^ . Cornell University Library 
 
 TJ 175.B261900 
 
 Kinematics of macliinery :a brief treatis 
 
 3 1924 016 001 004 
 
Kinematics of Machinery. 
 
 A BRIEF TREATISE ON CONSTRAINED 
 MOTIONS OF MACHINE ELEMENTS. 
 
 JOHN H. BARR, M.S., M.M.E., 
 
 Professor of Machine Design, Sibley College, Comel! UniversHy 
 Member of the American Society of Mechanical Engineers. 
 
 EJFitf) ofaer Etoo jKunHrftt jF<Surps. 
 
 FIRST EDITION'. 
 FIRST THOUSAND. 
 
 NEW YORK: 
 
 JOHN ^AAILEY -& SONS. 
 
 London: CTHAPM'AN & IJALL, Limited. 
 
 1900. 
 
m loS 
 
 Copyright, 1899, 
 
 BY 
 
 JOHN H. BARR. 
 
 ROBERT DRUMMOND, PRINTER, NEW YORK. 
 
PREFACE. 
 
 This book is the outgrowth of a somewhat smaller treatise which 
 was prepared and printed by the writer in 1894 for the use of the 
 classes in mechanical and electrical engineering at Sibley College, 
 Cornell University. 
 
 After having used the original for several years, it was decided 
 to issue the work in revised form, making such corrections and 
 changes as experience suggested. 
 
 The present volume was prepared especiajly to bring together, 
 and to present to the students in a condensed text-book, those prin- 
 ciples and methods which are deemed most important in a general 
 CDurse on Kinematics. This is the only excuse offered for another 
 book on a subject about which so much has been written. No pre- 
 tension is made to originality except in the arrangement and manner 
 of presenting a few subjects. Neither is the present work offered as 
 in any sense a complete treatise on the Kinematics of Machinery. 
 The treatment of many topics has been much abi'idged; particularly 
 the portion relating to toothed gearing, a subject which is exhaus- 
 tively treated in numerous available works. On the other hand, 
 the discussions of the applications of such important conceptions 
 as instantaneous centres, velocity diagrams, etc., are rather fuller 
 than are found in many of the shorter works on Mechanism. 
 
 The treatment of these subjects follows closely that given by 
 Professor Kennedy in his admirable work on the Mechanics of 
 Machinery. 
 
 It is believed that the presentation of principles and methods, 
 with illustrations of their applications, is the proper line to adopt 
 
IV PREFACE. 
 
 in. a text-book intended for a short general course on such a subject 
 as Kinematics. The detailed description of usual forms, and the 
 discussion of the innumerable considerations with which the expert 
 in any line must be familiar are to be sought in special treatises. 
 
 Messrs. A. T. Bruegel, D. S. Kimball, and "VV. N. Barnard, all 
 of whom have given instruction in the course to which it applies, 
 have rendered valuable assistance in the preparation of the present 
 book. Mr. Bruegel contributed most of the problems, which were 
 developed during his six years as instructor in Kinematics at Cor- 
 nell University. Professor Kimball kindly wrote the articles on 
 "Acceleration Diagrams" and " Epicyclic Trains," and he and 
 Mr. Barnard have cooperated in other ways in the revision. 
 
 ilany earlier works have been consulted and drawn on in the prep- 
 aration of the present book. The following, especially, should be 
 mentioned : Principles of Mechanism, by Professor Willis ; .Machin- 
 ery and Millwork, by Professor Eankine; Kinematics of Machinery, 
 by Professor Reuleaux ; Mechanics of Machinery, by Professor Ken- 
 nedy; Kinematics, by Professor MacCord; Machine Design, by 
 Professor Unwin ; Elementary Mechanism, by Professors Stahl and 
 "Woods; Teeth of Gears, by Mr. George B. Grant; A Practical 
 Treatise on Gearing (Beale), piiblished by the Brown and Sharpe 
 Manufacturing Company. 
 
 The writer desires to acknowledge his obligations to all who 
 have in anyway aided in the preparation of this little book. 
 
 John H. Baek. 
 
 Ithaca, New Yobk, 
 October 1899. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 PAGE 
 FCKDAMENTAL CONCEPTIONS OF MOTION. THB NATURE OP A MACHINE 1 
 
 CHAPTER II. 
 Genekal Methods op Transmitting Motion in Machines , . . . . 37 
 
 CHAPTER III. 
 
 Puke Rolling in Dikect-contact Mechanisms. Fkictional Gbak- 
 ing : 78 
 
 CHAPTER IV. 
 OuTLixEs OP Gear-teeth. Systems op Tooth-gearing 110 
 
 CHAPTER V. 
 Cams and Other Direct- contact Mechanisms; 153 
 
 CHAPTER VI. 
 
 LiNKWOKK 170 
 
 CHAPTER VII. 
 Wrapping-connectors. Belts, Ropes, and Chains 205 
 
 CHAPTER VIII. 
 Trains of Mechanism..' 217 
 
 Problems and Exercises 238 
 
 Index 341 
 
 V 
 
The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924016001004 
 
EINEMATIOS OF MACHINERY. 
 
 CHAPTEE I. 
 
 T-UNDAMENTAL CONCEPTIONS OF MOTION. THE NATURE OF A 
 
 MACHINE. 
 
 1. Motion is a change of position; and it ig measured by the 
 space traversed. Time is not involved in this conception. A train, 
 in running between two stations fifty miles apart, has the same 
 motion, whether the time occupied be one, two, or three hours. 
 The motion of a crank-pin in making a revolution is independent 
 •of the time required . 
 
 2. Linear Velocity, or simply velocity, is the rate of motion of a 
 
 point along its path in space. It is a function of both space and 
 
 time, and is measured in compound units of these fundamental 
 
 quantities ; as feet per second, feet per minute, miles per hour, etc. 
 
 ds 
 In mathematical terms, velocity = v = -^, in which s = the 
 
 space passed over in the time t. 
 
 If, in the illustration of the preceding article, the time of the 
 run between the stations is one hour, the train has an average, or 
 mean, linear velocity of fifty miles per hour; if the time be two and 
 a half hours, the mean velocity, or speed, as it is often called, is 
 twenty miles per hour, etc., or 1760 ft. per min., or 29' 4' per sec. 
 
 3. Acceleration, or linear acceleration, is the rate of change of 
 velocity. Acceleration is expressed in the same system of space- and 
 time-units as the velocity itself (as feet and seconds, feet and min- 
 
2 KINEMATICS OF MACRINEIiT. 
 
 Tites, miles and hours, etc.); but acceleration involves one space- 
 factor and two time-factors. The mathematical expression for 
 
 , ,. . dv d's 
 
 acceleration is » = ~^rr = -^Tj- 
 at at 
 
 If a velocity is uniformly increased from 10 feet per second to 
 18 feet per second, the change of velocity is 8 feet per second. If 
 this change takes place in 2 seconds, the rate of change, or the 
 acceleration; is 4ieet per second per second, or -4 foot-seconds per 
 second, or 4 feet per square second. If the increase of velocity is. 
 not uniform, the mean acceleration is 4 feet per square second in. 
 the above illustration, although the actual increase of velocity in. 
 any one second is not necessarily 4 feet per second. 
 
 4. Uniform and Variable Velocity. — If the motion of a body is. 
 uniform (that is, if all .equal increments of space are traversed in 
 equal increments of time) the velocity is uniform, and is equal to 
 the space traversed in any time divided by that time. If the veloc- 
 ity is uniform, the acceleration is zero. If a body moves 120 feet 
 in 10 seconds, with a uniform velocity, the velocity is 12 feet per 
 second, equivalent to 720 feet per minute. 
 
 If the velocity is not uniform, the space divided by the time 
 gives only the mean or average velocity, and the velocity may vary 
 between the widest limits during the motion. If the law of the 
 motion is known, the velocity at any instant may be determined 
 from the space and time; otherwise, only the mean velocity can be 
 determined from these data. 
 
 The velocity of a body may vary uniformly, the velocity increas- 
 ing or decreasing by equal increments with each equal increment of 
 time, in which case the acceleration is constant; or it may vary 
 according to any other law. For our present purposes it is only 
 necessary to discriminate between uniform, or constant, and vary- 
 ing velocity. 
 
 Although the velocity may be constantly changing, it is cus- 
 tomary to speak of a body as moving at a certain velocity, as 25 feet 
 per second, 30 miles per hour, etc. ; and such expressions are per- 
 fectly correct, even though the velocity does not remain constant 
 for a single instant. For example: a train of cars in getting up 
 speed passes through every velocity from zero to the maximum 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. B 
 
 Telocity attained; at a certain stage the velocity may be, say, ten 
 miles per hour, and in coming to rest the velocity again passes, 
 through this same value. • Perhaps the train does not maintain this, 
 particular velocity for a single foot; yet, for the instant, it is said 
 to have this velocity; meaning that if it continued to move with th& 
 velocity that it has at this instant it would move 10 miles in one 
 hour. 
 
 5. Relative and Absolute Motion. — All known motions are rela- 
 tive, for change of position can only be noted with reference to 
 objects at rest (or assumed to be at rest), or by reference to objects 
 the motion of which is known (or assumed to be known). We 
 know of no body absolutely at rest, nor do we even know the abso- 
 lute motion of any body in the universe. 
 
 In treating of the motion of a body, only its change of position 
 with regard to some other body, or its motion relative to that other 
 tody, can be considered. 
 
 In ordinary problems of terrestrial mechanics the earth is takea 
 as the standard from which to reckon, and a body which does not. 
 change its position relative to the earth is said to be at rest, station- ■ 
 ary, or fixed ; of course recognizing that it partakes of the motion 
 which the earth has about its axis, around the sun, and in common, 
 with the sun through space. 
 
 In problems of machinery the motions of the parts are usually 
 most conveniently taken with reference to the frame of the machine; 
 as a standard. In "stationary" or "fixed" machines this is: 
 equivalent to referring these motions to the earth, for the frame 
 has no appreciable motion relative to the earth; but in such cases 
 as locomotives and marine engines, for example, the parts have very 
 different motions relative to the frame and to the earth. In these 
 latter cases we are usually concerned with the motion of the parts, 
 relative to the frame, or with the motion of the machine as a whol& 
 (including everything. connected with it) relative to the earth. 
 
 The function of the machine, in these cases, is to impart motion, 
 relative to the earth, to the attached train or ship, and incidentally 
 to itself; but this motion of the entire system, and the motion of 
 the parts, as members of a machine, may generally be treated as-, 
 quite distinct, though related, problems. A marine engine can ba 
 
4: KINEMATICS OF MACHINERY. 
 
 studied as an engine just as a mill engine can be treated, without 
 considering the application of the energy beyond the engine itself. 
 
 As we know nothing of the absolute motion of a body, and can 
 only know its motion relative to other objects, it can have as many 
 relative motions as there are objects with which to compare its 
 changes of position. 
 
 A pair of locomotive drivers, for example, rotate on their axle 
 Telative to the frame; they roll along the track (each point tracing 
 a curve of the cycloidal class) relative to the track or the earth ; 
 they rotate about the axis of their plus I'elative to the attached side 
 rods; and have still different motions relative to the wheels on the 
 other axles, to the piston, etc. 
 
 It is important to get a clear conception of relative motion, for 
 in the study of mechanism the treatment may often be much sim- 
 plified by referring a motion to some member other than the frame 
 of the machine, as to other moving parts. 
 
 Throughout this work it will frequently happen that the motion 
 •of a part relative to some other moving part will be discussed ; but 
 it is to be understood, unless distinctly indicated to the contrary, 
 that the word "motion^' refers to the change of position relative 
 to the frame. Likewise, when a member is said to be at rest, fixed 
 or stationary, it is to be understood that its position relative to the 
 frame remains unchanged. 
 
 Two portions of a rigid body can have no motion relative to 
 €ach other; for a change of the relative positions of such parts 
 involves a change of form, and this is not consistent with the con- 
 ception of a rigid body. It will be evident, upon brief reflection, 
 that two separate bodies which have no relative motion could be 
 rigidly joined without affecting any motions that they may have; 
 ior as they do not change their position relative to each other they 
 must have identical motions relative to all other bodies, and may 
 be treated as parts of the same body so far as their motions are con- 
 cerned. 
 
 Bodies wMch have no motion relative to each other have the same 
 ■motion relative to any other body. 
 
 The converse of this statement, that all bodies which have the 
 same motion relative to another body have no motion relative to 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 5 
 
 each other, is not, however, generally true. Take the example of 
 the locomotive driving-wheels, again; each set of wheels has the 
 same motion relative to the frame, as well as to the track, but the 
 wheels on the different axles do, nevertheless, have motions relative 
 to each other; for these different sets of wheels could not be rigidly- 
 fastened together as one piece without preventing motion relative to 
 the frame. 
 
 6. Velocity Ratio. — In many problems of machine motions, the 
 actual velocity of the parts is not of so much importance as the 
 ratio of the velocities of two or more parts. In another class of 
 problems, the actual velocity (relative to the earth, or other stand- 
 ard) must be treated. The present work is concerned very largely 
 with the former class, and it is necessary to get a clear conception 
 of the term velocity ratio. This may perhaps be best accomplished 
 by a few illustrations. 
 
 Take, as an example, an ordinary simple hand-windlass, in 
 which a rope is vrrapped around a drum of known size, and a crank 
 of given radius is attached to the axis of the drum. If the crank 
 be turned through one complete revolution, the load attached to the 
 rope will be raised a height equal to the circumference of one coil. 
 For any number of turns of the crank, or fractional turns, the load 
 will be raised a proportional height; and it matters not whether the 
 crank be turned fast or slowly, the ratio of its motion, and of its 
 velocity, to that of the load is the same, depending entirely upon 
 the proportions of the device. The ratio of the velocities, or the 
 velocity ratio, is independent of the actual velocities, and of the 
 forces transmitted. In the case cited, the ratio is the same whether 
 a load of one ton be hoisted ten feet in one second, or one pound be 
 hoisted one foot in one minute. The same point is illustrated ia 
 the action of most of the common machines. In an ordinary steam- 
 engine, for every revolution of the crank the connected parts go 
 through certain definite motions; while the time of one such revo- 
 lution of the crank may be a tenth of a second or ten minutes, all 
 the parts (with the exception of such parts as the members of the 
 governor, to be mentioned later) go through the same relative 
 changes of position ; and though the actual velocities with which. 
 
€ KINEMATICS OF MACHmERY. 
 
 such changes take place are yery different in the two cases, the ratio 
 of these velocities remains the same. 
 
 7. Path. — A point in changing its position traces a line called 
 its path. 
 
 The statements in the preceding articles on the motions and 
 velocities of todies apply equally to every point in a moving body, 
 -whether the path of the point be rectilinear or otherwise. This is 
 •consistent with the definitions of motion and velocity; for these 
 definitions state that motion is measured by space traversed (not 
 restricted to space in a right line), and that velocity is the rate of 
 motion. 
 
 The path of a point may be of any form whatever, in a plane or 
 in space; it may be of finite length, the point traversing it, then re- 
 versing its direction of motion, returning through the same path ; it 
 may form a continuous loop, the point moving in one direction con- 
 tinually ; the point may change its direction of motion and retrace 
 the path; or it may be that the path does not cross itself, or return 
 upon itself, the point never twice occupying the same position. 
 There are many cases, however, and nearly all motions of mecha- 
 nisms are of this class, in which the path is definite and limited in 
 both form and extent. 
 
 If a point moves in a path which forms a closed loop, plane or 
 otherwise, the motion of the point may be constant in direction and 
 Telocity, or either or both of these elements may change. 
 
 8. Cycle; Period; Phase. — In most mechanisms the members go 
 through a series of relative motions, at the end of which they occupy 
 the same relative positions as at the beginning. 
 
 The completion of such a series of relative motions, with the re- 
 turn of the members to the relative positions which they had at first, 
 constitutes a cycle. 
 
 In the ordinary steam-engine, for example, the cycle corresponds 
 to one revolution of the crank, whatever the time occupied by the 
 revolution. In a common type of gas-engine, the cycle corresponds 
 to two complete revolutions of the crank, for the four strokes of the 
 piston during these two revolution are: a suction stroke; a com- 
 pression stroke ; a working stroke (impulse) ; and an exhaust stroke. 
 The valve-gear, in this case, is so arranged that valve, piston, etc.. 
 
CONCEPTIOm OF MOTION. NATURE OF A MACHINE. 7 
 
 only return to their initial relative positions after the completion of 
 four strokes of the piston, or two revolutions of the crank. 
 
 The time elapsing during a cycle is called the period. 
 
 The simultaneous positions occupied by the members, at any 
 instant duritig the cycle, is called a phase. 
 
 9. Continuous, Reciprocating, and Intermittent Motion. — If the 
 direction of motion does not change, the motion is sometimes said 
 to be continuous (using the word somewhat differently than in the 
 strict mathematical sense, in which all motion is continuous). 
 
 Motion is said to be reciprocating if its direction reverses. 
 
 Motion is called intermittent when it is interrupted by intervals 
 of rest. 
 
 Motion in a closed path may be continuous, reciprocating, or 
 intermittent ; and it may vary as to velocity in any manner what- 
 soever. 
 
 Motion in a path of finite extent, not forming a closed figure, 
 must be reciprocating, and may or may not be intermittent. 
 
 10. Plane Motion ; Rotation, Translation. — Of the great num- 
 ber of motions available in machinery, a very large proportion are 
 included in three classes of comparatively simple nature, viz. : 
 Plane Motion, Helical Motion, and Spherical Motion. 
 
 Plane Motion is by far the most common, and it is the simplest 
 class as well. 
 
 If any plane section of a body moves in its own plane, all points 
 in this section move in this plane, and all points outside of this sec- 
 tion move in planes parallel to the given section. Such a motion 
 ■constitutes a plane motion. Any point in a body having plane 
 motion may trace any path in its plane; but all points similarly 
 located in the other parallel planes, that is all points lying in a 
 common perpendicular to the different planes of motion, have paths 
 of identically the same form. Thus, in Pigs. 1 or 3, if the section 
 shown shaded always moves in its own plane, the successive positions 
 •of the perpendicular through any point as^ must always be parallel, 
 and therefore all points in this perpendicular move in similar paths. 
 
 The property of plane motions, just- discussed, greatly simplifies 
 the treatment of these motions, as the motion of one point (or of a 
 set of points) in any section represents the motion of all similar 
 
8 
 
 KINEMATICS OF MACEINEBT. 
 
 points in other sections ; or the motion of a single section (a plane 
 figure) in its own plane represents the motion of the entire body. 
 This can be extended even farther, for the motion of a point not in 
 the particular plane represented can be replaced by its correspond- 
 ing point on that plane (its projection on that plane), and thus a 
 single plane figure represents all the motions of all the points in the 
 
 Fig. 2 
 
 body. For example, the motions of the points p, s, and q in Figs.. 
 1 or 3, are in similar paths, and the motion of any one of these 
 points may be taken to represent that of any other. The motion 
 of an engine crank and of the eccentric can be, and often are, con- 
 veniently shown together, as if actually in one plane. 
 
 In case of all other than plane motions, however, it is necessary 
 to show the various positions of the members by two or more pro- 
 jections, or by some equivalent system, if it is desired to completely 
 represent the motion. 
 
 Plane Motion is either a Rotation, a Translation, or a motion 
 which can be reduced to a combination of these. The reduction of 
 the general motion to a combination of rotation and translation i& 
 not always to be desired, however, and such motion will often be 
 treated as a class of itself, without relation to the simpler and more 
 special classes to which it can be reduced. 
 
 If a body moves, as in Pig. 1, so that all points travel in paral- 
 lel planes and at constant distances from a fixed right line, it has 
 a plane motion of rotation. Examples: pulleys, cranks, levers, 
 etc. It is not necessary that the motion be continuous ; the rota- 
 tion may be continuous, reciprocating, or intermittent. 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 
 
 9 
 
 If a body moves, as in Fig. 2, so that all points move with equal 
 velocities in similar paths, the motion is a translation. If these 
 paths are parallel right lines, the motion is & rectilinear translation. 
 Examples : the carriage of a lathe, piston or cross-head of an engine, 
 platen of a planer, etc. If, however, the paths of all the different 
 points are equal curves, the motion is a curvilinear translation- 
 Example : the side rods of a locomotive. 
 
 Kectilinear translation is always to he understood when the word 
 translation is used without qualification. 
 
 A rectilinear translation may be treated as the special case of 
 rotation in which the distance to the axis is infinity, or as rotation 
 in a circle of infinite radius. 
 
 It has been shown that the plane motion of a body is completely 
 represented by the motion of any section taken in a plane of 
 motion, or by the change of position of a plane figure. Two points 
 suflBce to locate a figure in a plane, and hence the plane motion of 
 a body is determined by the motion (successive positions) of any 
 two of its points not in the same perpendicular to the plane of 
 motion. In the general case of motion in space, the motion is 
 determined only by the motions of at least three points, not in one 
 right line. For if the motions in space of two points are known, 
 the body may, in the general case, have a motion of rotation about 
 the line connecting these two points; but the motion of a third 
 point, outside of this line, deter- 
 mines the motion of the body 
 completely. 
 
 In general, the motion of a 
 body in a plane may be reduced to 
 an equivalent rotation and a trans- 
 lation. Thus, Fig. 3, the motion 
 of the body A, which is complete- 
 ly determined by the motion of 
 two points such as a and h, or by 
 the motion of the line connecting 
 these points, corresponds to a change of position from A to A'. 
 This change of position can be conceived as made up of a transla- 
 tion, a-b to a'-l', and a rotation, about b' through the angle 
 
10 
 
 KINEMATICS OF MACSINBRY. 
 
 a' V a". Or, the rotation can be conceived to take place first, fol- 
 lowed by the translation. 
 
 As this motion is a perfectly general case of plane motion, the 
 same reasoning applies to all such cases, no matter how large or 
 Jiow small the motion may be. 
 
 11. Helical Motion.— If all the points in a body have a motion 
 •of rotation abont an axis, combined with a translation parallel to 
 that axis, the motion is a Helical Motion (see Fig. 4). In nearly 
 all cases the helical motions met with in machines are regular hel- 
 ical motiotis, in which there is a constant relation between the 
 rotation and the translation; that is, the ratio between the transla- 
 
 O- 1 
 
 lA 
 
 
 ■<x 
 
 
 ,-o 
 
 ,-o 
 
 i- 
 
 o- 
 
 -Gr 
 
 ,C( 
 
 Fig. 4 T "-O 
 
 Fig. 5 
 
 tion component and the angular component is constant. The pitcli 
 of the lielix is the translation along the axis corresponding to one 
 complete rotation, and in a regular helical motion the pitch is con- 
 stant. 
 
 12. Spherical Motion. — If the motion of a body is such that 
 all points in the body remain at constant distances from a fixed 
 point (see Fig. 5), the motion is spherical. All points in the body 
 move in the surfaces of spheres, having the fixed point for a com- 
 mon centre. 
 
 13. Relation between Plane, Helical, and Spherical Motions. — ^ 
 If the translation component {pitch) in a helical motion be in- 
 creased till it equals infinity, the motion reduces to a plane trans- 
 lation. On the other hand, if the translation component be re- 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 11 
 
 duced to zero, the motion reduces to plane rotation. It is thus 
 seen that both of the limits of helical motion are plane motions, 
 and that plane motion of rotation or of translation may be treated 
 as special cases of helical motion. 
 
 If the distance from the fixed point to the moving body in a 
 spherical motion be increased to infinity, the surfaces of the spheres 
 in which the points of the body move are reduced to planes, and 
 we thus see that plane motion may be treated as a special case of 
 spherical motion. The much greater frequency of plane motion 
 and its simplicity makes its consideration, in practical cases, as a 
 
 Fig. 6 
 
 Fig. 7 
 
 Fig. 8 
 
 special form of these more complex motions undesirable, though 
 this view of the case is not without interest. 
 
 Motions more complicated than the classes just mentioned are 
 sometimes met with in machinery, and some of these will be dis- 
 cussed in the following statement; but they are comparatively so 
 
12 KINEMATICS OF MACHINERY. 
 
 few, and are so varied in character, that a classification of them is 
 not practicable. Figs. 6, 7, 8, and 9 show practical examples of 
 plane rotation, plane translation, helical motion (regular), and 
 spherical motion respectively. 
 
 14. Graphic Representation of Motion. — A rectilinear motion 
 is naturally represented by a right line, the direction of which cor- 
 responds with the direction of the motion, and the length repre- 
 senting the velocity to some convenient scale. 
 
 If, for instance, it is desired to represent the velocities: 20 feet 
 per second, 35 feet per second, 55 feet per second, and 40 feet per 
 second, by lines on a drawing, or diagram, a scale can be adopted 
 which will give convenient lengths (say 10 feet per second to the 
 inch); and, to this scale, these velocities will be represented by 
 lines 2 inches, 3.5 inches, 5.5 inches, and 4 inches long, respec- 
 tively. In a similar way, velocities in other units, as feet per 
 minute, miles per hour, etc., can be indicated to suitable scale. 
 In Kg. 10 the motion of the point, p, moving with velocity of 30O 
 feet per minute, is represented on a scale of 200 feet per minute to 
 the inch, by the Yvaa p-v, 1.5 inches long. 
 
 Fig. 10 Fig. II 
 
 If the motion is in any other path than a right line, it may still 
 be represented by a straight line passing through the point and 
 lying in a tangent to the path; for whatever the path of the mov- 
 ing point, the direction of the curve, and hence of the motion along 
 such path, is the same at any place as the tangent to the curve at 
 that place. 
 
 In Fig. 11, the motion of the pointy, moving in the path ;«-;e, 
 with a velocity of 45 feet per second, is represented on a scale of 40- 
 feet per second to the inch by the line jo-w, 1.125 inches long, lying 
 along the tangent to the path and through p. 
 
 This graphic representation of motion is of the greatest impor- 
 tance, as it makes many solutions possible on the drawing-board 
 
CONCEPTIONS OP MOTION. NATURE OF A MACHINE. 13 
 
 without the use of calculations; giving the results required directly 
 in connection with the regular process of designing, and permitting 
 the easy determinations of results that could only be arrived at 
 otherwise by tedious algebraic methods. 
 
 As to the accuracy of these graphic methods, it may be said that 
 they are as close as can be used in a drawing itself; so, for the ordi- 
 nary purposes of designing, they are all that can be desired in tliis 
 respect. Furthermore, the graphic method has the advantage of 
 showing a number of connected quantities in their true relation, 
 appealing to the mind through the eye much more efEectively than 
 do numerical quantities. A limited experience with such problems 
 as follow in this work will impress upon one the value of this 
 method. 
 
 15. Newton's Laws of Motion. — Starting with the statement of 
 Xewton's Laws, which enunciate fundamental relations between 
 force and motion, and with the familiar Parallelogram of Forces, 
 we can readily develop the theory of the very important subject of 
 llesolution and Composition of Motions. 
 
 Newton's Laws. 
 
 I. — Any material point acted upon by no force, or by a system 
 of balanced forces, maintains its condition as to rest or motion ; if 
 at rest it remains at rest; if in motion it moves uniformly in a 
 right line. 
 
 II. — Any material point acted upon by a single force, or by a 
 system of unbalanced forces, moves with an accelerated motion 
 proportional to, and in the direction of the force, or resultant of 
 the system. 
 
 III. Action and reaction are equal, opposite, and simultaneous. 
 
 16. Parallelogram of Forces. — The resultant of two or more 
 forces applied at a point of a body is the single force which, if ap- 
 plied at the same point, will have the same effect on the body, as 
 to rest or motion, as the given forces themselves. These forces 
 which act together are called components of the single force, which 
 is equivalent to their combined action. 
 
 Forces may be represented graphically, in a similar manner to 
 that already explained in connection with the representation of mo- 
 
u 
 
 KINEMATICS OF MACHINERY. 
 
 tions; the direction of the line indicating the direction of the 
 force, and the length of the line representing the magnitude of the 
 force. If two forces, acting on a point, are represented in this 
 .way, the resultant of these forces is similarly represented by the 
 diagonal of the parallelogram formed on the components as sides. 
 For the proof of this, see Mechanics of Engineering-, by Professor 
 I. P. Church, page 4. 
 
 This proposition can be extended to cover the case of any num- 
 ber of forces acting at a point; for the resultant of any two of such 
 a system of forces can be found, then the resultant of this first re- 
 sultant (which exactly replaces the two original forces), and an- 
 other of the forces can next be found, the resultant of this last 
 resultant and another component can then be found, and so on till 
 all of the original forces have been combined. The last resultant 
 is the resultant of the system. By the reverse of the process just 
 outlined, a single force can be replaced by two or more components. 
 
 The process of finding the resultant of several forces is called 
 the Composition of Forces ; the reverse process of finding the com- 
 ponents of a force is called the Resolution of Forces. 
 
 17. Resolution and Composition of Motions. — While a point 
 may be acted upon by any number of forces, simultaneously, it can 
 have but one motion at any time. However, the motion that the 
 point actually does have is in the direction of the resultant of all 
 forces acting; and, as this force may be considered as made up of a 
 number of component forces, so the resulting motion may be 
 looked upon as made up of, or equivalent to, two or more compo- 
 nent motions. 
 
 According to Newton's second law, a single force, F^, acting on 
 
 p 
 
 Fig. 12 
 
 Fig. 13 
 
 a point p (Pig. 12), would impart a motion represented by v, (to 
 scale). If (Pig. 13) F^ acts alone on p, it imparts the motion v, ; 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 15 
 
 or if F, were to act alone on^, it would impart the motion v,. If 
 these forces act together upon p, the resultant motion would be v„ 
 in_ the direction of, and proportional to, the resultant of F^ and 
 F„ or Fr. 
 
 But as F^: F^: F^:: v^: v, : Vr, the parallelogram on i», and 
 v^, with Vy as the diagonal, is similar to the parallelogram of forces, 
 and therefore we would find the resultant of any given system of 
 motions by a construction similar to that used in the parallelogram 
 of forces. 
 
 From the preceding discussion, the following, proposition can 
 be drawn : ' \ 
 
 Parallelogram of Motions. 
 
 If two component motions of a point be represented, to scale, 
 by the adjacent sides of a parallelogram, the diagonal of the paral- 
 lelogram will represent the resultant motion to the same scale. 
 
 Conversely, a motion represented by a line, to scale, may be re- 
 solved into any pair of component motions, which are represented 
 to the same scale by the sides of any parallelogram of which the 
 first line is the diagonal. 
 
 As in the case of forces, the reduction of more than two com- 
 ponent motions to one resultant can be effected by an extension of 
 the above principles. 
 
 This method can be applied to any number of motions, whether 
 in one plane or otherwise. In Fig. 14 the resultant of v, and v.^ 
 = v„; the resultant of v^ and t>, = v^; the resultant of v^ and v, = 
 v^, = the resultant of the system. 
 
 In Fig. 15 the resultant of f, and v, — v^; resultant of «„ and 
 
 If the single motion v^ (Figs. 13, 14, or 15) is given, it can be 
 replaced by the motions of which it is the resultant ; for they, 
 combined, are its equivalent. 
 
 Determining the resultant of a system of motions is called Com- 
 position of Motions ; finding the components of given motions is 
 called Eesolution of Motions. 
 
 A system of motions can have but one resultant ; but a given 
 motion can have an infinite number of sets of components. The 
 
16 
 
 KINEMATICS OF MACHINERY. 
 
 motion v (Fig. 16) may have for components v, and t;,; r/ and 
 v^, or any number of sets of components; or the resolution is in- 
 definite, because an infinite number of parallelograms can be 
 drawn with the line v for a diagonal. 
 
 Fig. 14 
 
 Fig. 15 
 
 If we know: {a) the direction of both, components; (5) the mag- 
 nitude of both ; or (c) the magnitude and direction of one, there is 
 a definite resohition [case (5) admits of a double solution]. For 
 illustration of these three cases see Figs. 17, 18, and 19, respec- 
 tively. 
 
 ' Fig. 16 
 
 Fig. 17 
 
 Case (a). The given motion p-v. Fig. 17, is to be resolved into 
 componenls in the directions ^-m and f-n. 
 
 From the point v draw line v-v^ parallel to p-n, and cutting p-m 
 in V,; also from point v draw line v-w,, parallel to/i-w, cutting ^-w 
 in v^ ; p-v^ and p-v^ are adjacent sides of a parallelogram meeting in 
 jP, and p-v is the diagonal of this parallelogram through this same 
 point p, hence the motions represented by p-v^ and p-v^ are the 
 components of p-v in the given directions, p-m and p-n. It is evi- 
 
GONGEPTIOUrS OF MOTION. NATURE OF A MACHINB. 17 
 
 dent that no other parallelogram can be formed on p-v as a diag- 
 onal with its sides in these given directions. 
 
 Case (5) : The given motion p-v, Fig. 18, is to be resolved into 
 two components of the magnitudes indicated by m and n, direc- 
 tions to be determined. 
 
 With radius m and centre p draw the arc m^-m^', and with same 
 radius and centre at v, draw the arc m,-m,'; also with radius n and 
 centre at v draw the arc n,-n/, and with same radius and centre at 
 
 Fig. 18 
 
 J), draw the arc n^n,'; this gives four intersections. Connect 
 these intersections with p by the lines p-v^, p-v^, p-v^', and p-v^'. 
 By drawing lines from v to each of these intersections of the arcs, 
 it is seen that two parallelograms are formed (p-v^-v-v^, and p-v/- 
 v-v,'), each having the given motion p-v for a diagonal, with sides 
 {p-v, and p-v„ and p-v/ and p-v/, respectively) equal to the re- 
 <iuired components ; hence there are two solutions to this case, both 
 satisfying the condition that the motion fj-w be resolved into two 
 components of values m and n. 
 
 Case (f) Fig. 19: The given motion, p-v, is to be resolved 
 into the component p-v,, known as to magnitude and direction, 
 and another component, entirely unknown. 
 
 Draw a line from v to v^, also draw a line from v parallel to 
 p-v„ then draw a line from p parallel to v-v^, cutting the line last 
 drawn in w,. p-v, is the required component; for the given com- 
 ponent p-v, and this line last found form adjacent sides of a 
 parallelogram with p-v, the given motion, as a diagonal. 
 
18 KINEMATICS OF MAOEINEBT. 
 
 It will be seen from the preceding discussion, that in the reso- 
 lution of a motion into two components, or the composition of two- 
 motions into one resultant, there are six elements involved, viz.: 
 Both the directions and magnitudes of three motions; and that if 
 four of these elements are known the other two may be determined, 
 (except for the double solution of case h, in which two values satis- 
 fying the conditions are obtained). 
 
 The first case (a) is by far the most common in practical ex- 
 amples. 
 
 18. Angular Velocity. — When a point is revolving about som& 
 axis, permanently or temporarily, it is frequently convenient to- 
 express its rate of motion in angular rather than in linear measure. 
 Such angular motion may be expressed in any system of time and 
 angular units, as revolutions per minute or per second, degrees per 
 second, radians per minute, etc. In many practical problems thfr 
 angular motion of a member is most conveniently stated in terms 
 of revolutions per unit of time; but in analytical expressions the 
 arc passed over is often more readily measured in other units. 
 
 The radian is an arc of a length equal to the radius r ] hence there 
 
 are = 27r = 6.283 radians to a circumference; or a radian is. 
 
 r ■ 
 
 equivalent to 360° H- 6.283 = 67.3°, nearly. 
 
 If a revolving point makes n revolutions about its axis per unit. 
 
 of time the space passed over in time unity, or its linear velocity, 
 
 is «; = 27irn; and the radius traversed in the same time will be- 
 
 CO = = 27in radians. 
 
 r 
 
 If the body A (Fig. 30) is revolving about the axis through 
 (which is perpendicular to the plane of the paper), at the rate of n 
 revolutions per unit of time, the point;?, at a distance r from the 
 axis, has a linear velocity of 27irn; another point at p', at a dis- 
 tance r' from the axis, has at the same time a linear velocity 
 27ir'n ; and any two points at different distances from the axis 
 have different linear velocities at any instant. But all points in 
 the same rigid body when revolving about an axis must describ& 
 equal angles in the same time, and the angle (or arc) is being de- 
 scribed at a rate, expressed in radians by 'Unn. This expressioa 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 1? 
 
 for the rate of angular motion is what is called the Angular Veloc- 
 ity of the body; and it is necessarily the same at any instant for all 
 points in the same rigid body. It 
 will be noticed that the only varia- ^ 
 ble in the expression for angular / 
 velocity as just derived is n, the num- / / 
 ber of revolutions per unit of time. I \ 
 
 Comparing the expression for \ 
 angular velocity with that for the 
 linear velocity of a revolving body, 
 it is seen that it corresponds with '^' 
 
 the linear velocity of a point in the body at a distance from 
 the axis equal to unity : from which we deduce the statement : 
 The angular velocity of a body is numerically equal to the linear 
 velocity of a point in the same body at unit distant from the axis. 
 
 The relation between the linear and the angular velocity of a. 
 point which is most frequently used and one that should be firmly- 
 fixed in the memory, is 
 
 T .. Linear velocity v 
 
 Angular velocity = jT^rr ' or ca. = — •. 
 
 If a point revolves about a fixed centre with a linear velocity of 
 60 feet per second (730 inches per second), and with a constant 
 radius of 18 inches (1.5 feet), its angular velocity is 
 
 An 
 GO = — - = 40 (radians per second), 
 
 720 
 or £w = — 5" =: 40 (radians per second). 
 
 lo 
 
 The space-units which measure the radius and the linear 
 velocity must be the same, and the angular velocity is then ex- 
 pressed in radians per second, or per minute, according to whether 
 the linear velocity time-units are seconds or minutes. 
 
 Angular velocity may be constant, or it may vary, uniformly or 
 otherwise. If the radius remains constant, as in a body rotating 
 about an axis to which it is rigidly connected, the angular velocity 
 
20 
 
 KINEMATICS OF MACHINERY. 
 
 
 must vary just as the linear velocity of any one point varies, as is 
 seen from the above relation; or it varies directly as n. 
 
 19. Instant Axis; Instant Centre.— It was shown in Art. 14 
 that the motion of a point, in any path whatever, may he repre- 
 sented by a right line the length of which indicates the linear 
 velocity, the position representing the direction of motion. If the 
 pointy. Pig. 21, has a motion in the plane of the paper represented 
 
 by the line jw-v, the point may 
 be considered at the instant as 
 having a motion in the path t-t, 
 a-h, a'h, a"-b", m-n, or in any 
 path whatsoever, passing through 
 p and tangent to p-v; for motion 
 in any such path would, at the 
 instant under consideration, be 
 represented by a line such as 
 p-v. The motion of p, whatever 
 its real path, maybe considered 
 for the instant as equivalent to a 
 rotation about such a centre as c, c' , or c" , or in fact as a rotation 
 about any point in the line n-n' passing through p and perpendic- 
 ular to t-t'. Such a point is called an Instantaneous Centre — or, 
 adopting a somewhat shorter term, an Instant Centre. < 
 
 Unless the real path of the point is the particular circular arc 
 whose centre corresponds with this instant centre, the rotation 
 about such centre is not maintained for any finite length of time, 
 which the name given to it implies; if the path is this particular 
 arc, the instant centre is a permanent centre as well. 
 
 It is strictly more exact to refer to rotation or revolution about 
 an axis than about a centre, though in treating of motion in a plane 
 it is often convenient to use the latter term and no misunderstand- 
 ing arises from such use. What is really understood in this special 
 class of motions (plane motions) by rotation about a centre is 
 rotation about an axis which passes through the point called a 
 centre and is perpendicular to the plane of motion. 
 
 In considering the more general case of motion in space, it is 
 necessary to refer the rotation to an axis. Kotation about an axis 
 
CONCEPTIONS OF MOTION NATURE OF A MACHINE. 21 
 
 must be, at any instant, in a plane perpendicular to the axis; and, 
 conversely, the axis must always be perpendicular to the plane of 
 rotation; hence if the plane of rotation is fixed there can be but 
 one axis through any point in that plane, about which rotation can 
 take place, viz., the line passing through this point and perpendic- 
 ular to this plane. If the plane of rotation is not fixed, however, 
 as in the general motion of a point in space, there are an infinite 
 number of axes passing through any given point, about which the 
 motion is equivalent to an instantaneous rotation. This will appear 
 as a result of the following discussion of the general motion of a 
 point in space. 
 
 If p-v (Fig. 22) is the motion of p in space and A-B- C-D is a plane 
 through p perpendicular to p-v, p may be considered as rotacing 
 about any line in A-B-G-D as an 
 axis. For, if a perpendicular be 
 dropped from p upon any such line 
 (as p-c upon a-V), p-v may be treated 
 as a rotation about any point of p-c, 
 in a plane containing p-v and p-c; 
 and, hence, about any line, as a-i, 
 through a point in p-c and perpen- 
 dicular to this plane of motion, c-p-v. 
 
 Similarly for any other line in A-B- C-D, as a'-b', etc.; for a 
 perpendicular passing through p can be drawn to any line lying 
 in the plane A-B- C-D. Such a line is an Instantaneuus Axis, or 
 Instant Axis. 
 
 The above motion of p-v (Fig. 22) has just been shown to be 
 equivalent to a rotation about any line in the plane A-B- C-D. 
 This motion is therefore equivalent to a rotation about any one of 
 an infinite number of lines passing through a point of A-B-G-D ; 
 hence we cannot definitely fix the motion of a point in space by 
 locating one point in its axis, as in the case of a motion in a given 
 plane, but the position of the axis itself must be fixed. 
 
 If two points are moving in any plane, their motions, at any 
 instant, are equivalent to rotations about one common centre. 
 
 Thus, Fig. 23, if the motions p-v and p'-v', in the plane of the 
 paper, of the points j» and p', are given, the two points have at the 
 
22 KINEMATICS OF MACEINERT. 
 
 instant rotations about the common centre 0, lying in the inter- 
 section of p-n and p'-n; for the motion of p is equivalent to a 
 rotation about any point in p-n (perpendicular to the motion p-v) ; 
 likewise the motion of p' is equivalent to a rotation about any 
 point mp'n'; and is a point common to each of these perpendic- 
 ulars, hence it is the coincident instant centre for both points. 
 
 There are two or three . special cases that require brief special 
 treatment. It has been seen that is the common instant centre 
 for the two plane-motions i?-y and ;?'-«' in Fig. 33; and if p and jj' 
 
 ._ P _ll' o— „,' 
 
 n'^ Fig. 23 v' 
 
 are two points in the same body, that body is rotating, at the 
 instant, about 0. If, however, the two normals coincide, as in 
 IPig. 24, their intersection is not definite ; but if the two points are 
 ■of the same rigid body the instant centre may be found by aid of 
 the principles discussed in Art. 18. All points in a rigid body 
 must have the same angular velocity at any instant, and the 
 angular velocity of a body equals the linear velocity divided by the 
 Tadius; or, the radii are proportional to the linear velocities of the 
 points of the body. 
 
 In the case of Pig. 24, the instant radii are found by the pro- 
 portion : p-v : p'-v' : : 0-p : 0-p', and the instant centre is located at 
 O, by the construction indicated. If, as in Fig. 25, the two 
 motions p-v and p'-v', are parallel, but if the normals p-n and p'-n' 
 do not coincide as in Fig. 24, the intersection of these normals, 0, 
 is at infinity. In this last case if the motions ^-v and. p'-v' are also 
 «qual, the two points may belong to the same rigid body, and the 
 motion of this body is equivalent to a rotation about an instant 
 centre at infinity, or it is a translation. If the two motions, 
 jPig. 25, are parallel but not equal, the two points cannot belong to 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 23 
 
 ^ 
 
 the same rigid body, as these motions imply a change in the dis- 
 tance between p and p', which is not consistent with the conception 
 of a rigid body. 
 
 Two points having any motions whatever in space have, at any) 
 time, a common instant axis about which both are rotating. _^_ -^ 
 ~ Thus, Pig. 26, let the motions of the points p and p' be p-v 
 
 1 
 
 P' n'i 
 
 > 
 
 ' > 
 
 ' 
 
 0-> 
 
 Fig. 25 
 
 and p'-v'. The motion of p is equivalent to a rotation about any 
 line in the plane A-B-C-D, perpendicular to p-v and passing 
 through p ; also, the motion pf p' is equivalent to a rotation 
 about any line in the plane A'-B'-O'-D', passing through p' and 
 perpendicular to p'-v'. The intersection of these planes, 00', is 
 a line common to both planes, and is therefore a common instant 
 axis of the two motions. In special cases the two planes may be 
 parallel, and then the intersection is at infinity. 
 
 20. Free and Constrained Motion. — It follows from the state- 
 ments of Art. 15 that if a point is to move in any prescribed path, 
 the resultant of all forces acting upon the point must always lie in 
 & tangent to that path. If the path be other than a straight line, 
 this involves a constant change in the direction of the resultant 
 iorce, caused either by a change in direction or magnitude (or 
 both) of at least one of the components of this resultant. This is 
 •exactly what takes place in every such case; but the method of 
 this readjustment of the resultant force affords the basis of a very 
 important division of motions into two classes, viz.: Free and 
 Constrained Motions. 
 
 A body which has no material connection with other bodies is 
 
24 
 
 KINEMATICS OF MACHINEBT. 
 
 called a free body; the planets are examples of this class of bodies. 
 A planet revolves around the sun in a path or orbit determined by 
 the resultant of all forces acting upon it; every disturbing action^ 
 or force alters its path. 
 
 A body which has a material connection with another body, 
 permitting motion relative to that body only in certain restricted 
 paths, is- said to be constrained ; the crank-pin of an engine is aii_ 
 example of this class. In this case, if motion takes place under 
 the action of any force it must be in the fixed path, and no force, 
 whatever its direction, short of one that will break or injure the 
 machine, can cause motion in any other path. 
 
 The primary actuating force in the case of the crank-pin is the 
 pressure or pull exerted upon the pin along the direction of the 
 connecting-rod (neglecting frictional influence). This primary 
 force does not, except at two instants in each revolution of the 
 crank, act tangentially to the path of the body acted upon ; there- 
 fore we must look for some other force which combined with this 
 primary force gives a resultant acting tai^ntially to the path of 
 the crank-pin. "While the study of such actions is not strictly 
 within the province of the present treatise, it is important to 
 clearly fix the nature of constrained motion, and for this purpose 
 the distribution of force acting through the connecting-rod upon 
 the crank-pin of the ordinary reciprocating steam-engine will now be 
 briefly considered. Pig. 27 indicates the mechanism of the engine 
 (without valve-gear). Pigs. 38, 39, 30, 31, 33, 33, 34 show the 
 connecting-rod and crank in different positions, or phases. The full 
 lines indicate the motions, and the dash- and dot-lines indicate 
 forces acting. 
 
 Fig. 27 
 
 Fig. 28 
 
 In Fig. 28 the connecting-rod is at right angles to the crank, 
 and therefore its centre line coincides with the tangent to the circle 
 in which the crank-pin must move. As the force P, acting on the 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 25 
 
 pin, is in the direction of the centre line of the rod, this force alone 
 would produce motion in the prescribed path, and no other force 
 need be considered as acting to produce such motion at this par- 
 ticular phase. The rod is under compression. 
 
 In Fig. 29 the condition is similar, except that the connecting- 
 rod is now under tension instead of compression, and the action on 
 the pin is a pull instead of a thrust, but, as before, the force acts 
 tangentially to the path. 
 
 In Fig. 30, however, the force P' exerted by the connecting-rod 
 on the pin (thrust) is not in the direction of the tangent to the 
 path, and hence it alone cannot produce motion in the required 
 
 Fig. 29 ^?^=* " m Fig. 30 
 
 direction. If, however, a force P", be introduced in the direction 
 of the centre line of the crank, of such magnitude that it, com- 
 bined with P', will have a resultant P in the line of the tangent to 
 the path of the pin, the conditions necessary to produce motion in 
 the required direction will be present ; and unless such a com- 
 ponent of force is acting in conjunction with P', the required 
 motion cannot take place. If the crank-pin were a free body this 
 force would be an external force, but it will be seen that it would 
 be very difi&cult to apply such an external force in the right direc- 
 tion and of the proper magnitude, for these requirements con- 
 stantly change. In case of constrained motion the material con- 
 nection (the crank in this case) supplies this force by its own 
 resistance to a change of form. The primary acting force, alone, 
 would impart motion in the direction of its own line of action, but 
 this motion could not take place without changing the form of the 
 crank, and the crank offers resistance to this change, by just the 
 necessary amount for constrainment. As action and reaction are 
 always equal, the force exerted on the crank to change its form is 
 met by a corresponding counter-action, or reaction, just sufficient 
 
36 
 
 KINEMATIC8 OF MACHINERY. 
 
 to give the required constraining force, and to cause motion in the 
 circle of which the crank is the radius. This external force tend- 
 ing to change the form of the crank calls out within the material 
 an internal molecular action known as stress, and this action is just 
 equal to the external force. In this particular phase both con- 
 necting-rod and crank are subjected to compression. 
 
 In Fig. 31 the condition is similar to that of the preceding, ex- 
 cept that the connecting-rod is under tension, the action on the pin 
 
 Fig. 31 
 
 Fig. 32 
 
 is a pull, and the resistance of the crank is necessarily reversed, 
 the stress now being a tension. This results of course in subjecting 
 the crank to tension also, and as it is of a material that will resist 
 this action, the motion in the required path is secured by the 
 combined action of the force exerted upon the pin through the 
 connecting-rod and of the secondary force called out in the crank. 
 Figs. 32 and 33 show phases in which the stresses in the connecting- 
 rod and crank are not similar. 
 
 "When the crank-pin is at one of the "dead centres" A or B, 
 as in Pig. 34, it will be noticed that the force exerted by the con- 
 
 Fig. 33 
 
 Fig. 34 
 
 necting-rod is at right angles to the direction of the pin's motion, 
 and hence no force combined with it can give a resultant in the 
 direction of the tangent to the path ; the whole effect of this force, 
 P', is now to compress or extend the crank (change its form), and 
 none of it is available in moving the crank-pin. If it were not for 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 27 
 
 the resistance of the crank at this time the pin would be impelled 
 in the direction of P', at right angles to its proper path, but the 
 resistance of the crank just balances the force received from the 
 rod, and, according to Newton's laws, the pin is, at the instant, 
 under a system of balanced forces, and it continues to move in 
 a tangent to its path, unaffected by these forces except as they 
 influence friction. Of course this is only an instantaneous con- 
 dition, and therefore the pin does not move through any finite dis- 
 tance under such a balanced system of forces. 
 
 Strictly speaking, the condition last considered is not equivalent 
 to the action of no force at all, although the forces are balanced, 
 for the pressure of the pin and of the shaft against the bearings 
 results in a frictional resistance tending to retard the motion. 
 The action of the fly-wheel also modifies the motion of the engine, 
 reducing the fluctuation of velocity that would be experienced 
 under the great variation of the resultant force throughout the 
 revolution; but neither of these causes need be treated in connec- 
 tion with the present discussion, which is simply intended to ex- 
 emplify the nature of constrained motion. 
 
 The distinguishing characteristic of a constrained motion is that, 
 in a body having such motion, all points in the body have definite 
 paths in which they move, if motion takes place under the action 
 of any force whatever. The stresses produced in the restraining 
 connections supply the components of force necessary to combine 
 with the primary force, or forces, to give a resultant in the direc- 
 tion of the prescribed path. If these connections are strong enough 
 to resist the maximum stress to which they are thus subjected, no 
 farther attention is required to secure the proper adjustment of 
 the resultant force to the prescribed path. The provision of the 
 necessary strength is in the province of another branch of me- 
 chanics,, and it may be assumed in the present work that such 
 strength is provided. 
 
 It will be shown that absolute constrainment is not possible 
 by the ordinary methods employed in machinery construction ; 
 because all materials are somewhat deformed under stress, but 
 practical constrainment may always be secured ; that is, the depar- 
 ture from the desired motion can be reduced to any required limit. 
 
28 KINEMATICS OF MACHINERT. 
 
 The nature of the constrainment depends upon the form of the 
 constraining members ; the degree of constrainment is determined 
 by their dimensions and material. 
 
 All motions used in machinery are either completely or partially 
 Constrained. 
 
 21. Mechanics is the science which treats of the relative motions 
 and of ihe forces acting between bodies, solid, liquid, or gaseous. 
 
 " The laws or first principles of mechanics are the same for all 
 bodies, celestial or terrestrial, natural or artificial." (Eankine.) 
 
 22. Mechanics of Machinery treats of the applications of those 
 principles of pure mechanics involved in the design, construction, 
 and operation of machinery. 
 
 Every problem of mechanics arising in connection with ma- 
 chinery is subject to the laws of pure mechanics, and we could con- 
 ceive of its solution by the general methods of the larger science; 
 but the operation would often be needlessly difficult, if not practi- 
 cally impossible, and more convenient special treatment has been 
 developed for the limited class of phenomena connected with prob- 
 lems of mechanism. It has been seen that constrained motions 
 are much more easily treated than are free motions; and all problems 
 of motions of machines are included under constrained, or partially 
 constrained, motions. It is mainly the distinction between free 
 and constrained motions, in fact, that separates the Mechanics of 
 Machinery from the more general science of Pure Mechanics. 
 
 23. A Mechanism, or train of mechanism, is a combination of 
 resistant bodies for transmitting or modifying motion, so arranged 
 that, in operation, the motion of any member involves definite, 
 relative, constrained motion of the other members. 
 
 24. A Machine consists of one or more mechanisms for modify- 
 ing energy derived from natural sources and adapting it to the per- 
 formance of useful work. A machine may consist of a single 
 mechanism according to this definition; but it has seemed best to 
 make the following distinction between a mechanism and a ma- 
 chine: the primary function of the former is to modify motion; 
 while that of the latter is to modify energy, and, of course, inci- 
 dentally motion. The term " mechanism" becomes more general, 
 and it includes the elements of a large class of instruments or appa- 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 29 
 
 ratus, such as clockwork, engineers' instruments, models, and also 
 most forms of governors, as well as some larger constructions, the 
 function of which is essentially the modification of motion, and 
 which only do work incidentally, such as the overcoming of their 
 own frictional resistance. There is a real, and vital, distinction 
 between machines and such apparatus; but so far as a study of 
 their motions is concerned, no such distinction need be made 
 usually. 
 
 From the above definitions of mechanisms and machines, we 
 may derive the following: 
 
 A Machine is a combination of resistd,nt bodies for modifying 
 energy and doing work, the members of which are so arranged 
 that, in operation, the motion of any member involves definite, 
 relative, constrained motion of the others. 
 
 The essential characteristics of a machine are : 
 
 (a) A combination of bodies. 
 {h) The members are resistant. 
 
 {c) Modification of energy (force and motion) and the perform- 
 ance of work. 
 
 (d) The motions of the members are constrained. 
 
 (fl) A machine must consist of a combination of bodies. 
 
 The lever does not, by itself, constitute a machine, nor even a 
 mechanism, and it only becomes such when combined with the 
 proper fulcrum or bearing. Without this complementary member, 
 properly placed and sustained, a definite, constrained motion is 
 impossible. 
 
 The fulcrum is just as important an element in the make-up of 
 the machine as is the lever itself. The screw is of no use in modi- 
 fying motion or energy unless it is fitted with the proper envelope, 
 usually called a nut. So with the wheel and axle. It makes no 
 difference whether made from a single piece of material or built up 
 from several pieces of stock, the wheel and axle is essentially one 
 piece when completed, as there is no relative motion between the 
 Tarious parts, and it can only be of use iu connection with appro- 
 priate supports or bearings. And so on with all other examples; 
 
30 KINEMATICS OF MACSINERT. 
 
 the simplest machine must haye at least two members, between 
 which relatiye motion is possible. 
 
 (h) The members of a machine are generally rigid, but not neces- 
 sarily so. Flexible belts, straps, chains, etc., confined fluids (liquid 
 or gaseous), and springs, often form important parts of machines. 
 The flexible bands can only transmit force when subjected to ten- 
 sion; the confined fluids transmit force only under compression; 
 springs may act under tonsign, compression, torsion, or flexure. 
 These bodies are not rigid, in the usual sense of the word, but they 
 are resistant under the particular action for which they are 
 adapted ; hence they can be used in special applications to great 
 advantage. In fact their value in such applications is due to the 
 absence of the property commonly designated as rigidity. 
 
 No material is absolutely rigid, and what is commonly and con- 
 veniently called a rigid body is one in which the distortions under 
 load are so small as to be negligible for many purposes. 
 
 The action of springs, when carefully analyzed, is found to be 
 identical in quality with that of the so-called rigid bodies. The 
 characteristic of springs is the magnitude of the distortions. Every 
 solid body possesses the property of yielding under a load to a 
 greater or less degree, following the same general law as springs, 
 within the safe working limit at least. The difference is one of 
 degree only, but in this difference of degree lies the special fitness 
 of springs for certain parts of machines. 
 
 (c) The machine is used to modify energy and do work. 
 
 It is interposed between some source of energy and the work to 
 be done, and it adapts this energy, as supplied by or derived from 
 natural sources, to the required work. 
 
 The conception of a machine involves the conception of some 
 source of energy, an effect to be produced, and a train of mecha- 
 nism suitably arranged to receive, modify and apply the energy 
 derived from this source to the desired end. 
 
 The nature of the source of energy and of the work to be done 
 determine the character of 'the machine, and the forms of the 
 members for receiving the energy, transmitting, modifying, and 
 applying it. The primary natural source of energy may be the 
 muscular effort of animals ; wind; water; heat, acting through. 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 31 
 
 such vehicles as steam, air, or other gases; etc. The secondary 
 sources maybe pulleys, gears, shafts, etc., deriving their energy, 
 directly or indirectly, from some of the primary sources. The 
 prime movers — ^windmills, water-wheels, steam-engines, etc. — are 
 driven from primary sources; while machinery of transmission — 
 machine tools, dynamos, etc. — are actuated from secondary 
 sources. 
 
 In_ajnachine-shop, for example,, the source of energy of the 
 tools is the line-shaft, or the counter-shaft, according as the latter 
 is, or is not, treated as a part of the tool; it evidently makes no 
 difference how this shaft is driven, so far as the study of the indi- 
 vidual tool is concerned. The source of energy being a rotating- 
 shaft, the member of the machine receiving the energy must be a 
 pulley, gear, sheave or other form capable of connection with such 
 rotating-shaft. Energy may be transmitted by compressed air or 
 by water under pressure ; then the receiving member may be a pis- 
 ton, reciprocating in a suitable cylinder, or a wheel with appropri- 
 ate vanes or blades attached. 
 
 In a similar way the desired result determines the motions and 
 forms of the members producing this effect. If this effect be the 
 planing of metal, a reciprocating motion is usually imparted to 
 the member to which the piece operated upon is attached, or to the 
 cutting tool. If the useful work is rolling rails, a rotation of the 
 rolls is to be obtained; and so in grinding grain, or minerals; in 
 pumping water or other fluids; compressing air or other gases; 
 weaving or spinning; cutting" woods, stones, or metals; for trans- 
 porting materials, etc. ; each class of work requires an appropriate 
 modification of the energy imparted to the receiving member. 
 
 In general, any of the sources of energy may be applied to pro- 
 duce any mechanical effect by means of proper trains of mechan- 
 ism; and this gives rise to a very great number of possible ma- 
 chines. The working membersof machines have been classified by 
 Willis as: 
 
 (a) Parts receiving the energy. 
 
 (S) Parts transmitting and modifying the energy. 
 
 (c) Parts pei-forming the required work. 
 
32 KINEMATICS OF MACEmEBY. 
 
 To these might be added : 
 
 {d) Auxiliary p^^rts, as regulators, etc. 
 
 (e) Frames for restraining the motions and sustaining the 
 machines. 
 
 Various classifications of the parts of machines have been made 
 by different writers, but that of Willis has perhaps been most gen- 
 erally accepted. From a kinematic standpoint, such classifications 
 are of doubtful value, and Reuleaux's masterly treatment of the 
 subject indicates that all such divisions are artificial and arbitrary. 
 This will be more fully discussed iinder Inversion of Mechanisms. 
 
 The working, or moving, ipembers of a machine may be levers, 
 arms, beams, cranks, cams, wheels with treads, blades, vanes, or 
 buckets, with teeth or with flat or grooved rims, etc., screws and. 
 nuts, rods, shafts, links, and other rigid members; as well as 
 belts, bands, ropes, chains (flexible members); and occasionally 
 confined fluids, as water, oil, air, etc. Many modifications of 
 these are used, and an indefinite variety of forms result, yet, kine- 
 matically, when reduced to the simplest forms, the variety of mech- 
 anisms is much less than would at first appear. 
 
 The frames which support the working parts and determine 
 their motions are almost as varied in form and materiails as the 
 moving members themselves, but are capable of similar simple 
 treatment. In fact, as will appear later, the frame may be 
 treated as exactly equivalent to any other member of the machine, 
 and so far as relative motion of the members is concerned, it mat- 
 ters not which particular piece is made " stationary." 
 
 The leading distinction between a machine and a "structure" 
 (such as a bridge) is thatihefdrme'r serves to modify and transmit 
 energy, or force and motion; while the latter modifies and trans- 
 mits force only. Some parts of machines, as the fixed frames, 
 are properly structures, while as a whole the construction is a 
 machine. 
 
 {d) The relative motions of the members of a machine are 
 constrained, or restricted to certain definite predetermined paths, 
 in which they must move, if they move at all, relatively. 
 
 The nature of constrained motion has been considered in the 
 preceding chapter. 
 
CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 33 
 
 The leading characteristic ql a mechanism or a machine is the 
 eonstrainment of its motion. A strncture^^es_n.ot permit relative 
 motion of its members, or at most it only allows the very limited 
 incidental motions, due to deformation of its members under loads, 
 the effect of changes of temperature, etc. Occasionally what 'are 
 ■usually classed as structures, or parts of structures, do have pre- 
 scribed motions, as the draw of a bridge, or a turn-table. These 
 are not properly machines, but they do come under the preceding 
 •definition of a mechanism. 
 
 All artificial combinations of bodies, to which may be given the 
 general name of constructions, and which have constrained motion, 
 may be classed as mechanisms; and if intended to be employed in 
 the performance of useful work, as machines. 
 
 The eonstrainment in mechanisms is sometimes 'partial, or in- 
 complete. Thus in the case of a crane, in which the load is sus- 
 pended by a chain or cable, the slightest horizontal force will sway 
 the load-hook, and therefore change its path from the right line 
 perpendicular to the earth's surface in which it normally moves. 
 This does not affect the useful operation of the crane, as the hook 
 is constrained against all undesirable motion, and for practical pur- 
 poses the action is just as good as if the eonstrainment were com- 
 plete. Often, in fact, this degree of freedom of motion is de- 
 sirable. 
 
 Again, consider the familiar fly-ball (conical-pendulum) gov- 
 ernor (Pig. 9) as used on many classes of steam-engines. The 
 balls of the governor are constrained to the extent that their cen- 
 tres always lie in the surface of a certain sphere, that is, they have 
 spherical motion ; but they may move in any path whatever (within 
 the limits of action), lying in the prescribed spheres. The path 
 in which they, actually travel depends upon the relations between 
 their mass, angular velocity, and radius relative to the axis 
 about which they revolve at the instant under consideration, and 
 their motion can be determined only in connection with these 
 forces. 
 
 In all but a few such cases as the latter we may study con- 
 strained motion quite apart from the forces involved in the opera- 
 tion of the mechanism. 
 
54 KINEMATICS OF MAOHINERY. 
 
 25. Machine Design; Kinematics. — The design of a new ma- 
 ihine or the analysis of an existing machine divides itself natu- 
 ally into two quite distinct processes, as will appear upon brief 
 eflection. In every machine energy is supplied from some source 
 .nd so modified as to produce some useful effect. A train of 
 nechanism is employed to secure the required transfer or modifi- 
 ation, and this intermediate mechanism must be adapted, first, to 
 ecure the motion demanded to produce the desired result ; and 
 econd, to transmit the necessary force without breakage or undu& 
 iistortion of the members of the machine. It will readily appear 
 hat the motion system can often be planned or studied without 
 onsidering the magnitude of the forces transmitted. As an illus- 
 ration, consider the lever of Fig. 35, in which the distance from 
 
 the fulcrum, 0, to A is, say 1 foot, 
 and distance from to ^ is 2 feet. 
 Now if B be moved through a dis- 
 tance of 3", A will move through 
 1". Suppose the resistance to act 
 at A, and the force which over- 
 comes it at B. The resistance at A 
 
 Fig. 35 
 
 may be 1 pound, 1 ton, or of any 
 ther amount whatsoever, then the force required at B to overcome 
 t will be of a corresponding magnitude; but in any case the ratio 
 f the motions or the velocity ratio of ^ to 5 will be determined by 
 he length of the lever-arms, independent of the actual forces in- 
 olved. Furthermore, if B moves 3" in one second, A will move 
 inch in one second; if B moves 6" in one second, A will move 
 " in one second ; or if 5 makes 4 strokes per second, A .will also 
 lake 4 strokes per second; but the (total) path described by A, or 
 istance moved through, will be but half the path of B. So for 
 ny motion whatever of B, A will have a definite, corresponding 
 lotion, and the ratio of the motions will remain invariably the 
 ime, whatever the actual motion and the forces acting may be. It 
 ppears then that the ratio of the motions which A and B have, 
 Blative to the fixed member, is determined purely by geometrical 
 onsiderations, and may be studied without taking into account 
 nything else. 
 
; CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 35" 
 
 The lever must not only give a required motion to one point 
 for a given motion of another, but it must transmit a certain force; 
 and having satisfied the motion requirement, it is necessary to give- 
 the lever, the pivot, and all parts subjected to load, sufficient' 
 strength to safely carry the loads. This second operation requires, 
 a knowledge and application of the physical properties of the mate- 
 rials used, and of other laws of mechanics than those relating to 
 simple motion. 
 
 A similar discussion would apply to all ordinary mechanisms, 
 but the foregoing is perhaps sufficient for the present purpose, 
 viz., showing how the design of a machine may be taken up as 
 two distinct processes, one of which can be completed, subject to 
 certain modifications, before the other is considered. 
 
 Frequently the actual motions, velocities, and accelerations, as 
 well as the external loading, is involved in the second part; for 
 the weight of a part and the changes in direction and velocity of 
 the motion of a body produce stress in the restraining members.. 
 An example is the stress due to "centrifugal force." In the 
 complete design of a machine, there are many other considerations, 
 affecting the durability, freedom from frictional and other losses, 
 etc., that are no less important than the preceding, and all of thesfr- 
 must be carefully weighed in their proper place ; but these consid- 
 erations are not within the province of the present work. 
 
 The two grand divisions of the Mechanics of Machinery out- 
 lined above are called : 
 
 (I) The Geometry of Machinery, Pure Mechanism, or Kine- 
 matics ; 
 
 (II) Constructive Mechanism, or Machine Design. 
 
 In beginning the study of machinery, it is both logical and 
 convenient to take up the above divisions, in the order given. 
 The first division. Geometry of Machinery, Kinematics, or Machina 
 Motions, will be the leading subject of the present work. 
 
 While, as in the illustration of the lever given above, the con- 
 sideration of the forces acting is not, generally, involved in the 
 study of machine motions, there are important classes of mechan- 
 isms the motions of which cannot be treated without taking cog- 
 nizance of certain forces. Examples of these are the centrifugal 
 
36 KINEMATICS OF MAOEINERY. 
 
 governors, already mentioned, so commonly used on steam-engines 
 and other motors, in which centrifugal force and the force exerted 
 by springs, or by gravity, can not be divorced from a treatment of 
 the motions; also escapements such as are used in clocks and 
 watches. 
 
CHAPTER II. 
 
 GENERAL METHODS OP TRANSMITTING MOTION IN MACHINES. 
 
 26. Transmission through Space without Material Connec- 
 tion. — In most mechanisms motion is transmitted and controlled 
 through actual contact of members of the mechanism; but there 
 are certain exceptions as, for example: electric motors, escape- 
 ments of clocks, governors, etc. The armature of the motor is 
 caused to be rotated by electromagnetic causes acting through 
 appreciable space; the pendulum or balance-wheel of the clock- 
 work is driven by the intermittent action of gravity or a spring, 
 though the resultant motion is affected by the length of the pen- 
 dulum or proportions of the wheel, independently of the intermit- 
 tent connection with the escape- wheel ; and the motion of the 
 governor-balls is determined by the combined effect of centrifugal 
 force and of gravity or of springs. In these instances, the motion 
 of the members is not fully constrained, and the motions of such 
 mechanisms cannot be treated by purely geometric methods, inde- 
 pendently of the forces involved iij, the actual operation. "With 
 the exceptions of these and similar mechanisms, motion is only 
 transmitted by direct contact of one material body with another. 
 We are at present only concerned with such cases as the latter. 
 
 27. Transmission by Actual Contact. — These cases may be 
 conveniently treated under two divisions; and the second division 
 may be subdivided into two classes. 
 
 In every mechanism we have one member, — ^frequently called 
 a link, whatever its form — driving another link- the former is 
 called the driver, and the latter the follower. 
 
 The driver may have a surface which bears directly upon the 
 follower, or there may be an intermediate piece serving to transmit 
 the force and motion. This intermediate connectoi: may be a rigid 
 
 37 
 
KINEMATICS OF MACHINERY. 
 
 r or block; it may be a flexible band (as a belt, cord, or chain); 
 it may be a confined fluid column. 
 
 These various modes of connection give rise to the following 
 iSsification : 
 
 ' Direct Contact. 
 
 Modes of 
 "ansmission ■; 
 if Motion. 
 
 ' Rigid Links. 
 
 Intermediate 
 Connectors 
 
 Bands, etc. 
 Fluid Connectors. 
 28. Higher and Lower Pairing.— Figs. 36 and 37 represent 
 
 Flexible 
 Connectors 
 
 Fig. 36 
 
 Fig. 37 
 
 amples of direct contact transmission, in which A will be con- 
 iered as the driver and B as the follower. The contact in these 
 amples is confined to a line (or a point), instead of being dis- 
 ibuted over a finite surface. Such contact — line or point con- 
 ct — constitutes what is termed higher pairing ; while contact 
 er a finite surface is called lower pairing. Higher pairing usu- 
 [y involves greater wear at the contact surfaces, and is generally 
 be avoided if it is possible to do so. Gear-teeth and most forms 
 cams necessarily form higher pairs. However, there are many 
 rses where it is perfectly practicable to introduce modifications in 
 le construction which distribute the contact over a surface, with- 
 it sacrifice of the kinematic motion. While this does not change 
 le relative motion of driver and follower, it is practically advan- 
 geous in reducing the intensity of contact pressure, and conse- 
 lently the wear of parts. It is usually desirable to substitute 
 wer pairs for higher pairs where practicable. In certain cases, 
 
TBAN8MITTING MOTION IN MACHINES 
 
 39 
 
 ■where pure surface contact is not possible, a modification, which 
 does not eliminate line contact, may be advantageously employed. 
 
 Figs. 38 and 39 show cases of transmission ffom the driver A 
 to follower B by direct contact, higher pairing being used. In 
 
 Fig. 38 
 
 Fig. 39 I 
 
 these cases (Figs. 38 and 39), the kinematic action is the same as 
 ■would result from contact between the point p of driver and the 
 dotted "pitch-line" of the follower, as indicated by Figs. 40 and 
 
 ^"^T 
 
 Fig. 40 
 
 Fig. 41 V, 
 
 41. These latter figures do not represent practical mechanisms, 
 for of course it is necessary to have the contact parts of sen- 
 sible size. 
 
 Figs. 43 and 43 show similar arrangements, with a suitable 
 block interposed between the element of the driver and fol- 
 lower; these intermediate pieces do not change the transmission 
 of motion in any degree, but it will be noticed that the driver now 
 acts upon the block, and the block upon the follower, eliminating 
 line contact entirely without sacrifice of the desired motion, and a 
 better practical mechanism is thus obtained. The mechanisms of 
 Figs. 38 and 39 are identical, kinematically, with those of Figs. 
 43 and 43. An intermediate connector, or a new link, has been 
 
40 
 
 KINEMATIOS OF MACHINER7. 
 
 introduced, and, in a sense, the mechanism comes under the second 
 division in the above classification j but this intermediate con- 
 nector, C, does not alter the transmission of motion. As we are 
 not concerned in the least with the motion of this block itself, it- 
 
 Fig. 42 
 
 Fig. 43 
 
 may be neglected in the kinematic analysis, and such substituted! 
 
 mechanisms will be treated as direct contact under Division I. If 
 
 desired, A could be treated as the driver of C; and (which is the 
 
 follower with regard to A) as the driver of B. 
 
 Except in cases where the contact surfaces of both links are- 
 
 either planes, surfaces of revolution, or regular helical surfaces, 
 
 such substitution cannot be made; for these are the only surfaces 
 
 possible in lower pairing. In gear- teeth, for example, it is not 
 
 possible to avoid higher pairing. 
 
 There are other cases, as in cams, wTiere it is practically of 
 
 great advantage — even though line contact is not thus eliminated — 
 to introduce an intermediate piece, replac- 
 ing one kind of line contact by another- 
 kind. Thus, in Pig. 44, the cam could act- 
 directly upon the end of the rod Bj but the 
 friction would be excessive and the action 
 would not be smooth, especially if the form 
 of the cam departs much from that of a 
 surface of revolution whose geometrical axis 
 coincides with the axis of rotation. By 
 fixing a roller, G, of suitable form, at the 
 ^'3- 44 end of the rod, the smoothness of action is 
 
 much enhanced. The roll does not nil on the cam, as in the- 
 
TRANaMITTING MOTION IN MACHINES. 
 
 41 
 
 direct sliding contact of the follower upon tlie driver, and the 
 sliding action is transferred to the pin which carries the roll, where 
 surface contact is procured. In this case, as in those of Figs. 42 
 and 43, the intermediate connector can be neglected kinematically. 
 The motion transmitted to the follower corresponds to the contact 
 of the centre of the pin, p, with a hypothetical surface — called the 
 pitch surface of the cam — indicated by the dotted line. The rela- 
 tion of this pitch surface to the actual working surface, and the 
 derivation of the latter from the former, will be treated later under 
 the head of Cams. 
 
 In the following discussions of the angular velocity ratio of 
 driver to follower, in direct-contact mechanisms, the auxiliary con- 
 nector — the block, cam-roll, etc. — will be neglected, as it has been 
 seen that its own motion is immaterial, and that it does' not affect 
 the velocity ratio of driver to follower. 
 
 It is interesting, in connection with the preceding discussion, to 
 note that it is sometimes advantageous to employ line or point 
 contact when the case will, kinematically, permit surface contact. 
 The familiar roller-bearings and ball-bearings are examples. In 
 these, friction and wear are reduced by the substitution of line or 
 point contact for the ordinary surface-bearing, becouse by this sub- 
 stitution the grinding effect of sliding is replaced by a rolling of 
 each member upon those with which it comes into contact. 
 
 29. Direct-contact Transmission. — The most general case of 
 direct contact is between two surfaces such as are shown in Figs. 
 
 Fig. 45 
 
 Fig. 46 
 
 36, 37, and 45. The surfaces may both be convex, or one may be 
 concave; as in Fig. 45; but in the latter event the radius of curva- 
 ture of the concave surface must always be at least as great as that 
 of any portion of the other member that can come in contact with 
 
42 
 
 KINEMATICS OF MAGHINERT. 
 
 it ; otherwise a certain part of the concave surface will not come 
 into contact and the action will be discontinuous or irregular, as in 
 Pig. 46, between e and/. Except for this limitation, the surfaces 
 may be of any form; but the present discussion will be confined to 
 those cases in which the contact faces are, in the general sense of 
 the word, cylinders, with the axes parallel to each other, the 
 motions therefore being plane. There are special cases not coming 
 under this class which will be treated later in the work, and the 
 necessary modification follows readily. 
 
 In treating these plane motions the simplification referred to in 
 Art. 10 can be applied, that is, the representation of these surfaces 
 and their motions by their projection on a plane parallel to the 
 plane of motion. 
 
 Motion can be transmitted by direct contract only by normal 
 pressure between the surfaces. The action of one piece upon the 
 other may be of the nature of rolling, sliding, or mixed rolling and 
 sliding. The last condition is the most general. The precise 
 nature of these actions and the method of determining them will 
 form the subject of a later section. 
 
 Keferring to Pig. 47, it is evident that all points in A must 
 rotate about 0, and, likewise, all points in B must rotate about 0'. 
 Consequently the motion of any point in either is represented by a 
 
 line through that point perpen- 
 dicular to the radius connecting 
 it with its centre, or 0', as the 
 case may be. 
 
 The point of contact is a point 
 common to both members for 
 the instant, and it may be con- 
 sidered as a double coincident 
 point. It is designated by P. 
 '^' Considered as a point in A, P 
 
 will be. called P„, and as a point in B, it will be called P^. Then 
 Pm and Pn represent the motions of P„ and Pj, respectively. 
 
 The pressure between A and B is transmitted in the direction 
 of the common normal to the two surfaces at the point of contact, 
 and whatever the actual motions of P„ and P^, the components of 
 
TRANSMITTING MOTION IN MACHINES. 
 
 43 
 
 these two motions along the line of this normal {NN') must be equal 
 "when they are in contact (as Ps). If the normal component of 
 Pj, were greater than that of P^, the rate of motion of P^ along 
 this normal wonld be greater than that of P„, and B would quit 
 contact with A. On the other hand, if the normal component of 
 P„ is greater than that of P^, A would enter the space occupied by 
 B, and this is inconsistent with our conception of a rigid body. 
 
 As we are concerned only with the velocity ratio of the two 
 members, and as this ratio is independent of the actual velocities, 
 the velocity — either angular or linear — of one member may be 
 assumed if not known, and this affords a means of determining the 
 velocity of the other member at that instant. The angular velocity 
 ratio of the driver to the follower is, in the general case, varying 
 continually. Simple methods of determining this angular velocity 
 ratio at any phase of the motion may be used, and a close analogy 
 exists in these methods as applied to the three different classes of 
 transmission. Each class will be discussed by itself, and the gen- 
 eral relation will be deduced afterward. 
 
 Fig. 48 
 
 If in Figs. 48 and 49, c», = the angular velocity of A, be 
 known, for the phase under consideration, the linear velocity of 
 P„ can be found from the relation : 
 
 ang. vel. = 
 
 linear vel. 
 radius. 
 
 or 00, 
 
 Pm 
 OP' 
 
 Represent the linear velocity of P„ by Pm, and resolve it into its 
 components along and perpendicular to the common normal NN'. 
 
44 KINEMATICS OF MAOHINERY. 
 
 Thus the normal component Ps, and the tangential component 
 Pt^ are obtained. The direction of the motion of P,, is known 
 (perpendicular to PO'), and its normal component must equal that 
 of P^, or it is Ps, hence the actual velocity of P,,, Pn, can be 
 found (Art. 17, Case (a)); and its tangential component, P^, may 
 be found from this if desired. Having found in this way the 
 linear velocity of Pj,, its angular velocity, ca^, may be obtained by 
 dividing this quantity by the radius PO' ; and the angular velocity 
 
 ratio of A to B, for this phase of the motion = — !, becomes known. 
 
 A similar method could be employed in determining this ratio for 
 any number of phases, and thus the motion of the follower, cor- 
 responding to the motion of the driver, whether uniform or other- 
 wise, could be derived. The following demonstration establishes 
 relations of the angular velocity ratio of driver and follower, in 
 direct-contact mechanisms, which are much more expeditious and 
 convenient in drawing-board practice. 
 
 In Figs. 48 and 49, let Pm and Pn represent the linear veloci- 
 ties of Pa and P^, respectively. Drop perpendiculars Of and O'ff 
 from and 0' upon the normal NN' . Ps is the common normal 
 component of Pm and Pn. Pms and OP/ are similar triangles; 
 also Pns and O'Pg are similar triangles. 
 
 Pm Ps 
 
 ojj = angular velocity of P„ about = -=rp = -yr-., . (1) 
 
 Pn Ps 
 CO, = angular velocity of B about C = —j-p = jy-. . (2) 
 
 w, _ Ps^ (Tg _ (yg 
 a>,~Of^Ps~0/ ^^^' 
 
 Prolong the normal and line of centres till they intersect at / ; 
 then 10/ and lO'g are similar triangles and 
 
 10^ _g^_ ^ 
 
 10 ~ Of w^ 
 
 (4> 
 
TRANSMITTING MOTION IN MACHINES. 
 
 45 
 
 It follows from the above discussion that': In direct-contact 
 motions the angular velocities of the members are, at any phase, 
 inversely as the perpendiculars let fall from their fixed centres iipcn 
 the line of the common normal of the two curves ; or inversely as the 
 segments into which the line of centres is divided by this normal. 
 
 30. Link-connectors. — A relation very similar to that just de- 
 rived can be deduced for angular velocity of a driver and follower 
 connected by a rigid link. In 
 this case we are not concerned 
 with the motion of the inter- 
 mediate connector itself. 
 
 Figs. 50 and 51 show two arms, 
 OA and O'B, free to turn about 
 the fixed centres and 0', and 
 connected by the link AB. The 
 velocity of the point A is rep- Fig. so 
 
 resented by Am, perpendicular to OA. The velocity of B is shown 
 by Bn, perpendicular to O'B; and its magnitude is determined by 
 
 the fact that its component in the direction of AB must equal the 
 component of Am in this same line ; for if these components are not 
 equal, the distance between A and B must change, which is incon- 
 sistent with the conception of a rigid body ; hence, if Pm is 
 assumed, Pn is thereby determined. 
 
46 KINEMATICS OF MAOHINBRY. 
 
 I 
 
 Let o^^ = angular velocity of A about = jrj, • • (1) 
 
 Bn 
 Let (w, = angular velocity of B about 0' = jy^. . (3) 
 
 Drop perpendiculars 0/and O'g upon AB; then triangles OAf 
 and Ams are similar; also O'Bg and Bnr are similar. 
 
 From 04/ and ^wzs, -^ = -r-j = ca,, . . . (3) 
 
 Br Bn 
 From O'Bg and 5wr, -^ = ^^^ = «»„ . . • (4) 
 
 . '^^-^^ ^91-91 U. As- Br\ {b\ 
 
 Produce AB and 00' (if necessary) to intersect in /; then 
 
 ^=^ = ^ (6) 
 
 10 Of GO, ^ ^ 
 
 From this reasoning the following statement is drawn : 
 
 In tiuo arms connected iy an intermediate linh the angular ve- 
 locities of the arms are to each other inversely as the perpendiculars 
 let fall from the fixed centres upon the line of the link j or inversely 
 as the segments into which the line of centres is cut iy the line of the 
 link (both of these lines produced, if necessary) . 
 
 These relations may be seen from direct inspection, by assuming 
 the system to be replaced by the two effective arms. Of and O'g, 
 connected by the link fg. This new system would evidently be 
 equivalent to the original system, for this particiilar position (bnt 
 for no other) ; and as the arms Of and O'g are perpendicular to the 
 link, the linear velocities of / and g are equal ; hence the angular 
 
 OD 0' (1 
 
 velocities of the arms are inversely as the radii, or -- = —^„. This 
 
 CO, Of 
 
 would apply to any phase; but the substituted arms. Of and O'g, 
 would not be of the same lengths for different phases. 
 
TRANSMITTING MOTION IN MACHINES. 
 
 4r 
 
 The relation deduced above may be arrived at also by means of 
 the method of instantaneous axis. In Figs. 53 and 53, cOj and c». 
 
 Fig. 52 
 
 have the same signification as before, and go = angular velocity of 
 the connecting-link about its instant centre. 
 
 A and B are two points in the connector AB ; therefore the 
 motions of these two points completely determine the motion of 
 this link. The velocity of A is Am, perpendicular to OA ; and the 
 velocity of B is Bn, perpendicular to O'B. Therefore Q, at the 
 intersection of OA and O'B, is the instant centre for the link AB 
 in the phase shown (see Art. 19). As the angular velocity equals 
 the linear velocity divided by the radius : 
 
 Am Bn 
 
 
 QA QB 
 
 (7) 
 
 Drop perpendicnlars Qh, Of, and O'g from Q, 0, and 0', 
 respectively, upon the line of the link AB. 
 
 OAf and QAk are similar triangles; also O'Bff and QBk are 
 similar. 
 
 
 Am QA 
 OA ^ Am 
 
 QA_Qk 
 OA ~ Of 
 
 (8) 
 
 GO _ Bn OB 
 w^~ QB^ Bn 
 
 O'B _ q^g 
 QB ~ Q¥ 
 
 (9) 
 
48 
 
 KINEMATICS OF MACHINEBT. 
 
 From equations (8) and (9), 
 
 CD '^c.o~oo~Of^Qk~Of~IO' 
 
 (10) 
 
 This last demonstration thus gives a result identical with equa- 
 tion (6). 
 
 '^ 31. Wrapping-connectors. — This term includes belts, bands, 
 ropes, chains, and all flexible members used to connect a driver and 
 follower, and transmitting force only under tension. The working 
 surfaces may be of any convex cylindrical form ; but concave forms 
 are excluded, as the wrapper would not follow the depressions of 
 such a form, and if used the action would not be smooth and con- 
 tinuous. 
 
 The term cylindrical as used above applies strictly in case of 
 flat bands. In case of round cords, ropes, etc., the contact surface 
 
 is usually grooved to correspond 
 "^ more or less closely to the form of 
 the wrapper, but the motion is in 
 this case equivalent to that which 
 would be obtained by the neutral 
 axis of the connector, wrapping 
 upon an ideal pitch line of the 
 member upon which it is carried 
 (see Fig. 54). The mathematical (and kinematic) wrapper, or the 
 pitch line, is the line xxx. 
 
 In case of flat bands, also, the true pitch surface, and line of 
 action, are at a distance from the physical face of the rigid mem- 
 ber or carrier, equal to about one-half the thickness of the band. 
 For the present purpose the connector will be treated as of no 
 sensible thickness, and the surfaces shown in Figs. 55 to 58 are to 
 be taken as the true pitch surface. The effect of thickness of con- 
 nector will be discussed in a later chapter. 
 
 The band is flexible, but is supposed to be practically inexten- 
 siblej and as it is subjected only to tension, the distance between 
 any two points of the band cannot change. This implies that, 
 whatever actual motions two such points may have, the components 
 
 Fig. 54 
 
TRANSMITTING MOTION IN MACHINES. 
 
 49 
 
 of these motions in the direction O''" the connector must be the same 
 at any instant. 
 
 In Pigs. 55 and 56, a is the driver and i is the follower. Either 
 
 Fig. 56 
 
 of the tangent points of the band and carrier {A or B) is a coinci- 
 dent point of the band and of the member which it meets at that 
 point; Gjj and c», are the angular velocities of the points A and B 
 respectively, and Am and Bn are the corresponding linear veloc- 
 ities. 
 
 Am 
 CIA' 
 
 Bn 
 WB' 
 
 Since A and B are two points in the inextensible band, their 
 ■components of motion in the direction of the connector are equal, 
 or As ^= Br. Drop perpendiculars from and 0' upon the line of 
 the connector ; then OAf and Ams are similar triangles ; also 0' Eg 
 and Bnr are similar. 
 
 A nyj A (J 
 
 00, = angular velocity of A about = jy-j = j-f. 
 
 co^ = angular velocity of B about 0' = -zyj, = -5 
 
 "9' 
 
 . *«. _^v^-^ (as As -Br) 
 
 (1) 
 (2) 
 (3) 
 
50 KINEMATICS OF MACHINERT. 
 
 Prolong the line of centres, 00', and the line of the band, AB^ 
 to meet in / ; then lOf and 10' g are similar triangles. 
 
 ' ' ID '^ Of ~ oD^ *• ^ 
 
 Prom equations (3) and (4) we can formulate the statement : 
 In wrapping-connectors, the angular velocity of the driver is to 
 that of the follower inversely as the perpendiciilars let fall from th& 
 fixed centres upon the line of the connector ; or inversely as the seg- 
 ments into which the line of centres is cut by the line of the connector 
 (both produced if necessary). 
 
 This relation may be shown by the instantaneous centre method 
 also (Figs. 55 and 56). A, as a point in the driver, has a linear 
 Telocity Am and oa, = Am -^ OA. B, as a point in the follower, 
 has a linear velocity Bn, and a?, = Bn -f- O'-S. The acting part 
 of the connector, AB, has an angular velocity about Q of 
 
 _ Am __ B?i 
 °°~QA- QB- 
 
 Let fall Qk perpendicular to AB ; then 0^/and QAk are similar; 
 also O'Bg and QBk are similar. 
 
 w^_Am QA_Q__Qk . 
 
 CO ~ OA ^ Am~ 0A~ Of ' ' ' ' ^^ 
 
 ^_Bn_ qB__o^_qi 
 
 oo^~ QB ^ Bn~ Q'B~ Qk ^^ 
 
 Multiply (5) by (6) : 
 
 ^'^Qk^q9__qg_io^ ,„. 
 
 00^ Of "^ Qk ^ Of ~ 10 ^^> 
 
 This result accords with that of equation (4). 
 
 32. Similarity of Expressions for Angular Velocity Ratio in the 
 Three Modes of Transmission. — By substituting the term line of 
 action for "line of the normal," "line of the link," and "line of 
 wrapping-connector," in the three cases of direct contact, link-con- 
 
TRANSMITTING MOTION IN MACHINES. 
 
 51 
 
 nector, and wrapping-connector, respectively; the following sttite- 
 ment will apply to all of these modes of transmitting motion : 
 
 The angular velocities of the members are inversely as the per- 
 pendiculars let fall from their fixed centres upon the line of action ; 
 or inversely as the segments into which the line of action cuts the 
 line of centres. 
 
 There are special cases in which the preceding theorems are not 
 available, because the expressions become indeterminate; though 
 these cases can be reconciled to the general statement. For ex- 
 ample: see the direct-contact mechanism of Pig. 57, or the link- 
 work of Fig. 58. In these mechanisms the centre about which B 
 
 o'at« 
 
 Fig, 57 
 
 Fig. 58 
 
 rotates is at co , hence the perpendicular from it upon the normal 
 = 00 (also the segment from its centre to the line of action = oo ) ; 
 
 and by the theorem of Art. 29 ; ^' = -r-. = =» . This is consistent^ 
 
 as the angular velocity of the follower B, is o ; hence that of the- 
 driver (A) is infinitely greater than that of the follower; but the 
 result does not enable us to derive the linear velocity of the follower,, 
 for this equals the angular velocity multiplied by the radius, = o- 
 X 00 , an indeterminate expression. The linear velocity of the fol- 
 lower is easily found by other means, however, as its normal com- 
 ponent must equal that of the linear velocity of the driver, and th& 
 direction of the follower's motion is known. From this data, the 
 linear velocity of the follower is derived (see Art. 17). A similar 
 course of reasoning applies to the mechanisms shown in Fig. 5R. . 
 
■52 
 
 KINEMATICS OF MACIITNERY. 
 
 In the linkwork shown by Fig. 59 the follower is not under 
 •■control of the driver at the particular phase there represented. As 
 
 '^4r- 
 
 i 
 
 ■^ 
 
 Fig. 59 
 
 b'-' 
 
 n 
 
 
 Fig. 60 
 
 A reaches the position shown (at either dead centre), it has no 
 component of motion in the line of the link AB, hence there is no 
 component compelling motion of B. As A passes this position, B 
 might be moved in either of the two directions indicated by Bn or 
 Bn'. If the shaft about which B rotates is provided with a fly- 
 wheel, or similar device, its direction of motion may be maintained, 
 and as soon as the dead centre is passed, A again exerts an influence 
 over its motion. In the case of Pig. 60 B comes to rest as A passes 
 the dead centre, Bn being zero at this phase. 
 
 33. Directional Relation. — It will be seen by reference to Figs. 
 48, 50, and 55 that the driver and follower both rotate in the same 
 direction; that is, iotJi members are turning in the right-handed, 
 clockwise, or negative direction as angles are usually reckoned ; or 
 iotJi rotate in the reverse direction. The follower is converted into 
 the driver when such reversal takes place in the case of direct con- 
 tact or of wrapping-connectors, though this is not necessarily the 
 case with link-connectors, or with direct contact if a double-sided 
 or slotted member, like Fig. 67 is used. It will be noticed further 
 that in the cases shown in Figs. 48, 50, and 55, the two fixed cen- 
 tres lie on the same side of the line of action. 
 
 In Figs. 49, 51, and 56, on the other hand, the fixed centres 
 lie on opposite sides of the line of action, and the two members ro- 
 tate in opposite directions; if one member has right-handed rota- 
 tion, the other has left-handed rotation, and vice versa. From this 
 .observation of all of these general cases, the following statement in 
 
TRANSMITTING MOTION IN MACHINES. 63' 
 
 regard to the Directional Relation is quite evident : In any of the 
 three ordinary modes of transmission of motion the directions of rota- 
 tioti of the driver and follotuer are the same if the fixed centres of both 
 lie on the same side of the line of action ; and the directions are 
 opposite if these centres lie on opposite sides of the line of action. 
 
 34. Condition of Constant Angular Velocity Ratio. — It has been 
 
 shown that with any of the three common methods of transmitting 
 
 motion the angular velocities of the members are inversely as thft 
 
 segments into which the line of centres is cut by the line of action. 
 
 Thus in any figure from 48 to 56 (except Pig. 54) ca, : co^ : : 10' : 10. 
 
 00 10' 
 If the angular velocity ratio is constant, ~ = ^p^- = a constant, 
 ^ •' w^ 10 ' 
 
 and as 00' — the distance between the fixed centres — is a constant, 
 
 / must be a fixed point in this line (or its extension) in order that 
 
 the above_ condition be realized. Therefore it may be stated that r 
 
 The Condition of Constant Angular Velocity Ratio is that the line- 
 
 of action must always cut the line of centres {produced if necessary) 
 
 in a fixed point. 
 
 This condition is fulfilled by an infinite number of pairs of 
 curves which may be used as the outlines of the acting faces of di- 
 rect-contact members. It will appear later that the proposition just 
 stated is of fundamental importance in the theory of teeth of gears. 
 
 The condition of constant velocity ratio is fulfilled in the case of 
 wrapping-connectors when the driver and follower have faces which . 
 are right cylinders with the axes of the cylinders as the axes of ' 
 revolution ; for example, in the case of ordinary pulleys with 
 crossed or open belts. 
 
 Constant velocity ratio is secured with link transmission when 
 the driving and the driven arms are equal, and the length of the- 
 connecting-link is equal to the distance between the fixed centres ; 
 and parallel to the line of centres, as in the parallel rods of loco- 
 tives. 
 
 35. Nature of Rolling and Sliding. — When two pieces act to- 
 gether by direct contact they may roll upon each other, they may 
 slide upon each other, or they may move with a combined rolling 
 and sliding action. 
 
 Fig. 61 shows two such members, in which p is the contact 
 
54: KINEMATICS OF MACHINEHT. 
 
 point at the phase shown. If r and s are any two points which 
 meet as the action continues (becoming coincident contact points), 
 the arcs pr and ps must be equal if the action is pure rolling. If 
 
 for any increment of motion the corresponding arcs of action of the 
 two curves are not equal, there must be some sliding between them. 
 In pure rolling action no one point of either body comes in contact 
 with two successive points of the other. 
 
 If a point of one of the bodies comes in contact with all suc- 
 cessive points of the acting surface of the other (within the limits 
 of the path), the action is purely sliding; for example, the piston 
 in the cylinder of an engine. 
 
 In some cases, as in many cams and all gear-teeth, the action 
 is mixed sliding and rolling. The sliding action must occur along 
 the common tangent at the point of contact of the two surfaces. 
 
 36. Rate of Sliding and Condition of Pure Rolling. — It has 
 been shown that in direct-contact mechanisms the normal compo- 
 nents of the motions of the points of contact must be equal; but 
 the tangential components may have any values, either in the same 
 or in opposite directions. When the tangential components of the 
 motions of the contact points are in the same direction and equal 
 there is no sliding, and the two motions are identical as both com- 
 jionents are equal. 
 
 The rate of sliding is the diiference of the tangential compo- 
 nents if they are in the same direction, or their sum if they are in 
 opposite directions ; or: The rate of sliding is the algelraic differ- 
 ence of the tangential components. 
 
TBAN8MITTING MOTION IN MACjHINES. 
 
 55 
 
 In direct-contact mechanisms the normal components of the 
 Telocities of the points of contact are always equal, and the tangen- 
 tial components are also equal when the action is pure rolling. 
 IFigs. 62 and 63 illustrate this condition, and the two motions, Pm 
 
 Fig. 63 B 
 
 IRl"' 
 
 and Pw, are identical. But P, as a point in A, moves at right 
 angles to OP; and, as a point in B, it moves at right angles to 
 O'P; therefore, when Pm coincides with Pn, OP and O'P are 
 iDoth perpendiculars to the same line at the same point and they 
 must therefore lie in one right line. In order that this may occur, 
 P must lie in the line of centres j or: The condition of pure roll- 
 ing is that the point of contact shall always lie in the line of 
 centres. 
 
 Any pair of direct-contact pieces bounded by curves which 
 satisfy the condition just stated act upon each other with a pure 
 rolling action; and any departure of the contact point from the 
 line of centres is accompanied by sliding action. There are many 
 sets of curves which may be employed to thus transmit motion by 
 ■direct contact and with pure rolling action, among which may be 
 mentioned: tangent circles (or circular arcs) rotating about their 
 ■centres; pairs of equal ellipses each rotating about one of its, foci 
 with a distance between the fixed centres equal to the common 
 major axis; and pairs of similar logarithmic spirals, rotating about 
 iheir foci. 
 
 As the common normal to two direct-contact members passes 
 ihrough the point of contact, and as this point always lies in the 
 line of centres if the action is pure rolling, the common normal 
 
56 
 
 KINEMATICS OF MACHINERY. 
 
 cuts the line of centres in the contact point when pure rolling 
 occurs. The angular velocity ratio is inversely as the segments- 
 into which the line of centres is divided by the normal; or in- 
 versely as the perpendiculars let fall from the fixed centres upon 
 the normal. In pure rolling these segments are the contact radii 
 themselves [OP and O'P of Figs. 63 and 63), and therefore in 
 such cases the angular velocities are inversely as the contact radii. 
 
 Drop perpendiculars. Oh and 7, from the fixed centres (Figs. 
 62 and 63), upon the common tangent, TT', and it will be seen 
 that O/y and OPTc are similar triangles; also that O'Pg and O'Pl 
 
 are similar. 
 
 O'l O'g CD, ^-, , , .^ 
 
 -r = -pTT^ — — ~ = tlie angular velocity ratio ot 
 I- Of '•-■' ^ •' 
 
 Ok Uf w^ 
 
 the members. We may then use, if convenient, the following 
 relation : The angular velocity ratio, iji direct-contact meclianisms 
 having pure rolling action is mverselg as the perpendiculars from 
 the fixed centres to the common tangent. 
 
 In the circular-arc forms (Figs. 64 and 65) the perpendiculars 
 from the centres to the tangent are the contact radii; thus the 
 
 well-known relation for.tangential wheels of circular section, — that 
 the angular velocities are inversely as the radii of the circles, — is 
 seen to agree with the more general relations deduced in this 
 article. 
 
 37. Constant-velocity Ratio and Pure Rolling Combinecl. 
 
 As stated in the preceding two articles, there are many paira of 
 curves which will satisfy either the condition of constant- velocity 
 
TRANSMITTING MOTION IN MACEINBS.' ST 
 
 ratio, or of pure rolling. There is but one class of curves, how- 
 ever, — viz., circular arcs rotating about their centres, — which can 
 have at the same time loth constant-velocity ratio and pure roll- 
 ing (see Figs. 64 and 65). For constant-velocity ratio, the normal 
 must cut the line of centres in a fixed point; for pure rolling, the 
 contact point (through which the normal passes) must lie in the 
 line of centres. If both of these requirements are met at the same 
 time the contact point must be a fixed point in the line of centres; 
 hence the contact radii must be constant; and therefore the out- 
 lines of the members are circular arcs. 
 
 -^ 38. Positive Driving. — Circular-arc members (right cylinders), 
 as shown in Figs. 64 and 65, do not transmit motion positively. 
 Actual physical bodies of the corresponding forms can transmit 
 motion from one to the other only through frictional action. In 
 the absence of friction, with such forms, no motion could be trans- 
 mitted against any resistance; with friction a limited resistance can 
 be overcome. There is no assurance that more or less slipping 
 may not occur, and if this does take place the velocity ratio be- 
 comes both variable and uncertain.* 
 
 In such forms as those shown in Figs. 62 and 63, on the other 
 hand, motion of the driver involves a positive and definite motion 
 of the follower. It is now in order to determine the conditions 
 necessary to insure positive or compulsory driving. It is some- 
 times stated that positive driving is only produced when the contact 
 radius of the driver increases as the action proceeds ; thus in Fig. 
 61 ^-1 can only drive B positively when Op is greater than any pre- 
 ceding contact radius, as Or, and less than any succeeding radius 
 as Or'. "While this is the case with many forms, it is not a general 
 
 * The tangential .component of the motion of eitlier the driving or driven 
 point represents its rate of sliding along the tangent. If the tangential com- 
 ponents of the motions of both these points are equal and in the same direction 
 there is no sliding between them. 
 
 V^ith. perfectly smooth suTtacos, one of the members could not move the other 
 against the smallest resistance. In the practical cases where motion is trans- 
 mitted by frictional action, the effectiveness increases as the departure from 
 ideal smooth cylindrical surfaces becomes greater. Perfect cylinders, if such 
 were possible, would not be of the slightest use in such cases. 
 
58 
 
 KINEMATIGS OF MAOHINERY. 
 
 requirement for positive driving. Figs. 66 and 67 show mechan- 
 isms in which A is the driver and B is the follower. It will be seen 
 that A can rotate indefinitely, causing continuous rotation of B 
 (though not with a uniform velocity ratio between A and B), and 
 
 Fig. 66 
 
 Fig. 67 
 
 at the end of each rotation the two members will return to the same 
 relative positions they had at the start. It is evident then that the 
 contact radius of the driver cannot increase indefinitely. It will 
 be seen, also, that B may be the driver, and that a similar remark 
 will apply in this case.* 
 
 Referring to Figs. 64, 65, and 68, it is seen that the motion, 
 Pm, of the contact point of the driver lies in the direction of the 
 common tangent, TT' ; hence the normal component of this motion 
 
 is zero. It is only the normal com- 
 ponent of the driving point's motion 
 which tends to impart positive mo- 
 tion to the follower ; and in the cases 
 of Figs. 64, 65, and 68, where the 
 motion of this point is wholly tan- 
 gential and the normal component 
 is zero, there is no tendency to pro- 
 duce positive driving. In other 
 words, positive driving is assured 
 only when the driving contact point has a component of motion in 
 
 * The presence of the intermediate roll or block is not essential, and the 
 above statement would be equally true if A were simply provided with a, pin 
 engaging the follower. 
 
 Fig. 68 
 
TRANSMITTING MOTION IN MACHINES. 59 
 
 the direction of the normal, and as the contact point moves per- 
 pendicularly to the contact radius, there can be no such normal 
 component of motion when this radius is perpendicular to the com- 
 mon tangent ; or, what is the same thing, when this radius coin- 
 cides with the common normal. Positive driving cannot occur 
 then if the common normal passes through the centre about which 
 the driver rotates. The contact radius may coincide with the tan- 
 gent, and in fact this is a very favorable position, as the motion of 
 the contact point is then entirely in the direction of the normal, 
 and there is no tendency to slide. If the common normal passes 
 through the fixed centre about which the follower rotates the driver 
 cannot impart positive motion to the follower; for in this position 
 the normal component of the motion of the driven point is per- 
 pendicular to the path, and any motion in this direction is prohib- 
 ited by the nature of constrained motion. A force directed toward 
 the centre does not tend to produce rotation, but only to exert 
 pressure against the bearings. 
 
 It is thus seen that positive driving cannot occur if the common 
 normal passes through either of the fixed centres. 
 
 If the normal passes through the centre of the driver only, the 
 driver can move, but motion is not transmitted to the follower. 
 If the normal passes through the centre of the follower, the driver 
 is locked, forjts motion can have no normal component; but the 
 follower may still move if under other influences, such as the action 
 of a fly-wheel. If the common normal passes through loth fixed 
 centres, as in Figs. 64 and 65, the motions of both contact points 
 are tangential and wholly independent, except for the frictional 
 action. 
 
 In conclusion the following statement may be formulated : 
 The condition of positive driving is that the common normal shall 
 not pass through the fixed centre of either the driver or the follower. 
 ~~' 39. Inversion of Mechanisms.— It was explained in Art. 5 that 
 one body may have at any time distinct motions relative to different 
 bodies, and that it is sometimes convenient to refer the motion of 
 a member to some other part than the fixed frame. Take, for 
 example, the crank and connecting-rod mechanism of the ordinary 
 reciprocating-engine, as shown in Figs. 27 and 69. There are 
 
60 
 
 KINEMATICS OF MACHINERY. 
 
 four members of this mechanism : the crank, connecting-rod, cross- 
 head and iDiston (the last two are kinematically one piece), and the 
 frame (including the cylinder).* 
 
 In Fig. 69 these parts are designated by the letters a, i, c, and 
 d respectively, and the shading of d is used to indicate that it is 
 
 the stationary member. The 
 crank, a, rotates about the centre 
 O^a relative to the frame, d, and 
 this centre, O^d-, is the instant 
 centre (also a permanent centre) 
 for the motion of a relative to d. 
 The frame, d, also rotates rela- 
 tive, to the crank, a, about this 
 same centre; for if we imagine 
 the crank to be the fixed mem- 
 ber, as in Fig. 70, d actually does 
 rotate about this centre as the 
 mechanism operates; but the change in the relative positions of the 
 members is simply that due to the mode of constrainment, which- 
 ever member is fixed, and we have not changed the form of the 
 mechanism in any way; hence the relative motions of the parts are 
 the same under both conditions. 
 
 The members in Fig. 70 are identical with those of Fig. 69, 
 and their connections with each other are the same as before. If 
 the member h, the original connecting-rod, be made the fixed mem- 
 ber, as in Fig. 71, a and d are both moving members; but they 
 still rotate, relatively, about Oad- This mechanism, as shown in 
 Fig. 71, is that of the oscillating steam-engine, in which a corre- 
 sponds to the crank, l to the frame, c to the cylinder, and d to the 
 piston-rod and piston. Fig. 72 represents another condition of 
 this same mechanism, in which c, the original crosshead, is the 
 fixed member. Under any of these four conditions the relative 
 
 * The notation used in the following; discussion is this : small letters, a, b, 
 c, etc., are used to designate the different members ; is used for all centres 
 (instant or permanent), and the subscripts of indicate the members which 
 rotate relatively about it. Thus Ono indicates that the member a rotates rela- 
 tive to c (or c relative to a) about the point designated as Oac. 
 
TMAJSSMIITIJfG MOIION IN MACHINES. 
 
 61 
 
 motion of a to d (or of d to a) is a simple rotation about the centre 
 Oad- In a similar way, the relative motion of a and J is a rotation 
 about the centre Oab', that of I and c is a rotation (or oscillation) 
 
 Fig. 71 
 
 Fig. 72 
 
 about Oic ; that of c and (? is a translation parallel to the centre 
 line of d, or a rotation about a centre, 0^^, lying at infinity and in 
 such a line as NN. 
 
 It is evident from the above illustration that apparently very 
 different mechanisms may be essentially the same thing under dif- 
 ferent conditions. This was referred to in speaking of the classifi- 
 cation of the parts of machines in Art. 24. 
 
 Such changes in the condition of a mechanism as are illustrated 
 in Figs. 69, 70, 71, and 72, and which are effected by making dif- 
 ferent members correspond to the stationary part, or frame, are 
 called by Professor Reuleaux The Inversion of Mechanisms. 
 
 Other examples of inversion will be given in a later part of this 
 work. 
 
 40. Relative Motion between Different Members of a Mechan- 
 ism. — The relative motions of the adjacent members of a mechan- 
 ism ai-e usually comparatively simple; thus in the mechanism of 
 Fig. 73 the relative motions of a to b, b to c, c to d, and a to d are 
 simply rotations or oscillations about permanent centres. The rela- 
 tive motions, in Fig. 69, of the corresponding adjacent members 
 were traced in the preceding article. 
 
 In any mechanism each link (member) has a distinct motion 
 relative to each of the other members. In four-piece mechanisms, 
 
62 KINEMATICS OF MACHINERY. 
 
 as those of Figs. 69 and 73, there are six distinct relative motions, 
 viz. : a to b, i to c,c to d, a to d, a to c, and i to d. Each of these 
 motions is equivalent to rotation or oscillation about a centre (per- 
 manent or instant). Pour of these are permanent centres in the 
 mechanisms referred to above, viz. : Oai, 0,,^, O^d, aiidO„s.* 
 ~ In Fig. 73, Ot,c rotates relative to d in an arc of finite radius. 
 In Fig. 69, the motion of 0^^ relative to d is equivalent to rotation 
 about the centre O,.^, in an arc of infinite radius. If it were pos- 
 sible to supply a link of infinite radius connecting the point O^d of 
 
 Fig. 73 
 
 d with the point O^c the mechanism of Fig. 69 would be similar 
 in character to that of Fig. 73 ; or the former may be considered 
 as a limiting form of the latter. The practical mechanism of 
 Fig. 69 is the exact kinematic equivalent of such an imaginary 
 mechanism. 
 
 The relative motions of the opposite links, a and c, and i and d, 
 of Figs. 69 and 73, are not so evident as are the motions of th& 
 adjacent members; but the instant centres for these motions are- 
 readily located from the principles of Art. 19. In the first place 
 it is to be noted that the instant centre for two members is a. 
 common or coincident point of both, for it is a point with regard 
 to which neither of them has any motion. If this point lies 
 
 * In the mecbaDism of Fig. 69 the centre Oad is at infinity. While it is not 
 an actual physical pin like the other permanent centres, still it is properly a 
 permanent centre rather than an instant centre, because it is equivalent to a^ 
 fixed centre of a link of infinite length. 
 
TRANSMITTING MOTION IN MACHINES. 63 
 
 outside of either actual body, this body may be imagined as. 
 extended to include this centre, for a body may haye rigid con- 
 nection with any point relative to which it has no motion, as stated 
 in Art, 5. 
 
 It is to be borne in mind, then, that Oab is a point in both a 
 and h; ()„„ is a point in both a and c, etc. This enables us to locate 
 the instant centres for the opposite links. In Fig. 73 0„6, as a 
 point in a rotating relative to d, must move in a line perpendicular 
 to the line joining the points Oa<j-0„.!,; hence its motion is equiva- 
 lent to a rotation about any point in this line or its extension (see 
 Art. 19). Likewise, the motion of Oi,o relative to d is equivalent 
 to a rotation about any point in the line Oea-Oto or its extension, 
 and therefore the instant centre for both of the points Oai, and 0^,^^ 
 relative to di is at the intersection of the lines Oad-Oat and 0^-0^^, 
 or at the point marked Oi,^. But Oab and 0^^ are two points in 
 the link 5, and, as such, their motions determine the plane motion 
 of I relative to d; therefore the point O^a is the instant centre for 
 i relative to d. In a similar way, all points of l rotate relative to a 
 about the centre Oab, and all points of d rotate relative to a about 
 Oad- The points Oj,, and 0^^ are points in I and d, respectively; 
 hence they move, relative to a, perpendicularly to the lines Obc-Oab 
 and OpijrOad respectively, and the intersection of these lines, or 
 Oac, is their common instant centre relative to a. But Obe and 
 Ocd are two points in the link c; therefore the point Oac is the 
 instant centre for c relative to a. We have thus located all of the 
 instant centres for the mechanism of Pig. 73. 
 
 Pour of the centres for the mechanism of Pig. 69 have been 
 located, viz.: the permanent centres Oad, Oab, Obc and 0„ci (the 
 last at infinity). The centres for the two pairs of opposite links, 
 b and d, and a and c, are yet to be found. The former, 0^ is 
 readily found; for Og^ (a point in S) moves perpendicularly to 
 the centre line of a, and, (\, (also a point in h) moves perpendicularly 
 to j^W; therefore the intersection of these lines, or Obd, is the 
 required instant centre for the motion of b relative to d. 
 
 The reasoning by which the centre for the relative motion of a 
 "to c is found is somewhat more involved. The point (?!,„ is a point 
 common to b and c. All points in b rotate relative to a about the 
 
64: KINEMATICS OF MACSINETIT. 
 
 centre Oab ', therefore O^c as a point in h rotates about this point, 
 •or it moves, relative to a, perpendicularly to the centre line of b 
 (tho line Ob^—Oab). The point Ocd (at infinity) is a point common 
 to c and d. As a point in d it rotates about Oad relative to a, 
 moving perpendicularly to a vertical line through Oad, or to N' W. 
 One point of c (Oic) moves perpendicularly to 0,,c-Oai„ and another 
 point of c (Ocd) moves perpendicularly to N'N'; hence the instant 
 centre for c relative to a is at the intersection of these two lines, or 
 at the point 0^. 
 
 By considering c, instead of a, as the stationary member, the 
 same result may be reached rather more easily. 
 
 It is possible to locate all of the instant centres of a mechanism 
 of four members by the principles already given; but with higher 
 numbers of members this cannot always be done, and a very im- 
 portant theorem given by Professor Kennedy affords a ready solu- 
 tion in these more difficult cases. This theorem is often advan- 
 tageous even in four-link mechanisms, and by its aid the rather 
 tedious reasoning employed above in finding Oac for th;s mechanism 
 of rig. 69 can be avoided. 
 
 The statement of this theorem, as given by Professor Kennedy, 
 is: " If any three hodies a, h, and c have plajie motion, their vir- 
 tual [instant] centres Oat, O^c, and 0„„ are three points upon one 
 straight line." 
 
 This theorem applies to any three bodies having plane motion, 
 whether they be members of the same mechanism or not. Two of 
 the three centres being known, or assumed, .the following demon- 
 stration proves that no point lying outside of the line connecting 
 the known centres can be the required third centre; hence this 
 third centre must lie in the line connecting the other two, as stated 
 in the theroem. 
 
 In Pig. 74, let a, b, and c be any three bodies moving in a 
 plane, members of a single mechanism as indicated by the broken 
 lines, or entirely independent bodies. 
 
 Suppose that a rotates relative to b about the centre O^^, and that 
 c rotates relative to b, about the centre Oio- Then 0„b is a point 
 common to a and b, and 0[,o is a point common to b and c. As- 
 sume that such a point as 0' is the instant centre for the relative 
 
TRANSMITTING MOTION IN MACHINES. 65 
 
 motion of a and c; then this point is a common point of a and c. 
 All points in a must rotate relative to b about Oat, and all points 
 in c ninst rotate relative to b about 
 Oj„. If 0' is a point common to 
 ■(I and c, it must have as a point in 
 ■a such a motion relative to b as 
 T'^^, perpendicular to Oab-0' ; and 
 as a point in c it has such a motion, 
 relative to b, as V^, perpen- 
 dicular to Otic-0'. A point can 
 liave but one motion relative to a '^' ^'^ 
 
 giYcn body at any time, and therefore F^ and V^ must coincide if 
 0' is a common point of a and c; hutF„ and F<, are perpendicular 
 to the radii O^b-O' and 0^0-0', respectively; hence, if F^ and V^ 
 coincide, these radii must lie in one straight line, and the three 
 centres Oab, Obc and Oac are in one straight line, viz. : the line 
 ■connecting the given centres 0„j and 0^^. It is thus seen that 
 the point 0' cannot be the centre required, unless it does lie in 
 such a line. 
 
 This theorem does not locate the centre ()„„ definitely; for it 
 may be any place along the line of O^b-O „, between these centres, 
 •or beyond either of them. This is as it should be, for in the ar- 
 rangement of Fig. 74 there is no prescribed connection between 
 a and c, and their relative motion is therefore not definitely con- 
 strained.* 
 
 If a fourth member, d, which will constrain a relative to c — 
 such as a link connecting the free ends of a and c — be introduced, 
 another combination of three members (as a, c, and d) may be 
 
 * It is to be noticed that the theorem discussed above has reference to 
 three members, and that these three members involve three instant centres ; 
 any member has a centre with reference to each of tlie other members. In 
 a combination of three bodies every letter which stands for one body is used 
 twice as a subscript to 0. If two of the three centres are given their symbols 
 ■will have one common letter in their subscripts, and the third (required) centre 
 ■will have for a subscript the two odd letters. Thus if Oab iiud Obc are the 
 given centres, Oac is the third. This is a convenient aid in applying the above 
 theorem to a mechanism. 
 
66 
 
 KINEMATICS OF MACHINEBT. 
 
 taken, in which Oaa and O^a, ^I's known, and, by the theorem, the 
 other centre for this combination {Oa^ will lie in the line of the 
 known centres. In this constrained four-link mechanism there are 
 
 ,to., 
 
 
 a 
 
 b 
 
 c 
 
 d 
 
 e 
 
 f 
 
 a 
 
 
 
 
 
 
 
 b 
 
 
 A^v 
 
 
 
 
 
 c 
 d 
 
 ■Z' 
 
 
 
 
 
 
 ;/,: 
 
 
 
 
 
 e 
 
 
 g^ 
 
 ^ 
 
 
 
 
 /I 
 
 o>. 
 
 two lines, each of which contain Oac, (viz. : Oab-Oic, and Oad-Ocd) ; 
 hence ()„<, is at their intersection, or its position is definitely deter- 
 mined. 
 
 By referring to Figs. 69 and 73 it will be seen that the loca- 
 tions of the centres, as already determined, agree with the state- 
 ment of the theorem. Thus, as to the links a, i, and c, the cen- 
 tres (9„(,, Obc, and Oac lie i^i one line ; also, as to a, c, and d, Oac, 
 Gad, and Oca lie in one line, and Oac lies at the intersection of these 
 two lines. 
 
 As an illustration of the application of the above theorem to 
 (nore than four members, the mechanism of the Atkinson Cycle 
 
TRANSMITTING MOTION IN MACHINES. 
 
 6T 
 
 Gas Engine, indicated in Fig. 75, will be taken.* In this 
 machine there are six members: — the stationary frame (including 
 the cylinder), / ; the crank, a ; the link, c ; the piston-rod, d ; 
 the compound-link connecting-rod, S, pivoted to a, c, and d; 
 
 and the piston, e. There are -^— ^r = — - — = 15 centres in this 
 
 mechanism. The permanent centres 0„f, Oab, 0^^, O^d, O^f Oae, 
 and 0^, (the last at infinity in line WI^) are readily located. The 
 other centres can be found by application of the preceding prin- 
 ciples, f 
 
 The following scheme suggests the solution of this problem : 
 
 Centre Kequired. 
 
 Lies at Intersection of the Lines. 
 
 Oao 
 
 Obc - Oai and 0,f - O^f 
 
 0,f 
 
 Oab - Oaf " Oba - 0,f 
 
 0.e 
 
 Oae-Oba " 0,f-Obf 
 
 0„ 
 
 Obe - Obc " 0,f - 0,f 
 
 
 
 0.e - Oao " 0,f - Oaf 
 
 Oaa 
 
 Oab - Om " 0^ - Oae 
 
 O^a 
 
 Oba-Obo ' 0,,,-ae 
 
 Oaf 
 
 Obf - Oba " Oaa - Oaf 
 
 The method of instant centres will be frequently used in the 
 later part of this work, especially in treating linkwork ; but it may 
 be well to give an illustration of its use at the present place. It is 
 to be remembered that the linear velocity of a point which is mov- 
 ing relative to any body is proportional to its distance from the 
 centre about which it rotates relative to that body. In Fig. 69, 
 for example, the body a rotates relative to the stationary member. 
 
 Ttfji ■ 
 
 — instant centres, some 
 
 ■■■ In a meclianism of n members, there are 
 
 of which are also permanent centres. 
 
 \ A diagram such as is shown in connection with Pig. 75 is convenient in 
 this work. The unshaded spaces indicate the centres to be located, and th« 
 memory is aided by checking ofE, in the proper place, each centre as it is found. 
 
t3S KINEMATICS OF MACEINERT. 
 
 d (the frame), about Oad- If the linear velocity, v,, of 0„6 is 
 known, the linear velocity, v^, of the crosshead (piston) is readily 
 found by the principles of instant centres. The point Oj<, is com- 
 mon to the connecting-rod, b, and the crosshead, c; while the point 
 Oab is common to the crank, a, and the connecting-rod, i. The 
 instant centre of i relative to d is O^a ; then, as all points of b must 
 have the same angular velocity about Of,ii their linear velocities are 
 proportional to their distances from this centre ; hence 
 
 «, : v, : : 0^^ C^^ : 0„,, 0^,,. 
 
 If w, be laid off from O^i, toward O^, along the line connecting 
 these points, and then a line, m?i, parallel to the connecting-rod, 
 be drawn till it cuts the normal NjV, the length on this normal 
 from Cjc to ?n equals v^, from the above proportion. As the motion of 
 •c relative to cZ is a translation, all points of c have the same velocity 
 relative to d ; hence w, is the velocity of the crosshead, or piston, 
 relative to the frame, or cylinder. 
 
 In an engine the crank rotates about the shaft with a velocity 
 which is usually taken as uniform ; while the velocity of the cross- 
 liead (or piston) is variable. The velocity of the piston can be 
 found for any phase by laying oft' the crank-pin velocity .along the 
 extension of the crank, drawing a line (as mn. Fig. 69) parallel to 
 the connecting-rod till it cuts the normal (i\"iV') through the cross- 
 head pin. 
 
 A modification of the preceding construction is often even more 
 convenient. Lay off the line AW (Fig. 69) through the centre of 
 of the shaft (0^^) and perpendicular to the line of the piston travel. 
 The connecting-rod (extended if necessary) cuts A'^'A^' in the point 
 Oa„ and, from similar triangles, 
 
 From the relations just deduced it is seen that the ratio of the 
 crank velocity to the piston velocity equals the ratio of the length 
 of the crank to the length of the perpendicular (to the line of piston 
 
TRANSMITTING MOTION IN MACHIISE8. 
 
 691 
 
 travel) erected from the centre of the shaft and terminating in the 
 line of the connecting-rod — the latter prolonged if necessary. 
 
 Fig. 76(a) 
 
 41. Velocity Diagrams.— It has been shown in the preceding 
 article how the method of instant centres can be used to determine 
 the linear velocity of one point from the known velocity of another 
 point. It is often desirable to represent, graphically, the velocities 
 of a point at various phases of a mechanism, and this is done con- 
 veniently by velocity diagrams. Pig. 76 shows the mechanism of 
 the reciprocating engine in outline. G is the crank-pin, c is the 
 crosshead-pin, Q is the centre of the shaft. The crosshead moves, 
 from to 9 and back again to during one complete rotation of 
 the crank. The simultaneous positions of crosshead-pin and crank- 
 pin are indicated respectively by 0, 1, 2, 3, etc., and 0', 1', 2', 3', 
 etc. As shown in the preceding article, if the linear velocity of the 
 crank-pin is represented by the length' of the crank, r, the velocity 
 of the crosshead for any phase is represented by the segment, s, of 
 the line N-N, which lies between Q and the line of the connecting 
 rod, 0-c. If the segment, s, is found for each of the crosshetui 
 positions from to 9, the corresponding lengths of s may be erected 
 ai ordinates to 0-9 at the corresponding crosshead positions. A 
 
70 
 
 KINEMATICS OF MAGEINERT. 
 
 ■curve passing through the upper ends of these ordinates gives a 
 velocity diagram of the point c with the path, 0-9, as a base.* 
 
 If a sufficient number of such ordinates have been determined 
 this diagram gives quite accurately the velocity of c for intermediate 
 positions. 
 
 Fig. 78 shows a method of constructing a velocity diagram upon 
 
 7i 8] 5i 4, 
 
 Fig. 78 
 
 Cm 
 
 a curved path as a base. The driving arm, or crank, a, imparts, 
 by its rotation, a reciprocating motion to the arm c in the arc 1-9. 
 The point 0^ occupies the positions 0, 1, 2, etc., when the point 
 Oab is at the corresponding points 0', 1', 2', etc. If 2'-2/ is laid 
 off equal to the linear velocity of Oat, upon the extension of the line 
 
 * The velocity diagram of Fig. 76 is called a ■velocity-space diagram, as the 
 coordinates represent velocity and space passed over. It is sometimes con- 
 venient to make one of the coordinates represent velocity and the other 
 represent time, when the diagram is called a velocity -time diagram. 
 
TRANSMITTING MOTION IN MACHINES. 71 
 
 of a, and 3,'-2, is drawn parallel to I, the segment of the extension 
 of c cut ofE by this parallel equals the linear velocity of the point 
 Obo. This is proven by reference to the instant centre of h and d 
 ( 0(,a) ; for the linear velocity of 0„6 relative to d is to the velocity 
 of Ofto as Oftjj-Oab is to Om-O^o (the linear velocities of two points 
 in I relative to d are proportional to their radii from 0^). But 
 2'-2/ (the velocity of 0^,,) is to 2-3, as OtgrOab is to OtcrOta, and 
 therefore 2-2, is the velocity of O^c. By a similar construction for 
 other phases, the corresponding velocities of the point O^c may be 
 obtained. If these velocities of the driven point are laid ofE as 
 Tadial ordinates at the corresponding points in its path, the curve 
 1,-2,-3,, etc., may be drawn, and it is the velocity diagram of 0„o 
 on its path as a- base. 
 
 If the motion of the driving point, Oab, is uniform, its velocity 
 diagram is a circle concentric with its path, as drawn in Fig. 78 ; 
 but the method applies equally well if the driving point has a 
 variable velocity. A velocity, diagram with rectangular coordinates 
 Tnay be constructed from the one just determined by rectifying the 
 path of Oftc, 1-2, etc., and erecting, at the various points, parallel 
 ordinates of lengths found as above. This derived velocity diagram 
 is shown in Fig. 78 a, but it is seldom necessary to construct it. 
 
 If on various positions of the crank (Fig. 76) the corresponding 
 velocities of the follower, c, are laid ofE radially from Q, as Q-1", 
 Q-2'', etc., and a curve is then drawn through 1", 2", 3", etc., a 
 Polar Velocity Diagram of the motion of c is obtained. This is 
 sometimes preferred to the rectangular diagram on the path of the 
 follower. 
 
 In the illustrations of Arts. 39, 40, and 41, linkwork mech- 
 anisms have been taken, as the methods developed in these articles 
 a.re especially useful in the treatment of this class; but the deduc- 
 tions are also applicable to other mechanisms. 
 
 42. Acceleration Diagram. — If the velocity-space or the veloc- 
 ity-time diagram of a body that moves with an accelerated motion 
 can be drawn, the value of the acceleration at any point can be 
 found graphically. 
 
 When the same length of ordinates and abscissas, respectively, 
 lepresents an equal number of velocity and space units on a veloc- 
 
72 KINEMATICS OF MACHINEBT. 
 
 ity-space diagram, or of velocity and time units on a velocity-time 
 diagram, the scales will be referred to as similar scales. 
 
 Thus a velocity-space diagram where 1 in. of ordinate repre- 
 sents 10 ft. per sec, and 1 in. abscissa represents 10 ft. of length, 
 will have similar scales. 
 
 Eeferring to Pig. 77 : 
 Let V — velocity of a body when it is at any point M, = PM to. 
 scale. 
 p = acceleration of the body when it is at same point M. 
 s = space passed over from rest, — OM. 
 t — time from rest. 
 PR = the tangent and PN the normal to the curve at point P- 
 9 — taWent o-f the angle made by PR with the axis of X. 
 Also let the scales for the full-line curve be similar, and the curve- 
 be considered as a velocity-space diagram. 
 
 As V = — , (is = vdt; also ^ = — (see Arts. 2 and 3). 
 
 Then tan (9 = — =-- = - x -r; = - (!)■ 
 
 as vat V at V ^ ' 
 
 Now the triangles MRP and MPN are similar. 
 . •. the angle MRP = angle MPN = 6. 
 
 , , n MN subnormal ;j „ ,^.-, 
 
 and tan = ^rp^- = ^ - Ftrom (1)1 
 
 MP V V ^ ^ '-' 
 
 (2) 
 
 . •. the subnormal = the acceleration to a similar scale to that 
 used for s and v. 
 
 The subnormals may therefore be used as ordinates of an accel- 
 eration curve, on a similar scale to that used for space and velocity. 
 
 Usually it is not convenient to use similar scales, and then the- 
 subnormal is proportional to the acceleration at the point consid- 
 ered. For in any curve if 
 
 2/=/(«) 
 -^ = tan (9 =f'{x) on which the length of the subnormal depends. 
 
TBAN8MI1TI1SG MOTION IN MACHINES. 73 
 
 If now a unit length of ordinate is made to represent n times as 
 many feet as the same length of abscissa, the ordinates will only be 
 
 — th as high as they would be if the scales were similar; or the 
 
 equation of the curve becomes 
 
 .'. -J- =. — f(x) — — tan 6 where n can be integral or fractional. 
 
 That is, tan 6 is increased or decreased as many times as the ordi- 
 nates are increased or decreased from their normal length. 
 
 The dotted curve (Fig. 77) shows the ordinates of the original 
 curve reduced to two-thirds their former length. If tan 6 had re- 
 mained constant the subnormal would have been reduced to two- 
 thirds its original length on account of the reduction of the ordi- 
 nate alone. But tan 6 has also been reduced, so that tan ^' = f tan 0, 
 and the total reduction of the subnormal is f X t = | of its origi- 
 nal length. Every subnormal will be reduced in like proportion, 
 but we may still use the subnormals for ordinates of an acceleration 
 curve. To read them correctly, we may either read them off in 
 space-units and multiply them by w", or read them off in velocity- 
 units and multiply .them by n, the result being obviously the same. 
 
 In a velocity-time diagram, when the scales are similar, and, 
 using the same notation as before, the acceleration is equal nu- 
 merically to tan 0, as the distance marked ds in Fig. 77 now be- 
 comes dt, 
 
 .'. » = — =: tan ft 
 
 -' dt 
 
 If different scales are used, so that the ordinates are increased or 
 decreased from their lengths, tan d will be increased or decreased 
 in like proportion as seen above. Eeferring to Fig. 76 («), equal 
 spaces on the X axis represent equal periods of time, but the 
 ordinates are drawn to a different scale. Drawing a tangent, as 
 shown at P. and using any number of time divisions as a base (two 
 
74 KINEMATICS OF MAGEINEBT. 
 
 in this example), the ratio -r; is found. This value multiplied by 
 
 n (or the number of times the ordinates have been increased or de- 
 creased) will give the true values of the acceleration. Or we may 
 plot the dv intercepts as a curve of acceleration, as shown, con- 
 structing proper scales to facilitate the reading. 
 
 Fig. 76 shows an acceleration curve constructed from the space- 
 velocity curve of the slider-crank mechanism for one stroke, with 
 scales to facilitate the reading. The ordinates of the velocity curve 
 represent the velocity of the slider to a scale at which the length of 
 the crank represents the linear velocity of the crank-pin. The 
 space scale (reduced in the reproduction) is l-j- in. = 1 ft. or 1 in. 
 = f ft. The velocity of the crank-pin is 9 ft. per second, and 
 since the crank as drawn is 1^ in.* in length, the velocity scale is 
 If in. = 9 ft., or 1 in. = 8 ft. . •. w = 8 -H | = 12, or the ordi- 
 nates are reduced to ^V their normal length. To find the scale by 
 which to read these ordinates, we may eithcvr scale them off in ve- 
 locity-units and multiply by 12, or scale them in space-units and 
 multiply by 12'. Thus 1 in. of acceleration ordinate = 8 X 12 = 
 I X 12^ — 96 ft. per second. If we divide 1 in. into 96 equal 
 parts and use 10 of them fdr our acceleration-scale unit, each unit 
 will represent 10 ft. of acceleration per square second. 
 
 Fig. 76 (a) shows the velocity diagram of Fig. 76 transposed to 
 a time base and the acceleration curve constructed from it. The 
 crank-pin has a linear velocity of 9 ft. per sec. and since it is 9 
 inches in length it rotates 1.91 times per second, or performs one 
 
 revolution in zr-^ = .52 sec. Since this time is divided into 18 
 
 equal parts on the time base, and we have used two of them in our 
 
 52 X 3 
 construction, dt = '- — --^ — = .058 sec. Now 1 in. of velocity or- 
 
 8 
 
 dinate = 8 ft. Therefore 1 in. of acceleration ordinate = 
 
 .058 
 
 dv 
 
 -- = 138 ft. acceleration. 
 
 at 
 
 * This figure is reduced to about one-lialf size in the reproduction. 
 
TRANSMITTING MOTION IN MACHINES. 75 
 
 It may be noted that the acceleration is indeterminate, graphic- 
 ally, on the velocity-space diagram, where the curve crosses the 
 axis of X. It can be found for several ordinates near that point and 
 extended to the end position ■without much error. On the time- 
 velocity diagram, however, it is wholly determinate. Both methods 
 are open to the objection that considerable error is necessarily in- 
 troduced in drawing tangents to curves which are not very well 
 defined themselves. 
 
 If the weight of the moving body is known the force required 
 
 to accelerate or retard it at any position can be found from the 
 
 acceleration curve. If F be this force, W the weight of the body, and 
 
 W 
 V the acceleration, F = — p. The acceleration can be read off 
 
 from the acceleration scale at any point and the force corresponding 
 
 W 
 may be found simply by multiplying the acceleration by — . Or 
 
 a force scale may be constructed, as can readily be seen. 
 
 43. Centrodes and Axodes. — The instant centre for two bodies 
 having plane motion may also be a permanent centre, in which 
 case it remains a fixed point in both bodies ; but in the general case 
 the instant centre does not occupy the same position in either body 
 for any two successive relative positions of these bodies, and the 
 locus of the instant centre upon each of the bodies is called a Cen- 
 trode. The instant centre is a point common to the two bodies for 
 the instant, and therefore the two coincident points of the bodies 
 which lie at this centre have for the instant no relative motion ; but 
 any other two points (one in each of these bodies) do move rela- 
 tively. The pair of centrodes traced on the bodies by the motions 
 of the instant centre are tangent to each other at the instant centre; 
 and as these contact points of the centrodes have no relative motion, 
 the pair of centrodes roll on each other with a pure rolling action. 
 Points in the pair of centrodes which previously coincided in the 
 instant centre are now — in common with other points of the two 
 bodies— rotating relatively about the present instant centre; and a 
 similar remark applies to a pair of such points which may coincide 
 in the instant centre at any succeeding phase. 
 
 Any plane motion between two bodies, whatever the mechan- 
 
76 
 
 KINEMATICS OF MAGHINEBY. 
 
 ism adopted for producing this motion, is exactly equivalent to 
 that resulting from the rolling upon each other of two members 
 whose contact lines conform to the centrodes for this motion. 
 The nature of this action may be made clearer by reference to 
 Fig. 79. Let a and I be two points in the body A, moving rela- 
 
 Fig. 79 
 
 tive to the fixed body M so as to occupy in succession the positions 
 a-l, a'-l' , a"-l)" , etc. From the middle of a-a' and l-d' erect per- 
 pendiculars to these lines, intersecting in ; then the motion from 
 (t-l to a'-V is equivalent to a rotation about the point as a centre 
 through the angle aOa' = iOb' = cf>. The motion «'-&' to a"-h" 
 is likewise equivalent to a rotation about 0' through the angle 0', 
 etc. 0', 0' , etc., are temporary or shifting centres, and they may 
 be connected by the polygon O-O'-O" , etc., which lies in the station- 
 ary body M. If the line 0-0 ' be laid off on the moving body A of 
 a length equal to 0-0' and making the angle with the latter, it 
 is evident that when a-h moves to a'-V , 0-0/' will fall along 0-0"^ 
 and 0/ will coincide with 0' . 
 
 From 0' lay off an angle with O'-O" equal to 0'; extend 0-0' 
 to g through this last angle; and let the angle gO'-O" = fi'. 
 This extension divides 0' into /3' and a', and «' = 0' — ft'. 
 Extend 0-0/ to the right; from 0/ lay off the line 0/-0/' equal 
 ,,to O'-O" andmaking an angle with O'-O/' equal to a'. When a'-d' 
 has moved to a"-b", 0/' will coincide with 0". By a continuation of 
 this process the polygon O-O^-O/'-O/" , etc., is constructed on the 
 
TRANSMITTING MOTION IN MACHINES. 77 
 
 face of the moving body A ; and the given motion of A {a-l, a'-b', etc. ) 
 is equivalent to the broken rolling action of this polygon of .1 upon 
 the polygon previously formed on M. If the positions of A {a-b, 
 a'-b', etc.) are taken closer together, the corresponding positions of 
 the temporary centres (0, 0', etc.) become closer, and the polygons 
 approximate more nearly to the centrodes for the given motion; and 
 at the limit these polygons reduce to a pair of centrodes, and the 
 temporary centres become true instant centres. 
 
 For other than a plane motion it has been seen (Art. 19) that 
 the motion must be referred to a rotation about an axis instead of a 
 centre. The locus of the instant axis is called an Axode. 
 
 Centrodes (or axodes) may be used in obtaining a motion which 
 is too complex to get directly by tlie usual methods. Several desired 
 positions of two points {a and I, Pig. 80) in a body A, relative to 
 
 the points m-n of a body M, may _ ,-- — "~-,^ 
 
 be laid .down, and the centrodes then 
 derivedby the process indicated above. 
 If the two bodies^ and J/ are attached 
 to figures having these centrodes for 
 contact surfaces, the simple rolling 
 upon each other of these surfaces will pig. oo 
 
 produce the required motion. As will be shown in a later chapter, 
 in treating the design of toothed gearing, it is possible to derive a 
 pair of gears which will produce a motion identical with the rolling 
 motion of these centrodes and free from any risk of slipping. It is 
 mathematically possible to secure very complicated motions by the 
 use of the principles given in this article ; but there are many prac- 
 tical limitations to the applications of such a process. 
 
CHAPTER III. 
 
 PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 
 
 GEARING. 
 
 PRICTIONAL 
 
 44. Nature of Rolling Curves. — Since the condition of roll- 
 ing action is that the contact point shall always lie in the line of 
 centres, the contact radii must both coincide in direction with the 
 line of centres to insure pure rolling, and as the contact radii lie 
 in one straight line they make equal angles with the common tan- 
 gent. In a pair of curves which roll upon each other (Figs. 81 or 
 83), any two radii, one to each of the pair of curves, as OM ana 
 
 Fig. 81 
 
 O'N, which may become simultaneous contact radii, must make 
 equal angles with the tangents to the curves at Jf and iV, respec- 
 tively; otherwise these radii could not lie in one straight line when 
 the two tangents coincide at contact of Jf and Ji. Furthermore, 
 the arcs PMand PiV^must be equal; and the sum of the radii OM 
 and O'iVmust equal the constant distance between centres 0-0'; 
 for if the first of these conditions is not satisfied, there must evi- 
 
 78 
 
PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 79 
 
 dently be some sliding action between the curves; if the second 
 condition is not fulfilled, the two points Jf and iV^ could not meet 
 on the line of centres. 
 
 Besides pairs of rolling circular arcs, in which the condition of 
 pure rolling (but not that of positive driving) is met, there are 
 many pairs of curves that satisfy the above conditions. Two of 
 these forms will be treated in detail, and a general practical method 
 will be given for deriving a curve which will roll with a given 
 curve, the two centres being fixed. 
 
 45. Rolling Circles. — Pigs. 64 and 65 show pairs of tangent 
 circles which may roll upon each other, for the contact point always 
 lies in the line of centres. The common normal passes through 
 both centres in these cases so motion is not transmitted positively; 
 but if we assume that there is no slipping between these curves 
 the linear velocities of the points P„ and P^ are equal. If A makes 
 Til revolutions, and B makes m, revolutions, per unit of time (call- 
 ing the radius of ^ = r^, and the radius otB = rj, the linear veloc- 
 ity of Fa = 2;rr,?j,, the linear velocity of F^ — 2;rr,w, and, from 
 the assumption of no sliding, 
 
 27mwi = 27rr,n, (1) 
 
 The angular velocity of ^ is c», = 2;rw, , the angular of 5 is t», = 
 37r«, , but from equation (1), 
 
 -1 = — — =a constant: (2) 
 
 Tj 2:tn, 00, 
 
 hence the angular velocities of A and B are inversely as their radii. 
 This familiar relation corresponds with the relations given in Art. 
 34, where it was shown that in any case of rolling curves the angu- 
 lar velocity ratio is inversely as the lengths of the contact radii, or 
 inversely as the perpendiculars from the fixed centres to the com- 
 mon tangent. This relation holds, whether the angular velocity is 
 constant as in the case of rolling circles, or otherwise. The general 
 theorem of Art. 29, that the angular velocity ratio is inversely as 
 
80 KINEMATICS OF MACEINEBY. 
 
 the perpendiculars from the fixed centres to the common normal, 
 or inversely as the segments into which the line of centres is cut 
 by the common normal is not applicable to the special case of tan- 
 gent circles, for this normal coincides with the line of centres, and 
 these ratios are indeterminate. Thus, the perpendiculars from 
 
 n 
 
 and 0' upon NN' are both zero, and their ratio gives — = -■ 
 
 ■^ '^ (i3j 
 
 The common point, P, of Figs. 64 and 65 may move either to 
 the right or the left along the common tangent. It is evident from 
 Fig. 64-, in which the centres lie on opposite sides of the path of 
 this point, that the rotations of A and B are opposite; if A has a 
 right-hand, negative, or clockwise rotation, B has a left-handed, 
 counter-clockwise, or positive rotation; or when the circles are in 
 external contact their rotations are opposite. On the other hand, 
 if the circles are in internal contact (one of them tangent to the 
 concave side of the other) the rotations are both in the same direc- 
 tion. 
 
 All the statements of this article apply to circular arcs rotating 
 about their centres as well as to complete circles; except, of course, 
 that unless the curves are full circles the action is limited, and must 
 be reciprocating. 
 
 46. Rolling Ellipses. — Two equal ellipses, each rotating about 
 one of its foci as a fixed centre, with a distance between centres 
 equal to the common major axis, will roll upon each other without 
 any sliding action. 
 
 In Fig. 83 two such ellipses are shown, with fixed centres at the 
 foci and 0', and free foci at Oi and 0/. 00' = AB = A' B'. 
 
 It is a property of the ellipse that the lines drawn from any 
 point {M) (Fig. 83) to the foci (0 and d) make equal angles {a), 
 with the tangent {t,n-tm)> and also that OM + Oi2I = AB. 
 
 If N is a point similarly located in an equal ellipse, O'N and 
 and 0^'JSf make an equal angle, a, with the tangent f„4^. Now 
 the two ellipses may be so placed together that iJf and iN'" will co- 
 incide at the contact point, when the tangents t^-t,„ and f„-t„ 
 will become the common tangent, and 0x1/ and 0'i\^ will lie in one 
 
PURE ROLLING IN DIRECT-CONTACr MECHANISMS. SI 
 
 Straight line, for they make equal angles with these tangents. If 
 O and 0' are made the fixed centres about which the ellipses 
 rotate the contact point lies in the line of centres; hence the action 
 is pure rolling. The distance 00' = 0M+ 0'N= AB, as already 
 stated. Also, 0,0,' = A' B' - AB = 00'. 
 
 As M and N are any points similarly located in the two equal 
 ellipses, the contact point will always be in the line of centres if 
 the conditions as to these centres given at the beginning of this 
 article be observed. 
 
 If there is no sliding between the two ellipses in acting through 
 the angles POM' and PO'N', respectively, (Fig. 82), the arcs PM' 
 and PN' must be equal. This equality can be shown as follows: 
 
 OP + 0,P = AB = A'B' = O'P + 0/P, . . (1) 
 
 also OP + O'P = AB = A'B' = 0,P + 0/P ; . . (2) 
 
 .-. OF + 0,P = 0P + O'P = O'P + 0,'P = 0:P + 0,'P. . (3) 
 
 From either the first and second, or the third and fourth mem- 
 bers of (3) we get: 
 
 O.P = O'P, (4) 
 
 from which it is seen that the arcs PB ' and PA are equal. 
 
 In a similar way it can be shown that OiM' = O'N,', and that 
 the arcs AM' and B'N' are equal; therefore the arc PM' = 
 AP - AM' is equal to the arc PN' = B'P - B'JV'. This 
 ■demonstration is general and will apply to any pair of points 
 "which can meet as contact points. If the points P and M lie 
 •on opposite sides of AB, and P and J^ lie on opposite sides of 
 A'B', the values of PM and PA^ become PB + BM, and PA' + 
 A'Ji^, respectively, but the equality of the arcs is maintained. 
 
 The driving will be positive in the direction indicated, until 
 the phase shown in Fig. 83 is reached, when the normal passes 
 through both fixed centres, and the driver might continue to rotate 
 without imparting further motion to the follower. To secure con- 
 
82 KINEMATIGS OF MAOHINEBY. 
 
 tinuous driving for the half revolution succeeding this phase ifc 
 must be provided for otherwise than by the simple contact of the- 
 two ellipses. It has been shown that the free foci 0, and 0/, are- 
 always at a distance apart equal to the major axis, A-B, and these- 
 foci could therefore be connected by a link. This system of link- 
 work alone would transmit motion exactly equivalent to that of 
 
 Fig. 83 
 
 the rolling ellipses; but in an actual mechanism the two pieces would 
 have to be at the ends of the shafts between which motion is to be- 
 transmitted, or the link would interfere with the shafts.* Another 
 obstacle to such a link connection, as a substitute for the rolling- 
 ellipses, is that at the phase shown in Fig. 83 (and at 180° from 
 this position) the linkwork would reach a " dead-centre " position,, 
 when it would not be effective in transmitting motion. 
 
 Teeth may be placed at the ends of the elliptical members (a& 
 indicated in Fig. 83), which would engage near the dead-centre 
 phases, and thus carry the follower past this critical position. If 
 such teeth were placed around the entire halves of the ellipses 
 which are in contact after the direct-contact driving ceases to be- 
 operative, the link could be omitted, and the necessity of placing 
 the ellipses at the ends of the shafts thus avoided. Where the 
 action is to continue through half a revolution, or more, such 
 teeth are usually placed entirely around the peripheries of the- 
 ellipses, and the result is a pair of elliptical gears such as is shown 
 in Fig. 84. The method of forming such teeth, to secure the 
 exact equivalent of the rolling ellipses, will be discussed in a later 
 chapter. "With the transmission through such elliptical members^ 
 
 * It will be noted that the pair of rolling ellipses correspond to the centrodes 
 of such a system of links as that just suggested. 
 
PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 83 
 
 as have just been discussed, the angular velocity ratio is inversely 
 as the contact radii at any phase. If the driver has a uniform 
 
 JCb' 
 
 Fig. 84 A 
 
 angular velocity, the angular velocity of the follower is a maximum 
 
 in the phase shown in Fig. 83, when -^ = y^r-^ = -p=-yz. When thfr 
 ^ ° 00^ OF OB 
 
 driver has made a half revolution from this position, the angnljir 
 
 r? O fi 
 velocity of the follower is a minimum, and — " = 77-;. These 
 •' ft5, OA 
 
«i 
 
 KINEMATICS OF MAOHINEBT. 
 
 extreme ratios are reciprocals of each other. Of course the driver 
 and follower both complete the half rotations from these two 
 positions (where the contact radii coincide with the major axes) in 
 ■equal times. If it is required to connect two shafts by rolling 
 •ellipses either the maximum or the minimum angular velocity of 
 the follower may be taken at will, but one of these being deter- 
 mined the other is fixed — the driver being supposed to have a 
 constant angular velocity. 
 
 Suppose it is required to construct a pair of rolling ellipses such 
 
 GO ^ 
 
 that the maximum value of — ^ = ^ . Divide the distance between 
 
 00, 1 
 
 centres 00' (Fig. 83) into such segments that OP : O'P :: 2 : 1. 
 Lay off PA and PB' each equal to 00' ; then lay off PO, and 
 B' 0' equal to PO' . PA and PB' are the major axes of the re- 
 quired ellipses, whose foci are and 0^ , and 0' and 0/, respec- 
 tively; from these data the curves can be constructed. 
 
 Sectors of ellipses can be used for transmitting a reciprocating 
 motion from the driver to the follower. In this case the angle 
 through which one of the members turns, and both the maximum 
 and minimum angular velocity ratios, can be assumed; but the 
 angle through which the other member rotates is not then subject 
 to control, for the two sectors are necessarily alike. Thus (Fig. 85) 
 
 -/ 
 
 the centres are at and 0' , and it is required to construct a pair 
 of elliptical sectors such that an angular motion, a, of the driver 
 ■will transmit motion to the follower pivoted at 0' , and w 
 to have for extreme values O'P -f- OP, and O'P' -¥ OP'. 
 
 ao„ IS 
 
PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 85 
 
 Draw a line from making the angle a with OP, and on this 
 line lay off OM = OP'. P and M are then points in the ellipse 
 which rotates about one of its foci at 0. The distance from P 
 to the free focns of this ellipse, 0,, equals the major axis minus 
 OP; or 0,P = 00' - OP = O'P. With this length as a radius 
 and P as a centre draw an arc, ee. Also, the distance from 0, to 
 M, or 0,3/, = 00' - 0M= O'P'. With this length as a radius, 
 and a centre at M, draw an arc ff. The intersection of the two 
 arcs fe and /^ is 0,. The foci being located and the major axis 
 known, the ellipse can be drawn. The elliptical arc, PN, of the 
 follower is equal to that of the driver. 
 
 The constructions just outlined apply either for actual rolling 
 elliptical members, or for finding the "pitch curves" for toothed 
 gears, or segmental gears. 
 
 Elliptical gears have been applied in many cases where a " quick- 
 return " action is required, as to shaping-machines, in order to get. 
 a quick-return motion to the tool with a slower stroke during the 
 cutting. They have also been used to actuate the slide-valve in a, 
 steam-stamp used for crushing rock, where it is desirable to admit 
 the steam above the piston throughout nearly the entire downward 
 stroke in order to cause a more effective blow ; while on the upward 
 stroke economy demands that only sufl&cient steam be used to return- 
 the stamp-shaft. 
 
 47. Rolling Logarithmic Spirals. — One of the properties of the. 
 logarithmic spiral is that the tangent to the curve makes a con- 
 stant angle with the radius vector at all points. Owing to this 
 property, the curve is also called the equiangular spiral. 
 
 The polar equation of this curve is 6* = log,, r, in which b is 
 the base of the system of logarithms. The angle made with the 
 tangent by. the radii vectores is different for different values of 5, 
 but it is constant for any one system of logarithms.* 
 
 * See Fig. 86, 6 = logsr. Let m = modulus of the system of logarithms, 
 
 dr , rdQ rmdr , ,, , , 
 
 .■. dO = TO— ; but tan d) = -=— = =- ar = to = the' modulus of the sys.- 
 
 r dr r 
 
 tern of logarithms; .'. = tan- i to = a constant. 
 
86 
 
 KINEMATIC8 OF MACEINEBY. 
 
 If two similar logarithmic spirals are placed tangent to each 
 other, as in Fig. 87 or 88, the tangents to the two coincident con- 
 tact points lie in the same line; and as the angles made by these 
 tangents ,with their radii vectores are equal, these radii lie in a 
 straight line. This holds for all tangent positions of the curves; 
 hence if the curves turn about fixed centres at their foci, the con- 
 tact point always lies in the line of centres, thus meeting the re- 
 quirement for pure rolling. 
 
 The sum of the contact radii if the foci are on opposite sides of 
 the contact point, and their difference if the foci are on the same 
 side of this point, is a constant and is equal to the distance be- 
 tween the fixed centres. Thus, in Fig. 87, OP + O'P = 00'; 
 
 Fig. 88 
 
 and, if r and s are two points which may become coincident con- 
 tact points, Or + O's = 00'. Also, in Fig. 88, O'P -OP =00'; 
 and, if r and s are two points which may become coincident con- 
 tact points, O's — Or = 00'. 
 In Fig. 87: 
 
 OP + O'P = 0r+ O's, 
 In Fig. 88: 
 
 O'P - 0P = O's - Or, 
 
 Or-OP= O'P - O's. 
 
 (1) 
 
 Or-OP = O's - O'P. . (2) 
 
 Equations (1) and (2) show that in either external or internal 
 contact the difference between two contact radii of one of the 
 
PUBE ROLLING IN DIREOT-CONTACT MECHANISMS. 87 
 
 spirals equals the difference between the corresponding contact 
 Tadii of the other spiral. It can be shown that any two arcs of 
 similar logarithmic spirals are equal in length when the difference 
 of the radii to the extremities of these arcs is the same. Hence in 
 J'igs. 87 and 88, Pr = Ps, as it should for pure rolling.* 
 
 A single pair of logarithmic spirals cannot transmit motion 
 oontinuously in one direction, but they may be used for a recipro- 
 ■cating transmission with pure rolling. The angular motion of the 
 -driver and both extreme angular velocity ratios may be assumed, 
 "when the angle through which the follower moves cannot be con- 
 -trolled. Thus, in Pig. 87, the driver may rotate about through 
 the angle POr = a, and the angular velocity ratio varies from 
 O'P -T- OP to O's -^ Or. These conditions determine the points 
 I" and r in the spiral which has its focus at 0. The focus 0' , the 
 point P, and the length of a second radius vector, O's =00'— Or, 
 are also fixed for the second spiral; but as this must be similar to 
 the first spiral, the angle PO's cannot be assigned in advance. It 
 is possible to fix the angles of motion of both driver and follower, 
 but with these conditions only one angular velocity ratio can be 
 taken arbitrarily. 
 
 48. General Case of Rolling Curves. — A general method will 
 now be given for constructing a pair of curves which will roll 
 upon each other in turning about two fixed centres. By this 
 method the angular velocity ratios at the beginning and end of 
 any angular motion of one member may be assigned; but the cor- 
 Tesponding angular motion of the other member cannot be pre- 
 determined. Or, if one of the curves is prescribed, a curve can be 
 iound which will roll upon it. The method gives only approxi- 
 
 * See Fig. 86. 9 = log;, r; to = modulus, (dsy = (rdSy + [drf; but 
 = VI- \ .: {d6y = Im—j , .: (ds)' = [rm—J + (drY = (m' + l)(dr)». 
 
 .'. ds = \/{m* + \)dr, .'. «■= |/(to' + 1) / dr = \/{in* + l)(r, — ra); lience 
 
 t/ra 
 "the length, of the arc s included between two radii vectores of the same difEer- 
 <.iice in length is constant. 
 
KINEMATICS OF MACHINERY. 
 
 Fig. 89 
 
 mate results inasmuch as it does rot absolutely insure tlieoretic.illy 
 perfect rolling between the points located; but the approximation 
 can be carried to any required limit by locating a sufficient number 
 of points. 
 
 Suppose the distance between the fixed centres, and 0\ 
 Fig. 89, to be given, and that it is required to construct a pair of 
 
 rolling curves such that the angular ve- 
 locity ratio oi B to A shall be OP-^OT, 
 OP.^O'P,, OP^^O'P,, OP.^O'P,, 
 etc., when the lines PO, m,0, mfi, mfi, 
 etc. respectively, lie in the line of cen- 
 tres; these last lings being drawn to cor- 
 respond with required angular motions, 
 of ^. 
 
 The first pair of radii are OP for A, 
 and O'P for B. With as a centre 
 and OP, as a radius, describe the arc 
 P^m^, cutting the line wi,0, then draw 
 a line from P to ?«,. With 0' as a centre and O'P^ as a radius, 
 draw the arc to the right of P, ; now take a radius equal to Pvi^ , 
 with P as a centre, and cut the arc Jrawu through P, with centre 
 0', at M,; and connect this point m, with P. It is evident that ?re, 
 and M, can meet in the line of centres when A has turned through 
 the angle ?«, OP. Next draw an arc through P, , from centre 0, 
 cutting the line m,0 in wj,, and connect m, and m^. Also draw 
 the arc to the right of P, with 0' as a centre and O'P^ as a radius; 
 now with a radius equal to m, m^, and with w, as a centre, cut this 
 last arc at wj then draw the line w,w,. Proceed in a similar way 
 with the points P^, P,, etc., locating the points m„ m„ etc., of ^4 ; 
 and W3, w„ etc., of B. It will be seen that the polygons P-m-m^ 
 . . . OTj, and P-n-n^ ■ • . «, may act together with a rough rolling 
 action, and that two curves can be passed through P-mi-m^ . . . 
 OTj, and P-n-n^ . . . w,, which will closely approximate pure roll- 
 ing if the points located are sufficiently close together; that is, if 
 the arcs approximate the chords. Evidently, if the outline of A 
 
PURE ROLLING IN DIRECT-CONTACT MECSANISMS. 89 
 
 had been given, the curve of B could have been derived by laying 
 off the lengths Om, Om^, etc., from upon 00', and then pro- 
 ceeding as before in the location of the points of the outline B. 
 
 Fig. 90 shows the derivation of a curve B to roll upon the 
 straight line which rotates about as a centre and constitutes the 
 acting line of A. The construction will be obvious from the pre- 
 ceding explanation in connection with Fig. 89. 
 
 This method cannot usually be applied where complete rotations 
 of both of the members is required; for, as appears from the con- 
 structions given, the angular motion of the follower for a given 
 
 Fig. 90 
 
 motion of the driver cannot be controlled ; hence it is not certain, 
 in the general case, that a complete rotation of one member will 
 correspond to a complete rotation of the other. But with con- 
 tinuous action in one direction, when one member has turned 
 through 360° the other must have turned through an angle of 300°, 
 or else some exact multiple or exact divisor of 360°. This require- 
 ment does not apply to rolling circles, but it holds for all other 
 pairs of rolling curves. 
 
 49. Lobed Wheels. — It has been seen that a pair of equal ellipses 
 can rotate continuously with rolling contact, and that the angular 
 velocity ratio passes through one maximum and one minimum 
 value for each revolution. It is sometimes desirable to have several 
 maxima and minima values of this ratio to a single revolution, and 
 
90 
 
 KINEMATIC 8 OF MACHINERY. 
 
 a class of rolling mechanisms called Lohed Wheels may then be 
 used. 
 
 Fig. 91 shows a pair of these wheels, each having three lobes. 
 The outlines are all logarithmic spirals. 
 
 If it be desired to have an unequal number of lobes on the two 
 
 Fig. 91 -— — • Pig 92 
 
 wheels these spirals cannot be used; but curves which are derived 
 from ellipses permit this condition. 
 
 Fig. 92 shows a set of three such wheels in series which roll 
 perfectly; there is a one-lobed wheel acting on a two-lobed wheel, 
 and this latter rolls with a three-lobed wheel. These figures are 
 drawn from McCord's Kinematics, to which the reader is referred 
 for a full treatment of Lobed Wheels. 
 
 In all of these wheels, as in the rolling ellipses, there are 
 periods during which the driving is not positive; but these outlines, 
 can be used as the pitch curves for toothed wheels, and teeth can 
 be formed upon these curves which will transmit. a positive motion 
 exactly equivalent to that of the pure rolling of such curves. In 
 these derived toothed wheels there is sliding between the teeth 
 themselves, but no sliding (if the teeth are properly formed) be- 
 tween the pitdh lines. 
 
 50. Rolling Surfaces. — The preceding chapter contained a dis- 
 cussion of plane curves which roll upon each other in rotating 
 about fixed centres. It was shown in Art. 10 that the plane motion 
 of any body can be represented completely by the motion of a plane 
 figure; thus the rolling curves of the last chapter represent corre- 
 sponding bodies which rotate about axes through the fixed centres 
 
PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 91 
 
 and perpeudicukr to the plane of motion. When two or more 
 such bodies can be represented by figures lying in the same plane, 
 it is evident that the axes of all of these bodies must be parallel. 
 The actual contact surfaces of such bodies are generated by a line 
 which travels along the curved outline, always remaining parallel 
 to the axes; hence these surfaces are cylindrical. The actual 
 bodies corresponding to Figs. 64 and 65 are figures of revolution or 
 right cylinders (see Fig. 93); while the bodies corresponding to 
 Figs. 82 to 90 are cylinders only in the general sense. Certain 
 other forms may roll together in rotating about fixed axes which 
 are not parallel, when the motion of each member about its axis is 
 still plane, but the planes of motion of the difEerent members do 
 not coincide. If the two axes intersect, tangent cones, or frusta 
 (as in Fig. 95), having a common contact element and a common 
 apex at the intersection of the axes, may act together with pure 
 rolling. These cones are not necessarily right cones, but the use of 
 cones of other than circular transverse sections is so rare that only 
 right cones will be treated in this work. 
 
 If the two axes are not in one plane {i.e., if they are neither 
 parallel nor intersecting) they may still be connected by two mem- 
 bers which will roll upon each other, with contact along a common 
 rectilinear element. Fig. 101 shows the general form of a pair of 
 such members; they are called Hyperholoids of Revolution. The 
 general method of generating these latter figures and the nature of 
 the action will form the subject of a later article, in which it will 
 be shown that there is, in a sense, a certain departure from pure 
 rolling in the action; however, this does not prohibit the use of 
 tJiese forms as pitch surfaces for toothed gears, owing to the pecul- 
 iar character of the sliding component. 
 
 51. Rolling Cylinders. — In rolling right cylinders the angular 
 
 03 T 
 
 velocities are inversely as the radii; or — - — —• Let d be the dis- 
 
 tance between the fixed axes. In external contact, r^-\- r^ = d; 
 and in internal contact r, — ?-, = d, (in this expression r, is taken 
 as the radius of the larger cylinder, inside of which the smaller one 
 
92 KINEMATICS OF MACHINEBT. 
 
 rolls). It is frequently required to find the diameters (radii) of 
 tangent cylinders whicli will connect two shafts and transmit mo- 
 tion (in the absence of slipping) with a given angular velocity 
 ratio. This ratio is the same as the ratio of the revolutions made 
 in a given time by the two cylinders, and in practical problems it is 
 usually stated in these terms. Thus, one shaft is to make Wi revo- 
 lutions imparting na revolutions to the other shaft, per unit of 
 
 time ; then — i = — = -?. In many cases the required radii, 
 
 »•, and r,, can be found by inspection, or by mental calculation ; but 
 it may be convenient to use the following expressions if m, and 
 w, are high numbers with no common divisor. 
 
 For Cylinders in External Contact : r, -\- r = a, .'. ri = d — rj, 
 and r, — d — r^. 
 
 — i = -?, .: r, = r,-^= (d — r,) —'; 
 
 
 or r 
 
 GO CO 
 
 Similarly : r, = r,-^ = (d - r,) --^•, 
 
 00. 00. 
 
 For Cylinders in Internal Contact : r^ — r^ = d (r^ being the 
 radius of the large cylinder). .•. r^ = d -{- r„ and r, = /•, — d. 
 
 CO, r' ' ' CO, ^ ' 'go. 
 
 or ri'^ - 1 = ^ "=-; .-. r. = -^^d; d = -^^ d. . (3) 
 ' \co, / 00/ oo^ — oo' n, — n, ^ "^ 
 
PURE BOLLINQ IN DIRECT-CONTACT MECHANISMS. 93 
 
 Similarly : 
 
 00„ 00, 
 
 or 
 
 or r 
 
 
 _^^rf; cl=^^^^d. . (4) 
 
 03, — CW, W. — »i. 
 
 The directions of the rotations of the two members are oppo- 
 site when they are in external contact, and the same when one is 
 tangent to the concave surface of the other, as previously pointed 
 out. 
 
 52. Rolling Eight Cones. — Two right cylinders, combined with 
 two right cones, are shown in Fig. 94. Bach cylinder has one 
 base in common with that of one of the cones, hence the axis of 
 this cylinder and cone must coincide. The bases of the two cones 
 (and of the corresponding cylinders) need not be equal, but the 
 
 Fig. 93 
 
 Fig. 94 
 
 Fig. 95 
 
 slant height of both cones is the same. The bases of the two 
 cones have a common tangent, in their plane (perpendicular to 
 the paper), passing through M. Now imagine the two axes, AA 
 and BB, to rotate in their common plane, about this tangent to the 
 bases through M as an axis (or hinge), till the apex a meets the 
 apex b at Q, as in Fig. 95 ; when the two cones become tangent 
 alon"- the element QM. It will be seen that the two base circles 
 
94: 
 
 KINEMATICS OF MAOEINEBT. 
 
 still have a common tangent through M and ihey can roll upon 
 each other in the new position, the two contact points having equal 
 velocities along their common tangent, as in the original position. 
 Any other corresponding transverse sections of the cones, equidis- 
 tant from Q along the elements, as m-m' and m-m" will also roll 
 together; or the two cones roll upon each other in a similar manner" 
 to the original rolling of the cylinders. 
 
 If it is required to connect two given intersecting shafts by 
 rolling cones, so that their rotations per unit of time shall be in 
 the ratio of m, to «, , it is only necessary to construct two tangent 
 
 right cones with these shafts for axes, 
 and with a common contact element 
 lying in such a position between the 
 axes that any pair of transverse sections 
 which roll together shall have radii in 
 - b' the inverse ratio of the required angular 
 motions. If A- A' and B-B', Fig. 96, 
 are the given axes, the position of the 
 contact element may be found by lay- 
 ing ofE from Q, on these shafts, the 
 ' distances Qa and Qb, directly propoi- 
 tional to the required numbers of rotations of these shafts ; thus 
 Qa : Qb :: n^ : w,. On Qa and Qb form a parallelogram, and the 
 diagonal of this parallelogram, Qc, or its extension, is the required 
 common contact element. 
 
 This can be proved as follows :' from c drop perpendiculars cd 
 and c/upou the axes^-^' and B-B'; the angle cbf= eac = a (sides 
 parallel) ; ce = ca sin a, cf = cb sin a. 
 
 .• . ce : cf :: ca : cb :: MM' : MM'; hence the cones with MM' 
 and MM" as the diameters of the bases, will roll together with the 
 required angular velocity. The frusta used for this transmission 
 may be taken from any part of the two cones, giving bases greater 
 or less than those indicated, if more convenient. 
 
 The parallelogram might have been drawn in any of the four 
 
 Fig. 96 
 
PURE ROLLING IN DIBECT-CONTACT MECHANISMS. 95 
 
 angles made by the intersection of A- A' and B-B'; thus if the 
 angle B'QA had been selected, the diagonal Qc' would have been 
 located for the contact element, and two such frusta as those 
 shown with M^M^' and MJd^" as bases would give the required 
 angular velocity ratio. Either of the other two angles, A'QB' or 
 A'QB, might have been taken if desired. It will be noticed that 
 the cones first found are not similar to those obtained in the second 
 construction; but the pairs constructed in both of the acute angles 
 are similar, as are the pairs in both of the obtuse angles. 
 
 If the driving-shaft A-A' rotates as indicated by the arrows, it 
 will be seen that the first construction (in the acute angle) imparts 
 rotation to B-B' in one direction ; while the second construction 
 (in the obtuse angle) causes B-B' to rotate in the opposite direc- 
 tion. The choice of angle for the location of the contact element 
 is governed by the required directions of the rotations, and the 
 locations of the actual shafts. It is evident that one of the ma- 
 
 terial shafts, but not both of them, can pass through Q. Fig. 97 
 shows a shaft A-A', from which four shafts (making equal angles 
 with A-A') are driven. One of the followers on either side oiA-A' 
 is rotated in one direction ; while the other followers (one on each 
 side of the driver) rotate in the opposite direction. 
 
 It may happen, as in Fig. 98, that one wheel cuts through the 
 
96 
 
 KINEMATICS OF MAGHINERY. 
 
 axis of the other wheel, when its shaft can be led oflE only in the 
 
 direction indicated by the full lines ; 
 for if it were to be carried through Q, 
 in the direction of the dotted lines, 
 the shaft and wheel would interfere. 
 This condition can only occur when 
 the contact radius is located in the 
 obtuse angles. The acute-angle con- 
 struction is to be preferred as avoiding 
 this difficulty in all cases, and also 
 because it gives smaller wheels ; but 
 there are conditions as to location of shafts and required directional 
 relation of rotation which may make the other construction desir- 
 able or necessary. The conditions of the problem may be such that 
 the contact element is perpendicular to one axis, when the cone on 
 this axis is of the special form (a flat disk) shown in Fig. 99. "With 
 somewhat different conditions, one of the rolling surfaces may be 
 the concave surface of a cone, as shown in Fig. 100. 
 
 Fig. 98 
 
 In a great majority of the cases requiring the construction of 
 rolling cones on intersecting axes, these axes are at right angles to 
 each other. With this condition the pairs of cones formed in any 
 of the four angles (for a given angular velocity ratio) have similar 
 inclinations. The location of the contact radius in one of these 
 angles, and the selection of the particular angle in which it lies, 
 
PURE ROLLING IN BIREOT-GONTAOT ME0HANI8M8. 97 
 
 are determined by the general relations previously treated in this 
 article. 
 
 53. Rolling Hyperboloids. — If one right line revolves about an- 
 other right line not in the same plane, and all points in these lines 
 remain at constant distances apart, the revolving line generates a 
 surface called the hyperboloid of 
 revolution. This will be a warped 
 surface ; for its elements are 
 straight lines corresponding to the 
 successive positions of the generat- 
 ing line. A meridian plane through 
 this figure cuts the surface in an 
 hyperbola, and it is evident that 
 this hyperbola would generate a 
 surface, in revolving about the axis, 
 identical with that generated by the 
 straight line; hence the name given 
 to these figures. Fig. 101 repre- 
 sents a pair of these hyperboloids 
 -of revolution tangent to each other 
 along a common element mm. Any 
 pair of these figures can be placed 
 in this position, as the elements are rectilinear; and if the axes are 
 fixed in the positions corresponding to such tangency, it is evident 
 that the two surfaces will remain tangent as the two figures rotate 
 about their axes; for each is symmetrical about its axis. 
 
 It is evident, also, that all points in the hyperboloid which 
 rotates about A- A, Fig. 101, must move in planes perpendicular 
 to this axis; likewise, all points in the other hyperboloid move in 
 planes perpendicular to B-B, and as the two axes are not parallel, 
 two contact points cannot have identical motions. Thus if F„ is 
 the motion of a contact point in the former figure, Fj is the 
 simultaneous motion of the corresponding point in the latter 
 figure. 
 
 These two motions must have, in rolling together, equal com- 
 
98 KINEMATICS OF MACHINERY. 
 
 ponents perpendicular to the contact element^ but their components 
 along this common line will not coincide. This is the characteristia 
 of the action of these bodies referred to in Art. 50, and, as stated 
 there, it does not affect the angular velocity ratio of the two mem- 
 bers, for this relative sliding along the common element cannot 
 transmit motion, nor can it affect the component of Va and V^ 
 perpendicular to the common element. 
 
 The axes and the common tangential element have a common, 
 perpendicular through B^P^A^, and therefore these lipes lie in 
 three parallel planes. If they be projected on a plane parallel to 
 all of them, the common tangent makes the angles a and y3 with 
 A A and BB respectively. 
 
 It can be shown that the angular velocity ratio of the figures- 
 (with no sliding perpendicular to the common elements) is 
 
 — -■ = - — - ; therefore the proiection of the contact element can 
 Cl?j sin /3 
 
 be found by constructing a parallelogram with the projections of 
 
 AA and BB, upon a plane parallel to them, as sides. If the sides 
 
 of this parallelogram. Pa and Pb, have the ratio of oo^ to c»„, the 
 
 diagonal, Pc, will locate the contact element and determine a and 
 
 yS. It can also be proved that the radii of the two hyperboloids 
 
 at the "gorge circles," viz., P,J.j and P,B,, are proportional to- 
 
 tan a and tan /?: thus „'-„' = ? T^. Or if a perpendicular, de, 
 
 P,B^ tan p ^ ^ ' ' 
 
 be drawn through any point, as c, in the common element, the 
 radii are respectively proportional to the segments of this per- 
 pendicular lying between the element and the two axes; thus 
 P,^, : P,5, ■.-.cd: cc. 
 
 To construct a pair of rolling hyperboloids to transmit motion 
 between two shafts with a given angular velocity ratio : — project 
 these shafts on a plane parallel to both of them, Pig. 101 ; lay of 
 Pa and Pb on AA and BB proportional to the required relative 
 revolutions; construct the parallelogram P-a-c-i, and draw Pc : 
 this locates the contact element. At any point on Pc erect a 
 perpendicular, cutting AA and BB in d and e respectively. Divide 
 
FRICTION AL GEARING. 99 
 
 the perpendicular distance {A^B^) between AA and BB, at P„ in 
 the ratio of the segments cd and ce; then P^A^ and P,i?, will be 
 the radii of the gorge circles of the required hyperboloids. For 
 full discussion of these figures, see McCord's Kinematics. 
 
 54. Frictional Gearing. — It has been shown that two axes, 
 whether parallel, intersecting, or neither parallel nor intersecting, 
 may be provided with contact members the surfaces of which will 
 roll upon each other. In many mechanisms it is necessary to 
 maintain, exactly, a prescribed relation between the motions of the 
 members throughout the entire cycle of operations. In other 
 instances this is not essential, a reasonable departure from the- 
 precise relative motions contemplated being permissible. Thus^ 
 in cutting a screw-thread in a lathe, it is essential that the relation 
 between the rotation of the spindle and the translation of the tool 
 shall be strictly constant, and the positive mechanism (gears and 
 the lead screw) insure this uniformity of action. But in plane 
 turning the feed may vary somewhat without serious results, and 
 the belt-driven rod-feed, depending upon friction, is often used, 
 thus saving unnecessary wear of the screw. It sometimes happens^ 
 as in machinery subject to severe shock, that a positive transmis- 
 sion is not desired; and in many cases this is not an absoluta 
 necessity. When a limited variation of the motion transmitted 
 may be permitted, and the two shafts to be connected are at a. 
 considerable distance apart, belting or rope transmission is most, 
 often employed. Occasionally, because the distance between the. 
 shafts is too small to employ these methods of transmission 
 advantageously, or for other reasons, the substitution of contact 
 members rolling upon each other is convenient. In all such trans- 
 missions having circular transverse sections the action is purely 
 frictional throughout the revolution, and these mechanisms are; 
 classed as Frictional Gearing. 
 
 If the sections are non-circular the action may still be pure 
 rolling, as shown in the preceding chapter; but the driving can- 
 not be positive during the entire rotation; for a critical phase is 
 reached at which the action is only frictional, and beyond this 
 
100 KINSMATICS OF MAOHINERT. 
 
 phase driving does not occur, even by friction, unless other ex- 
 pedients (as teeth) are introduced (see Fig. 83). It is evident, then, 
 that frictional gears must have circular transverse sections in order 
 to transmit continuous rotation. 
 
 The force that can be transmitted through frictional gearing 
 depends upon the physical character of the surfaces in contact and 
 on the normal pressure between the two surfaces. Some slipping 
 or "creeping" almost inevitably occurs; its magnitude depending 
 upon the character of the surfaces, the normal pressure between 
 them and the resistance to be overcome. 
 
 In certain applications this liability to slip is desirable rather 
 than otherwise. For example, in hoisting, where it is not essen- 
 tial that the load raised shall move through precisely the same 
 distance for each increment of motion of the driver. If any ob- 
 struction to motion of the load be met, the slip prevents the sud- 
 den strain (shock), that would be thrown upon the entire train of 
 mechanism if this elasticity (using the word in a somewhat popular 
 sense) were absent. If a car, or " skip," in being hoisted from a 
 mine leaves the track, meets an obstruction, or is overwound, the 
 yielding through the slipping of friction gears (or of belts) lessens 
 the danger of breakage over that encountered with a positive connec- 
 tion. Furthermore, these friction mechanisms are much simpler 
 in design and construction, and quieter in running than toothed 
 gears; and, owing to such considerations, the employment of fric- 
 tional gears, or "frictions " as they are frequently called for brev- 
 ity, is not uncommon, under proper circumstances. 
 
 Frictional gearing is important in itself, and the study of it also 
 affords a good basis for investigation of toothed gearing. 
 
 Kinematically, any of the figures of revolution which will roll 
 together, as pairs of right cylinders, right cones, or hyperboloids of 
 revolution, might be used as friction gears; but, practically, rolling 
 cylinders (Fig. 93), and the disk and plate (" brush-wheel " ) (Fig. 
 102), are by far the most common as the basis of such gearing. 
 EoUing cones are also used, but less frequently. 
 
 Two cylinders (Figs. 64, 65, and 93) may be used to transmit 
 
FBIOTIONAL GEARING. 
 
 101 
 
 motion and energy, up to the limits fixed by the friction at the 
 contact element. Supposing no slip to occur, any two contact 
 points have the same linear velocity, and the angular velocities of 
 the two members, A and B, are inversely as their radii. 
 
 If it is required to impart to a shaft a given number of revolu- 
 tions per unit of time, from a shaft of given rotative speed, the 
 distance between centres being also determined; the required radii 
 can be found by the expressions of Art. 51. For example, d = 48", 
 Wj — 210 rev. per min. ; n, = 270 rev. per min. 
 
 n,d 
 
 270 X 48 
 
 = 27" 
 
 n, + n, 270 + 210 
 
 = d- r, = 4:8-27 = 21". 
 
 The solution of the kinematic part of this problem is extremely 
 simple. 
 
 55. Grooved Frictions. — The consideration of the force that can 
 be transmitted by friction-gears involves the normal pressure and 
 the coefficient of friction between the contact 
 surfaces. This consideration often modifies the 
 forms of the members, without altering the kine- 
 matic action; and in many cases it may be advan- 
 tageous to use certain derived forms, known as 
 grooved frictions or "V" frictions, in place of 
 the fundamental rolling cylinders. Fig. 103 
 shows a pair of these derived forms in contact. 
 It will be seen that the original, or ideal, rolling 
 cylinders are replaced by rolls with circumferen- 
 tial grooves, the sections of which (in planes pass- 
 ing through the axis) are triangular, or more usually, trapezoidal. 
 The actual contact surfaces are frusta of cones of equal slant and 
 on parallel axes. 
 
 In order to discuss the action of these grooved rolls, and to 
 understand clearly their advantage over the simple rolling cylin- 
 ders, it will be necessary to treat briefly the action of the forces in- 
 volved in frictional transmission. 
 
102 KINEMATICS OF MACHINERY. 
 
 If two bodies are in contact, with a force F acting in thu direc- 
 tion of their common normal, there is a resistance to the sliding of 
 one body iipon the other, and this resistance, called friction, is what 
 .makes frictional transnaission possible. The resistance is a function 
 of this normal pressure and of the physical character of the sur- 
 faces. If the surfaces are very smooth, the resistance under any 
 normal pressure becomes comparatively small. If rough, the pro- 
 jecting particles of one member interlock with those of the other 
 and the friction increases. As absolutely perfect surfaces are not 
 attainable, absolute freedom from friction (absence of this resist- 
 ance to sliding) is impossible ; and the greater the departure from 
 ideal perfection of surface (smoothness), the greater is the friction 
 between any given pair of bodies. The friction varies inversely as 
 the smoothness, and this varies both with the nature of the 
 materials in contact and with the degree of "finish." In every 
 case the friction is greater than zero; and the ratio of this resist- 
 ance, /, to the normal force, F, is called the coefficient of friction, 
 /(. This coefficient can only be derived from experiment, directly 
 or indirectly. 
 
 Let the normal pressure between the surfaces of the two cylin- 
 ders (Fig. 93) be represented by F. According to ISfewton's third 
 law, action and reaction are equal and opposite; hence, the pressure 
 of A towards B is met by an equal and opposite pressure of B 
 towards A. These pressures can only be brought to bear upon the 
 contact surfaces through the bearings of the wheels (neglecting 
 weight), and the action and reaction at the bearings are equal ; 
 therefore a pressure F must be exerted by the bearings upon the 
 axle supported by them. In other words, the pressure between the 
 bearings and journals equals the pressure between the contact sur- 
 faces of the two wheels. As the bearings themselves, however per- 
 fectly formed and lubricated, are not frictionless, the normal force, 
 F, necessary to transmit energy from A to B, involves a frictional 
 action at the bearings, resulting in a prejudicial resistance to be 
 overcome, and also, incidentally, in wear of these parts. It is there. 
 iore desirable to reduce the pressure at the bearings as much as 
 
FBICTIONAL GEARING. 
 
 lOE 
 
 possible; but the friction at the contact surfaces must be sufficient 
 for driving, and 'the normal pressure at these surfaces is one of the 
 •elements which determine this friction. It is in order, then, to 
 investigate the relation between the bearing pressure and the 
 normal pressure at the contact surfaces, and to see if the former 
 «an be reduced without undue sacrifice of the latter. "With simple 
 cylindrical rolls (Fig. 93) the total bearing pressure for each wheel 
 €quals the normal pressure, F, at the con- 
 tact point. In the case of "V" frictions, 
 however, the normal pressure between the 
 contact surfaces may be much greater than 
 the bearing pressure. This can be shown 
 in connection with Fig. 104, in which the 
 wedge of A is inserted in the corresponding 
 groove of B. The common normals to the 
 contact faces of A and B through the 
 centres of the faces are Pn and Pn' , and 
 the normal forces between these faces may 
 be taken as acting in the lines of these normals (such normal forces 
 are really the resultants of systems of parallel forces, uniformly 
 distributed over these faces). The force F, acting in the centre 
 line of A and B as indicated, passes througli P, and it can be re- 
 solved into components along Pji and Pn' by the parallelogram of 
 forces. These components are represented by F^ and F/. The 
 effect of the initial force, F, is equivalent to the combined effect of 
 its components, and it may be replaced by them ; therefore, the 
 effect of F is equivalent to the normal actions, F^ -\- F^ . 
 
 F^ sin e = ^F; .: F, = ^^; similarly, F/ =. ^. If 
 
 S — 6' (the usual condition), F^ = F^', and the total normal action, 
 
 F 
 F, 4- F/ ^= 2F, = -. — 7,. It is seen, from this last expression, that 
 ' ' sm a 
 
 the normal pressure increases, for a given value of F, as d becomes 
 
 smaller. "When = 90°, the total normal pressure equals F, as it 
 
 should; for in this case the groove and wedge have disappeared 
 
 Fig. 104 
 
104 KINEMATICa OF MACHINERY. 
 
 and the contact surfaces are flat and perpendicular to the line of 
 F. For any value of 6 less than 90°, the total normal pressure is 
 greater than F. 
 
 The action between the grooved faces of the "V" frictions is 
 exactly like that of this wedge. The normal pressures between 
 the sides of the acting ridges and the grooves correspond to the 
 normal pressures in the wedge, and the initial force F equals the 
 radial force exerted between the bearings and journals of the 
 wheels. It follows from this discussion that any necessary normal 
 pressure at the acting contact surfaces of the "V" frictions can be 
 maintained by a force less than itself at the surface of the journal. 
 Hence, the prejudicial resistance is decreased by the substitutiori 
 of the "V" frictions for the fundamental rolling cylinders; or, to- 
 state it somewhat differently, for a given pressure at the bearings,, 
 a greater resistance can be overcome at the rim by "V" frictions 
 than by cylindrical rolls. As there is a practical limit to the press- 
 ure that can be safely carried at the bearings, and as excess of 
 bearing pressure means waste through friction, the importance of 
 the wedge-like action in frictional transmission is apparent. 
 
 The angle between the sides of the grooves [Zd) is usually from 
 
 40° to 50°. Assuming 40° as this angle, =z2Q , and sin 6 — 0.342> 
 
 F 
 Then 2 Fi ^ -— = S.93 F ; or the total resulting normal pressure 
 
 is nearly three times the force at the bearing, and the coefficient of 
 friction and bearing pressure remaining the same, nearly three 
 times as great a resistance can be overcome with these grooved rolls 
 as with corresponding true cylindrical rolls. 
 
 Grooved frictions are frequently so mounted that one of the 
 shafts can be moved slightly, relative to the other. This makes it 
 possible to throw the wheels out of gear, so that the follower can 
 be stopped without checking the driver. It also permits control- 
 ling the bearing pressure, so that it need not be any greater than 
 required to prevent serious slipping at the driving surfaces. This 
 adjustment also affords ready means of taking up the wear of the- 
 
FRICTIONAL GEARING. 105 
 
 <( 
 
 V " or of the bearings, so that good contact (without which driv- 
 ing is impossible) is maintained. 
 
 When in gear (close contact) there is, as usually constructed, a 
 small clearance at the bottoms of the grooves, as indicated in Fig. 
 104. If this were not provided, the edges of the rings might 
 "bottom; " that is, the contact might be entirely or mainly at the 
 bottoms of the grooves, instead of at the inclined sides. Such a 
 condition would defeat the object of the grooves, and to render it 
 impossible, even after considerable wear at the sides, this clearance 
 is provided. 
 
 The depth of the grooves of either wheel is the difference 
 between the radii to the tops and bottoms of the grooves or 
 rings. This distance minus the clearance at the bottoms may be 
 called the working depth, and the faces of the " Vs " above the 
 clearance may be called the working surfaces. The nominal radius, 
 or pitch radius, of a grooved friction may be taken as the mean 
 radius of the working surface, and the hypothetical cylinder corre- 
 sponding to this radius will then be the pitch cylinder, or pitch 
 surface. 
 
 The angular velocity of two V friction wheels, when in full con- 
 tact and working properly, may be taken, for most practical pur- 
 poses, as that corresponding to the rolling together of the pitch 
 cylinders, or, inversely, as the pitch radii. The relative sliding or 
 creeping of the wheels along the common tangent to the pitch sur- 
 faces may usually be neglected in well-constructed frictions; for 
 these wheels are only employed where some variations in the angu- 
 lar velocity ratio is admissible. Assuming that no sliding of this 
 character takes place — that is, that the angular velocities of the 
 two wheels are inversely as their pitch radii — there is nevertheless 
 some relative motion between the two surfaces when they are in 
 contact, causing a grinding action. The nature of this action may 
 be seen in connection with Pig. 105, in which and 0' are tlie 
 fixed centres of A and B, p is the contact point at the pitch circles, 
 and s and t are two coincident points in the working surfaces, one 
 on each side of p. The linear velocity of p, pv is assumed to be the 
 
106 
 
 KINEMATICS OF MACHINERY. 
 
 same for the coincident points of both wheels which lie at p. It is 
 evident that all points in either wheel which lie outside of its pitch 
 circle have linear velocities greater than pv, and all points of either 
 wheel lying inside of its pitch circle have linear velocities less than 
 jw, but those points of the working surface of one wheel which are 
 inside of the pitch curve come in contact with points of the other 
 wheel which are outside of its pitch circle; consequently, if the 
 points at the pitch circles have equal linear velocities, all contact 
 points not in these circles have different velocities, and there must 
 be some relative motion or sliding between any pair of such points. 
 Thus, in Pig. 105, the linear velocity of s, as a point in A, is .sv' ; 
 
 and, as a point in B, the velocity is 
 sv' ; therefore the rate of sliding of 
 these points equals sv" — sv'. Simi- 
 larly, the rate of sliding at t is 
 <«/— tv^". An expression for the 
 greatest sliding is derived below. Let 
 R and r be the two pitch radii, ^A^ 
 and n be the numbers of revolutions 
 per unit of time of the correspond- 
 ing wheels, and h the working depth 
 of the grooves. Then the velocity of 
 a point in the pitch circle of either wheel is 27tRX = 'Znrn .•. RX = 
 rn. An extreme outer point of the working surface of the first 
 wheel has a radius R -\- \li, and it comes in contact with a point of 
 the other wheel having a radius r — ^li; hence the sliding at these 
 points per unit of time equals 
 
 S= 27t{R + ^h)X - 27r{r - ih)n = 2n 
 
 IRN^ piX~ rn+ih7i] = 7th{N-[- M),since RN= rn. 
 
 By taking extreme contact points on the other side of the pitch 
 circles, having radii R — \li and r -|- -I^A, the same result can be 
 reached by a similar process. The grinding action just noted tends 
 to wear the working faces, even if no slipping occurs at the pitch 
 circles. Such action does not take place in simple cylinder friction- 
 rolls, but it cannot be avoided if the grooves have sensible depth. 
 
FRICTION AL GEARING. 
 
 107 
 
 The rate of this sliding action is directly proportional to h ; there- 
 fore the working depth should be as small as practicable. This 
 dimension is limited in practice, without sacrifice of the necessary 
 total contact surface, by using several grooves, side by side, as indi- 
 cated in Fig. 103, instead of fewer and deeper ones. 
 
 56. Brush-wheels. — Fig. 102 shows a mechanism sometimes 
 used where it is desired to vary the angular velocity of a shaft 
 which is driven by another shaft of constant angular velocity. 
 Suppose the plate on the shaft AA 
 to be the driver, and the disk, or 
 " brush wheel," on BB to be the fol- 
 lower. A long keyway, or spline (or 
 its equivalent), permits the disk to 
 be placed at different positions along 
 a line parallel to a diameter of the 
 plate, as indicated by the dotted 
 locations. The disk is imagined to 
 bo of no sensible thickness ; heiice 
 it touches the plate at a single point, 
 p. This point, ^j, is at a distance from BB equal to the radius of 
 the disk, r' ; and at a distance from the axis AA equal to r, 
 which may vary from zero to R (plus or minus). Assuming no 
 slipping at p, the angular velocity ratio of AA to BB is r' -^ r 
 (inversely as the radii). "When the plane of the disk is in the axis 
 AA, the velocity of p is zero, hence the follower is at rest. If the 
 disk is carried beyond this position (to the opposite side of AA), 
 the direction of the rotation of the follower is reversed. This 
 mechanism is not well adapted for heavy forces; but is very con- 
 venient in many cases for light work, as in feed mechanism and for 
 similar purposes requiring considerable change in the rotative speed 
 of a follower, or reversal of direction of rotation. The disk must 
 have sensible thickness in practical applications, and this gives rise 
 to a grinding action somewhat similar to that mentioned with " V " 
 frictions. If the disk is a cylinder, with contact between one of its 
 elements and a radius of the plate, it is evident that all points in 
 
 Fig. 102 
 
108 ■ KINEMATICS OF MACHINERY. 
 
 this cylindrical element must have the same linear velocity (being 
 points in a body at the same distance from the axis); while the cor- 
 responding contact points in the radius of the plate have different 
 linear velocities (being at different distance from the axis of this 
 plate). The disk should therefore be as thin as practicable, and 
 its edge is sometimes rounded to approximate the point contact of 
 the ideal disk. The plate should usually be the driver ; for if this 
 is not the case, when the disk is in contact with the centre of the 
 plate, the latter is at rest, and the edge of the disk is compelled to 
 slip on the contact surface. 
 
 The working disk is often made of leather, wood, or other yield- 
 ing material, held between metal washers of slightly smaller diam- 
 eter. This construction increases the adhesion, and makes it easier 
 to maintain the required normal pressure at the contact point as 
 slight wear takes place. In other friction mechanisms one of the 
 members is frequently made of a non-metallic substance for a 
 similar reason, and this member should usually be the driver; for 
 if any slip occurs, by reason of the resistance being greater than 
 the friction can overcome, the tendency is to wear off the edge of 
 the rotating driver evenly, and to wear a depression, or notch, in 
 the stationary follower. If the driver is made of the softer mate- 
 rial the more irregular and objectionable wear of the follower is 
 thereby reduced. In the brush-wheel mechanism it is not so easy 
 to support the soft face on the driver (the plate), and there is not 
 the same reason for doing so; because, even if the wear were all 
 concentrated on this face, it would not be worn off evenly all over,, 
 for the follower only covers a small portion of its working surface 
 in any position. When the follower (disk) is at the centre of the- 
 plate there is a tendency to wear a small flat place on the edge of 
 the former. This may be avoided in many cases by cutting a slight 
 depression at the centre of the plate, so that contact does not take 
 place in this position of the disk. 
 
 57. Cone Friction. — Intersecting axes are sometimes connected 
 by rolling conical friction wheels similar to the arrangements indi- 
 cated in Figs. 96 to 100; but these are not so satisfactory as the 
 
FRICriONAL GEARING. 109 
 
 frictions on parallel axes, as it is more difficult to adjust the posi- 
 tions of the shafts to maintain the requi)-ed normal pressure. If 
 the force to be transmitted between intersecting axes is consider- 
 able, it may be better to use positive connections, as bevel gears, to 
 connect the intersecting shafts, and to introduce the friction ele- 
 ment, if necessary, by means of a supplementary shaft parallel to 
 one of these. 
 
 Two cones, as shown in Fig. 106, are sometimes used to connect 
 parallel shafts, where changes in the angular velocity of the fol- 
 lower are required. These two cones 
 are similar in inclination, and placed '^~ 
 ■with the adjacent elements parallel, 
 
 but not touching. An intermediate ^ 
 
 disk, C (or its equivalent), capable of 
 being moved along the lengths of the ^'^' '"^ 
 
 cones, is in contact with both of them. Assuming no slipping at 
 either contact, the linear velocity of the edge of this disk will be 
 that of the part of the driver with which it is in contact, and this 
 same linear velocity will be imparted to the follower ; hence the 
 linear velocities of the contact points of the two cones will be equal, 
 and the angular velocities will be inversely as the contact radii of 
 the cones at these points. If the disk is placed nearer the large 
 base of the driver it acts on a smaller section of the follower, and 
 the angular velocity of the latter is correspondingly increased. 
 
 This device, or modifications of it, is now on the market, for use 
 as a countershaft. Provision is made for maintaining proper con- 
 tact between the disk and the cones. In this case, as in that of the 
 brush-wheel, the disk must have appreciable thickness; hence its 
 contact element engages with points on the cones which must have 
 somewhat difEerent linear velocities, and a corresponding grinding 
 action occurs. Similar remarks as to the means of reducing the 
 practical efEect of this action apply to both cases. 
 
CHAPTER IV. 
 
 t 
 
 </ 
 
 OUTLINES OP GEAR-TEETH. 
 
 Systems of Tooth-geaking. 
 
 58. Pitch Surfaces. — It has been shown that many pairs of 
 bodies (as cylinders, cones, etc.) may transmit motion from one to 
 the other with pure rolling, while these bodies rotate abont axes 
 fixed in the proper relative positions; but that the action of the 
 driver upon the follower is not continuously positive. 
 
 The application of these rolling bodies as frictional gearing has 
 already been treated. There are many cases, however, where it is 
 desirable to secure a motion equivalent to one of these rolling 
 actions, but where it is absolutely essential that no practical vari- 
 ation from this prescribed motion shall occur. These conditions 
 are frequently met by using the surfaces of the appropriate rolling 
 members as bases, and attaching interlocking teeth to them for the 
 prevention of slipping. These rolling surfaces, when so used, are 
 called pitch surfaces; and sections of them perpendicular to the 
 axis are called pitch lines, or pitch curves. 
 
 Toothed gearing may be classified according to the pitch sur- 
 faces, relation of the axes, and character of the elements as 
 follows : 
 
 Kind. 
 
 Relation of Axes. 
 
 Pitch Surfaces. 
 
 Spur 
 
 Parallel 
 
 Cylinders 
 
 Bevel 
 
 Intersecting 
 
 Cones 
 
 Screw 
 
 Not in one plane 
 
 Cylinders 
 
 Skew 
 
 " " '.' " 
 
 Hyperboloids 
 
 Twisted 
 
 Any 
 
 Any of above 
 
 Face 
 
 
 None, strictly 
 
 110 
 
OUTLINES OF OEAR-TEETH. 
 
 Ill 
 
 The action of these various classes will be treated in detail in 
 later articles. 
 
 The two tangent circles of Fig. 107, representing rolling cylin- 
 ders, may have their circumferences divided up into arcs of equal 
 
 Fig. 107 
 
 length, p = Pa = ab = he, etc., = Pa' = a'b' — b'c', etc. This 
 length of arc, />, must be a common divisor of both circumferences, 
 and the numbers of divisions on the two circles are proportional 
 to their circumferences, diameters, or radii. Let the radii be repre- 
 sented by r and r', and the numbers of divisions of the respective 
 circles be called t and t' ; then as 
 
 = t, and 
 
 P 
 
 2nr' „ 'Znr 2Ttr' 
 
 As t and t' are directly proportional to r and r' , it follows that the 
 angular velocity ratio is inversely as the number of the divisions 
 of the two circumferences. It is to be noticed that a and a', b and 
 b', c and c', etc., are pairs of points which become coincident con- 
 tact points as the circles roll together. 
 
 E"ow if we bisect the arcs Pa, ab. Pa', a'b', etc., and place pro- 
 jections and corresponding notches on the alternate subdivisions, 
 as indicated by the shaded outlines of Pig. 107, it will be seen that 
 the wheels resemble, somewhat, the familiar toothed gears. The 
 part of the tooth outside of the pitch circles is called the adden- 
 dum OT pomt ; the portion inside of the pitch circle, between the 
 spaces, is called the root. The acting surface of the point, or adden- 
 dum, is called the face, and the acting surface of the root is called 
 
112 KINEMATICS OF MAGHINERY. 
 
 the flank. By the formation of such teeth the pitch circles have 
 lost their physical identity, but they are, nevertheless, important 
 kinematically as the basis of the toothed wheels. The distances 
 Pa, ab. Pa', etc., from any point on one tooth to the corresponding 
 point of the next tooth of the same wheel, measured on the pitch 
 curve, is called the circumferential pitch, circular pit el t, or simply 
 the pitch. It is evident that the pitch must be the same for both 
 wheels. 
 
 If these wheels are " meshed " (that is, placed with the tooth 
 of one in the space of the other, and with the pitch curves tan- 
 gent), as shown in Fig. 107, it is apparent that the rotation of one 
 of them will cause the other one to rotate, and that the transmission 
 is now positive. As this rotation goes on, the successive pitch 
 points of the teeth of the two wheels come into contact on the 
 line of centres, and the mean angular velocity ratio for complete 
 rotations, or for angular motions of the wheels measured by their 
 pitch arcs, is identical with that due to the pure rolling of the 
 pitch circles. This might be sufficient for some purposes ; but we 
 have, as yet, 7io assurance that this angular velocity ratio is strictly 
 constant throughout the angular movements corresponding to the 
 pitch angles. That is, the mea)i angular velocity ratio during such 
 an angular motion agrees with that of the rolling circles ; but at 
 any phase intermediate between contact at two pitch points the 
 angular velocity ratio may be either greater or less than this mean. 
 It is imperative in many cases, and desirable for smoothness of 
 action and quiet running in nearly all cases, that the angular 
 velocity ratio be constant for all phases, 
 
 59. Conjugate Gear-teeth. — The condition of constant angular 
 velocity ratio in direct contact is that the common normal to the 
 acting faces, through the point of contact, shall always cut the line 
 of centres in a fixed point; hence the desired constancy of this 
 ratio in such wheels as those of Fig. 1,07 demands that the common 
 normal shall always pass through the point marked P. If the 
 teeth are of such form that this condition is met, the motion trans- 
 mitted is exactly equivalent to the rolling of the pitch circles. 
 
OUTLINES OF OEAR-rEBTH. 113 
 
 Otherwise there is some departure from the required relative 
 motion. 
 
 In general, the form of the teeth of one wheel may be taken 
 •quite arbitrarily, and an outline can be found for the teeth of the 
 other wheel which will give the required angular velocity ratio at 
 all phases; but this statement is subject to practical limitations. 
 A pair of teeth which work together properly are called conjugate 
 teeth. 
 
 A practical mechanical method of finding a conjugate tooth 
 outline, when both pitch curves and the form of the tooth to be r^ 
 mated are known, will be explained before treating the formation of 
 teeth geometrically. This method is applicable when it is required 
 to construct a wheel to mesh with an existing gear, whether the 
 form of tooth on the latter has lost its original form through wear 
 or not; or whether the pitch curves are circles or not. 
 
 Cut out two segments of wood, A and B (Pig. 108), correspond- 
 ing to the two pitch curves, and mount them on centres properly 
 located. Upon the segment A, repre- 
 senting the existing gear, attach, in 
 proper position, a sheet metal templet 
 corresponding in form to one of its 
 teeth, and have this slightly raised 
 above the surface of the wooden seg- 
 ment by inserting a piece of thick paper 
 
 or cardboard between them, so that aw ^ ..■ 
 
 " ^^ ^'3- 108 
 
 piece of drawing-paper attached to the 
 
 •segment B can pass under the templet. ° 
 
 Now roll the segments, without slipping, and trace the outline of 
 the templet on the paper attached to B in several positions quite 
 close together; a curve tangent to all of these tracings of the tem- 
 plet is the required tooth outline for B. A thin strip of metal be- 
 tween the edges of the two segments, one end of which is attached 
 as indicated to each of the segments, will prevent slipping during 
 the operation. It is evident that as B is rolled back and forth 
 upon A the outline just derived on B will always be tangent to the 
 
114 KINEMATICS OF MAGHINERT. 
 
 tooth of A, and if B is provided with a tooth of this form such a. 
 tooth in acting upon the given tooth of A will transmit motion 
 identical with that due to the rolling of the pitch curves. 
 
 The method Just explained is convenient for use in the shop^ 
 and it suggests a corresponding process for the drafting-room. 
 
 Draw the given tooth and its pitch curve upon a piece of heavy 
 paper, and then draw the pitch curve of the other member upon 
 tracirLg-paper, thin celluloid, or other transparent material. Place 
 this last drawing above the other, with proper tangency of th& 
 pitch curves, and trace the outline of the given tooth upon th& 
 tracing-paper; roll the curves through a small arc, being careful to- 
 avoid slipping, and trace the tooth outline in its new position ; re- 
 peat this operation until the entire arc of action of the teeth has. 
 been covered, and then draw on the tracing-paper a curve tangent 
 to all of the tracings of the given tooth. This tangent curve is the- 
 required tooth outline. 
 
 From what has preceded, it will be seen that two cylinders may- 
 be provided with teeth such that the positive motion transmitted 
 from one to the other will be identical with that of the two cylin- 
 ders when rolling upon each other without sliding. This applies 
 to cylinders other than those of circular cross-section ; for the- 
 methods of finding a conjugate tooth, as given above, apply to any 
 pair of rolling curves, such as rolling ellipses, logarithmic spirals, 
 etc. 
 
 60. General Method of Describing Teeth Outlines. — The general 
 method of describing gearrtooth outlines by means of an auxiliary 
 rolling curve, or generator, will be developed in this article. 
 
 Suppose A and B (Fig. 109) to be any two rolling plane figures 
 upon the outline of which a pair of gear-teeth are to be described. 
 As the pitch lines are rolling curves their point of contact is- 
 always on the line of centres. In the phase shown by the full lines,, 
 the angular velocity ratio of J. to -S is O'P -^ OP; in the phase 
 indicated by the broken lines, this ratio O'P' -h OP'; or for 
 any phase of these rolling curves, the angular velocities of the 
 members are inversely as the contact radii. If a pair of teeth give 
 
OUTLINES OF OEAR-TEETH. 
 
 115 
 
 Fig. 109 
 
 a motion identical with that due to rolling of the pitch curves, it is 
 evident that the common normal 
 to the two teeth in contact must 
 always pass through the point on 
 the line of centres at which the 
 pitch curves are tangent to each 
 other; for these- teeth are exam- 
 ples of direct contact members, in 
 which the angular velocities are 
 inversely as the segments into 
 which the line of'the normal cuts 
 the line of centres. 
 
 If such a figure as G be rolled 
 upon the convex side of the 
 pitch curve of A, the point g of the figure G will trace the curve 
 ga on the plane of A. Likewise, by the rolling of G on the con- 
 cave pitch curve of B, the point g will generate the curve gb on 
 the plane of B. 
 
 The curve G is the generating line of the teeth outlines, and it 
 may be any line capable of rolling on the convex side of A and the 
 concave side of B. The point, g, in this generating line is the 
 describing point of the teeth. Now suppose the pitch curves and 
 the generating line to be in the positions shown by the "broken 
 lines, with the generating point at P', the common point of tan- 
 gency of the three lines. If A is turned to the right, as indicated 
 by the phase shown in full lines, B will turn to the left in rolling 
 upon it, and G can be rolled upon the pitch curves so that it remains 
 tangent to both of them at their contact point in 00'. When the 
 pitch lines have reached such a position as is shown by the full lines, 
 G will lie in the position shown by the full line, and the original 
 contact points of A, B, and G will be at a, i, and g, respectively.. 
 The arcs Pfa, FV), and P'g must be equal, as the action has been 
 pure rolling. During this rotation the point g describes a curve 
 upon the surface of A (this surface being supposed to rotate with 
 A about 0) such as ag, as noted above; g has, in a similar way» 
 
116 KINEMATICS OF MACHINERY. 
 
 ^generated a curve bg on the surface of B (rotating about 0'), and, 
 «,t the instant under consideration, g is the contact point common 
 ■to ag and bg. Now as G is rolling u-pon the pitch, curves of A and 
 B, and is in contact with them at P, P must be the instant centre 
 •of G relative to both A and B; therefore the point g (a p)oint in 
 G) is, at the instant, rotating a.bout P, and its motion must 
 'be in the line gv, perpendicular to gP. As the jDoint g is generat- 
 ing the curves ag, and bg, the commoia tangent of these curves 
 must coincide with the line of motion of g {gv), and gP, perpendic- 
 ular to gv, is, therefore, the common normal to ag and bg. The 
 ■curves ag and bg are described upon the surfaces of A and B, 
 respectively; and it is evident that teeth upon these members, 
 having the outlines ag and bg, will transmit a motion exactly 
 corresponding to that of the rolling pitch lines; because their 
 common normal passes through the point in the line of centres at 
 which these rolling pitch curves are tangent to each other. 
 
 The reasoning of the foregoing discussion is perfectly general. 
 It applies to any phase, if the condition that the three curves roll 
 together with a common contact point is met at every instant of 
 the action; hence the curves derived by this construction fully 
 satisfy the kinematic requirements of teeth outlines. 
 
 The discussion immediately following will be confined to wheels 
 having circles (or circular arcs) for pitch lines. 
 
 61. Usual System of Gearing. — There are a great many curves 
 that can be used for generating lines of gear-teeth, but only two 
 are commonly used; viz. : circles and right lines. 
 
 The curve traced by a point in a circle as it rolls upon the con- 
 Tex side of another circle is called an epicycloid; if it rolls upon 
 the concave side of another circle, the curve traced is ahypocyvloicl; 
 ■and if it rolls along a straight line a cycloid is described. When a 
 right line rolls upon a circle any point in this line traces a carve 
 called an involute. 
 
 The common systems of gearing in which the teeth are generated 
 by circular or rectilinear describing lines are called, respectively, 
 the Epicycloidal System and the Involute System. 
 
OUTLINES OB' OEAR-TEETJS. 
 
 iir 
 
 Fig. no 
 
 62. Epicycloidal Gearing.— Fig. 110, A and B are two pitch 
 circles, with centres at and 0', and tangent at the point P. 
 The generator, 0, has its centre 
 at 0, on the line of centres 00'. 
 If these circles all turn about 
 their respective centres (rolling 
 upon each other) points in each 
 will describe equal linear paths in 
 any period. Suppose three such 
 points, originally coinciding at P, 
 to pass over the equal paths Pa, 
 Ph, Pg; now, if g be the gener- 
 ating point, it will during this 
 motion generate an epicycloid' by rolling on A, and hypocycloid by 
 rolling in B. At any instant these two curves will be in contact at 
 g, in the circumference of the generating circle. As the instant 
 centre of the generator, relative to either of the pitch circles, is 
 always at P, g moves perpendicular to Pg, and Pg is the normal to 
 the curves at their point of contact. This normal always passes 
 through P, hence the angular velocity ratio is constant. 
 
 The curves just discussed are suitable for the outlines of gear- 
 teeth, and if the driver. A, has teeth with epicycloidal faces, andl 
 the follower, B, has teeth with hypocycloidal flanks, generated by- 
 the same circle, G, the action would begin at the pitch point, P 
 and continue through a period depending upon the length of the 
 teeth. 
 
 The locUs of the point of contact is the generating circle. 
 
 The angle that each wheel describes while one of its teeth is in. 
 action is called the angle of action. In the case considered, ins 
 which the driver has faces only, and the follower has flanks only,, 
 the action takes place entirely as the contact point recedes from 
 the line of centres, or during the angle of recess. If the driver has 
 flanks and the follower faces only, the action takes place entirely 
 during the angle of approach. 
 
 Occasionally wheels are made in this way, and then the teeth 
 
118 KINEMATICS OF MAGEINEBT. 
 
 ■with the faces should drive, as the action during recess is smoother 
 than it is during approach. Such gears are confined to mechan- 
 isms requiring great smoothness, and not transmitting heavy press- 
 ures. Usually teeth of gears have both faces and flanks. 
 
 It is evident that a generator G' could be made to describe 
 flanks for A and faces for B, as shown by the curves a'g', and 
 b'(i', respectively, which would satisfy the conditions of constant 
 velocity ratio, and that the action of this pair of curves is entirely 
 independent of the first pair ; hence G and G' may be any two 
 circles. 
 
 At the sides of the figure are shown complete teeth of A and 
 B, the outlines of which correspond to the curves traced by the 
 describing circles G and (?'- The faces of A and the flanks of B 
 are the epicycloid and hypocycloid generated by G, and are identical 
 in form with arcs of the curves ag and ig, respectively. The 
 faces of B and flanks of A are of the forms generated by G', as 
 shown by h'g' and a'g', respectively. The teeth are symmetrical; 
 therefore either side may be the acting side, and either wheel may - 
 drive. 
 
 If the common pitch, p, is an exact divisor of both circumfer- 
 ences; if the lengths of the teeth are such that at least one pair 
 shall always be in contact; and if the spaces are deep enough to 
 
 allow the points to clear in passing the 
 
 centre line, these wheels will answer all 
 
 : / essential requirements. 
 
 / 63. Length of Teeth.— In Pig. Ill 
 
 two teeth are just beginning action at 
 
 the left, and another pair at the right 
 
 arejust quitting contact. The angles of 
 
 approach are aOP and JO'P, andthe 
 
 angles of recess are POa' and PO'b', 
 
 for A and B, respectively. The angle 
 
 '^' of action equals the angle of approach 
 
 plus the angle of recess, and it is evident that with shorter teeth 
 
 than those shown, the angle 'of action would be less. For con- 
 
OUTLINES OF GEAR-TEETH. 119 
 
 tinuous action, one pair of teeth must come into contact before 
 the preceding pair quits contact ; therefore,, the angle of action 
 ■cannot be less than the angle subtended by the pitch arc; or 
 the arc of action aPa' (or bPb') must at least- equal the dis- 
 tance between similar points (on the pitch line) of two adjacent 
 teeth of either wheel, '^his condition fixes the minimum length 
 of the teeth. \ If the given pitch of the two wheels (Fig. Ill) 
 is aa' = bV, this determines the minimum arc of action. This arc 
 may be distributed in any way between the approach and recess 
 arcs, though these are commonly nearly equal. Lay off Pa = Pb 
 .= Py, and Pa' = Pb' = Pg' equal to the desired arcs of approach 
 and recess, respectively; then g and g' are the extreme points in the 
 faces of B and A, respectively ; or circles drawn with the radii O'g 
 and Og' are the boundaries of the teeth of the two wheels. The 
 •strength of the teeth depends upon their thickness, and the pitch 
 is twice the thickness of the teeth at the pitch circle, or slightly 
 greater to allow clearance at the sides, which is called "backlash;" 
 thus the pitch is a function of the force to be transmitted. As has 
 been shown, the arc of action must at least equal the pitch; it is 
 ■often made great enough to insure that two teeth shall always be 
 in contact ; or that as one pair is in contact at the centre line, the 
 preceding pair shall be quitting contact, and the succeeding pair 
 shall be beginning contact. This requires an arc of action equal to 
 twice the pitch arc, and correspondingly longer teeth, for a given 
 pitch. 
 
 The force acting between the teeth is transmitted in the direc- 
 tion of the common normal (neglecting the effect of friction), or in 
 a line through P and the contact point of the teeth. This contact 
 point always lies in the describing circle G during approach, and 
 in G' during recess ; hence it appears that the force transmitted is 
 more oblique as the contact point is removed from P. The effect 
 of this obliquity is to increase the pressure between the teeth and 
 at the bearings, with a corresponding increase in the energy wasted 
 through friction. Wheels of smaller pitch have shorter teeth, other 
 things being equal, and their action is smoother under the ordinary 
 
120 KINEMATICS OF MAOB INERT. 
 
 conditions because the contact point is always nearer the line of 
 centres, where the rate of sliding of the teeth upon each other i» , 
 less. 
 
 It will be seen that, for a given pitch, the length of teeth re- 
 "quired for a given arc of action is less as the describing circles used 
 ' is larger in diameter.* 
 
 64. Determination of Describing Circles. — During contact tha 
 faces of the teeth of A act only upon the flanks of the teeth of Z?; 
 similarly, the faces of 5 act only on the flanks of^; hence the- 
 form of the faces of one wheel does not afEect that of its own flanks^ 
 nor of the faces of the mating wheel. There is no necessary fixed 
 relation between the two generating circles G and G'. 
 
 If the describing circle has a diameter equal to the radius (^ the- 
 diameter) of the pitch circle within which it rolls in tracing a hypo- 
 cycloid, this special hypocycloid is a right line passing through th& 
 centre of the latter circle, or a diameter of it. Hence if the describ- 
 ing circles, G and G' (Fig. 110), have diameters equal to the radii of 
 i? and ^4, respectively, both wheels will have radial flanks; but 
 these will operate properly in conjunction with the corresponding- 
 epicycloidal faces. The faces would not, in this case have the forms- 
 shown 'n Fig. 110, as the faces of one wheel and the flanks of the- 
 other oae must be derived from equal describing circles. The 
 radial fltink forms are simple in construction and describing circles- 
 are Bom >,times used for a pair of gears which will give such teeth. 
 If the describing circle has a diameter less than the radius of the- 
 pitch circle within which it rolls in tracing the hypocycloid, the 
 flanks lie outside of radii through the pitch point ; while if the 
 diameter of the describing circle is greater than the radius of this 
 pitch circle, the hypocycloidal flanks lie inside of the radii to the 
 pitch points. The first of the forms gives spreading flanks which 
 are much stronger than the converging or undercut flanks of the 
 latter form. The i-adial flank is intermediate between these fornis 
 in strength. Except in small gears (frequently called pinions) for 
 
 * As sho-wn in Art. 36, there is always some sliding in direct-contact trans- 
 mission unless tlie point of contact lies in the line of centres. 
 
OUTLINES OF QEAR-TEETH. 121 
 
 light work, undercut flanks are seldom used; the radial flank usu- 
 ally being the weakest form allowed. While it is desirable for 
 strength of the teeth to have spreading flanks, and therefore to use 
 a small describing circle, large describing circles give teeth which 
 act upon each other with less obliquity. 
 
 In exceptional oases, when a single pair of gears are to work 
 together, it may be good practice to choose the largest pair of de- 
 scribing circles which will give the necessary strength of flanks, 
 and flanks of a comparatively vieak form may be used by giving a 
 small excess to the pitch (thickness of teeth). In such cases of 
 single pairs of gears, for reasons already given, radial flanks will 
 sometimes be used for both wheels. In making a set of patterns 
 (or cutters for cut gears), however, it is desirable on the score of 
 economy to provide for the working of any wheel of the set with 
 any other wheel of the same pitch. If this is possible the set is said 
 to be interchangeable. Suppose that in the two gears, A and B (Fig. 
 110), the faces of the former and the flanks of the latter are described 
 by a generating circle G, and that the faces of B and the flanks of A 
 are described by another circle G'. It has been shown that these 
 two wheels will work together. A third wheel, C, of the same 
 pitch, cannot work properly with both A and B; for if the faces of 
 C are described by G, and its flanks are described by G', it may 
 engage with B; but it cannot act correctly with A, for the faces of 
 A and the flanks of C are not generated by the same circle ; neither 
 are the flanks of A and the faces of C, and the conditions of con- 
 stant velocity ratio are not met by this construction. 
 
 If G — G' , C would work correctly with either A or B, or 
 with any other wheel of the same pitch, the faces and flanks of 
 which are epicycloids and hypocycloids generated on its pitch line 
 by G' = G' . We may then state that: The conditions necessary 
 in an Interchangeable Set of Gears are that all of the wheels of the 
 set shall have the same pitch, and that the teeth of all of them shall 
 have faces and flanks generated by the same describing circle. 
 
 It is common to assume that the smallest wheel that will prob- 
 ably be required will be a pinion of either 12 or 15 teeth, and to 
 
122 
 
 KINEMATICS OF MACHINEBT. 
 
 take a describing circle which will give radial flanks to such a 
 pinion; that is, a describing circle with a diameter half that of the 
 pitch circle of this smallest pinion. If t is the number of teeth in 
 the smallest pinion, its pitch-circle' radius, or the diameter of the 
 
 describing circle, = -^y—. ' ~ 
 
 65. Annular Wheels. — Fig. 65 shows two rolling circles, one of 
 which is tangent to the concave side of the other. The correspond- 
 ing rolling cylinders may be used as pitch surfaces of gears. The 
 larger of these is called an annular gear. 
 
 The method of describing the teeth of such gears is indicated 
 in Fig. 113, and it is similar to that explained for external gears. 
 
 Fig. 112 
 
 except that the faces of B and flanks of A (described by 0) are 
 both epicycloids, and the faces of A and the flanks of B (described 
 by G') are both liypocycloicls. 
 
 66. Rack and Pinion. — If one pitch liije is a right line (a circle 
 of infinite radius), as shown in Fig. 113, teeth may be formed by 
 a method similar to that given for the more general case of spur 
 gearing. Such a gear is called a rack, and the wheel which meshes 
 with it is usually called a pinion. The faces and flanks of the 
 rack are loth cycloids ; and they are alike in an interchangeable 
 set of gears, where but one describing circle is used. In such a 
 set,. any wheel will engage properly with the rack. The construe- 
 
OUTLINES OF GEAR-TEETH. 
 
 123 
 
 tion of teeth for a rack and pinion is shown at the left of Fig. 113 ; 
 and at the right, the complete teeth are shown in the acting 
 positions. Of course the rack is necessarily of limited length, 
 
 and the motion transmitted between a rack and pinion must be 
 reciprocating. 
 
 67. Pin Gearing. — If the describing circle equals one of the 
 pitch circles, the hypocycloid in this pitch circle becomes a mere 
 point; and this point acting on an epicycloid generated on the 
 other wheel by this same describing circle will transmit a motion 
 identical with the rolling of the two 
 pitch circles. Fig. 114 shows such a 
 point in B acting on the epicycloidal 
 faces of A. In an actual gear a pin of 
 sensible diameter must be used, and Fig. 
 114 shows such a pin, and dotted line 
 curves parallel to the original epicycloid 
 of A and at a distance from this epicy- 
 cloid equal to the radius of the pin. 
 This pin and the dotted outline will 
 transmit motion similar to that due to the point and epicycloid. 
 
 With the point and the epicycloid the angle of action is entirely 
 on one sid.e of the line of centres, and the pin gear should always 
 be the follower, in order that the action shall take place during 
 recess rather than approach. With a pin of sensible diameter the 
 
124 KINEMATICS OF MACHINEBT. 
 
 action begins at a distance, practically equal to the radius of the 
 pin, before the line of centres is reached, and there is consequently 
 also an angle of approach. The derived curve of the driver gives 
 shorter teeth than the full epicycloids, and the height of the 
 driver's teeth, above the pitch line, is therefore diminished, 
 thus decreasing the angle of recess. These gears were formerly 
 much used, when teeth were commonly made of wood, as the pin 
 form is easily constructed; but this class of gearing is now used 
 but little, except for light gearing, such as clockwork, etc. 
 
 68. Involute Teeth. — If the diameter of the generating circle is 
 increased to infinity, the generator becomes a right line rolling on 
 the base line, and the curve traced is an involute. This corre- 
 sponds to a limiting form of the epicycloid, but there can be no 
 such curve corresponding to the hypocycloid. Two circles upon 
 
 which involutes are described could 
 roll in contact, but these tooth out- 
 lines, both lying outside of the pitch, 
 circles, could then only act on each 
 other in passing the pitch point, and 
 the angle of action would be zero. 
 From the nature of an involute, it is 
 seen that every normal to the curve 
 Fig. 115 represents one position of the gener- 
 
 ator, and is, therefore, a tangent to 
 the base circle. Now if these base circles do not touch, as a and b 
 (Pig. 115), the involute tooth outlines can act through a finite angle. 
 The two curves have a common normal at the point of contact, and 
 as every normal to an involute is a tangent to its base circle, the- 
 common normal, ^^', is a tangent to both base circles. OE ivadi 
 O'E' are radii of these circles to the points of tangency; hence- 
 they are both perpendicular to EB' ; and the triangles OPE and 
 O'PE' are similar. 
 
 Therefore OP : O'P -.-.OE: O'E ' ; and circles A and B, having 
 radii OP and O'P are pitch circles, tangent at the point through 
 which the common normal to the curves always passes. It follows 
 
OUTLINES OF OEAB-TEETH. 125 
 
 that the motion resulting from the action of these involute teeth 
 is equivalent to the rolling of the pitch circle of A upon that of B. 
 
 The line EE ' is the locus of the point of contact. If a pair of 
 teeth have sufficient length to begin contact at E and to continue 
 in action until their contact point reaches^', the driving will 
 bo continuous, provided the pitch arc does not exceed this arc 
 of action. 
 
 To secure such continuous action in a pair of gears having 
 oqual arcs of approach and recess, the angle POE = PO'E' = cf> 
 
 should not be less than 90° , in which n is the number of 
 
 n 
 
 teeth in the smaller wheelj because the maximum pitch angle 
 
 = ^-^° ^ 2(90^ - 0). .-.^^go^-l^. 
 
 It is desirable to have more than one pair of teeth in action ; or 
 
 to have the pitch arc less than the sum of the angles of approach 
 
 and recess. The pitch arc is often made only about one-half the 
 
 total arc of action, so that two pairs of teeth shall be in contact at 
 
 all times. 
 
 180° 
 The relation, = 90° gives the limiting value of 0. It 
 
 is usual to assume = 75°, which makes the obliquity (or the 
 angle between the force transmitted and the tangent to the pitch 
 circle) = 90° — 75° = 15°. In making standard gear-cutters for 
 involute wheels, the sine of the angle of obliquity is taken as .25, 
 which corresponds to an angle of rather less than 14^°. 
 
 The involute rack (Fig. 116) has teeth which are bounded by 
 plane surfaces inclined to the pitch surface at an angle equal to 0; 
 or the transverse sections of the teeth are trapezoids, in which 
 the slant sides of the working surfaces are perpendicular to the 
 straight locus of the point of contact. Fig. 116 indicates the 
 form of an involute rack and pinion in mesh. 
 
 Involute teeth are sometimes called teeth of single curvature, 
 as there is not a reversal of the outline curve at the pitch circle. 
 In fact, teeth of this system do not have a definite pitch circle 
 
126 KINEMATICS OF MAOEINBRT. 
 
 upon wliicli the teeth are formed as in the other systems; but the 
 teeth are generated on the base circle {aa or hb, Fig. 115), inside 
 the acting pitch circle, instead. The portions of the teeth from 
 
 Fig. 116 
 
 the base circle to the points are bounded by the involute curves; 
 and the flanks are made radial inside of the base circles, with 
 fillets at the bottoms as these parts of the tooth are not working^ 
 surfaces. 
 
 69. Interrerence in Involute Teeth. — Draw circles through E^ 
 and E with centres at and 0', respectively, as shown by the 
 broken lines in Fig. 115. If these are the addendum circles for 
 the teeth of A and B, respectively (that is, the circles which pass 
 through the extreme outer points of the teeth), the teeth will 
 work together properly, and the contact will begin at E and end 
 at E' (or vice vevsa). If the teeth are shorter than this the arc of 
 contact will be less, but the action will still be correct. If the 
 radii of the addendum circles are greater than OE ' a.\\di O'E for 
 the teeth of A and B, respectively, the teeth of A will come into 
 contact with the portion of the teeth of B which lie inside of the 
 base circle b; likewise the teeth of B will act on the portion of the 
 teeth of A which lie inside of the base circle a. The portions of 
 the teeth inside of the base circles are not involutes, and if these 
 portions are radial, as is usual, the action will depart from the con- 
 
OUTLINES OF GEAB-TEETS. 12T 
 
 stant angular velocity desired. This is called the inte7-ference of 
 the teeth, and to avoid it the flanks must be hollowed out, or the 
 points of the teeth rounded. The latter is the usual remedy. 
 
 If the portion of the face which comes in contact with the 
 radial base of the mating-tooth is given the form of an epicycloidal 
 arc generated on the pitch circle by a describing circle of half the 
 pitch diameter of the mating-wheel, the action will be correct, for 
 the radial flank is equivalent to a hypocycloidal flank formed by 
 this same describing circle. 
 
 The separation of the base circles is arbitrary, and the curves 
 will work properly whatever the distance between centres be; 
 except that it must not be so great as to cause one pair of teeth to 
 quit contact before the next pair meet, and it must not be so small 
 as to give a value of less than that required by the considerations 
 discussed in the preceding article. Between these limits the action 
 will be continuous and the velocity ratio constant, with changes in 
 the distance between me centres of the two gears; but the backlash 
 will vary. This property is peculiar'to the involute system, and is 
 exceedingly valuable, especially in gears connecting roll-trains, in 
 change gears, etc., where exact spacing of the centres cannot be 
 maintained. In a pair of rolls connected by involute gears (if con- 
 siderable backlash is originally given), when the rolls are worn, or 
 in adjustment for different thicknesses of material passing between 
 them, the centre distance can be changed considerably without 
 affecting the angular velocity ratio. Unless this centre distance is 
 made so great as to prevent one pair of teeth from engaging before 
 the preceding pair quits contact, or so small as to reduce the back- 
 lash" to zero, the angular velocity ratio remains constant and the 
 action is continuous. 
 
 70. Comparison of the Systems. — The property just mentioned 
 fits the involute teeth for, cases where the centre distance varies, 
 and permits of smaller backlash; very exact setting is not so neces- 
 sary, and wear of the bearings does not disturb the action as it does 
 in the epicycloidal system. 
 
 The line of action is always in the same direction, and the force 
 
128 KINE STATIC 8 OF MAGIIINERT. 
 
 between the teeth is nearly constant in the invohite sjBtem; while 
 the acting force is variable both in direction and magnitude in the 
 epicycloidal system. The former teeth wear more evenly as a con- 
 sequence. 
 
 The thrust on the bearings is slightly greater, but more uniform, 
 ■with involute teeth. 
 
 All involute teeth have the same generator; hence the gears are 
 interchangeable if of the same pitch. Epicycloidal teeth are better 
 for low-numbered pinions, but otherwise have no great advantage 
 iind many disadvantages. They are, however, commonly used for 
 gears having cast teeth; while the involute system has largely sup- 
 planted the epicycloidal system for cut gears. 
 
 V 71. Clearance and Backlash. — The term backlash has already 
 been explained as the clearance at the sides of the teeth; it is 
 equal to the width of a space minus the thickness of a tooth, both 
 measured on the pitch curve. 
 
 The backlash provides for any irregularity in the form or spac- 
 ing of the teeth. It may be very small in accurate cut gears; but 
 must be larger in cast gears. 
 
 The spaces are always made deeper than is required to allow the 
 points of the teeth to pass, this allowance is called bottom clearance, 
 or simply clearance; it also provides a lodging-place for a moderate 
 quantity of dirt or other foreign substance which may get between 
 the teeth. 
 
 i/ 72. Dimensions of Teeth. — The action of the teeth is smoothest 
 ■when the contact point is near the line of centres; hence a large 
 number of small teeth gives more uniform action than fewer and 
 larger teeth. The teeth must be thick enough to safely sustain the 
 load, however; and the pitch is determined by this consideration. 
 For epicycloidal gears of an interchangeable system, in which the 
 describing circle has a diameter equal to the radius of the 12-tooth 
 pinion, the circular pitch (jo), required for strength, may be found 
 by the following formula: 
 
 p = (0.23 - /o.05-i^)A 
 
OUTLINES OF OEAR-TEETH. 
 
 129 
 
 in which W = the load applied to the tooth per inch of its face, in 
 pounds; Z> = the pitch diameter of the gear in inches; and/ = 
 the working stress allowed in the teeth, in pounds per square inch. 
 This expression was derived from the investigation of Mr. Wilfred 
 Lewis. Involute teeth are usually somewhat stronger than the 
 •ordinary forms of epicycloidal teeth. 
 
 For many purposes the following rough formula may be used: 
 
 p = 
 
 WW 
 IT' 
 
 in which b equals the face of the gear, in inches, and the other 
 notation is as above. 
 
 It is usually desired that two pairs of teeth shall always be in 
 contact; or when one pair is at the centre line, one pair will be 
 quitting contact, and another pair will be beginning action. This 
 would determine the length of teeth for given wheels of given 
 pitch, but arbitrary proportions are generally used. 
 
 Fig. 117 will serve to explain the terms used for the parts of 
 teeth. 
 
 Fig. 117 
 
 p = t -{- s = circular pitch. 
 
 t = s — b — space — backlash = ^p — ^b = thickness of tooth. 
 
 3 = t -\- b — thickness + backlash — ip -{-^b = space. 
 A = whole depth of tooth. 
 d = working depth of tooth. 
 
130 KINEMATICS OF MACEINEBT. 
 
 a = addendum or point = ^d. 
 
 r = root or flank = id -\~ c. 
 
 c = 7i — d = clearance. 
 
 In cast gears the backlash usually equals the clearance; in cut 
 gears no appreciable allowance is made for backlash. As noted 
 above, the pitch is determined by the load to be transmitted, and it 
 is the unit for the other dimensions; these may be varied consider- 
 ably, but the following values are suitable for cast gears : 
 
 t = 0A8]}; s = 0.52p; c—b = Mp; a = 0.33p; r = 0.37p; 
 h = 0.74p; d = 0.7p. 
 
 The rim of the wheel should be about as thick as the base of 
 the tooth (without the fillet). The hubs of gears are about 2 or 2^ 
 times the diameter of the shaft. 
 
 73. Circular or Circumferential Pitch and Diametral Pitch. — 
 Circular pitch is the distance, p, from centre to centre of adjacent 
 teeth, on the pitch line. The pitch circumference = pn = pitch 
 X number of teeth; and the pitch diameter = j^n -i- tt. This gives 
 awkward fractions either for the radii and distance between centres 
 or for the pitch. 
 
 As calculations are commonly based upon diameters instead of 
 circumferences, it is more convenient to give' the pitch in terms of 
 
 the pitch diameter. The circumference — pn = nd; '. - = -z ^^ p''- 
 
 p d -^ 
 
 = the number of teeth for each inch in diameter; and this ratio is 
 
 called the diametral pitch. If the circular pitch = jo = 1.57", the 
 
 diametral pitch, p' = n ~ l.bl = %. 
 
 pin = Ttd .•. d = ~n — —; or pitch diameter = number of 
 
 teeth -7- diametral pitch. Also n — dp' ; or number of teeth = 
 diameter X diametral pitch. 
 
 The addendum, a, with this system, is made 1" -^ p' ; e.g., if 
 p' — 4: {4: teeth to each inch of diameter), equivalent to a circular 
 pitch of .785", a = 1" -^ 4: = i inch. 
 
 The outside diameter =^ d -j- 2a = (n -\- 2) -r- p'. 
 
 The clearance, c, = 0.1 thickness of tooth. 
 
OUTLINES OF GEAR-TEETH. 
 
 131 
 
 74. Unsymmetrical Teeth. — If gears are always to turn in one 
 direction, the opposite sides of the teeth may have different out- 
 lines. Fig. 118 shows such teeth. 
 The working sides may belong to 
 any system ; the backs being so 
 formed that they will not interfere, 
 simply. Involutes, with an angle 
 of the normal greater than would 
 be practicable lor working faces of 
 teeth, are suitable for the backs. 
 Stronger teeth, for any pitch, are 
 obtained by this construction. 
 
 Gear-teeth are seldom made un- ^^ ^'9- "^ 
 
 symmetrical. Such forms will generally be more expensive, and 
 the necessary strength may be obtained by increasing the pitch. If 
 the force transmitted is excessive, and driving is always in one 
 direction, teeth of this form may be used to avoid excessive pitch. 
 
 75. Stepped and Twisted Gearing. 
 — The action of gear-teeth is smooth- 
 est when the contact point is at the 
 line of centres; for in this phase 
 there is pure rolling between the = 
 teeth, and, in the epicycloid system,, 
 the obliquity is also zero at this in- 
 stant. It is desirable to have the teeth 
 as short as the required arc of action 
 permits; but, as has been shown, this 
 is governed by the pitch, which is a 
 function of the force transmitted". 
 It is therefore unsafe to reduce the 
 pitch beyond a certain limit in a 
 given case ; but it is possible by 
 " stepped " teeth to retain the re- 
 quired pitch, and still have two teeth 
 always in contact near the line of centres. Suppose a spur gear 
 
 Fig. 120 
 
132 KINEMATICS OF MACHINERY. 
 
 to be cut by a series of equidistant planes, perpendicuhir to the 
 axes ; then let the slices into whieh the gear is divided be placed 
 
 as in Pig. 119. If there are JSf of these slices, each may be — tli 
 
 the pitch ahead (or behind) the adjacent one. In this arrangement, 
 the maximum distance of the nearest contact point from the line 
 of centres is the corresponding distance with ordinary spur-gears 
 of the same pitch and length of teeth, divided hy JV. 
 
 The thickness of the teeth has not been reduced by this modi- 
 fication, hence the strength has not been sacrificed. Large gears 
 are sometimes constructed on this principle, with two sets of teeth, 
 stepped one-half the pitch. 
 
 As the number of slices into which the gear of Pig. 119 is cut 
 increases (their thickness decreasing correspondingly) the teeth 
 approach those of Pig. 120, which represents the limiting form of 
 the stepped wheels. That is, when the number of slices becomes 
 infinite, the stepped elements become spirals. Gears with teeth of 
 this kind are called hoisted gears. It is to be noticed that the 
 action of these twisted teeth is similar to that of the corresponding 
 spur-gears, and they must not be confused with screw-gears which 
 they resemble in form, but which are not constructed for parallel 
 axes. The distinction will be considered more fully in a later 
 article. AVith twisted gears there is a component of the pressure 
 transmitted which tends to slide the wheel along the axis, or to 
 crowd the shaft to which the wheel is attached against the bearing. 
 This thrust against the bearing can be taken up by a collar, and 
 axial motion thus prevented, but such an expedient results in an 
 undesirable frictional loss, with risk of heating, etc. By twisting 
 
 the teeth on the opposite sides of the 
 \1 central section in opposite directions, 
 as shown in Pig. 120 {a), the axial 
 efforts due to these two halves 
 Fig. 120(a) b balance each other, and there is no 
 
 such thrust imparted to the shaft. In actual gears of this form, 
 the two halves may be cast in one piece, if the teeth are not to be 
 
OUTLINES OF OEAR-TEETH. 13S 
 
 machined ; but if cut gears are used, the two halves are made as. 
 two separate gears of opposite inclinations (tlie elements of one 
 half are right-handed helices, and of the other are left-handed 
 helices), and these two gears may be attached firmly to the shaft, 
 side by side, thus constituting practically one wheel. 
 
 It will be seen that these twisted gears always have one contact 
 point on the line of centres, if the twist within ihe width (or the 
 half width, with the double form of Fig. 120 {a)) of the gear is at 
 least equal to the pitch, and the action at this one point is pure 
 rolling. Now if the addenda of 
 these teeth are relieved as indi- 
 cated in Pig. 121 by the dotted 
 lines, the faces will not act upon 
 the flanks of the mating gear till 
 the pitch point of any section 
 comes into contact; that is, till 
 the line of centres is reached, when the action is pure rolling. If 
 the mating gear is similarly treated, the two gears only touch at a 
 point, and this point is always in the line of centres. Thus contact 
 for any tooth begins when the foremost section reaches the line of 
 centres ; it travels along the pitch element of the tooth ending as 
 the last section passes the line of centres, the most advanced sec- 
 tion of the next pair of teeth taking up the action in turn. This 
 concentrates the force transmitted at a single point, theoretically, 
 which may result in too intense a pressure in heavy work ; but it 
 has the effect of producing pure rolling between the teeth in con- 
 tact at all times. This is "probably the only example of combined 
 pure rolling, constant angular velocity ratio, and positive driving. 
 
 76. Non-circular Gears. — In Arts. 46, 47, 48, and 49 the ac- 
 tion of rolling ellipses, rolling logarithmic spirals, general rolling 
 curves, and lobed wheels was briefly explained. It has been seen 
 that cylinders (in the general sense) corresponding to these curves 
 may roll together, and it has been stated that such surfaces may be 
 used as pitch surfaces for non-circular gears. 
 
 The general method of describing gear-teeth, as given in Art, 
 
134 KINEMATICS OF MAGEINERT. 
 
 59, may be applied in designing teeth for such gears, but a con- 
 venient approximate method will be indicated, using the rolling 
 ellipses for illustration. Circular arcs can be drawn which closely 
 approximate the ellipse at any point, and the methods for circular 
 flitch line gears can then be used for the teeth. Or accurate 
 elliptical curves may be drawn ; then lay ofE the pitch upon them 
 and apply the method given in Arts. 59, 62, or 68 in generating 
 the teeth on these arcs. Of course, with ii given describing curve, 
 the teeth on portions of the pitch line which have different curva- 
 ture will not have the same form. 
 
 This approximate method can be applied to other rolling curves 
 as well as to the ellipse, and it is thus possible to form teeth for 
 iiny of the non-circular forms (including the lobed wheels of article 
 49) which will transmit motion similar to that due to the rolling of 
 the pitch curves. The general method of Art. 60 may be used if 
 preferred. 
 
 77. Approximate Methods of Constructing Profiles. — The exact 
 construction of the tooth profiles is somewhat tedious, and in many 
 practical applications simpler approximate outlines may be substi- 
 tuted. Gears with cast teeth, especially if the pitch is small, de- 
 part somewhat from the ideal form, however carefully the patterns 
 may be made ; and, therefore, some one of the approximate methods 
 is generally used for laying out the patterns. In making cutters for 
 cut gears, the exact method is usually employed.* 
 
 The arcs of the curves (epicycloids and hypocycloids, or invo- 
 lutes) used in gear-teeth are so short that circular arcs can be 
 found which very closely approximate these curves ; and most of 
 the approximate constructions are circular-arc methods. 
 
 A method given in Unwin's Machine Design has the merit of 
 requiring no tables or special instruments, and it will be described 
 first. In Pig. 122, A and B are the two pitch circles, and G and 
 
 * A little book published by tbe Pratt & Wliitney Co. gives a description of 
 tbe machine used by this company for accurately making these cutters auto- 
 matically. This treatise was written by Professor McCord, and is reproduced 
 in his Kinematics. 
 
OUTLINES OF OEAR-TEETH. 135 
 
 G' are the describing circles. Ph is the height of the addendum 
 ■of B, and Pe represents two thirds this height. If a circle be 
 drawn through e with a centre at the centre of the pitch circle B, 
 it cuts the describing circle Ging; and if the arc Pb, on the pitch 
 circle B, be laid off equal to the arc Pg on the generating circle G, 
 
 i and g are two points in the tooth outline, as in the exact con- 
 struction. Draw gP, the normal to the tooth profile at g. Now 
 find by trial a circular arc with a centre on Pg, or its extension 
 beyond P, which will pass through g and b. This arc passes through 
 two points of the exact tooth outline, and its tangent at g also corre- 
 sponds in direction with that of the true epicycloid, as the normal 
 aX this point is Pg for both the exact and the approximate curves. 
 If the arc Pa is laid off on the pitch circle A, equal to the arc Pg, 
 another circular arc, with a centre on the normal Pg, will pass 
 through a and g, and it will approximate the flank of ^. By a 
 similar method, Pe' is laid off equal to two thirds the height of the 
 addendum of A, and a circular arc with a centre on Pg', and passing 
 through a' and g', is an approximation to the exact face profile of 
 A. The approximate outline for the flank of B is an arc passing 
 through b' and g', with a centre on g'P, or its extension. The point 
 e need not necessarily be two thirds of Ph; but this fraction gives 
 a good distribution of the error. 
 
 A method due to Professor Willis has been more widely used, 
 perhaps, than any other. It may be briefly explained as follows: 
 
136 
 
 KINEMATICS OF MACHINEBT. 
 
 Lay off from P (Fig. 123) the arcs Pa = Pa' — one-half the pitch, 
 and draw radii Oa 'and Oa'; then draw the lines mm and m'm' 
 through a and a', making an angle 4> with Oa and Oa' respect- 
 ively. At a point c (on mm) take a centre, and draw a circular arc 
 Pf through P; also with a centre at c' (on m'm') draw the arc Pf 
 through P- The angle and the centres c and c' may be so chosen 
 that these arcs will have radii of curvature equal to the mean radii 
 of curvature of the proper epicycloidal and hypocycloidal faces and 
 
 Fig. 123 
 
 flanks of the tooth. The angle was found by the originator of 
 this method to give the best results when taken at 75", and the 
 radii are given by the following formulas, in which p = pitch, and 
 
 = number of teeth in the wheel : Radius for faces 
 
 2 \r: 
 
 Radius for flanks = -^ , 
 
 2 \n — 13, 
 
 \n + 12/ 
 An instrument known as 
 
 Willis's Odontograph facilitates these operations. The form of 
 this instrument is indicated by the dotted lines of Fig. 123. It is 
 graduated along the edge m'm-' each way from the point a', and a 
 table which accompanies the instrument gives the positions of the 
 centres c and c' in terms of these graduations for wheels of givea 
 numbers of teeth and pitch. 
 
OUTLINES OF QEAB-TEETH. 
 
 137 
 
 Mr. George B. Grant has improved upon this odontograph by 
 tabulating the radii for faces and flanks, and also tabulating the 
 radial distances of the centres c and c' from the pitch circle. Mr. 
 Grant has completed a similar set of tables, called the Grant Odonto- 
 graph, which aroused in precisely the same way; but the values are 
 different, giving somewhat different tooth profiles from those of the 
 Willis system. The Willis method gives circular arc tooth outlines 
 which are correct for one point, and which have radii equal to the 
 mean radius of curvature of the exact curve. The approximate 
 faces derived by this system lie entirely loithm the true epicycloids. 
 Mr. Grant's system gives arcs which pass through three points of 
 the exact profiles of the faces, and thus more closely approximate 
 the correct curves. 
 
 The application of Grant's Cycloidal Odontograph, or, as its 
 author calls it, from the method of deriving it, the Three-Point 
 Odontograph, is shown in Fig. 124. The accompanying table is 
 
 LINE OF FLAWK CEW TFgf; 
 
 given, together with his table for constructing approximate involute 
 teeth. This matter is taken from Odontics, or the Teeth of Gears, 
 by George B. Grant, and is reproduced by permission of the author. 
 To use the table in drawing approximate epicycloidal teeth pro- 
 ceed as follows: Draw the pitch line and set off the pitch, dividing 
 the latter properly for thickness of tooth and space. The table 
 gives values both in terms of One Diametral Pitch (equal 3.14" 
 
138 
 
 KINEMATICS OF MAOEINEBT. 
 
 circular pitch), and of One Inch Circular Pitch. Use the part of 
 the table corresponding to the system of pitch employed. 
 
 THREE-POINT ODONTOGRAPH. 
 
 STANDARD CYCLOIDAL TEETH. INTBEGHAN6EABLE SERIES. 
 (From Geo. B. Grant's " Odontics.") 
 
 
 
 For One Diametral Pitch. 
 
 For One Inch Circular Pitch. 
 
 
 
 For any other pitch divide 
 
 by that 
 
 For any other 
 
 pitch multiply by 
 
 Numbe 
 
 r of Teeth. 
 
 pitch. 
 
 
 thai pitch. 
 
 
 
 Faces. 
 
 Flanks. 
 
 Faces. 
 
 Flanks. 
 
 Exact. 
 
 Intervals. 
 
 Rad. 
 
 Dis. 
 
 Had. 
 
 Dis. 
 
 Rad. 
 
 Dis. 
 
 Rad. 
 
 Dis. 
 
 10 
 
 10 
 
 1.99 
 
 .02 
 
 - 8.00 
 
 4.00 
 
 .63 
 
 .01 
 
 -3.55 
 
 1.37 
 
 11 
 
 11 
 
 3.00 
 
 .04 
 
 -11.05 
 
 6.50 
 
 .63 
 
 .01 
 
 -3.34 
 
 3.07 
 
 12 
 
 13 
 
 3.01 
 
 .06 
 
 00 
 
 00 
 
 .64 
 
 .03 
 
 00 
 
 00 
 
 13i 
 
 13-14 , 
 
 2.04 
 
 .07 
 
 15.10 
 
 9.43 
 
 .65 
 
 .03 
 
 4.80 
 
 3.00 
 
 15i 
 
 15-16 
 
 2.10 
 
 .09 
 
 7.S6 
 
 3.46 
 
 .67 
 
 .03 
 
 3.50 
 
 1.10 
 
 17* 
 
 17-18 
 
 2.14 
 
 .11 
 
 6.13 
 
 2.20 
 
 .68 
 
 .04 
 
 1.95 
 
 0.70 
 
 30 
 
 19-21 
 
 2.30 
 
 .13 
 
 5.13 
 
 1.57 
 
 .70 
 
 .04 
 
 1.63 
 
 0.50 
 
 23 
 
 22-24 
 
 2.26' 
 
 .15 
 
 4.50 
 
 1.13 
 
 .73 
 
 .05 
 
 -1.43 
 
 0.36 
 
 37 
 
 25-29 
 
 2.33 
 
 .16 
 
 4.10 
 
 0.96 
 
 .74 
 
 .05 
 
 1.30 
 
 0.29 
 
 33 
 
 30-36 
 
 3.40 
 
 .19 
 
 3.80 
 
 0.72 
 
 .76 
 
 .06 
 
 1.30 
 
 0.23 
 
 42 
 
 37-48 
 
 3.48 
 
 .23 
 
 3.53 
 
 0.63 
 
 .79 
 
 .07 
 
 1.12 
 
 0.20 
 
 58 
 
 49-73 
 
 2.60 
 
 .35 
 
 3.33 
 
 0.54 
 
 .83 
 
 .08 
 
 1.06 
 
 0.17 
 
 97 
 
 78-144 
 
 2.83 
 
 .38 
 
 3.14 
 
 0.44 
 
 .90 
 
 .09 
 
 1.00 
 
 0.14 
 
 290 
 
 145-300 
 
 3.9s! 
 
 .31 
 
 3.00 
 
 0.38 
 
 .93 
 
 .10 
 
 0.95 
 
 0.13 
 
 00 
 
 Rack 
 
 2.96 
 
 .34 
 
 8.96 
 
 0.34 
 
 .94 
 
 .11 
 
 0.94 
 
 0.11 
 
 The example of Pig. 124 is a wheel of 20 teeth, 2 diametral 
 pitch, hence the pitch circle is 10 inches diameter (this figure is 
 not reproduced full size). Opposite 20 teeth in the table we find 
 .13 in the column, of distances for faces ("dis."); divide this by the 
 diametral pitch (2), giving .06" a/ the distance of the circle of face 
 centres from the pitch circle. Lay this distance off inside the 
 pitch circle, and draw a circle through this point, concentric with 
 the pitch circle. In a similar way the distance for flanks (1.57) is 
 divided by 2, giving .78", which is laid off outside the pitch circle, 
 imd a circle is drawn through this point. All tooth /aces are to be 
 
OUTLINES OF GEAR-TEETH. 139 
 
 drawn with circular arcs having centres on the first of these lines 
 of centres, and the flanks are drawn by arcs having centres on the 
 last found line of centres. The tabular radius of faces for 20. teeth 
 is given as 2.20, and dividing this by the diametral pitch, Ave get 
 1.10" as the radius for the faces of the 3-pitch wheel. With this 
 radius and centres on the face centre line, draw arcs through the 
 proper points in the pitch circle, of course having the concave 
 sides of the arcs toward the body of the teeth. In a similar way, 
 the tabular radius tor flanks (5.12) is divided by the diametral pitch, 
 giving 2.56" as the corrected radius. With centres on the flank 
 centre line, draw arcs with this radius meeting the face arcs already 
 drawn at the pitch poiht, and with the concave sides towards the 
 spaces. Terminate the tooth profiles by the addendum and root 
 circles, determined as in Arts. 73 and 73, and put in fillets at the 
 bottoms of the spaces. 
 
 If the circular pitch is used the construction is .similar, using 
 the appropriate portion of the table, but multiplying the tabular 
 values by the circular pitch in inches instead of dividing. 
 
 As explained above, this table is calculated for arcs which pass 
 through three points in the true curve. It is recommended that 
 the stitdent construct tooth profiles on a large scale by the exact 
 method, and then draw the approximate profiles (superimposed), 
 for comparison. 
 
 Grant's involute Odontograph given below is used as follows: 
 Lay ofE the pitch circle, addendum, root and clearance lines, as in 
 the preceding case. " Draw the base line one sixtieth of the pitch 
 diameter inside the pitch line. Take the tabular face radius on the 
 dividers, after multiplying or dividing it as required by the table, 
 and draw in all the faces from the pitch line to the addendum line 
 irom centres on the base line. Set the dividers to the tabular flank 
 radius (corrected), 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." [Grant's 
 Teeth of Gears, p. 30.] 
 
140 
 
 KINEMATICS OF MACEINEUY. 
 
 INVOLUTE ODONTOGRAPH. 
 Standabd Interchangeable Tooth, Centkes on the Babe Line. 
 
 Teeth. 
 
 10 
 
 n 
 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 
 Divide 
 
 by the 
 
 Multiply by the 
 
 
 Diametral Pitoh. 
 
 Circular Pitch. 
 
 
 
 
 
 
 Teeth. 
 
 Face 
 
 Flank 
 
 P'ace 
 
 Flank 
 
 Badius. 
 2.28 
 
 Radius. 
 
 Radius. 
 
 Radius. 
 
 28 
 
 .69 
 
 .73 
 
 .22 
 
 2.40 
 
 .83 
 
 .76 
 
 .27 
 
 29 
 
 ---A.^51 
 
 .96 
 
 .80 
 
 .31 
 
 30 
 
 2.6-3 
 
 1.09 
 
 .83 
 
 .34 
 
 81 
 
 2.72 
 
 1.22 
 
 .87 
 
 .39 
 
 32 
 
 2.82 
 
 1.34 
 
 .90 
 
 .43 
 
 33 
 
 2.92 
 
 1.46 
 
 .93 
 
 .47 
 
 34 
 
 3.02 
 
 1.58 
 
 .96 
 
 .50 
 
 35 
 
 3.12 
 
 1.69 
 
 .99 
 
 .54 
 
 36 
 
 3.22 
 
 1.79 
 
 1.03 
 
 .57 
 
 37-40 
 
 3. 3 J 
 
 1.89 
 
 1.06 
 
 .60 
 
 41-45 
 
 3.41 
 
 1.98 
 
 1.09 
 
 .63 
 
 46-51 
 
 3.49 
 
 2.06 
 
 1.11 
 
 .66 
 
 52-60 
 
 3.57 
 
 2.15 
 
 1.13 
 
 .69 
 
 61-70 
 
 3.64 
 
 2.24 
 
 1.16 
 
 .71 
 
 TT?^ 
 
 3.71 
 
 2.33 
 
 1.18 
 
 .74 
 
 91-120 
 
 3.78 
 
 2.42 
 
 1.20 
 
 .77 
 
 121-180 
 
 3.85 
 
 2.50 
 
 1.23 
 
 .80 
 
 131-360 
 
 Divide by the 
 Diametral Pitch 
 
 Multiply by the 
 Circular Pitch. 
 
 34 
 
 48 
 1.61 
 1.83 
 2.07 
 3.46 
 3.11 
 ^4.26 
 6.88 
 
 Grant's special directions for drawing the teeth of the involnte 
 racli are substantially as follows: To draw the teeth for the involnte 
 rack, draw lines at 75° with the pitch line of the rack; the outer 
 quarter of the tooth length (one half the addendum) is to be 
 rounded off bj- an arc with a radius equal to 3.1u" divided by the 
 diametral pitch, or .67" multiplied by the circular pitch. This is 
 to avoid interference. 
 
 78. Bevel-gears. - It was shown (see Art. 52) that a pair of cones 
 can be placed on intersecting axes in such a manner that they will 
 transmit motion with a given angular velocity ratio if they roll to- 
 gether without slipping. Such rolling cones may be used for pitch 
 surfaces of bevel-gears just as rolling cylinders are used for the 
 pitch surfaces of spur-gears and teeth can be formed on these 
 conical pitch surfaces which will transmit a positive motion equiva- 
 lent to that of the rolling cones. 
 
OUTLINES OF QEAR-TEETH. 
 
 141 
 
 In. treating of spur-gearing plane sections at right angles to the 
 axes were used to represent the gear, and the tooth outline was 
 considered to be developed by the rolling- of one plane curve 
 (the describing line) upon another plane curve (the pitch line). 
 U'he real teeth are, of course, solids bounded by ruled surfaces, 
 all transverse sections of which are exact counterparts of the 
 l)lane curves discussed. That is, the actual teeth are not lines 
 generated by a point in the describing curve as it rolls upon the 
 pitch line, but they are really surfaces' generated by an element of 
 the describing cylinder as it rolls upon the pitch cylinder. 
 
 Spur-gears coming under the head of plane motions permit 
 representation by plane sections, as explained in Art. 10. This 
 simple treatment cannot be applied to bevel-gears, for although 
 each sepai-ate gear has a plane motion (rotation) about its axis, 
 taking two cones rolling together, the relative motion is a spherical 
 motion (see Art. 12). To construct bevel gear-teeth two projections 
 are required. 
 
 Just as an element of one cylinder in rolling upon another cylin- 
 
 Fig. 125 
 
 der generates a tooth surface, so an element of one cone in rolling 
 upon another cone sweeps up a surface which can be used as the 
 basis of a bevel-gear tooth. Fig. 125 shows a pitch cone ^ with 
 the generating cone (of equal slant height) in contact with it 
 
142 
 
 KINEMATICS OF MAGHINERT. 
 
 along a common element PQ. All points in the bases of both 
 cones are at the same distance from the common apex, hence these 
 bases are small circles of a sphere which has a radius equal to the 
 common slant height. If the generating cone be rolled upon the 
 pitch cone, a point in the base of G, as g, 'will describe a curve on 
 the surface of the sphere relative to the pitch cone. This curve is. 
 analogous to the epicycloid, the derivation of which was treated in 
 Art. 62, and the curve now under consideration may be called a 
 spherical epicycloid. In a similar manner another generating cone 
 G' can roll inside the pitch cone, a point g' in its base tracing a. 
 spherical hypocycloid on the surface of the sphere. Points on other 
 transverse sections of these generating cones would trace similar 
 curves on spheres of different radii. A line passing through the 
 centre of the sphere (the common apex of the cones) and moving 
 along the spherical epicycloids and hypocycloids described as above, 
 would give surfaces portions of which could be used as tooth 
 boundaries. In other words, the elements of the describing cones 
 
 which pass through g and g' would 
 sweep up these surfaces. If two pitch 
 cones of equal slant height have teeth 
 generated in the manner just outlined, 
 they will work together properly, trans- 
 mitting a positive motion equivalent to 
 the rolling of the pitch surfaces, pro- 
 vided the pitch of the teeth agree, and 
 that the faces of each wheel are de- 
 scribed by the same generating cone 
 which describes the flanks of the other 
 wheel. 
 
 Pig. 126 indicates two pitch cones A 
 and B, with axes QO and QO', respect- 
 ively, and a common element of contact 
 PQ. The describing cones are G and 
 G', with axes Qo and Qo' ; and if all 
 four axes lie in one plane (a meridian plane of the sphere), they 
 
OUTLINES OF GEAR-TEETH. 143 
 
 can roll together about fixed axes, always having a common con- 
 tact element in PQ. As the rolling proceeds, such an element of G 
 as gQ sweeps up the faces for £ and the flanks for A; and, at the 
 same time, the element g'Q of G' sweeps up the faces of A and 
 the flanks of B. 
 
 The faces of £ and the flanks of A always have gQ f or £( 
 common element, and a plane through the three points FgQ, 
 the common normal plane at the element of contact of these 
 surfaces, always passes through the contact element of the 
 pitch cones PQ. Likewise, the normal plane to the faces of A 
 and the flanks of B, Pg'Q, always passes through PQ; hence teeth 
 bounded by these swept-up surfaces will transmit motion with a 
 constant angular velocity ratio. The analogy between this case and 
 that of the spur-gear teeth, treated in Art. 62 (Fig. 110), is so close 
 that further .discussion is hardly necessary. 
 
 The describing surfaces are not necessarily right cones of circu- 
 lar cross-section, though these are the figures which correspond to 
 the epicycloidal class of spur-gears, and are the only forms com- 
 monly employed. Any cone with an apex at the common apex of 
 the pitch cones, and tangent to them along their common element 
 might be used, as it would satisfy the kinematic requirements. 
 
 It is diflBcult to construct spherical epicycloids and hypocycloids, 
 and to represent them on paper and in practice a method known 
 as Tredgold's approximation is always employed. 
 
 79. Tredgold's^Approximate Method of Drawing Bevel-gear Teeth. 
 — Fig. 127 shows the projection of two cones (with bases PJf and 
 PJSf) on a plane parallel to both axes QO and QO'. The line 00' 
 is drawn perpendicular to the contact element PQ, then OM is 
 perpendicular to MQ, and O'iVis perpendicular to NQ. A cone 
 can be constructed on the axis OQ with OP and OJf as elements, 
 and another on O'Q with O'P and O'iVas elements. These cones 
 are called normal cones to A and B, respectively, as any elemeTit of 
 one of these cones is perpendicular to an element of the pitch cone 
 having the same axis and the same base. The surfaces of these 
 normal cones approximate the spherical surface for a short space 
 
144 
 
 KINEMATIC a OF MACHINERY. 
 
 either side of the pitch circles, and the conical surfaces have the 
 practical advantage that they can be developed upon a plane for 
 the construction of tooth profiles. Tredgold's approximate method 
 consists in describing tooth outlines on these developed surfaces of 
 the normal cones, and then wrapping these surfaces back to their 
 original positions. The development of the normal cone surfaces 
 is indicated in Fig. 127 by PM'O, and PN'O'. Upon the de- 
 
 Fig. 127, 
 
 veloped bases of these cones {PM' and PW) as pitch lines, tooth 
 outlines can be drawn by any of the methods used for spur-gears, 
 just as if these were the pitch lines of such gears; and when these 
 surfaces are rolled back into the normal cones the ends of the teeth 
 are given by the profiles constructed in this way. A straight line 
 passing through Q and following such a profile would sweep up 
 tooth surfaces, all elements of which are right lines converging at 
 Q. Within the limits of practice, such teeth, if properly con- 
 structed, agree quite closely with the exact forms. 
 
 The application of this method is shown in detail in Fig. 128 
 for the teetli of a single wheel. The pitch cone is shown in side 
 elevation by MQM, and in plan by the circle Jf,Jf,. The side ele- 
 vation of the normal cone is projected in MOM, and the develop- 
 ment of the pitch circle is shown by MM'. The pitch, which must 
 
OUTLINES OF OEAR-TEETR. 
 
 145 
 
 be an aliquot part of the pitch circle J/jif, ( = MM times tc), is 
 laid off on MM', and the addendum and root circles (AA' and 
 EE', respectively) are drawn to give the proper length of teeth. 
 The tooth outline is now constructed on MM' as for a spur-gear. 
 
 It is evident that all of the pitch points will fall upon the line 
 MM, in the side elevation, when the normal cone surface is re- 
 
 Fig. 128. 
 
 turned to its original position; that the outer ends of the teeth will 
 fall upon A A ; and that the bottoms will fall upon EE. In the 
 ■other projection, the pitch points will all lie on the circle MiMi; 
 the tops will fall on the circle through A,A^; and the bottoms will 
 be in the circle EiEi. In this last projection (plan) the teeth will 
 all appear the same, and they will have their true thickness at all 
 parts; but the height {AE) will be shortened to A^Ei. Divide up 
 the circle M,Mi into the proper number of divisions for teeth and 
 
146 EINEMATIGS OF MACHINERY. 
 
 spaces, and draw radii through the middle of the teeth divisions. 
 Lay off half the thickness of the tooth at the pitch line (as obtained 
 from the construction on developed pitch line MM') each side of 
 these middle radii upon the circle M^M^; then lay off half the 
 thickness of the top in a similar way on the circle A^A/, and half 
 the thickness at the bottoms on the circle -B,i?,. The half thick- 
 ness at positions intermediate between the pitch circle and the ad- 
 dendum or root circles can also be laid' off on the corresponding 
 circles in the plan, taking as many such thicknesses as the desired 
 accuracy requires. The method of finding these intermediate 
 points is indicated in Fig. 128. Through the points thus found, the 
 curves A^M^R^ can be drawn, giving the plan of the large ends of 
 the teeth. To complete the plan of the wheel we may proceed as 
 follows: Draw radii from A,, J/,, i?,, etc., to Q^; then lay off Aa, 
 on the side elevation, equal to the desired length of the tooth, or- 
 the face of the gear, and draw ainr parallel to AMR; from a, m, and 
 r, points lying in elements from Q to A, M, and R, respectively, 
 carry lines across parallel to QQ/, then circles with Q^, as a centre 
 and tangent to these several parallels are the projections in the 
 plan of. the addendum, pitch, and root circles for the small end of 
 the teeth; , As all, elements of the teeth converge in plan at Q^,the- 
 intersections of radii through J„ Jf„ andi?, with these circles last 
 drawn locate points a„ m^, and r^ in the plan of the small ends of 
 the teeth. Curves through these intersections, with the portions of 
 the radial elements intercepted between them and the outer 
 curves, will complete this projection of the teeth. Eeturning to 
 the side elevation, the lines aa, mm, and rr, are the projections of 
 the smaller addendum, pitch, and root circles. To complete th& 
 side elevation of the teeth project across (parallel to Q^Q) from the 
 various points on A^A^ to A A; from M^M^ to MM; from R^R, tc^ 
 RR, etc., and draw curves through the intersections thus made. 
 This will give the side elevation of the large ends of the teeth by 
 passing curves through the corresponding intersections. In a 
 similar way the side elevation of the smaller ends is obtained, and 
 elements through Aa, Mm, Rr, etc., completes this view. It is 
 
OUTLINES OF GEAR-TEETH. 147 
 
 evident that each of the teeth appears in this projection with its 
 own distinctive form. 
 
 The ring, or rim, which supports the teeth usually has a thick- 
 ness equal to the roots of the teeth at the large ends, and this rim, 
 with the hub, arms, etc., can now be drawn. 
 
 80. Peculiar Smoothness in Operation of Bevel-gearing. — Among 
 spur-gears of an interchangeable system, those with the larger pitch 
 circles will drive more smoothly, other conditions being the 
 same. By referring to the bevel-gear of Fig. 128 it will be seen 
 that there are 16 teeth on a pitch circle of radius Q^Mi (diameter 
 = MM); but these teeth have profiles similar to those of a spur- 
 gear with a pitch radius OM, equal to the slaiit height of the 
 normal cone, and therefore the action of the bevel-gear would cor- 
 respond to that of this larger spur-gear instead of to a spur-gear of 
 diameter MM. 
 
 The actual pitch diameter of the bevel-gear is to that of the 
 equivalent spur-gear (so far as smoothness of running is con- 
 cerned) as sin ^0^: 1; thus if the pair of bevel-gears are equal, 
 angle AOQ = 45°, when sin AOQ : 1 :: lV2 : 1 :: .707 : 1. 
 
 81. Non-interchangeability of Bevel-gears. — Bevel-gears are 
 almost always made to work together in pairs, and it is not there- 
 fore of great importance to adopt a standard describing circle for 
 all pairs of the same pitch. If two intersecting axes approach 
 each other at a fixed angle, there is but one bevel-gear which will 
 work properly with any other gear;* for a change in the angular 
 velocity ratio involves a change in the direction of the contact ele- 
 ment {Ac of Fig. 96), and hence a change in loth pitch cones. A 
 given bevel-gear could work with more than one other wheel if the 
 inclination of the axes varied correspondingly, but this is a con- 
 
 * Mr. Hugo Bilgram has produced sets of bevel-gears, by his gearsbaper, 
 in which several different sizes of gears work correctly with a single gear, and 
 all the axes make the same angle with the common driver. However, the 
 pitch cones all of these gears do not have a common apex, although the teeth 
 elements all converge to a common point. These gears are not, properly, of 
 the common type of bevel-gears. 
 
148 KINEMATICS OF MACHINERT. 
 
 dition seldom met, and so these gears may generally be designed 
 to work in pairs without regard to other gears. 
 
 The describing circles (if the epicycloidal system is used) may 
 be so taken that both wheels will have radial flanks, which gives 
 a simple form of teeth to construct, though this is not always de- 
 sirable. For convenience of manufacture it is desirable to have a 
 uniform system, usually; and when bevel-gears are cut with rotary 
 milling cutters of the common type, standard cutters are used for 
 various gears of the same pitch. This will be discussed in the 
 next article. 
 
 In a great majority of cases requiring bevel-gears the axes 
 are at right angles to each other, and " stock-gears " can frequently 
 be obtained from gear-makers for such cases if the proportions are 
 not unusual. These stock-gears are generally much less expensive 
 than gears made to order; but special gears are almost always 
 required when the angle of the axis is other than 90°. When the 
 two gears are equal (angular velocity ratio 1 : 1), the gears are 
 called Mitre Gears. 
 
 82. Manufacture of Gears. — Gear -teeth are either cut in a 
 machine or are cast. For the rougher classes of work, and very 
 large gears, it is common practice to use gears with cast teeth; but 
 cut gears are now used almost exclusively for the better grades of 
 work if of moderate size, and their use is becoming more extended. 
 When gears are cast, it is important to form the patterns very care- 
 fully, and especially to space the teeth accurately. With the 
 utmost care, however, it is impossible to get very smooth and accu- 
 rately spaced teeth, so clearance between the sides of the teeth, or 
 backlash, must be provided. With small gears the enlargement of 
 the mould, due to " rapping " the pattern, more than compensates 
 for the shrinkage, and unless this is looked after in the pattern- 
 shop and foundry, the teeth may be too thick when cast. 
 
 A convenient device for forming the teeth of the pattern is 
 shown in Fig. 129. A block of hard wood (preferably of a color 
 quite distinct from the wood of the pattern) is shaped as shown, so 
 that the sections at amr and AMR correspond to the two ends of 
 
OUTLINES OF QEAR-TEETH. 149- 
 
 the teeth (for a spur-gear these sections are, of course, alike). 
 The middle portion is cut out, so that the distance L equals the 
 length of a tooth; that is, the part removed corresponds to the 
 form of a tooth. The stock for the teeth of the pattern is gotten 
 out in lengths equal to L, and large enough in cross-section to 
 make a tooth. The block of Fig. 129 is screwed in the vise, and 
 it may have two pointed brads projecting upward through the 
 bottom. Then a piece of the prepared stock is forced down into 
 
 1 . ,■ >! 
 
 Fig. 129. 
 
 the space in this "form " and is then planed up with "hollow and' 
 round " planes. By working down to the form it is quite easy to 
 produce a large number of teeth very uniform in shape. 
 
 Gear-cutters are of two general classes: those which operate on 
 the milling-machine principle, cutting the teeth by milling-cutters, 
 of a cross-section corresponding to the shape of the space between 
 two teeth, and those which act on the planer principle, cutting 
 the teetli element by element. 
 
 •For spur-gears of moderate size the former method is most 
 usual. For absolute accuracy it requires a different cutter for 
 each number of teeth of each pitch. Practically this is not neces- 
 sary, as the form of the teeth, for a given pitch, does not vary- 
 greatly with small changes in the number of teeth of a wheel,, 
 except with the lower-numbered wheels. The following tables,, 
 taken from the catalogue of the Brown and Sharpe Manufacturing 
 Co., gives a list of the different cutters for one set of involute and 
 epicycloidal cutters, covering all wheels from a pinion of 12 teeth 
 to a rack. 
 
160 
 
 KINEMATICS OF' MACHINERT. 
 
 INVOLUTE SYSTEM. 
 
 Eight Gutters are made for each pitch, as follows : 
 
 No. 1 will cut wheels from 135 teeth to a rack. 
 
 3 " 
 
 3 " 
 
 4 " 
 
 5 " 
 
 6 " 
 
 7 " 
 
 8 " 
 
 55 " 
 
 " 134 tee 
 
 35 " 
 
 ■' 54 " 
 
 36 " 
 
 " 34 " 
 
 21 " 
 
 " 35 " 
 
 17 " 
 
 " 20 " 
 
 14 '■ 
 
 " 16 " 
 
 13 ■' 
 
 " 13 " 
 
 EPICYCLOIDAL SYSTJIM. 
 Twenty -four Gutters foi- each pitch. 
 
 Cutter 
 
 A cuts 
 
 
 13 teeth 
 
 B '■ 
 
 
 13 " 
 
 C " 
 
 
 14 " 
 
 
 
 15 " 
 
 16 " 
 
 F " 
 
 
 17 " 
 
 G " 
 
 
 18 " 
 
 H " 
 
 
 19 " 
 
 I " 
 
 
 30 " 
 
 J " 
 
 21 
 
 to 33 " 
 
 K " 
 
 33 
 
 " 34 " 
 
 L " 
 
 35 
 
 " 26 '■ 
 
 Cutter M cuts 27 to 29 teeth. 
 
 N 
 
 ' 30 ■' 33 
 
 
 
 ' 34 " 37 
 
 P 
 
 ' 38 " 43 
 
 Q 
 
 ' 43 " 49 
 
 R 
 
 ' 50 " 59 
 
 S 
 
 ' 60 " 74 
 
 T 
 
 ' 75 " 99 
 
 U 
 
 ' 100 " 149 
 
 V 
 
 ' 150 " 349 
 
 W 
 
 ' 350 or more 
 
 X 
 
 ' Rack. 
 
 The cutters are so formed for each group that the error is dis- 
 tributed as uniformly as possible, and a set of cutters so arranged 
 is called an equidistant series. A greater number of cutters in the 
 series will reduce the maximum error, while a smaller nitmber 
 would increase this error. The exact number to be used is there- 
 fore a practical matter involving a compromise between accuracy 
 and outlay for cutters. 
 
 Bevel-gears are frequently cut by these milling cutters; but 
 they can, of course, not be so cut with extreme accuracy, for the 
 elements of a bevel-gear tooth should all converge towards the apex 
 •of the pitch cone; and as the teeth cut by such a cutter must have 
 parallel elements, it is impossible to make a cutter which would 
 give the correct form except at one section of the tooth. By limit- 
 
OnXLINES OF GEAR-TEETH. 151 
 
 ing the face of the gear (length of teeth along the elements), and 
 by cutting each tooth side with a separate cut, bevel-gears can be 
 made by this method which work reasonably well. Bevel-gear 
 teeth, to be perfectly accurate, should be planed by a tool the 
 ■cutting point of which is always directed towards the apex of the 
 pitch cone, or they should be formed by some equivalent process. 
 Such gear-planers may also be used for spur-gears. With this type 
 of machine, the cutting tool is guided by templets, or it is otherwise 
 Klirected so that the desired form of tooth is secured. With either 
 -class of machines, the "blank" is carried on an arbor with. a 
 ■'■■ dividing head " for spacing the teeth. 
 
 A blank for cutting a spur-gear is essentially a cylinder of 
 ■diameter equal to the addendum circle, and a thickness or face 
 ■equal to the length of tooth. A bevel-gear blank consists of a pair 
 ■of conical frusta with a common base, as a-A-B-B . A-a (Pig. 128). 
 As the backs of the teeth, AR, are perpendicular to the pitch 
 -element, MQ, the angle aAB is less than 90° ; because the angle 
 AQO is greater than the angle of the pitch cone. 
 
 Where many large cast gears are made, a gear moulding 
 machine is sometimes used, as it produces accurate work and 
 reduces the cost of patterns. A stake, or arbor, is set upright at 
 the centre of the mould, which may be swept up in loam to 
 approximately the outside form of the wheel. The pattern for the 
 teeth, simply a block with a few teeth attached which corresponds 
 to a segment of the entire rim, is fastened to the stake by an arm. 
 This arm holds the segmental pattern at the proper distance from 
 the axis of the wheel, and the arm and pattern can be turned about 
 this axis. The pattern can also be withdrawn towards the centre 
 or upward. In moulding the gear, this segment is placed in 
 position and a few teeth are moulded by filling in about the pattern 
 with the sand. The pattern is then drawn, rotated about the axis 
 through a small angle (one or two teeth less than the number in 
 the pattern), and a few more teeth are moulded. In this way the 
 ■entire rim is moulded by sections. An index plate, or ring, is used 
 to insure accurate spacing, very much as is done in making cut 
 
152 KINEMAriOS OF MACHINERT. 
 
 gears. After completion of the rim the pattern (or the cores) for 
 the arms, hub, etc., is used to complete the mould. 
 
 83. Other Classes of Gearing. — In the table at the beginning of 
 this chapter six classes of gearing are mentioned. Screw-gearing, 
 will be discussed in the next chapter under the general head of 
 Cams. Twisted Spur-gears have already been treated, in Art. 75 ; 
 and Twisted Bevel-gears may also be derived by a similar process. 
 Skew Bevel-gears, or skew-gears, are those based on rolling 
 hyperboloids as pitch surfaces. These rolling hyperboloids were 
 very briefly treated in Art. 53 ; but the theory of the teeth of these 
 wheels is so complex, and their application is so very rare, thiit a 
 discussion of them is hardly warranted in a short general treatise. 
 
 Pace-gearing is an almost obsolete class, formerly used v,'hen 
 wooden gears were the rule, because the teeth (mere pegs or pins) 
 were easily made. The consideration of this class will also be 
 omitted. 
 
 The Practical Treatise on Gearing, published by the Brown and 
 Sharpe Manufacturing Co., gives a great many valuable points on 
 the actual construction of gearing; such as directions for laying 
 out blanks for cut gears, etc. 
 
 Grant's Teeth of Gears is a most excellent concise treatise on 
 tooth outlines; and MacOord's Kinematics, which is devoted mainly 
 to gearing, is a very complete work, covering many points not 
 ordinarily taken up and containing much original matter. 
 
 The discussion of this chapter has been very much abbreviated, 
 as the subject is exhaustively treated in a large number of available 
 books, and no attempt has been made to give more than a general 
 treatment of fundamental principles. 
 
CHAPTEE V. 
 CAMS AND OTHER DIRECT-CONTACT MECHANISMS. 
 
 84. Cams. — The term cam is applied to a large and varied class 
 of machine members which are often used to impart a more or less 
 complex motion to a follower. Cams are most commonly employed 
 for motions which cannot be easily produced by other simple forms 
 of mechanism. A cam consists of a piece so shaped that its motion 
 (which is usually a rotation, but often an oscillation or translation) 
 imparts a definite, and ordinarily variable, motion to another mem- 
 ber upon which it acts by direct contact, or through an auxiliary 
 roll or block. 
 
 Figs. 36, 44, and 57 show common forms of cams. Such mech- 
 anisms as those illustrated in Figs. 38, 39, e'tc, and even gear-teeth, 
 might be treated as special forms of cams : but it is more convenient 
 to consider them by themselves. 
 
 There are two principal classes of cams: those with a curved 
 edge or groove, which impart motion to a follower moving iu the 
 plane of the cam motion, and those which cause the driver to move 
 in a different plane, usually perpendicular to the plane of the cam's 
 motion. The latter class may be derived from the former, as will 
 appear later. Figs. 130 to 137 represent cams of the first kind. 
 
 85. Cams moving the Follower in the Plane of the Cam by a 
 Point or Roller. — Fig. 130 represents a simple case, iu which the 
 path of the follower is a straight line passing through the fixed 
 centre about which the driver rotates. Suppose to be the fixed 
 centre of the cam', PM to be the path of the follower, and that the 
 follower is to have positions corresponding to 0, 1, 2, 3, etc.; for 
 
 153 
 
154 
 
 KINEMATICS OF MACHINERY. 
 
 the angular motions of the driver o, a^, a, > etc. These angles may 
 be equal or otherwise; and the follower may have a period of rest, 
 or a "dwell," as indicated by the coincidence of the positions 4, 5. 
 Lay off the radii 1', 2', 3', etc., making the desired angles a^ , a^, 
 etc., with PO, and locate the corresponding positions of the fol- 
 lower, 0, 1, 2, 3, etc. With a centre at and radius 01, draw an 
 arc cutting 01' at 1'; with radius 02, cut 02' at 2', etc. Through 
 
 4-5iM 
 
 Fig. 130. 
 
 these intersections pass a smooth curve. Then this curve as it ro- 
 tates will impart the required motion to the point P, which is sup- 
 posed to be guided in the line PM, for when the cam has moved 
 through the angle a^ , for example, the radius 01'' will coincide with 
 the line PM, and 1' will be at 1. The same reasoning applies to 
 all the points found above. 
 
 If the real cam is a solid (cylinder) of which the curve shown 
 is one of the equal transverse sections, the follower would have 
 to be a mere edge, to satisfy the conditions given. While such a 
 combination satisfies the kinematic requirements, it would work 
 with unnecessary friction and would wear rapidly; hence a derived 
 form is used which gives the same motion to the follower. A 
 
CAMS AND OTHER BIBECT-CONTAOT MEGEANISMS. 155 
 
 roller may be attached to the follower, as shown by the small 
 circle at P, for reducing friction, and a profile may be adopted for 
 the cam (parallel to the original cam outline which is called the 
 pitch line), giving the required motion as this modified cam acts 
 upon the roll. To draw this actual cam outline, take a radius equal 
 to the radius of the roll, and with centres along the pitch line draw 
 arcs inside the pitch line. A curve tangent to these arcs is the 
 required outline for the real cam; because at all positions this 
 derived line acting upon the roll will cause the centre of the latter 
 to lie on the pitch line of the cam. 
 
 If the path of the follower, which may be either straight or 
 curved, as in Fig. 131, does not pass through 0, the construction 
 
 Fig. 131. 
 
 differs slightly from that of Fig. 130. The radii of the cam are 
 laid off as before at angles corresponding to the required successive 
 angular motions of the cam. When the radii 01', 03', etc., lie in 
 the line OS, the follower centre is not on these radii, but in the 
 path PM, at 1, 2, etc. With as a centre and a radius 01, draw 
 the arc 1-1"-1"'-1'. Now lay off on this arc, from its intersection 
 with the radius 01', a. distance equal to 1-1", locating the point 
 1'". This is a point in the required pitch line, for when 1' is at 
 
156 
 
 KINEMATICS OF MAOHINERY. 
 
 1", the point V" coincides with 1. The other points in the pitch 
 line are located in a similar way. The distances I'-l'", 2'-2'", 
 etc., are laid off to the right or left of V, 2', etc., according as the 
 positions 1, 3, etc., of the follower are to the right or left of OS. 
 
 The actual cam outline to act properly with a roll of sensible 
 diameter is obtained precisely as in the other example. 
 
 It is usually desirable to have the path of the follower as nearly 
 in line with a radius of the cam as possible, as this condition gives 
 less obliquity of action especially with small cams. 
 
 86. Cams acting on a Tangential Follower to move it in the 
 Plane of the Cam. — It is often desired to have the cam act tan- 
 
 gentially upon a flat or a curved follower, as in Figs. 132, 133, or 
 134. In this case there is a limitation to the motion which it is 
 possible to impart to the follower. Thus in Pigs. 132 and 133, in 
 which the acting surface of the follower is a fiat face (plane 
 surface), it is evident that no portion of the acting surface of the 
 cam can be concave, for such portions could not become tangent 
 to the follower. 
 
 With the curved (convex) follower as shown in Fig. 134, it is 
 
CAMS AND OTHBR DIBECT-CONTAGT MEOHANiaMS. 157 
 
 possible to have a concave portion of the driver, but this portion 
 must have as great a radius of curvatui'e as any part of the follower 
 >vith which it acts. General methods of designing these tangential 
 
 "cams are shown in Pigs. 132, 133, and 134. 
 
 In Fig. 132 the various positions of the follower are parallel to 
 each other, and the acting face is preferably, but not necessarily, 
 perpendicular to the direction of the follower's motion. Lay ofE 
 radii of the cam, 01', 02', etc., marking desired angular motions 
 from the original position corresponding to the given positions of 
 the followers, 1-1, 2-2, etc. With a centre at 0, draw arcs through 
 the intersections of the various positions of the follower with the 
 reference line OM, and cutting the corresponding radii of the cam, 
 as shown, at 1', 2', etc. At 1', 2', 3', etc., draw lines making the same 
 angle with the respective radii that the follower makes with PM. 
 Draw a smooth curve tangent to all these lines last drawn.* This 
 curve is the required cam outline; for when any radius, as 3', 
 lies on OM, 3' will lie at 3 and the tangent through 3' coincides 
 with the required position of the follower. 
 
 \ If the successive positions of the follower are not parallel to 
 each other, as in Fig. 133, the solution is as follows: 
 
 3^ |M 
 
 Fig. 133 
 
 With a centre at 0, draw arcs through the intersections of O'-l, 
 .0'-2, etc., with the reference line OM; and cutting the respective 
 radii 01', 03', etc., at 1', 2', etc. At 1', draw a line making an 
 
 * If it is found impossible to draw a curve tangent to all these lines, a 
 condition as to the successive positions of driver and follower has been im- 
 posed which cannot be met with this type of cam. 
 
158 
 
 KINEMATICS OP MACHINEBT. 
 
 angle with the radius OV equal to y3, ; at 2' draw a line making an 
 angle with the radius equal to /3,, etc., then proceed as in the 
 former example by drawing a curve tangent to these last lines. 
 
 When the follower is curved as in Fig. 134, the following modi- 
 fication of the above method may be used. Connect 0' with the 
 
 points 1, 2, etc., where the curved 
 edge of the follower cuts OiL 
 Draw circular arcs about with 
 radii 01, 02, etc., cutting the radii 
 01', 02', etc., which correspond to 
 the desired angular motions of the 
 driver. Draw a circle through 0' 
 with centre at 0; then with radii 
 O'l, 0'2, etc., and centres at 1', 2', 
 etc., cut this last circle in the points 1", 2", etc. Now form a templet 
 with the curve of the follower for one edge, and a centre corre- 
 sponding to 0'. Place this centre at 1", with the edge of the templet 
 passing through 1', and trace along the edge. In a similar way 
 with the templet centre at 2", and the edge passing through 2', 
 trace the curve ; and so on for all the phases required. A smooth 
 curve tangent to all these traced positions of the follower is the re- 
 quired cam outline ; for when 2" rotates with the cam to 0', 2' will 
 fall at 2, and the corresponding tracing of the templet will coincide 
 with the required position of the follower, and hence the cam will 
 be tangent to this position of the follower. Similar reasoning ap- 
 plies to the other phases. 
 
 87. Positive Return of Follower. — In the forms of cams con- 
 sidered so far, no means has been shown for insuring a return of 
 the follower after it reaches its position farthest from the centre of 
 the cam. This is often accomplished through the action of gravity, 
 a spring, or some other external force ; but it is necessary under 
 many conditions to completely constrain the motion of the follower 
 by the mechanism itself. This is sometimes done as in Pig. 13.5 by 
 providing a groove in which the roll of the follower acts. In this 
 arrangement there are two defects, which may be serious, es- 
 
CAMS AND OTHER DIRECT-OONTAOT MEOHANISMS. 159 
 
 pecially at high speeds. The slot must have a width somewhat in 
 
 excess of the diameter of the roll, 
 
 for both faces of the cam groove 
 
 move in the same direction, and if 
 
 the roll is in contact with the inner 
 
 face of the cam it tends to rotate 
 
 in one direction, while if it touches 
 
 the outer face this tends to rotate 
 
 the roll in the opposite direction. 
 
 The figure shows the action when \^ ^~ ^v^'^'s- '3S. 
 
 the inner face is acting on the roll. 
 Owing to the necessary clearance there is some lost motion which is 
 taken up when the follower reaches either of its extreme positions, 
 and is acted upon by the other face of the cam. If the speed is high 
 the taking up of this " slack " may result in a sharp blow, or knock, 
 which makes a noise and may injure the mechanism. The other 
 defect is due to the fact that when the roll changes contact from 
 one face to the other, there is a tendency to instant reversal of its 
 motion, but the inertia of the rapidly revolving roll resists this, re- 
 sulting in a temporary grinding action which wears both cam and 
 roll. Under slow speeds these actions may not be serious. 
 
 A method of returning the follower which overcomes these de- 
 fects is shown in Fig. 136. Two rolls on opposite sides of the cam 
 
 shaft are mounted as shown, or in 
 some similar mannei'. A cam is de- 
 signed, by the method given in arti- 
 cle 85, to impart the required mo- 
 tion to one of these rolls. Every 
 position of this roll causes the other 
 to occupy a definite position, due to 
 the connection between them, and a 
 '^' ■ complementary cam is designed cor- 
 responding to these various posi- 
 ^///y/^>.-/yy>/y..^///^^^ tions of the second roll. This return 
 
 cam is placed beside the first one on the same shaft, and of course 
 
160 
 
 KINEMATICS OF MAGIIINERT. 
 
 its follower must lie in its own plane. If the two rolls are mounted 
 on a sliding-piece so that their line of centres passes through the 
 centre of the shaft, the complementary cam is easily drawn because 
 the radius of its edge at all points plus the opposite radius of the 
 forward cam is a constant, equal to the distance between the surfaces 
 of the two rolls ; or equal to the distance between the roll centres 
 minus the sum of their radii. 
 
 In some cases, as when the motion of the follower is symmetrical 
 either side of the middle position, such a cam as is shown in Fig. 
 137, a cam of constant diameter may be used. In this case the cam 
 may work upon two parallel faces of the follower, and the motion of 
 the latter is thereby completely constrained by the action of a single 
 cam. 
 
 A cam like the one shown in Fig. 137 can be designed very 
 easily, as it is bounded by circular arcs. The follower is shown in 
 
 its extreme position to the right. 
 There is a " dwell," or period of rest, 
 at both extreme positions of the 
 follower. The parts ab and cd are 
 arcs with a common centre 0, and 
 
 I \ ^j< 7^ . the sum of their radii equals the 
 
 ^ — >l K To^^. h distance between the parallel work- 
 
 ing faces of the follower, = D. To 
 draw the arcs be and ad take a 
 radius equal to D, and draw an arc 
 be with a centre on cd. This arc 
 must be tangent to ab at b. Now 
 ■with the same radius, D, and a centre at c draw the arc ad, thus 
 completing the outline. In the position shown the follower is at 
 rest ; when a comes in contact with the left-hand face of the 
 follower, on the line of centres, c is in contact with the other face 
 directly opposite ; then ad acts upon the left-hand face of the yoke 
 and the follower moves to the left ; this is followed by a period of 
 rest while do acts on the left face, and a then comes in contact 
 with the right face, returning the follower to the right. This cam. 
 
 Fig. 137. 
 
CAMS AND OTREB DIRECT-CONTACT MECHANISMS. 161 
 
 or a modification of it, is used to actuate the valves of the engines 
 on the stern-wheel steamers of the upper Mississippi and its tribu- 
 taries. 
 
 88. Translation Cams. — A form of cam which by its translation 
 imparts motion to a driver is shown in Fig. 138. If it be required 
 
 5" 6,',' 
 
 Fig. 138. 
 
 that the follower shall be at the points 1, 2, 3, etc., as the points 
 1', 2', 3', etc., of the driver coincide with 0, design the cam as fol- 
 lows: Erect perpendiculars at 1'^ 2', etc., as I'-l", etc. Draw a 
 line from 1, parallel to the motion of the driver, cutting I'-l" in 
 1"; through 2 draw a parallel, cutting 2'-2" in 2", etc. A line 
 through these successive intersections gives the pitch line of the 
 cam. If the follower is provided with a roll at P, the actual cam 
 -outline is formed as indicated in Fig. 130. 
 
 If the sucessive motions of the follower are to be proportional 
 to those of the driver, the cam becomes an inclined plane, as shown 
 in Fig. 139. In this case the flat shoe as shown may act directly on 
 
 Fig. [39. 
 
 the cam, as it will fit the surface of the driver at all points. This 
 will provide a larger contact surface and reduce wear, though it 
 results in greater friction than when a roll is used. 
 
 89. Motion imparted to the Follower Perpendicular to Plane of 
 Cam. — The cam of either Figs. 138 or 139 may be wrapped upon a 
 right cylinder, as shown in Figs. 140 and 141, and then the rotation 
 of these cylinders about their axes will impart a motion to the fol- 
 
162 
 
 KINEMATICS OF MACHINERT. 
 
 lower parallel to these axes and exactly equiyalent to tlie motion 
 due to the translation of the original cams. The base lines, or lines 
 parallel to the translation of the original cams, become circles when 
 the cams are wi-apped upon the cylinders. 
 
 If these cams are provided with grooves in which a roll acts, as 
 in Fig. 140, clearance must be provided; for the opposite faces of 
 the cam surface tend to rotate the roll in different directions; 
 hence these cams are subject to the defects of the ring cam men- 
 
 Fig. 140 
 
 Fig. 141. 
 
 tioned in the preceding article. There is another peculiarity of 
 the action of such a cam as that of Fig. 140, if the roll is a cylinder, 
 which results in a grinding action between the cam face and the 
 roll; but this is overcome by using a properly proportioned conical 
 roll, as shown in the figure. If the outer radius of the cam is r^ and 
 the radius at the bottom of the working part of the groove is r^, 
 points at the outer edge have a linear velocity of STrriW, while 
 points at the inner portion have a smaller velocity, ^nr^n. If 
 the roll is a cylinder, the faces of the cam being perpendicular to 
 the axis, all points on the contact element of this cylinder must 
 evidently have equal linear velocities; hence the velocities of con- 
 tact points of the driver and follower can only be the same at one 
 point along the element of contact. If, however, the roll is the 
 frustum of a cone with its apex at the axis of the cam and the 
 sides of the groove be given the corresponding slant, this difficulty 
 is practically overcome. The shaded portion of the face in the 
 section at the right of Fig. 140 shows the development of the acting 
 surface of such a conical roll, or of a portion of such surface. 
 
CAMS AND OTHER DIRECT-CONTACT MECHANISMS. 16^' 
 
 ,66'" 
 
 >! J'' 4' 
 Fig. 142. 
 
 If the cam is to rotate contimaously, instead of vibrating upon 
 its axis, the original cam of Fig. 138 must have its first and last 
 ordinates of equal lengths (measured from the hase line, or from 
 any line parallel to this); otherwise the groove would not be con- 
 tinuous when. formed on the cylinder, and it could only drive the 
 cam by reciprocation about its axis; 
 wlien the forward and return mo- 
 tions would be similar. The path 
 of the follower may be other than a 
 straight line, and the construction 
 of the pitch line of such a cam is 
 shown in Fig. 142. The points in 
 the pitch line are on the parallels 
 to the path of the driver; but to 
 the right, or left, of the intersections of such parallels with the- 
 corresponding perpendiculars by distances equal to the given 
 simultaneous distance of the follower from the reference line 0-M^ 
 The construction of the cam from this pitcli line is exactly similar- 
 to the cases already treated. 
 '/I 90. The Screw. — The cam of Fig. 141, as derived from the in^ 
 clined plane of Fig. 139, will be recognized as the common screw^ 
 in which the sliding block of the follower corresponds to a portiom 
 of the thread of the nut. 
 
 The ordinary nut and screw differs essentially from this only i» 
 the length of the block of the follower, which is made to include 
 several coils of the cam (threads of the screw) in order to distribute 
 the pressure, reduce wear, and- increase the strength. The groove 
 in which the nut works may be rectangular, triangular,, or of any- 
 one of many possible sections, without modifying the relative motion; 
 transmitted, which is governed entirely by the relation between the; 
 axial and circumferential components of the threads. 
 
 If the cam of Fig. 139 be wrapped upon a cylinder the circum- 
 ference of which is L = nD, each revolution of the cam or screw 
 will move the follower through a distance h, which is in this case- 
 the pitch of the helix or screw-tliread. If this cam were wrapped. 
 
164 KIJSEMATICS OF MACHINERT. 
 
 upon a cylinder of half the above diametei', two revolutions of the 
 screw vpould be required to move the follower the distance li, or the 
 base line of length L, as shown, would make two complete coils 
 of the helix, and the pitch of the screw (the distance from one 
 thread to the next, parallel to the axis) would be \h. In any such 
 case the inclination of the helix to the plane of motion of any point 
 in it is the angle whose tangent is h -f- L, and this inclination 
 determines the velocity ratio of driver to follower, which is L -=r h, 
 at a contact point on the pitch line. Of course the groove has sen- 
 sible depth in an actual screw, and then points on the screw-thread 
 at different distances from the axis have difEerent linear velocities 
 relative to the nut. If the screw is driven by a crank or pulley of 
 larger radius than the acting surface, the velocity of the "actual 
 point of application of the driving force is correspondingly greater, 
 relative to the nut, than L -H h, but is still proportional to this 
 quotient. 
 
 If the pitch of the thread is great enough to permit it, another 
 similar thread may be cut between the grooves, as indicated by the 
 dotted lines on Pig. 141, and corresponding extra threads of the 
 nut may work in this groove. This does not alter the motion trans- 
 mitted, but it gives more bearing surface for a given length of nut. 
 Such a screw is called a double-threaded screw. There may be any 
 number of such threads, if the proportions will permit, giving a 
 multiple-threaded screw. 
 
 91. The Endless Screw. Screw-gearing. — Fig. 143 shows a screw 
 with an angular thread and a small block (indicated by the shaded 
 tooth) for a follower. Rotation of the screw in the direction indi- 
 cated moves this follower to the left. If this block is pivoted at 0' 
 instead of being guided parallel to the axis of the screw, it will 
 move in a circle about 0' (provided it is relieved so as to avoid 
 binding), and it soon passes out of action. Now if a series of such 
 pieces be arranged in a continuous circle about 0' , all connected 
 together and properly spaced, as each piece passes out of action at 
 tlio left another will engage at the right. If these blocks consti- 
 tute a complete circular rim, the action will be continuous, and 
 
CAMS ANB OTHER DIRECT-CONTACT MECHANISMS. 165 
 
 indefinite rotation of the driver will result in indefinite rotation of 
 the follower. This arrangement constitutes, in a rudimentary 
 manner, the co;nmon worm and wheel, or endless screw, as it is 
 sometimes termed. If the teeth of the wheel in this figure were 
 given a transverse section exactly fitting the spaces between the 
 
 Fig. 143. 
 
 threads of the screw, as is the case in an ordinary nut and screw, 
 the teeth would bind or interfere; but by giving a properly modi- 
 fied form to the teeth the action may be made smooth and unifoi'm. 
 Suppose that the screw, or worm, as it is commonly called, be 
 translated axially without rotation. It will then cause the wheel to 
 rotate Just as a pinion is rotated by a rack ; tiud if the section of the 
 teeth of the wheel and the screw-threads be correct forms for teeth 
 of a pinion and rack, the velocity ratio will be constant. It will be 
 apparent that the rotation of the screw is equivalent to this trans- 
 lation of the screw when acting as a rack ; for by this rotation suc- 
 cessive equal meridian sections of the worm are brought into action 
 on the middle section of the wheel. It follows from this that these 
 sections should correspond to the forms proper for a rack and 
 pinion. The sections of an involute rack-teeth are trapezoids, (sec- 
 tions of the acting faces being inclined straight lines perpendicular to 
 the line of action, or the common normal). Then if the screw-tli reads 
 of Pig. 143 have meridian sections bounded by inclined straight. 
 
166 KINEMATICS OF MACHINEHY. 
 
 lines on the acting sides, the transverse section of the teeth should 
 he corresponding involutes. The sectioned view at the right of the 
 figure indicates this form. 
 
 It is evident that if the worm of Fig. 143 is a single-threaded 
 screw, each revolution of the worm causes the wheel to rotate 
 through an arc equal to the pitch arc of the teeth ; and to make a 
 complete revolution of the wheel the worm must he given as many 
 turns as there are teeth on the wheel. The pitch angle, as in spur- 
 gears, is the angle between two teeth. If the worm is a double- 
 threaded screw, the helical pitch is twice the distance from one 
 thread to the corresponding point on the next thread; and if each 
 of the threads bears upon a tooth of the wheel, one revolution of 
 the worm moves two teeth of the wheel past the line of centres. 
 This arrangement distributes the pressure over more teeth, and it 
 usually gives a higher efQciency (but a lower velocity ratio) than 
 single-thread worms. It is apparent that in this case one revolu- 
 tion of the wheel is accomplished by a number of turns of the 
 screw equal to the number of teeth of the wheel divided by two. 
 
 In general the angular velocity ratio of the worm to that of the 
 wheel equals the number of teeth of the wheel divided by the 
 number of separate threads of the screw. The number of threads 
 of the worm may be increased indefinitely, and if it is made equal 
 to the number of teeth on the wheel the velocity ratio becomes 
 unity. In this last case the two wheels become similar. The 
 driver is called a worm only when it has but few threads. 
 
 In designing a worm and wheel, let T = threads (teeth) on the 
 wheel; i = the threads on worm; p = circular pitch of wheel 
 = axial pitch of worm-threads; D = the pitch diameter of wheel; 
 d = the pitch diameter of worm; zJ = distance between axes; go 
 and cOj = angular velocities of wheel and worm, respectively; 
 <p = inclination of threads to transverse section of the worm. 
 
 This relation usually fixes T and i. 
 
CAMS AND OTHER DIRECT-CONTACT MECHANISMS. 167 
 
 Tp=7tD; D-Tp^n; or, p = nD ^ T. The strength of 
 the gear depends uponjs, and hence j9 should be fixed and D made 
 to agree, if the conditions -will permit. If, however, A is fixed, D 
 is limited, for 1{D -\- d) — A; but D and d may have any values 
 consistent with this requirement. The value of p gives the num- 
 ber of threads to the inch on the worm, and hence (when d is 
 fixed) the inclination ; for tan (p = p -^ nd. 
 
 One of a pair of equal spiral gears resemble a twisted spur-gear; 
 but the action is very different. In the case of the twisted spur- 
 gears, the velocity ratio depends upon the ratio of the radii, as in 
 ordinary straight spur-gears, and there is no sliding of the teeth 
 ■along their elements. In the case of a pair of spiral or screw- 
 gears, the action is a true screw-like action, and is not determined 
 by the relation of the diameters. 
 
 Two equal screw-gears (with axes at right angles) are both 
 right-handed or both left-handed; while in a pair of equal twisted 
 gears one must be right-handed and the other left-handed. A 
 twisted gear may work as a screw-gear with another similar gear, 
 the two shafts being at right angles to each other. 
 
 If the teeth of the wheel are ordinary screw-threads (all trans- 
 averse sections of either wheel being identical in form) upon a 
 cylindrical pitch surface, this pitch cylinder and that of the worm 
 are tangent at a single point, and the teeth only have point con- 
 tact. That is, the worm always engages with points on the central 
 transverse section of the wheel. The worm-wheel may be made of 
 t'le form shown in Fig. 144, when it is called a close-fitting wheel. 
 The teeth of this wheel may be drawn by passing a series of planes 
 through the worm, parallel to the axis and to each other and per- 
 pendicular to the axis of the wheel. Each of these sections of the 
 ■worm will be a rack section, but they are not all alike. Then 
 make the corresponding sections of the wheel those appropriate for 
 wheels to work with such racks.* This process is tedious, and is 
 
 * See Unwin's Machine Design, Part I, Art. 234, for a full description of 
 this method of drawing worm-wheel teeth. 
 
168 
 
 KINEMATICS OF MACHINERY. 
 
 seldom required in practice, as by the method of cutting the' 
 wheels it is not necessary to lay out the teeth. If a cast worm and 
 wheel are to be made, it is of course necessary to lay out the teeth. 
 92. Hobbing Worm-wheels, — A worm-wheel may be accurately 
 cut by the following process: Turn up the blank to correspond to 
 the outside of the teeth (Fig. 144 *). Next cut a screw of tool steel 
 to the exact form of the worm, then make a milling-cutter of this 
 tool steel worm by cutting flutes across the threads, and " backing. 
 
 Fig. 144 
 
 off " the teeth thus formed for clearance. This is called a " hob "" 
 (see Fig. 144 a), and it is hardened, tempered, and then used as a 
 milling-cutter. The wheel-blank is mounted in the gear-cutter, 
 with the distance between its axis and that of the hob a little 
 
 * Figs. 144, 144^, and 1445 are taken from Brown & Sharpe's Treatise oa 
 Gears. 
 
CAMS ANB OTEEB DIRECT-CONTACT MECHANISMS. 16& 
 
 greater than the distance it is finally to be placed from the axis of 
 the worm. As the hob rotates it cuts teeth in the blank, and the 
 latter is fed towards the former gradually, till the correct distance 
 is attained. The wheel is sometimes caused to rotate simply by 
 the driving action of the hob; but better results are attained when 
 it is driven positively from the cutter-spindle with the required 
 velocity ratio, from the cutter-spindle, through a suitable train of 
 gearing. It will be seen that the teeth thus formed on the wheel 
 will work correctly with a worm which is an exact reproduction of 
 the hob, except that the cutting-teeth are omitted. 
 
 The worm and hob may be cut like any screw in a lathe, with a 
 tool which will give the desired form of threads. 
 
 Pig. 144 (S) shows a method of cutting an approximate close- 
 fitting worm-wheel with an ordinary gear-cutter, the diameter and 
 section of which corresponds to the worm. If the cutter, as shown 
 in this figure, is fed diagonally across the wheel-blank, a straight 
 (point contact) wheel will be produced. 
 
 If the cutter is fed radially inward, toward the axis of the 
 wheel, a "drop-cut" worm-wheel is produced. Such a drop-cut 
 wheel resembles a hobbed wheel in form ; but the method does not 
 give a truly close-fitting wheel, such as is obtained by the bobbing 
 process. 
 
CHAPTEE VI. 
 
 LINKWORK. 
 
 93. General Scope of Linkwork. — The simplest form of a con- 
 strained link-mechanism consists of four links, each pivoted at two 
 points to adjacent links. A link with but two pivots, and joined 
 to two adjacent members, is called a simple link. If a link has 
 more than two such pivots and is joined directly by them to more 
 than two separate members it is called a compound link. 
 
 A complete linkwork " chain," as link-mechanisms are some- • 
 times called, cannot have less than four links; for if three links are 
 connected in a closed chain they form a triangle, which is a rigid 
 construction not permitting relative motion between the members. 
 If more than four simple links are connected in a closed chain, 
 forming a jointed polygon of more than four sides, a given motion 
 of one member does not compel the others to move in a definite 
 manner. Link-mechanisms of more than four members are used ; 
 but, in these cases, one or more of the members must be a compound 
 link. Linkwork can be used to convert : 
 
 [a) Continuous rotation into reciprocation (rectilinear or circu- 
 lar) or the reverse. 
 
 {h) Keciprocation into reciprocation with a constant or a vari- 
 able angular velocity ratio. 
 
 (c) Continuous rotation into continuous rotation, with a con- 
 stant or a variable velocity ratio. 
 
 Certain modified linkwork chains in which one or more links 
 of the common form is replaced by a sliding-block, or similar piece, 
 are of very great importance in practical machine construction, and 
 these may be readily derived from the fundamental forms. 
 
 170 
 
LimnvoRK. 
 
 171 
 
 94. The Four-link Chain. — The general form of the four-link 
 chain is shown by Figs. 50 to 53 and other figures already given. 
 It is now in order to examine the influence of the proportions of 
 the members of the four-link chain upon the motion transmitted. 
 
 The following notation will be used : The driver will be desig- 
 nated as a; the follower, J; the connector, c; and the stationary 
 link, or frame, d. 
 
 Fig. 145 shows a mechanism in which circular reciprocation of 
 a produces circular reciprocation of i. The phase shown by the 
 light lines a', V, c' , is a limiting phase, and a can move no farther 
 to the left (left-hand rotation).* 
 
 The driver (Fig. 145) can move through the arc a'-a-a"-a"', 
 causing the follower to move from V through b to b" , and then 
 return over this path to b'". Within these limits circular recipro- 
 cation can produce circular reciprocation, and either member might 
 be the driver; but, practically, the action is not smooth when the 
 
 &-t-c .: a-\-d 
 ;.c—a< d—b 
 
 follower is near a dead-point; and if b is the follower, the range of 
 action should be somewhat less than the maximum given. In this 
 figure it will be seen that b + c < a -\- d ; and it will be seen that 
 
 * This position of the follower, V , is called a dead-point position; for there 
 can be no component of the motion of the connector in the direction of its length, 
 and hence no positive transmission of motion. When the connector and either 
 the driver or follovrer lie in one straight line, a dead-point is reached. If the 
 two members lie on opposite sides of their common pivot, as 6' and c' in Fig. 
 145, the condition is called an outer dead-point position. If the links coincide 
 in direction and are on the same side of their pivot, as V and c' in Pig. 146, an 
 inner dead-point Is reached. 
 
172 
 
 KINEMATICS OF MACHINERY. 
 
 a cannot make a complete rotation unless 5 -)- c is equal to, or 
 greater than, a -{- d; o^ c — a>_d — b. 
 
 In Fig. 146 the follower reaches an inner dead-point when it is 
 in the position h', and a can lotate no farther to the right than the 
 position a'. In this case the driver can vibrate through the angle 
 a'-a-a"-a"', causing the follower to reciprocate from h' through 
 h to h" and back to V". It is evident that d < than a -\- c — b; 
 and the driver cannot make a complete rotation unless d is equal 
 to, or greater than, a -\- c — i ; or a -\- c <_ d ^ I. It will be 
 noticed that the follower might pass (but cannot be positively 
 driven by a) beyond the position V, in either Fig. 145 or Fig. 146; 
 if this should occur in any way, the motion transmitted would be 
 completely changed. 
 
 To sum up these two cases, we find that the driver cannot 
 make a complete revolution unless these conditions are present : 
 c — a^d — i; and c -\- a ;^ d -\- b. It c — a = d — b, the driver 
 and follower have simultaneously inner and outer dead-points, 
 respectively, as shown by Fig. 147 (in the phase a', b'), and the 
 
 c—a = d 
 
 Fig. 147, 
 
 motion of the follower may be towards either b" or i'", as the 
 driver passes this position. If c + a — d ^ b (Fig. 148) tlje 
 driver reaches the outer dead-point as the follower reaches tlie 
 inner dead-point {a' and b', respectively); and the follower may 
 either return to b" or pass on to b'". If c — a > d — b, and 
 
LINEWOBK. 
 
 173 
 
 c-\-a<d-\-b, &S. in Fig. 149, the motion of the follower is fully 
 constrained, and the driver can make a complete rotation. 
 
 o-a> d-h, 
 c+a <d+b 
 
 Fig. m.--^/-'"' 
 
 95. Continuous Rotation of both Driver and Follower. — (See Fig. 
 150.) If a single four-link chain is used to transmit positive con- 
 tinuous rotation to a follower from a rotating driver there must be 
 no dead-points ] for if the driver, a, reaches a dead-point, b will 
 come to rest; and if h reaches a dead-point, its motion will not be 
 fully constrained, and it will generally lock the driver, preventing 
 complete rotation of the latter. With the proportions of Fig. 150, 
 neither a nor b reaches a dead-point, and either of these members 
 may be used as a driver, compelling the other to rotate continuously, 
 but the velocity ratio will be variable. If rotation of both a and b 
 is to be continuous, they will have simultaneous dead-points if either 
 reaches a dead-point. If c = inn = d -\- b ~ a, a will have an outer 
 dead-point, and b will have an inner dead-point at the same instant. 
 If c = mn' = a -\- b — d, a and b will both have inner dead-points 
 at one phase. Hence, for continuous positive rotation of both a 
 and b the following conditions must be fulfilled : c > d -\- b — a, 
 and c < a + b — d; hence, d-{-b — a<a->rb — d . . d < a.* 
 
 The mechanism of Fig. 150 is called a "drag-link"; and it is 
 sometimes used to connect the two arms of a centre-crank or double- 
 throw crank. In this case a and b are equal, and d equals zero in 
 the proper adjustment, that is, the fixed axes of a and b coincide. 
 
 * If dead-points are permissible (as in the parallel rods of locomotives), 
 other provision is made for insuring that the dead-points shall be passed; then 
 d may be, and usually is, greater than a. 
 
174 KINEMATICS OP MACHINEBT. 
 
 As long as this condition is maintained, the link forms the equiva- 
 lent of a rigid connection, and as the mechanism is reduced to a 
 three-link chain, the motion transmitted is exactly similar to that 
 of a solid crank. If either axis is shifted, through improper align- 
 ment, springing of the shaft supports, or wear, the motion is trans- 
 mitted from one section of the shaft to the other with a slight 
 variation in their angular velocity ratio during the revolution, and 
 the wrenching action on the shaft is much less than it would be 
 with the usual form of rigid crank-shaft. 
 
 If a and b are equal the angular velocity ratio is constant when 
 d equals zero, or when t^ = c; for with these proportions the two 
 perpendiculars from the fixed centres to the line of t])e link (c) are 
 always equal for any phase. The former condition {cl = 0) is that 
 of the drag-link as applied to engine-cranks in proper alignment. 
 The second condition (d = c) is one met in the locomotive side-rod 
 connection; but in this case the driver and follower have simulta- 
 neous dead-points, and special means must be resorted to for com- 
 plete constrainment of the follower. 
 
 The essentials of the locomotive side-rod connection are shown 
 in Fig. 59, in which and 0' correspond to the centres of the 
 connected wheels, A'£' is the side rod (the dotted circles represent 
 the paths of the pins by which the side rod 'is pivoted to the wheels); 
 OA' and O'B' (radii of the pin circles) are the driver and follower 
 between which it is desired to transtaiit rotation with a constant 
 velocity ratio; and 00' (the frame) is the fourth link. The full 
 lines of Pig. 59 show a phase at which the driver and follower both 
 lie on the line of centres. As the driver passes its dead-point po- 
 sition, the follower might move in either of the directions indicated 
 by tlie arrows at B. Means of overcoming this defect in the con- 
 strainment will be shqwn later. 
 
 In the mechanism under consideration it is necessary that the 
 four links shall form a parallelogram in all phases; that is, in Fig. 
 59, A'B' must equal 00'; and OA' must equal O'B'. When this 
 condition is fulfilled the angular velocity ratio must always be unitv, 
 for the perpendiculars from the fixed centres {0, 0') to the con- 
 
LINKWORE. 
 
 176 
 
 nector {A'B') are equal in any phase (see Art. 30). In order to 
 insure continuous rotation of the follower when the dead-points are 
 passed, the simple mechanism of Fig. 59 must be supplemented. 
 Tlie method used on locomotives is shown in Pig. 151. 
 
 Each axle has two driving-wheels secured to it; the two wheels 
 on either side being coupled by a side rod. The pins on the two 
 
 ^ c 
 
 /¥TF? ! 
 
 \ 
 s 
 
 152." 
 
 wheels of each axle are placed so that they are not in line, but one 
 of these pins is ahead of the other, as shown in Fig. 151 by the 
 angle CfiG, = C.'O'C^'. 
 
 This angle is commonly 90°, so that when the system is at a 
 dead centre phase on one side, the complementary system on the 
 other side is in the best phase for transmission of motion. 
 
 Other possible arrangements to secure complete constrainment 
 are shown in Pig. 152. In this case three equal cranks, not neces- 
 sarily having their centres in one straight line, are connected by a 
 rigid member (a compound link) which has a bearing for the pin 
 of each. These bearings must be spaced to agree with the spacing 
 of the fixed centres, and the cranks are always parallel to one an- 
 other. The middle crank (shown with the arrow) should be the 
 driver. 
 
 96. Combined Linkwork and Sliding-block Mechanisms. — The 
 preceding articles of this chapter have been devoted to the four- 
 link cljaiii, and it was seen that by the mechanism of Fig. 149 a 
 
176 KINEMATICS OF MACHINERT. 
 
 circular reciprocation of the follower may be imparted by the re- 
 ciprocation or rotation of the driver. Fig. 153 shows a mechanism 
 in which one member of the four-link chain is replaced by a curved 
 block h, sliding in a corresponding circular arc groove in an exten- 
 sion of the fixed link d. It is evident that the motion transmitted 
 
 to this block is exactly equivalent to the motion which it would re- 
 ceive if it were connected to d by the dotted link h'. "Whatever 
 the length of h', it may be replaced by a block and groove, the 
 centre line of which corresponds to the/path of the moving end of 
 the link h' , without altering the character of the motion. Any 
 other link might be replaced in a similar way by the equivalent slot 
 and block. When any one of the links is very long this substitution 
 of a sliding-block for a link may be convenient, the radius of curva- 
 ture of the slot being equal to the length of the link replaced. If 
 the link is of infinite length the centre line of the slot becomes a 
 straight line, and the motion of the block is then a rectilinear 
 translation. 
 
 The mechanisms shown in Figs. 69, 70, 71, and 72 represent 
 this modified form of the four-link chain, or what Eeuleaux has 
 called the " slider-crank chain." This mechanism is so prominent 
 in practical machine construction that it will be treated in detail. 
 
 97. Crank and Connecting-rod. — The connection between the 
 piston and crank of the ordinary direct-acting engine, as shown 
 in Fig. 27, is one of the most important examples of the modified 
 four-link chain. In this case the motion of the piston and cross- 
 head relative to the frame is equivalent to that of a link of infinite 
 length. The piston and cross-head, being rigidly connected, are 
 kinomatically one piece; though we may be only concerned with 
 
LINEWORK. 
 
 177 
 
 the motion of the piston, it is often convenient to speak of the mo- 
 tion of the cross-head, which is identical with that of the piston. 
 
 With the usual arrangement the path of the cross-head is in a 
 line which passes through the crank-centre, and this line will be 
 spoken of as the centre line. This is not a necessary condition, 
 and it is sometimes departed from with a result that will be dis- 
 cussed in a later article. Unless otherwise stated it is to be -under- 
 stood that this ordinary arrangement is meant. 
 
 In the steam-engine the reciprocating piston is the driver and 
 the crank is the follower. In case of an air-compressor or power- 
 pump, the reverse is the case; but the relative motion of the mem- 
 bers is not affected by this relation, as it depends simply upon the 
 proportions of the mechanism. 
 
 The crank usually has a uniform motion, or approximately 
 «ucb, and this condition will be assumed in the following discus- 
 sion. It is evident that the cross-head (piston) must come to rest 
 as the crank-pin passes the line of centres (dead-centre positions). 
 Its motion is accelerated as it leaves either extreme position, attain- 
 ing a maximum velocity near the middle of the stroke, followed by 
 retardation (negative acceleration) through the latter part of 
 the stroke. The motion of the reciprocating parts is approxi7 
 mately harmonic, departing from true harmonic motion more as 
 the ratio of connecting-rod to crank lengths becomes smaller. 
 
 The position of the crosshead, c (Fig. 154), for any crank posi- 
 
 tion C, is obtained graphically by taking a radius equal to the 
 length of the connecting-rod Gc, and, with a centre at the given 
 <;rank position, cutting the path of the cross-head by an arc of this 
 
178 KINEMATICS OF MACHINERT. 
 
 radius. In a similar way the crank position corresponding to any 
 cross-head position is found by taking a centre at the given cross- 
 head position and cutting the crank-circle with an arc of the above 
 radius. This last process gives two intersections, one above and 
 one below the line of centres, as it should ; for the cross-head passes- 
 the same point in its path during the forward and return strokes, 
 both of which are accomplished during a single revolution of the 
 crank. If a series of equidistant cross-head positions are taken, it 
 is evident that the corresponding crank-pin positions will not be- 
 equidistant, and vice versa. That is, equal increments of cross- 
 head (piston) motion do not impart equal increments of motion ta 
 the crank. 
 
 In drafting-room practice these graphic methods of finding- 
 simultaneous crank and cross-head positions are usually most con- 
 venient; but sometimes it is desirable to use analytical expressions- 
 for the relations between the crank and connecting-rod. 
 
 The more important kinematic relations will be derived, trigo- 
 metrically, using the following notation (see Fig. 154) : 
 
 Centre of crank-circle = Q. 
 
 Centre of crank-pin = C. 
 
 Centre of cross-head pin = c. 
 
 Length of connecting-rod = I. 
 
 Length of crank = r. 
 
 Ratio of connecting-rod to crank (Z -f- r) = n. 
 
 Crank dead-centres = A and B. 
 
 Corresponding cross-head positions (ends of stroke) = a and b.. 
 
 Mid-stroke cross-head position = q. 
 
 Mid-crank positions (quarters) = Jlf and M'. \ 
 
 Simultaneous cross-head position = m. 
 
 Crank angle, ahead of A, = 0. 
 
 Corresponding angle of connecting-rod with line of centres = 0» 
 
 Drop a perpendicular. Ok, from G upon the centre line; then 
 Ck = r sin 6=1 sin ; Qk = r cos 6; ch = VP - (Cip 
 = Vl' - r' sin' e. 
 
LINKWORE. 179> 
 
 For any crank angle, 6, the distance from c io Q = ck -\- Qk = 
 
 VP — f sin" e -\-r cos d. 
 
 When C is at the quarter {M or M'), c is at m, a distance from 
 its mid-position = niq = Qq — Qm = I — VP — r" = ?8r — 
 r Vn" — \ = r {n — V n' — 1), & quantity which increases as n de- 
 creases, and equals zero when n = infinity. 
 
 It is seen from the above expression that when the crank has 
 rotated through 90° from A, the cross-head has moved through more 
 than half its stroke; while for the next 90° crank rotation the 
 cross-head moves through less than half its stroke. It follows that, 
 with uniform rotation of the crank, the half-stroke, aq, is made in 
 less time than the half-stroke, qb, this variation decreasing as the 
 connecting-rod length increases. The influence of this angularity 
 of the rod on steam distribution will be seen to be important, when 
 the subject of valve motions is studied. 
 
 In the illustrations of velocity diagrams (see Art. 41 and Figs. 
 69 and 76) it was shown that the ratio of the linear velocities of 
 the crank and cross-head is equal to the ratio between the length 
 of the crank and that segment of a perpendicular to the line of 
 centres through the shaft which lies between the centre line and 
 the line of the connecting-rod, the latter prolonged if necessary. 
 Thus, in Fig. 154, it QC represents the velocity of the crank, to 
 some scale, « = Qt is the velocity of the piston (to the same scale) 
 when the crank is at C. If the crank velocity cannot be represented 
 conveniently to this scale, lay oS. Qv, along the line of the crank, 
 to represent its velocity, and draw vt' parallel to cG; then Qt' = s^ 
 is the required velocity of the piston to the scale assumed ; and 
 the value thus obtained for the piston velocity can be used, as in 
 Fig. 76, for constructing the velocity diagram. 
 
 Another method of determining the ordinates of the velocity 
 diagram is shown in Fig. 155. With this method, Cv is laid off 
 on the extension of the line of the crank to represent its velocity 
 to a convenient scale ; an ordinate is erected at c, and this is- 
 cut by drawing the line vv' parallel to the connecting-rod; then. 
 
180 
 
 KINBMATIC8 OF MACHINERY. 
 
 €v' gives the velocity of the cross-head for this phase. The linear 
 '^-'^ velocities of Cto c are in the 
 
 ratio of OC to Oc ; as G and c 
 are two points in the connecting- 
 rod, which at the instant has a 
 motion equivalent to a rotation 
 
 ^ about the instant centre (9; hence 
 
 /-. ""'s- 155. the velocity of C is to that of 
 
 c as OCis to Oc. The construction of the complete diagram will 
 readily be seen from the figure. The method is the same as that 
 used for the four-link chain in Fig. 78. 
 
 From Fig. 154 it will be seen that the velocity of the piston is 
 equal to that of the crank when s = r. There are two positions 
 of the piston in each stroke where this condition is fulfilled. The 
 first of these positions is shown in Fig. 156, where cC, produced, 
 passes through M; hence s = Q2I — r. This equality of velocities 
 can be seen directly by locating the instant centre ; for OCc 
 is similar to QCM at this phase; hence 00 = Oc, and the linear 
 velocity of c = linear velocity of 0. The same relation can be 
 shown by resolution of the motions. 
 
 fc R 
 
 Fig. 157. 
 
 When C (Fig. 157) coincides with M, s= Q0= QM, and the 
 crank and the piston have the same velocity. Between these two 
 positions of equal crank and piston velocity, the piston moves /as^er 
 than the crank; for s is greater than r. (See Fig. 159.) 
 
 The second of these positions of equal velocity is always that 
 
LINKWOBE. 181 
 
 at which the crank is perpendicular to the line of centres, and 
 is independent of the ratio of connecting-rod to crank. The first 
 position is a function of this ratio ; and the crank-angle corre- 
 sponding to this phase is found as follows (Fig. 156) : Let fall 
 Qe perpendicular to C2I, then as QMG is isosceles, QMe and QCe 
 are equal triangles, and the angle eQG = eQM = 0, .". MQC ■= 
 20. .-. (9 = 90 — 20. Gh=l sin = r sin 6/ = ?-sin (90-20) = 
 
 r cos 20, .". — sin = ;; sin = cos 20; =1—2 sin" 0, .-. 2 sin" 
 r 
 
 -t- ?i sin = 1; dividing by 2 and completing the square ; 
 
 n m" " 
 
 sin" 04-2 sin 0+-=i+^ 
 
 A=±i- 
 
 n' 
 
 .: sin0 + -= ±y i+jQ^ ±i V8-f «% 
 
 , n 
 
 and sin = ± i V 8 + n' — y 
 
 The double sign of the radical may be dropped, for if the 
 minus sign be taken, with any value of n greater than 1, we 
 would get a value for sin numerically greater than 1, which is, 
 impossible. Taking the plus sign: 
 
 sin = i I V(8 -I- n") - u \ 
 
 As sin 6 =■ n sin 0, 
 
 n , ,,„ . ,, , n 
 
 4/(8 + w") — w} = ^[ ■/(2.828)" + n" - n\. 
 
 This form is convenient for graphical solution as follows (Fig. 158): 
 Lay off distance AB = w to a scale of r = 1, and erect a perpen- 
 dicular BC = 2.828 to the same scale. Connect A and C, then the 
 hyrothenuse AC = 4/(2.828)" + n\ "With ^ as a centre and AB 
 (= n) as a radius, describe arc BD, cutting AG m D. 
 
 DG = |/(2.828)" 4- n' - n. 
 
 11 7 
 
 DC X ^ = sin 0; or DG Y - = r sin = Gk. (Pig. 156.) 
 
182 
 
 KINEMATICS OF MACEINERT. 
 
 Between the two positions of the crunk at whicli its velocity 
 equals the velocity of the reciprocating parts, the velocity of the 
 latter is greater than that of the crank, as noted above. These 
 
 r 
 
 ^ 
 
 / 
 
 ^D 
 
 
 
 r 
 
 / 
 
 / 
 / 
 
 
 \^^Fig. 158. 
 
 
 1 
 
 
 
 
 \a 
 
 B 
 
 < 
 
 
 n 
 
 —4 
 
 reciprocating parts (piston, piston-rod and cross-head) have very 
 nearly the maximum velocity at the position where the connecting 
 rod and crank form a right angle at C (Fig. 159). The true phase 
 for the maximum velocity of the piston is a little later than the 
 above position; but it is difficult to locate this exact position, and 
 Tvith the proportions of crank and connecting-rod used in ordinary 
 «ngines (Z -^ r = « = from 4 to 6 usually), this error is of no prac- 
 tical account, and the approximation is much more conveniently 
 used. To find the crank position (Fig. 159) at which the crank is 
 perpendicular to connecting-rod, erect the perpendicular, Ac', to 
 the line of centres at A, equal to the length of the rod, and connect 
 c' with Q. The intersection of c'Q with the crank-circle locates 
 the required position of the crank, ; iov Ac' = Cc ; AQ = C'Q; 
 and in the two triangles AQc' and CQc, the angle ^^C'= is 
 common. When two sides and the corresponding angle of two tri- 
 angles are equal the triangles are equal; therefore, as QAc' is a right 
 angle by construction, cCQ is also a right angle, and C is the posi- 
 tion of the crank-pin required. The phase at which the piston has 
 its maximum velocity is of importance in certain problems relating 
 to the mechanics of the steam-engine, for it is the phase at which 
 the acceleration of the reciprocating parts is zero. In high-speed 
 engines the acceleration of the reciprocating parts has a very im- 
 portant bearing upon pressures transmitted from the piston to the 
 crank. 
 
LINKWOBK. 183 
 
 98. The Eccentric. — The eccentric is a modified crank, and all 
 that has been said in the preceding article applies to the eccentric 
 and rod. If the crank-pin be grad- 
 ually enlarged, its throw remaining 
 unchanged, the motion transmitted 
 io a given connecting-rod is un- 
 altered. Fig. 160 shows such a 
 crank-pin enlarged in diameter 
 until it includes the shaft, and it gives the familiar eccentric 
 and rod. 
 
 The throw of the eccentric is the radius of the equivalent crank, 
 QC; or it equals the distance from centre of eccentric figure to 
 ■centre of the shaft about which it turns. 
 
 The enlargement of the pin increases the friction, although it 
 has no kinematic effect. The eccentric is a useful expedient when 
 a crank of small throw is required which cannot be conveniently 
 located at the end of the shaft, for under such conditions the orili- 
 nj.ry connecting-rod would "interfere" with the shaft unless a 
 double-throw crank were used, and this latter form would weaken 
 the shaft by cutting into it, besides being a more expensive con- 
 struction. For these reasons the eccentric is very commonly eni- 
 ployed for operating the valves of engines, imparting a reciprocat- 
 ing, and nearly harmonic, motion to them. 
 
 99. Connecting-rod of Infinite Length. — It has been seen that 
 the stroke of the cross-head (Fig. 154) equals the diameter of the 
 ■crank-pin circle, = 2r ; and that the obliquity of the connecting- 
 rod distorts the cross-head motion from a true harmonic motion, 
 causing the half-stroke farthest from the shaft (at the head end of 
 the cylinder) to be made in less time than is taken by the half- 
 stroke nearest the shaft (the crank end). It was shown in Art. 97 
 that the displacement of the piston from mid-stroke, when the 
 crank is at either "quarter," or d — 90° (measured, in Fig. 154, by 
 ^m) is less as the connecting-rod is made longer, relative to the 
 crank; or as ? -j- r = « becomes greater. 
 
 If the rod were of infinite length, the cross-head would be at the 
 middle of its stroke when the crank is at the quarter {9 = 90°); 
 
184 
 
 KINEMATICS OF MACSINEBT. 
 
 for it was shown that mq = I — VP — r" ; hencej mq = I — I = o, 
 when the length of the rod is infinity. It is, of course, impossible 
 
 to have a rod of infinite length ; 
 but it was shown in Art. 96 that 
 the cross-head and guides give the 
 equivalent of an infinite length of 
 link as to one member of the four- 
 link chain; and the slotted rod 
 and block of Pig. 161 may be in- 
 troduced as an equivalent to ark 
 infinite connecting-rod. That is,, 
 this mechaTiism is the equivalent of the four-link chain with two 
 links of infinite length. 
 
 With the mechanism of Fig. 161 the crank is acted upon by the 
 slotted rod through the block. The component of the motion of 
 the crank-pin, which is normal to the acting faces of the yoke, equals, 
 the motion of the rod. This normal component is seen to equal 
 the motion of the crank-pin multiplied by cos <p ; and as = 90° — 
 6, cos = sin 0, hence the velocity of a piston attached to the 
 slotted rod is equal to sin 0, when v is the velocity of 'the crank. 
 The piston velocity is a maximum when sin 6^ is a maximum (assum- 
 ing the crank to rotate uniformly) ; or when = 90°. At this 
 phase the piston and crank have the same velocity, since sin 6—1, 
 This agrees with the statement in Art. 97 that the velocity of the 
 piston always equals that of the crank when = 90°. With the 
 finite rod there is another crank position, for a smaller value of 6,. 
 at which this equality also exists, and between these two crank posi- 
 tions the piston velocity is greater than that of the crank. As the 
 rod is increased in length, these two positions for equality approach 
 each other, the first one more nearly corresponding to (^ = 90°. 
 With the infinite rod the two phases for equality coincide, and the 
 phase for maximum velocity, which in the general case lies between 
 them, also falls at ^ = 90°, as seen above. These conditions will 
 be found to harmonize with the general relations deduced above. 
 Fig. 161 indicates the application of this mechanism, as it is some- 
 
LINEWORE. 185 
 
 times made to steam fire-engines and other steam-pumps. P indi- 
 cates the pump-cjlinder and 8 the steam-cylinder. Tlie crank- 
 shaft caiTies the fly-wheel. 
 
 The practical effect of this " rod of infinite length," or the 
 Scotch yoke, as it is frequently called, is to make a more compact 
 mechanism than would be obtained with a finite rod of ordinary 
 length; for the distance between the "glands" of the stuffing- 
 boxes on the two cylinders needs be only equal to the stroke plus 
 the outside width of the slotted yoke, with a small allowance each 
 side for clearance. 
 
 The sliding-block is not an essential, kinematically, as the 
 crank-pin could act directly on the faces of the slot ; but, as shown 
 in Art. 28, it is generally desirable to use surface contact to distrib- 
 ute the pressure transmitted, instead of line contactj when the 
 conditions will permit. 
 
 The sliding of the block in the slotted member produces friction 
 and resultant wear, which is not so easily overcome as in a pin con- 
 nection ; and the ordinary form of connecting-rod is therefore pre- 
 ferred as an engine connection when the utmost compactness is not 
 a leading consideration. 
 
 100. Connecting-rod of Length Equal to Crank. — If the connect- 
 ing-rod is of a length equal to the throw of the crank, as in Fig. 162, 
 these two members always form an 
 isosceles triangle, with the inter- 
 cept on the centre line between the 
 cross-head and shaft as a base. The 
 distance Qa = r ^-1= 2r, and b, 
 the end of the stroke next to shaft, 
 coincides with Q. In this arrange- 
 ment, c would be drawn from a to 
 Q during a crank movement AM 
 and the displacement from the 
 centre of stroke, due to angularity 
 of the rod, =AQ=r. If the cross-head comes to rest at Q 
 when C reaches M, with any farther motion of the crank the con- 
 
6 EINBMATICB OF MACHJNERT. 
 
 cting-rod would simply rotate around Q with the crank. If the 
 OSS-head continues to move in the line aQ, produced beyond Q, 
 e crank movement MB would drive the cross-head to d, a distance 
 Dm § = 2r, and the total stroke of the cross-head would = 4r. 
 :e inertia of the cross-head as it approaches Q would tend to pro- 
 ,ce this effect ; but such a motion can be made positive by the 
 sclianism shown by the extension of cC to c'. In this form the 
 ;id rod cc' (= 2r) is pivoted at its centre to the crank-pin, and 
 5ss-heads at the ends of the rod move in guides at right angles to 
 ch other which intersect in Q. When C is at A, c is at a, and c' 
 at Q. As G moves to M, c moves to Q,&n6. c' moves to a' ; then, 
 the motion of C continues to the position B, c passes Q, moving 
 d, and c' returns to Q. As C passes B and moves to M', c' passes 
 )m Q to d', and c returns to Q. During the completion of the 
 ink revolution, C moves from M' to A, c moves from Q to a, and 
 returns to Q, completing the cycle. 
 
 At any phase the distance of c' from Q corresponds to Qt, of 
 g. 154, and hence is proportional to the velocity of c ; likewise, 
 ; is proportional to the velocity of c' at any phase. In this mech- 
 ism there is a transverse stress, as well as tension or compression 
 
 the rod cc'. 
 
 101. Path of Cross-head Passing Outside of Shaft-centre. ^ — If the 
 le of cross-head motion, g . . h (Fig. 163), does not pass 
 rough Q, the motion is modified as follows. 
 
 M'.--^ b 
 
 Fig. res. 
 
 To find the ends of stroke a and I : first, take a radius ^=1 -\- r, 
 th a centre at Q, and cut gh in a ; second take a radius = I — r, 
 th the same centre, and cut gh in i ; the required points are a 
 
LINK WORE. ' 187 
 
 and i, and the corresponding crank positions are QA and QB. 
 The stroke from a to 6 is made while the crank moves through the 
 arc AMB ; and the return stroke takos place as C moves through 
 the arc BM'A. If the crank motion is uniform, the forward and 
 return strokes are made in unequal times, and this mechanism 
 gives one form of " quick-return motion." If it is required to de- 
 sign such a quick-return motion, the relative times of forward and 
 leturn strokes being given: draw the crank circle and divide its 
 circumference into two arcs having the required ratio, AMB, 
 BM'A. Extend the radii through these points of division A and 
 B, jn directions QA and BQ; then lay off from Q on the exten- 
 sion of QA, I + r, locating a; and on BQ lay oS. I — r from Q, 
 giving b; a and h are the ends of the stroke, and a . . J is the 
 line of cross-head motion. 
 
 In the Westinghouse engine the above construction is applied ; 
 that is, the line of piston travel passes to one side of the shaft- 
 centre. Two cylinders are placed side by side, with connecting-rods 
 acting on cranks which are opposite each other (180° apart). This 
 engine is single-acting, steam acting on each piston only during its 
 downward stroke ; therefore, by giving the quick return to the up- 
 ward stroke, one piston makes its exhaust-sti-oke and takes steam 
 again before the other piston has quite completed its "working" 
 stroke ; thus, there is no period at which the rotative effort is abso- 
 lutely zero. Furthermore, the greatest angularity of the connect- 
 ing-rod occurs on the exhaust-stroke, and for a given length of con- 
 necting-rod, the maximum obliquity of action is reduced for the 
 stroke during which steam-pressure is acting on the piston. Or, to 
 state the case somewhat differently, the length of the rod can be re- 
 duced ior a given maximum obliquity during the period of heavy 
 pressure, thus permitting a more compact construction. 
 
 102. Motion "of a Point in the Connecting-rod between the 
 CroBS-head and Crank. — In certain valve-mechanisms, motion is 
 taken from some point in a connecting-rod (or eccentric rod) other 
 than either of the pin-Centres previously considered. Let P (.Pig. 
 164) be such a point, the motion of which it is desired to find. 
 
!8 KINEMATICS OF MACHINERY. 
 
 ind the instant centre for any chosen phase of the rod Cc. All 
 points of the rod, at the instant, rotate 
 about with the same angular velocity, 
 and with linear velocities proportional 
 to their radii. Hence, the linear velocity 
 of Pis to that of C as (9P is to OC. The 
 direction of the motion of P is perpen- 
 dicular to OP, as indicated by Pi\. 
 A similar method can be used if the point P lies beyond either 
 e crank or cross-head in an extension of the connecting-rod. 
 
 This problem can be solved by the resolution and composition of 
 lative motions also, but not so readily. 
 
 103. Inversion of Crank and Connecting-rod Chain. — It was 
 own in Art. 29 that a kinematic chain may give apparently differ- 
 t mechanisms by making different members of it the stationary 
 ik. Thus, Figs. 69, 70, 71, and 72 show the four possible inver- 
 )ns of the crank and connecting-rod chain. The case of Fig. 69 
 ,s been treated in preceding articles of this chapter. Pig. 70 rep- 
 sents the condition when the former crank is made the fixed 
 ember ; this case is next in practical importance to the ordinary 
 ank and connecting-rod mechanism. This form may be used to 
 3ure a variable angular velocity of a continuous rotating follower 
 )m a uniformly rotating driver, resembling the drag-link in its 
 tion. In conjunction with another linkage this mechanism is 
 3quently used to produce a slow advance and a quick return of 
 e cutter-bar of a shaping-machine. 
 The condition shown in Fig. 71 is, as already'pointed out, the- 
 'chanism of the oscillating steam-engine. The case of Fig. 72 
 ,s comparatively little practical application. Any of these can be- 
 idily analyzed by the instant centre method. The form in which 
 s fixed (Fig. 70), will be treated in some detail, on account of ita 
 tended practical use ; the others will not be taken up as special 
 •ms. 
 
 Fig. 165 shows a crank a which rotates about and is pivoted 
 a sliding block by the pin P. This block fits a slot in the arm. 
 
LINKWORE. 
 
 189 
 
 b, which rotates about 0'. The stationary member d supports the 
 fixed centres and 0'. The point P rotates in the circle AB ; 
 lience, its motion at any instant is perpendicular to the radius PO 
 (the centre line of the crank a). This motion, which is usually 
 uniform but not necessarily so, is designated by v,. We may con- 
 sider the point P to act upon the centre line (pitch line) of the 
 slotted member, as the block does not affect the kinematic problem. 
 
 The point in a which lies at P has the velocity Pvi, and the 
 point in the slotted bar h, which is also at P for the instant, has the 
 velocity Pv^. As these two velocities have equal components along 
 the common normal to the contact surfaces, the normal component 
 of PVi, — Pv^. As a point in a, P is fixed at the distance OP from 
 the centre 0. As a point in i, it travels back and forth along the 
 pitch line of the slot, its distance from 0', or its effective radius, 
 Tarying from O'A to 0' B as the driver moves from A to B, 
 During the next half-revolution of the driver {B to A) the effective 
 radius of the follower decreases from O'B to O'A, thus completing 
 the cycle. 
 
 Only the component of the motion of the driving-point which 
 is normal to O'P can impart rotation to the follower. This com- 
 ponent is represented by v^ = v, cos <p (in which expression is 
 «qual to the angle OPO'); because v^, and v, are perpendicular to 
 
190 KINEMATICS OF MAOHINSRY. 
 
 OP and O'P, respectively. If a circle be drawn on 00' as a diam- 
 eter, the intercept, Pn, of the centre line of the follower (extended 
 through 0' if necessary), which lies between P and this circle 
 is to v^ as the constant radius of the driver is to w,. Or, : v^ : v^ :: 
 PO : Pn; for, as OnO' is an angle subtended in a semicircle. 
 Oil is perpendicular to O'P, hence Pn = PO cos 0. The velocity 
 of the driver may be represented to a scale which will make it equal 
 to PO, when Pn becomes the velocity v^. If this velocity scale is 
 not convenient, v, may be laid off from P towards 0, as Ph, and 
 a line kl drawn perpendicular to O'P will give PI = v^, to this 
 latter scale. 
 
 104. Guick-return Motions. — If (Fig. 165) a sliding block, c, 
 travels in the path ef, which passes through 0', and is connected 
 to a point in an extension of the slotted follower by the rod Cc, 
 it will reach one end of its stroke when the driving-point P is at 
 B, and this block c reaches the other end of its stroke when P 
 is at F. While P is moving through the arc FEE, c moves from 
 e to/; while P moves through the arc EAF, c makes its return 
 stroke from / to e. Now if the driver rotates uniformly the times 
 of these forward and return strokes are in the ratio of tlie arcs 
 FBF to EAF. This is, in principle, the Whitworth quick-return 
 mechanism, as it is frequently applied to shapers. The slow stroke 
 gives the cutting stroke of the tool; while the return stroke may 
 be made more rapidly, thus economizing time and increasing the 
 capacity of the machine. 
 
 In designing such a mechanism the circle in which P rotates 
 may be drawn with as a centre; then divide its circumference 
 by E and F into two arcs having the ratio desired for the times of 
 the forward and return strokes. Draw a line through EF, ex- 
 tended to one side, and the path of c lies in this line. Drop a 
 perpendicular from upon EF and its foot will locate 0', the 
 fixed centre for the slotted arm. Take at a distance from 0', 
 which will give the required length of stroke, and choose a suitable 
 length for the connecting-rod Cc, 
 
 In practice is a pin which can be set at different distances 
 
LINKWORK. 
 
 191 
 
 along a radius to 0', so that the length of stroke of c can be 
 varied to suit the work. The pin C might be placed on the same 
 side of 0' as the slot ; but it is usually more convenient to locate 
 it as in Fig. 165. 
 
 The velocity of Cis to v, as O'C is to O'P, since these are the 
 velocities of two points in one piece which rotates about 0'. 
 The motion of O is perpendicular to O'C, as shown by Cv^. To 
 find its velocity lay ofE v, (found as above) and draw the line 
 v^O'v, cutting the perpendicular to O'C at v„ and giving Cv, as 
 the velocity sought. To find the velocity of c, erect a perpendic- 
 ular to the path of this point through it; lay off Cv,' = Cv^ on 
 the extension of O'C, and draw a line v^'v/ parallel to the rod 
 Cc; v/ is an ordinate of the velocity diagram of c. The student 
 should complete this diagram for both strokes, by the method in- 
 dicated. 
 
 The practical construction of the Whitworth quick-return motion 
 is shown in Fig. 166, in which the letters correspond to those of 
 Fig. 165. The pin P is attached to a gear which rotates about 0, 
 
 JJlESUiv, 
 
 Fig. 166. 
 
 the centre of a large fixed stud. The centre 0' is a pin secured 
 in the fixed stud, and the slotted member rotates about this centre 
 0'. The pin C can be clamped at different points along its slot 
 to secure corresponding lengths of stroke of c. 
 
 Another quick-return mechanism, also much used for shapers, 
 is indicated by Fig. 167. The slotted bar is pivoted to the frame at 
 
192 KINEMATICS OF MACEINEBT. 
 
 0', imd is driven by tlie crank pin P, whicli rotates about as in 
 the preceding case. The slotted bar vibrates between the positions 
 O'c and O'f, reaching an end of its stroke when its centre line is 
 tangent to the crank-pin circle; or when the crank is at either E 
 or F. It will be seen that the driver passes over the arc FBE for 
 the forward stroke, and through the arc EAFtov the return stroke. 
 The former arc is greater than the latter; hence the times of the 
 strokes are in the ratio of these arcs, if the driver rotates uni- 
 formly. 
 
 The normal component only (wj of the crank-pin motion {v^) 
 transmits motion to the follower; and v^ = v^ cos 0, in which cp is 
 the angle OPO'. If a semicircle be drawn on 00' as a diameter, 
 cutting O'P at n, Pn = OP cos (p j hence v^:v,:: OP : Pn; or if 
 OP represents the velocity of the crank-pin, Pn represents the 
 Telocity of the driven point of the slotted arm to the same scale. 
 
 The upper end of the slotted arm drives the cutter-bar of the 
 shaper as indicated, through a pin, 0, which is between two parallel 
 projections attached to the cutter-bar. The velocity of C is v^ , 
 normal to O'C, and v, : v, :: O'C: O'P. To find this velocity 
 draw a line through 0'. . v^, extended till it cuts Cv^ in v,. The 
 motion of the tool, v^, is the horizontal component of v,. It dif- 
 fers little from v, ; but can be easily found by the graphical con- 
 structioii shown. 
 
 The fundamental portion of this mechanism is a modified form 
 of the one used in the Whitworth motion ; the only difEerence be- 
 ing that 0', in this case, lies outside of the crank-pin circle; while 
 in the other case it lies inside this circle. This difference in the 
 proportions causes the slotted bar to vibrate through a definite 
 angle in one case while it rotates continuously in the other case. 
 The methods of connection with the ram of the shaper are quite 
 different in these two cases, as is the means of changing the length 
 of the stroke, also. In the second form this change is made by 
 changing the length of the driving crank-arm, means being pro- 
 vided for moving the pin nearer to, or farther from, its centre, 0. 
 This device is extensively used on shapers. 
 
LINE WORK. 193 
 
 By proper means, the throw of the crank is set to give the de- 
 sired stroke. The adjustment can usually be made without stop- 
 ping the machine. 
 
 With the Whitworth device, the relative time of forward and 
 return strokes is not varied by changing the length of stroke. With 
 the second mechanism the ratio between the times of the forward 
 iiiid return strokes is greatest with long strokes. The angle through 
 which the driver passes for the forward stroke is 180° + 8, where 6 
 is the angle of vibration of the slotted bar; and during the return 
 stroke the driver passes through 180° — 6. The sine of ^ ^ = OP 
 -H 00', and as 00' is a constant, varies with changes of OP- 
 
 To design this machine, decide upon the ratio of the times to 
 be occupied in the forward and return strokes for some particular 
 length of stroke. Draw the crank-circle for this particular stroke 
 (Fig. 167) and divide it into the arcs FBB and EAF, having this 
 ratio. Draw tangents to this circle at B and F, and their intersec- 
 tion locates 0'. 
 
 The velocity diagram is readily constructed for both strokes by 
 :finding the velocity = ?», for various positions of the ram, by the 
 method given. This diagram should be drawn as an exercise. 
 
 The crank and connecting-rod when arranged so that the centre 
 line passes outside of the crank-circle centre (as discussed in Art. 
 101), may be used for a quick return. Elliptical gears (see Art. 46) 
 are also used for quick-return mechanisms. 
 
 105. Bell-cranks.— Fig. 168 shows the method of designing a 
 bent lever, or bell o-ank, to transmit motion from line OA to line 
 OB, with linear velocity m. -f- n. Lay 
 off Oa — in on OB, and Oi — n on 
 OA ; complete the parallelogram 
 Obqa by drawing aa and bh parallel to 
 OA and OB, respectively, and inter- 
 secting at q. Through and q draw p. ^^^ 
 a line. Any centre, as Q, on this line 
 may be taken as the bell-crank centre. From Q, drop perpen- 
 diculars QP and Qp on OA and OB; these are centre lines of the 
 
194 
 
 KINEMATICS OF MAOHINERT. 
 
 arms of the bell-crank. The angular motion of the arms should be 
 the same on each side of QP and Qp. 
 
 It will be seen that any motion, either side of the central posi- 
 tion, will shorten the effective arms. To reduce the deviation of tlie 
 connected points to a minimum, the lengths R and r should be- 
 greater than QP and Qp, respectively, by one-half the versed sine- 
 of the angle described each side of the central position multiplied 
 by the respective radii. 
 
 106. The Beam. — Fig. 1G9 indicates the beam of an engine - 
 CO is the line of piston motion; (^ii^= distance of beam centre- 
 
 from this line, = d ; AB = stroke ; 
 AE = \ stroke = s ; DP should = 
 EF, to minimize the angularity of the- 
 connecting-rod. The beam lengtli to 
 fulfil this requirement is found thus: 
 QD = QA=d +EP; but 'QA' — 
 .-. {d + EFf ={d- 
 
 Fig. 169. 
 
 QE' +AE' 
 
 EPy + s\ .: d' + 2d . EF + EF^=^d' - 2d . EF + EF' + s\ 
 .: U . EF=. s". .: EF = DP ^ s' -^ 4:d ; .: the length of the 
 
 beam = d + PF=d + 
 
 id- 
 
 
 ^^-^' 
 
 107. Rapid Change in Angular Motion of the Follower. — A 
 
 means of imparting rapidly changing angular velocity to a follower 
 by the use of the linkwork is shown 
 by Fig. ITO. As reaches the vari- 
 ous positions in its path marked 0, 1, 
 2, 3, etc., c lies at 0', 1', 2', 3', etc., 
 respectively, in its path. The prin- 
 ciple of this arrangement is of fre- 
 quent use. It is applied, as indicated 
 in the figure, in the " wrist-plate " of the Corliss valve-gear, to give 
 a rapid opening of the valves (and quick closing of the exhaust- 
 valves) with a small motion when they are closed. In this applica- 
 tion the rotating (vibrating) valve is attached to the arm 0' c, and 
 
LINKWORK. 195: 
 
 the mechanism is so proportioned that the required port opening is 
 given quickly to a valve at one end of the cylinder, while the valve- 
 arm at the other end moves but little during this period. 
 
 In general, the motion of the follower c is small compared to a 
 given motion of the driver G as the angle 0' c C approaches a right 
 angle and the angle OCc approaches or 180°. On the- other 
 hand, the relative motion of c to C is great as the angle 0' c C ap- 
 proaches or 180° and the angle OCc approaches 90°. 
 
 108. Straight-line Motion.— A large number of linkages have 
 been devised to make a point move in a straight line independently 
 of any planed guides. 
 
 The term Parallel Motions is usually given to such mechanisms, 
 but straight-line motions is a more appropriate term. Fig. 171. 
 shows what is known as Watt's par- 
 allel motion. R and r are arms cen- 
 tred at Q and q ; Aa is a link con- 
 necting the free ends of R and r, and 
 P is a point in Aa which traces an 
 approximately straight line, within 
 certain limits of the motion. 
 
 If R moves from its central posi- 
 tion, A is drawn to the right, while '^" ' 
 the accompanying motion of r carries a to the left. The path of 
 P is a function of both of these motions and the result is that P, if 
 properly located in Aa, moves very nearly in a straight line, pro- 
 vided the angular motion of R and r does not exceed about 20°. 
 The complete path of P is the "figure 8 " shaped curve Pmii. 
 
 If R = r, AP = aP- In general, the segments AP and aP' 
 are inversely as the length of the adjacent arms. 
 
 Watt used this mechanism to guide the piston-rod in place of 
 the slides now generally employed ; but the principal application 
 of " parallel motions " at present is on steam-engine indicator 
 pencil motions. The Eichards indicator, the earliest of the modem 
 type, has the Watt mechanism. 
 
 The Tabor indicator has a motion in which a curved guide is 
 
196 
 
 KINEMATICS OF MACHINEUY. 
 
 Fig. 172. 
 
 it is, therefore, of a different type from the pure liiikwork 
 mechanisms usually classed as j^ar- 
 allel motions. Pig. 172 indicates this 
 y pencil movement. It is desired that 
 the pencil point P shall move in a 
 right line, 7nm. It is evident that 
 the curved guide wm can be given 
 such a form that this will occur, and this curve can be found by 
 moving P along mm, tracing the curve nn by the point p. Hav- 
 ing found nn, a circular arc may be found whicli agrees closely 
 with it, within the range of motion ; and if the centre of this arc 
 be at d, a link dp can be substituted for the curved guide nn. 
 An arrangemejit similar to this substitution is used on the Thomp- 
 son indicator. If a moved in a straight line, instead of in the arc, 
 yy ; ii p were at the centre of Pa ; and if dp = Pp = pa, the 
 mechanism would be the same as that shown in Pig. 162, and the 
 result would be an exact straight-line motion ; requiring a straight 
 guide, however, for the point a. 
 
 The Crosby indicator has a pencil mecha.nism similar to that of 
 Pig. 173. If P be moved in a straight line mm, p (a point in the 
 link l)c) traces a curve ; the bridle link dp is one that will give an 
 arc most nearly approaching this curve. 
 
 Fig. 174. 
 
 Peaucellier's straight-line motion is exact, and it is a pure link- 
 work. It is shown in Fig. 174. Two equal links (( and i have a 
 fixed centre, Q. The links d, e,f, g are equal ; and c, with a fixed 
 centre at q, equals the distance Qq. P is constrained to move in 
 the straight line 7nm. 
 
LINKWORK. 
 
 197 
 
 109. Pantographs. — There are various linkages in which if one 
 point is made to travel in any path, some other point will be con- 
 strained to describe a similar path, enlarged or reduced. 
 
 Fig. 175 shows one such arrangement in which Pa — IQ; and 
 «C = Pi- These Knks form a parallelogram which has a fixed 
 centre at Q. A bar cd is attached to 
 aQ and Pb parallel to Pa, and the 
 point p, in cd, which lies on the line 
 connecting P and Q, will move in a 
 path similar to that traced by P- 
 Suppose P to move to P', then p 
 moves to p', and from similar tri- 
 angles, QPiQp-.-.Qa: Qc; also QP' : Qp' :: Qa' : Qc'; but Qa= Qa', 
 and Qc — Qc' .: QP : Qp :: QP' : Qp', hence the 'distance of p 
 from Q is proportional to the distance of P from Q. As p always 
 lies in the line QP (because QaP and Qcp are similar triangles), 
 the angular motion of p about Q is equal to the angular motion of 
 P about Q. Any path of P is determined by its angular motion 
 about Q and its radius vector to § as a pole; as the angular motion 
 of P and of p about Q are seen to be equal for any motion of either 
 of these points, and as the radius vector of p bears a constant ratio 
 to that of P, the path of ^ is similar to that of P 
 
 A form of pantograph, called the "lazy-tongs," is shown in 
 Fig. 176. It is frequently used to reduce the piston motion of an 
 
 Fig. 176. ' Fig. 177, 
 
 engine, in using the indicator. P is attached to the cross-head, and 
 the indicator cord is attached at jo. The practical objection to this 
 contrivance is the great number of joints, and consequent liability 
 to lost motion from wear. 
 
198 
 
 KINEMATI08 OF MAGHINEBT. 
 
 Fig. 177 shows another pantograph for the same use. P is at- 
 tached to the cross-head, and the cord is attached at ^ as before. 
 With either of these arrangements the point p must lie in the line 
 ■connecting P and Q, and the cord must be led off parallel to the 
 cross-head motion. 
 
 Watt combined the pantograph with his straight-line motion so 
 that the piston-rod, air-pump rod, and feed-pump rod were all 
 guided in straight lines by means of one combination of links. 
 
 110. Hooka's Coupling, or the universal joint, is used for con- 
 necting two shafts which intersect. It is equivalent to what Eeu- 
 leaux calls the four-link conic chain — that is, to a four-link chain 
 in which the pivots are not parallel as in the ordinary case already 
 treated, but their axes lie in radii of a sphere. Every point moves 
 in the surface of a sphere, instead of in a plane. In its typical form 
 ■(Fig. 178), each shaft has a forked end, and the two forks are 
 united by an equal armed cross ab, cd, or its equivalent. The 
 
 Fig.' 179. 
 
 shafts Mm and Nn and the arms of the cross (the axes of the pivots) 
 intersect in a common point 0. If only one half of each fork be 
 considered, as ma of Mm and nc of Nn, and these are assumed to 
 be connected by the spherical link ac equal to the fixed distance 
 between the two adjacent points of the cross, a four-link conic 
 chain is produced in which the axis of all the turning pairs inter- 
 sect in 0. With this arrangement the fork could be omitted, and 
 
LINK WORK. 199 
 
 T?e would have the kinematic equivalent of the original mechan- 
 ism. 
 
 The driver Mm and the follower Nn make complete revolutions 
 in the same time; but the velocity ratio is not constant througliout 
 the revolution. 
 
 If a plane of projection be taken perpendicular to the axis of 
 Mm, the path of a and h will be the circle ABCD in Pig. 179. If 
 the angle between the shafts is /?, the path of c and d will be a 
 circle which is projected on the ellipse AB'CD', in which OB' = 
 OD' = OB cos /3 = OA cos /3. 
 
 If one of the arms of the driver is at A an arm of the follower 
 will be at B'; and if the driver-arm moves through the angle d to P 
 the following arm will move to Q; 0§ will be perpendicular to OP, 
 hence B'OQ — 6. But B'OQ is the projection of the 7-cal angle 
 •described by the follower. Qn is the real component of the follow- 
 er's motion in the direction parallel to AC, which line is the inter- 
 section of the planes of the driver's and follower's paths. The true 
 angle (p, described by the follower, while the driver describes the 
 angle 6, can be found thus: draw QR parallel to OB so that Rm = 
 Q?i, then OR equals the radius of the follower, and BOB = cp =; 
 the true angle in plane AB'CD' which is projected as B'OQ = 6. 
 
 Now tan (p = Rm -t- Om, and tan 6 = Qn -i- On; but Qn — Rm, 
 
 4- Om, 
 
 and tan 6 = Qn 
 
 -^ On; 
 
 but 
 
 Qn^ 
 
 tan e 
 
 Om 
 ~ On ~ 
 
 OB 
 
 1 
 
 
 
 tan 
 
 ' OB' 
 
 cosyS' 
 
 
 tan <p 
 
 = cos/? 
 
 tan e. 
 
 • • 
 
 , 
 
 • . • 
 
 (1) 
 
 The angular velocity ratio of follower to driver is therefore found 
 as follows by differentiation of Eq. (1), remembering that /? is a 
 constant in this equation: 
 
 a' _ d4> _ cos /3 sec' S _ cob /3 sec' 6 _ 
 
 a ~de sec' ~ 1 + tan' 0' ■ • - K'^f 
 
200 KINEMATICS OF MACHINERY. 
 
 Eliminating 4> by 
 
 means of (1) 
 
 
 
 
 
 
 
 
 
 cos ft 
 
 
 
 a' 
 
 cos /? sec" 
 1 + cos' yStan" d ~ 
 
 
 cos" 6 
 
 
 
 a 
 
 cos" 
 
 e -f- sin" 8 
 
 cos 
 
 'ft 
 
 
 
 
 
 cos"& 
 
 
 
 
 
 cos ft 
 
 
 
 cos 
 
 ft 
 
 cos" 6* + sin" ^ (1 - sin" ft) 1 - sin" sin" ft' 
 By a similar process 6 could be eliminated, giving 
 a' 1 — cos" <p sin"/? 
 
 (3> 
 
 (4> 
 
 a cos ft 
 
 It is seen from (3) that a'-i- a is a minimum when sin ^ = 0^ 
 or when 6 — 0, n, etc., which corresponds to a value of = 0, n^ 
 etc. The same thing is seen from (4), which gives a minimum 
 value of a' ^ a when cos = 1, or = 0, tt, etc. Also, a' ■— a is. 
 a maximum when sin ^ = 1, or cos 0=0, corresponding to ^ = 90°, 
 |-7r, etc. ; = 90°, ^n, etc. 
 
 To summarize the foregoing, the follower has a minimum angu- 
 lar velocity, if the driver has a uniform velocity, when the driving- 
 arm is at A or C and the following arm is at B' or D'. The fol- 
 lower has a maximum angular velocity when the driving-arm is at 
 £ or D and the following arm is at A or 0. 
 
 By using a double joint the variation of angular velocity be- 
 tween driver and follower can be entirely avoided. To do this an 
 intermediate shaft is placed between the two main shafts, making- 
 the same angle, ft, with each. The two forks of this intermediate 
 shaft must be parallel. If the first shaft rotates uniformly, the 
 angular velocity of the intermediate shaft will vary according to the 
 law deduced above. This variation is exactly the same as if the lasi 
 shaft rotated uniformly, driving the intermediate shaft ; therefore,, 
 as uniform motion of either the first or the last shaft imparts the 
 same variable motion to the intermediate shaft, uniform motion of 
 either of these shafts will impart (through the intermediate shaft) 
 uniform motion to the other. This is the combination used in th& 
 feed-rod of the Brown & Sharpe milling machines and elsewhere. 
 
LINKWORK. 201 
 
 111. Ratchets. — The ratchet-wheel and pawl (Fig. 180) resembles 
 both the direct-contact motions and linkwork. The driving-pawl 
 GP acts by direct contact ; but dur- 
 ing driving the action is similar to 
 that of a four-link chain, consisting 
 of QC, qP, PC, and the fixed link 
 Qq. Such mechanisms are sometimes 
 termed intermittent linhworlc. 
 
 The two centres Q and q may co- 
 incide, the pawl-lever vibrating about 
 the axis of the wheel. In this case 
 there is no relative motion between ^'^' '*°" 
 
 the members during the forward (working) stroke. 
 
 The supplementary pawl,cp, has a fixed centre, c, and its object is 
 to prevent the backward motion of the wheel when CP is not driving. 
 If pn is the common normal to the end of cp and the tooth with 
 which it engages, there is no danger of the pawl becoming dis- 
 engaged under the reaction of the tooth upon it ; for the centres c 
 and q are on the opposite sides of the line of action, and the ten- 
 dency is for the wheel to run backward (right-handed rotation) and 
 for the pawl to turn with a left-handed rotation, which only forces 
 them together. If the direction of the common .normal is, pn' , the 
 centres both lie on the same side of the line of action, when the 
 tendency is for both pawl and wheel to rotate in the right-handed 
 direction, and the pawl would be forced out of contact, unless held 
 by friction. In a similar way the normal Pm of the driving-pawl 
 CP and the tooth on which it acts should pass between C and Q. 
 
 The pawl cp only prevents backward motion of the wheel after 
 the wheel has moved back far enough to come in contact with the 
 pawl. The amount of backward motion possible may vary from 
 zero to the pitch of the teeth. This action could be limited by 
 making the teeth small ; but this would weaken the teeth, and the 
 expedient is sometimes adopted of placing several pawls side by side 
 on the pin, c, the pawls being of different lengths. With this 
 arrangement the maximum backward motion may be reduced to the 
 pitch divided by the number of pawls provided. 
 
302 
 
 KINEMATICS OF MACHINEBT. 
 
 Sometimes, for feed-motions, etc., the pawl and wheel are made 
 as shown in Fig. 181. This pawl can be reversed for driving in the 
 opposite direction. 
 
 Fig. 182 shows a double-acting ratchet by which both strokes of 
 the lever drive the wheel. The locking-pawl may be omitted in 
 this case. 
 
 Fig. 182 
 
 Frictional pawls (Fig. 183) are sometimes used, in which the 
 wheel is made without teeth. The pawl grips the wheel by fric- 
 tion during one stroke and releases it on the return stroke. These 
 have the advantage of being noiseless, and the angular motion of 
 the wheel for each stroke is not restricted to some multiple of the 
 
 Fig. 184. 
 
 arc between two teeth, but the driving is not positive. Another 
 frictional pawl with a fixed centre at c can be used to prevent 
 -" overhauling " of the wheel. The letters of Fig. 183 correspond 
 Avith those of Fis;. 180. 
 
LINKWOHK. 
 
 203 
 
 In the form of frictional ratchet shown by Fig. 184, the wheel is 
 surrounded by a ring, which can be vibrated iibout the axis of the 
 wheel. One of these members (either) has teeth of the form 
 shown; and in the depressions formed by the teeth, rolls, or balls, 
 are placed. Motion of the driver in one direction causes these 
 rolls to bind the follower, while they release it on the return. 
 Positive " silent" ratchets have been made with a special device for 
 holding the pawl clear of the teeth on the return stroke. 
 
 The forms of ratchets shown by Figs. 180 to 185, and numerous 
 modifications of them, are suitable for many cases requiring the 
 conversion of a reciprocating action into an intermittent rotation. 
 They are especially convenient in feed-mechanisms when the vibra- 
 tions of the driver are not too rapid. At high speeds the shock 
 between the pawl and tooth, as the driving begins, may be objec- 
 tionable, and the inertia of the wheel is liable to make it move 
 farther than desired, or to cause " overtravel." This last tendency 
 prevents the employment of the ordinary ratchet when, as in revo- 
 lution registers or continuous counters, a definite motion of the 
 follower must be insured. A device for such purposes is shown by 
 
 Fig. 185. V2/ Fig. 186. 
 
 Fig. 185; The lever to which the pawl is attached has a tooth or 
 beak so formed and placed that over travel is impossible. When 
 the pawl first acts on a pin, another pin passes close to the point of 
 this beak; the beak then follows in behmd this pin, crossing the 
 path of pin-motions, and thus limiting the motion of the next pin. 
 The outline ab should be a circular arc with g as a centre, so that 
 
204 KINEMATICS OF MACEINEBY. 
 
 the pin which it stops will rest against it during the return stroke 
 of the driver. Another device much used for counters is shown 
 by Pig. 186. The " star " wheel is driven through half of its pitch 
 arc by the action of the projection b upon the tooth a during one 
 stroke of the driver, and b' acts upon the opposite tooth a' during 
 the return stroke, thus moving the wheel an equal distance in the 
 same direction. It will be seen that the motion of the wheel for a 
 double stroke of the driver is equal to the angle' between two teeth, 
 and if the wheel has ten teeth, it will make a complete rotation for 
 ten double strokes of the driver. 
 
 112. Escapements. — The mechanism of Fig. 186 resembles the 
 escapements used to control the motion of a train of clockwork,' 
 and it might, with slight modification, be used for sucli a purpose. 
 If the wheel A is acted upon by a spring or weight which tends to 
 rotate it continuously in the left-hand direction, this wheel would 
 tend to produce reciprocation of the piece B. If i? is a pendulum, 
 it has a normal period of vibration corresponding to its length, and 
 if the pendulum is so heavy that the rota- 
 tive effort of A cannot alter this period, 
 the pendulum in swinging will control 
 the motion of the wheel. The tendency 
 of the wheel to produce vibration of 
 the pendulum may be made sufficient to 
 overcome the frictional resistance which acts 
 to stop the pendulum, and thus the ampli- 
 tude of the vibrations is maintained. Other 
 
 Fig. 187. 
 
 outlines of teeth for the wheel and pendu- 
 lum are better, practically, and one common form is shown in 
 Fig. 187. The teeth of the piece which vibrates with the pendu- 
 lum are cd^ed. pallets. 
 
 Many modifications of the escapement have been devised to 
 meet special requirements. In watches and other portable time- 
 pieces a balance-wheel is used instead of the pendulum to regu- 
 late the period of the vibrating member, but all are similar in their 
 general action. 
 
CHAPTER VII. 
 
 WRAPPING -CONNECTORS. BELTS, ROPES, AND CHAINS. 
 
 113. Belts, Eopes, Chains, etc.— Flexible members are frequently- 
 used for transmission of motion between two pieces provided with 
 properly formed surfaces upon which the connector wraps or un- 
 wraps as the action takes place. The connector may be a flat belt 
 or band, a rope, or a chain composed of jointed members each one 
 of which :s itself rigid. 
 
 The great majority of the practical applications in which -bands 
 are used for transmitting motion are those in which the velocity 
 ratio is constant. Pigs. 188 and 189 show pairs of wheels of 
 
 T M 
 
 Fig. 188 
 
 circular section connected by bands. These evidently fulfil the 
 condition of constant velocity ratio, for the segments (^i^^and qF) 
 into which the line of the band cuts the line of centres (or its ex- 
 tension) are constant; also, the perpendiculars {R and r) let fall 
 from the fixed centres upon the line of the band are constant (see 
 Art. 31). In case exact motion through only part of a revolution 
 (or at most through a limited number of revolutions) is to be trans- 
 mitted, the ends of the bands may be fastened to the wheels. The 
 action, with this condition, is positive, provided the direction of the 
 
 205 
 
206 KINEMATICS OF MACHINERY. 
 
 motion is such that the baud is always kept iu teusiou. Thus in 
 Figs. 55 and 56, the piece which rotates about must be the driver, 
 while the one rotating about 0' is the follower, for transmission of 
 motion in the directions indicated. The motions of two pieces con- 
 nected in this way are necessarily of a reciprocating character, for 
 when the band is all unwrapped from the follower the mechanism 
 comes to rest, and any farther motion most be in the reverse direc- 
 tion. Such motion can only be secured when the former follower 
 becomes the driver. An example similar to this case is seen in a 
 hoisting-drum which pulls a car up an incline. "While hoisting, the 
 drum is the driver relative to the car; but in lowering, the action 
 of gravity on the car causes it to turn the drum backward. 
 
 In most common applications of flexible connectors the ends of 
 the band are joined together and not fastened to the wheels, and 
 the motion is continuous; this is commonly called an endless band. 
 
 In these cases the motion is not positive, as the bands may slip 
 (except when chains are used), but usually very exact motion is not 
 essential where these devices are employed. 
 
 It follows from the demonstration of Art, 31, referred to above, 
 that if wheels of circular transverse sections are connected by a flex- 
 ible band their angular velocities are inversely as their radii. 
 
 The effective radii are greater than the radii of the wheels by 
 about one half the thickness of the band; but generally the correc- 
 tion for thickness of the band need not be made with thin flat 
 belts. 
 
 The exact effective diameter is the length of band that will just 
 encircle the wheel divided by n. When a round cord or rope or a 
 chain is i;sed this affords a convenient way to get the effective or 
 pitch diameter. Wheels for such ropes or cords have grooves cut in 
 the rims to keep the band on the " sheave." For hemp or cotton 
 rope transmissions the grooves are given such a form that the rop& 
 is wedged into them slightly, thus increasing the tractive force. 
 With wire rope this wedging is inadmissible, as it would injure the 
 rope, and the bottom of the groove has a somewhat larger radius 
 than that of the rope. The bottom of the groove in wire-rope sheaves 
 
WBAPPINQ. CONNECTORS. 
 
 20' 
 
 is usually lined -with rubber, leather, wood, or some such material 
 to increase the adhesion and save wear of the cable. 
 
 Fig. 190 shows the section of the rim of a sheave as commonl; 
 designed for hemp or other fibrous ropes. Fig. 191 shows a sectio] 
 of rim employed with wire ropes. If sup- 
 porting sheaves or tighteners are required 
 in a hemp-rope transmission the groove 
 is made similar to that shown for wire 
 rope but without the soft lining; for as 
 these sheaves are not intended to transmit ^'^' "°' ^'9- '''• 
 
 power, the increased adhesion due tb the wedging of the rope is no 
 required, and unnecessary wear of the rope is avoided by making th 
 groove larger. 
 
 "With chain-bands the wheels, called " sprocket " wheels, hav 
 projections fitting the links of the chain (more or less closely) t 
 prevent slipping. With flat belts the pulleys have flat or near! 
 flat faces. The forms of sprocket-wheels and of the faces of pulley 
 for flat belts will be treated in later articles. 
 
 114. Shifting Belts. — If a pressure is brought to bear upon th 
 advancing side of a belt (Fig. 192) the belt is deflected in the direc 
 
 =T— , tion of this force. As the belt passes upon the pullej 
 each successive portion of it passes upon a part of th 
 pulley farther from the side from which the sbiftin, 
 J force acts, and the belt takes up a new position, a 
 shown by the dotted lines. A pressure upon th 
 receding side of the belt does not have this effecl 
 unless the force is great enough to overcome th 
 adhesion of the belt and pull it over bodily. It mus 
 be remembered, however, that the receding side o 
 the belt relative to one pulley is the advancing sid 
 ' — Fi0. f92. relative to the other pulley. 
 
 115. Crowning Pulleys. — If a flat belt is placed upon a con 
 (Fig. 193) the edge nearest the base of the cone is stretched mor 
 than the other parts, and the belt tends to take the position sliowi 
 by the dotted line. The effect of this is to shift the belt toward 
 
 c 
 
208 
 
 KINEMATICS OF ilAGHINEBT. 
 
 Fig. 193. 
 
 the base of the cone, as the advancing portion of the belt runs on 
 nearer to the base. 
 
 If a similar cone is so placed that its base coincides with that 
 of the first one, when the centre line of the belt 
 has mounted to the common base it will remain in 
 / that position, as any displacement from such position 
 (-1 i hn would bring about the condition tending to return 
 
 it. Pulleys are, therefore, usually made "crowning" 
 to keep the belt on the centre. If the pulley is 
 crowned about ^ inch for each foot in width, the belt 
 will ordinarily evince no tendency to run off, provi- 
 ded the axes of the connecting shafts are parallel. 
 If the shafts are out of alignment, the belt tends 
 to run toward the edges at which the belt is tightest, 
 unless the shafts are very much "out." 
 It is frequently desirable to stop the driven shaft without stop- 
 ping the driver, and a common method of doing this is by means 
 of " tight-and-loose " pulleys. Two pulleys are placed side by side 
 on the driven shaft, one of which is fastened to the shaft, while the 
 other is free to rotate relative to this shaft, but is prevented by 
 collars from moving axially. The hub of the tight pulley usually 
 serves as one of these collars, and the rims should not quite touch. 
 A pulley is secured on the driving-shaft, opposite the tight-and-loose 
 pulley, having a width (or face) equal to the combined width of 
 both of the latter. A belt of about the width of either of the single 
 pulleys connects one of them and the wide-faced driving pulley. 
 When this belt is on the tight pulley, the follower is driven; but if 
 it is shifted to the loose pulley the follower will stop, although the 
 belt continues to run. The belt is easily shifted by applying a 
 lateral pressure to the advancing edge, as explained in Art. 114. 
 It is usual with tight-and-loose pulleys to make them both crown- 
 ing, so that the belt will remain upon either when it is shifted; but 
 to facilitate shifting the wide driving pulley is generally made with 
 a straight face (cylindrical surface). 
 
 116. Length of Belt.— The length of belt is usually determined 
 
WBAPPING-CONNEGTOBS. 209 
 
 by direct measurement if the pulleys are constructed and mounted, 
 or by measuring a drawing if the work is not built and erected. 
 This length may be calculated for either an open or a crossed belt 
 (Figs. 188 and 189, respectively). This calculation is seldom of 
 practical value simply for the determination of the length, but it 
 plays an important part in the correct design of " stepped-cone 
 pulleys," such as are used on the countershafts and spindles of 
 lathes and other machines for securing changes of speed. The 
 importance of this calculation will appear from the discussion of 
 the next article. 
 
 The open band of Fig. 188 causes the follower to rotate in the 
 same direction as the driver, while the crossed band (Fig. 189) 
 gives the follower a rotation in an opposite direction. This will 
 be seen to agree with the general statement of Art. 33; for with 
 the open belt both fixed centres are on the same side of the line of 
 action (the driving side of the belt) ; while, with the crossed belt 
 these centimes are on opposite sides of the line of action. Owing to 
 the rubbing of the sides of the belt where they cross, the open 
 band is used when it is feasible. The crossed band has the advan- 
 tage of a larger arc of contact, which has an important effect on the 
 adhesion, especially on the smaller pulley; but with wide, stiff belts, 
 particularly when the distance between centres is small, the warp- 
 ing of the belt may largely destroy this advantage. 
 
 It is evident that the length of belt is different in the two cases, 
 other conditions being the same. The following are the algebraic 
 ■expressions for the length of belts: 
 
 The angle MQT ^ mqt ^ SqQ = (f>. 
 
 For crossed belts (Fig. 189), 
 
 sin = ^-p^> ^^^ Tt^ qS= VW - {R + r)\ 
 
 p „ 
 
 Foi-open belts, sin — — ^,and Ti — qS = V d' — (B —r)'. 
 The length of the crossed belt 
 
 = -L = 2v'd'-{E + ry+ 7rR+ 2R8m-'^~ + 7i:r+2rsm-^-^ 
 = 2 Vd'-(R + ry+ iR + r){7r + 2 sin-^^±^). ... (1) 
 
210 KINEMATICS OF MACHINERY. 
 
 The length of the open belt 
 
 = L = 2 1/rf ' - (^ -r )" +7r^+ ^R sm-'^^-{-nr-2r sin- ^^^ 
 = 2 Vd' -{R- rY + {R +r) 7C+{R- rfU sin-' ^^\ ■ (2> 
 
 It follows from (1) that a crossed belt which is of proper length 
 for any pair of pulleys, R and r, will be of correct length for any 
 other pair of pulleys, R' and r' (on the same shafts) if i? + r = 
 R' -f- r', that is, if the sum of the radii is constant; for {R -\- r) i& 
 the only variable quantity. 
 
 It will be seen, however, in (2) that if R' -\- r' = R = r; 
 R' — r' cannot equal R — r, unless R' = R and r' = r. 
 
 An open belt of the correct length for two pulleys, R and r, on 
 fixed shafts would not, therefore, be of exactly the right length 
 for another pair of pulleys, R' and r', on these same shafts, if 
 R' -\- r' = R -{- r, unless the two larger pulleys are equal, and the 
 two smaller pulleys are also equal. Such a belt might be made to 
 run if the distance between shafts were quite great and the change 
 in sizes of pulleys were small ; but it would not be equally tight on 
 the different sets. 
 
 117. Stepped Cones. — It is often important to change the speed 
 of a machine which is driven from a shaft having uniform speeds 
 Cones, as shown in Pig. 194, might be placed upon the counter- 
 shaft and on the spindle of the machine. If a crossed belt is used, 
 it would be equally tight at all corresponding positions on these 
 cones, but an open belt would not be; and in order to have it so, 
 " swelled " cones, as shown (exaggerated) in Fig. 195, would be re- 
 quired. Such conical drums have the advantage of permitting- 
 every possible variation in speed within limits; but the belt tends 
 to mount towards the large ends of both, which increases the strain 
 upon the belts and the pressure upon the bearings. 
 
 The stepped cones. Fig. 196, are more compact than conical 
 drums, and they avoid the objection just mentioned. It follows 
 from the preceding discussion that for a crossed belt the sum of 
 
WRAPPING-CONNECTORS. 
 
 211 
 
 the radii of any mating pair of steps should be a constant. But 
 the sum of the radii of the intermediate pairs of steps should be 
 greater than the sum for the outside steps when using an openi 
 belt. Eankine's Machinery and Millwork gives a method of deter- 
 mining the swell of the cones (Fig. 195) from which the radii of 
 the intermediate steps of a stepped cone can be derived. A much 
 
 Fig. 194. 
 
 3-t 
 
 3--E- 
 
 ^ 
 
 Fig. 195. 
 
 Fig. 196. 
 
 more convenient approximate graphical method is described by Mr. 
 C. A. Smith, in the Trans, of the A. S. M. E., Vol. X, page 269. 
 
 Lay off Qq (Fig. 197) equal to the distance between shafts; draw 
 the circles with radii R and r, equal to the radii of the known pul- 
 leys; at 0, half way between q and Q, erect the perpendicular 
 CG = .314^$', and with (r as a centre, draw the arc vim tangent to^ 
 tT. The belt line of any other pair of steps should be tangent to- 
 mm. R' and r' are radii of two such steps; and the velocity ratio 
 when using these steps will be R' -¥ r' = FQ -¥■ Fq. Let Qq = d;. 
 let Fq — x; and call the desired ratio a. 
 
 ^^ d + x 
 Now = a. 
 
 X — 
 
 d 
 
 Lay off Fq equal to this 
 
 value of x; draw FT' tangent to mm. Circles with Q nnd q as the 
 respective centres, and tangent to FT', give the required wheels with 
 
^12 
 
 KINEMATICS OF MAGHINEBT. 
 
 radii R' and r'. This method, as here outlined, only applies when 
 the belt angle, 4> of Fig. 197, is less than about 18°. 
 
 Fig. 197. 
 
 The original paper, referred to above, gives a modified method 
 for use when (p is greater than 18°. 
 
 The student should check this graphical method by using it in 
 laying out a set of stepped cones, and then calculating the belt 
 length for each pair of wheels by eq (3) of the preceding article. 
 
 118. Twisted Belts. — It is sometimes desired to connect two 
 shafts which are not parallel by a belt. This can often be done by 
 the use of a twisted belt. (See Fig. 198.) Suppose two pulleys 
 in the plane of the paper (the lower one shown 
 by the dotted circle) to be on parallel shafts, Q 
 and q, Fig. 198. 
 
 Draw Tt tangent to each pulley at the centre 
 of its face, and on the side at which the belt 
 leaves it. Then, if the lower pulley and its 
 shaft be turned about Tt as an axis to the posi- 
 tion shown by the full lines, the planes of the 
 two pulleys will intersect in Tt. The line Aa, 
 in which the belt advances upon the lower 
 pulley will lie in the plane of this pulley. The 
 line, bB, in which the belt advances upon the 
 upper pulley, will also lie in its plane. It has 
 been shown (Art. Ill) that the direction of the 
 receding side of the bolt does not affect the ac- 
 tion ; therefore this belt will remain upon the pulleys and con- 
 tinue to drive. If the motion of the pulleys be reversed, however, 
 
 Fig. 198. 
 
WBAPPINO- CONNECTORS. 
 
 213 
 
 the belt will at ouce mn off, because its advancing side does not lie 
 in the plane of the pulley under this new condition. If the angle 
 through which the lower shaft, q'q', is turned is 90°, the term 
 quarter-turn belt is applied. 
 
 119. Guide-pulleys. — The only condition necessary in order that 
 a belt shall run on a pulley is that the centre line of its advancing 
 side shall lie in the central plane of the pulley. By use of guide- 
 pulleys, or idlers, two shafts either intersecting at any angle or not 
 in one plane can be connected by a belt. If desired, the belts can 
 be made to run in either direction by so placing guide pulleys that 
 hotU sides of the belt lie in the planes of the pulleys. Pig. 199 shows 
 a few of the possible applications of guide-pulleys in connecting 
 shafts which are not parallel. In the arrangement of Figs. 199 (a) 
 
 Fig. r99. ^*— ^ (C) 
 
 and 199 (c) the belt may run in either direction ; but in Pig. 19^ 
 (1)) the belt will only remain on the pulleys when it is run m tha 
 direction indicated by the arrows. 
 
214 
 
 KINEMATICS OF MAOIIINERY. 
 
 120. Belt-tighteners. — It is sometimes desirable to provide for 
 variation in distance between shafts, to secure a greater arc of belt 
 contact, to take up stretch of belt, or to avoid the use of clutclies 
 and tight-aud-loose pulleys, by employing a belt- tightener. This 
 is simply an idle pulley, mounted on a suitable frame in such a 
 way that it can be moved by screws, levers, weights, or springs, to 
 change or maintain the tension of the belt. The only condition 
 necessarily complied with is that the centre line of the advancing 
 side of the belt shall lie in the central plane of the pulley to which 
 it runs. 
 
 121. Sprocket-wheels for Chains. — One form of transmission- 
 chain and sprocket-wheel is shown in Fig. 300. The true pitch 
 
 Fig, 200, 
 
 line of the wheel is a closed equal-sided polygon, each side being 
 equal to the length of a link from centre to centre of the pins. Or 
 if a circle be drawn about Q passing through the centres of all the 
 pins that lie on the wheel, the centre lines of the corresponding 
 links form cords of this circle.- As each link approaches or recedes 
 from the wheel, one of its pin centres rotates, relative to the wheel, 
 tibout its other centre, describing a circular arc relative to the 
 wheel. Thus, Fig. 300, h describes the arc hp relative to the wheel 
 as the link ab wraps upon the wheel. In order that the teeth of 
 the wheel shall allow the links to drop smoothly into place, the 
 
WRAPPING- CONNECTORS. 
 
 215 
 
 actual tooth outline may be an arc parallel to pi, as shown by the 
 arc mn. Adjacent sides of two teeth may be joined by an arc about 
 p, the radius of which is equal to the radius of the pin, or bushing, 
 which joins the links. By making the outer portions of the teeth 
 lie somewhat inside the arcs mn, the pin does not rub upon the 
 tooth as it approaches the wheel, but it will fall into place and 
 reach a bearing at the end of its approaching action. The backs 
 of the teeth are sometimes relieved more than the fronts or driv- 
 ing sides when the rotation is to be in one direction only. Since 
 the true pitch line of the wheel is a polygon instead of a true 
 circle, the velocity ratio is not exactly constant with sprocket- 
 wheels. The irregularity is usually not important with wheels of 
 i\. considerable number of teeth. 
 
 If two sprocket-wheels are connected by a chain, their angular 
 velocity ratio is inversely as their numbers of teeth, as in toothed 
 gearing. This is a more convenient measure of the velocity ratio 
 than the radii of the pitch circles, or the circles inscribing the pitch 
 polygons. 
 
 Modifications of the construction shown in Fig. 200 permit the 
 ■employment of chains with various forms of links, or of the special 
 chains called "link-belts,^' etc. 
 
 A wheel frequently used in cranes for the common chain, with 
 oval links of round iron, is shown 
 by Fig. 301. Every other link 
 lies on the wheel with its plane in 
 the central plane of the wheel ; while 
 the intermediate links lie in planes 
 normal to these. Pockets, as shown, 
 prevent slipping, and the flanges 
 tit the side strengthen the projec- 
 ing teeth greatly, so that there is no 
 difficulty in getting a wheel stronger 
 than the chain itself. ^'9- 201. 
 
 122. ■Wrapping-connectors with Varying Angular Velocity 
 Hatio. — As already shown, flexible connectors can be used to trans- 
 
216 
 
 KINEMATICS OF MACHINERY. 
 
 Fig. 202 
 
 mit a variable angular velocity ratio, for instance, by using such 
 
 forms as are shown in Figs. 54, 55, 
 and 56. A somewhat different appli- 
 cation is shown in Fig. 203. It has. 
 been employed in chronometers and 
 watches to secure a more uniform 
 driving action to the mechanism as the spring runs down. The^ 
 spring is placed in the cylindrical piece, called the barrel, and as it 
 uncoils the small chain is wound upon the barrel and unwound 
 from the conical piece, called a "fusee." It will be seen that as; 
 the spring runs down the pull on the connector diminishes, but the 
 "leverage" of the connector upon the follower is increased corre- 
 spondingly, and, therefore, the driving effort transmitted to th& 
 mechanism may be kept quite uniform. 
 
 The elliptical sprocket, which was exploited a few years ago for 
 a bicycle driving-gear, is another application of a wrapping-connec- 
 tor which transmits motion with a varying angular velocity ratio, 
 and, unlike the cases referred to above, these elliptical sprocket- 
 wheels permit of a continuous transmission of motion in one di- 
 rection. 
 
CHAPTER VIII. 
 TRAINS OP MECHANISM. 
 
 123. Substitution of a Train for a Simple Mechanism. — It is 
 
 kinematically possible to transmit motion between two parallel 
 shafts with any required angular velocity ratio by either a single 
 pair of gears or of pulleys; but there are practical conditons 
 which often make it desirable to effect the required transmission 
 of motion by a series of mechanisms, or a compound mechanism, 
 instead of by a single pair of gears, of pulleys, etc. Such an arrange- 
 ment constitutes a train of mechanism. The train may contain 
 pulleys with belts, ropes, chains, gears, screws, and linkwork, any 
 or all; and it may be used to transmit motion between other 
 members than parallel shafts. If two shafts are to be connected 
 by gears, and the required velocity ratio is high, the difference in 
 the size of the gears may be inconveniently great if a single pair is 
 used. That is, the large wheel may occupy too much room, or be 
 diflBcult to swing, or the small gear may have so few teeth that it 
 would be objectionable. For example : suppose the velocity ratio 
 is 25 to 1, and that strength requires wheels of 2 (diametral) pitch. 
 Then if the pinion be given only 12 teeth, it will be 6 inches in 
 diameter, and the large wheel will be 25 X 6 = 150 inches in 
 diameter ( = 12-J feet). Now, suppose that an intermediate shaft 
 be introduced. This intermediate shaft can be connected to the 
 slower of the original shafts by using a pair of gears which will cause 
 it to rotate 5 times to 1 rotation of this primary shaft, and it can 
 be connected to the faster of the original shafts by a pair of gears 
 ■which will give it 1 rotation to 5 of the latter shaft; then as each 
 
 317 
 
218 KINEMATICS OF MACHmERT. 
 
 revolution of the first shaft corresponds to 5 revolutions of this 
 intermediate shaft, and as each of its revolutions corresponds to 5 
 revolutions of the last main shaft, it is evident that the velocity 
 ratio between the first and last shaft is 5 X 5 to 1, equal 25 to 1, 
 as required. 
 
 The velocity ratio of first shaft to intermediate and of intermc 
 diate to last shaft are not necessarily equal. They may be any- 
 thing whatever if the product of the separate angular velocity 
 ratios equals the required ratio between the first and the last shaft. 
 Furthermore, the three axes need not lie in one plane; that is, the 
 centres need not be in one straight line. It is thus seen that the 
 use of a train in place of a simple mechanism permits considerable 
 flexibility in the arrangement; this will be clearly seen from an 
 examination of various actual trains. 
 
 In a similar manner to that of the preceding illustration, an in- 
 termediate shaft may be used in a belt transmission when the 
 velocity ratio is high. Such an arrangement is frequently seen 
 when a slow-speed engine drives a dynamo. The engine is belted 
 to a " jack-shaft," which in turn drives the dynamo. This may 
 be desirable either to avoid an excessively large pulley or to avoid 
 an extremely wide angle between the sides of the belt. The effect 
 of a large belt angle is to reduce the arc of contact on the smaller 
 pulley; this reduces the adhesion of the belt and increases liability 
 of slip of the belt. 
 
 Other considerations than a high velocity ratio may make it 
 desirable to substitute a train for a simple mechanism; for instance, 
 to secure a required directional relation, for compactness, etc. A 
 familiar example of such a train is seen in the back-gear mechanism 
 (Fig. 206), as used on lathes and other machine tools. 
 
 A shaft which carries intermediate gears of a train may itself 
 drive some member which requires a motion different from that of 
 the last member. Thus, in clockwork, the gear on the shaft to 
 which the minute-hand is fixed drives the hour-hand through a 
 reducing pair of gears, and it may also drive a second hand at a 
 higher rate. 
 
TMAIJfS OF MECHANISM. 
 
 219 
 
 124. Value of a Train.— Suppose four axes, I, II, III, and IV 
 (Fig. 303) to be arranged as shown and connected by toothed 
 gears of whicli the circles a, b, c, 
 etc., are the pitch lines. The 
 wheel a meshes with b; c meshes 
 with d, and e meshes with/. Both 
 of the wheels b and c are secured 
 to the shaft II; hence they must 
 rotate as one piece, having the 
 same angular velocity at any in- 
 stant. Likewise, d and e are 
 both secured so shaft III, and they have the same angular velocity. 
 Let the angular yelocities of the shafts I, II, III, and IV be repre- 
 sented by ffj, or,, a^, and a^, respectively. Two gears which mesh 
 together must have the same pitch ; hence the numbers of teeth 
 are proportional to the circumferences, to the diameters, or to the 
 radii. But their angular velocities are inversely as the radii, and 
 therefore inversely as the numbers of teeth on the wheel. It 
 follows that if a, b, c, etc., are the numbers of teeth on the wheels 
 designated by these letters, that. 
 
 /. 
 
 «. _ ^ . 
 
 a^ d _ 
 
 a> 
 
 a, a ' 
 
 — 
 
 «3 ~ c ' 
 
 a, 
 
 a a, a, a, b d ^f b .d .f 
 a. a„ a, a a c e a .c .e 
 
 (1) 
 
 In this train a is the driver and b is the follower in the first pair; 
 c is the driver and d the follower in the second pair; and e is the 
 driver and /is the follower in the third pair. It will be seen from 
 the above expression for a, -H a, that the angular velocity ratio of 
 the first driving-shaft I to the last driven shaft IV equals the con- 
 tinued product of the numbers of teeth in the driven wheels di- 
 vided by the continued product of the numbers of teeth in the 
 driving wheels. The angular velocity ratio between two wheels is 
 the direct ratio of the numbers of revolutions they make in a unit 
 
220 KINEMATICS OF MACEINERT. 
 
 of time, as a minute. In finding the value of a train, any of the 
 
 (X (X 
 
 factors, —1, — ^, etc., may be expressed in terms of the numbers of 
 
 teeth, radii, diameters, or revolutions per unit of time of the pair 
 of wheels involved; but if the latter relation is used the ratio is 
 direct, while with the other terms the inverse ratio is to be taken. 
 It is not necessary that these different factors be all given in the 
 same terms. Thus if a has 60 teeth and l has 16 teeth; c is 24 
 inches in diameter, and d is 8 inches in diameter; e makes 75 revo- 
 lutions and /makes 250 revolutions per minute, 
 
 a,_16 _8^ 75 _4 1 l._^ 
 a. "" 60 ^ 24 ^ 2"50 ~ 15 ^ 3 ^ 10 ~ 75' 
 
 For every revolution of I, IV makes 37^ revolutions; hence if I 
 makes 10 revolutions per minute, IV will make 375 revolutions per 
 minute. 
 
 A train is shown by Fig. 204 in which the shaft I drives the 
 shaft II through pulleys connected with a crossed belt ; III is 
 driven from II by an open belt ; and IV is driven from III by 
 gears. An expression similar to that given above can be used to 
 
 find the ratio of the angular velocities of I to IV. Thus suppose 
 that the pulleys a, h, c, and d are, respectively, 8, 20, 10, and 24 
 inches in diameter; and that the gears e and /have 18 and 70 teeth 
 respectively; then 
 
 a, _ 20 34 70 _ 70 
 a~ 8 ^ 10 ^ 18 ~ T" 
 
TRAINS OF MECHANISM. 
 
 221 
 
 The shaft I makes TO revolutions to 3 of the shaft IV (or 2^ 
 to 1). Or, if I makes 175 revolutions per minute, IV makes 
 
 1;;^^ = 7-J revolutions per minute. 
 
 In general, if there are m shafts connected by gears or pulleys, 
 the angular velocity ratio of the first shaft to the last is 
 
 oil a. a„ a, 
 — = — X — ' X - 
 a™ a„ as a. 
 
 (2) 
 
 If the numbers of revolutions per unit of time of the first and 
 lust shaft are JS\ and JVm, respectively, Wi : Nm ■■ oii : a^ (as the 
 angular velocity of a member is proportional to its revolutions per 
 unit of time); hence 
 
 iV™ = iV.- 
 
 (3) 
 
 Belt connections are usually preferred when the speeds of the 
 shafts are high, the distance between centres is great, and a moderate 
 amount of slipping is not seriously objectionable. When the speed 
 is slow, the distance between shafts is comparatively small, or when 
 
 Fig. 205. 
 
 positive transmission is essential, gears are better. When this last 
 condition is not a requisite, and the distance between shafts is too 
 small to use belting advantageously, frictional gears are occasionally 
 employed. When the distance between two shafts is very great, 
 rope transmission (wire or hemp) may be used. 
 
 A train is shown in Fig. 305 in which the axes are not all 
 
222 KINEMATICS OF MACHINERY. 
 
 parallel. A pinion a on shaft I drives the spur-gear b on II; a 
 pair of bevel-gears c and d connect II and III, and a ■worm e on III 
 drives the worm-wheel / on IV. If the numbers of teeth on a, h, c, 
 d, e, and /are 15, 45, 25, 35, 1, and 50, respectively, 
 
 or, _ 45 35 50 _ 
 «, 15 25 1 
 
 or the first shaft makes 210 revolutions to every revolution of the 
 last shaft. 
 
 It will be seen that the expression for the value of a train, as- 
 deduced above, is general, and applies to all cases when the proper 
 substitutions are made. 
 
 125. Directional Relation in a Train.- — When two spur-gears 
 mesh together they rotate in opposite direction ; hence, if the train 
 is made up entirely of spur-gears the adjacent axes rotate in oppo- 
 site directions, and the alternate axes (first, thii-d, fifth, etc., or 
 second, fourth, etc.) have rotations in the same direction. If such 
 a train has an odd number of axes the first and last axes will 
 rotate in the same direction; while if there is an even number 
 of axes the first and last will rotate in opposite directions. Thus- 
 in the train of Fig. 203 the shafts I and IV rotate in opposite 
 directions. 
 
 If one of the gears is an internal (or annular) gear the shaft 
 to which it is attached rotates in the same direction as the pinion 
 which meshes with this gear. 
 
 If an open belt connects the pulleys on two shafts these 
 shafts rotate in the same direction, while a crossed belt connect- 
 ing two shafts causes them to rotate in opposite directions. Tlius 
 in Fig. 204, I and II rotate in opposite directions ; II and III 
 rotate in the same direction,' and III and IV rotate in opposite 
 directions. In this example there is an even number of shafts, but 
 there is one open-belt connection ; hence, the rotations of the first 
 and last shaft are in the same direction, as will appear from an in- 
 spection of the figure. 
 
 126. Back Gears. — The common screw-cutting lathe and many 
 
TRAINS OF MECHANISM. 
 
 223 
 
 
 m 
 
 
 B 
 
 
 fl 
 
 
 
 == 
 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 6 
 
 
 
 d 
 
 
 
 
 
 
 
 
 
 a 
 
 F= 
 
 
 -1 
 
 _ 
 
 1- 
 
 F= 
 
 
 
 L- 
 
 ^ 
 
 Fig. 206. 
 
 
 D 
 
 -^ 
 
 other machine tools have a gear-train through which the stepped 
 cone can be connected with the 
 spindle. This is shown in Fig. 
 206. The cone is driven by a belt 
 from another cone on the counter- 
 shaft. When the back gears are 
 thrown out and the cone of the 
 headstock is locked to the spindle 
 these two members (the cone and 
 spindle) move as one piece. If 
 the cone has three steps the 
 spindle can be given three different speeds from the uniformly re- 
 volving countershaft. By means of the back gears the number of 
 speeds of the spindle is doubled without adding more steps to the 
 cone. "When the back gears are "in" the cone is not secured 
 directly to the spindle, but is free to rotate upon it. A pinion, a, 
 attached to the cone, engages with the first back gear, b, which is 
 mounted on the shaft B. This shaft has another gear, c, secured, 
 to its other end, and c engages with the gear d, which is attached 
 to the spindle. The angular velocity of the cone may be designated 
 by «i; that of the two back gears by a,, and that of the spindle by 
 a,; then the angular velocity ratio of the cone to the spindle is: 
 
 — = — ' X — ; or — = the product of the numbers of teeth on b 
 Us «s a, a, 
 
 and cl divided by the product of the numbers of teeth on a and c. 
 
 The cones on both the spindle and the countershaft are com- 
 monly equal with engine lathes; but on wood lathes (which do not 
 use back gears) the countershaft cone is usually the larger, to secure 
 the requisite high speed of the spindle from a moderate speed of 
 countershaft. 
 
 A countershaft runs at 90 revolutions per minute, the four 
 steps of the (equal) cones are 12", 9^", 7", and 4^" in diameter; 
 the numbers of teeth on the gears a, b, c, and cl are 28, 100, 24, 
 and 88, re.'^peetively; the following speeds of the spindle may be 
 obtained. Direct driving (back gears out): With the belt on 
 largest step of countershaft cone and smallest step of spindle cone r 
 
224 
 
 KINEMATICS OF MACEINERT. 
 
 12 
 
 (1) Spindle speed = 90 X j^ = 340. 
 
 Belt on next smaller step of countershaft cone and next larger 
 step of spindle cone: 
 
 (2) Spindle speed = 90 X y = 122.14. 
 Belt on next pair of steps : 
 
 (3) Spindle speed : 
 
 90 X ^ = 66.32. 
 
 Belt on smallest countershaft step and largest spindle cone 
 step : 
 
 (4) Spindle speed = 90 X ^ = 33.76. 
 
 Driving through back gears: Value of back-gear train: 
 
 a, 100 X 88 13.1 n-v^ / i \ ■ • -p j -^t, 
 
 — i- — -— = = .076 (nearly), givmg four speeds with 
 
 «j 28 X 24 1 ^ .1)' b b r 
 
 back gears which may be found by multiplying the four speeds as 
 
 ■calculated above by ^ ; or 
 
 (5) = 240 X .076 = 18.24. 
 
 (6) = 122.14 X .076 = 9.4 -. 
 
 (7) =66.32 X .076:= 5.0 +. 
 
 (8) =33.76 X .076 = 2.56+. 
 
 The student should take the required data from an actual lathe 
 
 and compute the various speeds. 
 
 127. The Idler. — It was shown in Art. 125 that one result of an 
 
 intermediate shaft in a spur-gear 
 train is to affect the direction 
 of rotation between the first and 
 third shafts. If these two shafts 
 were connected directly by a pair 
 of spur-gears they would rotate 
 in opposite directions ; but when 
 
 conuected through an intermediate shaft they rotate in the same 
 
 direction. 
 
 Fig. 207. 
 
TSAIJ^S OF MECHANISM. 225 
 
 In Fig. 207 three shafts, I^ II, and III, are shown connected 
 " in series " by the gears a, b, and c. If these letters designate the 
 numbers of teeth on the corresponding wheels : 
 
 ^-i;^=^;andi^=:-x4 = A 
 a, a «j «, a 6 a 
 
 or the intermediate wheel does not affect the ratio between the 
 angular velocities of the first and third shafts, but it does cause them 
 to rotate in the same direction. 
 
 Such an intermediate wheel in a train is called an idler. That 
 the idler does not affect the ratio between the times of revolu- 
 tions of a and c can be seen directly by inspection, for the linear 
 velocity of a point in the pitch circle of a must equal that of a 
 point in the pitch circle of h, and also points in the pitch circles of 
 & and c must have the same linear velocities ; therefore, as all 
 points in the pitch circle of i have the same velocity, the linear ve- 
 locity of points in the pitch circles of a and b are the same, and the 
 mgular velocities of these two wheels are inversely as their radii, 
 just as if they engaged directly. 
 
 Fig. 208 shows the " tumbling-gears " usually placed in the head- 
 stock of the screw-cutting lathe to enable the operator to easily- 
 3hange the direction of feed, or to 
 3ut either a right- or a left-handed 
 screw. The gear a is connected 
 to the lathe-spindle and d is on 
 the stud through which the feed- 
 rod or lead screw is driven. In 
 
 the position shown, a drives b, b *^ ^^.~_^^ °^"^ig. 208. 
 Irives c, and c drives d. It will be 
 
 ieen that a and d rotate in opposite directions, and as b and c are 
 30th idlers, the action is equivalent to direct engagement of a and d. 
 The gears b and c are carried on a support which can be swung 
 ibout the centre of (Z by a suitable handle extending through the 
 'ront of the headstock, and when this handle is dropped down, c 
 !an be meshed directly with a, b being thrown out of mesh with a. 
 
226 KINEMATICS OF MAOniNEET. 
 
 In this position b simply rotates, as it remains in mesh with c ; but 
 -a drives c directly, and c drives d. There are but three axes in the 
 train in this condition ; hence a and d rotate in the same direction. 
 128. The Screw-cutting Train. — In the screw-cutting lathe a 
 long screWji called the lead screzv, or leading sci;ew, is placed par- 
 allel to the bed, and the carriage which holds the lathe tool may be 
 connected to this screw by a clamp-nut. When this nut is closed 
 upon the screw the carriage will be fed alpng the bed as the screw is 
 turned. If the screw has four threads to the inch (^-inch pitch), 
 
 ivery turn of the screw will feed the 
 tool ^ inch parallel to the axis of the 
 lathe. If the screw has the gear g 
 (Fig. 209) mounted upon it at one 
 end, the screw will make one revolu- 
 tion for each revolution of this gear. 
 F'3- ^"^^ ^ Now suppose the gear e to rotate 
 
 with the lathe-spindle ; then if e is equal to g, and is connected 
 with it by the idler /, each revolution of the spindle corhpels the 
 screw to make one revolution. If a cylindrical piece of stock is 
 mounted in the lathe so that it rotates with the spindle, and a thread 
 tool in the tool-post is fed (transversely) till it enters this cylin- 
 drical piece, it will be seen that the feed-mechanism will cause the 
 tool to cut a thread on the stock which is a reproduction (as to 
 pitch) of the leading screw ; for the tool has a longitudinal motion 
 of J inch for each revolution of the work, and a proportional motion 
 for any fraction of a revolution. The idler, /, is carried on a slotted 
 piece which can be swung about the axis of the screw, VI, and the 
 stud upon which /rotates can be set at different distances from VI, 
 along the radial slot. The gear g could then be replaced by one 
 of a different size, / could be moved along to engage with it, and 
 ' by swinging the support of / it could also be. made to engage with e, 
 in which position it can be clamped. By this means the velocity 
 ratio between the spindle and the gear can be varied. 
 
 Suppose it is desired to cut a screw of 8 threads to the inch 
 (y pitch). By placing a gear {g) on the screw twice as large as e. 
 
TRAINS OF MECHANISM. 22 1 
 
 «ach revolution of the spindle will cause g to make but half a 
 revolution, and the tool will be fed only half the pitch of the lead- 
 ing screw along the stock during one complete revolution of the 
 latter. To cut a screw of 6 threads per inch, g must be 1^ (|) the 
 size of e, then a revolution of the spindle and of the stock would 
 occur for | of a revolution of the screw; or the feed per revolution 
 of the spindle would be ^ X | = ^ of an inch. It will appear that 
 screws of difEerent pitches may be cut from a given lead screw, each 
 of which is, in a sense, a reproduction, reduced or enlarged, of this 
 screw. 
 
 The screw-cutting lathe is provided with a set of gears to be 
 Tised as indicated above, for cutting all the whole number (even) 
 threads throughout a rather wide range. Such a set is called a set 
 ■of change gears. 
 
 A typical arrangement is a combination of the trains shown by 
 Figs. 208 and 209. The gear a (Kg. 208) is on the spindle, and it 
 •drives d in either direction, through the tumblers, as explained 
 in the preceding article. The stud (IV) to which d is attached 
 passes through the end of the headstock and e (Fig. 209) is fastened 
 upon its outer end. Then, by means of the change-gears any re- 
 ■quired thread within the range of the lathe can be cut, either right- 
 or left-handed. 
 
 The gear on the outer end of the stud may be fixed, g only 
 being changed ; but provision is usually made, for changing either e 
 or g (or both). Sometimes a and d are not equal {d being usually 
 twice as large as a in such cases) ; then the ratio between e and g 
 must be taken accordingly. More often, however, a and d are 
 equal. 
 
 A certain lathe of 16" swing has a lead screw of 4 threads per 
 inch, and change gears of the following numbers of teeth : 
 -24, 30, 36, 42, 48, 48, 54, 60, 66, 69, 72, 78, 84; with the 24 gear 
 ■on the stud it will cut: 5, 6, 7, 8, 9, 10, 11, 11^, 12, ]3, and 14 
 threads per inch, with the following gears, respectively, on the 
 screw : 30, 36, 42, 48, 54, 60, 66, 69, 73, 78, 84. 
 
 The 1 1| thread corresponds to a standard pipe thread, and it is 
 
228 KINEMATICS OF MACniNERT. 
 
 conseqneiitly convenient to be able to cut this pitch in a lathe. 
 To permit cutting this thread in the lathe, it is now not uncommon 
 to provide a gear for it. It will be noticed that the above list of 
 change-gears includes two 48-tooth gears. These are nsed for cut- 
 ting a 4-thread screw, one of them being placed on the stud and the- 
 other on the screw. For cutting 3 (or 3) threads, one of the 48 
 tooth gears is put on the stud, and the 24 j[g g ^ ^ ^-gSaj must be 
 used on the screw; as the screw must make 2 (or l|^)yevolutions,. 
 as the case may be, for each revolution of the spltrd-le. 
 
 When the stud-gear e makes the same number of revolutions 
 as the spindle, the following formula may be used for finding the 
 change-gears, in which e equals the teeth in the ■ stud-geai-, ff 
 equals the teeth in the screw-gear, t equals the threads per inch 
 of the lead screw, and n equals tlie threads per inch to be cut : — 
 
 — = — . If the stud-gear is fixed, g= -e. Any two gears of 
 
 the set may be taken which have numbers of teeth in the ratio of 
 n to t. If the stud-gear e does not make the same number of revolu- 
 tions as the spindle, that is, if a and d of Fig. 208 are not equal, 
 
 - — — X — . The idlers, b, c, and f do not enter into the calcula- 
 
 tions, for they do not affect the velocity ratio. 
 
 The above screw-cutting train is given as an example of the 
 ordinary arrangement; but among lathes of various makes there are 
 many modifications in detail to be found. All ordinary screw- 
 cutting lath«s have a mechanism which is fundamentally tiiat given 
 above. 
 
 It will be noticed that in the series of gears given above there 
 is a constant difference of 6 teeth between the successive gears 
 (neglecting the gear for 11^ threads and the extra 48-tooth gear). 
 In any such system, for whole numbers of threads to the inch, this 
 constant difference equal the number of teeth on the stud-gear 
 divided by the threads per inch of the lead screw (=6-4- t), when 
 the spindle and stud have the same number of revolutions per unit 
 of time. If the stud is geared to make only one revolution to two 
 
TRAmS OF MECHANISM. 229 
 
 Teyolutions of the spindle, the difference between successive wheels 
 is half of that given by this rule. 
 
 The change-gears should always be constructed on the involute 
 system, as this is the orily system in which the centre distances can 
 vary without affecting the constancy of the velocity ratio. 
 
 Many lathes are provided with a screw-cutting train which 
 ■can be "compounded." In this arrangement the simple idler/ 
 (Fig. 209) is replaced by two gears of different diameters, secured 
 together and rotating on the stud V as one piece. The gear e 
 meshes with one of these intermediate gears, and the gear g (which 
 must be correspondingly displaced laterally along its axis, VI) 
 meshes with the other. This pair of intermediate gears (unlike 
 the idler/) affects the velocity ratio between the spindle and the 
 screw, because of the difference in the diameters of the two inter- 
 mediate gears. The velocity ratio as found by the preceding 
 method must be multiplied by the ratio of the two intermediate 
 gears. This latter ratio is 3 to 1, or 1 to 2, depending upon 
 whether the larger of the compounding-gears engages with e or 
 with g. 
 
 129. Epicyclic Trains. — It was shown in Art. 39 that any mem- 
 l)er of a linkage could be considered as the fixed link, and different 
 mechanisms would apparently be thus obtained. This is true of 
 gear-trains as well as for linkages. If one of the gears of a gear- 
 train is made the fixed member, instead of the bar-supporting 
 gears, the mechanism is called an epicyclic train, because in its 
 action one or more of the wheels revolves around the fixed one, so 
 that points in the revolving-gears describe epicycloidal curves. 
 
 Generally in these mechanisms we are most concerned with the 
 relative angular velocity of the last rotating-gear and the arm that 
 carries it. In Pig. 210 let a and h be two gears mounted on an arm 
 c, so that if c were fixed, a and 6 would form a simple gear-train. 
 Nov/ suppose that a is made fast to some fixed body, so that it 
 really becomes the fixed member of the train. Then c can rotate 
 around 0, carrying i with it, h itself rotating relative to c around 
 its axis at 0'. It is required to find the number of revolutions that 
 
230 
 
 KlNEMATiaa OF MACHINEBT. 
 
 h ■will make around its own axis, relative to the fixed member^ for 
 every revolution of c around 0. 
 
 First, let a be disconnected from the fixed body, so that a, l, 
 and c can make one revolution as one piece in the direction indi- 
 cated around 0. Then b tvill mahe one revolution around 0' , solely 
 because of its motion around 0. This can be seen by noting th& 
 
 Fig. 210 
 
 positions of any point, as P, relative to 0' during different phases 
 of the revolution, as shown. Now if a be rotated backward one 
 revolution, l will occupy the position it would have had if a had 
 been held stationary all the time. Let r be the angular velocity 
 ratio between I and a. Then, when a is rotated backward one 
 revolution, b must receive r turns forward, and the total number 
 of revolutions which b will make for one revolution of c around o- 
 = w = 1 + '', and its direction of rotation will be the same as that 
 of c. It is evident that c can be rotated in either direction, and 
 the result obtained above will still hold. 
 
 If we place an idler between a and b (Fig. 211), the direction of 
 motion of b is reversed so that it will make one turn in the direc- 
 tion of rotation of c and minus r turns in the opposite direction, or 
 n = \ — r, for tliis arrangement; and the direction of rotation of b^ 
 relative to the fixed member, may or may not be in the same direc- 
 
TBAINS OF MECHANISM. 
 
 2?1 
 
 tion as that of c, depending on the value of r. A special case is that 
 when r = 1, whence n = o, and b does not 
 rotate around 0', relatiye to the fixed mem- 
 ber, but has a simple motion of circular 
 translation. 
 
 In general, then, if the train has an even 
 number of axes, the last gear makes 1 + r 
 revolutions for one of;the carrying-arm ; and 
 if an odd number of axes are used it makes 
 1 — ?• revolutions for one of the arm. 
 
 It is evident that a compound train can be 
 used between a and b. as shown in Fig. 212 ; 
 and the results will be the same as with the 
 arrangement shown in Fig. 211, since we are concerned only with the 
 angular velocity ratio of 5 to a and the direction of rotation of i 
 relative to a, regardless of how these are obtained. 
 
 Further, the axes need not lie in a straight line, but can occupy 
 any position relative to each other as long as the gears mesh prop- 
 
 Fig. 2ir 
 
 Fig. 212 
 
 Fig. 213 
 
 erly. A common arrangement is that shown in plan and elevation 
 at Fig. 213, where the axis of b is made to coincide with that of tho 
 fixed gear a. The gears d and e are fast together, and are carried 
 
232 EINEMATIGS OF MACEIITEBT. 
 
 by c, which is free to rotate on the spindle /. It is, therefore, a 
 ■compound chain, having three axes, and is called a reverted train. 
 
 In this form it is used extensively for obtaining great velocity 
 ratios between the arm c and the last gear h. 
 
 For example : 
 
 Let a have 99 teeth 
 " I " 100 " 
 " d " 101 " 
 " e " 100 " 
 
 Then _ 99 X 101 _ 9999 ^ 
 
 ''" 100 X 100 ~ 10000' 
 
 and since there are three axes, n = l— r= 1— iVWir — roirir ^'^v., 
 or c must make 10,000 revolutions in order to make b rotate once. 
 
 The application of epicyclio gears to hoisting devices will be 
 obvious from the above. They are also used as a feed-mechanism 
 on large boring-bars, in machines for making wire ropes, etc., etc. 
 
PROBLEMS AND EXERCISES. 
 
 Note. — A large number of exercises on Kinematics have been arranged by 
 Mr. A. T. Bruegel formerly of Sibley College, Cornell University, now of Pratt 
 Institute, wlio lias kindly consented to tlie use of some of tbem in the present 
 work. 
 
 The original set contains three classes of exercises, intended: — to illustrate 
 the principles treated; to drill the student on the application of these principles 
 in the solution of definite problems, and to extend the range of the text. The 
 exercises given below were selected mainly from those of the second class, and 
 tbey include a few additional ones by the writer. 
 
 The references in brackets are to^^e articles in the text which relate most 
 directly to the particular problem. ^ ^ 
 
 ^/^ J. H. B, 
 
 1. [Art. 3.] A train has attained a speed of 112 miles per hour for a 
 short distance. Express its velocity in feet per minute^ in feet per second, 
 and in inches per second. 
 
 3. [Art. 3.] The stroke of an engine is 18", and the crank-pin makes 
 250 vevs. per minute. Express the linear velocity, or rate of motion, of 
 this pin in feet per minute ; in inches per second ; in feet per second. 
 
 3. [Art. 4.] The drivers of a locomotive are 5 feet in diameter, and the 
 stroke of the piston is 34 inches. Calculate the mean, or average, piston 
 speed (linear velocity) in feet per minute when the locomotive runs at the 
 rate of 40 miles per hour. 
 
 4. [Art. 4.] An engine with a stroke of 5 feet makes 65 revs, per min. 
 "What is the mean piston speed ? 
 
 5. [Arts. 4, 5, 6.] A train runs 110 miles in 3 hours and 40 minutes. 
 Drivers, 64 inches in diameter. Stroke of piston, 22 inches. Eequired : 
 
 (a) Mean velocity of engine, in feet per minute, relative to the earth. 
 (6) Mean velocity of piston relative to engine-frame. 
 
 (c) Mean velocity of crank-pin relative to engine-frame. 
 
 (d) Mean velocity ratio between piston and crank-pin. 
 
 333 
 
234 KINEMATICS OF MAGEJNERY. 
 
 (e) Mean velocity of point iu tread, relative to frame. 
 (/) Path of point in tread relative to frame. 
 {g) Path of point in tread relative to earth. 
 {h) Kind of motion of crank-pin and piston. 
 
 6. [Art. 14.] Eepresent, graphically, the mean velocity of the crank-pia 
 of Prob. 5 (c). Use scale of 1000 feet per minute to the inch. 
 
 7. [Art. 14.] Represent, graphically, mean velocity of piston in Prob. 5 
 (6). Scale 700 feet per min. to the inch. 
 
 8. [Art. 17.] An engine with stroke of 18 inches makes 2^0 revolutions 
 per minute. Find, graphically, the vertical and horizontal components of 
 the crank-pin velocity when the crank makes angles of 30°, 120°, and 210° 
 respectively, with its initial position on centre line of engine. Write the 
 results in feet per second upon the lines which represent them. 
 
 9. [Art. 17.] A resultant pv (Fig. 18) equals 70 feet per second ; the 
 components pvi and pvi equal 64 feet and 48 feet per second, respectively. 
 Find, graphically, the directions of the components. Two solutions are 
 possible. 
 
 10. [Art. 17.] A velocity of 450 feet per minute is to be resolved into- 
 two components making angles with it, on opposite sides, of 30 and 60 
 degrees, respectively. 
 
 11. [Art. 17.] Three component motions in one plane have velocities of 
 60, 80, and 100 feet per minute, respectively; the first is vertically upward; 
 the second makes an angle of 80 degrees to the right with it ; and the third 
 an angle of 45 degrees with the second, also to the right. Find the value of 
 the resultant, graphically. 
 
 12.' [Art. 17,] A point moving upward and to the right, at an angle of 
 60 degrees with the horizontal, has a velocity of 40 feet per minute. 
 
 (a) Besolve it into a vertical and a horizontal component. 
 
 (J) Resolve it into two components, one of which makes an angle of 45 
 degrees with the horizontal towards the right and has a velocity of 30 feet 
 per minute. 
 
 (c) Resolve it into two components of 25 and 50 feet per minufe, re- 
 spectively. Graphical solutions required. 
 
 13. [Art. 17.] An engine of 24 inches stroke makes 160 revolutions per 
 minute. The connecting-rod is four times the length of the crank. Find 
 (graphically) the rate of motion of the cross-head when the crank is at 45 
 degrees and at 90 degrees with the centre line of engine. 
 
 14. [Art. 18.] A locomotive running at the rate of 35 miles per hour 
 has 63-inch driving-wheels and 24-inch stroke. Find the linear and the 
 angular velocity of the crank-pin relative to the frame. Give results, 
 in feet per minute and in inches per second. 
 
 15. [Art. 18.] An engine makes 600 strokes per minute. Fly-wheel is 
 
PROBLEMS AND EXEBGISES. 235 
 
 on the crank-shaft. Find the angular velocity of the fly-wheel, the linear 
 velocity of a point 3 feet from the centre of the shaft, and also of a point 
 4 inches from the centre. Express results in feet per minute. 
 
 When is the angular velocity of a point expressed by a number greater 
 than that of the linear velocity ? 
 
 16. [Art. 18.] A wheel 10 feet in diameter makes 100 revolutions per 
 minute. What are the linear and the angular velocity of a point in the rim ; 
 of a point 6 inches from the axis; of a point 13 inches from the axis? 
 Give all the results in feet per second. 
 
 17. [Art. 18.] (a) A body moves in a straight line with a linear velocity 
 of 25 feet per second. What is its angular velocity ? 
 
 (J) A governor-ball is 8 inches from the axis of rotation when revolv- 
 ing at the rate of 300 revolutions per minute. Express its linear and its- 
 angular velocity in units of feet and minutes and in inches and seconds. 
 
 18. [Art. 19.] Locate all the instant centres for the mechanism of Prob. 
 13, at the phases specified. 
 
 19. [Art. 20.] Same engine as Prob. 13, pressure on piston taken at 
 10,000 lbs. Draw the parallelograms of the forces acting upon the crank- 
 pin and which constrain it to move in a prescribed path. Make a separate 
 sketch for each of the following phases, the crank rotating clockwise 6 — 
 45°, 150°, 210°, 300°7 and the two positions at which the crank makes a 
 right angle with the connecting-rod. [9 is the angle which tlie crank 
 makes with the centre line of the engine.] , 
 
 Also state whether the connecting-rod and crank are under tensile or 
 compression stresses at each of the above positions. 
 
 20. [Art. 30.] An arm 13 inches long, rotating uniformly at 30 rev. per 
 minute, drives an arm 30 inches long through an intermediate link 3& 
 inches in length ; distance between fixed centres 48 inches. Find, by 
 method of instant centres, mean velocity of follower when driver-pin is on 
 the line of and between the fixed centres, and 90, 180, and 270 degrees^ 
 ahead of this position (4 phases). Also state the directional relation in 
 each case. 
 
 Express velocities in feet per minute and tabulate results. Graphical 
 solution. 
 
 31. [Art. 40.] Prove that, in the mechanism of Fig. 74, O^c must lie in 
 the intersection of the lines 6 and d (prolonged). 
 
 32. [Art. 41.] Draw velocity diagram for cross-head of an engine hav- 
 ing stroke of 16", and connecting-rod 40" long. Engine makes 150 revs. 
 per min. Prove for one ordinate. Also construct velocity diagram of 
 cross-head with a connecting-rod 48" long. 
 
 23. [Art. 41.] Fig. 78; a = 6", c — 14", 6 = 18", d = 30", and a makes. 
 
236 KINEMATICS OF MACHINMRT. 
 
 30 ruvs. per min. Construct velocity diagram for point Oj^; (a) upon its 
 jxitli lis a base; (b) upon a rectilinear base. Prove for one ordinate. 
 
 24. [Art. 43.] Draw the pair of rolling centrodes for the relative motion 
 <if the cross-head and crank (Fig .70). Also draw the pair of centrodes for 
 the relative motion of the connecting-rod and- frame. 
 
 25. [Art. 46. J Two shafts are 6 inches apart; driver makes 50 rev. per 
 min. Construct a pair of rolling ellipses for connecting the shafts, such 
 that follower shall have a maximum rate of 75 rev. per min. What is the 
 minimum rate of follower ? Give major and minor aies of ellipses. 
 (Draw pitch lines one-half size.) 
 
 26. [Art. 46.] Distance between fixed centres (opposite foci of ellipses) 
 13 8" Construct two rolling elliptical arcs, such that the velocity ratio 
 will vary between the limits; 3: 3, and 4: 8, for an angular motion of the 
 driver of 60 degrees. 
 
 27. [Art. 48.] Fig. 90. Take . . . 0' = 5", and Op = H"- Draw 
 Ap perpendicular to Op, and construct the curve which will roll upon Ap; 
 Ap and this curve to rotate about the fixed centres and 0', respectively. 
 
 28. [Art. 51.] Two parallel shafts, 24" between centres, are to be con- 
 nected by rolling cylinders. One shaft is required to make 350 revs, 
 clockwise ; while the other makes 500 revs, counter-clockwise. What are 
 the proper diameters ? 
 
 29. [Art. 51.] Same data as Prob. 29, except that the shafts are both to 
 turn in the same direction. Required, the diameters. 
 
 30. [Art. 51.] Design rolling conical frusta to transmit motion between 
 two shafts which intersect at an angle of 60 degrees. Driver to make 
 300 rev. to 400 rev. of follower. How may directional relation be changed 
 without affecting the velocity ratio? 
 
 31. [Art. 55.] A pair of grooved friction-wheels have pitch diameters 
 of 8 feet and 2 feet ; working depth of groove equals 1^ inches. The 
 pinion makes 180 rev. per min. Find maximum sliding action, in feet and 
 in inches per min.; assuming no slip at the pitch lines. 
 
 32. [Art. 62.] Epicycloidal gearing. Data : — pitch diameter of driver = 
 12"; of follower = 8"; 1 diametral pitch; addendum length of large 
 wheel = 1"; of small wheel = f"; backlash = 0; bottom clearance = 
 0.1"; ratio of arc of approach to arc of recess = f ; arc of action = 
 circular pitch. 
 
 Required : — Diameters of describing circles; full construction of three 
 teeth of each wheel ; angles of maximum obliquity during both approach 
 and recess. 
 
 33. [Art. 63.] Epicycloidal gearing. Data: — Pitch diameters 14" and. 
 10"; diameter of describing circles equal to radius of smaller wheel ; back- 
 
PROBLEMS AND EXERCISES. 237 
 
 lash = clearance = ^V'; angles of approach and recess equal; IV circular 
 pitch. 
 
 Required : — Least addenda which will insure contact between two 
 pairs of teeth at all times ; angle of action in terms of the pitch. Test 
 accuracy of the construction by rolling a tracing of one set of teeth upon 
 the other. 
 
 34. [Art. 68, 69.] Annular involute gears. Oonstruot several teeth of 
 annular gear and pinion complying with the following conditions : 
 
 Diameters of pitch circles 13" and 30"; 3 diametral pitch; clearance = 
 tV" ; backlash = ; addendum = J" ; root = addendum + clearance ; 
 profiles to be involutes line of action at 75° with line of centres), as far as 
 possible; roots of pinion to be radial inside its base circle, outline of 
 annular wheel teeth to be continued from the proper point by hypocloid of 
 suitable form. Mark the point where this hypocloid joins the involute. 
 
 35. [Art. 77.] Approximate tooth outline. Data :^Oircular pitch = 
 4' ; number of teeth = 18 ; diameter of describing circle = radius of 12- 
 tooth pinion; addendum = 33 pitch ; root = .37 pitch. 
 
 Required : — (a) Construction of tooth outline by the exact method ; (&) 
 approximate (circular arc) outlines by the Willis, Grant, aud Unwin 
 methods, for comparison. All of these outlines should pass through the 
 same point on the pitch circle, and should be very carefully drawn with 
 fine lines. 
 
 36. [Art. 77.] Dra'w outline Of an involute tooth for a wheel 18" diam. 
 with 27 teeth. Compare this with Grant's approximation for involute teeth, 
 by method similar to that outlined in Prob. 36. 
 
 37. [Art 79.] Design a mitre-gear (one of a pair of equal bevel-gears) 
 with greatest pitch diameter = 10" ; 20 teeth, epicycloidal outlines ; length 
 of teeth along the elements equal to 2^ times the circular pitch, and other 
 dimensions with customary proportions. Thickness of rim equal to roots 
 of teeth. Draw two views of one quarter of the wheel. 
 
 38. [Art. 82.] A No. 5 Brown & Sharpe cutter is used for involute 
 wheels having from 21 to 35 teeth. Construct, accurately, the outline for 
 one tooth of an involute gear of 1 diametral pitch, 21 teeth ; then with the 
 same pitch and pitch points draw the outline for a wheel of 25 teeth. This 
 comparison will show double the necessary maximum error in using one 
 cutter through this range. 
 
 39. [Art. 83.] A No. 3 B. and S. cutter is used for wheels having 35 to 
 54 teeth. Make a construction (1 diametral pitch) for one tooth of each of 
 these ex:treme sizres of wheels, and compare the difference with that found 
 in Prob. 39. 
 
 40. [Art. 83.] Compare the maximum error in using an " M" cutter fol 
 
238 KINEMATICS OF MACHINERY. 
 
 epicycloidal gears of 27 and 29 teeth, with the error in cutting 50 and 59 
 teeth by an " K" cutter. Use 3" circular pitch. 
 
 41. [Art. 85.] Construct a cam on a base circle 3" diam., to make one 
 revolution per minute, and to impart to a roll 1" diam., whose straight line 
 of motion passes through the centre of the axis, a stroke of 2". The roll is 
 to rise uniformly during 25 seconds, remain at rest for 20 seconds, and 
 descend during the remainder of the revolution with a uniformly acceler- 
 ated motion (spaces passed over in equal times in the ratio of 1, 8, 5, 7, 
 etc., as in falling bodies). 
 
 42. [Art. 85.] Draw a cam which by oscillating through an angle of 60° 
 shall give a uniformly ascending and descending motion to a sliding-bar 
 the line of motion of which passes 4" to the right of the axis. Stroke of 
 bar = 3" ; base circle = 10" Cam acting on a roll IJ" diam. at end of 
 the bar. 
 
 43. [Art. 85] The follower of a cam is a rocker, 22 inches long (roll 2" 
 •diam. at free end), with fixed centre 4 inches above and 24 inches to the 
 right of cam-shaft. Lowest position of follower is horizontal and cam 
 rotates uniformly, moving follower through 30 degrees. During 90 degrees 
 of rotation of cam follower describes angles in the ratio of 1, 3, 5, 3, 2, and 
 1, and then rests during the next 90 degrees of ratation of cam, and de- 
 scends with uniform angular velocity during the remainder of rotation of 
 cam. 
 
 44. [Art. 86.] A cam is to act upon a straight tangential follower, with 
 the working face of the latter perpendicular to its line of metion. (See Fig. 
 132.) The follower is to be moved uniformly upward, a total distance of 
 ly, wliile the came rotates through 120* ; then follower is to rest during 
 an angular motion of the cam of 90° ; and to descend with a uniformly 
 accelerated motion during the completion of the rotation. Make base circle 
 of cam = 4". 
 
 45. [Art. 91.] Design a worm and wheel such that 
 
 5^ = 50 ' ^ = ^ '°'- ' ^ = *"- 
 
 Eequired : T, t, D, d, <p. 
 
 Give the teeth the involute rack-and-pinion outline at middle section^ 
 and mark contact points of teeth. 
 
 46. [Art. 95.] If, in the four-link chain of Fig. 149, a = 3" ; 6 = 4" : 
 d — 10" ; find the limits between which the length of c must lie in order 
 to permit continuous rotation of a. 
 
 47. [Art. 95.] Taking same data as problem 47, is it possible to give c 
 such a length that a drag-link mechanism results ; that is, so tjiat both a 
 
PROBLEMS AND EXERCISES. 239 
 
 and 6 shall rotate continuously ? Test this by finding the limiting values of 
 c for the drag-link chain, with given values of a, b, and d. 
 
 48. [Art. 95.] If, in the drag-link chain of Fig. 150, a = 7" ; 6 = 6" ; 
 d = 3" ; find the limiting values of c which will permit continuous rotation 
 of both a and 6. 
 
 49. [Art. 97.] (See Fig. 154.) An engine with a stroke of 18" has a 
 connecting-rod 45" long. 
 
 (a) Calculate the distance of the piston (or cross-head) from the end of 
 stroke (a — e) when the crank angle (& measured from A) is 60*. 
 
 (6) Calculate the distance from the other end of the stroke when 
 e = 120°. 
 
 (c) Calculate the distance from cross-head to middle of the stroke 
 {q— m), when 6 = 90°, or 370. 
 
 50. [Art. 97.] With data as in Prob. 50, except that connecting-rod is 
 54" long, calculate (a), (6), (c). 
 
 51. [Art. 97.] Data as in Prob. 50. Calculate crank angles at which 
 Telocity of cross-head (piston) equals velocity of crank-pin. Also find ratio 
 of piston velocity to orank-pin velocity when crank and connecting-rod form 
 a right angle at C. 
 
 53. [Art. 101.] (Fig. 163.) The perpendicular distance from Q to the 
 line of stroke, h — g, is 3"- ; radius of a crank = 3" ; crank makes 20 rev. 
 per minute ; connecting-rod C — c = 9". Find length of stroke of c ; and 
 construct velocity diagram of c for forward and return strokes on a — 6 as 
 a base. 
 
 53. [Art. 104.] fFig. 165.) Design a Whitworth quick-return mechanism 
 such that length of stroke c shall be 10" ; ratio of times of forward and re- 
 turn strokes = 2:1; radius of driving-crank (OP) = 4" ; length of con- 
 necting-rod = 13". 
 
 Construct velocity diagram of c for both strokes. 
 
 54. [Art. 106.] (Fig. 169.) The stroke of a beam-engine is 4 feet ; dis- 
 tance from line of piston motion to beam centre (d) = 5 feet. Find proper 
 length of beam for minimum obliquity of connecting-rod. 
 
 55. [Art. 117.] A countershaft runs at 100 rev, per minute. This 
 countershaft is to drive a spindle through stepped cones and an open 
 belt at 150, 100, or 75 rev. per minute. Largest step on countershaft 
 = 14" diam. Distance between centres = 7 feet. Find, graphically, the 
 diameters of all the steps. Check the accuracy of the method by calculat- 
 ing the lengths of belts for each of the three pairs of steps. 
 
 56. [Art. 124.] (See Fig. 204.) The diameter of a = 24" ; 6 = 40" ; 
 c = 36" ; d = 54" ; e has 15 teeth ; and / has 48 teeth. Find velocity ratio 
 and the directional relation between a and /. 
 
 57. [Art. 124.] (Fig. 303.) Data :— a has 60 teeth ; 6 has 16 teeth ; 
 
240 KINEMATICS OF MACHINBRT. 
 
 diam. of c = 24" ; diam. ot d = 8" ; e makes 75 rev. per min., and/ 250 
 rev. per min. How many rev. per min. does a make ; and what is the 
 directional relation between a and/? 
 
 58. [Art. 124.] (Fig. 205.) The number of teeth on a, 6, c, d, e, and / 
 are, respectively, 15, 45, 25, 35, 1, and 50. Determine velocity ratio be- 
 tween axes I and IV. 
 
 59. [Art. 126.] (Fig. 206.) The cone-pulley is driven by an equal cone 
 on a countershaft which makes 90 rev. per min. The steps have diame- 
 ters of 12", 9f", and 7". The gear a is keyed to the cone-pulley, and it has 
 28 teeth ; gears 6 and c are fast to the shaft B, and have, respectively, lOO 
 and 24 teeth ; d is keyed to the spindle and has 88 teeth. Calculate the 
 various possible speeds of the spindle. 
 
 60. [Art. 128.] The lathe has a lead screw with 4 threads per inch. 
 The change-gears include wheels with the following numbers of teeth : 
 24, 30, 36, 42, 48, 48, 54, 60, 66, 69, 72, 78, 84. The "stud" makes the 
 same number of revolutions as the spindle in a given time. With the 
 24-gear on the stud what gears should be used on the screw to cut 9, 10, 11, 
 11^ and 12 threads, respectively ? What arrangement would be used ta 
 cut 4 threads per inch ? What for 2 threads ? 
 
 61. [Art. 128.] Same data as Prob. 61. Arrange table showing what 
 gears to use on the stud and screw to cut threads from 2 per inch up to 14 
 per inch. 
 
INDEX. 
 
 A 
 
 PAOB 
 
 Absolute motion 3 
 
 Acceleration 1 
 
 Acceleration diagrams 71 
 
 Angular velocity 18-53 
 
 Angularity of connecting-rod 179 
 
 Annular wheels 122 
 
 Approximate tooth profiles,. .134^-140 
 
 Axis, instant 30-33 
 
 Axodes 75 
 
 B 
 
 Back- gear s 233 
 
 Baelilash and clearance, gears 128 
 
 Bands - 305 
 
 Beam motion 194 
 
 Bell-cranks 193 
 
 Belts 205 
 
 Belt-tigbteners 214 
 
 Bent levers ■ 193 
 
 Bevel-gears 140 
 
 Brush- wheels 107 
 
 C 
 
 Cams 153 
 
 Cast gears 148 
 
 Centre, instant 20-23 
 
 Cen trodes ■ ''!> 
 
 Chain wheels 214 
 
 341 
 
43 INDEX. 
 
 PAGE 
 
 bange gears 227 
 
 ircular pitcli 130 
 
 ircuinf erential pitcli 130 
 
 learance and backlash, gears 128 
 
 lose-fitting worm-wheel 167 
 
 ommon methods of transmitting motion 37 
 
 omparison of systems of gearing 1 27 
 
 omposition and resolution of motion 13, 14 
 
 ondition of constant angular velocity ratio 58, 56 
 
 positive driving 57 
 
 pure rolling 54 
 
 one frictions 108 
 
 one pulleys 223 
 
 ones, rolling 91, 93 
 
 onjugate teeth 112 
 
 onnectors, link, wrapping 37 
 
 onstant angular velocity 53 
 
 onstant velocity ratio and pure rolling 56 
 
 onstrained motion 28, 28 
 
 jntact transmission , direct 37 
 
 antinuous motion 7 
 
 orliss wrist plate motion 194 
 
 rank and connecting-rod 176 ■ 
 
 rossed belts 205, 209 
 
 rowning pulleys 207 
 
 urvilinear translation 9 
 
 ut gears 1 48 
 
 utters, gear 149 
 
 ycle 6 
 
 ylinders, rolling 92 
 
 ycloids 116 
 
 D 
 
 ead-points, or centres 171 
 
 escribing circles in gears 120 
 
 iametral pitch 130 
 
 imensions of gear-teeth 128 
 
 irect-contact transmission 87, 41 
 
 irectional relation in rotation 52 
 
 trains 221 
 
 istance of centres in involute gears 127 
 
 rag-link 173 < 
 
INDEX. 243 
 
 E 
 
 PAGE 
 
 Eccentric , jgg 
 
 Ellipses, rolling 55 go 
 
 Epicyclic trains __ 229 
 
 Epicycloid \\q 
 
 Epicycloidal system of gears 116 
 
 Escapements 304 
 
 P 
 
 Four-link chain , I7I 
 
 Free motion 23 
 
 Frictional gearing , 99 
 
 Gear-cutters 149 
 
 Gearing, tooth 110, 114 
 
 Gear moulding machines 1 51 
 
 Gear-planers 151 
 
 Generating circles, epicycloidal gears 120 
 
 Grant's odontographs 137 
 
 Graphic representation of motion 13 
 
 Grooved friction-wheels 101 
 
 Guide-pulleys , 313 
 
 H 
 
 Helical motion 7, 10 
 
 Higher pairing 88 
 
 Hobbing worm-wheels 168 
 
 Hooke's coupling 198 
 
 Hyperboloids, rolling 91, 97 
 
 Hypocycloid 116 
 
 Idler gear 324 
 
 Indicator pencil motions 195 
 
 Instant axis, centre 20, 23 
 
 Instant centre theorem 64 
 
 Interchangeable set of gears 131 
 
 Interference in involute gears 126 
 
244 INDEX. 
 
 [iitermediate connectors <i7 
 
 [ntermittent motion • . • ■ 7 
 
 [nversion of mecbanism 32, 59, 1S8 
 
 [n volute gearing 116, 1 34 
 
 [nvolute teeth interference 136 
 
 K 
 S^inematics, definition 34 
 
 L 
 
 Lazy -tongs 197 
 
 [,engtli of belts 208 
 
 Length of connecting-rod. 179-183 
 
 Length of teeth 118 
 
 Links 37 
 
 Link- onnectors 45 
 
 Linkwork 170 
 
 Lobed wheels . . . , 89 
 
 Logarithmic spirals, rolling 55, 85 
 
 Lower pairing 38 
 
 M 
 
 Machine, definition 28, 33 
 
 Machine design, definition, 34 
 
 Manufacture of gears 148 
 
 Mechanics, definition 28 
 
 Mechanism, definition 28 
 
 Methods of transmitting motion 37 
 
 Mitre gears 148 
 
 Motion, absolute and relative 3 
 
 , definition 1 
 
 , free and constrained 23, );8 
 
 , graphic representation of 12 
 
 , Newton's laws of 13 
 
 , resolution and composition 13, 14 
 
 N 
 
 Newton's laws of motion 13 
 
 ■^on-circular gears 133 
 
 STon-interchangeability of bevel-gears 147 
 
INDEX. 245 
 
 O 
 
 PAGE 
 
 Obliquity of connecting-rod X79 
 
 Open belts 205_ 209 
 
 Oscillating-engine meclianism j^gg 
 
 Outlines of conjugate gear-teeth II3 
 
 Outlines of gear-teeth, general method 114 
 
 P 
 
 Pantographs 197 
 
 Parallel motions I95 
 
 Parallel rods, locomotive I74 
 
 Pencil motions, indicator I95 
 
 Piston, velocity ratio to crank IgQ 
 
 Positive driving in direct contact 57 
 
 Positive return cams 153 
 
 Q 
 
 Quick-return motions 186 
 
 Quarter-turn belts 212 
 
 R 
 
 Rack and pinion 122 
 
 Rapid change in angular motion of link 194 
 
 Ratchets 201 
 
 Rate of sliding in direct contact 54 
 
 Ratio, velocity 5 
 
 Reciprocating motion 7 
 
 Rectilinear translation 9 
 
 Relation of direction of rotation 52 
 
 Relative motion 8, 61 
 
 Resolution and composition of motion 13 
 
 Reverted train 232 
 
 Rolling circles , 79 
 
 cones 91, 98 
 
 curves ^ 78, 87 
 
 cylinders 92 
 
 ellipses 55, 80 
 
 hyperboloids 91-97 
 
 logarithmic spirals 55, 85 
 
 , pure, condition of 54 
 
246 INDEX. 
 
 F^GE 
 
 Rolling and sliding 53 
 
 Rope transmission 205 
 
 Rotation , 7 
 
 S 
 
 Screw , 163 
 
 Screw-cutting train 326 
 
 Sbaper quick-return motion 190, 192 
 
 Sheaves for ropes 207 
 
 Shifting helts 207 
 
 Side rods, locomotive 174 
 
 Slider-crank mechanism 176 
 
 Sliding, rate of , 54 
 
 Sliding and rolling 53 
 
 Slip in friotioiial gearing 106 
 
 Spherical motion 7, 10 
 
 Sprocket-wheels 214 
 
 Stepped cones 209 
 
 Stepped gearing 131 
 
 Straight-line motions. . , 1 95 
 
 Strength of gear-teeth 128 
 
 Subnormal, in acceleration diagrams 73 
 
 Systems of gearing, usual 116 
 
 T 
 
 Teeth, conjugate ^ 113' 
 
 Teeth of bevel-gears. 143 
 
 Teeth of gears, length of 11& 
 
 Tight-and-loose pulleys 208 
 
 Tooth-gearing 110 
 
 Tooth outlines, general methods „ , . 114 
 
 Trains of mechanism 217 
 
 Translation, rectilinear and curvilnear 7, 9 
 
 Tredgold's approximate method for bevel-gear teeth, 143 
 
 Tumbling gears 225 
 
 Twisted gearing 131 
 
 IT 
 
 Universal joint igg 
 
 Unsyrametrical teeth 131 
 
 TJnwin, approximate method for gear-teeth 1,34 
 
INDEX. 247 
 
 V 
 
 PAQQ 
 
 " V " frictions-gears 101 
 
 Value of a train of meclianism 219 
 
 Varying angular velocity, wrapping connectors 315 
 
 Velocity, angular 18 
 
 diagrams 69, 71 
 
 , linear 1 
 
 ratio 5 
 
 , uniform and variable 2 
 
 W 
 
 Wbitwortli's quick-return meclianism 190 
 
 Willis' odontograpb 135 
 
 Wrapping connectors 48 
 
 "Wrifit-plate motion , 194 
 
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 16