INTERFERENCE. When, as in Fig. 8, the pitch circle _p Ms so small with respect to the line of action O C G" W, that two tangent circles C c' and G" c" can be drawn to the line of action from the center G of the pitch line, we shall have a troublesome convolution in the resulting flank curve o d. This convolution will be formed of two cusps, a first cusp c' on the inner tangent arc, the "base circle" G' c' , and a second cusp c" on the outer tangent arc G" c". This happens with any form of odontoid, although sometimes in disguised form, and creates a practical difficulty that can be avoided only by stopping the tooth curve at the first cusp c' . Furthermore, any odontoid OD that is to work with the odontoid o d, must be cut off at the point k on the "limit line" G' k through the point G' from the center c. If the odontoids, when the pitch line is so small that the cusps occur, are not cut off as required, the action will still be mathemati- cally perfect, but, as the contact changes at a cusp, from one side of the curve to the other, the action is no longer practicable with solid teeth. The curves will cross each other, and there will be an interference. 17. — THE SMALLEST PITCH CIRCLE. To determine the smallest pitch circle that can be used, and avoid the cusps altogether, find by trial the point G, Fig. 9, from which but one tangent arc G' c' can be drawn to the line of action G' W. This point will be the center of the smallest pitch circle, and all points outside of it will avoid interference, while all inside of it will be subject to it. 18. — THE TERMINAL When a tangent arc can be drawn, from the pitch point as a center, to the line of action at any point T, except the vertex TT, Fig. 10, there will be a corresponding cross- ing of the normals to the odontoid commenc- ing at the point t, and a termination of the action when the point t reaches the point T. As the action approaches the terminal point 2' there will be two points of action, POINT, Terminal point Secondary Action. since the odontoid crosses the line of action at two points— one point of direct and ordi- nary action at 8, and another point of retro- grade and unusual action at Y. These two points of action will come together at T, the odontoid will leave the line of action, and all tooth action will then cease. The retro- grade action is theoretically and actually correct, but it is so oblique that it is of no practical value, and therefore the odon- toid may as well be cut off at its terminal point t. 19. — SPEED OF THE Lay off 8, Fig. 5, to represent the speed of the pitch lines, and draw S A at right angles with the common normal p. Draw p C tangent to the line of action at the point of action p. Lay o&p B equal to ^, and draw B G at right angles to B. Then p G will be the speed of the point of action along the line of action. POINT OF ACTION. When the line of action is a circle the angle 8 Ais always equal to the angle Bp G, and therefore the speed of the point of action is uniform, and equal to that of the pitch lines. If the line of action is a straight line the angle Bp (7 will be constant — always zero — and therefore the speed of the point of action will be uniform and always equal to ^. 2(X— NATURE OF THE TOOTH ACTION. The nature of the action may be deter- '■^Ined by a study of the normal intersections; the intersections of the normals with the odontoid being at uniform distances apart, their intersections with the pitch lines will indicate the action of the teeth. If the nor- mal intersections, as in Fig. 3, are quite regu- lar, the action of the teeth will be smooth and regular, while if they are crowded with- in a narrow space the action of the tooth will be crowded and jerky. 21. — THE SECONDARY LINE OP ACTION. From the universal law of tooth contact stated in (11) we can reason that any point on the tooth curve is in position for contact whenever its normal passes through the pitch point 0, and therefore that the point will then be upon a line of action. In Fig. 11 the normal to the point p must cross the pitch line twice — at a primary in- tersection a, and at a secondary intersection b, and therefore there will be a point of action on a primary line of action Jf' at q, when the curve has moved so that the pri- mary point of intersection a is at the pitch point 0, and a point of action w on a second- ary line of action, when the secondary point of intersection b has reached the pitch point. Therefore there will generally be not only the primary line of action q M or q' M' , but also a secondary line w T or w' Y'. The secondary line of action must have the same property as the first, as a locus of con- tact, and therefore if we can so arrange two pitch lines with their odontoids that their secondary lines of action coincide, there will be secondary contact between the odontoids. Seconrlary actio' t When it so happens that both primary and secondary lines coincide, we shall have double contact. Two points of contact will exist at the same time, one on the primary and the other on the secondary line of action. The secondary lines of action cannot be made to coincide when the contact is exter- nal, but when it is internal they sometimes can be, so that the matter has an application to internal gears. 8 Interchangeable Tooth. It is to be noticed that the primary line is independent of the pitch line, while the sec- ondary is dependent upon it. Secondary contact is an interesting feature of tooth action, but it is of small importance, and has been studied but little. 22. — THE INTEKCHANaEABLE TOOTH. The simple odontoid so far studied is the perfect solution of the problem from a mathematical point of view, for it will trans- mit the required uniform motion as long as it remains in working contact. But from a mechanical point of view it is still incom- plete, as it works in but one direction, through but a limited distance, and, although the odontoids are interchangeable, the gears are not. In order that the gears shall be fully in- terchangeable, it is necessary that the teeth shall have both faces and flanks, and that the line of action for the face shall be equal to that for the flank; that is, the tooth must have an odontoid on each side of the pitch line, the face o d. Fig. 12, outside, and the flank o d' inside of it, and the line of action I a for the faces must be like the line of action I a' for the flanks. If so made, any gear will work with any other, without re- gard to the diameters of the pitch lines. But such a gear will run in but one direc- tion, and to make it double-acting it must have odontoids facing both ways, as in Fig. IS. Gears so made will be both double-act- ing and interchangeable, and it is not neces- sary that both sides of the tooth shall be alike. Again, the unsymmetrical gear of Fig. 13 fails when it is turned over, upside down , for then the unlike odontoids come together, and, to avoid this last difliculty, all four of the lines of action must be alike, producing the complete and practically perfect tooth of Fig. 14. We can therefore define the completely in- terchangeable tooth, as the tooth that is formed from four like lines of action. Fig. 12. Unsymtnetrical teeth Fig. 14. Complete teeth 23. -rNTEKCHANGEABIiE BACK TOOTH. When the pitch line is a circle the flanks of the tooth are not like the faces, but when it is a straight line there is no distinction be- tween face and flank. We then have the im- portant practical fact that the four odontoids of the interchangeable rack tooth are alike. Construction by Points. 24. — CONSTRUCTION BY POINTS. Wlien we have an odontoid and its pitch line given, it is a very simple matter to con- struct either the line of action or the conju- gate odontoid for any other pitch line. We know, for example, the odontoid s p. Fig. 15, on the pitch line p I, and it is re- quired to construct an odontoid on the pitch line P L that is conjugate to it. As the odontoid is given we know or can construct its normals. Construct the normal p a from any chosen point p, draw the radial line da C, lay off J. equal to a 0, draw the radial line A C, lay off the angle NAD equal to the angle n a d, lay off P J. equal to pa, and P will be a point in the required conjugate odontoid 8 P. P A will be a normal to the curve. Construct a number of points by this process, and draw the required cv^ le through them. The tangents s t and 8 ITmake equal angles with the pitch lines, so that the required curve can often be fully determined by drawing its tangent and one or two points. To construct the line of action, make the angle m e equal to the angle n a d, and lay off g equal to p a. The point g- is on a circle from either p or P drawn from the centers C, and is the point at which p and P will coincide when the two curves are in working contact, the normals p a and P A then coinciding with the radiant q. Fig. 15. Construction hy points When the line of action alone is given, the odontoids for given pitch lines are fully de- termined, but there seems to be no simple graphical method for constructing them ex- cept for special cases. They can be obtained by the use of the calculus (33), or drawn by the integrating instrument of (34). The two tooth curves thus constructed are paired, and are said to be *' conjugate" to each other. 25. — THE ARC OF ACTION The action between two teeth commences and ends at the intersections m and iV of the line of action with the addendum lines of the two gears, a I and A L, Fig. 16. The arc of action is the distance ah on. the pitch line that is passed over by the tooth while it is in action. The arc a passed over while the point of contact is approaching the pitch point, is called the arc of approach, and b, that passed over while the action is receding from that point, is the arc of recess. With a given line of action the arcs of ap- proach and recess can be controlled by the addenda. If it is desirable to have a great xecess and a small approach, the addendum of the gear that acts as a driver is to be in- creased. When there is a limit line (16), it limits the addendum and the arc of action. 10 Molding Processes. 26.— OBLIQUITY When a pair of teeth bear upon each other, the direction of the force exerted be- tween them is that of the common normal Op, Fig. 17, and passes through the pitch point 0. Except when the point of contact Is at the pitch point the direction of the pressure will deviate from the normal to the line of centers by the angle of obliquity Z p, and with many forms of teeth the angle is never zero. The force exerted between two teeth at their point of contact is found by laying off the tangential force IT with which the driv- ing gear J) is turning, and drawing the line H V parallel to the line of centers, to find the force V — P JS^. It is proportional to the secant of the angle of obliquity, and in- creases rapidly with that angle. The chief influence of the obliquity is upon the friction between the teeth, and con- sequent ineflSciency of the gear, and upon the destruction by wearing. It is par- ticularly important upon the approaching action, and a gear that is otherwise perfect may be inoperative on account of excessive obliquity. Although the direct pressure of the teeth upon each other at their point of contact OF THE ACTION K Fig* 17. will vary with the obliquity, the tangential force exerted to turn the gear is always uniform. Leaving friction out of the calcu- lation, the two gears of a pair always turn with the same force at their pitch lines. The obliquity of the action has an effect upon the direction and amount of the pressure of the gear upon its shaft bearing, but the usual variation is of little conse- quence. It is desirable that the pressure between the teeth should be as uniform as possible, not only in amount, but in direction, and excessive obliquity is to be carefully avoided. 27.— CONSTRUCTION BY MOLDING. The mode of action of the conjugate teeth upon each other, suggests a process by which a given tooth can be made to form its conju- gate by the process of molding. The given tooth, all of its normal sections being of some odontoidal form, is made of some hard substance, while the blank in which the conjugate teeth are to be formed is made of some plastic material. The shafts of the two wheels are given, by any means, the same motions as if their pitch surfaces were rolled togeiher. The hard tooth will then mold the soft tooth into the true conju- gate shape. It matters not what shape is given the molding tooth, if its sections are all odon- toidal, and a twisted or irregular shape will be as serviceable as the common straight tooth. This process is continually in operation be- tween a pair of newly cut teeth, or between rough cast teeth, until the badly matched surfaces have been w^orn to a better fit, but it is too slow for ordinary purposes, and is of little practical value. Gears can be formed by this process, by rolling a steel forming gear against a white hot blank, but the process can hardly be called practical. 28.— MOLDING PLANING PROCESS, Although the described molding process is of limited practical value, having but one •direct application, it leads to a process of great value when the tooth is straight or of such a shape that it can be followed by a planing tool, its normal sections being alike. The originating tooth is fixed in the shape of a steel cutting tool C, Fig. 18, which is ADVERTISEMENT. SPOKED. WEBBED. PLAIN. READY MADE GEARS READY MADE IRON GEARS WITH CUT TEETH. READY MADE IRON GEARS WITH CAST TEETH. READY MADE BRASS GEARS. SPUR GEARS, MITER GEARS, BEVEL GEARS, INTERNAL GEARS, CROWN GEARS, PINION WIRE, RATCHETS, RACKS. A ready made gear can be obtained immediately and costs not nearly as much as a similar gear made to order, frequently not a half or a third as much. I liave no agents and sell only from my shops at Lexington, Mass., and Philadelphia, Pa. GEORGE B- GRANT. ADVERTISEMENT ,-addendum ,-clearance: - I OF INCREMENT PINION 1 ■; I I GRANT'S GEAR BOOK. A PAMPHLET OF FrPTY PAGES FULL OF INFORMATION ON GEARING. PRICE FIFTEEN CENTS. WIJH EACH ONE I SEND A COUPON THAT IS GOOD FOR FIFTEEN CENTS IN TRADE. PLEASE CALL THIS PAMPHLET BY THE NAME "GRANT'S GEAR BOOK." GEORGE B. GRANT, LEXtNGTON, MASS. PHILADELPHIA, PA. Pla7iing Processes. 11 rapidly reciprocated in guides O, in the direction of the length of the looth, as the two pitch wheels A and B are rolled together. Although the tool has but a single cutting edge, its motion makes it the equivalent, of the molding tooth, and it will plane out the conjugate tooth Z> by a process that is the equivalent of the more general molding process. A simple graphical method is founded upon this molding process, the shaping tool taking the form of a thin template G, Fig. 19, that is re- peatedly scribed about as the pitch wheels are rolled together, the marks combining to form the conjugate tooth curves D. This mechanical process has the decided advantage over the process of construction by points (24), that the tooth is formed with a correct fillet (44), and is much stronger, i^he dotted lines show the tooth that would be constructed by points. The only practicable method for forming the line of action when this method is used is by observing and marking a number of points of con- tact between the teeth. This method is applicable to all possible forms of spur teeth, either straight, twisted or spiral. It can be practically applied only to the octoid form of bevel tooth. On account of the fillet (44) that is formed by this process, the tooth space cannot be used with a mating gear having more teeth than that of the forming gear, although it belongs to the same interchangeable set. The tooth space of the figure will not run with a tooth on a pitch line larger than the pitch line A. Therefore the rack tooth must be usea as the forming tooih, to allow of the use of all gears of the set up to the rack . Gears of the set thus formed will not work with internal gears. Molding planing tnethod Fig. 18. Graphical molding method 29 -LINEAR PLANING PROCESS A second planing process, quite distinct from the molding process of (27), is founded upon the fact that the tooth curves are in contact at a single point which has a progressive motion along the line of action. Therefore if a single cutting point p. Fig. 20, is caused to travel along the line of action with the proper speed relatively to the speed of the pitch Hue, it will trim the tooth out- line to the proper odontoidal shape. The figure shows the application to Linear planing' method 12 Particular Forms. the involute tooth, the path of the cutting point being the straight line I a, and its speed being the speed of the base line h I, When the cutting point follows the circu- lar line of action with a speed equal to that of the pitch line, it will plane out the cycloidal tooth curve. This process is applicable to all possible forms of gear teeth, either spur or bevel, in either external or internal contact. When the curvature of the odontoid will permit, the milling cutter may take the place of the planing tool, and is the equivalent of it. 30. — THE RACK The molding planing, process of (28) sup- plies a means for easily and accurately pro ducing an interchangeable set of gears or cutters for gears, and it is best applied by means of the rack tooth as the originator. All four curves of the rack tooth being alike, the tooth is easily formed, particularly for the involute or the segmental systems, and it is a matter of less consequence that the curves ORIGINATOR. shall be of some particular form, if care i« taken that it is odontoidal. It has been taught, and it is therefore some- times considered, that any * ' four similar and equal lines in alternate reversion" will an- swer the purpose, but it is necessary that the four similar curves shall be odontoids. Four circular arcs, with centers on the pitch line, will answer the definition, but are not odontoids. 31,— PARTICULAR FORMS OF THE ODONTOTD. The odontoid, as so far examined, is un- defined except as to one feature of the ar- rangement of its normals, and to bring it into practical use it is necessary to give it some definite shape. This is most easily ac- complished by choosing some simple curve for the rack odontoid, and from that making an interchangeable set. A more correct but much more difficult method would be to choose some definite line of action, and from that derive the odontoids. If the rack odontoids are straight lines. Fig. 21, the common involute tooth system will be produced, and the line of action will be a straight line at right angles with the rack odontoid. For bevel teeth, as will be shown, the straight line odontoid produces the octoid tooth system, while to produce the involute system it is necessary to define the line of action as a straight line, and derive the system from that. If the rack odontoids are cycloids, as in Fig. 22, the resulting tooth system will be the cycloidal, commonly misnamed the " epicycloid al " system. The line of action will be a circle equal to the roller of the cycloid. If the rack odontoids are segments of cir- cles from centers not on the pitch line, but inside of it, as in Fig. 23, the tooth system Fig. 2U Segmental Fig. 23. Rolled Curve Theory. 1:3 will be the segmental, and its line of action will be the loop of the " Conchoid of Nico- medes." If we choose a parabola for the rack tooth, as in Fig. 24, the parabolic sysiem will be formed with its peculiar "hour glass" line of action. Only three of these tooth systems are in actual use, the involute and the cycloidal for spur gears, and the octoid for bevel gears only, and we will therefore confine the ap- plication of the theory to them. Only one of the systems in common use for spur gears, the involute, should be in use at all, and we will pay priocipal attention to that. I*arabolic Fig, The segmental system would be superior to the cycloidal, and in many cases to the in- volute; but as there is already one system too many, we will not attempt to add another. 32,— THE ROLLED CURVE TIIEOLY If any curve B, Fig. 25, is rolled on any pitch curve p I, & point p in the former will trace out on the plane of the latter a curve s pz, called a rolled curve. The line pq, from the tracing point p to the point of contact q, is a normal to the curve s p z, and, as all the normals are ar- ranged in "consecutive" order, that curve must be an odontoid. The converse of this statement is also true, that all odontoids are rolled curves ; but the fact is generally ery far fetched and of no practical imp'^rtance. It is also a property of all such curves that are rolled on different pitch lines, that they are interchangeable. This accidental and occasionally useful feature of the rolled curve has generally been made to serve as a basis for the general theory of the gear tooth curve, and it is re- sponsible for the usually clumsy and limited treatment of that theory. The general law is simple enough to define, but it is so diffi- cult to apply, that but one tooth curve, the cycloidal, which happens to have the circle for a roller, can be intelligently handled by it, and the natural result is, that that curve has received the bulk of the atten- tion. For example, the simplest and best of Moiled curve Fig. 26. all the odontoids, the involute, is entirely beyond its reach, because its roller is the logarithmic spiral, a transcendental curve that can be reached only by the higher mathe- matics. No tooth curve, which, like the involute, crosses the pitch line at any angle but a right angle, can be traced by a point in a simple curve. The tracing point must be the pole of a spiral, and therefore the trac- ing of such a curve is a mechanical impossi- bility. A practicable rolled odontoid must cross the pitch line at right angles. To use the rolled curve theory as a base of operations will confine the discussion to the cycloidal tooth, for the involute can only be reached by abandoning its true logarithmic roller, and taking advantage of one of its peculiar properties, and the segmental, sinusoidal, parabolic, and pin tooth, as well as most other available odontoids, cannot be discussed at all. 33. — MATHEMATICAL RELATION OP ODONTOID AND LINE OP ACTION. In Fig. 26 the odontoid on the pitch line i by the relations P T = p t = y, and 2' 8 ~ p Z is connected wiih the line of action I a,\t — x, where P 8 Ss, the normal to the 14 Mathematical Relation. odontoid at the point P, T 8\s,ia, tangent to the pitch line at the intersection of the nor- mal, and P y is a normal to the tangent. When any odontoid is given by its equa- tion, that of the line of action can be found by a process of differentiation, and when the line of action is given by its equation, that of the odontoid can be found by a process of integration. These processes, for the general case vs^here the pitch line is curved, are quite intricate, but when the pitch line is a straight line, they are simple, and may be worked as follows. To get the equation of the line of action from that of the given rack odontoid, ar- range the equation of the odontoid in the form X = f(y), and put its differential co- efficient - — equal to — . Thus, the equation of the straight rack odontoid of the involute system is y = a; tan. ^, from which dx _ 1 _ y - X dy ~ ~ and y = taa. A ~ x' '^ tan. A the equation of the straight line of action at right angles to the odontoid. Again, the equation of the cycloid being x = vtr. sin.-i . X = ver. sin, -' V^ry — d X Ty y and 2! 2 -[_ 2^2 _ 2ry is the equation of the circular line of action. To get the equation of the odontoid when that of the line of action is given, arrange the equation of the line of action in the form v d X — —f iy), put it equal to -, — , and inte- X d y grate. Thus, the equation of the straight line of action being X lan. A we have dx d~y' and X tan. A = X tan. A is the equation of the straight odontoid at right angles to the line of action. Again, the equation of the circu- lar line of action being x^ -\- y^ = 2ry, we have X \/2ry—y'^ and X = ver. sin.-^ ; y — ^2ry cyclcidal odontoid. d X dy' is the ! 34. — THE ODONTOIDAL The form of the odontoid to correspond to a given line of action and a given pitch line can be determined only by the integral cal- culus (33), it evidently being impossible to contrive a general graphical or algebraic method. But it can be directly drawn by an instru- ment, the principle of which is analogous to that of the well-known polar planimeter for integrating surfaces. The bar R, Fig. 27, moves at right angles to the line of centers, and it moves the pitch wheel A, with the same speed at the pitch line. The bar G has a point p, that is confined to move in the given line of action p W, and it is so guided that it always passes through the pitch point 0. The two bars bear upon each other by friction, and we must suppose that there is no other friction to oppose the motion of the bar G. INTEGRATER. Odoutoidal Integrater Then the point p will trace out the odontoid spsupon the pitch wheel ^, or upon any other pitch wheel B rolling with the bar R on either side of it. 2. THK SF^UR. OEAR IN OErsTERAIv. 35. — THE CIRCULAR PITCH. The distance a 0, Fig. 1 4, covered by each tooth upon the pitch circle, is commonly called the "circular pitch," and often the "circumferential pitch." The term "pitch arc" is the most appropriate but is not in common use. This was formerly the measurement by which the size of the tooth was always stated, a tooth being said to be of a certain " pitch," and all of its other dimensions being expressed in terms of that unit, but it is fast being replaced, and should be entirely replaced, by the more convenient "diametral pitch " unit. The circumference of a circle is measured in terms of its diameter by means of an in- commensurable fractional number 3.14159, called TT (pi), and, therefore, if the tooth is measured upon the arc of the circle by means of the circular pitch, one of two inconveni- ences must be tolerated. Either the pitch must be an inconvenient fraction, or else the pitch diameter must be as inconvenient, for the gear cannot have a fractional number of teeth. The fractional calculations are so clumsy that a table of pitch diameters cor- responding to given numbers of teeth should be used, and errors in the laying out of the work are of constant occurrence. Again, outside of the liability of error in making calculations, the circular pitch sys- tem is a constant source of error in the hands of lazy or incompetent draftsmen or work- men, for there is a constant temptation, often yielded to, to force the clumsy figures a little to produce some desired result. For example, a millwright has to make a gear of fourteen inches pitch diameter with fourteen teeth. He finds by the usual computation that the circular pitch is 3.14 inches, and, as his odontograph has a table for three-inch pitch, he uses that with the remark that it is "near enough," laying the blame on the odontograph or on the iron founder if the resulting gear roars. His next order is for a gear of one-inch pitch to match others in use, and to be fourteen and a half inches diameter. The circumference of the pitch line is 45.53 inches, and he has his choice be- tween 45 and 46 teeth, both wrong. Per- haps the most frequent cause of error is that the workman is apt to apply a rule directly to the teeth of a gear he is about to repair or match, to get the circular pitch, and the result is more likely to be wrong than right. The best plan when using this unit is to get convenient pitch diameters and let the pitch come as it will, provided that gears that work together are of the same pitch, and that is simply a roundabout way of using the diametral pitch unit. When the circular pitch must be used the following l;:ible will greatly assist the work and save calculation. For example, the pitch diameter of a gear of three-quarter-inch pitch and 37 teeth is three-quarters the tabu- lar number 11.78, or 8.84 inches. PITCH DIAMETERS. For Onj; Inch Circular Pitch. for ant other pitch multiplt bt that pitch. T. P. D. T. P. D. T. P. D. T. P.D. 10 3.18 33 10.50 53 17.83 79 25.15 11 3.50 34 10.82 57 18.14 80 25.47 12 3.82 35 11.14 58 18.46 81 25.79 13 4.14 36 11.46 59 18.78 82 26.10 14 4.46 37 11.78 60 19.10 83 26.42 15 4.78 38 12.10 61 19.42 84 26.74 16 5.09 39 12.42 62 19.74 85 27.06 17 5.41 40 ]2.73 63 20.06 86 27.38 18 5.73 41 13.05 64 20.37 87 27 70 19 6.05 42 13.37 65 20.69 88 28.01 20 6.37 43 13.69 66 21.01 89 28.33 21 6.69 44 14.00 67 21.33 90 28.65 22 7.00 45 14.33 68 21.65 91 28 97 23 7.32 46 14 64 69 21.97 92 29.29 24 7.64 47 14.96 70 22.28 93 29.60 25 7.96 48 15.28 71 22 60 94 29.92 26 8.28 49 15.60 72 22.92 95 30.24 27 8.60 50 15.92 73 23.24 96 30.56 28 8.91 51 16.24 74 23.56 97 30.88 29 9.23 52 16.55 75 23.88 98 31.20 30 9.55 53 16.87 76 24.19 99 31.52 31 9.87 54 17.19 77 24.51 100 31.83 32 !0.19 55 17.51 78 24.83 36. — THE DIAMETRAL PITCH. This is not a measurement, but a ratio or proportion. It is the number of teeth in the gear divided by the pitch diameter of the gear. Thus, a gear of 48 teeth and 12 inches pitch diameter is of 4 pitch. The advantages of the diametral pitch unit are so apparent 16 Pitches and Addendum. that it is fast displaciDg the circular pitch unit, and has almost entirely displaced it for cut gearing. It is so simple that a table of pitch diameters is entirely useless, although such useless tables have been published. The diametral pitch is sometimes defined as the number of teeth in a gear of one inch diameter. It is a common, but bad practice, to designate diametral pitches by numbers, as No. 4, No. 16, etc. 37. — RELATION OF PITCH UNITS. The product of the circular pitch by the diametral pitch is the constant number 3.1416, so that if one is given the other is easily calculated. The following tables of equivalent pitches will be convenient in this connection. 38.— ACTUAL SIZES. Figs. 28 and 29 show the actual sizes of standard teeth of the usual diametral pitches, and give a better idea of the actual teeth than can be given by any possible description. They are printed from cut teeth, and may be depended upon as accurate. Diametral Circular Pitch. Pitch. 2 1.571 inch 2^4 1.396 •• 2/^ 1.257 " 2M 1.142 " 3 1.047 '^ 3)^ .898 " 4 .785 " 5 .628 '' 6 .524 '• 7 .449 '' 8 .393 " 9 .349 '• 10 .314 '' 11 .286 " 12 .262 " 14 .224 '' 16 .196 '• 18 .175 " 20 .157 '' 22 .143 " 24 .131 '' 26 .121 " 28 .112 '' 30 .105 " 32 .098 ^' 36 .087 " 40 .079 '' 48 .065 " Circular Diametral Pitch. Pitch. 2 1 571 1% 1 676 I'M 1.795 1% 1.933 iJ^ 2.094 1t5 2 185 1% 2.285 m 2.394 2.513 ^tS 2.646 13^ 2.793 ItV 2.957 3.142 II 3 351 3.590 T5 3.867 M 4.189 11 4.570 5.027 ft 5.. ^85 6.283 s 7 181 8 378 IB 10 053 /4 12.566 § 16.755 25.133 T^H 50.266 39. — ADDENDUM AND DEDENDUM. The tooth is limited in length by the circle a I, Fig. 30, called the addendum line, and drawn outside the pitch line at a given distance, called the addendum. Its depth is also limited hyaline ?• I, called the dedendum or root line, drawn at a given distance inside of the pitch line. The addendum and the dedendum are both arbitrary distances, but, for convenience in computation, they are fixed at simple fractions of the unit of pitch that is in use. When the circular pitch is used the ad' dendum is one-third of the circular pitch. When the diametral pitch unit is used the addendum is one divided by the pitch. It is customary to make the addendum and the dedendum the same, except in certain cases where some special requirement is to be satisfied. Addenda Clearance Backlash Actual Sizes. Actual Sizes. 2 \ Pitch. 3 Pitch. Fig, 29, Items of Construction. 19 40.— THE CLEARANCE. To allow for the inevitable inaccuracies of workmanship, especially on cast gearing, it is customary to carry the tooth space slightly below the root line to the clearance line c I, Fis:. 30. The clearance, or distance of the clearance line inside of the root line, is arbitrary, but it is convenient and customary to make it one- eighth of the addendum . 41. — THE BACK-LASH. When rough wooden cogs or cast teeth are used, the irregularities of the surface, and inaccuracies of the shape and spacing of the teeth, require that they should not pre- tend to fit closely, but that they should clear each other by an amount h. Fig. 30, called the back-lash. The amount of the back-lash is arbitrary. but it is a good plan to make it about equal to the clearance, one-eighth of the addendum. Skillfully made teeth will require less back-lash than roughly shaped teeth, and properly cut teeth should require no back- lash at all. Involute teeth require less back- lash than cycloidal teeth. 42. — THE STANDARD TOOTH. The tooth must be composed of odontoids, preferably of odontoids of which the proper- ties are well known, and an advantage is gained if it is still further confined to a par- ticular value of that odontoid. If the teeth are to be drawn by an odontograph some standard must be fixed upon, since the method will cover but one proportion of tooth. For example, the standard involute tooth is that having its line of action inclined at an angle of obliquity of fifteen degrees. For the cycloidal system the standard agreed upon is the tooth having radial flanks on a gear of twelve teeth. 43 . — ODONTOGRAPHS. The construction of the tooth is generally not simply accomplished by graphical means, as it is generally required to find points in the curve and then find centers for circular arcs that will approximate to the curve thus laid out. It is sometimes attempted to construct the curve by some handy method or empirical rule, but such methods are generally worth- less. An odontograph is a method or an instru- ment for simplifying the construction of the curve, generally by finding centers for ap- proximating circular arcs without first find- ing points on the curve, and those in use will be described. 44. — THE Wh%n the teeth are laid out by theory there will be a portion of the tooth space at the bottom that is never occupied by the mating tooth. Fig. 31 shows a ten-toothed pinion tooth and space with a rack tooth in three of its positions in it, showing the un- used portion by the heavy dotted line. If this unused space is filled in by a "fillet "/the tooth will be strengthened just where it needs it the most, at the root. The fillet is dependent on the mating tooth, and is therefore not a fixed feature of the tooth. If a gear is to work in an inter- changeable set, it may at some time work with a rack, and therefore its fillet should be fitted to the rack ; but if it is to work only FILLET. with some one gear it may be fitted to that. The light dotted line shows the fillet that would be adapted to a ten-toothed mate. The fillet to match an internal gear tooth would be even smaller than that made by the rack. The fillet 20 Equidistant Series. When the tooth is formed by the molding process of (27), or by the equivalent planing process of (28), the fillet will be correctly formed by the shaping tool, but not so when the linear process of (29) is used. When the tooth is drawn by theory or by an odonto- graph the fillet must be drawn in, and can be most easily determined by making a mating tooth of paper, and trying it in several posi- tions in the tooth space, as in the figure. Except on gears of very few teeth the strength gained will not warrant the trouble of constructing the fillet. 45. — THE EQUIDISTANT SERIES. When arranging an odontograph for drafting teeth, or a set of cutters for cutting them, we must make one sizing value do duty for an interval of several teeth, for it is impracticable to use different values for two or three hundred different numbers of teeth. The object of the equidistant series is to so place these intervals that the necessary errors are evenly distributed, each sizing value being made to do duty for several numbers each way from the number to which it is fitted, and being no more inaccurate than any other for the extreme numbers that it is iorced to cover. This series is readily computed for any case that may arise, and with a degree of ac- curacy that is well within the requirements of practice; by the formula , a s in which a is the first and z is the last tooth of the interchangeable series to be covered; n is the number of intervals in the series, and s is the number in the series of any interval of which the last tooth t is required. For example, it is required to compute the series here used for the cycloidal odonto- graph, having twelve tabular numbers to cover from twelve teeth to a rack. Putting a = 12,z = infinity, and n = 12, the formula becomes 12 X 12 12 X 12 144 t = 12 — 5 -f 12 12 -f 0~ 12 and then, by putting s successively equal to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, we get the series of last teeth, ld^\, 14|, 16, 18, 20^, 24, 28|, 36, 48, 72, 144, and infinity. These give the required equidistant series of inter- vals. 12 13 to 14, 15 to 16, 17 to 18, 19 to 21, 23 to 24, 25 to 29 30 to 36, 37 to 48, 49 to 72, 73 to 144, 145 to a rack ; and the method is as easily applied to any- other practical example. This formula and method is independent of the form and of the length of tho tooth, and therefore is applicable to all systems under all circumstances. This is proper and con- venient, for these elements can be eliminated without vitiating the results or destroying the "equidistant" characteristic of the series. The formula is an approximation based upon an assumption, but nothing more convenient or more accurate has so far been devised by laboriously considering all the petty elements involved. The sizing value, or number for which the tabular number is computed, or the cutter is accurately shaped, can best be placed, not at the center of the interval, but by considering the interval as a small series of two intervals, and adopting the intermediate value* The sizing value for the interval from ctodia given by the formula 2 cd c-\-d Thus, the sizing value for the interval from 37 to 48 teeth should be 41.8, and that for the interval from 145 to a rack should be 290. It is sometimes the practice to size the cut- ter for the lowest number in its interval, on the ground that a tooth that is considerably too much curved is better than one that is even a little too flat. This makes the last tooth of the interval much more inaccurate than if the medium number was used. Friction of Approach. 21 46. — THE HUNTING COG. It is customary to make a pair of cast gears 'with incommensurable numbers of teetb so that each tooth of each gear will work with all the teeth of the other gear. If a pair of equal gears have twenty teeth each, each tooth will work with the same mating tooth all the time; but if one gear has twenty and the other twenty-one teeth, or any two num- bers not having a common divisor, each tooth will work with all the mating teeth one after the other. The object is to secure an even wearing action; each tooth will have to work with many other teeth, and the supposition is that all the teeth will eventually and mysteriously be worn to some indefinite but true shape. It would seem to be the better practice to have each tooth work with as few teeth as possible, for if it is out of shape it will dam- age all teeth that it works with, and the damage should be confined within as narrow limits as possible. If a bad tooth works with a good one it will ruin it, and if it works with a dozen it will ruin all of them. It is the better plan to have all the teeth as near perfect as possible, and to correct all evident imperfections as soon as discovered. 47. — THE MORTISE WHEEL. Another venerable relic of the last century is the • * mortise " gear. Fig. 32, having wooden teeth set in a cored rim, in which they are driven and keyed. Where a gear is subjected to sudden strains and great shocks, the mortise wheel is better, and works with less noise than a poor cast gear, and will carry as much as or more power at a high speed with a greater dura- bility. But in no case is it the equal of a properly cut gear, while its cost is about as great. In times when large gears could not be cut, and when the cast tooth was not even ap- proximately of the proper shape, the mortise wheel had its place, but now that the large cut gear can be obtained the mortise gear should be dropped and forgotten. JHortise wheel Fig. 32. 48. — THE FRICTION OF APPROACH When the point of action between two teeth is approaching the pitch point, that is, when the action is approaching, the friction between the two tooth surfaces is greater than when the action is receding. This extra fric- tion is always present, but is most trouble- some when the surfaces are very rough, as on cast teeth, giving little trouble when the teeth are properly shaped and well cut. When the roller pin gear (93) is used, the friction between the teeth is rolling friction, and is no greater on the approach than on the recess. Hon of roach 22 Efficiency, The difference in the friction is probably- due to the difference in the direction of the pressure between the small inequalities to which all friction is due. When the gear B, Fig. 33, is the driver, the action between the teeth is receding, and the inequalities lift over each other easily, while if F is the driver, the action is approaching, and the Inequalities tend to jam together. In the exaggerated case illustrated, it is plain that the teeth are so locked together that ap- proaching action is impossible, while it is equally plain that motion in the other direc- tion is easy. The same action takes place in a lesser degree with the small inequalities of ordinary rough surfaces. The action of the common friction pawl,, which works freely in one direction and jams hard in the other, is upon the same principle. A weight may be easily dragged over a rough surface that it could not be pushed over by a force that is not parallel to the surface. The extra friction of approaching action can be avoided by giving the driver the long- est face. When the driver has faces only, and the follower has only flanks, the action is particularly smooth. Teeth that are subject to excessive maxi- mum obliquity, such as cycloidal teeth, should not be selected for rough cast gearing, for it is the maximum rather than the average obli- quity that has the greatest influence. 49. — EFFICIENCY OF GEAB TEETH. Much has been written, but very little has been done to determine the efl3ciency of the teeth of gearing in the transmission of power, and therefore but little of a definite nature can be said. The question is mostly a prac- tical one, and should be settled by experi- ment rather than by analysis. The only known experiments upon the fric- tion of spur gear teeth are the Sellers experi- ments, more fully detailed in (112), and but. one of these relates to the spur gear. From that one it is known that a gear of twelve teeth, two pitch, working in a gear of thirty- nine teeth, has an efficiency varying from ninety per centum at a slow speed to ninety- nine per centum at a high speed. That is, an average of five per centum of the power received is wasted by friction at the teeth and shaft bearings. This result is probably a close approximation to that for any ordinary practical case. Although theory can do nothing to de- cide such a question as this, it can do much to indicate probable results. If a pair of involute teeth, for example, move over a certain distance, w, either way from the pitch point, the distance being mea- sui-ed on the pitch line, they will do work that is theoretically determined by the formula : work done = *^-^r- . — — =— w^ 2 k h in which / is the coefficient of friction, P is the pressure, and k and h are the pitch radii of the gears. The positive sign is to be used for gears in external, and the negative sign for those in internal contact. The loss by friction, as shown by the for- mula, decreases directly as the diameters in- crease, the proportion of the diameters being constant. The loss increases rapidly with the distance of the point of action from the pitch point. When the contact is at the pitch point the teeth do not slide on each other, and there is no loss, but away from that point the loss is as the square of the distance in this case, and in a vStill greater proportion in the case of the cycloidal tooth. Therefore a short arc of action tends to improve the eflSlciency. It has been satisfactorily determined that the loss is greater during the approaching than during the receding action. This is not shown by the formula, but it may be laid to a variation in the coefficient /. The formula shows that the loss is inde- pendent of the width or face of the gear, and therefore strength can be increased by widening the face, without increasing the friction. If the work of internal gearing is com- pared with that of external gearing of the same sizes, the losses are in the proportion, k— h k+ h' Strength. 23 BO that the internal gear is much the more economical, particularly when the gear and pinion are nearly of the same size. If the gear is twice the size of the pinion the loss is but one-third of the loss when both gears are external. Small improvement can be effected, by put- ting a small pinion inside rather than outside of a large gear. A six-inch pinion working with a six-foot gear has but 1.18 times the loss by the same gears, when the gear is in- ternal. Theoretical efficiency is discussed at great length in the Journal of the Franklin Insti- tute, for May, 1887. Also by Reuleaux, and again by Lanza, in the Transactions of the American Society of Mechanical Engineers for 1887, and the discussion has been carried far enough. A series of experiments with gear teeth oi various sizes and forms, of various metals, would add greatly to our knowledge of this important matter. A true determination of the efficiency of the rough cast gear, as compared with that of the cut gear, would tend to discourage the use of the former for the transmission of power, for experiment would undoubtedly show that the power wasted by the cast gear would soon pay the difference in cost of the better article. 5C. — STRENGTH The strength of a tooth is the still load it will carry, suspended from its point, and is to be carefully distinguished from the horse-power, or the load the gear will carry in motion. The strength of a substance is not a fixed element, but will vary with different samples, and with the same sample under different circumstances ; allowance must be made for the amount of service the sample has seen, concealed defects must be provided against, and therefore nothing but an actual test will surely determine its character. Although no possible rule can be depended npon, the ultimate or breaking strength of a standard cast-iron tooth, having an addendum about equal to a third of the circular pitch, will average about three thousand five hun- dred pounds multiplied by the face of the gear and again by the circular pitch, both in inches. But a tooth should never be forced up to its ultimate strength, and the best practice is to give it only about one-tenth of the load it might possibly bear, so that the following rule should be used : Multiply three hundred and fifty pounds by the face of the gear, and again by the circular pitch, both in inches, and the product will be the safe working load of one tooth. Example : A cast-iron gear of one inch pitch, and two inches face, will safely lift 350 X 2 X 1 = 700 pounds, although it would probably lift 7,000 pounds. OP A TOOTH. When there are two teeth always in work- ing contact^ it is safe to allow double the load, but care must be taken that both teeth are always in full contact. A hard wood mortised cog has about one- third of the strength of a cast-iron tooth: steel has double the strength ; wrought-iron is not quite as strong. A small pinion generally has teeth that are weak at the roots, and then it will increase the strength to shroud the gear up to its pitch line, but shrouding will not strengthen a tooth that spreads towards its base, like an involute tooth, and when the face of the gear is wide compared with the length of the tooth the shroud is of rittle assistance. It does not increase the strength of a tooth to double its pitch, for when the pitch is increased the length is also increased, and the strength is still in direct proportion to the circular pitch, while the increase has reduced the number of teeth m contact at a time. Cut gears and cast gears are about equal as to actual strength, with the advantages in favor of the cut gear, that hidden d 3f ects are likely to be discovered, and that it is not as liable to undue strains on account of defective shape. The rules for strength must not be used for gears running at any considerable speed, for they are intended only for slow service, as in cranes, heavy elevators, power punches, etc. 24 Horse- Power , Although no rule can be called reliable, the one that appears to be the best is that given by Box, in his Treatise on Mill Gear- ing. Box's rule, which is based on many actual cases, and which gives among the lowest, and therefore the safest results, is by the formula: Horse-poM^er of a cast gear = 12 c-^f ^J dn J, 000 in which c is the circular pitch, /is the face, d is the diameter, all in inches, and n is the number of revolutions per minute. Example : A gear of two feet diameter, four inches face, two inches pitch, running at one hundred revolutions per minute, will transmit 12X3X2X4X a/ 24 X 100 ^ , ^ = 9.4 and 51. — HORSE-POWEB OF CAST GEARS. The horse-power of a gear is the amount of power it may be depended upon to carry in continual service. It is very well settled that continual strains and impact will change the nature of the metal, rendering it more brittle, so that a tooth that is perfectly reliable when new may be worthless when it has seen some years of service. This cause of deterioration is particularly potent in the case of rough cast teeth, for they can only approximate to the true shape required to transmit a uniform speed, and the continual impact from shocks and rapid variations in the power carried must and does destroy the strength of the metal. There are about as many rules for com- puting the power of a gear as there are manufacturers of gears, each foundryman having a rule, the only good one, which he has found in some book, and with which he will figure the power down to so many horses and hundredths of a horse as con- fidently as he will count the teeth or weigh the casting. Even among the standard writers on en- gineering subjects the agreement is no bet- ter, as shown by Cooper's collection of twenty-four rules from many different wri- ters, applied to the single case of a five-foot gear. See the ' ' Journal of the Franklin Institute" for July, 1879. For the single case over twenty different results were ob- tained, ranging from forty-six to three- hundred horse-power, and proving conclu- sively that the exact object sought is not to be obtained by calculation. This variety is very convenient, for it is always possible to fit a desired power to a given gear, and if a badly designed gear should break, it is a simple matter to find a rule to prove that it was just right, and must have met with some accident. 1,000 For bevel gears, take the diameter pitch at the middle of the face. It is perfectly allowable, although it is not good practice, to depend upon the gear for from three to six times'the calculated power, if it is new, well made, and runs without being subjected^ to sudden shocks and varia- tions of load. The influence of impact and continued service will be appreciated when it is con- sidered that the gear in the example, which will carry 9.4 horse-power, will carry seventy horse-power if impact is ignored, and the ultimate strength of the metal is the only dependence. • A mortise gear, with wooden cogs, will carry as much as, or more than a rough cast- iron gear will carry, although its strength is much inferior. The elasticity of the wood allows it to spring and stand a shock that would break a more brittle tooth of much greater strength. And, for the same reason, a gear will last longer in a yielding wooden frame than it will in a rigid iron frame. 52. — HORSE-POWEB, OF CUT GEARS. data upon which a reliable rule can be founded. Admitting, as we must, that impact is the chief cause of the deterioration of the We know a little, and have to guess the rest, as to the power of a cast gear, but with respect to that of a cut gear we are not as well posted, for there are no experimental The Involute Tooth. 25 cast gear, we are at liberty to assume that a properly cut and smoothly running cut gear is much more reliable. No definite rule is possible, but we can safely assume that a cut gear will carry at least three times as much power as can be trusted to a cast gear of the same size. The great reliance of those who claim that a cast gear is superior to a cut gear is upon the hard scale with which the cast tooth is covered. This scale is not over one-hun- dredth of an inch thick, is rapidly worn away, and is of no account whatever. From that point of view it is difficult to explain why a wooden tooth will outwear an iron one, although it is softer than the softest cut iron. Assuming that a cut gear is about three times as reliable as a cast gear, we can com- pute its power by the formula : Horse-power of a cut gear = -^^— g^ d n in which c is the circular pitch, / is the face, and d is the pitch diameter, all in inches, and n is the number of revolutions per minute. 3. THK INVOIvUTE SYSTTKNl. 53. -THE INVOLUTE TOOTH, The simplest and best tooth curve, theo- retically, as well as the one in greatest prac- tical use for cut gearing, is the involute. The involute tooth system is based on the straight rack odontoid, (31) and Fig. 21, and it is illustrated by Fig. 34. If the four odon- toids of the rack outline are equally inclined to the pitch line, the resulting tooth system will be completely interchangeable; but if, as in Fig. 35, the face and flank are inclined at different angles of obliquity, T 8 K and T 8 K' y the system is not interchangeable, although otherwise perfect. The rack odontoid cannot have a comer or change of direction anywhere except at the pitch line, without causing a break in the line of action. As the normals p q are parallel, the line of action is a straight line WO TT at right angles to the rack odontoid. The inter- changeable line of action is continued in a straight line on both sides of the pitch line, bUi, the non- interchangeable line changes di- rection at that line. In accordance with the universal custom we will considei that the involute tooth is always iri'erchangeable, having a single angle ot obliquity. Fig. Tlxe involute tooth interchangeable h 34. Non-^i nterchangea b l^ involtite Fig. 35, 26 Involute Interference. 54. — THE CUSP. As a circle i e. Fig. 34, can always be drawn tanjf;ent to the line of action at an interfer- ence point i, from the center b of any pitch line B, there will always be a cusp in the curve at the point c (16), and at that point the working part of the curve must stop. The working part of the rack tooth must end at the limit line i L through the interference point i. The working curves of any two teeth that work vnth each other must each end at the line drawn through the interference point of the other, Fig. 43, being limited by limit lines J I and L L. r The second branch c m' of the curve is equal to the first branch c m, but is re- versed in direction. The second cusp is at infinity, and therefore has no practical ex- istence. The tangent circle i c, through the inter- ference point and the cusp, is called the "base line." It is customary to continue the flank of the tooth inside the base line by a straight radial line, as far as may be necessary to allow the mating gear to pass. 55. — INTERFERENCE. When the point of the tooth is continued beyond the limit line it will interfere with and cut away a portion of the working curve of the mating tooth. Fig. 36 shows a rack tooth working with the tooth of a small pinion, and cutting out its working curve. This cut is not confined to the flank, but extends across the pitch line into the face, as shown by the line qmn. The rack tooth of the figure will not work with the pinion tooth unless it is cut off at the limit line 1 1 through the interference point i. The mathematical action still continues, and the figure shows the rack tooth in action at k with the second branch of the curve. Effect of Jiiierference Fig, 86. 56. — ADJUSTABILITY. An interesting and in many cases a valua- ble feature of the involute curve, and one that is confined to it, is the fact that its posi- tion as a whole with regard to the mating curve is adjustable. Two involutes, each with its base line, will work together in perfect tooth contact when they are moved with respect to each other, as long as they touch at all. The lines of action and the pitch lines will shift as the curves are moved, and will accommodate themselves to the varying position of the base lines. But this valuable feature of the involute curve is not always available, and involute gears are not, as commonly supposed, neces- sarily adjustable, for the conditions are often such that the teeth will fail to act when the centers are moved, except within very narrow limits. Care must be taken that the arc of action is not so reduced by separating the centers of the gears that it is less than the cir- cular pitch, for the former arc is variable and the latter is fixed. Care must also be taken that the working curve is not pushed over the limit line when the centers are drawn to- gether. In any limiting case, such as in Fig. 43, the centers are not adjustable. The gears of the standard set are either not adjustable at all or are so within very narrow limits, on ac- count of the correction for interference. Involute Construction. 27 57.— CONSTRUCTING THE USTVOLUTE BY POINTS. The simple involute curve can be con- structed by points by the general method of <24), but it is much better to take advantage of the property that it is an involute of its base circle, and construct it by the rectifica- tion of that circle. As in Fig. 37 any convenient small dis- tance ^ G^ is taken on the dividers, and the points on the curve located by stepping along the circle and its tangent from any given point to any desired point. This method is so accurate, if care is taken to step accurately on the line, that the curve seldom needs correction; but, when great ac- curacy is required, correction can be applied at the rate of one- thousandth of an inch to the step, if the length of the step is regulated by the diameter of the circle according to the following table: Diameter of Circle : 1 2 3 4 5 6 7 8 9 10 11 12 length of Step : .17 .26 .37 .46 .53 .60 .67 .73 .76 .79 .82 .84 For example: If the circle of Fig. 37 is Constriictioti points Fifj. 37. four inches in diameter, and the dividers are set to .46 inch, the true curve, A h' d' , will be outside of the constructed curve J. 5 «? by .002 inch at h and .005 inch at d. From the table we can form the handy and sufficiently accurate rule that the length of the step should be about one-tenth of the di- ameter of the circle, for a correction of about one-thousandth of an inch per step. Having thus found several points of the in- volute, we can draw it in by hand, or by con- structing a template, or by finding centers from which approximately accurate circular arcs can be drawn. 58. — THE STANDARD INVOLUTE TOOTH. Tlie tooth that is selected for general use, and the one that is the best for all except a few special cases and limiting cases, is the in- terchangeable tooth having an angle of ob- liquity of fifteen degrees, an addendum of one-third the circular pitch, or one divided by the diametral pitch, and a clearance of one- eighth of the addendum. The standard to which involute cutters are made is slightly different, having an angle of 14° 28' 40", the sine of which is one-quarter, and a clearance of one-twentieth of the circu- lar pitch. If the obliquity is 15° the smallest possible pair of equal gears have 11.72 teeth, and therefore 12 is the smallest gear of the inter- changeable set. The base distance, the distance of the base line inside of the pitch line, is about one-tifty- ninth of the pitch diameter, and one-sixtieth is a convenient fraction for practical use. The limit points of the whole set must be determined by that of the twelve-toothed gear, for any gear of the set may be required to work with that one, and the working curve of each tooth must end at the point thus de- termined. As the limit point is always in- side of the addendum line there must always be a false extension on the tooth, the point being rounded over outside of the limit point. 59. — THE INVOLUTE ODONTOGRAPH. As the base line must always be drawn, it is advisable, to save work, to locate the cen- ters of the approximate circular arcs upon that line. It is also necessary that the points of the teeth shall be rounded over, to avoid interference. These requirements made it impracticable to compute the positions of the centers, and an empirical rule had to be adopt- ed instead. Teeth were carefully drawn by the stepping 28 Ten and Eleven Involute Teeth. method of (57) on a very large scale, one- quarter pitch, giving a tooth eight inches in length. These teeth were corrected for inter- ference by giving them epicycloidal points that would clear the radial flanks of the twelve-toothed pinion. Then the proper centers on the base line were determined by repeated trials, and tooth curves obtained that would agree with the true involute up to the limit point, and still clear the corrected point. The odontograph ! table is a record of these radii, which are be- i lieved to be as nearly correct as the given conditions will permit. It was found that separate curves were required for face and flank up to thirty-six teeth, but that one curve would answer for teeth beyond. It was found necessary to devise a separate method for drafting the rack tooth. Theoretically the twelve-toothed pinion is the smallest standard gear that will have an arc of action as great as the circular pitch, but ten and eleven teeth may be used with an error that is not practically noticeable. Fig. 38 shows a pair of ten-toothed gears in 60. — TEN AND ELEVEN TEETH. action. They can be in correct action only when the point of contact is between the two interference points i and /, but they will be in practical contact for a greater and suffi- cient distance Fig, 38. Odontoyrapliic pair 61.— A BAD RULE. There is a simple and worthless rule for involute teeth that deserves notice only be- cause it is considerably in use. It constructs the whole tooth curve, face and flank, for all numbers of teeth, as a single arc from a center on the base line, and with a radius equal to one-quarter of the pitch radius, Fig. 39. This is wonderfully convenient, but the convenience is purchased at the expense of The Involute Odontograph. 29 ordinary accuracy, for the rule is not even approximately correct. It is handy, and nothing else. Figs. 38 and 40 show the kind of teeth that are constructed by this rule on gears of ten and twelve teeth, where its error is the greatest, and it is reasonable that the invo- lute tooth should not be in great favor with those who have been taught to draw it thus. The error gradually decreases, until, for more than thirty teeth, it is tolerably correct, but it gives the rack with the straight, uncor- rected working face that would interfere, as shown at g, Fig. 40. As it is tolerable only for thirty or more teeth, and not good then, it may well be dropped altogether. A. bad rule Fig\ 39. 62. — USING THE INVOLUTE ODONTOGRAPH. INVOLUTE ODONTOGKAPH. Standard Interchangeable Tooth, Centers on Base Line. {For Table of Pitch Diameters see 35.) Divide by the Multiply by the Circular Pitch. Diametral Pitch. Teeth. Face Flank Face Flank Radius. Radius. Radius. Radius. 10 2.28 .69 .73 .22 11 2.40 .83 .76 .27 13 2 51 .96 .80 .31 13 2.62 1.09 .83 .34 14 2.72 1.22 .87 .39 15 2 82 •1.34 .90 .43 16 2.92 1.46 .93 .47 17 3.02 1.58 .96 .50 18 3.12 1.69 .99 .54 19 3.22 1.79 1.03 .57 20 3.32 1.89 1.06 .60 21 3.41 1.98 1 09 .63 22 3.49 2.06 1.11 .36 23 3.57 2 15 1.13 .69 24 3.64 2.24 1.16 .71 25 3.71 2.33 1 18 .74 26 3.78 2.42 1 20 .77 27 3.85 S.50 1.23 .80 28 3 92 2.59 1 25 .82 29 3 99 2.6T 1.27 .85 30 4.06 2.76 1.29 .88 31 4.13 2.85 1.31 .91 32 4.20 2.93 1.34 .93 33 4.27 3 01 1 36 .96 34 4.33 3 09 1.38 .99 35 4 39 3 16 1.39 1.01 36 4.45 3 23 1.41 1.03 87^0 4 20 1.34 41-45 4.63 1.48 46—51 5 06 1.61 52-60 5 74 1.83 61-70 6 52 2.07 71-90 7.72 2.46 91-120 9.78 3.11 121-180 13.38 4.26 181-360 21 62 6 88 Draw the rack tooth by the special method. The Involute Odontograph. To draft the tootli lay off the pitch, ad- dendum, root, and clearance lines, and space the pitch line for the teeth, as in Fig. 40. Draw the base line one-sixtieth of the pitch diameter inside the pitch line. Take the tabular face radius on the divid- ers, after multiplying or dividing it as re- quired by the table, and draw in all the faces from the pitch line to the addendum line from centers on the base line. Set the dividers to the tabular flank radius, and draw in all the flanks from the pitch line to the base line. Draw straight radial flanks from the base line to the root line, and round them into the clearance line. Odotvtoijraphic exatnylc 63.— SPECIAL. RULE FOR THE RACK. Draw the sides of the rack tooth, Fig. 40, as straight lines inclined to the line of centers c c at an angle of fifteen degrees, best found by quartering the angle of sixty de- grees. Draw the outer half a b of the face, one- quarter of the whole length of the tooth, from a center on the pitch line, and with a radius of 2.10 inches divided by the diametral pitch. .67 inches multiplied by the circular pitch. 64. — DRAFTING INTERNAL GEARS. When the internal gear is to be drawn, the odontograph should be used as if the gear was an ordinary external gear. See Fig. 41. But care must be taken that the tooth of the gear is cut off at the limit line drawn through the interference point ^ of the pin- ion. The point of the tooth may be left off altogether or rounded over to get the appear- ance of a long tooth. The pinion tooth need not be carried in to the usual root line, but, as in the figure, may just clear the truncated tooth of the gear. The curves of the internal tooth and of its pinion may best be drawn in by points (57), The Involute Odontograph. P)1 for the odoniographic corrected tooth is not as well adapted to the place as the true tooth, and no correction for interference is needed on the points of the pinion teeth or on the flanks of those of the gear. Care must be taken that the internal teeth do not interfere by the point a striking the point t, as they will if the pitch diameters are too nearly of the same size. Internal involxites 65. — INVOLUTE GEARS FOR GIVEN OBLIQUITY AND ADDENDA, When the obliquity and addenda, as well as the pitch diameter and number of teeth in a gear are given, as is generally the case, we can proceed to draft the complete gear as follows: Draw the pitch line i? I, Fig. 42, the ad- dendum line a I, the root line r I, and the clearance line c I, as given. Draw the line of action I adit the given obliquity W Z — K. Draw the base line h I tangent to the line of action. Find the interference point i by bi- secting the chord v. Draw the involutes i a m and i" a" m" , and a a" will be the maximum arc of ac- tion. If the given arc of action a a' is not great- er than the maximum arc, the pitch line is to be spaced and the tooth curves drawn in from the base line to the addendum line. These tooth curves, when small, are best drawn as circular arcs from centers on or near the base line, one center x for the flank from the base line to the pitch line, and another center i/ for the face from the pitch line to the addendum line. One involute i a m should be carefully constructed by points, and then the required centers can be found by trial. One center and arc will often answer for the whole curve, and it is only when great accuracy is required that more than two centers will be necessary. Continue the flanks of the teeth toward center by straight radial lines, and round these lines into the clearance line. If the interference point for the gear that the gear being drawn is to work with is at /, within the addendum line, the limit line 1 1 must be drawn through it, and the points of 32 Involute Special Cases. Fig. 42, Given obliquity and addendum the teeth outside of this limit must be slightly rounded over, to avoid interference (55), If a fillet / is desirable, to strengthen the tooth, it can of (44). be drawn in by the method 66. — INVOLUTE GEARS FOR GIVEN NUM- BERS OF TEETH. When the numbers of teeth and the pitch lines are the only given details, the shape and action of the tooth depends upon the obli- quity, and the action v^ill fail if the angle is too small. The principal object is to deter- mine the least possible angle that is permitted by the given pitch diameters and numbers of teeth. Draw the pitch lines P L and p I, Fig. 43, lay off the given pitch arc, as a straight line c d or G JD, at right angles to the line of centers, and draw the line C d or c D. Then the required line of action will he I a pass- ing through at right angles to c D or C d. The complete teeth can then be drawn in as previously directed. In this case, the obliquity W Z being the least possible, the limit lines and the adden- dum lines must coincide, but the addenda may be reduced by increasing the angle. Given numbers of teeth Limiting Involute Teeth. 83 67. — INVOLUTE GEARS FOR GIVEN OBLIQUITY. When the pitch diameters and the obliquity are the only given details, the lines G I and c i, Fig. 43, drawn from the centers at right angles to the line of action, will determine the limit lines. The maximum arc of action a a' may be found either by drawing the involutes i a and la', or by continuing the line C 7 to the line c d, and measuring the required distance c d. Any arc of action less than a a' may be used. The drawings should always be made to a scale of one tooth to the inch radius, so that the pitch arc will be 2n. If the scale is one tooth to the inch of diameter, the pitch arc will be tt. -INVOLUTE GEARS WITH LESS THAN FIVE EQUAL TEETH. The method of Fig. 43 and (66) will be found to apply to any given numbers of teeth not less than five, and to fail, if either gear has but three or but four teeth. Any external gear of five or more teeth will work with any external gear of five or more teeth, and with an internal gear of any number of teeth unless stopped by internal interfer- ence (64). For example, if a pair having four and five teeth, Fig. 44, is tried, the four-toothed pinion will fail, because its tooth will come to a point upon the line of action before it has passed over the required pitch arc. The diSiculty cannot be remedied by increasing the obliquity, for an angle that would allow the four-toothed pinion to act would also cause the five-toothed pinion to fail. The practical limit is five teeth, but the mathematical limit is the pair having the fractional number 4.62 teeth. Fig. 45. The four-toothed pinion will not work with any external gear, not even with a rack, but it will work with an internal gear that has about ten thousand teeth, and is practically a rack. It will work with any internal gear having less than ten thousand teeth, and Fig. 46 shows it working with an internal gear of six teeth. Internal interference will prevent its working with an internal gear of five teeth. The three-toothed pinion has no practical action. It has a mathematical action with in- ternal gears of 3.56 or less teeth, as shown by Fig. 47, but as its limit is less than four, it cannot work vdth any whole number. The figure shows the interference at a. The extreme mathematical limit may be said to be the gear of 2.70 teeth, which has a theoretical action with an internal gear of the same size, coinciding with it. Fig, ^4. c \ '^< \ 1 \ 'k ) \ c Failin fj case 4.62 X 4.G2 limif for equal ieeih Fig, 45. Lifniting Involute Teeth. 69. — INVOLUTE GEARS WITH LESS THAN FIVE UNEQUAL TEETH. If we drop the condition that the pitch line must be equally divided into tooth and space arcs, we can make gears of three and of four teeth work with external gears by the method of (65). The failing case of Fig. 44 may be corrected by widening the failing tooth until it acts, and narrowing the other tooth to correspond, as shown in broken lines. ■ In this way a four-toothed pinion will work with any number of teeth not less than 5.57, at which limit both gears have pointed teeth, as in Fig. 48. The three-toothed pinion will work with any gear having 10.17 or more teeth. Fig. 49 shows the 3x10.17 limiting pair, and Fig. 50 shows the three-toothed pinion working with an internal gear of five teeth. It will not work with an internal gear of four teeth, on account of internal interference, and there- fore the combination shown by Fig. 50 may be said to be the least possible symmetrical in- volute pair. A gear of 2.70 teeth will work with a rack, but there seems to be no way to make a pinion of two teeth work under any circum- stances. 4 teetU Fig. 48, LiniUing' Iiivolute Teeth. 35 Fig. J:9 3 X 10.17 Unequal teeth Fig. 50 70. — THE MATHEMATICAL, LIMITS. The above results for low numbered pinions can be obtained by graphical means, but that method is not accurate enough to determine the limits with great precision, and in any case is tedious and laborious. The mathematical process is not particu- larly difficult, and consists in repeated trials with given formulce. To determine the obliquity at which a limiting pinion will be pointed on the line of action, for tooth equal to space, we use the formulae : 2nM tan. h = M-^n 90 in which n is the given number of teeth in the pointed gear, Fig. 51, Jf is the number in the gear having the radius M, and h is the angle c I. Knowing n, we assume a value for M, and from that find a value for h by means of the first formula. This value of h, tried in the second formula, will give an error. A second assumption for M will give a second error, and if the two errors are not too great a comparison will nearly locate the true value of M. Knowing n and M, we find the obliquity from 27T tan. K = M-\-n JPointed j^lnion. Fig. 51. In this way the f ollov ring values were de- termined : n M K 2.695 1.26 57° 49' 3. 1.51 54° 20' 4. 2.86 42° 29' 4.62 4.62 34° 11' 5. . 6.75 28° 8' 5.58 00 Having determined tl le obliquity for the pointed pinion. we can determine the least number of teeth it will ^ vork with by means of the following formuh B : Angle B = 180 n ^ tan.K—-^ -^K tan. B = ^i^ tan. K-\- tan. K in which N is the required least number. 36 Limiting Involute Teeth. In this way it was found that a gear of four teeth will not work with a rack, hut will work with an internal gear having a number of teeth not easily calculated with existing loga- rithmic tables, but which is approximately ten thousand. Also that a pinion of three teeth will not work with an internal gear having more than 3.56 teeth. For unequal teeth we can use the formulae, 27riVr tan. h = 7i(i\^+7l) + 4 7r2 tan. H = 2 TT 7J, N{N-\-n)-\r^T^^ in which N and n are the numbers of teeth in the pair of pointed gears. By these form- ulae the following results were determined, n N K 2.695 00 3. 10.17 25° 27' 4. 5.57 33° 17 4.62 4.62 34^ 11 71. — MINIMUM NUMBERS FOR UNSYMMETRICAL TEETH. If we drop the condition that the fronts and backs of the teeth shall be alike we have an unimportant case that is similar to that already studied, but much more intricate. If we carry this case to its extreme, and adopt single acting teeth, we have no mini- mum numbers at all, for any two numbers of teeth will then work together. Fig. 52 shows one tooth working with three teeth, and any other combination can be obtained. The minimum obliquity for a given pair is obtained, as in (66), by laying off the known pitch arc, G D, at right angles to G c, and drawing the line of action at right angles to the line D c The obliquity is also given by the formula : tan.K-= -^TT-j , N-\-n in which n and N are the numbers of teeth. When the obliquity is as great as is often JFig. 52 Unsymmetrical teeth the case for very low numbers of teeth the action may be impracticable on account of the great friction of approach (48). The gears of Fig. 52 will not drive each other on the approach, unless the tooth surfaces are very smooth, and the power transmitted is almost nothing. 72. -MINIMUM NUMBERS FOR GIVEN ARC OF RECESS. It has generally been assumed, although no good reason for the assumption has ever been given, that the minimum numbers of teeth occur when the tooth of one of the gears. Fig. 53, is pointed at the interference point I, and at the same time has passed over an arc of recess a that is a given part of the whole pitch arc a' a. The solution is simple enough, graphically by repeated trials, or by a formula that can be applied directly without the usual process by trial and error. But, as involute teeth have a uniform ob- liquity, there is no necessity for assuming Fig.\ 53„ a definite arc of recess, and the condition on Involute Efficie7tcy, 37 which the problem is based is unwarranted. No real limit is reached, and the matter is not worth examination at any length. The problem is investigated, for both bevel and spur gears, in either external or internal con- tact, in the Journal of the Franklin Institute for Feb., 1888, and it has received more atten- tion than its slight importance entitles it to. 73. — EFFICIENCY OF INVOLUTE TEETH. But little can be said in addition to the matter in (49), for both forms of teeth in common use are substantially equal with re- spect to the transmission of power. From the formula of (49), which is the formula for the involute tooth, it is seen that the loss from friction is entirely independent of the obliquity, and, therefore, all systems of involute teeth are independent of the ob- liquity in this respect. This is contrary to the accepted idea that a great eflSciency re- quires a small Obliquity. It has been stated on high authority that the involute tooth is inferior to the cycloidal tooth in efficiency, but the statement is not true. The difference in efficiency is minute, a small fraction of one per centum, but what little difference there is is always in favor of the involute tooth. 74. — OBLIQUITY AND PRESSURE. The involute tooth action is in the direction of the line of action, and the obliquity is a constant angle. It is variable only when the shaft center distance is varied. As the pressure is always equal to the product of the tangential force at the pitch line multiplied by the secant of the obliquity, (26), it is constant for the involute tooth. Involute teeth, therefore, have a steady ac- tion that is not possessed by other forms ; particularly by forms which, like the cy- cloidal, have a pressure and an obliquity that varies between great extremes. -THE ROLLER OP THE INVOLUTE. The involute odontoid, like all possible odontoids, can be formed by a tracing point in a curve that is rolled on the pitch line, and this roller is the logarithmic spiral with the tracing point at its pole, (32). This feature is, however, more curious than useful, and it is not of the slightest im- portance in the study of the curve. Neither is the operation of rolling the involute me- chanically possible, for the logarithmic roller has an infinite number of convolutions about its pole, and the tracing point would never reach the pitch line. The involute is often considered to be a rolled curve, because it can be formed by a tracing point in a straight line that rolls on its base line; but, although that is the fact, it is a special feature and has nothing to do with the rolled curve theory. The rolled curve theory requires that the odontoid shall be form( d by a roller that rolls on the pitch line only. A. THK CYCLOIDAIv SYSTKNl. 76. — THE CTCLOIDAL SYSTEM. If the curve known as the cycloid is chosen as the determining rack odontoid, (31), the resulting tooth system will be cycloidal. It is commonly called the * ' epicycloidal " system, because the faces of its teeth are epicycloids, bQt, as the flanks are hypocy- cloids, it seems as if the name ''epihypo- cycloidal " would be still more clumsy and accurate. There is no more need of two different kinds of tooth curves for gears of the same pitch than there is need of two different kinds of threads for standard screws, or of two different kinds of coins of the same value, and the cycloidal tooth would never be missed if it was dropped altogether. But it was first in the field, is simple in theory, is easily drawn, has the recommendation of many well-meaning teachers, and holds its position by means of "human inertia," or the natural reluctance of the average human mind to adopt a change, particularly a change for the better. 77. — THE CYCLOIDAL TOOTH. The cycloid is the curve A that is traced by the point p in the circle C that is rolled on the straight pitch line p I, Fig, 54. The normal at the point p is the line p q to the point of tangency of the rolling circle and the pitch line. The line of action is the circle I a, of the same size as the roller C. As no tangent arc can be drawn to tlie line of action from the pitch point as a center, no terminal point (18) exists. As there is no point upon the line of centers from which a circle can be drawn tangent to the line of action, there will be no cusps, (16) except on the pitch line. The cycloidal tooth can be drawn by the general method of (24), but there are several easier methods which will be described. There are numerous empirical rules and short cuts to save labor and spoil the tooth, which will not be de- scribed. When the pitch line is of twice the diame- ter of the line of action, the flank of the tooth is a straight line. If the pitch line is less than twice as large as the line of action, the flank of the tooth will be under-curved. as shown by Fig. 55, and it is customary to avoid the resulting weak tooth by limiting the line of action to a diameter not greatei than half that of the smallest gear to be used. Cycloidal Secondary Action. 39 78. — SECONDARY ACTION. The secondary line of action (21) is a circle. Fig. 56, differing from the pitch circle by the diameter of the primary line of action, either inside or outside of it. When the internal secondary line of action of an internal pitch line coin- cides with the external secondary line of action of its pinion there will be secondary contact between the gears, the face of the gear working with the face of the pinion at a point of contact upon the combined secondaries. Fig. 57 shows this for the cycloidal tooth, the two faces working together at the point a. As both secondaries are cir- cles they must coincide, and the sec- ondary action will be continuous. When the teeth are also in contact at h on the primary line o^ action, there will be double contact. Undercurved flanhs Fig, 55, Secondary lines of action Fig\ 50, 40 Cycloidal Interference. 79. — INTERNAL INTERFERENCE. If the secondary lines of action do not come together the teeth will not touch each other at all, but if that of the gear is smaller than that of the pinion the teeth will cross each other and interfere. The line c. Fig. 57, is the face of the gear tooth, and the line d is the face of the pinion tooth having a primary line of action equal to the difference between the pitch lines. The secondary line of each gear coincides with the pitch line of the other, and the faces interfere with each other the amount shown by the shaded space. The only remedy for internal interference is to reduce the diameter of the primary line of action to half the differenjce between the diameters of the pitch lines, or else to leave off one of the faces of the teeth. The discovery of the law of internal cycloid- al interference is due to A. K. Mansfield, who published it in the "Journal of the Franklin Institute" for January, 1877. It was afterwards re-discovered by Professor MacCord, and most thoroughly applied and illustrated in his " Kinematics." When interference is avoided by omitting one of the faces of the teeth the primary line of action may be enlarged, but it must not then be larger than the difference between the pitch diameters. Fig. 58 shows on the right the action when the face of the gear is omitted, and on the left the action when the face of the pin- ion is left off. The teeth will just clear each other, each one touching the other at a single point a in its pitch line. As the contact at a is not a point of practi- cal action, care must be taken that the arc of action at the primary line of action is as great as the circular pitch, for otherwise, as in the figure, the gears will not be in continu- ous primary action. The rule for internal interference, simply stated, is that the diameters of the pitch lines must differ by the sum of the diameters of the lines of action if the teeth have both faces and flanks, and by the diameter of the acting line of action if the face of either gear is omitted. For the standard interchangeable system the gears must differ by twelve teeth Internal interference Fig, if both teeth have faces, and by six teeth if one face is omitted. Fig. 62 shows the secondary contact in the case of a standard internal gear of twenty- four teeth working with a pinion of twelve teeth, and it is to be noticed that the teeth nearly coincide between the two points of contact. Where there is secondary contact the teeth practically bear on a considerable line instead of at a point. Cycloidal Odontograph. 41 80. — THE STANDARD TOOTH. The standard tooth (42), selected for the cycloidal system, is by common consent the one having a line of action of half the diame- ter of a gear of twelve teeth, so that that gear has radial flanks. The standard adopted by manufacturers of cycloidal gear cutters is that having radial flanks on the gear of fifteen teeth, but it is not and £hoi)ld not be in use for other pur- poses. If any change is made, it should be made in the other direciion, to make the set take in gears of ten teeth. It must be borne in mind that the standard adopted does not limit the set to the stated minimum number of teeth, but that it sim- ply requires that smaller gears shall have weak under-curved teeth. 81. — THE ROLLED CURVE METHOD. It happens in this case, and in this case only, that the rolled curve method, which theoretically applies to all odontoids, can be actually put into practical use, for the generating roller is here the circle, the sim- plest possible curve. As in Fig. 59, roll a circle of the diameter of the circle of action upon the outside of the pitch line for the faces, and upon the inside for the flanks, and a fixed point in it will trace the curve. The method can be used by actually con- structing pitch and rolling circles, but the same result can be reached more easily and quite as accurately by drawing several cir- cles, and then stepping from the pitch point along the pitch line, and back on the circles to the desired point. If the length of the Construction by rolling Fig. 59. step is not more than one-tenth of the diam- eter of the circle, the error will not be over one-thousandth of an inch for each step. This method is the best one to adopt, ex- cept for the standard tooth. 82. — THE THREE POINT ODONTOGKAPH. It is a simple matter to draw the tooth curve by means of rolling circles, but such a method requires skill on the part of the draftsman. It is, moreover, nothing but a method for finding points in the curve for which approximate circular arcs are then determined. The "three point" odontograph is sim- ply a record of the positions of the centers of the circles which approximate the most closely to the whole curve of the standard tooth. The positions of two points, a at the center of the face or of the flank. Fig. 60, and b at the addendum point or root point of the curve, were carefully computed, and then the position of the center C of the circle which passes through these two points and the pitch point 0, was calcu- lated. The circle that passes through these three points is assumed to be as accurately approximate to the true curve as any pos- sible circular arc can be. The odontograph gives the radius "rad." of the circular arc, and the distance " dis." of the circle of centers from the pitch line, for the tooth of a given pitch, and their values for other pitches are easily found by simple multiplication or division. The advantages of this method lie in the facts that the desired radius and distance are given directly, without the labor of find- ing them, and that as they are computed they are free from errors of manipulation. In point of time required, the advantage is 42 Cycloidal Odontograph. with the odontograph in the ratio of ten to one. The greatest error of the odonto- graphic arc, shown greatly exaggerated by the dotted lines, is at the point c on the face, and it is greater on a twelve- toothed pinion than on any larger gear. For a twelve-toothed pinion of three- inch circular pitch, a large tooth, the actual amoufit of the maximum error is less than one one-hundredth of an inch, and its average for eight equidistant points on the face is about four-thousandths of an inch. Any error that is greater than Q^ ftori fc cpvter s that stated will be due to manipulation, and not to the method. 83. — USING THE ODONTOGRAPH. To apply the odontograph to any particu- lar case, tirst draw the pitch, addendum, root, and clearance lines, and space the pitch line, Figs. 60 and 61. Then draw the line of flank centers at the tabular distance "dis." outside of the pitch line, and the line of face centers at the distance "dis." inside of it. Take the face radius ''rad."on the dividers, and draw in all the face curves from centers on the line of face centers; then take the flank radius "rad."and draw all the flank curves from centers on the line of flank centers. THREE POINT ODONTOGRAPH. Standakb Cycloidal Teeth, interchangeable series. From a Pinion of Ten Teeth to a Rack. For One For One Inch BER OP For 8 DIAMETRAL PITCH. eby CIRCULAR PITCH. NtTM my other pitch divic For any other pitch multiply by TEETH that pitch. that pitch. IN TB in GEAR. Faces. Flanks. Faces. Flanks. Exact. Intervals. Rad. Dis. Rad. Dis. Rad. ns. Rad. Dis. 10 10 1.99 .02 — 8.00 4.00 .62 .01 —2.55 1.27 11 11 2.00 .04 — 11.05 6.50 .63 .01 —3.34 2.07 12 12 2.01 .06 oo 00 .64 .02 00 00 13/^ 13—14 2.04 .07 15.10 9.43 .65 .02 4.80 3.00 15^ 15—16 2.10 .09 7.86 3.46 .67 .03 2.50 1.10 17V^ 17-18 2.14 .11 6.13 2.20 .68 .04 1.95 .70 20 19-21 2.20 .13 5.12 1.57 .70 .04 1.63 .50 23 22-24 2.26 .15 4.50 1.13 .72 .05 1.43 .36 27 25-29 2.33 .16 4.10 .96 .74 .05 1 30 .29 33 30-36 2.40 .19 3.80 .72 .76 .06 1.20 .23 42 37-48 2.48 .22 3.52 .63 .79 .07 1.12 .20 58 49—72 2.60 .25 3.33 .54 .83 .08 1.06 .17 97 73—144 2.83 .28 3.14 .44 .90 .09 1.00 .14 290 145-300 2.92 .31 3.00 .38 .93 .10 .95 .12 00 Rack 2.96 .34 2.96 .34 .94 .11 .94 .11 Cycloidal Odo)itograph. 48 The table gives the distances and radii if the pitch is either exactly one diametral or one inch circular, and for any other pitch multiply or divide as directed in the table. Fig. 61 shows the process applied to a practical case, with the distances given in figures. Fig. 63 shows the ssame process applied to an internal gear of twenty-four teeth work- ing with a pinion of twelve teeth. It illus- trates secondary action and double contact. It also shows the actual divergence of the Willis odontographic arc from the true n-ne^oj£— ^ 1 ''"''" Internal teeth Fig. 02, 44 Willis Odo7itogi'aph. 84. — THE WILLIS ODONTOGRAPH. This is the oldest and best known of all the odontographs, but it is inferior to several others since pro- posed, not only in ease of operation, but in accuracy of result. To apply it, find the pitch points a and a' half a tooth from the pitch point 0, Fig. 63, draw the radii a c and a' c\ lay off the angles c ah and c' a' h\ both 75°,, and lay off the distances a b and a' b' that are given by table. The centers b and b' thus found are the centers of circular arcs that are tangent to the tooth curves at d and d'. The dividers are set to the radius b or b' to draw the curves. The Willis arc touches the true curve only at the pitch point 0, and its variation else- where is small, but noticeable. On the face of the tooth of a twelve-toothed pinion of three inch circular pitch, its error at the ad- dendum point is four-hundred ths of an inch, and it will average three times that of the three point method (82). The error is shown by Fig. 62. The greatest error of the method is due to manipulation. The angle is usually laid off by a card, and the center measured in by a scale on the card. The circle of centers is The Willis odontograjph Fig. 03. then drawn through the center, and unless great care is used the chances of error are great. The. angle 90°— c ab = Tr=— , and the distance ab =. —-• 27r sin. W, in which s t is the number of teeth in the gear of the same set which has radial flanks, usually 12 ; c is the circular pitch, and t is the num- ber of teeth in the gear being drawn. The positive sign is used for the face radius, and the negative for the flank radius. 85. KLEm'S CO-ORDINATE ODONTOGRAPH. This is a method of finding the positions of several points on the tooth curve by means of their co-ordinates referred to axes through the pitch point. Any point on the curve is found by laying off a certain dis- tance on the radius Y, Fig. 64, and then a certain distance at right angles to it, the distances being given by a table for a certain standard tooth. As many points as required are found by this method, and then the curve is drawn in by curved rulers, or by finding the approxi- mating circular arc. This odontograph is to be found in Klein's Elements of Machine Design. Coordinate odontograph Fig. 64. Obliquity of Action, 45 -THE TEMPLET ODONTOGRAPH. Prof. Robinson's templet odontograph is an instrument, not a method. It is a piece of sheet metal, Fig. 65, having two edges shaped to logarithmic spirals. It is laid upon the drawing, according to directions given in an accompanying pamphlet, and used as a ruler to guide the pen. It can be fastened to a radius bar, and swung on the center of the gear, to draw all the teeth. See Van Nostrand's Science Series, No. 24, for the theory of the instrument in detail. The templet odontograph Fig. 65. 87. — OBLIQUITY OF THE ACTION. When the point of contact between two teeth is at the pitch point 0, Fig. 66, the pressure between the teeth is at right angles to the line of centers, but, as the point of con- tact recedes from the line, the direction of the pressure varies by an angle of obliquity which increases from zero until the point K, at the intersection of the addendum circle with the line of action, is reached. The angle K = K W, of the maximum obliquity, can be found by solving the trian- gle G c K, and for the standard set we have, 2?i + 17 3 7i -f 18' in which n is the number of teeth in the gear. For the smallest gear of the set, the one having twelve teeth, K is 20° 15', and for the rack it is 24° 5', so that it will always be be- tween those two limits for external gears, and greater for internal gears. The friction between two gear teeth in- creases with the angle of obliquity, but not COS. 2K Obliquity Fig. 66. in direct proportion. With the involute tooth the work done while going over a cer- tain arc from the line of centers is propor- tional to the square of the arc, and for cycloidal teeth the increase with the arc is still more rapid. Therefore it is the maxi- mum obliquity of the action that principally determines the injurious effects of friction. -THE CUTTER LIMIT, When the number of teeth in the gear is less than that in the gear having teeth with radial flanks, the flanks will be under-curved, and when too much so they cannot be cut with a rotary cutter. The teeth of Fig. 55 could not be cut with a rotary cutter beyond the points where the tangents to the two sides are parallel. The limit is reached when the last point that is cut by the rotary cutter is also the last point that is touched by the tooth of the rack in action with it, not allowing for in- ternal gears. The diameter of the gear when this limit is reached is found by the formula, c 1) = M — 2 sin. \/l 4G Limithig Cycloidal Teeth. in which B is the diameter of the gear, d is the diameter of the circle of action, c is the circular pitch, and a is the addendum For the common addendum of unity divided by the diametral pitch this may be put in the shape, ^ n — s \ |/- in which s is the number of teeth in ciie radial flanked gear, and n is the number in the required cutter limit. For the common series, where s = 12, we have n = 8.26; and for the cutter standard of s = 15, we have n = 10.80, so that cutters could easily be made to cut gears with less than s teeth. -RADIAL FLANKED TEETH. When the rolling circle for the faces is of half the diam- eter of the pitch line of the mating gear, the flanks of both gears will be straight radial lines, as in Fig. 67. Such gears are fitted to each other in pairs, and are not irterchangeable with other sizes. Their teeth are more easily made than those of standard gears. The maxi- mum obliquity is less, but the strength of the teeth is also less than usual. There is no reason for making such teeth in preference to the though, for that reason probably, they are used to a considerable extent. It would be standard, al- Jtadial flanks Fig. 67. difficult to devise a form of tooth so whimsi- cal that it would find no one to adopt and use it. 90. — THE LIMITING NUMBERS OP TEETH. When the number of teeth in a driving gear is small, the point p. Fig. 68, of its pointed tooth may go out of action by leav- ing the line of action g before a certain definite arc of recess r has been passed over, and the problem is to find the smallest num- ber of teeth in the following gear that will jast allow the given recess. This question, which is not a particularly important one, is discussed at length, and applied to both bevel and spur gears, in either external or internal contact, in an article in the "Journal of the Franklin Institute*' for Feb., 1888, and we will here consider only the case of the common spur gear. The recess r is given as a times the cir- cular pitch, and the thickness a r of the looth is given as b times the same. The diameter of the circle of action is q times Limiting tvtfth Fig. 68, that of the pitch line of the following gear. The number of teeth in the driving gear is d, and the number in the following gear is /. The Pin Tooth. 47 1. 360 a M IS an auxiliary angle equal to , and W is an angle 360 (a- d \ 2 j" Then the required number / can be found by a process of trial and error with the formula, sin. (3/ -I- W) d_ _ -^ ^ q sin. W qf ~ ' For an example, let the recess be f of the pitch, the tooth equal to the space, and the flanks of the follower to be radial. Let the problem be to find a follower for a driver of seven teeth. This gives a = f, 5 = i,g = i, d = l, and the formula becomes L'^ /540' sin. I + 25° 43' \ / sin. 25° 43' i'-... If we put / at random, at 20, we shall get, + .134 = 0. Next, trying / = 10, we get, — .132 = 0, and the opposite signs show that / is between 20 and 10. Trying 12 the result is positive, and for 11 it is negative, showing that 12 is the required value of /. That is, 7 teeth will not drive less than 12 teeth with radial fl.anks, unless it is allowed an arc of recess greater than f of the pitch. For another example, test MacCord's value of 382 as the least driver for a follower of 10 teeth, when recess equals the pitch and the follower has radial flanks. Trying d = 382, the error is negative ; for 383 it is also negative, but for 384 it is positive, and there- fore the latter is the true number. Extensive and suflBciently accurate tables of limiting values are given by MacCord in his "Kinematics." 5. XHE^ PIN TOOTH SYSTKNl. 91. -THE PIN GEAR TOOTH. The theory of the pin gear tooth is en- tirely beyond the reach of the " rolled curve" method of treatment, and, therefore, writers who have adopted that method have had to depend more on special methods adapted to it alone than on general principles. The re- sult is that its properties are often given in- correctly, or with an obscurity and complica- tion that is bewildering to the student. Although the tooth is one of the oldest in use, its theory is so difficult that its defect was not discovered until within a very few years, by MacCord, about 1880, and it was not until it was examined by means of its normals that a remedy for that defect was discovered. By treating the curve on the general prin- ciples here adopted, as a special form of the segmental tooth, it can be studied with ease, and its peculiarities developed in a complete and satisfactory manner. The method, in general terms, is to find the conjugate tooth curve of the gear, for the given circular tooth curve of the pinion, and it presents no new features or difficulties. 92. -APPROXIMATE FORM OF PIN TOOTH CURVE. Considered roughly, but accurate enough for teeth of small size, the form of the gear tooth b, Fig. 69, is a simple parallel to the epicycloid B, formed by the center e of the pin, and is to be drawn tangent to any convenient number of circles having centers on the epicycloid. The action is practically all on one side of the line of centers, the face of the gear tooth working with the part of the pin that is 48 The Pin Tooth. inside of its pitch lice. It is, theiefore, all approaching action when the pin drives and all receding action when the gear drives, and it is best to avoid the increased friction of the approaching action by always putting the pins on the follower. Juantevn u^Jieel Fig. 70. Pin gearing Fig. 69, . — KOLLEB TEETH. The pin gear is particularly valuable when the pins can be made in the form of rollers. Fig. 70, for then the minimum of friction is reached. The roller runs freely on a fixed stud, or on bearings at each end, and can be easily lubricated. The friction between the tooth and pin, otherwise a sliding friction at a line bearing, is, with the roller pin, a slight rolling fric- tion, and the sliding friction is confined to the surface between the roller and its bear- ings. When the roller pin is used there can be no increased friction of approach, and the pin wheel can drive as well as follow. For very light machinery, such as clock work, there is no form of tooth that is su- perior to the roller pin tooth, and, with the improvement to be explained, there is no better form for any purpose. 94.— CUTTING THE PIN TOOTH. The pin gear tooth can be very easily and accurately shaped by mounting a revolving milling cutter M, Fig. 71, of the size of the pin, upon a wheel A, and causing it to roll with a wheel B, carrying the gear blank G. The mill will shape the teeth to the correct form. Pin gear cutter Fig. 71. 95. — PARTICULAR FORMS OF PIN GEARS. When the pins are supported between two plates, as in Fig. 70, the wheel is called a "lantern" wheel, and is the most common form of clock pinion. The pins are some- times called " staves," and are sometimes known as "leaves." Defect of Pin Tooth. 49 When the diameter of the pin is zero, Fig. 72, it being merely a point, the correct tooth curve will be a simple epicycloid. When the pin gear is a rack, Fig. 73, the tooth bears on the pin only at a single point on the pitch line, and the action is therefore very de- fective unless the roller form of pin is used. This form is more properly a particular case of the involute tooth, for the shape of the pin is immaterial if it does not interfere with the gear tooth. The circle with center on a straight line is not an odontoid at all, for, although it coincides as a whole and for a single instant with a cir- cular space in the gear, it has no proper and continuous tooth action. The gears of Fig. 74, sometimes classed with pin gearing, are not pin gears at all. An epicycloidal face working with a radial flank is a very common combination. When the diameter of the pin wheel is half that of the internal gear with which it works. we have the combination of Fig. 75. The pins may run in blocks fitted to the straight slots. JPoint gears Fig. 72. I*in rack Fig. 73. 2fot pin gears Fig. 7.4:. Radial pin teeth Fig. 75. 96. — COKRECT FORM AND DEFECT OF PIN TEETH. Although the pin tooth is apparently of a very simple form, a close examination will show that it is really quite complicated, and that its practical action is incomplete and de- fective. There is a cusp (16), and conse- quent failure in the aciion, that is of small importance when the teeth are small, but which is troublesome when they are large. This defect need not be considered when pinions for clock work are in view, but if pin wheels are to be used for large machinery and heavy power it is important. If the pin a, Fig. 76, is examined as an odortoid, it will be seen that it is a true odoDLoid only within the line TeT that is tangent to the pitch line at the center of the pin, for all normals, as pe, from points out- side of that line, intersect the pitch line at the center. Drawing the normals, which are radii of the pin, we can ea:ily construct the line of action and the conjugate tooth curve. The line of action, commencing at the pilch point 0, Fig. 77, is there tangent to the line eOin, which passes through the center e of the pin, curves toward Oil, the tangent to the pitch line at the pitch point, and touches it at the point Ti, at the distance Oh, equal to the ra- dius of the pin. From the point h it follows the circle hjh' to the point h' , thence return- ing to the pitch point and forming the loop OKL. From the center c of the gear, Fig. 78, we can always draw a tangent arc FN to the line of action at the point F, and therefore there will always be a cusp at N on the tooth curve. The tooth curve must end at the cusp, and, to avoid interference, the pin must be cut off at the arc W, drawn through the point F, from the center G. The whole pin is generally used, and when it is a roller it must be whole, and 50 Improved Pin Tooth. then interference can be avoided only by cutting away the tooth curve until it will al- low it to pass. The complete tooth curve has a first branch NOM, Fig. 78, which is the only part that can be used, an inoperative second branch from the first cusp N to the second cusp Q, on the arc EQ,, and thence an inopera- tive circle OB,(c^ . Z,ine of action Fig. 77. Correct action Fig. 78. 97. — AN IMPROVED PIN TOOTH. The cause of the broken action of the pin tooth is the cusp, which is always present when the center of the pin is on the pitch line, and it can be avoided by placing the center back, as in Fig. 79, to such a distance inside the pitch line that the cusp does not occur. "When the center of the pin is inside the pitch line, the whole circle of the pin is a true odontoid, and the distance en of the center from the pitch line can be so chosen that the cusp is not formed. This distance does not appear to be sub- ject to any simply stated rule, but in the single case of the pin rack it is determined by the formula: in which x is the required distance en, D is the diameter of the gear, and d is the diame- ter of the pin. If the angle CeO, Fig. 79, is not less than a right angle, there will be no cusp on the Fig, 79, Corrected pin gear gear tooth if the diameter of the gear is greater than that of the pin. B. TWISTED, SPIR.AL, AND ^?VOR.NI QEARS. 98. "When two or more gears, Fig same pitch diameter, are placed in contact on the same shaft, they will evidently act as in- dependently of each other as if they were some distance apart, while they appear to act together as a single gear with irregular teeth. They are known as " Hooke's Gears," It matters not how many different kinds or numbers of teeth the several gears may have, or in what order they are arranged, if those that work together on opposite shafts are matched. They may be given an irregular arrangement, as in Fig. 80 ; a spiral arrange- ment, as in Fig. 81 ; a double spiral, or "her- ring-bone " arrangement, as in Fig. 82 ; a cir- cular arrangement, as in Fig. 83, or other- wise at will. -STEPPED GEARS. of the Fig. 80 Stepped (/ear Spiral arrangement Fig. 81, Double spiral arrangement Fig. 82. Circular arrangement Fig. 83. . — TWISTED TEETH. The thickness of the component gears has nothing to do with the theoretical action of the stepped gear as a whole, and therefore we can have them as thin as required. If the thickness is infinitesimal the component character of the gear is not apparent, and it is known as a twisted gear. Fig. 84. When the teeth are twisted there may al- ways be one or more points of contact at the line of centers, where the theoretical fric- tion is nothing, and therefore they are par- ticularly well suited for rough cast teeth! Furthermore, if the teeth are badly shaped Xwisted arrangement Fig. 84. 52 Twisted Teeth, the twisted arrangement tends to distribute the errors so that they are not as noticeable. The oblique action of twisted teeth tends to produce a longitudinal motion of the gears upon their shafts, which must be guarded against. This end thrust may be avoided by so forming the twist that there are aiways two oblique bearings between the teeth, acting in opposite directions, as in the herring-bone arrangement. The twisted form of tooth is seldom found in practice, except in the form of spiral and double spiral teeth, for the difficulty of form- ing other twists is great. 100. — EDGE TEETH. If the twist of the twisted tooth is such that some part of the twist at the pitcli cylin- der is always upon the line of centers, the gears will always be in action whether there are full teeth or not, and they will work with theoretical accuracy if they are reduced to edges in the pitch cylinder, as in Fig. 85. The friction of the edge tooth is theoret- ically nothing, as there is no sliding of the teeth on each other. There is but one point of contact, and that is always upon the line of centers; but if any power is carried the pressure will soon destroy the single point of contact. Fig. S5, ^d and d are the diameters of the gears, A and a are their spiral angles, and V and 'd are their velocities in revolutions. If the angles are equal, the velocity ratio is the same as for spur gears of the same diameters. Fig. 88 shows a pair of gears B and G that are of the same size and have the same angle in opposite directions, requiring the shafts to be parallel. See also Fig. 89. The pair of gears A and B are exactly alike, with equal angles in the same direction, re- quiring the shafts to be at an angle equal to Fig, SO Spiral Sjiur Gears Equal Gears Fig. 00 twice the spiral angle. See also Fig. 90. The statement that like spiral gears will not run together is founded on the Willis theory of spiral gear contact, and is wrong. 111. — SPIRAL WORM AND GEAR. When the shafts are at right angles, and the angle on one is so great that it is a screw, the combination is known as a worm gear and worm. Figs. 91 and 93, and is much used for obtaining slow and powerful motions. It is also too much used for wasting power and wearing itself out, for its friction is very great and consumes from one-quarter to two-thirds of the power received . When the screw has a single thread, the velocity ratio is simply the number of teeth in the gear, and if th re are two or three threads it must be modified accordingly. The spiral worm is adjustable in its gear both laterally and longitudinally, so that it will change its position as required by wear in the shaft bearings. It is an excellent substitute for the hobbed worm and gear, and in most cases will serve practical purposes quite as well. Worm Gearing". 57 Sptral Worm Gear and Worm. Fig. 91, Worm Gears 112. — EFFICIENCY OP SPIRAL AND WOKM GEARING. Unless the shafts are parallel the teeth of a pair of spiral gears are moving in different directions, and therefore they cannot pass each other without sliding on each other an amount that increases rapidly with the angle of divergence of the directions of motion, that is, the shaft angle. This sliding action creates friction and tends to wear the teeth, and to a very much greater extent than is generally supposed. The friction is so great, in fact, that such gears, particularly worm gears, should be used only for conveying light powers. They are extensively used, or rather misused, for driving elevators, and are even found in mill- ing machines, gear cutters, planers, and similar places, in evident ignorance that they waste from a quarter to two-thirds of the power received. The most extensive experiments on the efficiency of spiral and worm gears ever made were made by Wm. Sellers & Co., and they may be found described in great detail in a paper by Wilfred Lewis in the Transactions of the American Society of Mechanical En- gineers, vol. vii. Space will not permit ex- tensive quotations from this valuable paper, but the general result of the experiments is shown by the diagram. Fig. 93. The diagram shows that a common cast-iron spur gear and pinion on parallel shafts have an efficiency of from ninety to ninety -nine per cent. , accord- ing to the speed at which they are working ;, that a spiral pinion of 45°, angle working in a spur gear, with shafts at 45°, has an effi- ciency of from 81 to 97 per cent. ; that the efficiency decreases as the angle of the shafts increases, until, for a worm of a spiral angle of 5°, at a shaft angle of 85°, it goes as low as 34, and does not rise higher than 77 per cent. This includes the waste of power at the shaft bearings as well as that at che teeth of the gears. The efficiency is lowest for slow speeds, and rises with the speed. The diagram may be relied upon to give its true value, under ordinary conditions^ within five per cent. The same experiments developed the fact that the velocity of the sliding motion of the cast-iron teeth on each other should not be over two hundred feet per minute in contin- uous service, to avoid cutting of the surfaces. It may be assumed that the efficiency will be higher when the worm is of steel, particu- larly when the gear is of bronze. Diagram, Fig. 94, shows the result of simi- 58 Worm Gearing: Telocity at Pitch Zine in feet per minute. o o o g M •* =,2:: o 5 % 5 < § i S g§2 ^ 2 § — - = = j= ^ c..,-..^; n,-.,;... ^r° .■,,.;. .95 p=^ = = b^^.,..-.-.,...,-,.-,..,. — — 5^ — -^ — =-■ i^Sz.-. Pn. i20" .90 — ^."^•^ 1 — — ^^ 1 ^ f^ -^ ^ ^5 ^ -^ ..^^^ ^ ,6>r-/^«^(//-W-&rm-7.i^ .80 ^ ,i — y^ ~F — ^ / -55- ^ ^ j/ ^ ..,S„_A,— m— M-^,»,-i;-° 1 ^ — ^^ 7^ -7 — ' 77^ i:^: ?, .70 • Z: / •2 / .60 -S; / N ;Z : y 'X / / / / / .45 2^ : < y 1 g§8g £ Sellers* Experiments Fig, 93, lar experiments by Prof. Thurston, with a worm of 6" diameter and one inch circular pitch running in a gear of 16" diameter, both cast-iron. It is to be observed that it is the shaft angle, and not the angle of the spiral, that deter- mines the efficiency. A pair of spiral gears on parallel shafts are practically as efficient as gears with straight teeth. The great friction of worm gearing is of advantage for one purpose, and for one only, to secure safety and prevent undesired mo- tion of the gears. The worm of Fig. 97 will easily move the gear, but the gear must be moved with great force to start the worm. When the angle of the worm is as small as the "angle of repose" for the metals in contact, it is impossible for the gear to drive the worm. This may be an excuse for the use of the worm gear in elevators, but it would seem that the safety of the cage should de- Revolutions of Driver per minute 50 100 150 200 250 300 350 400' --"Pr .::_: Pft ■ffi f^^ I;: ■+K m w- m *4i^: : J. i tt^^ S J iFftn .::: 4 ::::::: r- li -t ::::::: :::: Yale & Tntviie Experimen^ts Fig, 94. pend on devices attached to the cage itself, rather than to the hoisting machinery or other distant part. Unless the friction of the gears must be depended upon for safety, the worm gear should be used only for purposes of adjust- ment, or when speed must be greatly reduced or power increased within a small compass, and not for conveying power. Worf?2 Gearing". 59 113. — THKUST OF SPIRAL TEETH. The oblique action of the teeth of spiral gears on each other tends to throw the gears bodily in the direction of their axes, and this tendency creates a thrust that must be opposed by thrust bear- ings. The end pressure on the shaft of a worm is greater than that exerted on the teeth of the worm gear it is driving. When the shafts are parallel the thrust may be completely avoided by the use of double spiral or "herring- bone " teeth, Fig. 82 or 83, which act in opposite directions, and neutralize each other. When the shafts are at right angles the thrust may be neutralized by op- posing a second gear in the manner shown in section by Fig. 95. The two worms with opposite spirals run in two spiral worm gears that also work with each other, and, as the pressure on one gear is opposite that on the other, there is no thrust on the shaft. When this combination is made with worm gears having concave teeth, the teeth can bear only at their ends. \I\J\f\J\f Artanffement to avoid tJirust Fig. 95. When the thrust cannot be avoided it should be taken by a roller bearing, rather than by the common collar bearing. The diagram. Fig. 94, shows the superior efficiency of the roller bearing as compared with the collar bearing, the gain being from ten to twenty per cent. 114. — THE HOBBED OR CONCAVE WORM GEAR. If a spiral gear is made of steel, provided with cutting edges by making slots across its teeth, and hardened, it will be a practical cutting tool called a spiral milling cutter or hob. Fig. 96 shows a spiral milling cutter, having a great spiral angle, and therefore called a worm. If this cutting spiral gear is mounted in connection with an uncut blank so that both are rotated with the proper speeds, and the shafts of the two gears are gradually brought together while they are revolving, the edge of the blank will be formed with concave teeth that curve upwards about the sides of the cutting gear. If the hob is then replaced with a spiral gear that is a duplicate of it, ex- cept that it has no cutting teeth, we shall have the familiar worm and worm gear of Fig. 97. The principle of the concave gear applies to any pair of spiral gears, on shafts at any Concave Worm Oear c Fig. 97. Worm, m Worm Gearing-. angle, but in practice it is confined to the worm and gear on shafts at right angles. The nature of the contact between the worm and the concaved worm gear has not yet been definitely determined, but there is ho reason to suppose that it is different from that between plain spiral teeth, a point con- tact on the normal spiral, but it is probably continuous. It is certain, however, that the contact is considerably closer, more nearly resembling surface contact, and being sur- face contact when the diameter of the gear is infinite. The worm is adjustable in the concaved teeth of the gear in the direction of its axis, A Hob. Fig. 96. and will change its position as required by the wear of the thrust bearing. It is not ad- justable laterally. 115. — HOBBING THE WORM GEAR. When the hob is provided it is a simple matter to cut the gear. The gear is generally provided with the desired number of notches in its edge, that are deep enough to receive the points of the teeth of the hob, and the hob will then pull it around as it revolves. It is a too common practice to make the hob do its own nicking, for, if it is forced into the face of the gear as it revolves, it will pull it around by catching its last teeth in the nicks made by the first. If luck is good these nicks will run into each other, and the gear will be cut with teeth that appear to be correct, but, as the outside diameter is greater than the pitch diameter, there will be one, two, or three teeth too many. The teeth of the finished gear are therefore smaller than those of the worm by an amount that is ruinous if the gear is small, although it is not noticeable when the diameter is large. If there are 12 teeth where there should be but 10, each tooth will be too small by two-twelfths of itself; but if there are 102 teeth where there should be but 100, each tooth is too small by but two- hundredths of itself. This handy makeshift process will do very well on large getirs, but not on small ones, unless the worm to run in the gear is made to fit the tooth, with a tooth that is smaller, and lead that is shorter than that of the hob. 116.— INVOLUTE WORM Ti::ETn. Worms are generally cut in the lathe, and as a straight-sided tooth is most easily formed, the involute tooth is generally adopted. Strictly, the form of the tool should be that of the normal section of the thread, and it should always be set in the lathe with its cutting face at right angles to the thread. But custom and convenience allow the tooth to have s raight sides, and to be set with its face parallel with the axis of the worm, and the real difference is not generally notice- able. The standard tool has its sides inclined at an angln of 30°, and has a length and a width dependent upon the pitch. 117. — INTERFERENCE OF INVOLUTE WORM TEETH. There is one difficulty that is seldom recog- nized, but which must be carefully guarded against if properly running gears are ex- pected, and that is interference. The teeth of worm gears will interfere with each other when the conditions are right for interference, Worm Geaj 'iiig Gl just as spur involute teeth will interfere, as shown by Fie:. 36. Fig. 98 shows the gear that would be formed by the usual process. The diflBculty can be remedied by rounding over the tops of the teeth of the hob and worm, as described in (55). It is also a simple matter to avoid the inter- ference by enlarging the outside diameter of Interfering Worm, Fig. 98. Interference Avoided. Fig. 99. Involute tvorin and (/ear twenty-one or more teeth Fig. 100. 62 Worm Gearing. tbe worm gear. If, as shown by Fig. 99, the tooth has but a short flank, or none at all, and the addendum of the gear is about twice that by the usual rule, the action will be con- fined to the face of the gear and the flank of the worm, and there can be no interference. By adopting an obliquity greater than 15°, interference can be avoided without changing the addendum. This method has the advantage that the same straight-sided worm and hob can be used for small gears as for large ones, and the disadvantage that the action is confined to the approach and subject to greater fric- tion (48). When the standard 30° tool is used, all gears of 26 teelh, or smaller, should be made in this way, but the correction is not strictly necessary for gears of more than 20 teeth, unless particularly nice work is required. Fig. 100 shows the proper construction of a gear of 21 or more teeth, and Fig. 101 shows that of a gear of less than 21 teeth. In the former case, the teeth of the worm should be limited by the limit line II, but the interfer- ence for 21 to 25 teeth is not noticeable. Dratu toorm teeth straight Draw gear teeth by points (57) 118. — CLEARANCE OF W^ORM TEETH. There is another practical ])oint that is sel- dom recognized, and that is that worm teeth should have clearance (40), for there is no reason for clearing spur teeth that will not apply quite as well to any other kind. The clearance is easily obtained by making the tooth of the hob a little longer than that of the worm, as shown by the tooth a of Fig. 100. For the same reason the hob should have no clearance at the bottom of its thread, so that the tops of the gear teeth will be formed of the proper length. The custom of making the hob and worm of exactly the same diameters will apply only when the worm ' ' bottoms " in the gear and the gear bottoms in the worm. 119. — CIRCULAR PITCH WORM TEETH. The old and clumsy circular pitch system is in universal use for worm teeth, for the reason that worms are generally made in the lathe, and lathes are never provided with the proper change gears for cutting diametral pitches. The error is so firmly rooted that it is useless to attempt to dislodge it. It is therefore necessary to figure the diam- eters of worm gears as if their throat sections were the same as those of common spur gears and racks on the circular pitch system. The table of diameters (35) will be of great assist- ance. One great objection to the use of the circu- lar pitch system for spur gears does not ap- ply to worm gears, that the center distance between the shafts will always be an incon- venient fraction, unless the pitch is as incon- Diametral Worm Gearing. 63 veuient. The worm can be made of any diameter, and can therefore be made to suit the pitch diameter of the gear and the center distance at the same time. The sides of the tool for circular pitches should come together at an angle of thirty- degrees, and the width of the point, as well as the depth to be cut in the worm or in the hob, should be taken from the following table. The diameter of the hob should be greater than that of the worm by the "in- crease" given. Make the tool with the proper width at the point to thread the worm, and then, after making the worm, grind off half the "in- crease" from the length of the tool, and use it to thread the hob. TABLE FOR CIRCULAR PITCH WORM TOOLS. Circular fiitch 2 .644 .620 1.416 .16b 1% .564 .542 1.240 .146 .483 .466 1.062 1.249 XX4 Pf unt of hob tool Point of worm tool... Depth of cut in worm or hob Increase .402 .388 .886 .104 .362 .349 .797 .094 Circular pitch Point of hob tool Point of worm tool.. Depth of cut in worm or hob .322 .310 .708 .ObS % .282 .271 .620 .073 M .241 .233 .531 .062 % .201 .194 .443 .052 .161 .155 354 Increase .042 Circular pitch Point of hob tool Pointof worm tool .. Depth of cut in worm or hob .141 .135 .310 .036 .121 .116 .265 .031 100 .097 .222 .026 .080 .078 .177 .021 .060 .058 133 Increase .016 120. — DIAMETRAL PITCH WORM TEETH. If the proper change gears are provided, it is as easy to cut diametral pitch worm teeth as any. The proper gears can always be easily calculated by the rule that the screw gear is to the stud gear as twenty-two times the pitch of the lead screw of the lathe is to seven times the diametral pitch of the worm to be cut. For example, it is required to cut a worm of twelve diametral pitch, on a lathe having a leading screw cut six to the inch. We have screw gear _ 22 X 6 _ 11 stud gear ~ 7 X 12 ~~ "7~ ' and any change gears in the proportion of 11 and 7 will answer the purpose with an error 1 of of an inch to the thread of the worm. 10,000 If 22 and 7 give inconvenient numbers of teeth, the numbers 69 and 22 can be used with sufficient accuracy, and 47 and 15, or even 25 and 8 may do in some cases. To save calculation and study, the table of change gears for diametral pitches is provided, and it will give the proportion of screw gear to slud gear to be used for all ordinary cases. The pair on the left will give the proper pitch within less than a thousandth of an inch, and that on the right will serve with an error always less than a hundredth of an inch, and sometimes less than two or three thou- sandths of an inch. Having the change gears, figure the pitch diameter of the gear as if the throat section is a spur gear on the diametral pitch system. The sides of the tool should come together at an angle of thirty degrees, and the width of the point of the tool, as well as the depth to be cut in the worm or in the hob, should be taken from the following table. The diame- ter of the hob should be greater than that of the worm by the "increase" given. TABLE FOR DIAMETRAL PITCH WORM TOOLS. Diametral pitch. 1 1.035 .968 2.125 .250 2 .517 .484 1.063 .125 3 .345 .323 .708 .083 4 Point of hob tool .258 Point of worm tool Depth of cut in worm or hob Increase .242 .532 .063 Diametral pitch 5 .207 .194 .425 .050 6 .173 .162 .354 .042 7 .148 .138 .304 .036 8 Pointof hob tool Point of worm tool . . Depth of cut in worm or hob .129 12-1 .266 Increase .032 Diametral pitch . . 10 .104 .097 .213 .025 12 .086 .081 .177 .021 14 .074 .069 .152 .018 16 Point of hob tool Point of worm tool Depth of cut in worm or hub Increase .065 .060 .133 .016 Make the tool with the proper width at the point to thread the worm; and then, after making the worm, grind off half the "increase" from the length of the tool, and use it to thread the hob. 64 Diametral Woi Gea ri ff/y". 3 4 5 6 7 8 10 12 14 16 Pitch of Leading Screw. 6 8 10 44 23 21 " 11 22 88 46 21" 11 22 7 ' 110 21 ' 55 U* 22 25 7 ■ 8 44 69 7 " 11 33 7 ' 132 15 35 * 4 21 3 * 11 2 * 176 92 21 'll 220 21 ' 11 7 ' 33 14' 66 15 35" 8 11 30 7 ' 19 44 7 ' 55 7 44 25 7 ■ 4 110 100 21 ■ 19" 44 5 35" 4 88 5 35* 2 44 40 21 ' 19 88 9 49* 5 11 8 7 ' 5 44 5 35' 4 22 35 5 * 8 176 5 35 ■ 1 88 80 21*19 22 21 ' 55 50 21 ' 19 110 9 49 * 4 55 2 28* 1 22 60 7 " 19 132 27 "ig'^io 33 12 14* 5 66 15 35' 8 11 30 7 ' 19 66 4 49* 3 33 6 28* 5 11 70 3 "19 22 63 7 '20 11 14 4 ■ 5 44 9 49' 10 66 27 49*20 33 6 28* 5 176 18 49 ' 5 22 16 7 ' 5 88 5 35* 4 , 220 9 49 * 2 11 4 14' 5 22 5 35' 8 55 4 14* 1 33 15 35*16 11 25 ,7 ' 16 55 25 42*19 55 10 49* 9 55 1 56* 1 11 35 5 ' 16 11 35 6 ' 19 22 25 7 " 8 55 50 21*19 11 10 21" 19 11 15 14' 19 22 20 2l'l9 44 8 49* 9 22 4 28* 5 44 40 21*19 88 16 49*7 22 4 49" 9 33 2 49 '¥ 33 3 56* 5 11 14 7 * 9 77 7 56* 5 110 20 49 ' 9 55 2 28* 1 11 2 28' 5 11 8 7 " 5 Exact numbers on the left. Approximate on the right. Table of Change Gears for Diametral Pitch Worms. 121. — WIDTH OP WORM GEAR FACE. The bearing between the tooth of the worm and that of the gear is near the center of the gear, and it is quite small (104). It is, therefore, useless to make the gear with a wide face. If the face is half the diameter of the worm it will have all the bearing that can be obtained, and any extra width will simply add to the weight and cost of the gear. The length of the worm need be no more than three times the circular pitch, for there are seldom more than two teeth in contact at once. If, however, the worm is made long, it can be shifted when it becomes worn, so as to bring fresh teeth into working position. This provision is wise, for the reason that the worm is always worn more than the gear. 122. — THE HINDLEY WORM AND GEA"1. If the cutting hob and the worm is shaped by the tool a, and the process indicated by Fig. 102, the resulting pair of gears is known as the Hindley worm and gear. The worm is often called the "hour-glass" worm. It is commonly but erroneously stated that this worm fits and fills its gear on the axial section, the section that is made by a plane through the axis of the worm and normal to the axis of the gear. It has even Hi?idlev Wor?n Gearing. r)5 been stated that the contact is between sur- faces, the worm tilling the whole ^ear tooth. The real contact is not yet certain, but it is certain that it is not a surface contact. It is also certain that it is on the normal and not on the axial section, and that the Hindle}'^ worm hob will not cut a tooth that will till any section of it. The contact may be linear, along some line of no great length, but it is probably a point contact on the normal sec- tion. The order of the contact is certainly very close, resembling that of two surfaces. The worm is limited in length, for the sides of the teeth cannot slant inward from the normal to the axis. The end tooth m in Fig. 102 cannot be used, for it will destroy the teeth of the gear as it is fed towards this axis in the operation of bobbing. It has the one great defect that it is not adjustable in any direction, and, therefore, cannot change its position when the shaft Tlxe Mindley Wortu Gear bearings wear, unless it is itself worn the same amount. It is doubtful if this form of gearing has any advantage over the plain spiral gearing, except when new and in per- fect adjustment. 133. — THE PIN WORM AND GEAR. If the hob and the worm are shaped by the pin-shaped revolving milling tool b of l^in ti-onn gear Fig, 103. Fig. 102, the gearing produced will have linear bearing between the teeth. The action will be the same as between a series of pin teeth like the milling tool, each pin being in the axial section of the worm, but having a linear bearing on the normal section of its teeth. This form of gearing, which is a modification of the Hindley form, may take the shape of pin gearing, the- teeth being round pins like the milling tool. If the pins are mounted on studs, so as to revolve, a roller pin worm gear will be produced. Fig. 103 shows a form of roller pin gearing in which the pins have been en- larged. 124. -THE WHIT WORTH HOBBING MACHINE. When the amount of work to be done will warrant the use of a special machine, the bobbing machine of Sir Joseph Whitworth m3y be used. It was invented in 1835, and has not been materially improved since then. although there are numerous patents relating to it. The worm gear to be hobbed is fixed upon the same spindle with a master worm- wheel. A driving worm runs in the master wheel, and it is connected by a train of gear- 66 Hobbing Machines. ing with a hob that is so mounted on a carriage that it can be fed towards the gear blank. The hob is slowly forced into the blank, while both are revolving with the proper speeds, and the gear is cut without the assistance of previously made nicks. See British patent 6,850 of 1835. IPig. lOG Spiral and Spur Gear, 125. — THE CONJUGATOR. This is a machine for cutting spur or spiral gears by means of a hob, and its principle is an extension of that of the Whitworth worm gear hobbing machine. If, when the hob in the Whitworth ma- Conjugator. Elevation m Fig, 104:, Plan I ^^y 105. chine has reached the full depth of the tooth, it receives a new motion in the direction of the tangent to its pitch spiral, it will continue the tooth to the edge of The gear, and form the plain spiral gear of Fig. 91, Fig. 104 is an elevation of the machine, and Fig. 105 is a plan. The hob h is mounted upon an arbor that is connected by a train of gearing with the spindle s that carries the blank gear ^ to be cut, so that the hob and blank revolve together with any definite proportionate speed. The hob is carried upon a carriage that is fed on a frame /. The hob swivels upon the carriage, so that the tangent to its pitch spiral can be set parallel with the direction of the feed, and the frame swivels so that the tooth can be cut at any angle with the gear spindle. As the blank and the hob are revolving, the latter is fed into the former, and it will cut a perfect tooth in the direction that the frame is set at. As the frame can be set in any direction, the machine will cut the com- mon straight tooth, as shown by Fig. 106. All gears cut by the same cutter will run together interchangeably, and if two spiral gears are cut at the same angle in opposite directions they will run together on parallel shafts. See U. S. patent number 405,030, June 11th, 1889. 7. IRREQUIvAR. AND ELLIPTIC QEARS. 126.— NON-CIRCULAR PITCH LINES. The consideration of pitch lines that are not circular, and of the teeth that are fitted for them, is an interesting but not particularly- important branch of odontics. Such pitch lines are largely used for producing variations of speed and power, but have no other prac- tical applications. 12'; -THE IRREGULAR PITCH LINE. The most general case is that of two indefi- nite irregular curves rolling together, Fig. 107, the only condition being that they shall be so shaped that they will roll together continuously. As the practical importance of the free pitch line is very small, we shall not ex- amine it in detail. Irregular pitch: lines Fig, 107. 128.— PITCH LINES ON FIXED CENTERS. When we attach the condition that the two pitch lines shall revolve in rolling contact on fixtd centers, we have a definite problem of more interest and importance than that of the free pitch line. If, as in Fig. 108, we have a pitch line A revolving upon a fixed center a, we can con- struct a pitch line B that will roll with it, and revolve on the given fixed center b, by the following process. From any pitch point 0, step off equal arcs Oc, cc, cc ; draw circular arcs cd from the center a; draw circular arcs dn from the center h; step off the same equal arcs Oe, ee, ee, then Oeee will be the required mat- ing pitch line. These curves will always be in rolling con- tact at a point on the line of centers ab, the pitch point and the angle of the curves with the line of centers continually changing. The velocity ratio of ihe curves will be Winced centers JFig. 108. variable, and always equal to the inverse pro- portion of any two mating radiants, ac and be. 68 Multilobes. 129. — CLOSED PITCH LINES. When one of the curves of Fig. 108 is a closed curve, the other will in general not be closed, but By trying different centers, a curve can be found that will be closed. If the closed curve x\ \ V ; N A 1 < O ^ y / / \ / / / ( \\ \ \ 1 1 > k lS ^' / \ //\ vJ ^^ / \ / \ /— / / > \// J / \\ o< \ c >^./ ' / / / / V -^\ ^ /j \ / ) n -• r >< Ci \^ Involute elliptic teeth. Fig, 129, 143. — myoLUTE elliptic teeth. As in the case of the circular gear, the best form of tooth for the elliptic gear is the involute, and for the same reasons. The base line of the involute tooth is any ellipse BE, Fig. 125, which is drawn from the same foci as the pitch ellipse ; the limit point i is the point of tangency of a tangent from the pitch point 0, and the addendum line a I of the mating gear must not pass beyond that point. The method of laying out the tooth and drafting it is so exactly like the process for the circular gear that the description need not be repeated. The centers of involute elliptic gears can be adjusted without affecting the perfection of the motion transmitted, but, as the focal distance remains fixed, the ratio of the axes will be altered. The work of drawing the teeth can be much abbreviated by the process illustrated by Fig. 129. Find the centers for approxi- mate circular arcs, preferably by the method of (140), and then consider the gear as made up of four circular toothed segments. It is then necessary to construct but two tooth curves, one for the major and one for the minor segment, and the flanks will be radii of the circular arcs. The line of action, la, Fig. 125, is not a straight line, and it is not the same for all the teeth. It is not fixed when the pitch point and the line of centers is fixed (134). 78 Cycloidal Elliptic Gears. Cycloidal elliptic teeth* Fig, 130. 144. — CYCLOIDAL ELLIPTIC TEETH. The cycloidal tooth is drawn, exactly as upon a circular pitch line, by a tracing point in a circle that is rolled on both sides of the pitch line. The line of action is not a circle, and it is not the same curve for all the teeth. That the flanks shall not be under-curved, the diameter of the rolling circle should not be greater than the radius of curvature at the tooth being drawn, and when, as usual, the same roller is used for all the teeth, its diameter should not be greater than the radius of curvature at the major apex, the distance Gh of Fig. 122. Fig. 130 shows a cycloidal gear drawn as four circular segments, by the methods of (140) and (83). 145. — IRREGULAIl TEETH. It is most convenient to draw all the teeth alike, with the same rolling circle, or from the same base line, and also to uniformly space the pitch line, but such uniformity is not The only requirement is that each tooth curve shall be conjugate to the tooth curve that it works with, and if that condition is satisfied the teeth may be of all sorts and 146. — FAILURE IN THE TOOTH ACTION. When the major axes are in line the action of the teeth on each other is nearly direct, but when the minor axes are in line the action is more oblique, as shown by Fig. 127. The teeth tend to jam together when the driver is pushing the follower, and to pull apart The Link. 79 when the follower is being pulled, and when the ellipse is very flat this tendency is so great that the teeth fail to act serviceably. At first glance it might appear that this diflBculty in the tooth action of very eccentric gears might be overcome by making the teeth radial to the focus, as shown by Fig. 131, but examination will show that but little can be gained in that way. The teeth on the gear C were obtained by the method of (28) from the assumed tooth on the gear c, and the effect of the defective shape of one side of the assumed tooth was to cut away the conjugate curve of the derived tooth. Such teeth would not work as well as the ordinary form, and their construction would be very difficult. Radial teeth mg.131. 14- -THE LINK. When the teeth of the elliptic gear fail to ])roperly engage, on account of the obliquity of the action, the difficulty can be entirely overcome by connecting the free foci by a link (141), as shown by Fig. 137. This link works to the best advantage when the teeth are working at the worst, and when it fails to act, as it passes the centers, the teeth are working at their best. There- fore gears that are connected by a link need teeth only at the major apices. When the tooth action is imperfect by rea- son of its obliquity, and the link is not avail- able or desirable, the difficulty can be over- come by using three or more gears in a train, as shown by Fig. 137, for then the same re- sult can be obtained by the use of gears that are much more nearly circular. 148. -VARIABLE SPEED AND POWER. If the shaft c, Fig. 132, turns uniformly, the slowest speed of the shaft G will occur when the gears are in the position of the figure, and the proportion between the two speeds will be the proportion between the distances cO and CO. The greatest speed of the driven shaft will occur when the shafts have turned through a half revolution from the position of the figure, and the relative speed will be the same, reversed. The ratio of speed, the ratio of the greatest speed to the slowest speed, is the square of the ratio between the speed of the driving shaft and the greatest or the least speed of the driven shaft, so that it requires but a slight Fig, 132. variation of the axes to produce a decided variation of the speed. The following table will give the propor- tion of minor to major axes that will give any desired ratio of speeds. 80 Elliptic ^uick Return Motiot tio of Speeds. 2 Katio of Axes 985 3 .962 4 952 5 924 6 . 907 7 892 8 878 9 868 10 854 11 844 12 834 13 14 824 817 15 807 16 800 The power is always inversely proportional to the speed. If the variable shaft is running twice as fast as the uniform shaft, it will ex- ert but one-half the force. When the gears are arranged in a train, as in Fig. 137, the speed ratio for the second, third, and following gears will be in the pro- portion of the first, second, third and follow- ing powers of the first ratio. Thus, the ratio for a pair of gears with axes in the proportion of .952 to 1 being 4 for the second gear, will be 16 for the third gear, 64 for the fourth gear, and so on. The use of gears of troublesome eccentric- ity can be avoided by this means. A train of three gears of .953 axes. Fig. 137, is equivalent to a single pair of very flat gears vrith .800 axes, Fig. 138, and, in general, three gears that are nearly circular are equiva- lent to a single very flat pair. 149.— ELLIPTIC QUICK RETURN MOTION. If the gears are arranged with respect to the piece to be reciprocated, in the manner shown by Fig. 133, the time of the cutting stroke will be to the time of the return stroke, as the angle PEK is to the angle PEF, where £^and ^are the foci of the ellipse. The following table will show the ratio of axes that must be adopted to produce a re- quired ratio of stroke to return. Quick Return. 2 to 1 .3 tol 4tol 5 tol 6 tol Ratio of Axes. 964 910 861 817 778 To determine the ellipse that v^ill give a required quick return, we lay off the angles PEK and PEF in the given proportion, and then find by trial a point P such that the length PE plus the length of the perpendicu- lar PF is equal to the known center distance Ee. F will be the other focus of the re- quired ellipse. When the driving gear has turned through the angle PEF, from the position of the figure at the middle of the return, the varia- ble gear will have turned through the angle P"eO — P'FO, and we can study the action of the tool by drawing equi-distant radii about E, and finding the corresponding radii about F. Quick return Fig. 133. Fig. 134 shows the arrangement of the radii {PF = P"e of Fig. 133) in the case of a four to one quick return, and it is seen, by the parallel lines, that the motion of the tool is very uniform, coming quickly to its maxi- mum speed, and holding a quite uniform speed until near the end of the stroke. Fig. 135 shows that the same motion derived from a simple crank is not as uniform. When the gears are arranged in a train, Fig. 137, the quick return ratios can be de- termined by the construction shown by Fig. 136. Draw Fc at right angles to AA\ and draw cEd through the other focus. The quick return ratio of the second gear will be the ratio of the angles a^ and 63. Draw dFe, and the ratio for the third gear will be Elliptic Trains. Quick return crank Ordinary crank Fig, 135, that of the angles a^ and Jg. Draw eEf, and a^ and 64 will give the ratio for the fourth gear. And so on, in the same man- ner, as far as desired, the ratio being greatly increased by each gear that is added to the train. If carefully performed, the graphical pro- cess is quite accurate. The case of axes in the proportion of .98 to 1 gave a quick re- turn of 1.6 for the second gear, and 2.8 for the third gear, while their true computed values are 1.66 and 2.74. The chart will solve quick return train questions involving gears not flatter than .80, as accurately as need be. For example, the ratio of axes of .95 will give a quick return of 2.25 for the second gear, 4.85 for the third gear, 9.80 for the fourth gear, and 19.70 for the fifth gear. Again, the proportion of axes to give a quick return of 5 for the third gear is .948. Quick return train Fig, 136, JElliptic train Fig, 137 1 Fig, 138. 82 Elliptic Gear Cutting MacJiine. Elliptic Quick Iteturn Chart \ \ y \, \ \ \ \ \ N \ \ \ \ \ s ^ \^. N \V N ^^ VL N w V V. ^^»i \^ \^ ^ k^^ S' \ S?^ \ \ '^ s. \ ■^ ^ \ V =^ "^ \^ \ — — Si ?coi *2^ ^ar' ^ \ \ ^^^ ^ \ y ^ ■^ ■^"^ A Fir stg 3nr ~~ ■ ^^ .so ,81 .82 .87, .88 .89 .90 .91 .92 .93 -94 .95 .96 .97 .98 .99 1.00 Froporfion of Axes 150. — THE ELLIPTIC GEAR CUTTING MACHINE. The conditions of the described operation of drawing the ellipse by- means of the trammel (138) may be reversed, the bar being held still while the paper and the cross are revolved, and it is evident that the result will be the same ellipse on the paper as if the bar is revolved as described. By thus reversing the process of describing the ellipse, and by adopt- ing the improved spacing device of (142), we can construct a machine for accurately cutting the teeth in an elliptic gear, the main features of which, omitting various unessen- tial details, are shown by Figs. 139 and 140. The blank to be cut is fastened upon a trammel stand, which cor- responds to the paper in the graphi- cal process, and revolves upon the fixed base. The adjustable trammel pins a and b are fixed in a slot in the bed, and they fit and slide in the slots M and JSf in the under surface of the stand. The cutter which corresponds to the tracing point is fixed with the pitch center of its flan Fig. 139, Cutter Elevation Fig. 14.0, Elliptic JBevel Gear. 83 tooth curve directly over the point P in the line of the pins. The index plate has a diameter equal to the sum of the axes of the ellipse, and it is held by an index pin p, which slides in the slot, and is always'in the line of the pins. Thus arranged, the machine will always cut its tooth in the true ellipse, and the teeth will be accurately spaced. The direction of the tooth will be sub- stantially at right angles to the pitch line, and a simple arrangement can be applied to make it exactly so. An index plate of a fixed diameter may be used for all sizes of gears, if the index pin is carried by an arm which swings about the center of the gear, and has an adjustable pin that slides in the slot. The tops of the teeth are trued by a cutter having a square edge, and the line of the tops will be substantially parallel to the pitch line. The blank is held by an arbor through its focus hole, and the arbor is held by a slide, which slides in a chuck upon the stand, so that the focus can be accurately set in the major axis at the proper distance from the center. 151. — CHOICE OF CUTTERS. Theoretically, the teeth are of different shapes, as they are in different positions upon the ellipse, and, therefore, each space should Idc cut with a cutter that is shaped for that particular space. But as this is impracticable, it is necessary to choose the cutter that will serve the best on the average. Strictly, the cutter should be the one that is fitted to cut a spur gear having a pitch radius equal to the radius of curvature of the ellipse at the major apex, but as that cutter will be much too rounding for the minor apex, it is better to choose the one that is fitted for the medium radius of cur- vature. The two radii of curvature are the dis- tances Ch and Gk, Fig. 122, and the cutter should be chosen for the radius half way between the two, approximately half the sum of the two. 152, — THE ELLIPTIC BEVEL GEAR. An ellipse may be drawn on the surface of a sphere by means of a string and two pins, according to the method of (138), and a pair of such spherical ellipses will roll on each other while fixed on their foci, their free foci moving at a constant distance apart. Therefore we can have elliptic bevel gears that are very similar to elliptic spur gears, as shown by Fig. 141. The two gears revolve on radial shafts through their foci, and the link connects radial shafts through the free foci. The velocity ratio is the ratio of the perpendiculars a h and a c. The elliptic bevel gear is the invention of Pro- fessor MacCord. The spherical ellipse cannot be drawn by the trammel method of (13S), and therefore the method of spacing of (142), as well as JEllipiic bevel gea'rs Fig. 141, the gear cutting machine of (150), does not apply. 84 Elliptic Calculations. 153.— MATHEMATICAL TREATMENT. If the major semi-axis is a, and the minor semi- axis is h, the equation of the curve from the origin at C is a'^ yi _|_ J8 a.3 ^ ^8 JyZ^ the major axis being the axis of X The distance CF from the center to the focus will be in which n is the ratio of axes = — . a The radius of curvature at the major apex is — , and that at the minor apex is — . a b There is no practicable formula for the recti- fication of the curve, as the length is express- ible only by a series. The special spacing method of (142) is true only at the instant of passing either apex, for the tracing point describes half the arc described by the line of the bar on the index circle only when the bar is at right angles with the curve. The error will be at its maximum when the bar is at the maximum angle with the normal, which is at about an angle of forty-five degrees with the major axis. The difference between an ordinary tooth space at the major apex, and that at the minor apex, is very minute. A very careful calculation of the length of the chord of a gear of seventy-two teeth, and eight and ten inch axes, gave a chord of .41433" at the major apex, a chord of .41495" at 45° for the maximum, and a chord of .41441" at the minor apex. The difference between the chords at the apices is .00008", but as the cur- vature at the major apex is greater than at the minor apex, the difference between the arcs would be less, perhaps not over .00004". The ratio of speeds (148), is 1 + Vi" / 1 + yi - 71^ \« \ 1 — Vi — ^' / The ratio of quick return being given as qr, the value of n is / 180 \° 2d\ in which d = tan. When the gears are in a train, there seems to be no simple method for computing the ratio of axes to produce a given quick return, but, when the ratio is given, the quick return for each gear can be computed best by trial and error with the formula sin. (M — JV) = V~i n'^ sin. M -\- sin. N in which M is any known angle h, Fig. 136, and iVis the angle 6 for the next following gear in the train. Thus, assuming n = .98, and Jfi = 90°, we find JSf^ = 67° 28'. Then putting J/g = 67° 28', we find W^ - 48° 5'. Knowing the angles, we compute the quick return ratio from which, for- n = .98 gives qr for two gears equal to 1.66, and for three gears equal to 2.74. The graphical process of Fig. 136 should first be employed to fix the angles approximately. 8. THE BKVKIv QE)AR. 154. — THE BEVEL GEAH. The theory of the bevel gear cannot be properly represented, and can be studied only with the greatest difficulty, upon a plane surface. It is essentially spherical in nature, and should be shown upon a spheri- cal surface, as in Figs. 143 and 144. This is best done upon a spherometer, which is simply a painted sphere fitted in a ring. The sphere rests upon a support, so that the ring coincides with a great circle upon it, and the ring is graduated to 360°. A very roughly made wooden sphere and plain ring will be found to answer the gen- eral purpose very well, and should be pro- vided if the study of the bevel gear is seriously intended. If painted, ink marks can be scrubbed off, and pencil marks re- moved with a rubber. The mathematical treatment is unapproach- able without a knowledge of the common principles of spherical trigonometry. A wide, interesting, and difficult field of study is offered, but space will permit but a brief examination of the more prominent and practical points. A careful examination would require ten times the available space. 155. — THE GENERAL, THEORY. When thus represented upon the spherical surface, the theory of the bevel gear is so similar 16 that of the spur gear, as repre- sented upon a plane surface, that any de- tailed description would be mostly a repeti- tion of what has already been stated. All straight lines of the spur theory are represented by great circles, the crown gear being the rack among bevel gears, and all distances are measured in degrees. Irregular pitch lines and multilobes are managed substantially as for spur gearing. The elliptic bevel gear has been described in connection with elliptic spur gears (152). The tooth surfaces of the bevel gear are generally formed by drawing straight lines from the spherical outline to the center of the sphere, as in Figs. 143 and 144, the pitch lines and tooth outlines being the bases of cones with a common apex. When limited in width, as is usually the case, it is by a sphere concentric with the outside sphere, so that a spherical shell is formed. These concentric spherical shells can be moved on their axes to form twisted and spiral teeth, Fig. 142, precisely as described for spur gears (99). The molding process of (27) will apply per- fectly, but it has but one practical applica- tion. Fig. 142, Twisted bevel gear. The planing process of (28) will fail, for practical purposes, except for one particular form of tooth, because the shape of the cut- ting tool cannot in the general case be 86 Involute Bevel Gears. changed in form as it approaches the apex, and therefore the tooth will not be conical. The planing process of (29) will apply per- fectly, the strokes of the tool being radial, and on this method we must depend for the accurate construction of all forms of bevel gear teeth except the octoid and the pin tooth. As the diameter of the sphere is increased, the radii become more nearly parallel, until, when the diameter is infinite, they are paral- lel. Therefore the spur gear is a particular case of the bevel gear, and all formulae and processes that. apply to the bevel gear will apply to the spur gear if the diameter of the sphere is made infinite. The most scientific method of study would be to develop the theory of the bevel gear, and fr«om that pro- ceed to that of the spur gear, but such a method would be diflScult to clearly carry out, and is best abandoned for the more con- fined process here adopted . 156. — PARTICULAR FORMS OF BEVEL TEETH. As in the case of spur gearing, there can be an infinite number of tooth curves for bevel gearing (31), each form having its own line of action, but as there are only four forms that are available for practical use by means of simple processes of construction, our attention will be confined to them. These four particular forms are, first, the involute tooth, having a great circle line of action; second, the cycloidal tooth, having a circular line of action; third, the octoid tooth, having a plane crown tooth, and a "figure eight" line of action; and, fourth, the pin tooth, for which one gear of a pair has teeth in the form of round pins. 157. — THE INVOLUTE BEVEL TOOTH. The spherical involute must be studied as a whole if its form is to be clearly seen. Its definition is that it is the tooth curve having a great circle for a line of action. In Fig. 143 the great circle line of action la ex- tends around the sphere at an aagle with the crown pitch line 'pl, and it is tangent to two base lines hi and hV , that are paral- lel with the crown line. The most convenient method of draw- ing the tooth curve is by rolling the line of action on the base line, while a point in it describes the curve on the surface of the sphere. The equivalent graphical process is to step along the base line and any two tangent great circles, from any point on the curve to any desired point. It will take the form shown by the dotted lines ; rising at right angles to the base line, it curves until the crown line is reached, there reversing its curva- ture and bending the other way until it meets the other base line. At the base line it has a cusp, and rises from it to repeat the same course indefinitely. Fig. 143 shows a crown gear or rack. The pitch line is the great circle 'gl. The line of centers c 0(7 is a great circle at right angles with the crown line pl. The line of action is the great circle la set at a given angle of obliquity with the crown line. The base Fig, ld3. The involute Tooth Bllgram Bevel Gears. 87 circles are the small circles hi and W . The spherical involutes have the same property of adjustability as have the spur involutes, the motion being confined to the sphere, and there- fore the gears are adjustable as to their shaft angle, the apex remaining common to both. 158. -THE CYCLOID AL BEVEL TOOTH. The definition of the cycloidal tooth is that it is that form which has a circular line of action. The rolled curve method of treatment (32) applies, and is the best means of studying the curve. There is no gear with radial flanks, the flank formed by a roller of half the angular diameter of the gear being nearly but not exactly a plane. The theory differs so little from that of the spur gear, that but little of interest can be found, and the curve will not be consid- ered further. 159. -THE OCTOID BEVEL TOOTH. The definition of this tooth system is that it is the conjugate system derived from the crown gear having great circle odontoids. In Fig. 144 the crown gear has plane teeth cutting the sphere in great circles, mOn, while a pinion would have convex tooth curves conjugate to the great cir- cles of the crown tooth. The line of action, from which the tooth derives its name, is the peculiar ' ' figure eight " curve la, which is at right angles to the tooth curve at the crown line 'pl, and tangent to the polar circles 8 and 8\ to which the great circle crown odontoids are also tangent. This tooth owes its existence to the fact that it is the only known tooth, and probably the only possible tooth, that can be practically formed by the mold- ing planing process of (28).* The cutting edge of the tool being straight, no change is required while it is in motion, except in its position, and that is accom- plished by giving it a motion in such a direction that its corner moves in the radial line of the corner of the bottom of the tooth space. The octoid tooth, together with an ingeni- * Since this statement was made, another bevel practically constructed b)' the process of (28). ous machine for planing it, was invented by Hugo Bilgram, but it has always been mis- taken for the very similar true involute tooth. Xlie Octoid Tooth Fig, 14L^. Bilgram' s machine is described in the American Machinist for May 9th, 1885, and in the Journal of the Franklin Institute for August, 1886. tooth, the " planoid " tootli, has been invented and 160. — the pin bevel tooth. If the tooth of one gear of a pair is a coni- cal pin. Fig. 145, with apex at the center of the sphere, that of the other will be conju- gate to it, and the combination deserves notice because it is one of the few forms that are easily constructed. It may be said that 88 Tredgold' s Method. its practical construction is simplei; and easier than that of any other form of bevel gear tooth except the skew pin tooth of (180). Tig. 14^5. ^"^^ *''''*^ '' preferably, but not necessarily, of the conical form, for other forms of circu- lar pins would serve the theo- retical p u r - pose. IBin hevel geara^ Its theory is, in the main, the same as that of the spur pin tooth. It has the same troublesome cusp, which can be avoided in the same way, by setting the center of the pin back from the pitch line. It is the only known form of tooth that can be formed in a practical manner by the molding process of (27). If the cutting tool is a conical mill, it will form the conjugate tooth while the two pitch wheels are rolled together. The pins may be mounted on bearings at their ends, forming roller teeth. They would be weak, but would run with the least possible friction, all the rubbing friction being confined to the bearings. 161. — tkedgold's approximation. The construction of the true bevel gear tooth curve upon the true spherical surface is impracticable with the-means in ordinary use, and the true method of computation by means of spherical trigonometry is equally unfitted for common use. But, by adopting Tred- gold's approximate method the difficulties can be overcome. By this method the tooth curves are drawn, not on the true spherical surface, but, as in Fig. 146, on cones A and B drawn tangent to the sphere at the pitch lines of the gears. The cones are then rolled out on a plane sur- face, and the gear teeth drawn upon them precisely as for spur gears of the same pitch diameter. Practically correct tooth curves could thus be drawn on the spherical surface by cutting the teeth to shape, and bending them down to scribe around them, but in practice the back rims of the gears are shaped to the tan- gent cones so that the teeth lie directly upon the conical surface. This method is called approximate, but its real error would be diflScult to determine, and is certainly not as great as the inevitable errors of workmanship of any graphical pro- cess. The tooth outline drawn by it upon the spherical surface may be considerably different from that which would be drawn directly upon it, but it does not follow that it is therefore incorrect. The only require- ment is that the engaging curves shall be Fig. IdO Tredgold' s method, conjugate odontoids, and it is a matter of very small consequence whether or not the curve on the sphere is the same kind of curve as that upon the cone. If the true plane in- volute curve is drawn upon the developed cone, the corresponding curve on the sphere will not be an exact spherical involute, but its divergence from some true odontoidal shape must be minute, even when the teeth are very large indeed . In ordinary cases it cannot be sufllcient to affect materially the constancy of the velocity ratio. What is sometimes given as its error is mostly the " difference in shape" between the plane and the spherical teeth. Draftiitg Bevel Gears. 89 162.— DRAFTING THE BEVEL GEAR. The practical application of Tredgold's i method is illustrated by Fig. 147. ' Draw the axes GA and CB at the given | shaft angle AGB. Lay off the given pitch j radii a and l, and draw the lines c and d in- tersecting at tbe pitch point 0. Dra the center line OG, and lay off the face Of. The pitch diameters are ON and OM, and NGO and MGO are the pitch cones. Draw the back rim line 0T> at right an- gles with the center line, lay off the addenda Oe and Og, and the clearance gli. Draw the front rim line parallel to the back rim line. The center angle is X, the face increment is F, and W is the face angle. The cutting decrement is J, and T is the cutting angle. Twice the distance mn is the diameter incre- ment, and em is the outside diameter. The pitch radius of the Tredgold back cone is OB, and the figure shows the con- struction of the gear teeth on this cone developed. The teeth are represented as drawn upon the figure, but it is better to use a separate sheet. The odontograph should be used, calculating the number of teeth in the full circle of the developed cone. ;Fig. 14:7, Drafting the bevel gear.. 163. — THE BEVEL GEAR CHART. diameters must be taken off for use at the lathe, and that is by no means a simple mat- The drafting of the bevel gear blanks by means of the method of (162) is simple, but the method requires drafting instruments, not always at hand, as well as the ability to use them accurately. The drawing must be carefully made, to give correct results, par- ticularly when the gears are small. After the drawing is made the various angles and ter. So great are the practical difficulties that any one who has a knowledge of simple arith- metic will find it not only easier, but more accurate to use the chart and method by means of the following rules. 90 THE BEVEL &EAR CHART. Shafts at 90° I'roportion. Center Angle. 1 ^ i > Shafts at 90° Proportion. Center Angle, ^ 5 5 S .10 1—10 5.72 11 2.00 10.00 10—1 84.28 114 .20 .11 1—9 B.33 13 2.00 9.00 9-1 83.67 114 .22 .13 1— 8 7.12 14 1.99 8.00 8—1 82.88 113 .25 .14 1— 7 8.13 16 1.98 7.00 7—1 81.87 113 .28 .17 1— 6 9.47 19 1.97 6.00 6-1 80.53 113 .33 .20 1 1— 5 11.32 23 1.96 5.00 5—1 78.68 112 .39 .22 2— 9 12.53 25 1.95 4.50 9—2 77.47 111 .43 .25 . 1— 4 14.03 28 1.94 4.00 4 1 75 97 111 .49 .29 2— 7 15.95 32 1 92 3.50 7—2 74.05 110 .55 .30 1 3—10 16.70 33 1.92 3.33 10-3 73.30 109 .58 .33 1 1— 3 18.44 36 1.90 3.00 3-1 71.57 109 .63 .38 1 3— 8 20.55 40 1.87 2 67 8—3 69.45 107 .70 .40 1 2— 5 21.80 43 1 86 2.50 5—2 68.20 106 .74 .43 3— 7 23.20 45 1.84 2.33 7—3 66.80 105 .79 .44 4— 9 23.97 46 1.83 2.25 9—4 66.03 104 .81 .50 1-2 26.57 51 1.79 2.00 2—1 63.43 103 .89 .56 5— 9 29.05 56 1.74 1.80 9-5 60.95 101 .97 .57 1 4— 7 29.75 57 1.74 1.75 7—4 60.25 99 .99 .60 1 3— 5 30.97 59 1 72 ^ 1.67 5—3 59.03 98 1.03 .63 5— 8 32.00 61 1.69 1.60 8-5 58.00 97 1.06 ^iTr .67 2— 3 33.68 64 1.66 1.50 3-2 56.32 95 .70 7—10 34.99 66 1.64 1.43 10—7 55.00 94 1.15 .71 1 5— 7 35.53 67 1.63 1.40 7-5 54.47 93 1.16 .75 3— 4 36.87 69 1.60 1.33 4—3 53.13 92 1.20 .78 7— 9 37.87 70 1.58 1.29 9—7 52.13 ■91 1.22 .80 4 5 38.67 72 1.56 1.25 5-4 51.33 90 1.25 .83 5— 6 39.80 73 1.54 1.20 6—5 50.20 88 1.28 .86 6— 7 40.60 75 1.52 1.17 7—6 49.40 87 1:31 .88 7— 8 41.18 76 1.50 1.14 8—7 48.82 86 1.32 .89 8— 9 41.63 76 1.49 1.13 9—8 48.37 86 1.33 .90 9-10 41.98 77 1.49 1.11 10—9 48.02 85 1.34 1.00 1— 1 45.00 81 1.41 1.00 1—1 45.00 81 1.41 specimen Chart Calculatiofis. 91 Fis. 148. Sample Computation. SHAFTS AT A RIGHT ANGLE. Pitch = 3 Prop. = 7 — 5 Shaft ang. 90 Teeth = 42) 93 (2.22 = 84 . .37 + 90 2.59 = 84 60 face incr. i i cut deer. Center angles = 54.47 + incr 2.22 35.58 2.22 Face angles 56.69 37.75 Center angles = 54.47 — deer 2.59 35.53 2.59 Cut angles 51.88 32.94 Pitch = 3) 1.16 3) 1.63 .54 10. Diam. incr. = .39 -\- p. diams. = 14. 0. diams. = 14.39 10.54 Fig. 149. Sample Computation. SHAFTS at ANY ANGLE. 1 Pitch = 5 Prop. = X Shaft ang. 52.8 Teeth — 20) 66 (3.30 .55 3.85 = face incr. = cut deer. Center angles = 35.80 -finer 3.30 17.00 3.30 Face angles = 39.10 20.30 Center angles = 35.80 — deer 3.85 17.00 3.85 Cut angles = 31.95 13.15 Pitch = 5) 1.66 5) 1.91 .38 2. Diam. incr 33 -f p. diams 4. 0. diams 4.33 2.38 92 Chords of Angles. 164. — SHAFTS AT RIGHT ANGLES, 1st. — Divide the pitcli diameter by that of the other gear of the pair, or else the number of teeth by that of the other gear, to get the proportion. Enter the table by means of the proportion. All numbers for that pair will be found on the same horizontal line in the two columns. 2d.— The center angles are given directly by the table at the proper proportion. 3d. — Divide the tabular angle increment by the number of teeth in the gear, to get the angle increment. This need be done for but one gear of a pair, as the increment is the same for both. 4th. — Add the angle increment to the cen- ter angle, to get the face angle. 5th. —Increase the angle increment by one- sixth of itself, to get the cutting decrement, and subtract this decrement from the center angle, to get the cutting angle. 6th. — Divide the tabular diameter incre- ment by the diametral pitch, to get the diameter increment, and add that to the pitch diameter, to get the outside diameter. Fig. 148 is a sample computation for shafts at right angles. 165.— SHAFTS NOT AT EIGHT ANGLES. The table cannot be entered by means of the proportion, and the numbers for the two gears of the pair will not be found on the same horizontal line, and it will be necessary to determine the center angles. As in Fig. 147, draw the axes, at the given shaft angle, and find the center angles, by the method described in (162). Then enter the table, for each gear by itself, by means of the center angles, and proceed as for shafts at right angles. The angle increment and decrement is the same for both gears of a pair. Fig. 149 is a sample computation applied to the case of Fig. 147, the center angles be- ing found by means of the table of chords. If preferred, the center angles can be found by means of the formula, tan. G sin. 8 -\- COS. 8 in which C is the center angle of the gear, P is the proportion found by dividing the num- ber of the teeth in the gear by the number in the other gear, and 8 is the shaft angle. Having found one center angle, subtract it from the shaft angle to get the other center angle. 166. — THE TABLE OF CHORDS AT SIX INCHES. When the lathesman is provided with a graduated compound rest which feeds the tool at any angle, nothing but the computa- tion is required; but when there is nothing but the common square feed, the faces must be scraped with a broad tool, A templet for guiding the work can easily be made by means of the table of chords at six inches. To lay out a given angle, draw an arc with a radius of six inches, draw a chord of the length given by the table for the angle, and then draw the sides oc and ob of the angle hoc, Fig. 150. For tenths of a degree use the small tables. The chord of 37.5° is 3.81 -f .05 = 3.86 inches. Fig. 151 shows the manner of using the angle templet at the lathe. This table of chords is very convenient for many purposes not connected with gearing, and it is more accurate than the common horn or paper protractor. 167. -BILGRAM S CHART. A graphical method for determining the angle and diameter increments, the invention of Hugo Bilgram, is described in the Ameri- can Machinist for November 10, 1883. It determines the required values by the inter- sections of lines and circles, and requires no computation. Chords of Angles. 98 (Jhord of an angle Fig. 150. Using the templet TABLE OF CHORDS OF ANGLES, AT RADIUS OF SIX INCHES. Degrees. Chord. Tenth.«. Degrees. Chord. Tenths. Degrees. Chord. Tenths. 1 .10 31 3.20 61 6.10 2 .20 32 3.31 62 6.19 3 .31 33 3.41 63 6.28 4 .42 34 3.51 64 6.36 5 .52 35 3.61 65 66 6.45 6 .62 36 3.71 6.54 7 .73 37 3.81 67 6.62 8 .84 38 3.91 68 6.71 9 .94 39 - 4.01 69 6.80 10 1.04 40 4.10 70 6.89 11 1.15 41 420 71 6.97 12 1.26 42 4.30 72 7.06 13 1.36 .1— .01 43 4.40 .1— .01 73 7.14 .1— .01 .2— .02 .3—02 .4_.03 U 1.46 .2— .02 44 4.50 .2— .02 74 7.22 15 1.57 .3— .03 .4— .04 45 4.60 .3— .03 .4— .04 75 7.31 l(i 1.67' 46 4.69 76 7.39 17 1.77 .5— .05 47 4.79 .5— .05 77 7.47 5 04 18 1.87 .6—06 48 4.88 .6— .05 78 7.55 .0 — .05 19 198 .7— .07 49 4 98 .7—06 79 7.63 .7— .06 20 2.08 .8— .08 .9— .09 50 5.08 .8-=-. 07 .9—08 80 81 7.71 7.79 .8—06 .9— .07 21 2.18 51 5 17 22 2.29 52 5.26 82 7.87 23 2.39 53 5.35 83 7.95 24 2.49 54 5.45 84 8.03 25 2.59 55 5 54 85 8.11 26 2.70 56 5.63 86 8.18 27 2.80 57 5.72 87 8.26 28 2.90 58 5.82 88 8.34 29 3.00 59 5.91 89 8.41 30 3.10 60 6 00 90 8.48 94 Te?nplet Planer. 168. — ROTAKY CUT BEVEL TEETH. The most common method of forming the teeth of the bevel gear is by cutting them from the solid blank by the use of the com- mon rotary cutter, ' The cutter should be shaped to cut the tooth of the correct shape at the large end, and the small end must be shaped either by an- other cut with a different cutter, or with a file. It is impossible to cut the tooth correctly at both ends, for the simple reason that the ehape of the tooth changes, while that of the cutter is invariable. Therefore the result must always be an approximation depending upon the personal skill and experience of the workman. It is a too common practice to make the teeth fit at the large ends, and to increase the depth of the tooth toward the point, so that the teeth will pass without filing, but such teeth can be in working con- tact only at the large ends. 169. — THE TEMPLET GEAB PLANEK. The most common method of planing the teeth of bevel gears is by means of devices adapted to guide the tool by a templet that has previously been shaped, as nearly as may be, to the true curve. The arm that carries the tool is hung by a universal joint at the apex of the gear, so that all of its strokes are radial, and a finger placed in the line of the stroke of the cutting point of the tool is held against the templet. There are many different arrangements for the purpose, but they are all founded on the same princi- ples, and differ only as to details. The invention of the templet gear planer is commonly credited to George H. Corliss, who patented it in 1849, and was the first to use it in this country. But it was patented in France, by Glavet, in 1829, and may be even older. It is largely used for planing the teeth of heavy mill gearing, but has not been, and cannot be, profitably applied to common small gear work. Its product is, in any case, superior to the rough cast tooth, but its accu- racy is dependent on that of the templet, and is therefore dependent on personal skill. 9. TOE SKEW BEVELv QEAR. 170. — THE SKEW BEVEL GEAR. When a pair of shafts are not parallel, and do not intersect, they are said to be askew with each other, and they may be connected by a pair of skew bevel gears, having straight teeth, which bear on each other along a straight line. Such gears are to be carefully distinguished from spiral gears, used for the same purpose but having spiral teeth bearing on each other at a single point only. We will endeavor to describe the skew bevel gear so that its general nature can be understood, bat it is impossible to do so in simple language. It is the most diflScult ob- ject connected with the subject. The theory cannot even be considered as yet settled, for writers upon theoretical mechanism do not agree upon it, and there are points yet in controversy. In the theory of the bevel gear the surface of reference is the spherical surface upon which the tooth outlines are drawn, and upon which the laws of their action may be studied, for spheres of reference of two sep- arate gears may be made to coincide so that the lines upon one will come in contact with those upon the other. For the spur gear, the spheres become planes and the process is the same. But for the skew bevel gear there is no analogous process, for it is impossible to imagine a surface of such a nature that it can be made to coincide with a similar surface when both are attached to revolving askew shafts. There are spiral surfaces which will approximately coincide, and are analogous to the Tredgold tangent cones of bevel gears (161), but any tooth action developed upon such approximate surfaces must, of necessity, be not only approximate, but also very diflS- cult to define and formulate. Of all the skew tooth surfaces that have been proposed, there is but one, the Olivier involute spiral oid, that can be proved to be theoretically correct. 171. — THE HYPOID. The pitch surface of the skew bevel gear lines d and d , Fig. 153, either one of which is the surface known as the " hyperboloid of is an element of the surface, and will form it revolution," and it is so intimately connected if used as a generatrix* A section by any with the subject that it must be thor- oughly understood before going further. The clumsy name may be abbreviated to ''hypoid." K a line J), Figs. 152 and 153, called a generatrix, is attached to a revolving shaft A, so that it revolves with it, it will develop or ' ' sweep up " the hypoid H in the space surrounding the shaft. A section of the surface by any plane normal to the axis is a circle. The com- mon normal to the generatrix and the axis is the gorge radius G, and circular section through that line is the gorge circle. A section by a plane B, Fig. 152, parallel to the axis, at the gorge distance I other • plane parallel to the axis will be a from the axis, will be the pair of straight | hyperbola, to which the elements d and d Sypoidal sections, FUj. 152. Hyperbolic sections. Fig, 153, 96 Rolling Hypoids. are assymptotes, or lines which the curves continually approach, but reach only at in- finity. Fig. 153 shows at Q the hyperbolas cut by the plane Q, of Fig. 152, and at B those cut by the plane B. The principal hyperbola H is the only one with which we are concerned. The hypoid is best studied as projected upon a plane parallel to the axis, as in Fig. 154, in which A is the projection of the axis, d is that of the generatrix, dQA is the skew angle, and H'w, the principal hyperbola. When the skew angle and the gorge radius are given, the hyperbola is easily constructed by points. Any line ab is drawn normal to the axis and the gorge distance le— Qg is laid off from h, the distance ah is made equal to ec, and a is then a point on the curve. The curve is to be drawn through several points thus constructed. The Tiyperbvla. Fig. 154. To draw a tangent to the curve at any point a, draw a line am parallel to the assymptote d, lay off mn equal to Om, and draw the tangent an. 172. — THE PITCH HYPOIDS. The utility of the hypoid as the pitch sur- face of the skew gear depends upon the pe- culiar property that any number of such surfaces will roll together, and drive each other by frictional contact with velocity ratios in the proportions of the sines of their skew angles, if their gorge radii are in the propor- tions of the taDgents of their skew angles. It is required to construct a pair of rolling hypoids that will transmit a given velocity ratio between two shafts that are set at a given angle with each other. In Fig. 155, A and 5 are the given axes, and AQB the given shaft angle. The directrix D is to be so drawn that the sines of the skew angles AOB and BGB are in the proportion of the given velocity ratio, and this is best done by drawing lines parallel to the axes, at distances from O that are in the given ratio, and drawing the directrix thr- ;agh their intersection B. In the figure the axes are situated one over the other at a distance (rS" called the gorge distance, and the directrix B is situ.aied be- tween them so as to pass through the gorge line and divide the gorge distance into gorge radii, (rTFand HW, which are in proportion to the tangents of the skew angles. This is Fitch hypoids. Fig. 155, best done by r" rawing cd normal to GB in any convenient position, laying off the gorge distance ce at any convenient angle with cd, and drawing 6^ and ^ parallel to it; cf will be the gorge radius G W for the axis GA^ Rollhig Uypoids, 0' and / 177 Beale's Treatise 4 Begin at the Beginning 3 Bevel Gears 6, 7, 8, 152, 154 to 169 Box's Mill Gearing 4, 51 Brown & Sharpe Co.'s Treatise 4 Cast Gearing, Friction 49 Chart for Bevel Gears 163 Chordal Protractor 166 Clearance 40 Complete Tooth 22 Conic Pitch Lines 131 Conjugator 125 Consecutive Action 12 Construction by Points 23, 57 Cusp 16, 17, 54 Cutter Limit 88 Cutter Series 45 Cycloidal System 31, 76 to 90, 158, 174 Dedendum 39 Demonstrations Avoided 2 Disputed Points • 9 Double Secondary Action 21 to 78 Double Terminal Action 18 Edge Teeth 100 Efficiency in Transmission 49, 73, 112 Elliptic Gears 136 to 153 Elliptic Gear Cutting Machine 150 Elliptic Pitch Lines 131 Elliptographs 138 Epicycloidal Teeth 76 Equidistant Series 45 Extent of the Subject 2 Fillet 28, 44 Friction 26 Friction of Approach . - - 48, 49 Gear Cutting Machines 27, 28, 29, 94, 102, 105, 122, 125, 150, 159, 168, 169 Gear Teeth, Theory 1 to 34 Spur 35 to 52 " Involute 53 to 75 Cycloidal 76 to 90 Pin 91 to 97 Spiral 98 to 110 Worm Ill to 125 Irregular 126 to 135 Elliptic 136 to 153 Bevel 154 to 169 Skew Bevel 170 to 181 Section. Gears, Beale's 177 Bilgram's 159 " Composite 133 " Elliptic Bevel 152 " Herrmann's 175 Hindley 122 Hooke's 98- " Hyperbolic 131 " Lantern 93 " Mortise 47 " Parabolic 131 Pin Bevel 160 " Skew Pin 181 " Stepped 98 Herrmann's Erroneous Law 176 Herrmann's Treatise 4 Hindley Worm Gear 122 Hobbing Machines 124, 125 Hobbing Worm Gears 114, 115 Hooke's Gears 98 Horse Power of Gears 51, 52 Hunting Cog 46 HyperboHc Pitch Lines 131 Hyperboloid of Revolution 7, 171 Hypoid 171 Integrater, Odontoidal 34 Interchangeable Odontoids 14, 22 Interchangeable Rack Tooth 22 Interference . . c 16, 55 Interference, Internal • 79 Interference, Worm 117 Internal Contact 15 Internal Double Action 21 Internal Gears 64 Internal Friction 49 Involute System 31, 53 to 75, 157, 175 Irregular Pitch Lines 126 to 128, 13a Kinematics Klein's Treatise Klein's Odontograph . o 4 85 Law of Tooth Contact 11, 12 Limiting Numbers of Teeth 66 to 72, 90 Limit Line 16 Line of Action 13 Literature 4 Logarithmic Pitch Lines 132 Logarithmic Spiral 32, 75, 132 MacCord's Treatise 4 Molding Construction 27 Mortise Gear 47 Multilobes 130 Natur iTooth Action 20 Normals 11 Normal Surfaces -8 Section. Obliquity of Action 26, 74, 87 Octoid Teeth 31, 159 Odontics 6 Odontographs 43, 59, 62, 82, 83 Odontoid and Line of Action 33 Odontoids 12 to 34 Olivier's Spiraloidal Teeth 175 Parabolic Pitch Lines 131 Parabolic System 31 Pin Tooth System 91 to 97, 160, 181 Pitch, Circular 35, 119 Pitch Cylinders 10 Pitch Diameters, Table 35 Pitch, Diametral 36, 120 Pitch Lines 11 Pitch Point 11 Pitch Surfaces 7 Pitch Table 37 Planing Construction 28, 29 Quick Return Motion 149 Radial Flank Teeth 89 Rankine's Treatise 4 Rack Originator 30, 31 Retrograde Action 18 Reuleaux's Treatise 4 Robinson's Odontograph 86 Rolled Curve Theory 32,75,81, 91 Roller Teeth 93 Section. Secondary Action 21, 78 Segmental System 31 Sellers' Experiments 49, 112 Skew Bevel Gears 6, 7, 8, 170 to 181 Smallest Pitch Circle 17 Speed of Point of Action 19 Spiral Gears 98 to 103 Spur Gears 6, 7, 8, 35 to 153 Stahl & Wood's Treatise 4 Standard Teeth 42, 58, 80 Stepped Teeth 98 Strength of Teeth 49 Systems of Teeth 31 Templets 38 Terminal Point 18 Twisted Teeth 99, 101, 102, 155 Unsymmetrical Teeth 22 Variable Speed 148 Willis' Odontograph 84 Willis' Treatise 4 Wooden Teeth 47 Worm Gears HI, 113, 114, 116 Yale & Townes Experiments 112 ^MICHIGAN BRICK AND TILE MACHINE CO. MoRENCi, Mich., Nov. 24th, 1891. George B. Grant. Dear Sir : — Two years ago you sent me one of your books on Teeth of Gears, and I have replaced all of the gears in our brick machinery with new ones from your in- volute odontograph table. I find that we now have the finest cast gears in the world. I do not understand why pattern makers don't catch on to your book. It is a sight to see the gear pat- terns that are made by some men who are called good pattern makers. O. S. STURTEVANT, Pattern Maker for M. B. &" T. M. Co. ADVERTISEMENT. IT'S NO USE TRYING TO GET ELLIPTIC GEARS OF ANY ONE BUT GEO. B. GRANT, LEXINGTON, MASS. PHILADELPHIA, PA. / LIBRARY OF CONGRESS III II 021 213 113 1