Cornell University Library The original of tliis 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/cu31924031203742 arV1673 '^°'""" ""'"^^"V Library ^I'lSIHSDlSfY mechanism: *C24 031 203 742 ELEMENTARY MECHANISM: ^ ^jext''l00l^ FOR STUDENTS OF MECHANICAL ENGINEEEING. ARTHUR T. WOODS, ASSISTANT ENSINBER, UNITED STATES NATT; MEMBEB OP THE AMEEIOAN SOCIETY OF MECHANICAL ENGINEERS; ASSISTANT PROrESSOE OF MECHANICAL ENGINEERING, ILLINOIS STATE UNIVERSITY, CHAMPAIGN, ILLINOIS, AND ALBERT W. STAHL, M.E., ASSISTANT ENGINEER, UNITED STATES NAVY; MEMBER OF THE AMEEI- CAN aOCIETT OF MECHANICAL ENGINEERS; PROrESSOB OP MECHANICAL ENGINEERING, PURDUE UNIVERSITT, LA TAYETTE, INDIANA. NEW YORK: D. VAN NOSTRAND, PUBLISHER, 23 Murray Street and 27 Warren Street. 1885. @ ^^CORNELL\, ,UfMiVER3iTY! LIBRARY..-;'^ Copyright, 1885, By D. van NOSTRAND. PEEFAOE. Quite a number of treatises have appeared on the subject of Kinematics, or Pure Mechanism, most of which are now in print, so that a few words of explanation as to the reasons for publishing this book seem necessary. In searching for a text-book on this subject for the use of our classes of Mechanical Engineering students, we were unable to find a book which met our requirements. Some were so vague and incomplete as to be almost useless, while others were large, exhaustive treatises, more valuable as books of reference than as text-books for the use of stu- dents. The following pages were therefore prepared in the form of lectures ; the object being, to give a clear description of those mechanical movements which may be of practical use, together with the discussion of the principles upon which they depend. At the same time, all purely theoretical discussions were avoided, except where a direct practical result could be reached by their introduction. These lec- tures were used in our classes ; and, having proved com- paratively satisfactory in that shape, it was thought best to publish them, after making such improvements as our class- room experience dictated. IV PREFACE. We make little claim to originality of subject-matter, free use having been made of all available matter bearing on the subject. There is, in fact, very little room for such origi- nality, the ground having been almost completely covered by previous writers. Our claim to consideration is based almost entirely on the manner in which the subject has been presented. Accuracy, clearness, and conciseness are the points that we have tried to keep constantly in view. AVhile much has been omitted that is of merely abstract interest, yet it is believed that nearly all that is of direct practical importance will be found in these pages. We have, in common with nearly all other writers on this subject, closely followed the general plan of Professor ' Willis' "Principles of Mechanism." Other works which have been consulted, and to which we are in a greater or less degree indebted for hints as to definition and arrange- ment, are Rankine's " Machinery and Millwork," Fairbairn's "Mechanism and Machinery of Transmission," Goodeve's "Elements of Mechanism," MaeCord's "Kinematics," Eeuleaux* "Kinematics of Machinery," Robinson's " Teeth of Wheels," Grant's " Teeth of Gears," Appleton's " Cyclo- psedia of Mechanics," and Unwin's "Elements of Machine Design." That a want exists for a clear, concise text-book on this subject, we know ; that we have in some measure filled this want, we can only hope. OOITTEE'TS. CHAPTER I. PAGE Introdtjction 1 CHAPTER II. Elementary Propositions 10 Graphic Representation of Motion. —Comppsition and Eesoiution of Mo- tions. — Modes of transmitting Motion. — Velocity Ratio. — Directional Relation. CHAPTER III. oommunication oy motion bt rolling contact. — velocity ratio Constant. — Directional Relation Constant 27 Cylinders. — Cones. — Hyperboloids. — Practical Applications. — Claseifi- catiou of Gearing. CHAPTER IV. Communication op Motion by Rolling Contact. — Velocity Ratio Varying. — Directional Relation Constant 52 Logarithmic Spirals. — Ellipses. — Lobed "Wheels. — Intermittent Motion. — Mangle Wheels. CHAPTER Y. Communication op Motion by Sliding Contact. — Velocity Ratio Constant. — Directional Relation Constant. — Teeth op Wheels 66 Special Curves. — Rectification of Circular Arcs. — Construction of Special Curves. — Circular Pitch. —Diametral Pitch. VI CONTENTS- CHAPTEK yi. PAGE c0mmt7nicati0n op motion ey sliding contact. — velocity ratio Constant. — Directional Relation Constant. — Teeth of Wheels (Continued) 85 Definitions. — Angle and Arc of Action.— Eplcycloidal System.— Inter- changeable AVheels. — Annular Wheels. — Customary Dimensions. — Involute System. CHAPTER VII. commitnication of motion by sliding contact. — velocity ratio Constant. — Directional Relation Constant. — Teeth op Wheels (Continued) 113 Approximate Forms of Teeth. — Willis' Method.— Willis* Odontograph. — G-rant's Odontograph. — Robinson's Odontograph. CHAPTER YIII. Communication op Motion by Sliding Contact. — Velocity Ratio Constant. — Directional Relation Constant. — Teeth op Wheels (Concluded) Pin Gearing. — Low-nuraberedPinions.-Unsymmetrical Teeth. — Twisted Gearing. — Non-circular Wheels. — Bevel Gearing. — Skew-bevel Gear- ing. — Face Gearing. CHAPTER IX. Communication op Motion ey Sliding Contact. — Velocity Ratio AND Directional Relation Constant or Varying 165 Cams.— Endless Screw. — Slotted Link. — Whitworth's Quick Return Motion. — Oldham's Coupling. — Escapements. , CHAPTER X. Communication op Motion by Linkwork. — Velocity Ratio and Directional Relation Constant or Varying 193 Classification. — Discussion of Various Classes. —Quick Return Motion. — Hooke's Coupling. — Intermittent Linkwork. —Ratchet Wheels. CHAPTER XI. Communication op Motion by Wrapping Connectors. — Velocity Ratio Constant. — Directional Relation Constant . . . 22T Forms of Connectors and Pulleys. — Guide Pulleys. — Twisted Belts. — Length of Belts. CONTENTS. Vll CHAPTER XII. PAGE Tkains of Mechanism 240 Value of a Train. — Directional Relation in Trains. — Clockwork. — Nota- tion. — Method of Designing TraiDS. — Approximate Nuaibers for Trains. CHAPTER XIII. Aggregate Combinations 263 Differential Pulley. — Differential Screw Feed Motions. —Epicyclic Trains. — Parallel Motions. — Tramnael. — Oval Chuck. Problems 289 Index 295 ELEMENTARY MECHANISM. CHAPTER I. INTRODUCTION. 1. A Machine is a conibiuation of fixed and movable parts, interposed between the power and the work for the purpose of adapting tlie one to the other. This definition presupposes the existence of two tilings ; namely, a source of power, and a certain object to be ac- complished. The source of power may be one of the forces of nature applied directly, such as the expansive force of steam in a steam engine, or it may be obtained by the indirect application of such natural forces ; that is, the latter may have been already modified by some other machine. Thus, when a steam engine drives the machinery of a shop by means of a line of shafting, the latter may properly be considered as the source of power of the individual machine. 2. Meclianism In designing a machine, we must take into consideration both the motions to be produced and the forces to be transmitted. But these two elements may most conveniently be discussed and investigated separately ; and such discussions and investigations constitute the two divisions of the general subject of mechanism ; namely, Pure Mechanism and Constructive Mechanism. 1 2 ELEMENTARY MECHANISM. Pure Mechanism, then, treats of the designing of ma- chines, as far as relates to the transmission and modification of motion, and explicitly excludes all considerations of force transmitted, or of strength and durability of parts. In order that the sense in which we shall use certain fundamental terms may be clearly understood, we shall now give an explanation of such words and phrases. 3. Motion and Rest. — These terms are essentially relative. When a body changes its position with regard to some fixed point, it is said to be in motion relatively to that point ; when no such change is taking place, it is said to be at rest relatively to that point. Two bodies may evi- dently be iu motion relatively to a third, and still be at rest with regard to each other. 4. Path. — When a point moves from one position to another, it describes a line, either straight or curved, con- necting the two positions. This line is called its path. But the path alone does not completely define the motion, for the point may move iu the path in either of two directions; as, up or down, to the right or to the left, in the direction of the hands of a watch or the reverse. 5. Kinds of Motion. — Motion may take place along either a straight or curved path ; in the former case it is termed rectilinear motion, and in the latter case curvilinear motion. In either case, when a moving point travels for- ward and backward over the same path, it is said to have a reciprocating motion. For example, the piston of a loco- motive has reciprocating rectilinear motion. In the par- ticular case where the reciprocating point moves in the arc of a circle, as, for example, the weight of a pendulum, it is said to oscillate, or, by some, to vibrate. When the motion of a point is interrupted by certain definite intervals of rest, it is said to have an intermittent motion. The motion of the escape wheel of a clock is of this kind. 6. Revolution and Rotation. — These terms are ordi- INTRODUCTION. 3 narily used synonymously, to denote the turning of a body about an axis ; and no ambiguity is usually likely to arise from so using them. Thus, the fly wheel of an engine is said to rotate or revolve. Bj' more strict definition, rotation should be applied only to the turning of a body about an axis which passes through it, while revolution is a more general term to include the motion of a body along a path which is a closed curve. Thus, the earth rotates about its axis and revolves about the sun. 7. Velocity In addition to the path and direction of a moving body, there is another element necessary to com- pletely determine its motion, and that is its velocity. Velocity is measured by the relation between tlie distance passed over and the tiine occupied in traversing that distance. Velocity may be uniform and unchanging, or it may become greater or less ; and then changes may take place quickly or slowly, regularly or irregularly. But, for our purposes, it is sufficient to consider only two kinds of velocity, constant or uniform, and variable. Velocity is expressed numerically by the number of units of distance passed over in one unit of time. The units of distance and time may be selected . at pleasure ; but, for mechanical purposes, the most convenient units are feet and minutes ; and these will, in general, be employed throughout this volume. When a body moves with a uniform velocity, the distance passed over varies directly with the time. Thus, if by V we designate the velocity, and by S the total distance passed over in the time T, we have S = VT. Again : if the velocity is given, we may find the time Tto traverse a given distance S, for T = —. When the distance and the time are given, we may deter- mine the velocity from the equation V = —. 4 ELEMENTARY MECHANISM. For example, if a body moves at a uniform velocity over a distance of 100 feet, and occupies 5 minutes in doing so, it has a velocity F = — = = 20 feet per minute. J. u In case the velocity is variable, these expressions do not give the velocity at any particular instant, but only the mean velocity for the whole time considered. The velocity at any particular instant is measured by the distance which the body would pass over in the next succeeding unit of time, were the velocity with which the body commences that unit to continue uniformly throughout it. Thus, if a railway train is slowing down in coming to a stop, its velocity is de^creas- ing, but may, nevertheless, be measured at any instant. If, for instance, we say that the train has a velocity of 20 miles per hour, we mean, that, if it were to continue in motion for one hour at the velocity which.it has at that instant, it would travel 20 miles. 8. Ang-ular Velocity. — The most natural way of ex- pressing the velocity of a rotating body consists in stating the angle through which it turns, or the number of revolu- tions which it makes, in the unit of time. When the number of revolutions is given, it must usually be expressed as an angle before it can be used in calculation ; and the angle may be stated in degrees or in circular measure. For con- venience of comparison with linear velocities, we shall define angular velocity to be the velocity of a rotating body thus expressed in circular measure; i.e., as the quotient obtained by dividing the length of the arc subtending the angle through which it turns in one unit of time, by the length of the radius of that arc. All the points of a rotating body move with the same angular velocity, but the linear velocity of each point varies directly with its radial distance from the centre of motion. Let a = angular velocity of a body, B = radial distance of some point in that body, and V = linear velocity of that INTRODUCTION. 5 point; in other words, the length of the arc which it describes in the unit of time. Then a = — . B Thus, if a locomotive having driving-wheels 5 feet in diameter is moving at a speed of 30 miles an hour, the linear velocity of a point on the rim of the wheel, relatively to the frame of the engine, is evidently F = — = ^^ ^ ^^^^ = 2640 feet per minute. The angular velocity of the wheel is therefore a = — = = 1056 feet per minute. The relation between the number of revolutions per minute and the angular velocity is readily found. Thus, let a wheel make N revolutions in T minutes. Let a point be taken at a radial distance B. Then this point will, in each complete revolution, describe a circle whose length is 2'7rB ; in T minutes it will describe ISF such circles, and travel a distance i-n-NB, and its linear velocity V = — -=- . Hence its angular velocity is a = — = ^„ = — =-• When T ° ■' B TB T is unity, that is, when N is the number of revolutions per minute, a = ^irN, and N ~ —. Hence, in the above ex- ample of the locomotive driving-wheel, we find that the wheel makes N = — = ^-^ = 168.07 revolutions 27r 2 X 3.1416 per minute. 9. Periodic Motion. — During the operation of a ma- chine, it usually happens that the various moving parts go through a series of changes of motion which recur per- petually in the same order. The interval of time which 6 ELEMENTARY MECHANISM. includes in itself one such complete series of cLanges is called a period, and the character of the motion is described by the term periodic. The complete series of changes of motion included in one period is called a cycle. In periodic motion, the general law of the succession of changes is the same in successive periods, but the actual time may vary ; that is, the periods may be unequal in length. As a rule, however, the periods are equal, and the duration, magnitude, and law of succession of the changes are identi- cal, in successive periods ; such motion is known as uniform 2xriodic motion. • lO. Classification of Parts of Machines. — As the work for which machines are designed varies so widely, and as they may be actuated by so many different kinds of power, we find great differences in them as to details of construction and manner of operation. But, in spite of these differences, every machine may be considered to consist of three classes of parts. At one end we have the parts which are specially designed to receive the action of the power; at the other we have those which are determined in form, position, and motion, by the nature of the work to be done. Between and connecting the former and the latter, we find the parts which are interposed for the purpose of transmitting and modifying the force and the motion ; so that, when the first parts move according to the law assigned them by the action of the power, the second must necessarily move according to the law required by the character of the work. These three classes of parts are so far independent of one another, that any kind of work may be done by any kind of power, and by means of various combinations of interposed mechanism. The motion of the parts which receive the action of the power must be transmitted to the working-parts ; and, as the action of the latter is usually very different from that of the former, it follows that the motion must be modified, during transmission, according to certain definite conditions. INTRODUCTION. 7 This modification is accomplished by means of the interposed mechanism above mentioned, and it is to the discussion of the methods by which motion may be transmitted and modi- fied that the following pages are devoted. 11. Elementary Combinations. — The motion of every point of a given piece of a machine being defined by path, direction, and velocity, it will be found that its path is assigned to it by the connection of the piece with the frame- work of the machine ; but its direction and velocity are determined by its connection with some other moving piece or pieces. Two pieces, so connected that, when a given motion is imparted to one, the other moves in a determinate manner, form an elementary combination. 12. Driver and. Follower. — The piece of an ele- mentary combination to which motion is imparted from some extraneous source is termed the driver; and the piece whose motion is received from and governed by the driver is called the follower. 13. A Train of Mechanism consists of a series of movable parts, each of which receives its motion from the preceding one, and transmits it to the one next in order. The train is therefore made up of elementary combinations ; and each piece is at once the follower with regard to the piece that drives it, and the driver of the piece which follows it. 14. Modes of Transmission of Motion. — The sim- plest means by which one piece can produce motion in another is evidently by direct contact ; the two pieces thus forming an elementary combination, as previously defined. But it frequently happens that motion is communicated from one piece to another through the medium of a third and con- necting piece, under such circumstances that the motion of the connecting piece is of no consequence whatever, the proper action of the whole depending entirely on the relative motion of the other two pieces. In this case, the latter may 8 ELEMENTARY MECHANISM. be properly regarded as forming an elementary combination. We thus see tiiat motion may be transmitted from driver to follower, — I. By direct contact. II. By intermediate connectors. 15. Velocity Katio and Directional Kelation. — It has been already shown that the paths of the pieces in an elementary combination are fixed, and depend on the con- nection of the pieces with the framework of the machine ; while their velocity and direction of motion may vary, and must be determined for each instant of action. Thus, in comparing the motions of the pieces for successive instants, we may find changes of velocity or of direction, or both. But, while the absolute velocities and the absolute directions of both pieces may be liable to continual variation, it is evident that there will exist, at each instant, a certain definite ratio between the velocities, and an equally definite relation between the directions, of the driver and follower. This velocity ratio and this directional relation will depend solely on the manner in which the two pieces are connected, and will be entirely independent of their absolute velocities or directions. The velocity ratio, and also the directional relation, may be constant during the entire period, or either or both may vary. For example, if two circular wheels turning on fixed axes gear with each other, their velocity ratio is constant. If one wheel is twice as large as the other, it will make only one-half as many turns in the same time, or its angular velocity will be half that of the smaller wheel. But during any changes in velocity whatsoever, as one wheereannot rotate without turning the other, and as the respective radii of contact do not change in length, the ratio of their velocities at any instant is the same ; that is, such wheels have a constant velocity ratio. And so, also, of the relative directions of the rotations. If the wheels are in external gear, they will turn in opposite directions ; if in internal gear, in the same direction : but in either case the INTRODUCTION. 9 directional relation will remain coiistant, without regard to any change of absolute direction of the driver. If the two wheels are elliptical, however, as those shown in Fig. 42, the directional relation will be constant, while the velocity ratio will vary according to the varying lengths of the radii of contact. If, then, in addition to the paths of both driver and fol- lower, we have determined their velocity ratio, and the directional relation of their motion, for every instant of an entire period, our knowledge of the action of the combination will be complete. 10 ELEMENTARY MECHANISM. CHAPTER 11. ELEMENTAEY PROPOSITIONS. Graphic Representation of Motion, t- Composition and Resolution of Motions. — Modes of Transmitting Motion. — Velocity Ratio. — Directional Relation. 16. Graphic Representation of Motion. — The prob- lems relating to the motions of points may be most readily sohcd by geometrical construction. It is evident that the rectilinear motion of a point may be represented by a straight line ; for tlie direction of the line may represent the direction of tlie motion, while the velocity may be indicated by its length. When a point moves in a curve, its direction of mo- tion at any instant is the same as tlie direction of the tangent to the curve at the point considered. Hence the curvilinear motion of a point may be represented in the same manner as the rectilinear motion, using the direction of the tangent as tlie direction of the straight line above mentioned, and making its length proportional to the velocity, as before. By thus representing the motion of properly selected points, we may establish certain relations, by purely geomet- rical reasoning, which will not only enable us to obtain the velocity ratio and the directional relation in the particular phase represented, but will lead to, and almost involve, the accurate construction on paper of the movements considered. Tlie latter is such an important advantage in practical work, that this method is greatly to be preferred, and has been adopted in this volume. ELEMENTARY PROPOSITIONS. 11 17. Composition of Motions. — If a material point receives a single impulse in a given direction, it will move in that direction with a certain velocity ; and, as above explained, its motion may be represented by a straight line having the same direction as the motion, and of a length proportional to the velocity. If a point receives, at the same time, two impulses in different directions, it will obey both, and move in an intermediate direction with a velocity differing from that due to either impulse alone. Such a point may receive, at the same instant, any number of impulses, each one tending to impart to it a motion in a definite direction and with a certain velocity. Now, it is evident that the point can move only in one direction and with one velocity; this motion is called the resultant; and the separate motions which the different impulses, taken singly, tended to give it, are called the components. 18. Parallelogram of Motions. — Given two com- ponent motions of a point, to find the resultant. In Fig. 1, let the 'point A be acted on at the same time by two impulses, tending to give it the motions represented, in direction and velocity, by the straight lines AB and AD respec- tively. Through B draw BO parallel to AD ; through D draw DC parallel to AB; join AO. Then AC will represent, in direc- tion and velocity, the motion which the point A will have as the result of the two im- pulses which separately would have produced the motions AB and AD respectively. The length of the resultant may be altered by varying the lengths of the components or the angle between them, but in no ease can it exceed their sum nor be less than their difference. This proposition is known as the parallelogram of motions, and may be thus stated : — 12 ELEMENTARY MECHANISM. If two component motions be represented, in direction and velocity, by the adjacent sides of a parallelogram, the resultant will be similarly represented by the diagonal passing through their point of intersection. 19. Polygon of Motions. — By a repetition of the above process, we may find the resultant of any number of simultaneous independent components. In Fig. 2, let AB, AD, AF, represent three such com- ponents. We first compound any two of them, as AB and AD, by completing the parallelogram ABCD, and find the Fis. 3 resultant AC. We next compound AG with AF in a similar manner, and find the resultant AE. The latter is evidently the resultant of the th7'ee components. ELEMENTARY PROPOSITIONS. 13 This process may be continued for any number of com- ponents, and it makes no difference in what order they are taken. In Fig. 3, for instance, we have the same com- ponents as in Fig. 2, and find the same resultant, AE, though the composition is carried on in a different order. 20. Resolution of Motion. — This is the inverse of the process just explained. It is obvious, that, if two or more independent motions can be compounded into a single equivalent motion or resultant, the latter can be again separated, or resolved, into its components. But it evidently makes no difference whether the single motion to be resolved is the resultant of a previous composition, or whether it is an original independent motion. Any single motion can be resolved into two others, each of these again into two others, and so on as far as desired ; these components being given any directions at pleasure. In Ij'ig. 4, let AC represent the_ given motion. Through A draw the indefinite lines AE and -4^ in the directions in which it is desired to resolve AC. Through C draw CB parallel to AH, and intersecting AE at B; also CD parallel to AE, and intersecting All at D. Then AB and AD will be the components required ; FigT and it is evident that by their composition (Art. 18) we would find their resultant to be AC, the given motion. 21. Communication of Motion by Direct Contact. — In Fig. 5, let AD and BC be two successive pieces of a train of mechanism, turning about the centres A and B respectively. Let AD be the driver, turning the follower BC, by contact, between the curved edges, as shown. Let c be the point of contact between the two pieces ; and let the driver move the follower, until they occupy the positions shown by dotted lines, the points a and b having come in 14 ELEMENTARY MECHANISM. IT'ig. 5 contact at d. During this motion, every point of the curved edge of the follower between b and c has been in contact with some point of the curved edge of thp driver between a and c. If be is not equal in length to ac, it is evident that sliding of one edge on the other must have taken place through a space equal to their difference ; but, if be = ac, there will have been no sliding. In the latter case the mo- tion is said to be communicated by rolling contact, and in the former case by sliding contact.* Motion, then, may be communicated by two kinds of direct contact : — 1 . By rolling contact, when each point of contact of the driver with the follower is continually changed, but so that the curve joining any given pair of points of contact of the driver shall be equal in length to the curve joining the respective points of the follower. 2. By sliding contact, when each point of contact of the driver with the follower is continually changed, but so that the curve joining any given pair of points of contact of the driver shall 7iot be equal in length to the curve joining the respective points of the follower. In contact motions, one or both of the curves must be con- vex ; and, in the former case, the convex edge must have a * More strictly speaking, sliding contact should be defined as that motion in which every point of contact of one piece comes into contact ■with all the consecutive points, in their order, of a line in the other. Thus, the piston of a steam engine moves in true sliding contact with the interior surface of the cylinder. When this definition of sliding contact is adopted, it is usual to class under the head of mixed contact those contact motions which partake of both rolling and sliding. But, for our purposes, it is sufficient to distinguish between contact which is rolling and that whicli is not; designating by the term " sliding" not only that which is purely so, as just defined, but also the cases just spoken of as mixed contact. ELEMENTARY PKOPOSITIONS. 15 sharper curvature than the concave edge. If this condition is not fulfilled, contact will take place at discontinuous points. 22. Communication of Motion by Intermediate Connectors. — Such intermediate connectors may be divided into two general classes : links, which are rigid, and must be jointed or pivoted to both the driver and follower ; and bands, or lorapping connectors, which are flexible. The former class includes all forms of rigid connectors which can transmit motion by pushing or pulling, such as connecting-rods, locomotive side rods, etc. ; the latter in- cludes all forms of connectors which can transmit motion by pulling only, such as belts, ropes, chains, etc; B'ig, In Fig. 6, let AP, BQ, be driver and follower, moving about the centres A and B respectively, and connected by the link PQ. If AP is turned so as to occupy another position, Ap or Ap', it will, by means of the link, move the arm BQ into the position Bq, or Bq'. If the driver jpiish tlie follower, the connector is necessarily rigid, and, as just stated, belongs to the general class of Ibiks- But the con- nector may be flexible, as in Fig. 7, where AGE is the driver, and BDF the follower, turning about the centres A and B respectively, and connected by a flexible but inextensible band which lies in the direction of the common tangent of the two curves. If the driver be moved in the direction of the arrow, it will, by means of this connector, turn the 16 ELEMENTARY MECHANISM. follower as indicated ; and the connector will unwrap itself Q ]i from the curved edge of i^km the latter, and wrap itself on that of the former. By means of this form of f intermediate connector, \ I which belongs to the \Jn general class of bands or wrapping connectors,. it is evident that motion can be transmitted by pulling or tension only. 23. Modes of Trans- mission of Motion. — Every elementary com- bination may be classi- fied according to one of the four modes of trans- mission of motion just defined ; namely, — Fig.^ 1. Rolling contact. 2. Sliding contact. 3. Linkwork. 4. Wrapping connectors. 24. Velocity Ratio in Liinkwork. — In Figs. 8 and 9, let AP, BQ, be two arms, turning on fixed centres A and B respectively, and connected by the rigid link PQ. Since the fig. a arm AP turns about the centre A, the point P will move in the arc of a circle, and hence its direction of motion at any ELEMENTARY PROPOSITIONS. 17 instant will be represented by the tangent to that arc ; that is, by a line perpendicular to the radius AP. Draw Pa perpendicular to AP, and of such a length as to represent the velocity of the point P in that direction. Resolve the velocity Pa into two components (Art. 20), Pc and Pd, along and perpendicular to the link PQ respectively. Similarly, let Qb represent the velocity of the point Q, and resolve it into the components Qf and Qg, along and perpendicular to r'ig. 9 the link. Since the link PQ is rigid, the component velocities of the points P and Q in the direction of the link must be the same ; that is, Pc must be equal to Qf. If Pc were greater or less than Qf, the distance between P and Q, that is, the length of the link, would be diminished or increased, which is impossible. Let fall the perpendiculars AN" and BM from the fixed centres of motion upon the line of the link. Let T be the point of intersection of the line of the link with the line of centres. By construction, we have the similar triangles APN aud Pac ; BQM and Qbf; and ^TiV^and BTM. Let a = angular velocity of P about the centre A, and a = angular velocity of Q about the centre B ; then (Art. 8) Pa But, from similar triangles. Pa ^ Pc PA AN QB and Qb_ QB Of . BM' Pc^, BM' 18 hence ELEMENTARY MECHANISM. a' ^ Pc_ AN ^ AN_ ^ AT, a BM Pc BM BT Hence, in the communication of motion by linkwork, — 1. The angular velocities of the arms are inversely pro- portional to the perpendiculars from the fixed centres of motion upon the line of the link. 2. The angular velocities of the arms are inversely pro- portional to the segments into which the line of the link divides the line of centres. 25. This proposition may also be proved by means of the instantaneous centre. In Figs. 10 and 11 the link PQ may be regarded as turning, during each instant of its motion, about some centre in space. This centre may be constantly changing its posi- tion in space, and also with regard to the line PQ itself; but at any given instant every point in PQ has the same ELEMENTARY PROPOSITIONS. 19 angular velocity about this centre, and moves in a direction perpendicular to the line joining it to the centre, and with a linear velocity proportional to its distance from it. As P moves perpendicularly to AP, the centre must lie in AP ^'ig. IX (produced if necessary) ; and as Q moves perpendicularly to BQ, it must also lie in BQ (produced if necessary) : hence it will be found at the intersection of these two lines at 0. Let V and v represent the linear velocities of P and Q respectively. As both P and Q have the same angular velocity about O, their linear velocities will be proportional to tlieir distance from that point ; that is, V : V :: PO : qO. Let a and a! be the angular velocities of P and Q about A and B respectively. Then V . v_ AP ' BQ PO . QO AP ■ BQ' 20 ELEMENTARY MECHANISM. Let fall the perpendiculars OR, AN, and BM upon tlie line of the link ; then, from the similar triangles ANP and OMF, BQMsiud OQR, B TM a,nd ATJSf, we have PO ^ OR . QO^OR AP AN ^^ BQ BM Hence ^ ^0R_ AN ^ AN_ ^ AT^ a BM OR BM BT' as before. 26. Directional Eelation. — From Figs. 8, 9, 10, and 11, it is evident that the directional relation of the rotations of the two arms depends on the position of the centres A and B with reference to the line of the link PQ. If they are on the same side of PQ, the rotations will take place in the same direction ; if on opposite sides, the rotations will be in contrary directions. 27. Velocity Ratio in Wrapping Connectors — In Figs. 12 and 13, let AG and BD be two curved pieces ■Fig. 13 moving about the fixed centres A and B respectively, and let them be connected by the flexible but inextensible band EPQF, fastened to them at E and F. li AG be turned in the direction of the arrow, it will cause BD to turn by means of the band, which will unwrap itself from the curved edge of BD, and wrap itself on that of AC. Let P and Q be the points at which the line of the band is tangent to the ELEMENTARY PROPOSITIONS. 21 curved edges. These points must move perpendicularly to the radii AP and BQ,\ and the action at any instant is precisely the same as that of two arms, AP and £Q, connected by a link, PQ, as discussed in the preceding articles. Hence, letting fall the perpendiculars AN and BM upon the common tangent, which is the line of the wrapping- connector, . and finding the intersection T of the latter and the line of centres, it follows, that, in the communication of motion by wrapping connectoi'S, — 1. The angular velocities of the pieces are inversely pro- portional to the perpendiculars from the fixed centres of motion upon the line of the wrapping connector. 2. The angular velocities of the pieces are inversely pro- portional to the segments into which the line of the wrapping connector divides the line of centres. 28. Directional Relation. — From Figs. 12 and 13, it is evident that the directional relation of the rotations of the two pieces depends on the position of the centres A and B with reference to the line of the wrapping connector PQ. If they are on the same side of PQ, the rotations will take place iu the same direction ; if on opposite sides, the rotations will take place in contrary directions. 29. Velocity Ratio in Contact Motions. — In Figs. 14 and 15, let AP, BP, be two curved pieces, moving on fixed centres A and B, and in contact at the point P. Now, when the lower piece moves in the direction of the arrow, 22 ELEMENTARY MECHANISM. the upper one will be compelled to turn. Draw Rr, the common tangent, and Hh, the common normal, at the point of contact. The point P, considered as a point of the lower piece, moves at any instant in a direction perpendicular to the radius PA. i^ig. li Draw Pa perpendicular to PA^ and of such a length as to represent the velocity of P m that direction. Resolve Pa into two components, PS and PC; the former in the direc- tion of the normal, and the latter in that of the tangent. PS is the component which causes the motion of the upper piece ; for PC, acting tangentially, can, of course, produce no motion in tlie latter whatever. Tlie direction of the motion of the point P, coasidered as a point of the upper piece, will be represented by the line Ph drawn perpendicularly to PB. It is evident that the length of Pb must be such that its normal component will be PS: for, if the normal component of Pb were greater ELEMENTARY PROPOSITIONS. 23 than PS, the curves would quit contact ; while, if it were less, the curves would intersect. Hence, draw Pb till it intersects aS (produced if necessary) at b. Resolving Pb, we now find its components to be PS and PD. Draw AN, BM, perpendicular to the common normal lUi, and call T the intersection of the line of centres with the common normal. We now have the similar triangles PAN and PaS, PBM and PbS, BTM and ATN. Let a = angular velocity of lower piece about A, and a! = angular velocity of upper piece about B ; then (Art. 8) Pa , Pb pa: pb But, from similar triangles. Pa_ PA PS AN' and Ph_ PB PS . BM' 24 ELEMENTARY MECHANISM. hence a' ^AN PS_^ AN_ ^ AT a PS BM BM BT Hence, in the communication of motion by contact, — 1. The angular velocities of the pieces are inversely pro- portional to the perpendiculars from the fixed centres of motion upon the common normal. 2. The angular velocities of the pieces are inversely propor- tional to the segments into which the common normal divides the line of centres. 30. Directional Relation. — From Figs. 14 and 15, it is evident that the directional relation of the rotations depends on the position of the centres A and B with reference to the normal Hh. If they are on the same side of Hh, the rotations will take place in the same direction ; if on opposite sides, the rotations will take place in contrary directions. 31. Condition of Constant Velocity Ratio. — The value of the velocity ratio (Art. 29) is a_^AT a BT' Now, in order that this expression shall have a constant value, the ratio of AT to BT must remain unchanged. But, as AT + BT — AB, which is itself constant, it follows, that, in order to preserve the constancy of the above ratio, the actual Lengths of AT and BT must not vary; in other words, the point T must remain fixed in position. Hence we see, that, in order to obtain a constant velocity ratio in contact motions, the curves must be such that their common normal at the poiutof contact shall always cut the line of centres at the same point. 32. Condition of Rolling Contact. — In Figs. 14 ELEMENTARY PROPOSITIONS. 25 and 15, while the normal component PS represents tbe velocity perpendicular to the two curves, the tangential components PC and PD represent the rate at which the respective curves are at any instant sliding over the common tangent. In Fig. 14, PC and PD lie in the same direction ; and consequently their difference, PC — PD, represents the velocity with which the two curves are sliding past each other. In Fig. 15, PC and PD lie in opposite directions, and the velocity of sliding is represented by their sum. AVhen PC and PD lie in the same direction, and are equal, the expression PC — PD becomes zero ; in other words, there is «o sliding between the cux'ves, and the motion is trans- mitted by rolling contact. Now, as PS is the normal component of both Pa and Pb, and as their tangential components PC and PD are to be equal, it follows, that, in this case. Pa and Pb must be the same in both magnitude and direction ; that is, they must coincide in one right line. And, as AP and BP are perpendicular respectively to Pa and Pb, it is evident that AP and BP must also coincide in one right line ; and this can be no other than the line of centres AB. The condition of rolling contact, then, for curves revolving in the same plane about parallel axes, is, that the point of contact shall always lie in the line of centres. In order that this condition may be fulfilled, the curves must revolve about their respective centres of motion in opposite directions when the point of contact lies between those ce?itres (Fig. 14), or in the same direction when the jKiint Qf contact lies on the same side of both centres. The curves may be of such a nature that this condition is continuously satisfied ; the point of contact travelling along the line of centres, and the velocity ratio varying accordingly. On the other hand, that point may travel across the line of centres, the action taking place partly on one side and partly on the other. In this case, the velocity ratio may or may 26 ELEMENTARY MECHANISM. not vary ; but, whether it does or not, there will be more or less of sliding between the curves, except at the instant when the point of contact crosses that line. As the point of contact approaches the line of centres, it diminishes the distance between itself and the point of inter- section of the common normal with this line. When the point of contact reaches the line of centres, the contact be- comes purely rolling, and these two points coincide. Hence, in rolling contact, the angular velocities of the pieces are inversely proportional to the segments into which the point of contact divides the line of centres. 33. Similai'ity in all Modes of Transmission It will be observed that the common normal in contact motions bears a very striking resemblance to the lines of the link and of the wrapping connector, previously discussed. In fact, by selecting any two points in this normal, and joining one to each centre by a straight line, we will have two arms and a link, by which, for that instant, we may produce the same velocity ratio as by the curved pieces in contact. If we designate by line of action the line of the link in link- work, the line of the wrapping connector, and the common normal in contact motions, we may express the laws govern- ing the action of any elementary combination in which the pieces rotate about fixed parallel axes as follows : — 1 . The angular velocities are inversely proportional to the perpendiculars let fall from the centres of motion upon the line of action. 2. The angular velocities are inversely proportional to the segments into which the line of action divides the line of centres. 3. The rotations have the same direction if the centres of motion lie on the same side of the line of action, and contrary directions if they lie on opposite sides of that line. MOTION BY EOLLIKG CONTACT. 27 CHAPTER III. COMMUNICATION OF MOTION BT ROLLING CONTACT. VELOCITY KATIO CONSTANT. DIRECTIONAL RELATION CONSTANT. Cylinders. — Cones. — Uyperholoids. — Practical Applications. — Classification of Gearinrj. 34. It lias been shown (Art. 32) that, in the rolling con- tact of curved pieces revolving in the same plane about fixed parallel axes, the point of contact will always lie in the line of centres, and that the angular velocities are inversely propor- tional to the segments into which the point of contact divides that line. Therefore, if the velocitiy ratio of two such pieces in roll- ing contact is constant, these segments must be constant, and the curves must have a constant radius ; in other words, the curves must be circular arcs turning about their centres, and no other curves will satisfy the conditions. Axes Parallel. 35. Kolling Cylinders. — In Fig. 16, let AC, BD, be parallel axes mounted in a framework, b}- which they are kept at a constant distance from each other. Let E and F be two cylinders, fixed opposite to each other, one on each axis, and concentric witli it ; the sum of their radii being equal to tlie distance between the axes. The cylinders will, therefore, be in contact in all positions, 28 ELEMENTARY MECHANISM. the line of contact being a common element of both. If one cylinder be made to rotate, it will drive the other by rolling contact, and compel it to rotate. The linear velocity of every point in the cylindrical surface of either wheel must evidently be the same. S'is. 16 Let R be the radius of the driver, and r the radius of the follower. Let the circumference of the driver be divided into JV" equal parts, and let the circumference of the follower contain n of these parts. Let P and p be the periods or times of rotation ; L and I the number of rotations in a given time, or the syndironal rotations of driver and follower re- spectively ; and, as before, let a and a' be their angular veloci- ties. Then ^ = :? = E = :?= 1. a r H p L^ and it is evident that these ratios will hold, withoat regard to the absolute velocities. 36. If the cylinders roll together by external contact, as in Fig. IC, they will evidently rotate in opposite directions. MOTION BY ROLLING CONTACT. 29 If it ia deaired to have them rotate in the same direction, one wheel is given the form of an anniilus, or ring, as in Fig. 17, to which the other wheel is tangent internally. The rolling surfaces are cylinders, as before ; the line of contact is an element of both cylinders ; and the relations stated in the last article are equally true for this case, the only change being, that the rotations now take place in the same direction. The difference of the radii is evidently the distances between cen- tres. Thus, if we have given the distance between two axes, and the velocity ratio of driver and follower, expressed in any of the above terms, we can readily find the radii of wheels which will answer the given conditions. If the axes of rotation are not parallel, they may or may not intersect ; and these cases will be cousidered separately. Axes Intersecting, 37. Rolling Cones The conclusions arrived at in Art. 34 follow directly from our proi>ositions concerning rolling contact; the circles in contact being in the same plane, and rotating about fixed parallel axes. A little con- sideration will, however, make it clear, that, if the axes be 30 ELEMENTARY MECHANISM. turned in their common plane about the point of contact of the two circles, the latter will, at any angle, have a common tangent at this point. This tangent will be the line of in- tersection of the planes in which the two circles lie. Both circles will be in true rolling contact with this common tan- gent, and hence with each other ; and their perimetral and angular velocities will be the same as before. I"ig. xs In Fig. 18, let OA, OB, be two axes which intersect at ; and let the two right cones OTD, OTF, be constructed on these axes, the cones having a common clement OT. If through any point Jf in OT vfe pass planes perpendicular to the axes OA and OB, the sections of the cones will be cir- cles which will be in contact at 3f; and a constant velocity ratio will be maintained between the axes by means of these circles. For the angular velocities of these circles are, as before, af_ _ MK ^ AT a Mil BT' a constant ratio ; therefore the two cones will rotate in true rolling contact, and their angular velocities will be inversely proportional to the perpendiculars from any point of the common element on the axes. The relations of angular ve- MOTION BY ROLLING CONTACT. 31 locities, periods, etc., will evidently be the same as for two cylinders whose radii are in the same proportion as the radii of the bases of the cones. 38. Having given the positions of the axes, and the ve- locity ratio, it is required to construct the cones. \X 1 \ \ OT^D i B ' T I"ig. 19 In Fig. 19, let OA be the driving axis, and OB the follow- ing axis : and let the velocity ratio of driver to follower be — = — ; in other words, Oxi is to make n revolutions while a n OB makes m revolutions. On OA lay off OC equal to n divisions on any convenient scale. Through 6' draw CD parallel to OB, and make it equal in length to m diA'isions of the same scale. Through D draw ODT, which will be the line of contact. From any point T of ODT, let fall the per- pendiculars AT and BT on the axes. If we now construct two right cones on these axes, having AT and BT as radii of their respective bases, these cones will roll together with the required velocity ratio ; for, from the figure, we have a' m sin COD sin GOD AT BT AT a n sin' ODG sin BOD OT ' OT BT 32 ELEMENTARY MECHANISM. In other words, the radii of the bases have the required rela- tion. 39. It is usual in practice to employ, not the whole cones, but only thin frusta of them, as shown in Figs. 19 to 24. ITig. 30 In Fig. 19, the common element is located in the acute angle between the intersecting axes ; but it may as readily be placed in the obtuse angle, the location depending on the exact data of the problem. Examples of different arrange- ments are shown in Figs. 20, 21, and 22. In these figures tlie angles of intersection of the axes are the same as in Fig. 19, MOTION BY ROLLING CONTACT. 33 but the velocity ratio and the directional relation may be varied at pleasure. In Figs. 19 and 20 the velocity ratio is different, and the direction of rotation of the follower is also Fig changed in the latter by moving the element of contact from the acute to the obtuse angle. In Figs. 21 and 22 the direc- Fig. tional relation is the same as that of Fig. 20 ; but, by altering the velocity ratio, one of the cones becomes a flat disc in one case, and a concave conical surface in the other. 34 ELEMENTARY MECHANISM. 40. Thus far we have considered only those eases in which the axes intersect obliquely ; but in practice the axes intersect most frequently at right angles, as in Fig. 23. In this case it will be noticed that the cones are in contact along two ele- ments, OM and ON, and that the followers will rotate in opposite directions. Thus, in Fig. 23, where A is the driver, the two followers, B and C, rotate in opposite directions, as shown by the arrows. But if, as in Fig. 24, the driving axis A be continued beyond the common vertex of the cones, and two other frusta be constructed, motion will be given to the two followers B and G in the same direction ; the velocity ratio of both pairs of frusta being the same. Asxs neither Parallel nor Intersecting. 41. Hyperboloid. of Revolution. — When the axes do not lie in the same plane, motion may be transmitted from the one to the other by means of surfaces, known as hyperbo- loids of revolution. The hyperboloid of revolution is the warped surface generated by a right line revolving about another right line not in the same plane with the first. Its form and the manner of constructing it are shown in Fig. 25, both vertical and horizontal projections being employed for the sake of clearness. Let the axis be taken vertical ; it will be horizontally projected at 0' and vertically at Cc. The revolving line, or generatrix, is, for convenience, taken in a position parallel to the vertical plane of projection, and is shown at MN, M'N'. As this line revolves about the axis, any point, P, P , of the line describes a circle, whose radius is projected vertically at OP, and horizontally wi its true length at 0' P'. Draw the common perpendicular to the two lines MN and Cc. It will be projected horizontally in its true length at O'D', and vertically in the point D. The circles described by different points of the line MN will evidently vary in size ; the largest being described by the points M and MOTION BY ROLLING CONTACT. 35 N respectively, and the smallest by the point D. To con- struct the projections of the curved surface, we must find the B'ig. 35 projections of the circle described by any point P of the line 'MN. Its horizontal projection will be the circle W'P'JR' ; while its vertical projection will be the straight line WPB, 36 ELEMENTARY MECHANISM. and B and W mil be points of the meridian curve. By re- peating this process for a sufficient number of points of the line MN, the meridian curve may be drawn ; and it will be found to be a hyperbola. The circle A'D'B', described by the point ly (which is the intersection of the generatrix with the common perpendicular O'jy) , is called the circle of the gorge; and the circles described by the points M' and N' are called the circles of the lower and upper bases respectively. If we take the line mn, parallel to the vertical plane of projection, intersecting MN at D, and making angle iwnc = angle NMc, and revolve it about Co, we will evidently generate the same surface as before ; for the paths of m and M coin- cide, as do also those of n and JV, and the point D is com- mon to both lines : hence any two points, one on each line, equidistant from D, such as P and Q, will describe the same circle. Through any point of the surface, then, two rectilinear elements, or generatrices, may be drawn ; and their projec- tions on a plane perpendicular to the axis will be tangent to the projection of the gorge circle on that plane. 42. Rolling Hyperboloids. — If through any point of a surface two lines of the surface be drawn, the plane which contains the tangents to both these lines will be tangent to the surface at that point. Hence, if through any point of the curved surface of a hyperboloid we pass two intersecting generatrices, the plane containing these two elements will be tangent to the surface at that point. The normal to the sur- face at that point must, of course, be perpendicular to that tangent plane ; and, as "the surface is one of revolution, it must intersect the axis. If a series of such normals be drawn through different points of the revolving line, they will lie in planes perpen- dicular to the latter, and therefore parallel to each other^ Suppose three planes to be drawn parallel to both the axis Co and the generatrix MN; one through the axis, another MOTION BY ROLLING CONTACT. 37 through the generatrix, and the third at any convenient dis- tance. Conceive a number of points to be laid off at definite and equal intervals on the line MN. Now, in passing along JOT from one point to the other, the normal, though always remaining perpendicular to MN, will still turn about the latter, so that its other end will describe on the plane through the axis a straight line ; viz., the axis itself. Now, as these three planes are parallel, and the normal moves so that its two ends trace straight lines on two of the planes, it is evi- dent that the prolongation of the normal will trace a straight line on the third plane. This straight line may be taken as a new axis ; and by revolving MN, the generatrix of the first hyperboloid, about this new axis, a second hj'perboloid will be generated : and these two surfaces will, by construction, have a common normal at every point of the element of con- tact MN, and will be tangent to each other all along that element. If one of these hyperboloids be now rotated about its axis, it will drive the other by a mixture of rolling and sliding contact ; the sliding taking place in the direction of the element of contact, and the rolling in a direction perpen- dicular to that element. 43. Velocity Batio of Rolling Hyperboloids In Fig. 26 we have two hyperboloids in contact along the line MN, and revolving about the axes Oo and Rr respectively. Let P be the point of contact of the gorge circles APL and BPK, and let the inclined hyperboloid be the driver. Then P, considered as point of the driver, will move in the direc- tion of the tangent to the circle APL. This motion, being parallel to the vertical plane of projection, may be repre- sented, in direction and velocity, by the line Pa. Eesolve Pa into two components, PC and PS, along and perpendicular to the element of contact MN respectively. PS will be the component giving motion to the follower, while PG will rep- resent the sliding motion to the point P in the direction of the common element MN. MOTION BY ROLLING CONTACT. 39 Tlie motion of P, considered as a point of the follower, will be similarly represented by the line Pb. The latter must be of such a lengtli that its normal component shall also be PS, its sliding component being then found to be PD. Let a = angular velocity of driver, and a' = angular velocity of follower. a Pa PA = Pa P'Q'' a' Pb PB = Pb P'O" t a a _ Pb Pa X P'Q' P'O' Draw the common normal to both surfaces at the point N. As the line MN is parallel to the vertical plane of projection, its vertical projection will evidently be perpendicular to that of the normal. Hence, for the vertical projection of the normal, draw ONB, perpendicular to MN at N. and B being points on the axes, we readily find its horizontal pro- jection O'N'B'. As 0, N, and B are respectively the ver- tical projections of three points on a straight line whose horizontal projections are 0', N' , and B', we have the ratio N'B' ^ NB O'N' ON' From the similar triangles O'P'N' and O'Q'B' we have P^^^N^. hence ^1^ = ^ P'O' O'N'' P'O' ON From the similar triangles OPN and NPE, BPN and NPH, we have PO ^ PN, . NB ^ NH ON EN' ^""^ PB PN' 40 ELEMENTARY MECHANISM. Multiplying these equations together, and combining the result with the preceding equation, we have P'q ^NR ^PR NH F'O' ON PO EN' Again : from similar triangles OPR and hPa we have Ph ^PO Pa PR Substituting the values found in the last two equations in the expression for velocity ratio, we have a;_ P6 P^^PO PR NH ^NH , a Pa P'O' PR PO UN EN' that is, the angular A'elocities are inversely proportional, not, as in the case of rolling cones, to the perpendiculars from any point of the common element on the axes, but to the projections of these perpendiculai's upon a plane parallel to both axes and the common element. The above determination is based on the relative motions of the gorge circles ; but the use of these circles is a mere matter of convenience, as the same may be proved for any other two circles in contact. 44. Percentage of Sliding. — The motion of the line MN of the vertical surface is one of revolution about Oo ; and, Pb being the linear velocity of P, every point in the line MN must have a component velocity along that line equal to PD, and in the same direction. Similarly, in revolv- ing about Rr, every point in MN must have a component velocity in the direction of and equal to PC. The motion, then, is transmitted by rolling, accompanied by sliding ; the latter taking place along the common element MOTION BY ROLLING CONTACT. 41 at a rate represented by the sum PG + PD. From this we see that the velocity of sliding is constant all along the rectilinear element, while the linear velocity of any point is evidently proportional to its distance from the axis of revo- lution. It follows, therefore, that the percentage of sliding will be greatest at the circle of the gorge, and will diminish as the distance from that circle increases. 45. Having given the positions of the axes, and the velocity ratio, it is required to construct the hyperboloids. In Fig. 27, let Rr, iJV, and Oo^ 0', be the projections of the driving and following axes respectively; the vertical plane of projection being taken parallel to both axes. Let 42 ELEMENTARY MECHANISM. the driver be required to make n revolutions while the follower makes m revolutions ; in other words, — = — . a 71. On PH lay off PV equal to n divisions on any convenient scale. Through V draw VN parallel to Oo, and equal to m divisions of the same scale. P'rom N let fall EN' aiul Nil perpendicular to Oo and Rr respectively. Through N and P draw the line NP- This will be the vertical projection of the element of contact ; for, from the triangles in the figure. m _ NV sin NPV sin NPV NH ^ n PV sin PNV sin EPN EN a Through N draw ONR perpendicular to NP. ONR is the vertical projection of the normal ; hence the horizontal pro- jection of the point R must be at R' on the horizontal projection, i?'}-', of the axis Rr. Joining O'R' . we have the horizontal projection of the normal. Projecting N hori- zontally at N' on O'R' -I and drawing N'P' parallel to S'/, we have the horizontal projections of the common element and the gorge circle radii O'P' and P'Q. We have thus determined all the data necessary to the construction of the hyperboloids, as explained in Art. 41. As in the case of cones, only thin frusta of these hyper- boloids (Fig. 26) are used in practice ; and their location is optional, except that, as already indicated, the percentage of sliding increases as they come nearer the gorge planes. 46. Analogy between Cones and Hyperljoloids. — As the radii of the gorge circles are made smaller, the meridian curves of the hyperboloids will become flatter, and the surfaces will begin to approximate to the conical shape. When the radii of both gorge circles reduce to zero, the axes ,will intersect, and the hyperboloids will become true cones ; the element of contact lying in the plane of the axes, and passing through their point of intersection. Cones, MOTION BY ROLLING CONTACT. 43 then, may be considered as the limiting case of hyper- boloids ; and it will be found, that, under similar conditions, they will present similar peculiarities of arrangement. From the similarity of the solutions in Arts. 38 and 45, it is obvious that we may use our discretion in locating the common element in the case of the hyperboloids, just as explained in Art. 39 for the case of cones. By changing the common element from the acute angle between the projections of the axes, to the obtuse angle (a change similar to that shown by Figs. 19 and 20), we will change the directional relation of the hyperboloids. Again : by varying the velocity ratio so as to divide the angle in the same ratio as in Figs. 21 and 22, we will reduce one hyperboloid to a flat disc in one case, and to a liollow hyperbolic surface in the other. 47. The ease of axes neither parallel nor iijtersecting may also be solved by means of two pairs of cones. Fig. as In Fig. 28, let Aa, Bb, be the driving and following axes respectively. Draw the line Oc intersecting the two axes in the points C and c, and let an intermediate axis be taken in this line. Now, a pair of rolling cones, d and e, having their common apex at C, will communicate motion from the axis Aa to the intermediate axis Cc ; and a pair of rolling 44 ELEMENTARY MECHANISM. cones, /and g, having their common apex at c,' will transmit motion from the intermediate axis Cfc to the axis Bb. By this means the rotation of Aa is transmitted, by pure rolling contact, to Bb. Let a, a', and a" be the angular velocities of the axes Aa, Bb, and Cc respectively, and li, r, and B' the radii of the bases of their cones, those of the cones e and / being the same. Then — = T^) also — = _ ; hence - = - ; a M a r u. r exactly as if the cones d and g were in immediate contact. Practical Applications. 48. We have now determined the theoretical forms re- quired to transmit motion by rolling contact with a constant velocity ratio, but the successful application of these forms in practice requires certain changes or substitutions to be made. It is impossible to transmit motion against any considerable resistance by means of such smooth surfaces, and hence various expedients are resorted to in order to obtain the necessary adhesion. 49. Friction Gearing. — For light machinery, and in cases where a constant velocit}' ratio is not imperative, the rolling pieces may be made of different materials ; for instance, one may be made of wood and the other of iron. In this case, the iron wheel should be the follower. Again : one of the wheels may be covered with leather, or rubber, or other elastic material. To secure the necessary amount of adhesion in such cases, the rotating pieces are kept in contact and pressed together by adjusting their bearings, or applying weights or springs. MOTION BY ROLLING CONTACT. 45 50. Grooved Friction Gearing. — Another method is shown in Fig. 29. The wheels are provided with angular grooves, shown in an enlarged section on the left. The angle between ah and cd is usually about forty to fifty degrees. The adhesion is greatly increased by this means, and is obtained, as before, by pressing the wheels together. Such wheels are widely used for hoisting-engines, and are generally made of cast-iron. Fig. 39 51. Gearing. — The method in most general use for the prevention of slipping between rotating pieces is, to form teeth upon them. Oearing is the general term which includes all forms of mechanistic devices in which the motion is transmitted by means of teeth. The contact surface of the rotating pieces is called the pitch surface, and its intersection with a plane perpendicular to the axis of rotation is termed the pitch line. This line is the basis of all calculations for velocity ratios and for the construction of teeth. The pitch line in the cases in which the velocity ratio is constant evidently becomes a pitch circle. 52. Classification of Gearing. — Gearing is divided into classes according to the form of the pitch surfaces for 46 ELEMENTARY MECHANISM. which the toothed wheels are the equivalents. There are five such classes ; namely, spur gearing, bdvel gearing, skew gear- ing, screw gearing, and face gearing. I^ig. 30 53. In Spur Gearing, illustrated by Fig. 30, the pitch surfaces are cylinders, and the teeth engage along straight mig. 31 lines which are parallel to the elements of the cylinders. A spur wheel having a small number of teeth is usually called MOTION BY ROLLING CONTACT. 47 a pinion. When the teeth are formed on the inside of a ring, as shown in Pig. 31, the wheel is termed an annular wheel. In this case, as before pointed out, the directions of rotation of driver and follower are the same ; while in the case of two spur wheels, the directions are opposite to each Other. As the diameter of the pitch circle of a wheel increases, its curvature becomes less and less, and finally disappears when the former becomes infinite. In this case the toothed piece is called a rack (Fig. 32) , and its pitch line is the straight line tangent to the pitch circle of the wheel with which it works. In Figs. 30, 31, and 32, the various pitch lines are shown dotted. I"?g.33 54. In Bevel Gearing, illustrated by Fig. 33, the pitch surfaces are cones, and the teeth engage along straight 48 ELEMENTARY MECHANISM. lines the directions of whicli must all pass through the com- mon vertex of the two cones. In actual wheels, th^ teeth are, of course, placed all around the frusta ; but in the figure they are drawn only on part of the wheels, in order to show more clearly the relation in which they stand to the pitch surfaces. When the axes are at right angles, and two bevel wheels are constructed on equal cones, the line of contact maRing an angle of forty-five degrees with each axis, or, in other wordsj the velocity ratio being unity, the wheels are termed 7nitre gears. 55. In Skew Gearing', illustrated by Fig. 34, the pitch surfaces are hyperboloids of revolution. The teeth of these wheels engage in lines which approximate, in their 3fig.34. general direction, to that of the common element of the two hyperboloids. This class of gearing is not often used, owing to the difHculty of forming the teeth ; the usual method for axes neither parallel nor intersecting being, to employ the intermediate cones described in Art. 47. MOTION BY ROLLING CONTACT. 49 56. In Screw Gearing, illustrated by Fig. 35, the pitch surfaces are cyliaders whose axes are neither parallel nor intersecting ; and hence the cylinders touch each other at E^ig. 35 one point only. The lines upon which the teeth are con- structed are helices on the surfaces of these cylinders. Motion is transmitted by a purely helical or screw-like motion. 67. In Face Gearing, illustrated by Fig. 36, the teeth are pins usually arranged in a circle, and secured to a flat ^ wuuuuLtm c T7ig. 36 circular disc fixed on the axis. Thus the contact is only between points of the surfaces of the pins. . In Fig. 36 50 ELEMENTARY MECHANISM. the wheels are in planes perpendicular to each other, and tlie perpendicular distance between the axes is equal to the diameter of the pins, which in this case are cylindrical. This class of gearing is best adapted to wooden mill ma- chinery, and has been used for that purpose almost exclu- sively. 58. Twisted Gearing. — In Fig. 38 is illustrated another form of gearing, sometimes called twisted gearing. It may be regarded as oTatained from the stepped wheel shown in Fig. 37. The latter may be produced by cutting an ordinary spur wheel by several planes perpendicular to yie-sy the axis, turning each portion through a small angle, and then securing them all together. By placing this wheel in gear with another, made in a similar manner, we combine the advantage of strength of large teeth with the smooth- ness of action of small ones. If the number of cutting planes be indefinitely increased, and each section be turned through an exceedingly small angle, it is clear that a twisted wheel, such as shown in Fig. 38, will be the result. But instead of ordinary spur teeth, whose elements are parallel to the axis of the wheel, we now have teeth whose elements have the directions of helices. The result is, tht^t, in addition MOTION BY ROLLING CONTACT. 51 to the pressure producing the rotation, there will be a pressure produced in the direction of the axis, tending to slide the wheels out of gear. There is, however, no screw- like action in the direction of rotation, in which respect there is a broad distinction to be made between such wheels and screw gearing. DffUg. 3S 3fiS.39 The endlong pressure on the bearings may be prevented by the use of a wheel such as is shown in Fig. 39. By this arrangement, there is no longitudinal pressure on the bear- ings whatever, and the wheels run in gear with a smoothness of action unsurpassed by any other kind of gearing. In fact, when the teeth of these wheels are accurately formed, and their axes are carefully adjusted in position, we have the perfection of spur gearing. 52 ELEMENTARY MECHANISM. CHAPTER IV. COMMUNICATION OF MOTION BY EOLLING CONTACT. VELOCITY EATIO VARYING. DIKECTIONAL EELATION CONSTANT. Logarithmic Spirals. — ElUpseSi — Lohed Wheels. — Intermittent Motion. — Mangle Wheels. 59. It has been shown (Art. 32) that, in the rolling con- tact of curves revolving in the same plane about fixed parallel axes, the point of contact always lies in the line of centres. The radii of contact coincide with this line ; and at the point of contact the curves have a common tangent which must make equal angles, on opposite sides of the line of centres, with the two radii of contact. 60. In the preceding chapter, the ratio of the radii of contact was constant, and hence the velocity ratio was con- stant. If the curves are such that the radii of contact vary, the point of contact moving along the line of centres, the velocity ratio must vai-y. - The sum of the lengths of each pair of the radii of contact must evidently be constant if the point of contact lies between the axes, or their difference must be constant if the axes lie on the same side of the point of contact. 61. The Logarithmic Spiral is a curve having the property, that the tangent makes a constant angle with the radius vector. Let two equal logarithmic spirals be placed in reverse positions, and turned about their respective poles MOTION BY ROLLING CONTACT. 53 as fixed centres until the curves are in contact. Each of the radii of contact is a radius vector of the curve in which it lies, and hence both radii make the same angle with the common tangent at the point of contact. But this can only be true if the radii of contact lie in one straight line, namely, the line of centres ; in other words, the point of contact lies on the line of centres, and equal logarithmic spirals are therefore rolling curves. 62. To Construct the Liog-aritlimic Spiral In Fig. 40, let be the pole of the spiral, and let A and B be two Fig. 40 points through which it is desired to draw the curve. From the property of the curve given above, namely, that the tan- gent makes a constant angle with the radius vector, it may readily be proved that, if a radius vector be drawn bisecting the angle between two other radii vectors, the former will be a mean proportional between the two latter. Draw the radii vectors AO and BO-, and the line OD bisecting the angle AOB. Then, if Z) is a point of the curve, OD must be a mean proportional between OA and OB ; in other words, ^ = ^. On the straight line AO lay off 00 = OB. On OD OB AOC as a diameter, describe the semi-circle AEC. Draw OP perpendicular to AOC. Then OE is a mean proportional 64 ELEMENTARY MECHANISM. between OA and OB. Therefore make OD = OE, and D will then be a point on the curve. In the same manner, bisect the angle AOD, make OF a mean proportional between OA and OD to find the point F, and so on. 63. Since the logarithmic spiral is not a closed curve, two such spirals cannot be used for the transmission of continu- ous rotation ; but they are well adapted for reciprocating circular motion. In Fig. 41, let the distance between the axes A and B be given ;- and let it be required, that, while the driving axis A turns through a given angle, the velocity ratio shall vary between given limits. B^ig. 41. Divide AB at T into two segments whose ratio is one of the given limits, and at G into segments whose ratio is the other limit. Lay off the angle DAC equal to the given angle, and make AD — AO. The problem is now simply to construct a logarithmic spiral (Art. 62) having the pole A, and passing through the points T and D. The follower is necessarily a portion of the same curve in a reverse position ; and the latter having been drawn about MOTION BY KOLLING CONTACT. 55 the pole JJ, draw arcs of circles about B with the radii BO aud BT. The portion of the curve betweeu the intersections of these arcs and the spiral will be the required edge of the follower. Let a = angular velocity of driver, and a' = angular velocity of follower ; then, while the driver turns from the position in the figure through the angle TAD, the velocity ratio will vary between the limits — = — - and — = — -. ^ a BT a BO 64. Rolling Ellipses. — In Fig. 42, let ETH and FTG be two similar and equal ellipses, placed in contact at a point Fig. 43 T, such that the arcs ET and FT are equal : E and F being the extremities of the respective major axes. It is a prop- erty of the ellipse that the tangent OTD makes equal angles with the radii BT and hT, or AT and aT. Therefore the angle DTA = angle CTB, and angle DTh = angle CTa; hence BTA and hTa are straight lines. Also, since the arc ET = arc FT by construction, TA and Th are equal; therefore BT + TA = BT + Th = FG ■= EH, a constant length whatever the position of the point of contact, T. Similarly, bT + Ta =^ FG = EH. Hence two equal and 56 ELEMENTARY MECHANISM. similar ellipses can transmit motion between parallel axes by pure rolling contact ; each ellipse turning about a focus as a fixed centre, and its major axis being equal to the dis- tance between those centres. The velocity ratio will in this 1 . « I- •. »■' ^H AH , a' case vary between the limits — = — — = — — - and — = ^ a BO AE a AE AE — — = — — , the two limits being reciprocals of each other. 65. Lobed Wheels. — By using rolling ellipses, as shown in the preceding article, we can obtain a varying ve- locity ratio having one maximum and one minimum value during each revolution. But it may be necessary that there shall be two, three, or more maximum, alternating with a-s many minimum, values of the velocity ratio during each revolution. Lobed tcheels which will roll together and answer these conditions can be produced by several methods from the logarithmic spiral and the ellipse. 66. Lobed Wheels derived from the LiOgarithmic Spiral. — In Fig. 43, let A and B be two fixed parallel axes, and let it be required to communicate motion between them by wheels so constructed that the velocity ratio will have four maximum and four minimum values. Let — = be a AT one limit : tlieu the other is necessarily the reciprocal of this, a' AT or — = . a BT Make the angles TAC and DBT equal to 45°. Make BD = AT and AC = BT. Construct (Art. G5) the por- tion CT of a logarithmic spiral having A as the pole, and passing through the points C and T. Draw CF, TE, and TD, similar curves symmetrically placed with regard to BT and AC. We have thus constructed one lobe of each wheel; and, as the angles TAF and DBE each include one-fourth of a circumference, the quadrilobes can be completed as MOTION BY ROLLING CONTACT. 57 shown, and will roll together with the varying velocity ratio required. Fig. 43 The angles TAF and DBE may include any aliquot part of a circle ; hence pairs of wheels with any desired number of lobes can be made in this way. They will roll together in similar pairs, unilobe with unilobe, bilobe with bilobe, and so on ; but dissimilar pairs, such as one bilobe and one trilobe, will not roll together. 67. Lobed Wheels derived from the Ellipse. — Lobed wheels may be derived from rolling ellipses by the method of contracting angles, as illustrated by Fig. 44. Let A and B be the fixed foci of two equal rolling ellipses in contact at T. Draw the radii A\, A2, etc., dividing the semi-ellipse T6 into equal angles about the focus A, and con- sequently into unequal arcs. If we describe arcs about T 58 ELEMENTARY MECHANISM. through the points 1, 2, 3, etc., cutting the other semi-ellipse 2^6' at the points 1', 2', 3', etc., it is evident that the arc Tl = TV, T'2 = T2', T3 = T3', etc. Therefore the points 1 and 1', 2 and 2', etc., will come in contact on the line of centres AB; and ^4_B = ^1+ -Bl' = ^2 + B2' = , etc. Bi- sect the angle TAl by the line AI, and bisect the angle TBI' by the line BI'. Make AI = Al, BI' = J31'. It is evident that these points, I and /', will come in contact on tlie line of centres when they have turned through the angles TAI (= I angle TAl) and TBI' {= ^ angle TBI') respectively. Thus, if we find the series of points 7, II, III, etc., and I', IT, III', etc., in the manner just described, and draw through them two curves, as shown in the figure, they will be quad- rants of two similar and equal bilobes, of which the remain- ing similar portions can then be readily drawn. From the above considerations, it is evident that these bilobes will roll together in perfect rolling contact. The velocity ratio will = ■ — •• By contracting the AT ^ o angles to one-third, we can form the outlines of a pair of trilobes, and so on. , , a' AT , vary between — = •— — and a BT MOTION BY ROLLING CONTACT. 69 The wheels thus outlined will roll together in similar pairs, as bilobe with bilobe, trilobe with trilobe, and so on ; but dissimilar pairs, such as one bilobe and one trilobe, will not roll together. 68. InterchangeaWe Lobed Wheels. — In Figs. 45 and 46 is illustrated a method of constructing lobed wheels from an ellipse, by which any two wheels of the set will roll together. The process of construction is simple and practi- cal ; but the rolling properties of the curves do not admit of simple demonstration, although they may readily be proved by graphical construction. In Fig. 45, let A and B be the Nl M L K ^^H^ ^q T^g. 45 foci of an ellipse, CGPV. Describe a circle about its centre with a radius equal to the semi- focal distance OA. Draw the indefinite tangent HN parallel to BA. With radius OC, equal to the semi-major axis, and centre 0, describe an arc CK, and lay off on the tangent the lengths KL, LM, and MN, equal to HK. From the centre lay off on OF the dis- tances OG = OK, OD = OL, OE = OM, and so on. With OC, OD, OE, etc., as semi-major axes, describe a series of concentric ellipses, having the common foci A and B. The primary ellipse is the curve required for the unilobe ; the second ellipse, DQ, is the basis for the bilobe ; the third, EB, for the trilobe ; the fourth, FS, for the quadrilobe ; and 60 ELEMENTARY MECHANISM. SO on. Draw a semi-circle about A, and divide it into any number of equal angles by equidistant radii. To form the bilobe (Fig. 46), divide a quadrant into the same number of equal angles as the semi-circle is divided, and on the equidistant radii in the quadrant lay off JBl' = Al, B2' = A2, etc. Through the points 1', 2', 3', etc., draw a curve : this will be one-fourth of the bilobe ; the remaining portion of which, being symmetrical, can readily be drawn. For a trilobe, an angle of 60° is -similarly divided, and the proper distances laid off on the equidistant radii in that angle. For a quadrilobe, we use an angle of 45°, and so on. The velocity ratio of any two of these wheels in gear will vary between two limits, one of which will be the longest radius of the driver divided by the shortest radius of the follower, and the other the shortest radius of the driver divided by the longest radius of the follower. 69. Compulsory Rotation of Rolling- Ellipses. — In the case of rolling ellipses (Fig. 42), it is evident that, when the motion takes place in the direction of the arrows, the radius of contact of the driver is incn-asing from AE to AH, and hence motion can be readily transmitted from the axis A MOTION BY ROLLING CONTACT. 61 to the axis B. But, when H has passed O, the radius of the driver is decreasing, and tl)e driver will therefore tend to E'ig,4,7 leave the follower. This can be prevented by forming teeth on the rolling faces of both pieces ; but, if this is done, we no longer have pure rolling contact. I^ig. 48 When the position of the pieces will allow it, we can con- nect the free foci by means of a link, as in Fig. 47, since 62 ELEMENTARY MECHANISM. (Art. 67) the distance between the free foci is constant in rolling ellipses. There will, however, be times during the revolution when the link will be in line with the fixed foci, aud hence cannot transmit motion. This necessitates the formation of teeth on a small portion of each ellipse, near the ends of the major axis, as shown in Fig. 47. Another method, when the revolution always takes plaee in the same direction, is, to form teeth on the retreating edge of the driver and the corresponding edge of the follower. In this case it is necessary to provide some means of insuring the proper contact of the teeth in order to prevent jamming. This may be done, as shown in Fig. 48, by placing a pin on the driver and a guide plate on the follower, which arrange- ment compels the first tooth to enter the proper space. 70. luterniittent Motion. — It may happen that the variation in the velocity ratio is to consist of an intermittent T'ig. 49 motion of the follower, while the driver revolves uniformly. In Fig. 49 is shown an intermittent motion formed from two spur wheels by cutting away the teeth of the driver on a MOTION BY ROLLING CONTACT. 63 portion of the circumference. Tiiere is the same objection to this method as before mentioned for elliptical, wheels ; namely, that the teeth are apt to jam after a period of rest of the follower. A partial remedy is the application of a pin and guide plate, similar to the arrangement shown In Fig. 48. A more complete motion is shown in Fig. 50. A portion of inig. 50 the driver is a plain disc of a radius greater than the pitch circle of the driver. A portion of the follower is cut away, to correspond to this ; so that, while there is a slight clear- ance between the two faces, the follower is prevented from turning until the pin and curved piece come in contact. Velocity Ratio Varying. Directional Relation Changing. 71. Mangle Wheels. — By combining a spur wheel with an annular wheel, we obtain a mangle wheel, as shown in Fig. 51. The direction of rotation is changed by causing the pinion, which always revolves uniformly in the same* direc- tion, to act alternately on the spur and on the annular portion. 64 ELEMENTARY MECHANISM. The velocity ratio is constant during each partial revolution of the mangle wheel ; but it is changed each time that the pinion passes from the spur to the annular portion, and vice I^ig. 51 versa. The pinion is mounted so that its shaft has a vibra- tory motion, working in a straight slot cut in the upright bar. The end of the pinion shaft is guided in the groove CD, the I^ig. 53 centre line of which is at a distance from the pitch lines of the mangle wheel equal to the pitch radius of the pinion. The pinion may also be mounted in a swinging frame, as indicated by dotted lines. MOTION BY ROLLING CONTACT. 65 If we construct the teeth of the spur and annular portions of the mangle wheel on the same pitch line, as in Fig. 52, we will obtain a combination in which the velocity ratio is con- stant ; the directional relation changing, as in the preceding arrangement. 72. Mangle Rack A rack can be made in a similar manner to the above, and a reciprocating motion obtained from continuous rotation. Such motion is, however, more (^ (5) r± So ITig. 53 simply obtained by means of the pinion and double rack, shown in Fig. 53. Pins are placed on a portion of the face of the pinion, which engage with the pins of the rack above and below alternately, driving the rack back and forth. 66 ELEMENTARY MECHANISM. CHAPTER V. COMMONICATION OF MOTION BY SLIDING CONTACT. VELOCITY RATIO CONSTANT. DIRECTIONAL RELATION CONSTANT. TEETH OF WHEELS. Special Cicrves. — Rectification of Circular Arcs. — Construction of Special Curves. — Circular Pitch. — Diametral Pitch. 73. General ProWem It has been shown (Art. 32) that, in order to obtain a constant velocity ratio in contact motions, the axes of the pieces being parallel, the curves must be such that their common normal at the point of contact shall always cut the line of centres at the- sapne point. The curved edge of one of the moving pieces lilEiy ulways be assumed at pleasure ; the problem then being to,find such a curve for the edge of the other, that, when motion is tijansmitted by the contact of these curved edges, the velocity ratio of the two axes may be constant. This problem is always capable of solution, theoretically at least ; and, as the assumed curve may be of any shape whatever, we can obtain an infinite number of pairs of such curves. For practical purposes, there are certain definite curves in almost universal use, and these will be first discussed. In Chap. III. has been explained the method of finding the diameters of two pitch circles, which by their rolling contact shall trans- mit motion with a given velocity ratio. We now propose to show how to describe certain curves, which, when substi- MOTION BY SLIDING CONTACT. 67 tuted for the circles, and caused to move each other by sliding contact, shall exactly replace the rolling action of the circles, so far as relates to the production of a constant velocity ratio. 74. Epicycloid and Hypocycloid. — In Fig. 54, let A and B be the centres of motion of the driver and follower respectively, and let — be the required velocity ratio. a AT Divide the line of centres at T, so that : -. Then, if with radii AT and BT we describe two pitch circles, MN and BS, as shown, these two circles will roll in contact with the required velocity ratio. Let a describing circle be taken of any radius, such as cT, and with it describe an epicycloid Td by rolling it on the ozitside of the pitch circle MN, and a hypocycloid Te by rolling it on the inside of the pitch circle RS. If these curves be used for the curved edges of two pieces whose centres of motion are A and B respectively, and the lower one be rotated to the position aa', it will drive the other so 68 ELEMENTARY MECHANISM. as to bring it to the position 66' ; for, by the known properties of the curves, thej' will have their point of con- tact, P, in the circumference of the describing circle when its centre c is on the line of centres, AB, and they will also have a common normal and a common tangent at that point. Draw the line TP from the point of contact of the two pitch circles to the point of contact of the two curves. Now, on whichever of the two pitch circles we regard the describing circle to be rolling at the instant, its instan- taneous centre of motion will evidently be the point T. For that instant, then, the point P revolves about T\ that is, it moves in a direction perpendicular to TP, and hence the line TP is the common normal to the two curves at that instant. Of course, this same argument may be applied to any other position of the curves in contact ; and, as their normal thus always cuts the line' of centres in the fixed point T, it is evident that these curves will transmit motion with a constant -s'eloeity ratio. Furthermore, as the arc Ta — arc TP, and as the arc Th = arc TP, we have arc Ta = arc Tb ; showing that the velocity ratio will be the same as that of the two pitch circles. By transmitting motion by sliding contact, then, between these two curves, we may exactly replace the rolling action of the two pitch circles, as far as the velocity ratio is concerned. 75. Epicycloicl and Radial Line. — In Fig. 54 the diameter of tlie describing circle is less than the radius BT. But this is not a necessary condition. If we change its diameter, we will change the shape of both curves ; but tlie two curves generated by the same describing circle will always work together. If we take the diameter of the describing circle just equal to the radius BT, we will get a special case of the hypo- cycloid. Under these conditions (Fig. 55) the latter will become a straight line passing through the centre B. All the arguments of the last article apply to> this cp,se as well ; MOTION BY SLIDING CONTACT. 69 and we thus see that in this case an epicycloidal curve turning about A, and a radial piece turning about B, will, by sliding contact, transmit motion with the same velocity ratio as the pitch circles. 76. Epicycloid and Pin. — In Fig. 54 the convexity of the two curves lies in the same direction, and they lie on tlie same side of the common tangent. In Fig. 65 the hypocyeloid has become a straight line coinciding with the tangent of the epicycloid. If we increase the diameter of the describing circle still more, the two curves will have their convexities in opposite directions, and they will lie on opposite sides of the common tangent. As the. describing circle becomes larger and larger, the hypocyeloid becomes more and more convex, and decreases in size, until, when the describing circle is taken with the same dianleter as the pitch circle BS, the hypocyeloid will degenerate into a mere point, the tracing point itself. If, then (Fig. 66), we assume a pin to be placed at the point P in the circum- ference of US (the diameter of the pin being so small that the latter may be considered as a mere mathematical line) , 70 ELEMENTARY MECHANISM. it follows that it may be driveo by the epicycloid Pa witii the same constant velocity ratio as the pitch cii'cles. 77. Involutes. — In Fig. 57, let A and B be the centres AT a of motion, make = — , and describe the pitch circles ST a MN and RS, as before. Through T draw the straight line DTE inclined at any angle to the line of centres ; from A and B drop the perpendiculars AD and BE upon DTE. "With these perpendiculars as radii, and A and B as centres, describe the circles M'N' and B'S', which will evidently be tangent to the line DTE. Through the point T describe the involute aTd on the base circle M'N', and the involute hTe on the base circle R'S'. If these curves be used for the edges of two pieces whose centres of motion are A and B respectively, and the lower one be rotated to the position a'Pd', it will drive the other to the position b'Pe'. For any line tangent to either base circle will evidently be normal to the involute of that circle. Now, when the curves are in contact, the normal to the involute of M'N' must be MOTION BY SLIDING CONTACT. 71 a line drawn from the point of contact tangent to M'N'^ and the normal to the involute of R'S' must be a line drawn from the point of contact tangent to R'S'. But, as the curves must be tangent to each other at the point of contact, they must have a common normal at that point. This common normal must evidently be tangent to both base circles, and must heilce be the line DTE. The point of contact, then, always lies in the straight line DTE ; and as the latter is the common normal, and cuts the line of centres in the fixed point T, the velocity ratio is constant, and is equal to that of the base circles. But, from similar triangles, that is, the BE ^ BT_ AD AT' velocity ratio of the pitch circles is the same as that of the base circles. Hence the involutes, -as described, will by sliding contact transmit motion with the same velocity ratio as the pitch circles would by rolling contact. 78. General Solution. — The four methods just de- scribed are the ones most generally employed in the practical solution of the problem of securing a constant velocity 72 ELEMENTARY MECHANISM. ratio in sliding contact motions. But we are not by any means limited to the curves above given. Instead of a describing circle, we may use a describing curve of any shape, provided only that its radius of curvature never exceeds in length the radius of the circle in which the curve is to roll, and thus generate an infinite number of pairs of curves that will satisfy the given condition. Thus, in Fig. 58, let A, B, and T be taken as before, and draw pitch circles MN and BS. Now, if we take any curve, such as HTP, and roll it on the outside of one pitch circle and on the inside of the other, any point of this describing curve will generate two curves which will give the desired velocity ratio by sliding contact. For, let the describing curve be in the position shown, being in contact with the pitch circles at T\ and let P be the describing point. The straight line TP will be the common normal to the two curves, because, on whichever of the two pitch circles we regard the describing curve to be rolling at the instant, the point of contact, T, is the in- stantaneous centre of motion ; so that the motion of P in MOTION BY SLIDING CONTACT. 73 either curve is perpendicular to TP. As the point of contact of the two curves is alwaj'S in the describing curve, the same argument is true for any point of contact. As tlie common normal will thus always pass through the same point, T, of the line of ceatres, AB, these curves will, by moving in contact, produce the desired velocity ratio, exactly replacing the rolling action of the two pitch circles. 79. Conjugate Curves. — Any two curves so related, that, by their sliding contact, motion will be transmitted with a constant velocity ratio, are called conjugate curves. Any curve being assumed at pleasure, we may proceed to find another curve, so that the two curves will be conjugate to each other. If, foV instance, in Fig. 58, the curve Pa be given, it is only necessary to find the shape of the curve, HTP, which, by rolling on the outside of MN, will generate Pa. By then rolling this describing curve HTP on the inside of RS, we will obtain the required curve, Ph. Again : had Pb been given, we could, by a similar process, have found Pa ; and Pa and Pb are conjugate curves. The labor of finding the shape of this describing curve, and using it in this manner, is, however, generally very considerable ; so that, for practical purposes,^ie following simple and satisfactory mechanical expedient, due to Pro- fessor Willis, is usually resorted to. In Fig. 59, A and B are a pair of boards, whose edges are formed into arcs of the given pitch circles. Attach to A a thin piece of metal, C, the edge of which is cut to the shape of the proposed curve a&, and to ^ a piece of draw- ing paper, D ; the curved piece being slightly raised above the surface of the board to allow the paper to pass under it. Roll the boards together, keeping their edges in contact, so that no slipping takes place ; and draw upon D, in a suffi- cient number of positions, the outline of the edge ab of C. A curve, de, which touches all the successive lines, will be the corresponding curve required for B. 74' ELEMENTARY MECHANISM. For, by the very mode in which it has been obtained, it will touch ab in every position ;~ hence the contact of the two curves ab and de will exactly replace the rolling action of the two pitch circles. To prevent the boards from slipping, a thin band of metal, such as a watch spring, may be placed between them, being fastened to B at g, and to 5'ig. 59 A at h. The respective radii of the circular edges of the boards must, in that case, be made less than those of the given pitch circles by half the thickness of the metal baud. 80. The solutions given above may be used to find the curved edges of any two pieces transmitting motion by sliding contact with a constant velocity ratio, but by far their most important application is in finding the proper shapes for the teeth of wheels. We shall now give the methods of laying out on paper the principal curves employed for that purpose, and then proceed to examine their practical application in the forma- tion of teeth. 81. Kectiflcation of Circular Arcs. — In construct- ing these curves, as well as in many other graphic operations, it becomes necessary to determine th? lengths of givea MOTION BY SLIDING CONTACT. 75 circular ares, as well as to lay off circular arcs of given lengths. Either of these problems may, of course, be solved by calculation ; but for our purposes it is much more satis- factory to employ the followiug elegant and surprisingly accurate methods of approximation, devised by Professor Eankine. I. To rectify a given circular arc; that is, to lay off its length on a straight line. In Fig. 60, let AT be the given arc. Draw the straight line JBT tangent to the arc at one extremity, T. Bisect tlie chord AT&tD, and produce it to C, so that TO = DT = AD. With C as a centre, and radius AC, describe the circular arc AB, cutting BT at B. Then BT is the length of the given arc AT, very nearly. II. To lay off, on a given circle, an arc equal in length to a given straight line. In Fig. 61, let T be the point desired for one extremity of the arc. Let BT, drawn tangent to the circle at T, be the given straight line. Lay off CT = ^BT. With C as a centre, and radius BO, describe the circular arc BA, cutting 76 ELEMENTAEY MECHANISM. the given circle at A. Then the arc ^T is equal in length to the given straight line BT, very nearly. It follows that, to lay off on a given circle an arc equal to a given are on another circle, we must first rectify the given arc according to I., and then lay off according to II. the required arc equal to the length so found. 82. Degree of Accuracy in Above Processes. — The error in each of these processes consists in the straight line being a little less than the arc. But this difference is very slight, amounting to only ^o of the arc when the latter is 60°. The error varies as the fourth power of the angle, so that it may be reduced to any desired limit by subdivision. Thus, for an arc of 30°, the error will be ttou X (fo)^ = TrJoTT- ^o long, then, as we use these processes for ai-es not exceeding 60°, the results will be abundantly accurate for all practical purposeg. When the arcs exceed 60°, sub- division should be resorted to. 83. Construction of the Epicycloid — In Fig. 62, let O be the centre and CT the radius of a circle rolling on the outside of the Jixed circle whose centre is A and whose radius is AT. Any point in the circumference of the rolling circle will describe a curve, which is known as an epicycloid. Let it be required to draw the curve described by the point T of the rolling or describing circle. Divide the semi-circumference of the latter into any number of equal arcs, ,n', 1'2', 2'3', etc., and through the points of division, 1', 2', etc., and also through C, describe arcs of circles about ^ as a centre. Lay off on the fixed circle (Art. 81) the arcs Tl = TV ; 1, 2 = 1', 2'; 2, 3 = 2', 3', etc. ; and through the points of division, 1, 2, 3, etc., draw radii from A, and produce them. As the describing circle rolls along the fixed circle, its centre will successively occupy the positions Cj, Cj, Cj, etc. If we draw the describing circle with its centre in any one of these successive positions, as c^, its intersection 6 with MOTION BY SLIDING CONTACT. 77 the circular arc through 2' will he a point of the epicycloid required. Similarly, we obtain the points a, d, e, /, g ; and the curve drawn through these points will be the epi- cycloid required. If greater accuracy is required, we need only increase the number of arcs into which we have divided the describing circle. E"ig. 63 This method of finding points of the curve is objectionable on account of the resultant obliquity of the intersections at a and /. This may be avoided, and the construction sim- plified, by laying off the arc lb = 7i2', md = k2>\ etc. In this case it is not necessary to construct the rolling circle in its various positions ; and, as this method gives the best results for points of the curve near T (which is the part of 78 ELEMENTARY MECHANISM. the curve employed in teeth of wheels) , it is greatly to be preferred for practical work. 84. Coustruction of the Hypocycloid The hypo- cycloid is the curve described by a point in the circumference of a circle rolling on the inside of a fixed circle. Its con- struction, shown in Fig. 63, is in every way similar to that of the epicycloid. When the diameter of the rolling circle is less than the radius of the fixed circle, the curve lies on the same side of the centre A as the successive points of contact of the two circles. When the diameter of the rolling circle is greater than the radius of the fixed circle, the curve lies on the opposite side of the centre A. When the diameter of the MOTION BY SLIDING CONTACT. 79 rolling circle is equal to the radius of the fixed circle, as shown ou the left in Fig. Go, the radii A 2, A 3, etc., pass through the points 2', 3', etc., and the points b and h, Tc and d, etc., coincide so that the curve becomes a straight line ; and this line is a radius of the fixed circle. 85. Construction of the Cycloid The cycloid is the special case af the epicycloid and hypocycloid, in which the radius of the fixed circle becomes infinite, and the circum- ference of the circle a straight line. The cycloid is thus described by a point in the circumference of a circle rolling on a straight line. Its construction is in all respects similar to that of the epicycloid and hypocycloid. 86. Construction of the Involute. — The involute is generated by a point in a straight line which rolls along a fixed circle ; or we may regard it as formed by the end of a thread which is unwound from about the circle, and kept taut. It will thus always lie in the direction of a tangent to the 83 ELEMENTARY MECHANISM. circle. Hence, to construct the curve, draw any number of tangents to the base circle, and on them lay off the rectified arc of the circle from the point of tangency to the point on the circle where the involute begins. In Fig. G4, then, make fl2 = arc 1,2; 13 = arc 1, 3, etc. The curve drawn through the points \, a, b, c, etc., will be the required involute. 87. Circular Pitch. — Having divided the line of cen- tres, in any given case, according to the assigned velocity ratio, and described the pitch circles, we must next divide the circumference of each pitch circle into as many equal parts as its wheel is to have teeth. The length of the circu- lar arc measuring one of these divisions is called the circular pitch, and often simply the pitch, of the teeth. Circular piitch, then, is the distance, measured on the circumference of the pitch circle, occupied by a tooth and a space. This pitch must evidently be the same on both pitch circles. The num- bers of the subdivisions, and hence the lyimbers of teeth, are proportional to the diameters of the pitch circles ; and, a fractional tooth being impossible, the pitch must be an aliquot part of the circumference of the pitch circle. Let P = circular pitch of the teeth in inches ; D = pitch diameter, i.e., diameter of pitch circle in inches ; N = number of teeth ; TT = ratio of circumference of a circle to its diame- ter = 3.141C. Then NF = ttD, and hence N=lD, I)=^N, P=%^. F IT N From the above relatione, we may evidently find any one of the three elements F, D, N; the other two having been given by the problem. MOTION BY SLIDING CONTACT. 81 For convenience in calculation, the following table is ap- pended, in whicli the pitch diameters are calculated for a pitch of one inch. PITCH DIAMETERS. FOR ONE INCH CIRCULAR PITCH. No. of Pitch No. of Pitch No. of Pitch No. of Teeth. Pitch Teeth. Diameter. Teeth. Diameter. Teeth. Diameter. Diameter, 9 2.86 32 10.19 55 17.51 78 24.83 10 3.18 33 10.50 56 17.83 79 25.15 11 3.50 34 10.82 57 18.14 80 25.46 12 3.82 35 11.14 58 18.46 81 25.78 13 4.14 30 11.40 59 18.78 82 26.10 14 4.40 37 11.78 60 19.10 83 26.42 15 4.77 38 12.10 61 19.42 84 26.74 10 5.09 39 12.41 62 19.74 85 27.06 17 5.41 40 12.73 03 20.05 86 27.37 18 5.73 41 13.05 64 20.37 87 27.09 19 0.05 42 13.37 65 20.69 88 28.01 20 6.37 43 13.69 66 21.01 89 28.33 21 6.08 44 14.00 67 21.33 90 28.65 22 7.00 45 14.32 68 21.05 91 28.97 23 7.32 40 14.04 69 21.96 92 29.28 24 7.64 47 14.90 70 22.28 93 29.60 25 7.96 48 15.28 71 22.60 94 29.92 26 8.2S 49 15.60 72 22.92 95 30.24 27 8.59 50 15.92 73 23.24 96 30.56 28 8.91 51 16.23 74 23.55 97 30.88 29 9.23 52 16.55 75 23.87 98 31.19 30 9.55 53 16.87 76 24.19 99 31.51 31 9.87 54 17.19 77 24.51 100 31.83 This table is used in the following manner : — 1. Given the circular pitch and the number of teeth, to find the pitch diameter. Take from the table the diameter 82 ELEMENTARY MECHANISM. corresponding to the given number of teeth, and multiply this tabular diameter by the given pitch in inches. The product will be the required pitch diameter in inches. 2. Given the pitch diameter and the number of teeth, to find the pitch. Take from the table the diameter correspond- ing to the given number of teeth, and divide the e/iven pitch diameter by this tabular diameter. The quotient will be. the required pitch in inches. 3. Given the pitch and the pitch diameter, to find the number of teeth. Divide the pitch diameter by the pitch ; and, taking the quotient as a tabular pitch diameter, find from the table the number of teeth corresponding to this tabular diameter. If the latter is not found in the table, the pitch assumed is not an aliquot part of the pitch circumference, and must be altered slightly so as to agree with the number of teeth corresponding to either the next larger or next smaller tabular diameter. 88. Diametral Pitch. — It has been shown in the last article, that the relation between the circular pitch, the pitch diameter, aud the number of teeth, introduces the incon- venient number 3.1416. As the number of teeth must be an integer, and as the pitch is usually taken some convenient part of an inch, it follows that the pitch diameter will very often contain an awkward decimal fraction. This may be obviated by the use of the diametral pitch, which is being rapidly introduced in this country. As the circular pitch is obtained by dividing the pitch circumference by the number of teeth, so another ratio may be obtained by dividing the pitch diameter by the number of teeth. In practice, it is found more convenient to invert this last ratio ; and, when so inverted, it is called the diametral pitch, though theoretically that designation would more prop- erly belong to the ratio as it stood before inversion. In other words, we define diametral pitch to be the number of teetli per inch of pitch diameter. Thus, a wheel which has 8 teeth MOTION BY SLIDING CONTACT. 83 per inch of pitch diameter, is spoljen of as an " 8-pitch " wheel. The chief merit of this system, and one which entitles it to great favor, is, that it establislies a convenient and manage- able relation between the pitch diameter and the number of teeth ; so that the calculations are of the simplest descrip- tion, and the results convenient and accurate. Let M= diametral pitch ; then we have MP =3.1416, or the product of the circular and the diametral pitches is the number 3.1416. In this system, the number of teeth and the pitch diameter are so related that the circular pitch is usually some decimal ; but this is of slight importance, as the circular pitch is rarely set off by actual measurement, but usually by dividing the pitch circle into the required number of parts. To find the number of teeth in any wheel, multiply the diametral pitch by the pitch diameter. For instance, an 8-pitch wheel of 12 inches pitch diameter has 8 x 12 = 96 teeth. Again : to find the pitch diameter, divide the number of teeth by the pitch. Thus, a 6-pitch wheel of 25 teeth has a pitch diameter of ^- = 6:^ inches. In the comparison of circular and diametral pitches, the following table will be found useful : — A B A B A B A B i 12.56 If 1.80 34- 0.90 7 0.45 * 6.28 ■ 2 1.57 4 0.78 8 0.39 * 4.20 2i 1.40 44 0.70 9 0.35 1 3.14 2* 1.25 5 0.63 10 0.31 n 2.50 2f 1.15 54 0.58 12 0.26 14 2.10 3 1.05 6 0.52 16 0.20 Find the given pitch, circular or diametral as the case may 84 ELEMENTARY MECHANISM. be, in column A ; then the equivalent pitch in the other sys- tem will be found opposite in column B. In this volume, circular pitch is always meant when the word " pitch " is used without further qualification. MOTION BY SLIDING CONTACT. 85 CHAPTER VI. COMMUNICATION OF MOTION BY SLIDING CONTACT. VELOCITY RATIO CONSTANT. BIRECTIONAL RELATION CONSTANT. TEETH OV WHEELS (CONTINUED). Definitions. — Angle and Arc of Action. — Epicycloidal System. — Inter changeable Wheels. — Annular Wheels. — Customary Dimen- sions. — Involute System. 89. Teetli. Definitions. — That part of the front or acting surface of a tooth which projects bej'ond the pitch surface is called the face., and that part which lies within the pitch surface is called the j^anZc. The corresponding portions of the back of a tooth may be called the hack face and the baxik flank. The face of a tooth in outside gearing is always convex ; \he flank may be convex, plane, or concave. By the pitch point of a tooth is meant the point where the pitch line cuts the front of the tooth. In Fig. 72, let the front or acting surface of the teeth be to the left. Then 6, A;, are the pitch points of the teeth; ab is the /ace; hm is the flank; de is the ba.ck face; en is the back flank. The depth, AD, of a tooth is tlie radial distance from root to top ; that portion of the top of a tooth which projects be- yond the pitch surface is called the addendum, AB ; and a line drawn parallel to the pitch line, and touching the tops of all the teeth of a wheel or rack, is called the addendum line, or, in circular wheels, the addendum circle, adA. The radius 8b ELEMENTARY MECHANISM. of the pitch circle of a circular wheel is called the geometrical or p«te/i radius ; that of the addendnm circle is called the 7-eal radius ; their difference is evidently the addendum. Clearance is the excess of the total depth above the work- ing depth ; or, in other words, the least distance between the top' of the tooth of one wheel and the bottom of the space between two teeth of another wheel, with which the first wheel gears. Bacldash is the excess of the space between the teeth of one wheel over the thickness of the teeth of another wheel, with which the first wheel gears. The amount of backlash depends on the accuracy with which the teeth are constructed, and should always be made as small as possible. For our present purposes we may neglect it altogether. 90. Angle and Arc of Action. — The angle through which a wheel turns, from the time when one of its teeth comes in contact with the engaging tooth of another wheel until their point of contact has reached the line of centres, is called the amgle of approach; the angle through which it turns from the instant that the point of contact leaves the line of centres until the teeth quit contact, is called the angle of recess. The sum of these two angles is called the angle of action. The arcs of the pitch circles which measure these angles are called the arcs of approach, recess, and action respectively. The corresponding arcs must evidently be the same in both pitch circles, while the corresponding angles are proportional to the velocity ratio ; in other words, in- ^versely proportional to the diameters of the pitch circles. In order that one pair of teeth may continue in contact until the next pair begin to act, the arc of action must be at least equal to the pitch arc, and in practice it ought to be considerably greater. Now, in practice, the friction which takes place between surfaces whose points of contact are approaching the line of centres is found to be of a mucli more vibratory and injurious MOTION BY SLIDING CONTACT. 87 character than that which takes place while the points of contact are i-eceding from the line of centres. It is therefore expedient to avoid the first kind of contact as much as possible. 91. Teeth with Faces or Flanks only In Fig. 65, let A and li be the centres of driving and following wheels Fig. 65 respectively. Let T be found as usual. Draw the pitch circles MN and RS, and assume a describing circle of any radius CT, less than ^. Let Ta, Tb, be the pitch arcs, and lay off on the describing circle the arc TP equal to arc Tb. If we roll the describing circle first on the outside of MN, and then on the inside of BS, the point P will de- scribe the epicycloid Pa and the hypocycloid Pb respectively 88 ELEMENTARY MECHANISM. (Art. 74) . Pa will then be the face of a tooth of the driver which will gear correctly with the flank Pb of a tooth of the follower. Bisect Ta in H, and draw through H a reversed face, HP, similar to Pa. The acting outlines of all the teeth of the driver may now be completed by laying off the distance Ha all around the pitch circle MN, and drawing through the points so found a series of equal and alternately reversed faces similar to Pa. Pb is all of the hypocycloid that comes into contact with the face Pa ; but, in order to provide room for the point of the latter, the hypocycloid is continued to D, making the depth of the teeth the same in each wheel. By laying off the half pitch arc all around the pitch circle RS, and drawing a series of equal and alternately reversed flanks similar to Dh, we will get the outlines of the follower's teeth. The entire outlines of the teeth of both wheels are then completed by connecting, by circular ares concentric with the respective pitch circles, the bottoms of the faces of the driver, and the tops and bottoms of the follower. In this construction the acting outlines first come into contact at T on the line of centres. The driver moving as indicated by the arrow, the point of contact travels along the describing circle in the arc TP, until it reaches P, where the teeth quit contact. The line of action at any moment is the straight line from T to the point of contact. There is no arc of approach, and the arc of recess is exactly equal to the pitch Ta-; that is to say, one pair of teeth quit contact at P at the same moment that another pair come into contact at T. This case is therefore a barely possible one ; the driver's tooth being pointed, and just high enough to make the arc of action equal to the pitch. Pa, then, is the neces- sary length of face to secure this arc of action. Drawing the radial line PA, and calling K the intersection of tliis radius with the pitch circle MN, we see that in this case Ka is just half the thickness of the tooth ; in other words, just MOTION BY SLIDING CONTACT 89 one-quarter of the pitch. Had Ka been greater than half ifo, HK must have been less, so that the reversed face through H would not have intersected the face Pa at P, but at some point between P and a, and the case would have been impracticable with the pitch assigned. But if Ka had been less than half the thickness of the tooth, as required by ;F'ig. 66 the pitch assumed, the teeth could have been made higher by extending the faces above P, or they might be left of tlie same height, and given some thickness at the top, as in Fig. 66. 92. Practical Example Teeth snch as shown in Fig. 65 may, of course, be constructed,- but with the slightest wear at the point of the driver's tooth the action will be deranged, for the face will no longer be of the requisite length ; and though its extremity will drive the flank of the 90 ELEMENTARY MECHANISM. follower's tooth before it, yet it will not do so with the con- stant velocity ratio required. In Fig. 64 this difficulty is obviated, and teeth formed as there shown are successfully used in practice. The diagram is drawn full size, and shows the practical solution of the following problem : — Distance between centres of pitch circles, 9 inches. Driver (lower wheel) to have 40 teeth ; follower, 50 teeth. Arc of recess = 1^ times the pitch. Divide line of centres AB at T so that = — = — = -. Hence the radius of the pitch JSr a 50 5 '■ circle MN = 4 inches, and that of the pitch circle BS is 5 inches. Let the driver move as indicated by the arrow. Take the diameter of the describing circle = f of that of the pitch circle BS of the follower = f X 10 = 3J inches. Find the pitch by dividing the circumference of MN into 40 equal parts, and lay off the arc Ta = 1^ x the pitch so found. Lay off the arcs TP = Th = Ta, also Ha = i pitch. Roll the describing circle on the outside of MN and on the inside of BS, describing the epicycloid Pa and the hypocycloid Pb respectively. Drawing a radial line from P to the centre of MN, we find Ka to be less than ^ Ha ; hence the case is a practicable one. Through H draw an epicycloid HE similar to Pa, but reversed in position ; through P draw an arc of a circle PE concentric with MN, and cutting HE at E. Lay off bF = Ha, through F draw a reverse hypocycloid similar to Pb, and join F and b by an arc of the pitch circle BS. Now, Pb is all of the hypocycloid that comes into contact with the epi- cycloid Pa ; but, in order to provide room for the point of the latter, the hypocycloid is continued to D, just as was done in Fig. 65. If the workmanship were accurate, the wheels would work properly, provided the depth of the space between two sue- MOTION BY SLIDING CONTACT. 91 cessive teeth of one wheel were just equal to the height of the teeth of the other. To provide against any accidental contact, however, both sets of teeth are given clearance; that is, the bottoms of the spaces between the teeth are formed by arcs of circles concentric with MN and RS respectively, and at such a distance as to leave a clearance of about one- tenth the pitch in both wheels. The outlines of the teeth are then completed by joining the bottoms of the epicycloids and hypocycloid previously drawn, to these arcs by means of small fillets, as shown in the figure. The teeth will come into contact at T, the point of contact travelling in the arc TP, until it reaches the point P, where the contact ceases. It is evident that, before any one pair quits contact at P, another pair will have been in contact while the wheels were moving over one-third the arc of action. 93. Use of sncli Wheels. — From the considerations concerning friction (Art. 89), it is advisable to employ as little arc of approach as possible ; and hence, when wheels with such teeth are used, the one whose teeth have /aces only should always be the drivar. At the moment of quitting contact at P, the line of action of these tooth surfaces is the straight line passing tln-ough T and P. As the arc of action increases, the face of the tooth becomes longer, and this line of action becomes more oblique ; that is, its inclination to the common tangent of the two pitch circles increases. Now, as these teeth must transmit pressure as well as motion, it follows that the greater this obliquity, the greater will be the component of the pressure in the line of centres producing wear and friction in the bearings. As the arc of action increases, the percentage of sliding also increases ; so that, upon the whole, such wheels are bet- ter suited for use in light mechanism, where smoothness of action is more important than the transmission of heavy pressures. 92 ELEMENTARY MECHANISM. 94. Teeth with hoth Faces and Flanks. — Where either wheel is to drive, or where heavy pressures are to be iT'ig. 67 transmitted, such wlieels as above described are evidently unsuitable. To produce their teeth, we rolled a describing circle so as to generate an epicycloid on the outside of J/iV, and a hypocycloid on the inside of US. If, now, in addition, we roll another describing circle so as to generate an epicy- MOTION BY SLIDING CONTACT. 93 cloid on the outside of BS, and a hypocycloid on the inside of MN, it is evident that the teeth of each wheel will have both faces and flanks, and that the faces of the teeth of each wheel will work correctly with the flanks of those of the other. The method of construction is shown in Fig. 67. Let MN and ES be the driving and following pitch circles, as before, drawn about the centres A and B. The epicycloid Pa and the hypocycloid Pb are generated, as in Fig. 65, by the point P of the describing circle whose radius is CT. To complete the teeth, the hypocycloid P'a' and the epicycloid P'b' are generated by means of the point P' of the describing circle whose radius is LT. Continue the hypocycloids Pb and P'a' to D and G', to give clearance, as previously ex- plained. Draw bQ = b'P'; aG = a'G'a'; V=Pa; and b'D' = bD. We now have the complete tooth outline for one side of each tooth, shown in two positions ; on the left at the moment of engaging in contact at P', and on the right at the moment of quitting contact at P. The action begins at P' ; the driver's flank pushing the face of the follower, and the point of contact moving in the arc P'T, until the points a' and b' come in contact at T. The face of the driver then pushes the flank of the follower, the point of contact now travelling in the arc TP ; and at P the action ends. P' and P may be assumed at pleasure on the circumfer- ences of the respective describing circles, and will fix the lengths of the arcs of approach and recess ; viz., Tb' = Ta' = TP'; and Ta = Tb = TP. The arc of approach evidently governs the length of face of the follower's teeth, and the arc of recess the length of face of the driver's teeth. If each wheel is to be used indiscriminately as driver or follower, the arcs of approach and recess are made equal; but, if the same wheel is always to drive, the arc of recess, for the sake of freedom from vibratory motion, is usually made the greater. 94 ELEMENTARY MECHANISM, 95. Size of Describing Circle. — The lengths and shapes of the faces and flanks of the teeth of the wheels, with given arcs of approach and recess, evidently depend on the relation between the diameters of the pitch and de- scribing circles. If, in Fig. 67, the diameter of the upper describing circle were increased, the face Pa would become sliorter, and the curvature of both Pa and Pb would decrease, until, when the diameter of the describing circle became just equal to the radius of RS, the hypocycloid Pb would become a straight line passing through the centre of the pitch circle RS (Art. 75). This fact is often taken advantage of in laying out teeth. When the diameters of both describing circles are thus taken equal to the radii of the pitch circle in which they roll, the flanks of the teeth of both wheels become radial lines, while the faces remain epicycloids. The consequent reduction in the labor of laying out the shape of such teeth has led to their extensive introduction ; though, in conse- quence of the convergence of their radial flanks, they have the disadvantage of being comparatively weak at the root. If the diameter of the describing circle be made still larger, the hypocycloidal flanks will converge still more as they recede from the pitch circle, making the tooth still weaker at the root. Though describing circles have been successfully used having a diameter five-eighths as great as that of the pitch circle in which they roll, yet it seems a good practical rule to make the radial flank the limit in this direction. The smaller the describing circles, the longer will be the faces of the teeth, and the greater will be the consequent obliquity of action ; but, on the other hand, the stronger will be the tooth. We thus have the two conflicting conditions of obliquity of action and strength of teeth, and the size of the describing- circle will be regulated in each ease by their relative impor- tance. A good general rule, which is found to work well in practice, is to make each describing circle of a diameter MOTION BY SLIDING CONTACT. 95 equal to three-eighths of the diameter of the pitcli circle in which it rolls. 96. Relation between Pitch and Arcs of Approach and Recess — To find the limiting values of the pitch which will secure a given arc of approach or recess, the diameters of pitch and describing circles l)eing given. In Fig. 67, let MN, RS, be the pitch circles, and CT the radius of the upper describing circle. Lay off Ta = Tb, the arcs of recess desired. Lay off the arc TP = Tb, thus fixiho- the position of P. Describe the epicycloid Pa, and draw PA, cutting MN in K. Now, as previously explained, if Kd is equal to or less than half the thickness of the tooth, • — in other words, if Ka is equal to or less than one-fourth the pitch, — the construction is possible. Hence the pitch of the teeth of the driver must be equal to or greater than four times Ka. If it is just equal to four times Ku, the teeth will be pointed ; if greater, they will have some thickness at the top. Let Ta' = Tb' be the given arc of appi'oach ; then, by a similar construction, we find that the pitch of the teeth of the follower must be equal to or greater than four times K'b'. Agam : it is evident that the pitch of the driver's teeth cannot be greater than the arc aa' ; for, if it were, one pair of teeth would quit contact at P before the next pair would come into contact at P'. Similarly, the pitch of the follower's teeth cannot be greater than the arc bb'. But aa' = bb' = total arc of action. The pitch of the teeth of both wheels must evidently be the same ; hence wc find, that, to secure the desired arcs of approach and recess, the pitch must not be greater than the total arc of action, nor less than either 4Ka or iK'b'. The pitch being given, to find the arcs of approach and recess, draw a radius of MJSf, and lay off on MJSf, from the point where the radius intersects the latter, an arc = ^ pitch. Through the point so found draw the epicycloid which would 96 ELEMENTARY MECHANISM. be formed by rolling the describing circle CT on MN, until it meets and intersects the radius at some point. Through this point of intersection draw a circular arc concentric with MJSf •, where the latter cuts the describing circle will be the point P, and the arc of recess will be determined on the sup- position that the teeth are pointed. If they are not pointed, let X be the addendum ; then a circular are with radius AT + X will cut the describing circle at the point of quitting con- tact, P, as before. The arc of approach is found in a similar manner. 97. Practical Example. — In Fig. 68 is shown the method of laying out a pair of such wheels. The diagram is drawn full size, and is the practical solution of the follow- ing problem : Distance between centres to be 9 inches. *The driver (lower wheel) to have 40 teeth, and the follower 50 teeth. Arc of approach to be equal to the pitch, and the arc MOTION BY SLIDING CONTACT. 97 of recess to be one and a half times the pitch. The condi- tions given are the same as those given in Art. 92, except that there is to be an arc of approach in this case. Tlie pitch radii of the wheels are 4 and 5 inches, as before ; and the diameters of the respective describing circles are 3 and 3f inches. The faces of the driver's teeth and the flanks of the fol- lower's teeth are found as in Art. 92, and are, in fact, iden- tical with those there found. In this case, however, we do not finish off the bottoms of the faces of the driver's teeth and the tops of the flanks of the follower's teeth by arcs of circles, as is done in Fig. 66. Lay off the arc TP' = Tb' = arc of approach. Using the describing circle of three inches diameter, and going through the process explained in Art. 94, we obtain flanks for the driver's teeth, and faces for those of the follower. By this construction, as shown in Fig. 66, there are three pair of teeth in contact ; one just quitting contact at P, another in contact at p, and a third pair at j)'- In practice, after we have determined that the given arcs of action may be secured with the given pitch (Art. 96), the four curves are usually laid down at T, as shown ( Td and Tg being epicycloids, and Te and Th hypocycloids) . The addendum circle bounding the tops of the teeth, and the root circle bounding the bottoms of the spaces, are next drawn. The pitch points of the teeth are then laid off on the respective pitch circles, and the re- spective curves are drawn through the successive pitch points in alternately reversed directions, being limited at the top by the addendum circle, and connected at the bottom by fillets to the arcs of the root circle. 98. Interchangeable Wlieels. — If the describing circle be made of a diameter bearing a fixed ratio to that of the pitch circle, any pair of wheels so laid out will work together ; but they cannot both work properly with a third wheel of different diameter. Thus, a given wheel having radial flanks 98 ELEMENTARY MECHANISM. cannot work properly with two or more otlier wheels of dif- ferent diameters, and also having radial flanks. If, however, we use the same describing circle for all tlie faces and all the flanks, we will obtain a series of inter- changeable wheels, any one of which will work correctly with any other of the same set. This suggestion is due to Pro- fessor Willis, and this method of laying out teeth is invaluable for such purposes as constructing the change- wheels of a lathe. As, with a constant describing circle, the outlines of the teeth will vary with the diameters of the wheels, so as to make the obliquity of action greater as the latter increases, it is usually advisable to employ as large a describing circle as possible. From the considerations discussed in Art. 95, the practical rule follows, that, for a set of interchangeable wheels, the diameter of the constant describing circle should be half the diameter of the pitch circle of the smallest wheel of the set. 99. Kack and Wheel. — "When a wheel works with a rack, the line of centres becomes a perpendicular to the pitch line of the rack, and passing through the centre of the wheel. The rack will travel through a distance equal to the circum- ference of the pitch circle of the wheel for each revolution of the latter, whatever the number of teeth. The pitch of the rack teeth, therefore, is found by rectifying the pitch are of the wheel, and laying off this rectified arc upon the pitch line of the rack. In Fig. 69 the two describing circles are made of the same diameter, so that any other wheel of the same pitch whose tooth outlines are formed by means of the same describing circle will also gear with the rack. In fact, the rack is merely a special case of the wheel ; and all the deductions of the previous articles as to tooth outlines, arcs of action, etc., apply, with obvious modifications, to this case as well. Both faces and flanks of the rack teeth are cycloids (Art. 85) : their tops and bottoms are straight lines. The clearance is obtained as usual. MOTION BY SLIDING CONTACT. V\) In the figure, which is drawn full size, the diameter of pitch circle of the wheel is four inched, and the wheel has forty teeth. The arcs of approach and recess are each made CO equal to the pitch. Assuming the -rack to drive to the right, the contact begins at P', the point of contact travelling along the arcs P'T and TP; and at P the action ends. 100 ELEMENTARY MECHANISM. The principle of making teeth with straight flanks may, of course, be extended to the case of a rack and wheel, as shown in Fig. 70. Tlie describing circle whose diameter is TB, the radius of the wheel, generates the cycloidal faces of the rack teeth and the radial flanks of the wheel teeth. The radius of the rack being infinite, the diameter of the other describing circle is also infinite ; in other words, it is a straight line. B'ig, 70 Hence the faces of the wheel teeth are evidently involutes of the pitch circle, while the flariks of the rack teeth are straight lines perpendicular to MJSf. The arcs of action and the addendum of the rack teeth are found as before. The rack driving to the right, the contact begins at P' (the point of intersection of the wheel addendum circle with the line MN) , travels along the straight line P'T, then along the arc TP to the point P (tlie intersection of the rack addendum line with the describing circle) , where the teeth quit contact. In this form of rack tooth the acting flank has degenerated into a mere point, which is consequently subjected to excessive MOTIQN BY SLIDING CONTACT 101 wear. This is a serious defect, and forms a grave objection to the use of this form of tooth for racks. 100. Auniilar Wheels — The construction explained in Art. 94 is applicable not only to the case of wheels in exter- nal gear, as there shown, but to that of wheels in internal gear as well. In Fig. 71 the smaller pitch circle lies within the greater ; two describing circles are used, as before. In fact, on comparing this diagram with Fig. 67, both figures 102 ELEMENTARY MECHANISM. being similarly lettered, we will see that all the details of construction are the same in both. The pinion is an ordinary spur wheel ; while the acting curves of the annular wheel are identical with those of a spur wheel, having the same pitch and describing circles, the tooth of the one corresponding to the space of the other. The principle of interchangeability (Art. 98) applies to annular wheels just as to spur wheels. Thus, a set of spur and annular wheels may be made in which each spur wheel will gear not only with every other spur wheel, but also with every annular "wheel. In this case, however, there must be a difference in the number of teeth of the spur and annular wlieels which are to gear together, at least equal to the num- ber of teeth on the smallest pinion of the set. lOl. Customary Dimensions of Teeth. — By the pre- ceding methods we may design the teeth of gear wheels so as to fulfil any proposed conditions as to the relative amounts of approaching and receding action. In the majority of cases, however, the precise lengths of the arcs of approach and recess are not a matter of importance ; and under these cir- cumstances it is customary to make the whole radial height of the tooth a certain definite fraction of the pitch, the part without the pitch circle being a little less than that within, by which clearance is provided for. There are a number of such arbitrary proportions ; but none of them can be considered absolute, as the proper amount of clearance and backlash evidently depends on the precision with which the tooth curves are laid out, in the first place, and on the accuracy with which the shapes of the teeth are made to conform to the curves so found. In the manufacture of the best cut gears at the present day, the backs of the teeth barely clear each other when the fronts are in contact ; but in the majority of cases a greater allowance is still made, depending for its amount on the accu- racy of the workmanship. In cast wheels backlash is abso- MOTION BY SLIDING CONTACT. 103 lutely necessary to allow for irregular shrinkage or accidental derangement of the mould. d. 1 Fig. 7» In Fig. 72, let bk = circular pitch = P. Then, accord- ing to several systems in general use for proportioning teeth, we have the following values : — Total depth . . AD -f-sP O.loP HP 0.750P Clearance . . . CD -^^P O.OoP AP 0.060P + 0.04 in. Working-depth . AC AC ■A-P 0.70P UP 5| 0.6S0P - 0.04 in. Addendum, AB = 2 ^\P 0.35P rs^ 0.34.5P — 0.02 in. Thickness of tooth , be t^-P 0.45P -hP 0.470P - 0.02 in Width of space . Ice AP 0.55P ■ftp 0.530P + 0.02 in. Backlash . . Ice -he T^fP O.lOP A-P 0.060P + 0.04 in. In the first three systems the percentage of backlash is constant, the actual amount of backlash thus increasing directly with the pitch. It seems more rational, however, to make the percentage of backlash greater for small pitches than for large ones ; for, the coarser the pitch, the smaller will be the proportion borne to it by any unavoidable error. The last system, that of Fairbairn and Rankine, is founded on this view of the proper proportion of backlash. In this 104 ELEMENTARY MECHANISM. system the percentage of backlash gradually diminishes as the pitch increases. The actual amount as given by this system is, however, rather larger than is generally used at present. Teeth proportioned by any of these systems will in general be of good shape, and answer the purpose desired. Should the wheel have less than about twelve teeth, or should the exact amount of approaching or receding action be of importance, no arbitrary system should be used. In all such cases the proper dimensions of the teeth should be found as previously explained. Tiie backlash and clearance should always be made as small as the character of the workmanship will permit. In our diagrams we have assumed no backlash to exist ; but its introduction would have no effect, except to diminish tlie thickness of the tooth. Instead of half the pitch, as it is in the diagrams, the thickness of the tooth would be half the pitch minus half the backlash. In using the diametral pitch, the working depth of a tooth is almost always taken at two pitch parts of an inch, and the addendum at one pitch part of an inch. That is, in a 4-pitch wheel, the working depth is J = ^ inch, and the addendum is ^ inch. The clear- ance and backlash are taken at from a fourth to an eighth of one pitch part of an inch ; thus, in a 4-pitch wheel, they would be taken at from ( = — I to ( = — | of an inch. 4 X 4V 10/ 8 X 4\ 32/ The simplicity of these proportions have led to their almost universal adoption whenever the diametral pitch is employed. 102. Involute Sy.stem. — It has been shown (Art. 80) that involutes of certain circles possess the property of trans- mitting motion by sliding contact with a constant velocity ratio, and the application of such curves to the formation of the teeth of wheels is shown in Fig. 73. Let AB be the line of centres, divided at T, so that A T' ^ Tj— = = — . Draw the pitch circles MN and JRS, and their J} 1 a common tangent t'Tt. Draw P'TF, making an oblique angle MOTION BY SLIDING CONTACT. 105 PTt with the tangent IfTt. From the centres A and B drop the perpendiculars AP' and BP on the line P'TP\ and, with these perpendiculars as radii, describe the circles M'N' and R'S', which will be tangent to the line P'TP. E'ig.ra With these circles as base circles, describe the involutes a"P and PQ in contact at P. Through T and P' draw invo- 106 ELEMENTARY MECHANISM. lutes equal to a"P and PQ respectively. The driver, MN, moving in the direction of the arrow, the involutes are shown in three positions : at P' at the moment of coming in con- tact, at T in contact on the line of centres, and at P at the moment of quitting contact. The tooth outline is evidently one continuous curve, there being no marked division into face and flank ; and, as the curves cannot extend inside their own base circles, they are in this diagram of their maximum length. From the-figure, ■we have the ratio Arc of approach ^ P^ ^ P^ ^ ITA. ^ AT Arc of recess Oa" TP PB BT In other words, if each involute be long enough to touch the other at its root, the arcs of approach and recess will be to each other in the direct ratio of the radii of the base circles, or pitch circles, of the driver and follower. In order that the arc of recess may be greater than that of approach in this case, the smaller wheel must evidently be the driver. But, by shortening the curves of driver or follower, we may adjust the ratio, of these arcs at pleasure, and thus secure just the proportion of approaching and receding action that we desire, whether the driver be the smaller or the larger wheel. The maximum values of these arcs is, of course, given by the construction in Fig. 73. 103. Given the pitch circles, the obliquity of the line of action, and the desired arcs of approach and recess, to find the limiting values of the pitch which will secure these arcs of action. The receding action evidently continues while the point of contact travels from 7" to P in the line TP, a dis- tance equal to the arc Oa" . The curves OT/Jaud a"aP being equal involutes of M'JSf', and the points T, a, lying in the circumference of the circle 'MN, concentric with that of M'N' which contains the points MOTION BY SLIDING} CONTACT. 107 0, a", it follows that the angle TAa = angle OAa", and are Oa" AP' „ ^ „ AP „ '^F' = T^- Hence Oa!' = — — - x Ta. are Ta AT AT On the tangent Tt lay off the distance Td = Ta ; from d draw a perpendicular to TP. Then, from the similar trian- gles TAP' and TOP, we will have TP = — x Ta = Oa", as required. Draw the radius PA, cutting MN in K. Now, the pitch cannot be less than 4/fa. If it is just equal to 4/ia, the teeth will be pointed ; if greater, they will have some thick- ness at the top. Similarly, the pitch cannot be less than 4K'b. Hence we find that the pitch cannot be greater than the total arc of action nor less than either 41{a or AK'b. 104. Given the pitch circles, the obliquity of the line of action, and the pitch, to find the arcs of approach and recess. From T lay off on the circle MN the arc TV = i pitch, and draw the radius AH. Through T draw the involute OTH of the base circle M'N', and prolong it till it cuts the radius at H. Through H draw the circular arc HP, cutting the circle R'S' at P. Then P will be the point at which the AT teeth will quit contact, and Ta = — — X TP will be the arc of recess. This is only true if the teeth are pointed ; if they are not, let x be the addendum. Then a circular arc struck about A, with radius AT + x, will cut R'S' in the point where the teeth quit contact, as before. The arc of approach is determiued in a similar manner. 106. Practical Example. — Fig. 74 is drawn full size, and is the practical solution of the following problem : — Distance between centres, 9 inches. Driver (lower wheel) to have 40 teeth ; follower, 50 teeth. The constant obliquity of the line of action to be 15°. Draw the pitch circles MN and RS with radii of 4 and 5 inches respectively, and their common tangent t'Tt. Draw the line of action DE, making 108 ELEMENTARY MECHANISM. the angle ETt = 15°. On DE drop the perpendiculars AD and BE, with which, as radii, describe the base circles M'N' and R'S'. The arcs of approach and recess in this problem are each to be equal to the pitch. Hence lay off the pitch arc Ta ; and lay off, on the line of action, the distance TP = — =- x AT Ta. Then P is the point at which the teeth quit contact. Fig. 74 As the arcs of approach and recess are to be equal, lay off TP' = TP. Then P' is the point at which the teeth first come in contact. Through P draw the involute Pa" of the base circle M'N', and through P' draw the involute P'h" of the base circle R'S'. Draw the addendum circles through F' and P, lay off the pitch points of tlie teeth around the pitch circles MN and liS, and draw through the points so found, in alternately reversed positions, the involutes Pa" and P'b" respectively. The tops of the teeth are bounded by arcs of the respective addendum circles. To provide clearance, continue the tooth MOTION BY SLIDING CONTACT. 109 outlines from the bottoms of the invohites by radial lines to the proper depth. The bottoms of the spaces are circular arcs concentric with the centres of motion, and joined to the tooth outlines by means of small fillets, as shown. In these wheels, there are evidently always two pairs of teeth in contact. In the position shown, there is one pair in contact at T on the line of centres, while a second pair is quitting contact at P at the same moment that a third pair is engaging in contact at P'. 106. Interference of Involute Teeth. — So long as the teeth are of such a length that the points P and P' {¥\g. 74) lie between E and D, they will work properly. In other words, the addendum circle of the teeth of the lower wheel must lie within a circle through E, and concentric with MN. Also, the addendum circle of the teeth of the upper wheel must lie within a circle through D, and concentric with liS. But when the dimensions of teeth are decided on by means of some arbitrary system, such as those of Art. 101, it fre- quently happens that the length of tooth so found will be great enough to cause the addendum circles to lie outside of the concentric circles through E and D respectively. It fol- lows, that the part of the tooth projecting beyond tiiis limiting circle will come into contact with that part of the tooth of the other wheel which lies within the base circle. As this inner part is always made radial, it cannot gear correctly with an involute face, and interference will take place. In case a tooth of such length is considered necessary, and the involute system is to be used, all that part of the face of the tooth of one wheel coming into contact with the radial part of tiie tooth of the other wheel must be an epicycloid whose de- scribing circle is half the diameter of the pitch circle of the second wheel. As, by this means, we forfeit one of the great advantages of the involute system (the power of varying the distance between centres without affecting tlie velocity ratio), this construction is not to be recommended, and the length of 110 ELEMENTARY MECHANISM. tooth should not be allowed to exceed the amount determined by the methods of the preceding articles. 107. Jiack and Wheel. —If, in Fig. 74, the radius AT of the driver were to increase, the curvature of MN, as well as that of the involute of M'N', would necessarily decrease; until, when MN became a straight line, M'N' would also become a straight line, and the involute of M'N' would be- come a straight line, which must be perpendicular to the line of action, DTE. Fig. 75 The method of constructing the teeth is exactly similar to that shown in Fig. 74. In Fig. 75 the rack is the driver, and the follower is the same wheel that was used as follower in Fig. 74. The teeth of the follower remain the same as in the other case, while those of the rack have straight sides. The tops and bottoms of the rack teeth ai-e straight lines parallel to pitch line MN of the rack. In order to drive the follower through one complete revolution, the rack will evidently have to travel a distance equal to the circumference of tlie pitch circle of the wheel. The construction of the teeth of annular wheels is also in all respects similar to that explained above for spur wheels. MOTION BY SLIDING CONTACT. Ill 108. Peculiar Properties of Involute Teeth In the preceding constructions of practical problems, the line of action was drawn at an angle of 15° with the common tan- gent of the two pitch circles. This angle is by no means fixed, and may be considerably varied ; but experience has shown that for general practice it should not be greater than fifteen degrees. As the magnitude of this angle has formed no part of the argument in the preceding cases, it follows that, by varying the obliquity of action, an infinite number of pairs of base circles may be used in connection with any giv^n pair of pitch circles. Conversely, with a given pair of base circles, we may, by altering the length of the line of centres, have an infinite number of pairs of pitch circles. The common tangent to the two base circles will always cut the line of centres into segments having the same ratio as their radii, which will be the same as that of the radii of any of the pairs of pitch circles ; from which follow two impor- tant practical deductions : — 1. Any two wheels with involute teeth of which the pitch arcs on the base circles are equal, will gear correctly with each other. 2. The velocity ratio will not be affected by any change in the distance between their centres. The peculiarity of interchangeability is also obtainable with epicycloidal teeth under certain conditions (Art. 98). The peculiarity of constant velocity ratio with varying dis- tance between the centres is not found in any other form of teeth, and is of special importance in Tnechanism requiring exceptional smoothness and uniformity of action. The shiifts may be at the proper distance apart, or not, as happens ; and they may change position by wearing, or by variable adjustment, as when used on rolls, or they may be brought closer together to abolish backlasli. In fact, the involute tooth is remarkably well adapted to such variable demands, and will accommodate itself to errors and defects that are difficult to avoid in practice. 112 ELEMENTARY MECHANISM. The line of action of epicycloidal teeth is perpendicular to the line of centres at the instant when the point of contact is on that line ; but that of involute teeth is constantly in the direction of the common tangent of the two base circles, and hence always oblique to the line of centres. The obliquity of involute teeth, then, is constant ; and it is, in general, greater than the mean obliquity of epicycloidal teeth having the same angle of action. The thrust on the bearings is therefore greater with involute than with epicycloidal teeth ; and though for heavy pressures this is sometimes a serious objec- tion to tlie use of involute teeth, yet for ordinary work it would scarcely be so considered. The involute tooth has a great advantage over the epicy- cloidal tooth in being of a much stronger shape, spreading considerably at the root, which in the epicycloidal form is often the weakest part. Though the epicycloidal tooth is still in much greater use than the involute tooth, yet the merits of the latter are being rapidly recognized by manu- facturers ; and, for light work at least, it is gradually coming into more general use to replace the epicycloidal form. MOTION BY SLIDING CONTACT. 113 CHAPTER VII. COMMUNICATION OF MOTION BT SLIDING CONTACT. VELOCITY EATIO CONSTANT. DIRECTIONAL RELATION CONSTANT. TEETH OP WHEELS (CONTINUED). Approximate Forms of Teeth. — Willis', Method. — Willis' Odonto- graph. — Grant's Odontograph. — Robinson's Odontograph. 109. Approximate Forms of Teeth. — In order to secure perfect sniootbness of action in toothed wheels, it is essential that the tooth outlines should be accurately laid out, as explained in the preceding pages, and that the teeth should be constructed so as to conform exactly with the outlines so found. If the teeth are to be cut, there is no reason why the exact curves should not be used, for it is as easy to form the cutter of the exact shape as of any approximate one ; and the cutter once formed, the exact curves can be cut as easily as any Other. When the teeth are to be simply cast, however, or when, for other reasons, perfect accuracy is not sought after, we may replace the exact curves by others which approximate to them more or less closely, but which are simpler to construct. When approximate forms of teeth are employed, some one of the arbitrary sets of proportions given in Art. 101 is generally followed. The two principal methods of approximation are by cir- cular arcs and by curved templets. 114 ELEMENTARY MECHANISM. 110. Willis' Method of Circular Arcs. — In Fig. 76, let A and B be the centres of motion, and T the point of contact of the pitch circles MN and JiS. Draw the line of action DTE, making any assumed angle with AB, and erect on it the perpendicular TO. On TO assume the point 0, and through this point draw the lines APO and BOQ. "We have now formed a system of linkwork, consisting of the arms AP and BQ, connected by the link PQ ; and as, by construction, is the instantaneous centre of PQ, it follows (Art. 25) that — = ^^-^ for that instant. If at any u B jL point b on DE we draw two curves, abc and Jibe, in contact, and of such shape that P and Q are their respective centres of curvature, these curves will, by revolving about centres A ' AT and B respectively, produce the constant velocity _ = — — , a BT the same as that of the pitch circles. In the preceding articles we have already discussed the tlieoretical shapes of such curves ; and, from the above, it is evident that, if circular arcs be drawn through 6, with centres P and Q, MOTION BY SLIDING CONTACT. 115 they will fulfil the required conditiou for that instant. If, however, the teeth are short, and the obliquity is not very great, these ares differ so slightly from the true curves that they may be substituted for the latter with very good results. In the figure the arc abc will be the face of the tooth of MN, while libe will be the flank of the follower. 111. Approximate Involute Teeth by "Willis' Method. — In this case the side of the tooth is made to consist of a single arc, and a very simple rule may be obtained. In Fig. 76, let TO = co \ then AP and BQ will become perpendicular to DE, and the points P and Q will fall at P' and Q' respectively. Let the circular arcs be struck through T\ let i? be the radius, AT, of the wheel, and <^ the angle which DE makes with AB. Then TP = R cos ^, which is independent of the wheel RS, as well as of the pitch and number of teeth of MN. If, therefore, the angle <^ be made constant in a set of wheels, and their teeth be described by this method, any two of them will work together. Assume <^ = 75° 30', which is a very convenient value, for which TP = R cos 75° 80' = 0.25038i2 = - very nearly. 112. Practical Example. — Let it be required to con- struct, by this method, the teeth of a wheel of 25 teeth ; diameter of pitch circle, 4 inches. Let AT (=2 inches) be the radius (Fig. 77), and JfJVthe pitch circle, of the proposed wheel. The pitch, as near as may be, is half an inch. We will make the teeth of the proportions given in the first sys- tem of Art. 101. This gives addendum = 0.15 inch, total depth = 0.35 inch, backlash = 0.04 inch. Hence draw the addendum and root circles at distances of 0.15 inch without, and 0.20 inch within, the pitch circle, respectively. Draw TP, making an angle of 75° 30' with the radius, and drop 116 ELEMENTARY MECHANISM. a perpendicular, AP, upon TF (or describe a semicircle upon AT, and set off TF = AT\ then will F be the centre from whicli an are, aTb, described through T, will be the side of the tooth required. To describe the other teeth, draw, with centre A and radius AF, a circle, mn, within the pitch circle MJSf; this will be the locus of the centres for the teeth. Set off around the pitch circle, arcs of 0.23 inch and 0.27 inch in length alternately, being the respective widths of tooth and space on the pitch circle. Take the constant radius in the compasses, and, keeping one point in the circle mn, step from tooth to tooth, and describe the arcs, as shown in the figure, joining them directly to the arcs of the adden- dum circle, and by small fillets to the arcs of the root circle. If aTb were an arc of an involute having mn for a base circle, TP would be its radius of curvature at T. These teeth, therefore, approximate to involute teeth ; and they MOTION BY SLIDING CONTACT. 117 possess, in common with them, the oblique action, the power of acting with wheels of any number of teeth, and the adjustment of backlash. But, as the sides of the teeth con- sist each of a single arc, there is but one position of action in which the angular velocity is strictly constant ; namely, when the point of contact is on the line of centres. The length of the teeth should always be kept within the limits shown in Art. 102, and in such cases the above method of approximation will give fairly good results. The larger the wheel, the more closely will the circular arcs obtained by this rule agree with the true involute curve. 113. Approximate Epicycloidal Teeth by Willis' Method. — By making the side of each tooth consist of two arcs joined at the pitch circle, and struck in such wise that the exact point of action of the one shall lie a little before the line of centres, say at the distance of half the pitch, and the exact point of the other at the same distance beyond that line, an abundant degree of exactitude wiU be obtained for all practical purposes. In Fig. 78, let A and B be the centres of motion, and T the point of contact of the pitch circles MN' and BS. Draw DE, making an angle of 75° with AB. This angle is, in fact, arbitrary ; but 75° has been found by Professor Willis to give the best form to the teeth. Draw OTCy perpendicular to DE, and set off the lengths TO and Tff, equal to each other, and less than either^ Tor BT. Through draw the lines BOQ and APO, and through 0' draw the liues BQ'O' and AO'P'. By this construction, which is merely an extension of that of Art. 109, we obtain four tooth centres. P will be the centre for the faces of MN, Q the centre for the flanks of BS, Q' for the faces, of RS, and P' for the flanks of MN. The flank of RS and the face of MN will be circular arcs, with centres Q and P respectively, and drawn in contact at a distance of half the pitch to the right of the line of centres ; the face of RS and 118 ELEMENTARY MECHANISM. theflanJc of MNmW be circular arcs, with centres Q'andP', and drawn in contact at a distance equal half the pitch to th^ left of the line of centres. /IB From the construction it appears that the teeth of one wheel are not changed in shape by any change in the i-adius of the other wheel. In short, if any number of wheels be described in the above manner, in which the angle DTA is constant, the distances TO and TO' being the same for the whole set of wheels, then any two of these wheels will work together. The distance TO' may be determined for a set of wheels by considering that if A approach T, the point C remaining fixed, AP' becomes parallel to DE, and the flank of the tooth of MN becomes a straight line. If A approach still nearer, P' appears on the opposite side of T, and the flank becomes convex, giving a very awkward form to the tooth. The greatest value, therefore, that can be given to TO and TO' must be one which, when employed with the MOTION BY SLIDING CONTACT. 119 smallest rnrlins of the set, will make AP' parallel to DE. By assuming constant values for this smallest radius, as well as for the angle DTA, in a set of wheels, the values of tlie radii of curvature of the faces and flanks which correspond to different numbers and pitches, maybe calculated and tabu- lated for use, so as to supersede the necessity of making the construction in every case. Thus, the values in the tables of Fig. 79 were obtained by assuming that the least radius was just great enough to give the wheel twelve teeth of the required pitch, and that the angle DTA was 75°. 114. Willis' Otlontograpli. — This instrument, repre- sented in Fig. 79, was contrived by Professor Willis for the purpose of laying out the approximate forms of teeth accord- ing to the principles of Art. 11.3. The figure represents the instrument exactly half the size of the original ; but, as it may be made of a sheet of bristol-board, this figure will enable any one to make it for use. The side NTM, which corresponds to the line DE in Fig. 78, is straight ; and the line TC makes an angle of exactly 75° with it, and corre- sponds to the radius AT of the wheel. This side, NTM, is graduated into a scale of twentieths of inches ; and each tenth division is numbered, both ways, from T. The instrument is often made of brass, and in that case is of the shape shown in Fig. 80 ; the tables not being on the instrument, but on a printed sheet accofnpanying tlie same. The manner of using the instrument is shown in Fig. 80. Let it be required to describe the form of a tooth for a wlieel of 29 teeth of 3 inches pitch. This determines the radius ^r of the pitch circle MN. Lay off the arcs TD and TE, each equal to half the pitch, and draw the radial lines AD, AE. To draw the flank, apply tlie instrument with its slant edge on AD, so that D is at the zero point of the scales. In the table headed "Centres for the Flanks of Teeth," look down the column of 3-inch pitch, and opposite to 30 teeth, which is the nearest number to that required, will be found TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALES. CENTRES FOR THE FLANKS OF TEETH. NUMBER OP TEETH. PITCH IN INCHES. 1 154 1V& 1« 3 2.V4. 2H 3 13 139 160 193 235 357 389 331 886 14 69 87 104 121 139 156 173 308 15 49 63 74 86 99 111 133 148 16 40 50 59 69 79 89 99 121 17 34 43 50 59 67 75 84 101 18 30 37 45 53 59 67 74 89 20 25 31 87 43 49 56 63 74 23 33 37 33 39 43 49 54 65 24 20 25 30 85 40 45 49 59 36 18 33 37 32 37 41 40 55 30 17 31 35 29 33 37 41 49 40 15 18 21 25 28 33 35 42 60 13 15 19 23 25 38 31 37 80 13 15 17 20 33 36 29 35 100 11 14 17 20 23 25 28 34 150 11 13 16 19 21 24 27 33 Back 10 12 15 17 20 32 35 30 CENTRES FOR THE FACES OF TEETH. 12 5 6 7 10 11 12 15 15 5 7 8 10 11 13 14 17 20 6 8 9 11 12 14 15 18 - 80 7 9 10 13 14 16 18 3] 40 8 9 11 13 15 17 19 33 60 8 10 13. 14 16 18 20 35 80 9 11 13 15 17 19 31 26 100 9 11 13 15 18 30 33 26 150 9 11 14 16 19 31 33 37 Rack 10 13 15 17 30 33 35 30 WILLIS' ODONTOGRAPH. Fig. 79 x 120 200" 1 go- ISO' 170- 160-"f m -g isn- -^ ■n — ^ ^5 n 140- ^ 2 'g 3 130- -^ m ^ U) 120- — ^ ■n — S 3D 110- ■— s H I 100- n ■n 90- 2 80- en 70- ■n H 60- n n H "in- Z 4G- 30- 20- 10- M MOTION BY SLIDING CONTACT. 121 the numhcr 49. The point g, indicated on the drawing-board by the position of this number on the scale marked "Scale of Centres for the Flanks of Teeth, ' ' is the centre required, from which the arc T^i must be drawn with the radius gT. The centre for the face Tn is found in a manner precisely simi- lar, by applying the slant edge of the instrument to the radial line jLE. The number 21, obtained from the lower table, will indicate the position, 7i, of the required centre on the lower scale. The arc Tn is then drawn, with h as a centre, iTig. so and radius TYi. We have now the complete tooth outline for one side of one tooth ; the curve pTii being limited at the top by the addendum circle, and at the bottom by the root circle. Having proceeded thus far, the simplest way of drawing the rest of the tooth curves is to describe two circles about A, one through g and the other through h. Then all the centres for the flanks will lie on the former, and all the centres for the faces on the latter, of these two circles. We may now find these centres by striking from each pitch point an are with radius equal to gT to cut the circle of centres for flanks, and an arc with radius Th to cut the circle of centres for faces. The curve nTp is also correct for an annular wheel of the same radius and number of teeth ; n becoming the root, and 122 ELEMENTARY MECHANISM. p the point, of tlie tooth. Numbers for pitclies not inserted in the table may be obtained by direct proportion from the column of sojne other pitch ; thus, for 4-inch pitch, by doubling those of 2-inch pitch. Also, no tabular numbers are given for 12 teeth in the upper table, because their flanks are radial lines. The variation in the contour, due to the addition of a single tooth, becomes less and less as the number of teeth increases ; so that the same curve will serve for wheels with nearly the same number of teeth. Consequently, if the num- ber assigned is not found in the tables, the nearest number found there is to be used instead. 115.* Improved Willis Odontograpli. — In Fig. 80 the points g, h, are found by drawing two radial lines, AD and AE, and applying the instrument to each of them, or by drawing two additional lines, gD and Eh, at an angle of 75° with AD and AE respectively, and setting off on them cer- tain lengths obtained from tables. Having found these points, circles of centres are drawn through them, and used as explained above. If, now, instead of proceeding in this manner, we could find from tables the radii of the two circles of centres, and the radii gT and Th, the construction would be much simplified. This improvement is due to Mr. George B. Grant, who has calculated the distances of the two circles of centres from the pitch circle, and also the radii of the arcs for the faces and flanks. His results appear in the following table, where "Dis." represents the radial distance between the circle of centres and the pitch circle, and "Had." the radius of the face or flank arc as the case may be : — * The taWes in Arts. 115 and 116, and the substance of the matter in those articles, are taken, by permission, from " A Handbook on the Teeth of Gears," by George B. Grant, Boston, Mass. MOTION BY SLIDING CONTACT. 123 IMPROVED WILLIS ODONTOGRAPH TABLE. (Copyright, 1886, by George B. Giant.) Fob One Diametkal Fob One-Inch Circular Pitch. Pitch. IN THE Wheel. For any Dther Pitch, For any other Pitch, divide Tabular Val le by multiply Tabular Value by that Pitch. that Pitch. Faces. Flanks. Fiices. Flanks. Ead. Dis. Ead. Dis. Ead. Dis. Rad. Dis. 12 12 2.30 0.15 _ _ 0.73 0.05 _ _ 13* 13- 14 2.35 0.16 15.42 10.25 0.75 0.05 4.92 3.26 15i 15- 16 2.40 0.17 8.38 3.86 0.77 0.05 2.66 1.24 m 17- 18 2.45 0.18 6.43 2.. 35 0.78 0.06 2.05 0.75 20 19- 21 2.50 0.19 5.38 1.62 0.80 0.06 1.72 0.52 23 22- 24 2.55 0.21 4.75 1.23 0.81 0.07 1.52 0.39 27 25- 29 2.61 0.23 4.31 0.98 0.83 0.07 1.36 0.31 33 30- 36 2.68 0.25 3.97 0.79 0.85 0.08 1.26 0.26 42 37- 48 2.75 0.27 3.69 0.66 0.88 0.09 1.18 0.21 58 49- 72 2.83 0.30 8.49 0.57 0.90 0.10 1.10 0.18 97 73-144 2.93 0.33 3.30 0.49 0.03 0.11 1.05 0,15 290 145-rack 3.04 0.37 3.18 0.42 0.97 0.12 1.01 0.13 This improved Willis process will produce exactly the same circular arc as the usual Biethod, with the same theo- retical error ; but its operation is simpler, and less liable to errors of manipulation. By this process the circles of cen- tres are drawn at once, without preliminary constructions, at the tabular distances from the pitch line ; and the table also gives the radii of the face and flank arcs. No special instru- ment is ]-equired, no angles or special lines are drawn to locate the centres, and heuce the chance of error is much less. 124 ELEMENTARY MECHANISM. 116.* Grant's Odontograpli. — If, in the method de- scribed in the preceding article, we use, instead of the cir- cular arcs employed by Professor Willis, arcs which shall approximate still more closely to the true epicycloidal and hypocycloidal curves, we shall evidently obtain more satis- factory results. Mr. Grant has computed and tabulated the location of the centre of the circular arc that passes through the three most important points of the true curve; viz., at the pitch line, at the addendum line, and at a point midway between. The Willis arc runs altogether within the true curve, while the Grant arc crosses the curve twice. The average error of the Grant arc is much less than that of the Willis arc, and it is hence to be preferred. The circles of centres are drawn at the tabular distances, "Dis.," iuside and outside the pitch line respectively; and all the faces and flanks are drawn from .centres on these circles, with the dividers set to the tabular radii, "Rad." The tables are arranged in an equidistant series of twelve intervals. For ordinary purposes the tabular value of any interval can be used for any tooth in that interval ; but for greater precision it is exact only for the given "exact" number, and intermediate values must be taken for inter- mediate numbers of teeth. When the number of teeth is twelve, the flanks are radial, and hence no tabular values are given for the flanks of that number. To illustrate the use of the following table, let it be re- quired to draw the tooth outline for a wheel of 24 teeth of l|-inch pitch. Draw the pitch circle with its proper radius of 11.46 inches, and mark off the pitch points of the teeth. Draw the addendum, root, and clearance circles, having fixed on the dimensions of the tooth by means of some system of proportions such as those given in Art. 101. * See note on p. 122. MOTION BY SLIDING CONTACT. 125 GRANT'S ODONTOGRAPH TABLE. EPICrCLOIDAL TEETH. (Copyright, 1885, by George B. Grant.) For One Diametral For One-Inch Circular Pitch. Pitch. Number of Teeth in the m'heel. For any 5ther Pitch, For any other Pitch, divide Tabular Val ne by multiply Tabular Value by that Pitch. that Pitch. Face3. Flanks. Faces. Flanks. Exact. Intervals. Ead. Dis. Ead. Dis. Rad. Dis. Rad. Dis. 12 12 2.01 0.06 0.64 0.02 13i 13- 14 2.04 0.07 15.10 9.43 0.65 0.02 4.80 3.00 15+ 15- 16 2.10 0.09 7.86 3.46 0.67 0.03 2.50 1.10 m 17-- 18 2.14 0.11 6.13 2.20 0.68 0.04 1.95 0.70 20 19- 21 2.20 0.13 .5.12 1.57 0.70 0.04 1.63 0.50 23 22- 24 2.26 0.15 4.50 1.13 0.72 0.05 1.43 0.36 27 25- 29 2.33 0.16 4.10 0.96 0.74 0.05 1.30 0.29 33 30- 36 2.40 0.19 3.80 0.72 0.76 0.06 1.20 0.23 42 37-48 2.48- 22 3.52 0.63 0.79 0.07 1.12 0.20 58 49- 72 2.60 0.25 3.33 0.54 0.83 0.08 1.06 0.17 97 73-144 2.83 0.28 3.14 0.44 0.90 0.09 1.00 0.14 290 145-rack 2.92 0.31 3.00 0.38 0.93 0.10 0.95 0.12 From the above table take the values given for the interval 22-24 ; and, as the pitch is 1 J inches, multiply these tabular values by 1|-. We then obtain Distance between pitch circle and circle of face centres = 0.07; face radius = 1.08. Distance between pitch circle and circle of flank centres = 0.54; flank radius =2.15. Draw the circle of face centres 0.07 inch inside the pitch circle, and the circle of flanlc centres 0.54 inch outside of 126 ELEMENTAEY MECHANISM. the pitch circle. With a pitch point as a centre, strike an arc with radius 1.08 inches to cut the circle of face centres, and an arc with radius 2.15 inches to cut the circle of flank centres. With these two points of intersection as centres, describe the face and flank through the pitch point, draw the same arcs in reversed position through a point on the pitch circle whose distance from the pitch point is the desired tooth thickness, connect the faces by an arc of the addendum circle, and join the flanks by fillets to the clearance circle, and the tooth is complete. This odontograph, as well as Willis', is arranged for an interchangeable set (Art. 98), from a wheel with twelve teeth to a rack. 117. Kotoinson's Templet Odontograph. — lu the use of this instrument, a method entirely different from those just mentioned is pursued. Instead of using circular arcs, the outlines of the teeth are drawn by means of a templet, which is the curved edge of the instrument itself, when the latter is brought into a proper position. As the epicycloidal curve is normal to the pitch line, and very nearly so to the tangent to the pitch circle drawn from the middle of a tooth, it is clear that if a curve of rapidly changing curvature be so placed as to be normal to the tan- gent, as above described, and at the same time intersecting the addendum circle at the same point that the epicycloidal curve required for the tooth does, it will represent the epicy- cloidal tooth face with great precision. The curve adopted as conforming most closely, in general, with limited initial portions of the epicycloid, is the loga- rithmic spiral. This curve 'appears to possess the highest degree of adaptation, because of its uniform rate of curvature, and also because this rate can be assumed at pleasure. In adopting the particular logarithmic spiral for the odontograph curve, inasmuch as this spiral may have an infinite variety of obliquities, it is evident that the selection is not a matter of MOTION BY SLIDING CONTACT. 127 indifference. "When the obliquity, or angle between the nor- iTinl and radius vector, is very small, the arc of this spiral changes curvature less rapidly than when the obliquity is great. When the obliquity is zero the spiral becomes a circle, and when it is 90° the spiral is simply a radius ; neither of which approximates to the desired curve. To find that obliquity which makes the spiral best fit the epicycloid, it will probably be most satisfactory to assume an epicycloid which represents an average of those likely to be used for both curves, and adapt the spiral to it, though any ordinary logarithmic spiral will evidentlj' conform more closely to it than the circle. The spiral which most closely osculates the epicycloid for a pair of equal pitch circles is therefore adopted, because the opposite wheel may be either larger or smaller, thus making a higher or lower epicycloid. By an elaborate mathematical investigation,* Professor Robinson has shown that this curve will produce the required results in all the various cases of epicycloidal and involute gearing. 118. Manner of using- Odontograpli The instru- ment is shown in Fig. 81 of full size,, and of suitable capacity for laying out all teeth below six inches pitch. The curved edge AB is the logarithmic spiral above spoken of ; and the curve AC is its evolute, in other words, an equal spiral. The instrument should be made of metal, because it is intended that it may be used directly for a scribe templet, in which use it will be subject to wear from the passes of the scribe. It has several holes in it, so that it may be attached by wood screws, or by bolts expressly prepared, to any con- venient wooden rod, in^ such a manner, that, when the rod swings around a centre-pin of the wheel, all the faces of the teeth may be described directly from the instrument itself. * For the complete mathematical discussion, see No. 24 Van Nos- trand's Science Series. 128 ELEMENTARY MECHANISM. O v^ CO o MOTION BY SLIDING CONTACT. 129 The desired result is thus obtained directly without the use of a pair of compasses. Accompanyiug the instrument are six different tables, varying according to the kind of tooth desired. One of the tables is for the teeth of wheels belonging to an interchange- able series ; the other tables are for variously curved flanks and for annular wheels. The manner of using all the tables is nearly the same, so it is simply necessary to indicate the method for any one of them. Fig. 82 shows the manner of- using this odontograph to lay out the teeth of a wheel belong- ing to the interchangeable series. The table for this system is arranged in four columns, headed respectively, 1, "Diameter in Inches;" 2, "Num- ber of Teeth ; " 3, " Face Settings ; " 4, " Flank Settings." The two settings are given for one-inch pitch. In the figure, let MN be the pitch circle. If it is not given, it may be found by multiplying the pitch by the num- ber in the column "Diameter in Inches" corresponding to the number of teeth. Assume the point T as the middle of a tooth, and lay off" TD = its half-thickhess. At T draw the tangent tTif , and at D the tangent Dd. Make TH = TD. Take from the column "Face Settings" the figure corresponding to the number of teeth, and multiply it by tiie pitch ; this will give the setting number. Then place the graduated edge of the odontograph at H, and in such position that the number and division of the scale shall come precisely on the tangent lino at H, while at the same time the other curved edge is tangent to the line tTt'. The tooth outline is then traced along the instrument from D as far as needed. By turning over the fnstrument, which is graduated on both sides, and repeating the operation, we get the opposite face of the same tooth. To draw the flank, find a similar setting number by using the column " Flank Settings." The instrument is to be set with the division at Z*, and the other curved edge tangent to 130 ELEMENTARY MECHANISM. Dd ; and the flank may then be drawn to the proper depth. When it is desired to repeat the operation of drawing the curves all around the wheel, the simplest way to locate the instrument is by drawing circles through the points A and C when it is once properly located. The instrument can then be readily placed at any tooth outline by placing the gradu- ated edge on the pitch point, and keeping the points A and G in the circles just mentioned. " For instance, let it be required to draw the teeth of a wheel having 50 teeth of 3-inch pitch. For this number of teeth we find the tabular values : — Biam. in Inches. No. of Teeth. Face Setting. Flank Setting. 15.917 50 0.42 0.66 The diameter of the pitch circle is 3 x 15.917 = 47.751 =- 47f inches. The proper setting to draw the face is 3 X 0.42 = 1.26, and the corresponding setting for the flank is 3 x 0.66 = 1.98. Hence, to draw the face, the odontograph is placed so that the number 1.26 on the scale is at the* point H (Fig. 80) ; and, to draw the flanks, it is placed so as to bring the number 1.98 at D. MOTION BY SLIDING CONTACT. 131 CHAPTER VIII. COMMUNICATION OF MOTION BY SLIDING CONTACT. VELOCITY EATIO CONSTANT. DIRECTIONAL r.ELATION CONSTANT. TEETH OF WHEELS (CONCLUDED). Pin Gearing. — Low-Numbered Pinions. — Umymmetrical Teeth. — Twisted Gearing. — Non-Circular Wheels. — Bevel Gearing. — Skew-Bevel Gearing. — Face Gearing. 119. Pin Gearing. — In Art. 76 it has been shown that an epicycloid traced on the pitch circle of the driver, by rolling on the latter a describing circle equal to the pitch circle of the follower, will drive a pin in the circumference of tbie following pitch circle with the same constant velocity ratio as if the pitch circles rolled together. In Fig. 83, let MN and BS be the pitch circles. Lay off the equal pitch arcs Ta and Tb ; and, with US as the describing circle, trace through a the epicycloid aD, which will, of course, pass through b. Draw the equal epicycloid TD in reverse position through T, and let D be the point of intersection of tlie two epicycloids. Then aDT is the com- plete outline of a tooth of JfiV which will drive a pin b (having no appreciable diameter) on US with the constant velocity ratio — = — — . Through D draw the arc DP con- a BT centric with MJV. The point P, where this arc intersects US, will evidently be the point at which the tooth aDT and 132 ELEMENTARY MECHANISM. the pin 6 will quit contact. Through P draw the epicycloicl Pa' equal to Da ; then TP = Ta' = arc of recess. The wheel MN moving as indicated by the arrow, the contact will begin at T, and the point of contact will travel to the right, along the arc TP, until it reaches the point P, where contact ceases. The contact is wholly on one side of the line of centres ; and when the teeth drive, as they should always do (Art. 90), there is no arc of approach. 120. With given pitch circles, to find the relation between the arc of recess and the pitch. " In Fig. 83, let TP, the arc of recess, be given. Through P describe the epicycloid Pa' by rolling RS on MN; draw the radius PA, intersecting MN in K. Then, in order to secure the desired arc of recess, the pitch must not be greater than Ta' = TP, nor less than 2Ka'. If Ta (= iKa') be the pitch, and the tooth be pointed, the arc of recess will be MOTION BY SLIDING CONTACT. 133 TP, as required. If, with the same pitch, the tooth be given some thickness at the top, the arc of recess will becoirie less ; and, when the latter has its smallest value (i.e., when it is just equal to the pitch), the top of the tootli will be cut off so as to give the tooth outline Tcha. 121. If the pitch be given, and it is required to find the arc of action which may be secured, lay off on MN the oiven pitch arc Ta, and, with BS as descri?jing circle, construct the epicycloids TD and aD. Through their point of inter- section, D, draw the arc DP concentric with MN. Then TP is the maximum, and Tb (= pitch) is the minimum, value of the arc of recess ; the tooth in the former case being pointed, and in the latter cut off at cb. 122. Pins of Sensible Diameter, — In the preceding articles we have treated the pins as mere mathematical lines ; but in practice they must, of course, be given some magni- tude, and they are usually made as cylinders of a diameter of about half the pitch. The form of the tooth must then be so modified, that, when it acts on the cylindrical surface of the pin, the latter shall move just as though its axis were being driven by the original epicycloid ; in other words, the constant normal distance from the latter to the new tooth outline must be equal to the radius of the pin. The manner of finding this desired curve is shown in Fig. 83. About successive points along the epicycloid, as centres, circular arcs are drawn, having the same radius as the pin ; a curve drawn tangent to this series of arcs will be the required tooth outline. 123. Character of Derived Tooth Outline,* — The driver's tooth has evidently been shortened, and conse- quently the arc of recess has been reduced. At first sight, it would seem as though an arc of approach had been * First investigated by Prof. C. "W. MacCord, from wliose "Kine- matics "the substance of Arts. 123 and 12i is talien by permission. 134 ELEMENTARY MECHANISM. obtained as a compensation for the loss of part of the arc of recess, for tlie tooth and pin seem to come into driving contact when the centre of the latter is at T ; and, if this wei-e so, there would be an arc of approach approximately equal to the radius of the pin. If, however, the nature of the derived curve be carefully examined, the fallacy of this assumption will become very evident. In Fig. 84, let MN and BS be the pitch circles of the toothed wheel and of the pin wlieel respectively. Let TDEHK be the epicycloid gen- erated by rolling RS as a describing circle on the outside of MN. Let Tc = radius of the proposed pin, which is in this' figure drawn entirely out of proportion, being greatly ex- MOTION BY SLIDING CONTACT. 135 aggerated in order to show the peculiarities of the derived curve more clearly. At successive points of the epicycloid draw a series of normals to the same, making their lengths, Tt, Dd, Ee, Hh, Kk, each equal to Tc, the radius of tlie pin. The curve tdehk will be the derived curve. The curve begins at t, the extremity of the normal Tt, which is perpen- dicular to AB; the curve then at first descends, forming a cu.'ij) within the circumference of the pin ; then, rising, it cuts this circumference at some point, j9, and passes to the outside of the pitch circle JfiV. Now, it is evident that the part tdep of the derived curve cannot form part of the tooth outline, as it lies wholly within the circumference of the pin. The part p/i/c of the derived curve will then be the part which will form the outjine of the iwoposed tooth, and 2'> will be the first point of the tooth to come into driving contact with the pin. It now remains to find the position of the pin when this first contact takes place. Draw pP normal to the epicycloid. Now, p will come into correct driving contact with the circumference of the pin at the same time that P would come into contact with the axis of the pin. But P will not be in contact with this axis until the latter is at Y, the point at which the pitch circle BS is cut by a circular arc passing through P, and concentric with MN. Hence p will not come into correct driving con- tact witli the circumference of the pin until the axis of the latter is at Y. About F as a centre, draw a circular arc of radius = Tc ; draw also through p a circular arc concentric with MN. These two arcs will intersect at pi, which will be the point at which correct driving contact begins. From A draw a radius through 2h to intersect MJST at V; then TV will be the arc of approach. If V falls on AB (that is, at T), the arc of approach will reduce to zero, and contact will begin on, the line of centres. If V falls to the right of AB, contact will not begin until after the respective points have passed the line of centres. 136 ELEMENTAKY MECHANISM. 124. Limiting: Diameter of Pin. — The two pitch circles and the pitch being given, let it be required to deter- mine the maximum radius that can be given to the pin. Evidently the arc of action cannot be less than the pitch. But this arc of action depends directly on the position of the point p (Fig. 84), and this again depends on the very thing we wish to determine ; i.e., the diameter of the pin. A direct construction being thus impossible, we are com- pelled to resort to the following tentative method, also de- vised by Prof. MacCord. In Fig. 85, let MN and BS be the respective pitch circles, as before, and let Ta = Tb = pitch MOTION BY SLIDING CONTACT. 137 arc. Rolling JiS as a describing circle on the outside of MjS', describe the epicj'cloids TD and aD, intersecting at L\ and forming the tooth outline to drive a pin of no sensible diameter (Art. 120). From A draw tlie radius Ax, bisect- ing Ta, and, of course, passing through D. Draw the chord Tb, cutting Ax at Q. Assume any radius, lb {less than QD), for the pin, and construct the derived curve. The . latter will cut Ax at C, which will be the point of the tooth. From C draw CII perpendicular to the epicycloid aD. Hav- • ing thus found the point H, draw through it a circular arc concentric with 3f2{, and cutting BS at L. Then L will be the position of the centre of the pin at the moment when the tooth and pin quit contact. To find its position at the moment when contact begins, we must proceed as in Art. 123. Drawing the pin of the assumed diameter (twice lb), describing the epicycloid Td (same as aD) , and finding the derived curve as in Fig. 84, we determine the points 2^, P, and finally Y, the point required. Hence contact will con- tinue while the centre of the pin travels along the arc YL. If this arc is less than Tb, the assumed radius lb is too great, and must be reduced. If YL = Tb, the case is just barely possible, the assumed radius of the pin is a maxi- mum, and the tooth is pointed. If YL is greater than Tb, the case is practicable ; and in this case the tooth may be given some thickness at the top. In practice, such wheels will be found to work more smoothly if the arc of action is made considerably greater than the pitch arc, in which case the usual rule of making the radius of the pin equal to one-fourth the pitch .will gen- erally give very satisfactory results. 125. Rack and Wheel. ^ As previously stated, the pins are always given to the follower, and hence this com- bination will present two cases according to whether the rack is driver or follower. In Fig. 86 the rack drives and the wheel carries the pins. The teeth of the rack are formed 138 ELEMENTARY MECHANISM. by curves parallel to tlie cycloids which would work correctly with the axes of the pins. lu Fig. 87 the wheel drives and Fig. se the rack carries the pins. The teeth of the wheel are formed by curves parallel to the involutes of its own pitch circle, which would work correctly with the axes of the pins. 126. Annular Wheels. — If the annular wheel drives, as in Fig. 88, the pins are given to the small wheel, and the teeth of the annular wheel are formed by curves parallel to the hypocycloids which would work correctly with the axes of the pins. If the annular wheel is the follower, as in Fig. 89, it carries the pins ; and the teeth of the small wheel are formed by curves parallel to the epicycloids which would work correctly with the axes of the pins. Wiien the annular wheel is the driver, and is twice as large as the wheel with which it gears, the hypocycloids become straight lines, and the parallel tooth outlines will evidently also be straight lines. MOTION BY SLIDING CONTACT. 139 Fig. 90 shows such an arrangement, in which the pin wheel has but three pins, while the wheel teeth are formed T^ig. 88 by cutting three straight grooves, intersecting each other at the centre of the wheel, at angles of sixty degrees, each Fig. 89 being of a width equal to the diameter of a pin. By placing rollers on the pins, and making the widths of the slots equal Fig. 90 to the diameter of these rollers, tliis arrangement of pin 140 ELEMENTARY MECHANISM. gearing can be used as a shaft coupling to drive in either direction. 127. LiOw-NumTjered Pinions. — As the number of teeth in a wheel decreases, the teeth themselves become longer, and both the obliquity of action and the amount of sliding rapidly increase. Pinions having very few teeth are, for these reasons, unsuitable for general use ; and accord- ingly we find that in practice no wheel of less than about twelve teeth is employed if it can possibly be avoided. In order to secure smoothness of action and a minimum obliq- uity of pressure, the number of teeth assigned to any given wheel is usually so great that no doubt exists as to their successful working. It occasionally happens, however, that it becomes imperatively necessary to employ wheels having as few teeth as possible ; and it then becomes a matter of importance to determine whether the desired numbers of teeth will work together. 128. Practical Example. — The practicability of any assumed case can be readily determined by the construction of a diagram, keeping in mind the limitations as to pitch, arc of action, etc., explained in previous articles. For example, let it be required to determine the practicability of using two equal pinions of six teeth, having radial flanks. In Fig. 91, let A and B be the centres of the equal pitch circles, and T their point of tangency. As the flanks are to be radial, the diameter of the describing circles will be equal to the radius of the pitch circles, as shown in the figure. Assume the arc of action to have its smallest value (namely, just equal to the pitch arc) , and let the arcs of approach and recess be equal. Constructing the teeth under these condi- tions (Alt. 97), we will obtain the wheels shown in Fig. 91. These wheels will just barely work, one pair of teeth quitting contact at P at the same instant that another pair are coming into contact at P'. It is evident that, by continuing the opposite faces until they meet, the arc of action can be some- MOTION BY SLIDING CONTACT. 141 yig. 91 what increased without making any other change. Two such wheels, then, can be made to work, though they will never run with the smoothness of action that characterizes wheels having a large number of teeth. 142 ELEMENTARY MECHANISM. 129. In the above example the maximum obliquitj,' of action (that is, the angle PTt) is thirty degrees. In the position represented, P'T is evidently the direction of the pressure between the teeth. Considering the line P'T to represent this pressure in magnitude as well as in direction, P'n and P'm Will represent its components parallel and per- pendicular to the line of centres respectively. We thus have P'm as the component which tends to produce rotation ; while P'n simply forces the axes apart, thus increasing the friction in the bearings. In the figure, P'n = PTsinSO" = ^P'T; that is, this objectionable component is equal to half the pressure between the teeth at P'. It is evident that there must be a practical limit beyond which the angle of maximum obliquity cannot go without increasing the prejudicial component P'n to an inordinate extent. This limit is usually placed, as in the foregoing example, at thirty degrees ; and fifteen degrees is usuallj' taken as the maximum allowable value for the mean obliquity. In the case of involute t^ 2 impossible impossible impossible 11 3 It 37 it rack II 4 ii 15 8 to 5 ti 11 53. i 6 21 10 rack 7 11 9 O O 8 8 8 ;h •< 135. Unsym metrical Teeth. — In all the figures of teeth hitherto given, the teeth are symmetrical, so that they will act whether the wheels be turned one way or the other. If the machine be of such a nature that the wheels are to be required to turn in one direction only, the strength of the teeth may be greatly increased by an alteration in form first suggested by Professor Willis. In Fig. 94 are represented two wheels, of which the lower is the driver, and always moves in the direction of the arrow. The describing circles are made large, thus reducing the obliquity of action. The right side of the driver's teeth and the left side of the fol- lower's teeth are the only portions that are ever called into action ; and they are made precisely as usual in the epicy- cloidal system. If the other sides were made the same, this would give a very weak form at the root. To obviate this, the back of each tooth is bounded by an arc of an involute. The bases of these involutes being proportional to the pitch circles, they will during the motion be sure to clear each other, because, geometrically speaking, they would, if the MOTION BY SLIDING CONTACT. 149 wheels moved the other way, wovk together correctly, though the inclination of their common normal to the line of centres is too great for the transmission of pressure. The effect of this shape is to produce a very strong form of tooth by taking away matter from the extremity of the tooth where the ordinary form has more than is required for strength, and adding it to the root. 136. Twisted Gearing. — In this class of gearing (Art. 58) the point of contact travels, during the motion of the wheels, from one side to the other. The outer planes of the wheel should be twisted through an angle equal to the pitch, so that a fresh contact is always beginning on one side as the last contact is quitting on the other. In the double wheel shown in Fig. 38, there are, of course, two points of contact, travelling in a symmetrical manner with respect to the mid-plane of the wheel. Tlie teeth must be so formed, that, when the angular velocity ratio is constant, contact shall only take place at the instant of crossing the line of centres. Otherwise, if the teeth were formed upon the usual priuci- 150 ELEMENTARY MECHANISM. pies, it is evident that the sliding contact before and after the line of centres would still remain. This may evidently be accomplished by making the flanks by any of the usual methods, and then making the faces so that they will He within the faces which would be proper for a spur wheel with the flanks assumed. The simplest mode of making such teeth is to give them radial flanks, and make the faces semi- circles whose diameter is the thickness of the tooth at the pitch circle. The motion is now transmitted by pure rolling contact, and the action of these wheels is exceedingly smooth and noiseless. They are, however, better suited for light work, because the pressure is couflned to a single point, instead of being distributed along a line. For heavy work it is preferable to employ the stepped wheels (Fig. 37) in which the teeth are of the usual forms for spur wheels. In this case, the motion is, of course, no longer transmitted by pure' rolling contact ; but the action is, nevertheless, much smoother than that of ordinary spur wheels. 137. Non-circular Wheels. — In all the preceding cases of toothed wheels the pitch curves of the wheels have been circles ; but the teeth may be just as well laid out when the pitch curves are not circular, though in the latter case the operation is much more tedious. The two pitch curves must, in any case, be capable of rolling together with a constant velocity ratio. For instance, let it be required to lay out the teeth of a pair of equal ellipses. Divide the perimeter of the ellipse for the location of the teeth and the spaces. Find, by trial and error, the centre of curvature of the ellipse at the point where it is desired to draw a tooth outline. The tooth outline may then be drawn by rolling within and without the pitch ellipse a describing circle in the usual manner ; the actual operation being performed by substituting for the pitch ellipse a circle whose radius is the radius of curvature of the ellipse at the point considered. By repeating this operation at successive MOTION BY SLIDING CONTACT. 151 pitch points, we can thus draw all the teeth. This method is perfectly general, and may be applied to rolling curves of any form, such as, for instance, the lobed wheels shown in Figs. 43 to 46. If the same describing circle be used throughout, its diameter should be such as to give radial flanks to the teeth in that part of the pitch line where the curvature is sharpest. Should other parts be very much flatter, the flanks of the teeth may spread too rapidly. This may be remedied by using different describing circles for the teeth in those parts, care being taken that the same one be always used for the face and flank that are to work together. If one of the wheels be made a pin wheel, its pitch curve is to be used as the describing curve to generate the teeth of the other. 138. Bevel Wheels. — In all the cases of wheels pre- viously considered, the pitch surfaces have been cylinders, all the transverse sections being consequently alike. Hence it was found most convenient to deal with one such section, so that the problems involved only lines instead of surfaces. But the pitch and describing curves employed, as well as the tooth outlines constructed, are merely transverse sections of surfaces whose elements are parallel to the axis of the wheel. Considering the cylinder as the special ease of the cone in which the vertex is removed to an infinite distance, it would seem, that, in the case of the cone, the elements of the analogous surfaces sliould converge to the vertex of the cone. In other words, just as we roll a describing cylinder withm and without a pitch cylinder to generate the tooth surfaces of spur wheels, so may we roll a describing cone within and without a. pitch cone to generate the tooth surfaces of bevel wheels. In both cases the line of contact of the tooth sur- faces will be a right line ; in the former it will be parallel to the axis of the cylinder, and in the latter it will pass through the vertex of the cone. 152 ELEMENTARY MECHANISM. 139. In Fig. 95, let CDTE be the pitch cone, and GPTH the describing cone ; the two cones having the common vertex C, and being in contact along the right line CT. Draw any element, such as CP, of the describing cone, and consider the latter to roll to the left, keeping its vertex at C, and remaining always in contact with the pitch cone. CT is at any moment the instantaneous axis about which the plane OPT revolves ; hence the surface CPa, generated by the line GP, will be normal to the plane OPT. We have seen that, with parallel axes, pitch curves may be selected which will produce a variable velocity ratio. Similarly, in bevel wheels, the bases of the cones might be so shaped as to produce changes in the velocity ratio. lu practice, however, this is never done ; any desired variation of velocity ratio being produced by some other means. We may, therefore, confine our attention to the case in which all the cones have circular bases. In this case, the point P, being at a constant distance from O, will move in the surface of a spliere of which G is tlie centre, and whose radius is equal to the slant height of the cones. The arc TP = arc MOTION BY SLIDING CONTACT. 153 Ta ; and the curve Pa, described by the point P, is a splier- ical epicycloid. Similarly, by rolling the describing cone within a pitch cone, a spherical hypocycloid will be generated. P^ollowing out the analogy between cylinder and cones, it is evident that, just as the tooth surfaces of cylindrical wheels are formed by moving a right line along the epicycloid and hypocycloid previously discussed, keeping the line always parallel to the axis of the pitch cylinder, so the tooth sur-' faces of conical wheels must be formed by moving a right line along the spherical epicycloid and hypocycloid, making the line in this case always pass through the common vertex of the pitch cones. 140. Construction of Tooth Outline. — The portion of the spherical surface occupied by the spherical epicj'cloid and hypocycloid, when they are used in the formation of teeth, is a narrow zone extending a short distance on both sides of the base circle of the pitch cone. For all practical purposes we may substitute for this narrow spherical zone a portion of the surface of a cone which is tangent to the sphere in the base circle of the pitch cone, and whose ele- ments are consequently perpendicular to the corresponding elements of the pitch cone. In Fig. 9G, let CA and CB be the given axes of the pitch cones. Dividing the angle ACB so as to obtain the required / q velocity ratio, which in this case is — = -, we And CP, the a 2 common element. The bases FGP and EHP are evidently small circles of the sphere whose radius is CP- Draw PA perpendicular to CP, and revolve it around the axis GA, generating the normal cone FPA. Similarly, draw PB perpendicular to CP, and revolve it about the axis CB, generating the normal cone PBE. These new cones comply with the conditions above men- tioned, and a narrow zone of their curved surfaces may be used upon which to describe the tooth outlines. 154 ELEMENTARY MECHANISM. If, now, we roll a describing cone witliout one of the pitch cones and within the other, we will generate the tooth sur- face for the faces of the former and for the flanks of the latter. In order to construct this surface, we must select some particular element of the describing cone, and find the curve which it describes on the surfaces of the normal cones. To do this, we need only draw this element in successive positions, and find the points in which it pierces the normal cones. The curve formed by joining these successive points will be the directrix of the tooth surface ; and the latter will be formed by moving a straight line along this generatrix, the line always passing through the common vertex of the pitch cones. This method will give the exact curves ; the error of using the surface of the normal cone, instead of that of the sphere, being so small as to be inappreciable. Its application to practical cases involves more labor, however, than that of the following approximate method, which is the one in almost universal use. 141. TredgoWs Method. — If we assume the curved surface of each of the normal cones to be cut along one of the elements, and spread out on a plane, we will have (Fig. 96) portions of two circles whose radii, A'P' and B'P' are MOTION BY SLIDING CONTACT. 155 the slant heights of the cones. If, now, these circles be taken as pitch circles, and teeth be constructed on them by any of the usual methods for spur wheels, we may then wrap these surfaces, with the teeth, back into their original conical shape ; and using the tooth curves, as they then appear on the normal cones, as directrices, we may generate the re- quired tooth surfaces by moving a right line in contact with the curves, and passing through the common vertex of the cones, as before. 142. The practical method of drawing such teeth is shown in Fig. 97. Let AC be the axis of the bevel wheel, let CDE be the pitch cone, and AED the normal cone ; DJSfE being the circular base common to both cones. In the side view, draw a line parallel to AD, and project the latter on it at A'D'. With centre A' and radius A'D', describe a circular arc which will be an arc of the pitch circle to be used. On this arc lay off a tooth by the usual method, being careful to make the pitch an aliquot part of the cir- cumference of a circle whose radius is MD. The tooth 156 ELEMENTARY MECHANISM. outlines may then be drawn by means of describing circles or by the approximate odontograph methods, according to the degree of accuracy required. Project A'K', the radius of the root circle, at AK, and A'H', the radius of the addendum circle, at AH. The points H, D, K, of the line AH will, by revolution, describe circles about AC, which will be repre- sented in the side view by the straight lines FH, ED, and LK, and which will be seen in their true size and shape in the end view. On the end view of the circles just mentioned we must next lay off, on each side of a radius, the half-thickness of the tooth at the top, at tlie pitch line, and at the bottom, as obtained from the development. If great accuracy is re- quired, any number of additional circles may be used in a similar manner. Having thus determined the end view of the tooth outlines, we must next project each one to the side view ; the points lying in each circle being projected to the straight line which is the side view of that circle. In prac- tice, only frusta of conical wheels are employed, and the teeth are limited at both ends by normal cones. It is evi- dent that in this case the shape of the teeth will be similar at both ends, except that the outer ones will be larger in proportion to tlieir greater distance from the vertex. The points of the inner tooth outlines are found by drawing radii through the principal points of the outer tooth outlines already determined, and finding the intersection of these radii with the circles corresponding to the inner normal cone. It may be required to describe the teeth by either the epi- eycloidal or the involute system, or so that they may be used for an annular bevel wheel ; but the modification of the general operation is in each case similar to the correspond- ing modification for wheels on parallel axes. 143. Relative Action of Bevel and Spur "Wheels The action of a toothed wheel, other things being equal, is always more smooth ia proportion as the teeth increase in MOTION BY SLIDING CONTACT. 157 number and decrease in size, because these conditions diminish the obliquity of action, as well as the amount of sliding. But in bevel wheels the action of the outer tooth outlines does not deviate much from the plane tangent to the two normal cones at P (Fig. 96), and hence they act the same as spur wheels having the radii J.P, BP, which are larger than the radii of the bevel wheels themselves in the ,. AP ,BP ^ , AP CP -.BP CP T ratios — — - and . But -— - = — — , and = . in GP HP GP CG HP CH other words, the action of a bevel wheel, so far as it is affected by the number of its teeth, is equal to that of a spur wheel of the same pitch whose radius is greater than that of the given bevel wheel iu..the sam^ ratio that the slant height of the pitch cone is greater than its altitude. In a pair of mitre wheels this ratio is 1^, so that the action of a mitre wheel having, say, fifty teeth is equivalent to that of a spur wheel of seventy teeth. 144. Skew Bevel Wheels. — ■ From the manner in which the tooth surfaces are generated in spur and bevel wheels by means of describing cylinders and cones, it would seem natural to suppose that, when the pitch surfaces become hyperboloids, the tooth surfaces might be generated in a similar manner by means of describing hyperboloids rolled without and within the pitch hyperboloids. This is, indeed, the solution which has usually been given of this problem ; but it is radically wrong, which fact was iirst demonstrated by Professor MacCord. There is, of course, no doubt but that, by rolling a describing hyperboloid within one pitch hyperboloid and without the other, we will obtain two sur- faces whose element of contact lies in the surface of the describing hyperboloid. But, in order that such surfaces should be available for tooth surfaces, it is essential that this line of contact should be a line of tangency. And it is just here that the whole construction fails. By an elegant method of demonstration, Professor MacCord has proved 158 ELEMENTARY MECHANISM. that this common line of contact is a line, not of tangency, but of intersection; so that these surfaces simultaneously generated by the rolling of the describing hyperboloid ai;e instantaneously destroyed by each other. This method, then, is inapplicable to the case of skew bevel wheels ; but the tooth surfaces of such wheels may be constructed by taking advantage of a peculiar property of the involute. 145. Teeth of Skew Bevel Wheels In Fig. 74 we have shown a pair of wheels with involute teeth, DE being the line of action. In the figure the wheels are in tha same plane, and the point of contact is always situated in the line DE. The upper wheel remaining fixed, suppose the plane of the lower wheel to be revolved through any given angle about the line DE, as on a hinge. The two wheels will now lie in different planes, their, axes being neither parallel nor inter- secting. The line DE will be the intersection of these two planes ; and tlie position of each wheel in its own plane, with reference to tliat line, is unaltered. But DE is the locus of contact ; and, as the position of neitlier wheel with reference to DE has been changed, it follows that the velocity ratio of the wheels will not be affected by the inclination of their planes. When the wheels are so inclined, they can, of course, move only in the direction which makes DE the locus of contact. If they are required to move in the reverse direction, they must be swung about a line similarly inclined to the line of centres in the opposite direction ; but it is evi- dent that in no case can they drive in both directions except when they are in the same plane. This property of invohUe teeth, of transmitting motion between axes neither parallel nor meeting, is only true when the wheels are very thin ; so that in practice the teeth of one wheel must be rounded so as to touch those of the other in points only, and not in lines. It is possible, however, to employ teeth of this kind in MOTION BY SLIDING CONTACT. 159 wheels of any thickness by retaining the involute form for the teeth of one wheel only, and modifying the shape of the other, so as to produce teeth which shall act together in right- line contact. The fronts of the teeth of one of the wheels are made of such a shape that all sections by planes perpendicular to the axis of the wheel shall be involutes ; the shapes of the cor- responding sections of the fronts of the teeth of the other wheel are then constructed by a modification of the method of Art. 78. 146. Approximate Method Teeth laid out by the above process will not be symmetrical, their fronts being single-curved surfaces, and their backs warped surfaces. Their theoretical construction, as well as their practical man- ufacture, are both complicated and laborious operations. The result is, that such wheels are rarely employed ; the most usual means of transmitting motion under such circumstances being the employment of two pairs of bevel wheels, as explained in Art. 47. A.n approximate method of laying out the teeth of skew bevel wheels consists in drawing two cones normal to the byperboloidal frusta, developing their curved surfaces, and after laying out teeth on them, as in Tredgold's method for bevel wheels, wrapping them back in their proper relative positions. The surfaces of the teeth will be formed by joining the corresponding points of these curves by right lines. The teeth should be small and numer- ous, and the frusta should be placed as far as convenient from the common perpendicular of the axes. 147. Face Gearing' Before the introduction of bevel gearing, the problem of transmitting motion between axes that were not parallel was usually solved by means of face gearing. Let two face wheels with cylindrical pins, exactly alike in every respect, be placed in gear, as shown in plan and elevation in Fig. 98, with their axes at right angles ; the latter not meeting in a point, but having their common per- 160 ELEMENTARY MECHANISM. pendicular equal to the diameter of the pins. Tlieh will these wheels revolve together with the same angular velocity. T'iU. 98 Let B be the driver, and let the pins c, g, he in contact. The distance between the axes of these pins is the sum of the radii of the pins ; that is, the diameter of a pin, or, wliat is equal to this diameter, the pei-pendicular distance between the axes of the wheels. Let the driver B turn, in the direction of the arrow, through one-sixth of a revolution ; the pin g moving to the MOTION BY SLIDING CONTACT. 161 position e, and driving before it the pin c to tlie position 6. Tlie distance between the axes of the pins is equal to the diameter of a pin, as before ; and consequently the length of the perpendicular let fall from g on Be must equal the length of the perpendicular let fall from c on Ah. In other words, Bg sin gBe = AcsinbAc; and, as Bg = Ac, we have sin gBe = sin hAc. Hence angle gBe = angle bAc, which proves the equality of the angular velocities. The driver was in this case supposed to turn through an angle of sixty degrees ; but this was merely a matter of con- venience, as the same proof could have been applied to any other angle. The pin g must not be so long that its end will come into contact with the piu h, as the wheels revolve in the directions of the arrows. This consideration fixes the max- imum length of the pins, which is the same in both wheels. 148. Axes Intersecting-. — As the common perpendic- ular to the two axes becomes less, the- diameter of the pins decreases ; so that, when the axes intersect, the pins become mere lines. In order to transmit any power, the pins must manifestly have some thickness ; but they cannot be cylin- drical on both wheels. The pins on one wheel may, how- ever, still be cylinders, in which case the shape of those on the other is found by a method analogous to that employed in pin gearing. In Fig. 99, let the axes intersect at right angles, and let A and B be two equal wheels, whose pins, consequently, have become reduced to mere lines. Instead of having the corresponding pins of the two wheels in contact, let them be separated by some arbitrary distance, as shown in the figure. Now, if both wheels be turned, in the directions of the arrows, with the same angular velocity, it is evident that the common perpendicular between any two corresponding pins will change according to the positions of the pins at any instant. If, still using mere lines for the pins of the upper wheel, we now expand the pins of the lower wheel into solids of revolution, the radius of whose 162 ELEMENTARY MECHANISM. cross-section shall at any point be equal to this common perpendicular, it is evident that the two wheels will work together with a constant velocity ratio. In order to obtain the meridian section of these solids of revolution, we pro- ceed as shown in Fig. 100. Let MN and R8 be the pitch circles, of which the former is to carry the cylindrical pins. Fig. 99 Let p and q be the positions of the axes of the pins at the moment when contact is to begin, the common perpendicular pC having been assumed at pleasure. This gives us the radius of the solid of revolution at its upper surface. The curve pr is the locus of the extremities of the perpen- dicular distances x found for a number of corresponding positions of the pins p and g as they move through the equal arcs pP and qQ ; and the curve p'r is the locus of extremi- MOTION BY SLIDING CONTACT. 163 ties of the perpendicular distances y similarly found while the same pins move through the equal arcs Pp' and Qq' . The curves pr and p'r' will work properly with pins that are mere lines, and whose projections are, consequently, mere points. On expanding these theoretical pins into practical cylinders, their projections will become circles, and the desired curves for the lower pins must be found, as shown in the figure, by the same process as that employed in pin gearing (Art. 122). W A Fig. lOO 149. From a consideration of the arcs involved, it can be readily proved that j)c is greater than p'K, and, in general, that the values of x are greater than the corresponding values of y. This difference is greatest at the top of the solid, and decreases as we descend, becoming zero at the bottom ; i.e., rD = r'H. This difference of corresponding radii is, of course, also true for the derived curve ; so that, if we desire the action to take place on both sides of the line of centres, the solid cannot be a solid of revolution, but must have the part which lies without its pitch circle constructed with the meridian curve derived from pir, and the part which lies within its pitch circle constructed with the meridian curve derived from p'r'. If it is desired to form the cogs in the lathe, which is usually the case, the axis of the solid of 164 ELEMENTARY MECHANISM. revolution will coincide with the centre ; and the latter curve, being the smaller, must necessarily be used. In this case the action will only be maintained while the cylindrical pin lies between the cog and the plane of centres ; and, as receding action is preferable to approaching action, it follows that the cylindrical pin must be given to the driver, and the cog to the follower. By similar methods we may find the shape of the pins when the wheels are unequal or when the axes are not at right angles.^ The principal advantage of face gearing is the facility of making the pins and cogs in a lathe ; but, on the other hand, we have the serious drawback, that the pressure between the teeth is only exerted at a single point. Where the pressure is very light, so that the teeth merely polish each other, this kind of gearing may often be employed to advantage ; but where the pressure is at all heavy, they are unsuitable, as the teeth cut each other, and soon wear out. In face gearing, a derangement in the relative position of the two wheels, if it take place in the direction of the axis of the wheel with cylindrical pins, will not interfere with the action of the gearing. MOTION BY SLIDING CONTACT. 165 CHAPTER IX. COMMUNICATION OP MOTION BY SLIDING CONTACT. VELOCITY EATIO AND DIRECTIONAL RELATION CONSTANT OR VARYING. Cams. — Endless Screw. — Slotted Link. — Whitworth^s Quick Me- turn Motion. — Oldham's Coupling. — Escapements. 150. In the last four chapters have been discussed the cases of sliding contact where both the velocity ratio and the directional relation were necessarily constant ; in the present chapter will be presented the various arrangements in which either or both of these may vary. 151. A Cam is a plate which transmits motion to its fol- lower by means of its curved edge, or by means of a curved groove cut in the surface of the plate. When the motion is small or intermittent, such plates are often called tappets, or wipers. In most cases which occur in practice, the conditions to be fulfilled in designing a cam or wiper do not directly involve the velocity ratio ; usually a certain series of definite posi- tions is assigned which the follower is to assume when the driver is in a corresponding series of definite positions. In cam motions, the motion of the follower is usually derived from the cam by means of a cylindrical roller turning about a smaller pin as an axis, the latter being rigidly fastened to the follower. This has the advantage that nearly all the wear is concentrated on this axis, which may readily be 166 ELEMENTARY MECHANISM. renewed when worn out. If the pin is to be driven by the cam in one direction only, being made to return by the force of gravity or the elastic force of a spring, the cam need only have one acting edge ; but if the cam is to drive the pin in both directions, it must have two acting edges, with the pin between them, so as to form a groove or a slot of a uniform width equal to the diameter of the pin, with clearance just sufficient to prevent jamming or undue friction. The centre of the pin may be treated as practically at all times coinciding with the centre line of such a groove, which centre line may be called the pitch line of the cam. The most convenient way to design a cam is usually to draw in the first place its pitch line, and then to find the acting edge or edges by the process of Art. 122 ; using a radius slightly greater than that of the pin in case two edges are employed. 152. Construction of the Cam Curve. — In Fig. 101, let A be the centre of motion of the proposed cam, and SH the path of the centre of the pin on the follower ; the con- dition being that the centre of the pin shall start from the point H, and assume in succession the positions E, D, C, and B while the cam revolves through successive angles of thirty degrees. With centre A and radius AH, describe tlie circle JUST; produce the radius AH to S, and draw the other radii (produced) , AK, AL, AM, and AN, at successive angu- lar intervals of thirty degrees. With centre A, draw circular arcs through the successive positions E, D, G, B, of the pin, and on these arcs lay off the distances Kk = Co, LI = Dd, Mm = Ee. Then will k, I, and m be points of the cam curve required. The curve nmlkB, drawn through these points and N and B, will be the curve which will fulfil the required conditions ; for, assuming m to be at H, and the cam to revolve in the direction of the arrow, it is evident that as the radii AM, AL, AK, and Ah successively come into the position AS, the joints m, I, k, and B of the cam MOTION BY SLIDING CONTACT. 167 curve will coincide with E, D, C, and B respectivelj', thus driving the pin as required. To And the curve for a pin of sensible diameter, we proceed as in Art. 122, drawing circles of the same diameter as the pin in a sufficient number of positions along the pitch line already found, and then draw- ing the acting edge tangent to these circles. When the path of the pin passes through the centre of motion of the cam, the distances -Be, Del, etc., all reduce to zero ; and the pitch line is drawn through the points of inter- section of the successive radii and the circular arcs through the corresponding positions of the pin. As the angle BUS increases, the action between the edge of the cam and the pin becomes more oblique, thus increasing the friction ; and it is hence advisable to make that angle as small as possible ; in other words, the path of the pin should point as near as possible to the centre of motion of the cam. In case the motion of the follower is required to be uni- 168 ELEMENTARY MECHANISM. form, the distances HE, ED, DC, and CB would all be equal, but no modification of the method of construction would thereby be introduced. 153. Another Example In Fig. 101 the path of the follower is a straight line, and the cam has uniform motion about a fixed centre. But none of these conditions need be adhered to. The path of the follower may be any curve whatever, and it may move in this path in either direc- tion, and with uniform or varying velocity. The cam usually revolves about a centre, or has rectilinear motion ; but its velocity may also be varied at pleasure. All these possible variations give rise to an endless variety of shapes for the cam curves, but the principles underlying their construction are always the same. Thus, in Fig. 102, let the path of MOTION BY SLIDING CONTACT. 169 the pin be the curved line HB, and let the i)in successively occupy the positions F, E, D, etc., while the cam revolves in the direction of the arrow through the unequal angles NAM, MAL, LAK, etc. The radii being drawn at the given angles, circular arcs are drawn through F, E, D, etc., and the points of the curve found, just as in Art. 153, by making Rr = Cc, KJc = Dd, etc. 154. Cam for Complete Revolutions In Figs. 101 and 102 the directional relation is constant ; in other words, the direction of rotation of the cam must be reversed in order to bring the pin down again to H. But this may ho accomplished by simply adding to the curve of the cam, in which case the latter may revolve continually in the same direction. The law of motion of the pin in one direction may be entirely different from that in the other direction, and the pin may be given an interval of rest at any eleva- tion by making the corresponding part of the cam curve an arc of a circle. In Fig. 103, let A be the centre of motion of the cam, and let the vertical numbered line be the path of the follower. The cam is to revolve uniformly at the rate of one revolution in twelve seconds. Each number on the vertical line shows the required posi- tion of the pin at the end of the second indicated by that number. Draw twelve equidistant radii, and draw circular arcs through the various positions of the follower. Making la = la', lib = 2b', Illf = 3/', IVk = 4k', etc., we find the points of the curve, as before. The interval of rest indicated by the coincidence of the numbers 7 and 8 is obtained by means of the circular arc cc. The cam in the figure is drawn in the position when the pin is at the point 012; and the cam is ready, by one complete revolution in the direction of the arrow, to cause the pin to go tlnough the cycle of motion required. 170 ELEMENTARY MECHANISM. MOTION BY SLIDING CONTACT. 171 1 55. Cam moving in Straig-lit Path. — In all the pre- ceding eases we have assumed the cam to revolve about some fixed centre of motion. But this is not a uecessary con- dition ; it may move in any path whatever. In practice, however, there is but one other path employed; viz., the straight line. In Fig. 104, let ABCD be a flat rectangular A VII VI V IV III II V P123 4 5 6 ■>N Fig. 104 plate moving in the direction of its length, and let the path of the follower be the line MN, at right angles with the direction of motion of the plate. Let the pin of the follower start from a position of rest at P, and move with a gradually accelerated velocity, so that it occupies the positions 1, 2, 3, 4, etc., at the end of equal successive intervals of time. Lay ofif on AB the distances MI, I II, II III, III IV, etc., through which the plate moves during the same intervals. In the figure the plate is supposed to be moving uniformly, and these distances are consequently equal ; but they may vary according to any assigned law. Draw the lines la, in, IIIc, IVd, etc., parallel to MJST, and the lines la, 26, 8c, 4d, etc., parallel to AB. Their intersections, a, b, c, cl, etc., will be points of the required cam curve. From this the working curves are derived in the usual manner, being the 172 ELEMENTARY MECHANISM. sides of a groove or slit of the proper width in the plate. The theoretical curve above found is the centre line of the groove. In the special case shown in the figure, the lines la, la, lib, 2b, etc. , are at right angles ; the angle between them is always the angle between the directions of motion of the plate and the follower. If in Fig. 104 we make the velocity of the follower also constant, the cam curve will become a straight line ; for instance, if we assume that the follower is to traverse the distance P~, with a uniform velocity, during the same time that the plate moves, also with a uniform velo- city, over the distance M VII = "ig, tlien the straight line PG must be the cam line required. This line will evidently be the hypothenuse of a right triangle, the other two sides of which are the lines representing the respective distances traversed by the plate aud the follower in the same time. The velocity ratio of the cam plate aud tlie follower in this special case is evidently constant, and is simply the ratio of the isochronous distances above mentioned. 156. The Screw If a plate with a straight slit or groove, as just described, be wrapped around a cylinder whose axis is parallel to the path of the follower, the slit in the plate will become a spiral groove in the cylinder. If the cylinder be revolved uniformly, this groove will impart pre- cisely the same motion to the follower as before. If the length of the plate be greater than the circumference of the cylinder, the spiral groove will "encompass its surface through more than one convolution, and may in this way proceed in many convolutions from one extremity of the cylinder to the other. Such a recurring spiral is called a screw. The inclination of the spiral to a line drawn on the surface of the cj'linder parallel to the axis is constant, and is the same as the inclination of the straight line in the flat plate to the path of the follower. The pitch of a screw is the distance between successive MOTION BY SLIDING CONTACT. 173 convolutions of the spiral measured along a rectilinear ele- ment of the cylindrical surface. Tlie screw is sometimes made in this elementary form, consisting of a simple spiral groove which gives motion to a slide, by means of a pin fixed to the latter, and lying in the groove ; but generally screws receive a more complex arrangement. In the first place the pitch is made comparatively small, the necessary motion of the follower being secured by a corresponding increase in the number of revolutions of the screw. The convolutions of the groove are brought so close together that the ridge which separates two contiguous grooves becomes the counterpart of tlie groove itself. This ridge is termed the thread of the screw ; and from its section the screw derives its distinctive title, such as square- threaded, V-threaded, and round-threaded. In the second place, instead of a single pin, other pins may be fixed to the follower opposite the other convolu- tions ; then, since each pin will receive an equal velocity from the revolving cylinder, the motion of the follower will be effected as before, with the advantage of an increased number of points of contact. But this series of pins may be replaced by a short comb or rack, the outline of which exactly fits that of the threads of the screw". This is the most ancient form in which the screw was employed. Most commonly, however, the piece which receives the action of the screw is provided with a cavity embracing the screw, and fitting its thread completely ; being, in fact, a hollow screw corresponding in every respect to the solid screw. Such a piece is termed a 7iut, and the hollow screw an inside screw, the solid screw being then called an outside screiv. These modifications are only introduced to distribute the pressure of the screw upon a, greater surface ; for, as the action of the thread is exactly alike upon every section of the nut, the result of all these conspiring actions is the same ; 174 ELEMENTARY MECHANISM. namely, that the piece to which the pin or comb or nut is attached advances in a direction parallel to the axis of the screw through "a distance equal to the pitch for every revolu- tion. 157. A screw may be right-handed or left-handed; the majority of screws are the former, the latter being used only when other conditions make it necessary. Supposing the nut to be fixed, a right-handed screw will enter its nut when turned in the direction of the hands of a clock ; a left- handed screw must be turned in the opposite direction. If the inclination of the thread of a screw to the recti- linear elements of the cylinder be very great, one or more intermediate threads may be added. In such cases the screw is said to be double-threaded, triple-threaded, etc., according to the number of separate spiral threads on the cj'linder. Screws whose pitch is an aliquot part of an inch are usually classified by mentioning the reciprocal of the pitch ; i.e., if a certain screw has a pitch of one-quarter of an inch, it is spoken of as having four threads to the inch. During one complete revolution of any screw, the follower will evidently move through a space equal to the pitch of the screw; i.e., through a space equal to the distance between successive convolutions of the same spiraL measured on a rectilinear element of the screw cylinder. When the comb or rack form above spoken of is used, the screw is frequently made short, and the rack lengthened. If it is essential that the screw shall always remain com- pletely in gear with the rack, then the maximum length of path described by the latter will be the difference between' their lengths. 168. Endless Screw. Worm and Wheel. — From the rack driven by a short screw, we readily pass to the so- called endless screw, shown in Fig. 105. In this combina- tion the screw, or worm,BB, gives motion, uot to a rack, but to the wheel C. The screw is so mounted that it can have MOTION BY SLIDING CONTACT. 175 no motion except that of rotation, and tlie wheel has teeth of the same pitch as the screw thread. If the screw axis be turned around, every revohition will cause one tooth of the wheel to pass across the line of centres ; and as this action puts no limit, from the nature of the contrivance, to the number of revolutions in the same direction, a screw fitted up in this manner is termed an endless screw, in oppo- sition to the ordinary screw, which, when turned around a certain number of times either way, terminates its own action by bringing the nut to the end of the thread. I^ig. 105 159. Shape of tlie Teeth. — If we make any meridian section of the screw, we will find it to be a rack ; in fact, the screw may be moved in the direction of its length, and will then drive the wheel precisely in the manner of a rack. Consequently, if the wheel be merely a thin plate, we need only make the meridian section of the screw a rack with teeth laid out in the usual manner to work correctly with the assumed wheel tooth. But as in practice the wheel must 176 ELEMENTARY MECHANISM. be given some thickness, it is necessary to determine the proper form of the teeth for that case. If we make a series of sections through Z), E, etc. , parallel to the mid-plane of the wheel, we will still find the section of the screw to be a rack, though the outlines of the teeth will change in shape in each section. If, now, we make the outline of any section of the wheel tooth of the proper shape to gear correctly with the outline of the corresponding section of the screw tooth, considered as the tooth of a rack, we shall evidently have a point of contact between the corresponding teeth in every section. In fact, if we make a wheel tooth whose shape is continu- ally changing in every section to correspond to the change in the same section of the screw tooth, we shall have the teeth in contact at each instant along a line, so that the wear will be distributed along a surface. Such a screw is called a close-fitting or tangent screw. 160. Practical Metliod of Cutting: Wlieel Teeth, — The practical difficulty of making the teeth of a wheel of which the form in every parallel section shall be different, is very simply overcome by making the screw cut the teeth. An exact copy of the tangent screw is made of steel, the edges of its threads are notched, and it is then hardened, so that it becomes a cutting tool. It is then mounted in a suit- able frame, so as to gear with the roughly formed teeth on the wheel, and turned so as to drive them ; in the course of which operation it cuts them to the proper figure. The axis of the cutting screw is placed at first at a distance from the axis of the wheel somewhat greater tlian the intended per- manent distance ; and, after each complete revolution of the wheel, the axes are brought nearer together, until the per- manent distance is attained ; and, by turning the screw ,in this last position, the shaping of the teeth is finished. An involute wheel tooth working with a screw tooth whose meridian section has straight, sloping sides, is the best MOTION BY SLIDING CONTACT. 177 combination, as the successive diminutions of the distance between the axes will not affect the velocity ratio (Art. 108). In order to secure a good arc of action, and diminish obliq- uity, such wheels should not be given less than about thirty teeth. In order to avoid weak corners in the wheel teeth, their sides are usually bounded by straight lines, BH and BK, radiating from the axis of the worm ; and the angle HBK usually varies between sixty and ninety degrees. 161. Hour-Glass Worm. — Instead of making the pitch surface of the worm a cylinder, we may make it conform to the curvature of the wheel. In that case its pitch surface will be the surface produced by revolving an arc of the wheel pitch circle about the axis of the worm, thus forming the shape from which the worm derives its name. This arrangement is also named, after its inventor, Hindley's screw. The acting surfaces of both the worm and the wheel are very peculiar ; but the arrangement may, neverthe- less, be very easily constructed in practice. Just as m the ordinary tangent screw, we must first pre- pare a cutting screw. To obtain this, a tool whose cutting edges are formed in the shape of the proposed wheel tooth is so clamped to a horizontal revolving plate of the size of the proposed wheel that the plane of its cutting edges passes through the axis of the worm. The plate and the worm blank being rotated at their proper relative velocities by means of some interposed mechanism, the distance between the two axes is gradually diminished until the permanent dis- tance is reached, during which operation the worm will be cut to the proper shape. By taking such a worm, notching Its edge to make a cutting tool of it, the wheel teeth can then be cut just as in the ordinary worm and wheel. Such teeth are in contact along a line in the meridian plane of tlie screw, but do not come in contact along the whole surface of a wheel tooth. 178 ELEMENTARY MECHANISM. 162. The endless screw falls under the case of two revolving pieces whose axes are not parallel and do not meet. It communicates motion very smoothly, and is equiv- alent to a wheel of a single tooth, because one revolution passes one tooth of the wheel across the line of centres ; but, generally speaking, it can be employed only as a driver, on account of the great obliquity of its action. A worm may be multiple-threaded, just like any other form of screw, and, in that case, will pass as many wheel teeth across the line of centres for every revolution of the worm as there are separate threads on the latter. The practical process of cutting the teeth is, however, the same as before. 163. Screw to produce VariaWe Motion In all the cases previously described, the screw has been supposed to have a uniform pitch, and hence to produce a uniform motion in the follower. But we may impart any. motion whatever ; the only condition being, that the pitch of the fol- lower must not deviate much from a straight line parallel to the axis of the screw. As the inclination of the spiral groove varies, the velocity of the follower changes ; a period of rest of the follower being obtained by making the inclina- tion zero. A small intermittent motion may readily be obtained by making the groove in the shape of a simple ring, except at a certain portion, where it deviates the necessary amount. If it be required that the follower shall move back and forth while the screw revolves continually in the same direc- tion, the spiral must be cut in both directions ; in which case the follower cannot be a rack or nut or wheel, but must be a single pin or similar piece. On the cylinder of the screw are cut two complete spirals, one right-handed and the other left-handed, joined together at their ends ; so that the two screws form one continuous path, winding around the cylinder from one end to the other and back again continuously. When the cylinder revolves, the piece which lies in this MOTION BY SLIDING CONTACT. 179 groove, and is attached to the followei', will be carried back- wards and forwards ; and each total oscillation will corre- spond to as many revolutions of the cylinder as there are convolutions in the compound screw. As the screw grooves necessarily cross each other, the piece that slides in them must be made long, so as to occupy a considerable length of the groove ; thus making it impossible for it to quit one screw for the other at the crossing places. Also, as the inclina- tions of the two screws are in opposite directions, it is neces- sary to attach that piece to the follower by means of a pivot, so as to allow it to turn through a small arc as the inclina- tion changes. By varying the inclination at different points, the velocity ratio may be varied at those points. 164. Pin and Slotted Cranlc. — In Fig. 106, let .1 be the centre of rotation of an arm, AP, carrying at its extrem- ity a pin, P, which slides freely in the slot in the piece BC. The latter lias its centre of rotation at B. If the arm AP be revolved uniformly, it will impart a variable velocity to the arm BO. Let Pa, perpendicular to AP, represent the lineai- velocity of the pin in the circle MN. Draw an indefinite line per- 180 ELEMENTARY MECHANISM. pendicnlar to PB at P, and let fall on it the perpendicular ah ; then will Pb be the linear velocity of the point P of the arm BC at that instant. Let a = angular velocity of the arm AP, and a! = angular velocity of the arm BG. Also let the constant length AP be designated by R, and the variable length BP by r. Then Pa , Ph a = — , a = ^ . it r But Ph = Pa cos APb = Pa cos APB, hence , Pa cos APB , a R , „„ a = , and — = — cos APB. "When APB = 0, that- is, when both the arms lie in the line of centres DE, the limiting values of the velocity ratio will be obtained., When P is at E, the velocity ratio — has a its maximum value, — = -—. The ratio' becomes r B — AB smaller as P leaves E and approaches D. ai which point — has its minimum value, — = r B -{- AB So long as AB is less than is!, we may by this means cause one arnj to revolve with a variable velocity by means of another arm revolving uniformly. When AB exceeds R, the second arm merely swings on each side of the line of centres through an angle whose sine is — — -. When it is at the end of its outward swing, the AB °' angle APB = 90° and - = :? x = 0; showing that for u. T that instant no motion is imparted to the arm BO by the rotation of AP. MOTION BY SLIDING CONTACT. 181 The necessary length of slot when B lies within the circle MN is the diameter of the pin + BD — BE ; when B lies without MJSf, the necessary length of slot is the diameter of the pin + 2AP. 165. Wliitworth's Quick Ketiirn Motion. — If in Fig. 106 we attach a connecting-rod to the end G of the arm BC, and compel the other end of the rod to move in a straight line perpendicular to AB at B, we will have a com- bination such as is represented by Fig. 107. The length of the stroke is evidently 2B0. If the arm AP be revolved uniformly, the forward and back strokes of Q will be made in times proportional to the arcs aclb and acb. We thus have a form of " quick return " motion. This has been applied in a modified form, as shown in Fig. 108, to a shaping machine, by Sir J. Whitworth. Fig. 107 In the figure, D is a plate spur wheel which turns about its centre A, upon the large fixed shaft. A pin, P, fixed in and projecting from the face of wheel D, corresponds to the point P in Fig. 107, so that AP is the arm which revolves uniformly. A pin, B, eccentric in the large shaft, is the centre about which the arm of varying length turns ; BP cor- 182 ELEMENTARY MECHANISM. responding to BP oi Figs. 106 and 107. A crank piece, E, turns al)oat 23, and has a slot in one end, in which P shdes. To tlie opposite end of this piece the connecting-rod is attached. Tlie end Q of the rod is attached to the sliding head which carries the cutting tool. B^ig. lOS As D revolves, motion is given to Q by means of the pin P and the crank piece, and the varying distance of P from B exactly replaces the arm of varying length. The length of stroke is adjusted by altering the position of C in that end of the crank piece, thus changing the length of the crank arm BC, but in no way affecting the ratio of the periods of advance and return. Thus, for example, if the arc ach (Fig. 107) is one-third of the circumference, adb being two-thirds, the period of advance is to the period of return as 2 is to 1, without regard to the actual length of stroke. 166. Pin and Slotted Sliding Bar. — In Fig. 109, let the pin P be fixed at the extremity of the uniformly revolv- ing arm AP, as before. The piece B is free to move in the direction CD or DG only, and has in it a slot perpendicular to the line DC, in which the pin slides. Let Pa = V = linear velocity of the pin in the arc of the circle ; then v, equal to linear velocity of the piece B in the direction AG^ will be found, as in the last article, by dropping the perpen- dicular ah on tlie line P&, the latter being perpendicular to the line of the slot. Hence MOTION BY SLIDIKG CONTACT. V Ph Tr— TT = sinPaS = smPAB, V Pa 183 hence V = Vsm PAB. When PAB is or 180°, i.e., when P is at D or E, we have V = Q\ when PAB = 90° or 270°, v = V. This motion of B, varying between and F, and going from to F and from F to twice in each revolution of AP, is called harmonic motion. ^^ The necessary length of slot is iAP + diameter of pin. The length of the path of B is 2AP. This arrangement is much used in some varieties of pumps to connect the ily-wheel shaft with the piston rod. 167. Cam and Slotted Sliding- Bar. — In Fig. 110 is shown an example of a cam which, by its uniform rota- tion, produces a motion similar to that of Fig. 109, but with intervals of complete rest. The cam consists of a triangular piece ; the sides of the triangle being three equal arcs, each described around the point of intersection of the other two. The cam revolves uniformly ahout one of its corners, as A. 184 ELEMENTARY MECHANISM. The. follower is the slotted bar BB, and the cam acts upon the two straight edges of the slot, the distance between which is equal to the radius of curved edges of the cam. Consequently the slot will be in contact with an angle and a side of the cam in every position, and the motion produced is as follows : Let the circle described by the outer edge of tlie cam be divided into six equal parts, as in the figure. Tracing the motion as the angle m of the cam goes round the circle'in the direction of the numbers, it appears that no motion will be given to the bar while m is moving from 1 to 2. While m travels from 2 to 3, the face Am drives the upper side of the slot with an increasing radius ; and hence the bar begins to move, and its velocitj' gradually increases. While m travels from 3 to 4 the action is similar to that of Fig. 109, and the motion of the bar will gradually be decreased until m reaches 4, when the bar will come to rest. As m moves from 4 to 5 the bar remains, at rest; from from 5 to 6 the bar begins to move with an increasing velocity; from 6 to 1 the bar moves with a decreasing velocity, coming to rest as m reaches 1. • 168. In case the direction of motion of the follower intersects the axis of motion of the cam, the latter may be made in the shape of a screw thread on a cone ; when the MOTION BY SLIUING CONTACT. 185 follower's direction neither intersects nor is parallel to the cum axis we may employ a screw thread ou a hyperboloid of revolution. lu fact, almost any kind of motion may be obtained by means of a suitably shaped cam ; but the general principles employed in the various cases above treated of apply equally well to any other special cases. 169. Oldham's Coupling. — In Fig. Ill is shown a method of communicating equal rotation by sliding contact between two axes whose directions are parallel. Aa and Bb are the axes, each of which is furnished with a forked end, terminated by sockets bored in a direction to intersect the respective axes at right angles. The whole is so adjusted that all four sockets lie in one plane perpendicular to both axes. A cross with straight cylindrical arms is fitted into the sockets in the manner shown iu the figure, and its arms are of a diameter that allows them to slide freely in their respect- ive sockets. If one of the axes be made to revolve, it will drive the other with the same angular velocity. Fig. Ill For let the sketch at the right be a section through the cross perpendicular to the two axes Aa and Bb, and let the large circles be those described by their respective sockets. Then, if C be a socket of Aa, the arm of the cross which passes through it must meet the centre A ; and in like man- 186 ELEMENTARY MECHANISM. ner, if 2) be a socket of Bb, the arm DB must pass througli the centre B. Also, if G move to C", the new position (clotted lines) of the cross will be found by drawing C'A through A, and BD' perpendicular to it through B. And it is evident that the angle C'AC = angle I/BD, hence the angular velocity is the same in both axes. In every position of the cross we will have the triangle APB, in which the side AB'is constant, and the angle APB opposite to it is always a right angle. Hence the locus of P must be the circle whose diameter is AB ; i.e., the centre of the cross will travel around the small dotted circle whose diameter is the distance between the axes. Also every arm will slide through its socket and back again during each revolution through a space equal to twice the distance between the axes. In practice this arrangement is usually made in the shape of two discs, with a bar sliding in a diametral slit in each ; the two bars being rigidly connected in the form of a cross. 170. An Escapement is a combination in which a toothed wheel acts upon two distinct pieces or pallets attached Fig. 113 to a reciprocating frame, so that, when one tooth ceases to act on the first pallet, a different tooth shall begin to act on the second pallet. A simple example is shown in Fig. 112. The wheel A revolves continually in the direction of the MOTION BY SLIDING CONTACT. 187 arrow. The frame has two pallets, o! and e, and can only move in the direction of its length. In the position shown, the tooth a is just escaping from the tooth d, and 6 is just ready to come in contact with e, by which the frame will be driven to the left. The shapes of the teeth may be designed as usual for a wheel and rack, and the point of quitting con- tact is found by the intersection of the addendum line of the wheel teeth with the describing circle of the pallets. The number of teeth on the wheel must evidently be odd. But the frame may be used as the driver, instead of the wheel, by moving it alternately in each direction. This will cause the wheel to revolve in the opposite direction to that in which it would itself produce the reciprocation of the frame. But, when the frame is the driver, there is always a short interval at the beginning of each stroke, during which no motion will be given to the wheel. 171. Crown-wheel Escapement The crown-wheel escapement is used for causing the vibration of one axis by means of the uniform rotation of another. The latter carries a wheel consisting of a circular band, with large teeth, like those of a saw, on one edge. The vibrating axis, or verge, as it is often called, is located immediately above the crown wheel, and in a plane at right angles to the wheel axis, the latter being vertical. The verge carries two pallets, project- ing from it in directions at right angles, and a sufficient distance apart so that they may engage alternately with teeth on opposite sides of the wheel. By this alternate action a reciprocating motion is set up in the verge. The rapidity of this vibration depends largely on the inertia of the verge, which may be adjusted by attaching a suitably weighted arm to the latter. This escapement, though but rarely used at the present day, is of interest as being the first contrivance used in a clock for measuring time. 172. Anchor Escapement. — In Figs. 113 and 114 are 188 ELEMENTARY MECHANISM. shown two forms of this escnpement. In Fig. 113 the wheel has long, slender teeth, and turns in the direction of the arrow. The vibrating axis B carries a two-armed piece having pallets G and D at its extremities, and resembling somewhat the form of an anchor, whence the name of the combination. When the tooth g presses against the pallet C, the normal at the point of contact passes on the same side of the centres A and B ; hence (Art. 30) the tooth will tend Fig. 1X3 to turn the pallet in the same direction as the wheel. BC will therefore turn upwards, and allow the tooth to escape from the pallet. At this instant the tooth h will begin to act on the pallet D ; and, as the normal here passes between the centres A and J3, BD will move in opposite direction to the wheel, and hence the tooth k will escape. The teeth in an anchor escapement are often replaced by pins, in which case the form of the anchor may be so altered that the action shall take place entirely on one side of the line of centres, as shown in Fig. 114. The rapidity of vibra- tion is controlled by the inertia of a weight or pendulum MOTION BY SLIDING CONTACT. 189 attached to the verge. This very inertia, however, prevent- ing the verge from being suddenly stopped and reversed in direction, causes a recoil action to be set up in the wheel, which materially diminishes the utility of this escapement ; for it is evident that, as the verge cannot be stopped sud- denly, the wheel must of necessity give way and recoil at tlie first instant of each engagement between a tooth and its cor- responding pallet. The greater the inertia due to the load attached to the verge, the more slowly will the escapement work, and the greater will be the amount of recoil. Fig. 114 173. Method of connecting Anchor and Pendu- lum. — There is one uniform method of connecting the anchor and the pendulum, which can be seen in almost any clock. The pendulum, consisting often of a compound metal rod with a heavy bob, is swung by a piece of flat steel spring, and vibrates in a vertical plane very near to that in which the anchor oscillates. To the centre of the anchor is attached a light vertical rod, having the end bent into a hori- zontal position, and terminating in a fork which embraces the pendulum rod. It follows that the anchor and the pen- dulum swing together, though each has a separate point of suspension. 190 ELEMENTARY MECHANISM. 1 74. Action of Escapement on Pentlulum, — In Fig. 113, let the escape wheel tend to move in the direction of the arrow, so as to press its teeth slightly against the pal- lets of the anchor ; the pendulum being hung from its point of suspension by a thin strip of steel, and vibrating with the anchor in the manner already stated. Let the arc abecd be taken to represent the arc of swing of the centre of the bob of the pendulum. As the pendulum moves from d to b, the point g of the escape wheel rests upon the oblique lower sur- face of the pallet C, and presses the pendulum onward until the latter reaches 6, when the point of the tooth escapes at the end of the pallet. For an instant the escape wheel is free ; but a tooth is caught at once upon the opposite side by the oblique upper surface of D, and the escape wheel then presses against the pendulum, and tends to stop it, until finally the pendulum comes to rest at the point a, and com- mences the return swing. During tlie latter the pendulum is similarly at first urged on, and then held back by the action of the escape wheel. This alternate action witJi and against the pendulum pre- vents the pendulum from being, as it should be, the exclusive regulator of the speed of revolution of the escape wheel ; for its own speed, instead of depending solely on its length, will also depend on the force urging the escape wheel round. Hence any variation in the maintaining force will disturb the rate of the clock. 1 75. Dead-beat Escapement. — This objectionable feature is obviated in Graham's dead-beat escapement. Fig. 115. It is, however, most worthy of note that the change in construction which abolishes the defects due to the recoil, and gives the astronomer an almost perfect clock, separates the combination entirely from its original conception ; viz., that of an apparatus for converting circular into reciprocating motion. The improvemept consists in making the lower surface of the pallet C aud the upper surface of the pallet D MOTION BY SLIDING CONTACT. 191 arcs of circles, whose centre is at B. The oblique surfaces qm, rqj, complete the pallets. As long as the tooth is resting on the upper surface of D, the pendulum is free to move, and the escape wheel is locked ; hence in the portion ba of the swing, and back again through ab, there is no action against the pendulum except the very minute friction which takes place between the tooth of the escape wheel and the surface of the pallet. Through the spa.ce be the point of the escape wheel tooth is pressing against the oblique edge np, and is urging the pendulum forward. Then at c this tooth escapes, and the tooth upon the oppo- site side falls upon the lower surface of C, and the escape wheel is locked ; from c to d, and back again from d to c, there is the same friction which acted through ba and ab. From c to 6 the point of a tooth presses upon gm, and urges the pendulum onward ; at b this tooth escapes, another one 192 ELEMENTARY MECHANISM. comes into contact, and so on. It follows that there is no recoil, and the only action against the pendulum is the very minute friction between the teeth and the pallets. The term "dead-beat" has been applied because the seconds hand, which is fitted to the escape wheel, stops so completely when the tooth falls on the circular portion of the pallet. There is none of that recoil or subsequent trembling which occurs in the other escapements. MOTION BY LINKWORK. 193 CHAPTER X. COMMUNICATION OF MOTION BY LINKWOEK. VELOCITY RATIO AND DIRECTIONAL RELATION CONSTANT OR VARYING. Classification. — Discussion of Various Classes. — Quick Setum Motion. — Hooke's Coupling. — Intermittent Linkioork. — Hatchet Wheels. 176. As has been shown by the general definition (Art. 22), linkwork derives its name from the rigid connecting piece or link. This connecting piece is known by various names under different circumstances, such as connecting-rod, coujjlmg rod, side rod, eccentric rod, etc. The arms are known as cranks when they perform complete revolutions ; and as beams, crank arms, rocker arms, or levers, when they oscillate. 177. Classiflcation of Linkwork. — Linkwork is used, I. To transform circular motion into rectilinear reciproca- tion, or the reverse. II. To transform continuous rotation into rotative recipro- cation, or the reverse. III. To transmit continuous rotation. Examples of the first class are seen in slotting and shap- ing machines, power pumps, and in the usual forms of the steam engine ; of the second class, in steam engine valve motions, where a rocker shaft is employed ; and of the third class, in locomotive side rods. 194 ELEMENTARY MECHANISM. Class I. Transformation of Circular Motion into Rectilinear Reciprocation, and the Reverse. 178. In Fig. 116, let AP be a crank revolving about tlie fixed centre A, and connected by a link PQ to a point Q, travelling in a straight line KL whose direction passes through the centre A. Let AP = E, and PQ = I. The length of the path of Q is evidently equal to 2E. When P is at C or Z>, the points A, P, Q, will be in one straight line. The points C and JD are called dead points ; since when P is at either of them, the revolution of AP will cause no motion whatever to, be transmitted to the point Q, for that instant. "When PQ overlaps AP, as when P is at D, we shall term the point D the inward dead point ; and when Q lies at the other extremity of its stroke, so that P is at C, we shall term the point O the outward dead point. In Fig. 116, let fall from P the line PE perpendicular to AQ. Then the distance of Q from A is at any instant, AQ = QE + AM = VP - iJ'sin^e + P cos 0, the last term of which will be essentially negative when 6 lies between 90° and 270°. If PQ were of infinite length, the motion of Q would be equal to that of the point E ; but as PQ is of finite length (usually from four to eight times AP) , Q is drawn toward A through the distance PQ — EQ =^ I - V^^ — B^ sin^ d. So MOTION BY LINKWORK. 195 that when AP has moved to its mid-position Ap or Ap' (or, as it is frequently expressed, AP is on the half-centre) , Q will have passed its mid-position M by the distance qM = AM — Aq = I — sjp — E\ Also when Q is at M, P will be at some point /S or S' (Fig. 117), intermediate between G and p or p'. These points may be readily determined, for in this case AQ = I ; hence AQS and AQS' are equal isos- celes triansles, and cos 9 = = — . ^ AQ 21 The velocity ratio of P and Q varies for each instant, but may be determined at any time by means of the instantaneous centre (Art. 25) or by resolving the velocities, as in Fig. 118. Let V be the linear velocity of P, and v that of Q. Resolve these along and perpendicular to the link PQ ; then, as shown in Art. 24, Pc must be equal to Qf. Draw AN 196 ELEMENTARY MECHANISM. perpendicular to AQ and intersecting the link (produced) at JSf; draw Ad perpendicular to FQ. Then, from similar triangles, we have V ^AP ^ B Pc Ad Ad' hence Also hence V = Pc X R Ad V Qf AN Ad' V = Qf X AN Ad' nations we get V V~ Qf AN AN Pc E R' a variable quantity. It is evident from this expression that when AN = R, the velocities of P and Q are the same. This will occur when AP is perpendicular to AQ,, as at Ap, Ap' (Fig. 116), in which case AP coincides with AN; and it will also occur when AP occupies such a position that the triangle APN is isosceles. To determine the angle 6 which will give this position of AP, we have, from similar triangles, AN-.PE-.-.AQ-.EQ. AN==R = PI!x^ = iJsinef^^^H^^±J.^2?i> EQ \ )/P -R^sin'e ) From this equation we deduce I sm0 = ^^Qj8R' + P-l). 179. The distance through which the point Q is drawn toward A by reason of the finite length of the link (Art. 178) increases rapidly as the link becomes shorter. If we MOTION BY LINKWORK. 197 m.ike the link of the sam,e length as the crank arm, as in Fig. 119, the point K (]?ig. 116) coincides with A, and the path inig. 119 of Q is AL = 2R, as before. But Q is drawn toward AP so rapidly on account of the angularity of PQ, that when Fig. ISO AP is perpendicular to AL, Q coincides with A and has completed its stroke. If we produce QP to V, making 198 ELEMENTARY MECHANISM. PV = PQ = AP (Fig. 120), it is evident that as AP revolves as indicated by the arrow, Q will move from L to A, and V will move from A to W. If now we continue the motion of AP, Q will be driven past A to ilf, and V will return to A. Thus, the revolution of AP will cause Q to move over the path LM, and V over the path WX. By this means, the arm AP can be made to move the two ends of a link of twice its length through paths at right angles, and each equal in length to A:AP. 180. We have thus far considered the end of the link, Q, to travel in a path of which the direction passes through the axis A, but this path may be a straight line not passing through A, as in Fig. 121. In this figure, let AP be a crank revolving about the centre A and connected by the link PQ to the point Q travelling in the straight line KL. An arc of a circle struck about centre A, with radius I — It, will cut the line KL at K, the end of the stroke ; and the inward dead point D will lie in the straight line KAD. Similarly, the other end of the stroke, L, and the outward dead point C, may be found by striking an arc about centre A, with radius AL = I + E. The position of the point Q corresponding to any given position of AP may be thus determined : — MOTION BY UNKWORK. 199 Let the perpendicular distance between A and tlie line KL be AM = e, and let the angle FAM = 0. Then MQ = MS + SQ, MS = Esine; SQ = sIP - Ps^ = V^'- (e ± RcoaOy. Hence we have MQ = Bsind + \JF - (e ± Bcosdy- Also ML = \/{l + By - e\ From these expressions, the distances QL and QK can be determined. 181. By comparing Figs. 116 and 121, it will be seen, that, in the former, the outward and inward dead points, C and Z), lie in one straight line ; while in the latter, they depart from a straight line by the angle DAV = KAL. This angle increases as AM is increased, and as the ratio — I is increased. The practical result is, that, supposing AP to revolve uniformly in the direction of the arrow, the point Q will move over its path from L to K in less time than from K to L, the times being proportional to the arcs DGC and DFC. 182. Eccentric A particular form of this class of link motions is the eccentric, shown in Fig. 122. A circular disc called the eccentric, or the eccentric sheave, has its centre at the point P, and is made sufficiently large to embrace the shaft at^, to which it is fastened. The eccentric is enclosed by a strap or band, FG, in which it revolves. This strap is rigidly connected to the rod or bar HN, by which motion is transmitted to the point Q. It will be seen, that, as the eccentric turns about A and slides within the strap, it will communicate exactly the same motion to the point Q as would be given by a crank arm AP and link PQ. In fact, 200 ELEMENTARY MECHANISM. it is used as a substitute for small cranks on account of the practical difficulties in the formation of the latter. The travel of the point Q will, as in Fig. 116, be equal to 2AP. The term, throw of an eccentric, is given, by various authorities, either to the arm AP, or to twice that distance ; and hence the meaning of tlie term is often ambiguous. Class II. TVansformation of Continuous notation into Rotative Beciprocation, and the Reverse. 183. In Fig. 123, let AP and BQ be arras turning about the fixed centres A and B respectively, and connected by the link PQ. If AP be rotated about A, it will compel BQ to oscillate between the positions Be and Be, or Be' and Be', according as the arm BQ has been previously placed above or below the centre B. Let AP = R, BQ = r. PQ = I, AB = d. To find the dead points : About -4 as a centre describe circular arcs with radii I + R and I — R. They will cut the circle about centre B, radius r, in the points c, c', and e, e', MOTION BY LINKWOEK. 201 respectively. These give the outward and inward dead points for R, and hence the limits of the oscillation of r. Drawing the pieces in these extreme positions, it will be seen that we obtain a series of triangles, of which the base is always the line of centres AB (= d), and of which the other two sides are r, &ud I + B or I - B. We will term these \ s / f \ / \ i / ■--j!. ^^ \ / dead-point triangles. As long as we can construct such tri- angles with a sensible altitude, it is clear that there can be no dead points for r, and hence the rotation of R will cause r to oscillate. But if with any assumed values of R, Z, and r, the triangle will reduce to a straight line, r will have dead points, and we can uo longer control the direction of its motion by the single combination shown. Thus, in order that the rotation of an arm R may produce oscillation of an arm r, we must have r greater than R, and also d + r>l + R. d - rrag Link. — In Figs. 128 and 129, let AP, BQ, be two arms turning about fixed centres A and B, and con- nected by the link PQ, as before. In order that the continu- ous rotation of one may produce a continuous rotation of the other, it is necessary that there shall be no dead points. If the link PQ (= I) is made equal to Cc, it is evident that we will have an outward dead point of jR at C with an inward dead point of r at c. Or, if I = Ce, we will have siraultane- MOTION BY LINKWOEK. 205 ous inward dead points at C and e. Therefore, in order tliat there may be no dead points, we must make I > Cc and < Ce ; T'ig. 12S that is to say, l> r — B + d, and l ^ = ^ = cosA and an ellipse constructed with CD as its major and OK as its semi-minor axis will represent the projection, on a plane perpendicular to the driving axis, of the path of the following I"ig. 14S Suppose a driving arm to move from the position OC to OP through the angle COP = d. Then a line OQ drawn perpendicular to OP will be the projection of a following arm which has moved from OK, while OP moved through the angle COP = EOQ = 6. OQ is perpendicular to OP, since the latter lies in the plane of projection ; and hence the angle POQ is shown in its true size. The point Q has moved through the actual vertical distance Qn, although the actual 216 ELEMENTARY MECHANISM. path of Q is a circle of radius equal to OE. Therefore, if through Q we draw a line parallel to OE, and connect OR, then EOB will be the actual angle through which the follow- ing arm has moved, while the driving arm has moved through COF = EOQ = e. Let angle EOE = ^. tan 4> mR nQ On SQ OK „ tan 61 Om On Om iSB OE Hence (1) tan^ = tan 5 cos/?. To obtain the velocity ratio, we must differentiate this ex- pression, whence (2) ^ = 52!l^cos^ = cos^Lii^n;^ = < ^ ^ dO cos^e 1 + tan> a Eliminating <^ and in turn from Equation (2) by means of Equation (1), we get (3) (4) a' cos j8 a 1 — sin^^sin'^jS a' _ 1 — cos^ sin^ /8 a cos/3 Starting with a driving arm at 00 and a following arm at OK, we measure the angles and rj> from these positions re- spectively. 199. The expressions (3) and (4) will have minimum values when sin 6 = 0, and cos (^ = 1 ; in that case — = cos /?, a and 6 and <^ both = 0, tt, 2 it, etc. That is, the minimum values of the velocity ratio occur when a driving arm is at 00 or OD, and the following arm is at OK or KO produced. MOTION BY LINKWOKK. 217 Maximum values occur when sin 6 == 1 ; cos ^ = 0. In '1 Q this ease — = , and 6 and at the barrel may revolve more slowly, consequently taking more time to uncoil all the cord. For example, let the clock be required to go 8 days with- TRAINS OF MECHANISM. 257 out re-winding ; then, with 16 coils of cord on the barrel, the 8 X 24 latter must revolve once in =12 hours. Then, as- 16 suming w = 100, and p = 8, we may use the train, — Periods. Barrel arbor, 96 12 hours. Hour arbor .8 — 90 1 hour. 12 — 96 8 minutes. Minute arbor . . . .12 — 30, swing-wheel . . 1 minute. It is often convenient to add to the notation the periods of the different arbors, as has been done in this case. 236. Month Clock Let the clock be required to run 32 days without re-winding, and let there be 16 coils on the barrel as before ; then the latter must revolve once in 32 X 24 16 = 48 hours. The train from the barrel to the hour arbor is — = 48, which will require an intermediate axis. Letting w = 100, and^ = 12, we may employ the follow- ing train : — Periods. Barrel arbor, 96 48 hours. 16 — 96 8 hours. Hour arbor . . 12 — 90 1 liour. 12 — 96 8 minutes. Minute arbor . . . . . . 12 — 30, swing- wheel, 1 minute. 237. Now, in the clock (Fig. 169), the arbor of A is made to revolve in one hour, because the wheels E and e are equal. By making these wheels of different numbers, we get rid of the necessity of providing, in the principal train, an arbor that shall revolve in one hour ; and we may thus, in many cases, distribute the wheels more equally. For ex- ample, in an eight-day clock let the swing-wheel revolve once 258 ELEMENTARY MECHANISM. per minute, and let the train from the barrel-arbor to this minute-arbor be D ^ 108 X 108 X 100 F 12 X 12 X 10 = 810, in which case the barrel will revolve once in 810 minutes, or 13^ hours. The second wheel of this train, which, in Fig. 169, cor- responds to D, will revolve in -^^ x 810 = 90 minutes, or 1^ hours. On its arbor must be fixed, as in the figure, the wheels E and F for the minute and hour hands ; and we may employ, for the two pairs of wheels, F 1 12" "" 8 10 J? _ 1| _ 3 _ 54 80 ' ^°*^ 7 ~ T = 2 "" 36* So that our train will be as follows : — Periods. Barrel, lOS 810 minutes. 12 - 108 64 10 ... 90 " 12-100 I : . . . . 10 10-30 s^'ng-j t wheel . ( minute- I hand 80 I hour- j I hand i 1 minute. 1 hour. 12 hours. 238. The above examples have been confined to clock- work, because the action is more generally understood than that of other machines. The principles' and methods are, however, universally applicable, or, at least, require very slight modifications to adapt them to particular cases. For instance, in a screw-cutting lathe, there is usually one intermediate axis between the leading-screw and the head- stock spindle. Let the leading-screw be right-handed, and TRAINS OF MECHANISM. 259 have two threads to the inch ; let tv = 130, p = 20 ; and let it be required to cut a right-handed screw of 13 threads to the inch. Here ^ ^ ^ ^ 13 ^ 130 X 90 F 2 20 X 90' which is a good train for the purpose. The wheels for form- ing a series of such trains, calculated for the different numbers of threads to be produced, are known as a set of change- wheels ; and tables for the use of such wheels are furnished by lathe-manufacturers with all screw-cutting lathes. 239. Frequency of Contact between Teeth. — It is sometimes a matter of interest to know how often any two given teeth will come into contact as the wheels run upon each other. We will take the case of a wheel of A teeth driving one of B teeth, where A is greater than B, and let — = - when reduced to its lowest terms. B b It is evident that the same points of the two pitch circles would be in contact after a revolutions of B, or h revolutions of A. Hence, the smaller the numbers which express the velocity ratio of the two axes, the more frequently will the contact of the same teeth occur. 1. Let it be required to bring the same teeth into contact as often as possible. Since this contact occurs after h revolutions of A, or a revolutions of B, we shall effect our object by making a and 6 as small as possible ; this is, by providing that A and B shall have a large common divisor. For example, assume that the comparative angular velocity of the two axes is intended to be as nearly as possible as 5 to 2. Now make J. = 80, 5 = 32 ; then J. 80 5 ,, — = — = - exactly ; £ 32 2 ^ 260 ELEMENTARY MECHANISM, or, the same pair of teeth will come in contact after 5 revo- lutions of B, or 2 of A. 2. Let it be required to bring the same teeth into contact as seldom as possible. Now change A to 81, and we shall have -— = — = - very nearly ; or, the angular velocity of A relatively to B will be scarcely distinguishable from what it was originally. But n Q-i the alteration will effect what we require, for now - = — . There will, therefore, be a contact of the same pair of teeth only after 81 revolutions of jB, or 32 revolutions of A. The insertion of a tooth in this manner was an old~ contriv- ance of millwrights to prevent the same pair of teeth from meeting too often, and was supposed to insure greater regu- larity in the wear of the wheels. The tooth inserted was called a hunting cog, because a pair of teeth, after being once in contact, would gradually separate, and then approach each other by one tooth in each revolution, and thus appear to hunt each other as they went round. Clockmakers, on the contrary, appear to have adopted the opposite principle ; though it has probably been partly forced on them, as the velocity ratio of the clock arbors must neces- sarily be exact. 240. Approximate KTumbers for Trains. — If — A = Tc, when A; is a prime number, or one whose prime fac- tors are too large to be conveniently employed in wheel- work, an approximation may be resorted to. For example, assume -^ = /c ± h. This will introduce an error of ±h revolutions of the last axis during one of the first, and the nature of the machinery in question can alone determine whether such a variation is permissible. For example, let e = — ^ = 269, which is a prime num- TKAINS OF MECHANISM. 261 ber. Take e = 269 + 1 = 270, which can readily be fac- tored into 6x5x9; and we may employ the train i5 72 X 60 X 90 ™ . , . .„ „ — = . This tram will cause an error of one F 12 X 12 X 10 revolution of the last axis for every revolution of the first axis, the altered value of e varying less than two-fifths of one per cent from the correct value. 241. But we may obtain a better approximation than this, without unnecessarily increasing the number of axes in the train ; for, determine, in the manner already explained, the least number m of axes that would be necessary if k were decomposable, and the number of teeth that the nature of the machine makes it practicable to give to the pinions, and let F be the product of the pinions so determined ; hence L^ F F' supposing the wheels to drive. Assume D _ FJc ± h F" F ' where li must be taken as small as possible, but so as to obtain for Fk ± h a. numerical value decomposable into factors. There will be, in this ease, an error of ±h revolu- tions in the last axis during F of the first, or an error of =— during one of the first. If the pinions are to be the drivers, then, in the same manner, assume and there will then be an error of — ^ revolutions in the first 262 ELEMENTARY MECHANISM. axis during one revolution of the last axis. Let us take, as in the previous example, e = 269. Let w = 90, and p = 10; then 269 = (9)"'-S whence we find the least number of axes to be four. Let us assume that pinions of 10 will be employed ; then D ' n„Q 269000 F 10 X 10 X 10 Now add 1 to the numerator, and we have D _ 269001 ^ 81 X 81 X 41 2?" ~ 10 X 10 X 10 10 X 10 X 10' This will give a good train with an error of only 1 revolution in 269000. As another example, let it be required to find a train that shall connect the twelve-hour wheel of a clock with a wheel revolving in a lunation (viz., 29 days, 12 hours, 44 minutes nearly), for the purpose of showing the moon's age on a dial. Eeducing the periods to minutes, we have ^L^ ^ 42524 * ii 720 ' of which the numerator contains a large prime ; viz., 10631 ; but 42524 + 1 ^ 60 X 63 720 8x8' giving a good train, with an error of one minute in a lunation. AGGREGATE COMBINATIONS. 263 CHAPTER XIII. AGGREGATE COMBINATIONS. Differential Pulley . — Differential Screw Feed Motions. — Bpicyclic Trains. — Parallel Motions. — Trammel. — Oval Chuck. 242. Aggregate Combinations is the term applied to those assemblages of pieces in mechanism in which the motion of a follower is the resultant of the motions it re- ceives from more than one driver. The number of drivers which impress their motion directly upon the follower is generally two, and cannot exceed three, since each driver determines the motion of at least one point of the follower, and the motion of three points in a body determines its motion. Such combinations enable us to produce by simple means very rapid or very slow velocities, and complex paths, which could not well be obtained directly from a single driver. These combinations may be divided into two classes, accord- ing as velocity or path is the principal object to be attained ; and we will consider these two classes separately. Aggregate Velocities. 243. By liinkwork In Figs. 170 and 171, let ^JB be a rigid link, and let the point A be given a velocity a, while the point B is given the velocity h. Then it is required to determine the motion of an intermediate pomt, C, which is affected by the motions of both A and B. These motions are generally perpendicular to AB, or so nearly so that the 264 ELEMENTARY MECHANISM. error in their comparative motions will not generally be prac- tically appreciable. - ng.i70 If we consider the motion of A alone, regarding B as BC stationary, C will move with a velocity = a. . If we con- sider the effect of the motion of B alone, regarding A as AO stationary, we have the velocity of C = b.— — . Considering motion in one direction as positive, and in the opposite direc- tion as negative, we have for the resultant motion of C from both A and B, c = — 7^ — '- , or the algebraic sum of the AB ^ two component velocities. A C This result may be represented graphically, as follows : Perpendicularly to AB draw AA' and BB' to represent in length and direction the velocities of A and B respectively. Draw AB'. Then CO' drawn through C perpendicularly to AB will represent in length and direction the resultant velo- city of the point O. Examples of aggregate motion by linkwork are to be. seen AGGREGATE COMBINATIONS 265 iu the several forms of " link motion " valve gears of revers- ible steam-engines. In these, motion is given by eccentrics or cranks to points such as A and B in the figures, and the steam-valve receives its motion from some intermediate point,' the distance of which from the ends can be varied. As will be seen from the figures, if C is nearer A than B, for instance, its motion will be derived to a greater extent from A than from B. If it is midway between these points, it will re- ceive an equal proportion from each. 244. Differential Pulley In Weston's differential pulley, illustrated by Fig. 172, the principle of aggregate 266 ELEMENTARY MECHANISM. velocities is made use of for lifting heavy weights by the ap- plication of a small amount of force. It consists of a single movable pulley, D, from the axis of which the weight to be lifted is suspended ; a fixed pulley, C, having two circum- ferential grooves, the diameter of one being somewhat less than that of the other ; and an endless chain passing around the pulleys, as shown in the figure. The combination is ope- rated by hauling upon the chain LN in the direction indicated by the arrow. The velocity of the pitch circle, EL, is evi- dently equal to that of the hauling part of the chain. Let I, k, denote the velocities of the pitch circles EL and UK respectively, and b the velocity of BP. Then, if the point JTwere stationary, hauling down upon LN would evidently raise B vrith a velocity = -. But K, being rigidly connected to L, moves downward with a velocity such that - = — — , or A = Z.— — . Considering E as fixed, h this would give to -B a downward velocity of -. Hence the it resultant velocity of B upwards will be 6 = 1-^= ; AL - AK 2 2 ■ -iAL ' or the velocity ratio = - = 4LL:rLAE, I 2AL 345. Compound Screws. — In Fig. 173 let SS' be a cylinder upon which two screw threads are formed. Let the portion ab have a pitch n, and be fitted in a fixed nut N; and let the portion cd have a pitch m, and be fitted with a nut Jf which is free to move in the direction SS', but which is prevented from turning. Then, if the bolt be turned in the nuts as indicated, it will move through the nut N, a dis- tance n, during each turn, while at the same time the nut M AGGREGATE COMBINATIONS. 267 will move along SS', a distance m, during each turn. There- fore, if the screws wind the same way, M will move relatively to the fixed nut iV, a distance equal to the difference between n and m for each turn of SS'. That is, if n is greater tlian m, M will move aivay from N the distance n — m for each Fig.X73 turn ; or if m is greater than m, M will move towards N the distance m — n. If the screws wind iu opposite ways, the motion of M relatively to JSf will be m + mi for each turn. 246. Automatic Drill Feed. — Fig. 174 illustrates a combination for the production of a slow endlong motion of a spindle, together with a rapid rotation such as is re- quired for the spindle of a drill-press. In the figure, AB is H K i Cl — 1L» ^ s = i N Al U\\\' \\\\\ 1^ =1 Ie F inig. XT^ the spindle to which is fastened tlie spur wheel E. A thread is cut on a portion of AB, to which is fitted a nut N mounted in the frame of the machine, so that it is free to rotate, but can have no other motion. To N is fixed a spur wheel F, E and F gear respectively with a long pinion H and a spur wheel K, both fixed to a driving-shaft CD. Let c be the 268 ELEMENTARY MECHANISM. number of revolutions made by CD, while F and E make / and e revolutions respectively. Also, let E, F, H, and K represent the number of teeth upon the respective wheels. Then, - = — ; and ■'-=—. Let » be the pitch of the screw, c E c F ^ ^ then c revolutions of CD will cause AB to travel through the distance {f -e)p = cpf- - ^\ For example, let^ H K i", ^ = T^, and ^ = I ! then, for one turn of CD, the spindle will travel i"(| — 1"^)= i"x ^V — X" — 8 ff 247. An Epicyclic Train is a train of mechanism, the axes of which are carried by a revolving, arm. Simple mraimiiiipi _] I f Fie. X75 forms of epicyclic trains are illustrated by Figs. 175 and 176. In both figures the train-bearing arm, A, revolves about a fixed centre, B, and carries the train of wheels shown. C, which is considered to be the first wheel of the i IIIETIIIIIIIIIIII i LTIT Fig. 176 train, is concentric with B, and may be fixed, or may receive motion from some external source. The wheel E, which is considered to be the last wheel of the train, may be carried by the arm, as in Fig. 175, or be concentric with it, as in Fig. 176. In the latter case it is carried by a separate shaft, AGGREGATE COMBINATIONS. 269 or turns loosely upon B. In either case its actual motion is the resultant of the motion derived from the revolution of the arm A and that received from C by means of the connecting train. It will be seen that the connection between C and E may be made by any of the modes of transmitting motion which have been discussed. Epicyclic trains are used : (1) To produce an aggregate motion of the last wheel by means of simultaneous motions given to the first wheel and the arm. (2) To produce an aggregate motion of the arm by means of simultaneous mo- tions given to the first and last wheels. 248. Velocity Katio in Epicyclic Trains. — In Fig. 177 let A be the train-bearing arm of an epicyclic train turning about B. Let C be the wheel concentric with B, Fig. 177 and E the axis of a wheel F carried by the arm and con- nected to C by a train of mechanism. Suppose that while A turns about B to some other position A', a point a, on wheel 270 ELEMENTARY MECHANISM. C, moves to 6 from any external cause, and that a point d, on wheel jP, moves to e by reason of its connection with G. For simplicity, all are supposed to turn in the same direction. Draw r^'F parallel to EB. Then aBh and liE'e are the absolute angular motions of C and F respectively, and cBb and gE'e are their angular motions relatively to the arm A. liE'g = aBc = angular motion of the arm. aBh = aBc + cBb. hE'e = JiE'g + gE'e = aBc + gE'e. Or, cBb = aBb — aBc; gE'e = liE'e — aBc. These equations are true for angles of any magnitude, and hence for complete revolutions since the velocity ratio is con- stant. Let a, m, and n be the synchronal absolute rotations of the arm, of the first wheel C, and of the last wheel irrespectively. Let e be the value of the train between C and F, that is the quotient which has been represented by — s = in Chap. XII. Then the rotations of the first wheel relatively to the arm = m — a, and the rotations of the last wheel relatively to the arm = n — a. Therefore e = , which is the m — a general equation for epicyclic trains. From this we derive me — n , n — a , , n a = — , m = a -|- ^ , m = a + «(m — a). e — 1 £ If the first wheel is fixed, m = 0. a -, a = , n =(1 — e)a. AGGREGATE COMBINATIONS. 271 If the last wheel is fixed, n = 0. a me .-. e=- , a= -, a — m £ — 1 .(.-!> In all of the above formulse, the arm, first wheel, and last wheel are assumed to rotate in the same direction ; but if tlie direction of rotation of any one is changed, the sign of a, m, or n should be changed accordingly. In applying the formulse, we first assume that the rotations take place in the same direction, and then, one direction for the arm being taken as positive, the + or — sign of m and n will show whether they are rotating in the same direction or the reverse. If the connecting train is such that the first and last wheels would rotate in the same direction, supposing the arm to be fixed, the sign of e is plus, but if they would rotate in opposite directions, it is to be taken as minus. For example, if the connection is by spur gearing, and there are an odd number of axes, t is positive; but if the number of axes is even, c is negative. 249. Ferguson's Paradox, illustrated by Fig. 178, will serve as a simple example for the application of these formu- H Fig.iTS Iffi. The wheel C has 20 teeth, and is fixed to the shaft B, about which the arm A rotates. This arm carries the axis of the wheel D, which gears with C and with three wheels E, F, and G, which turn loosely on the shaft H also carried by the arm. E has 19 teeth, F 20, O 21, and D any num- ber. Since there are three axes, e is +, and has the three , 20 C 20 ^. (7 _ 20 values,- =-,-=-'^■^^^^-21- C is fixed ; there- fore, ??i = 0, and w =(1 — t)a. 272 ELEMENTARY MECHANISM. In the three cases we have That is, when the arm revolves the wheel F will have no absolute rotation, while, for each revolution of the arm, E will make -^^ of a turn in the opposite direction, and O will make ^\ of a turn in the same direction. 250. Watt's Crank Substitute, otherwise known as the Sun and Planet Motion, belongs to the general class of epi- cyclic trains. In Fig. 179, AB is one end of the main beam of an engine, C is a spur wheel fastened to the main shaft, and ^ is a spur wheel fastened to the connecting-rod BD, and gearing with C. E is held in gear with G by means of a connecting link OD, or by a circular groove concentric with C in which a pin at D slides. As E is raised and lowered by the motion of the beam, and forced to revolve about C, since it cannot rotate its own axis, it causes C to rotate. E has a vibratory motion due to the varying angle of the connecting-rod, but as this is periodic, it may be neglected for complete revolutions. Considering the combination as an epicyclic train, OD will be the train-bearing arm, C the first wheel, and E the last wheel. The latter has no absolute rotation ; hence, applying the general formula, and letting Ji = 0, we have m = a(l j. A-lso, since there are but two axes, e is negative. Let = E, then e = — 1, m = a( 1 ) = ea, {^-^) AGGREGATE COMBINATIONS. 273 Or, for one revolution of the train arm OD corresponding to an up-and-down stroke of ttie piston, C malies two revolu- tions. Thus by this a,rrangenient the shaft rotates twice as fast as it would with the ordinary crank connection. If G lias twice as many teeth as .B, £ = —2, and m = all = - a, or C revolves three times while OD revolves twice. :^) If E has twice as many teeth as C, e = —-J, m = a(l -|- 2) = 3a, or G revolves three times for one revolution of OD. 261. Epieyclie trains are used in some forms of rope- making machinery. In order that a rope shall not untwist, it is necessary that the separate strands shall either be laid together without any twist, as in wire rope, or that they shall 274 ELEMENTARY MECHANISM. have a slight twist in the opposite direction to the apparent twist of the rope. In Fig. 180, let B be the bobbins from which the wire or strands are unwound as the rope is formed. These bobbins are cairried by wheels D, which are connected to a centre wheel A by intermediate wheels C. The axes of all the wheels excepting A are carried by a frame which turns about the axis of A. If the bobbins were fixed in this frame, as the frame revolved, each strand would be twisted as it was unwound, but if we arrange it so that the axes of the bobbins shall always lie in the same direction, there will be no twist. This is accomplished by fixing the axes of the bobbins to the wheels D, fixing the wheel A, and making D = A. We then have an epicyclic train in which m = 0, and « = -^H — = — x — = 1, .• . n — a = —a, n— a O I) and- n = 0, or the wheels D have no absolute rotation, and consequently there is no twist given to the strands. By giving D a few more teeth than A, the strands will be given a slight twist in the opposite direction to tlie twist of the rope. AGGREGATE COMBINATIONS. 275 232. Epicyclic trains may be used to transmit velocity ratios wiiich could not be conveyed by direct trains except by using a large number of axes or inconveniently large wheels. The necessity for such ratios rarely arises except in astronomical machinery, and for explanations of such applications the student is referred to Willis' "Principles of Mechanism," and the works there referred to. Aggregate Paths. 253. Parallel Motions. — The most important applica- tion of aggregate combinations in which the path is the immediate object sought, is to give motion to a piece such tliat a point in it shall move in a straight liue. Such combi- nations are commonly called "parallel motions," although "straight-line motion" would be a more correct and de- scriptive name. Some of these combinations give an exact straight-line mo- tion, but in most of them the motion is only approximate. We have seen an example of exact straight-line motion in the case of a point on the circumference of a circle roll- ing within another circle of twice its diameter, being in fact a special case of the hypocycloid. By means of accurately cut gears, this could, of course, be applied to machinery. In the parallel motions in general use, the straight-line path is produced by combinations of links, and such combinations will be now considered. 254. Peaucellier's Exact Straight-Line Motion. — In Fig. 181 is sliown the general arrangement of Peaucel- lier's exact straight-line motion. It consists of seven mov- able links connected as shown. Two long links AD, AE, oscillate about a fixed centre A, and are jointed at the ends D and E to opposite angles of a rhombus, CDPE, composed of four shorter links. At C is connected a link BO, oscil- lating about a fixed centre B, so located that AB = BC. 276 ELEMENTARY MECHANISM. Then the point P will describe a straight line perpendicular toAB. From the symmetrical construction of the combination it is evident that the points A, C, and P must always lie in one straight line. Let the combination be moved, Fig. 182, from the central position shown dotted, to some other posi- tion, such as that shown in full lines, the point P occupying the position P'. Draw AP, AP', and CC ; also DL per- AGGREGATE COMBINATIONS. 277 pendicular to AP, and D'K perpendicular to AP'. From the construction, P'K = KG', and PL = LC. Tlien, AD' = AK + Kiy = AK +(D'C" - KC) ; AD" - D'C = AK" - ifC * = {AK - KC) {AK + KC) = AC x AP'. Similarly, A& = AV + DL' = AL' + (DO^ - lO') ; Zd' - DG" = 3i' - 1:0' = {AL - LC) {AL + LC) = AC x AP. . • . AC X AP= AC X AP' ; or AP ^ AC AP' AG ' AC is a diameter of the circle ACC ; hence CCA is a right angle, and P'P is perpendicular to ABP. And P' having been assumed as any position of P, it follows that the above relation is true for all positions, or P moves in a straight line perpendicular to AB. 255. In applying this motion to engines, the point P is connected to the end of the piston-rod, and thus takes the place of the usual cross-head and guides. It is to be par- ticularly noted, that, as stated above, the arm BG is equal in length to the distance AB. If this is not so, instead of a straight line, circular arcs will be described by P. If the AB ratio is less than one. the arc described will be concave BC towards A ; if the ratio is greater than one, the arc described will be convex towards A ; and if the ratio is equal to one, the circular arc becomes a straight line. 278 ELEMENTARY MECHANISM. There are other exact parallel motions * formed by combi- nations of liukwork, most of which are derived from the Peaucellier cell; but they are of so little practical impor- tance that they will not be discussed in these pages. 256. Watt's Approximate Straight-L,lne Motion. — The most widely used of the approximate straight-line mo- tions is that invented by James Watt. It is shown in its simplest form in- Fig. 183. AC and BD are two arms r'ig.183 turning about fixed centres A and -B, and connected by a link CD. When in the mid position the arms are parallel, and CD is perpendicular to them. If the arms be made to oscillate, a point in CD, such as P, will describe a figure similar to that shown. But we can so arrange the propor- tions of the links, and the position of P, that for a limited motion it will not deviate much from a straight line. 257. Let the arms AC and BD be turned to some other positions, as Ac and Bd in Fig. 184. Then the link CD will be moved to cd. The end C has been moved to the right, and the end D to the left, so there will be some point P, of cd, which will lie in the continuation of the line CD. Let * For description of parallel motions referred to, see A. B. Kempe's 'How to Draw a Straiglit Line." AGGREGATE COMBINATIONS. 279 AO =Ii,BD = r, CAc = 6, DBb = 0, CD = e, and cP = x. Drawing ce and dg parallel to AG, we have cP X _ ce _ B(l — cos 6) dg r (1 — cos ^) e — X 2i?sin'^- 2 r 2 2 2 In practice, ^ does not exceed about 20°, the inclination of the link cd is small, and Bd is very nearly equal to ?-. As these ansles are small, we may assume B sin - = r sin *, •' 2 2 Fig.lS4= hence = — , or the segments of the link are inversely e — X B proportional to the lengths of the nearest arms, which is the usual practical rule. 258. Amount of Deviation. — The deviation of the point P from the line SS can be calculated, but will not generally exceed about -^^ inch. This may be greatly re- clnced by the arrangement shown by Fig. 185, which should always be used. In the mid position the arms are perpen- dicular to the line SS in which the point P should lie, and 280 ELEMENTARY MECHANISM. which in an engine should coincide with the centre line of the cylinder or pump. This line should bisect the distances Ce and Df which are the versed sines of the maximum values of the angles B and . The ends C and D of the link will then evidently deviate equal amounts on each side of SS. Drawing d'h and dg perpendicular to SS, and Gn parallel to S8, we have three equal triangles, c'd'h, CDn, and cdg. Therefore, c'p' = GP = cp, or the mid and extreme posi- tions of the guided point P are exactly on SS. S «' The greatest deviation of the guided point from SS occurs when GD is parallel to SS, and is best determined in any case by drawing the combination to a large scale, and find- ing the parallel position by trial. AGGEEGATE COMBINATIONS. 281 259. ProWem. — In Fig. 186, let CA be an arm as before, cA its extreme position, and SS the line of stroke. Bisect Ce. Join Cc, and draw AN perpendicular to it. N bisects Cc, since the latter is the chord of the angle CAc, and hence is on the line SS. Also MN = ^ ec, or, since ec may be taken as |- the stroke, MJSf = |- the stroke. Therefore, if we have given the length of stroke and direc- tion, SS, the centre of one arm A, and mid position of the guided point P, we can construct the remainder of the mo- tion as follows : Draw AR perpendicular to SS, lay off MJSf = J stroke, draw AN, and perpendicular to the latter draw NO. Where this line intersects AR at 0, will be the end of the arm AC. CP will be the direction of the link in mid position. If we assume, or have located, the point J/ where the mid position of the second arm cuts SS, draw an indefl- 282 ELEMENTARY MECHANISM. nite straight line, FH, througli this point perpendicular to 8S. The point D, where CP produced cuts FH, is the ex- tremity of the second arm. Then, since HD must be -J the versed sine of the arc through which D moves in either direc- tion, we can find the centre B by laying off HT = \ stroke, and drawing TB perpendicular to TD. 260. Practical Form of TVatt's Motion We have thus found the proper proportions for the simplest form of the motion ; but, as usually constructed, the motion is of the form shown in Fig. 187. AE is one arm of the main beam of an engine, and turns about the centre A. EF is the main link, ^connecting AE with the piston-rod FS. CD is the back-link equal and parallel to EF. FD is the parallel- bar equal and parallel to EC. BD is the radius bar, or bridle. The point P, in CD, is the guided point whose mo- tion we have discussed. If we draw AP, and produce it until it cuts EF in F, the latter point will have a motion similar to P. This will be clear when we consider that in all positions EF is parallel to CP; then, since AE and AC are fixed lengths, we have for any position two similar tri- A P AC angles ACP and AEF ; hence ==— = =i^ = constant. So AF AE that, if P describes a straight line, F will also move in a straight line parallel to the path of P. 261. Scott Russell's Motion. — A combination due to Mr. Scott Russell, similar to that of Fig. 120, is usually AGGREGATE COMBINATIONS. 283 classed as an exact straight-line motion. In that figure, if the point Q be compelled to move in straight guides along AL, the point Fwill move in a straight path AV, the arm AP oscillating instead of performing complete revolutions. This would scarcely seem to be entitled to the term " exact motion," since it depends upon the accuracy of the guides at Q, the necessity of which it is the object of straight-lipe motions to avoid. 262. Grasshopper Motion A form of the above motion in which the guides are replaced by a comparatively long radius-rod perpendicular to AL in mid position, and con- nected to Q, is approximate, and is known as the "Grass- hopper Motion." In Fig. 188, let p, P, and p' be the extreme and middle positions of the guided point, lying in one straiglit line. Draw the straight line DPB, perpendicular to pPp' ; and lay off pa = p'a = PA = the proposed length of the guid- 284 ELEMENTARY MECHANISM. ing bar, so as to find the extreme positions A and a of its farther end. Tiiis end is to be guided by a lever centred at C; that lever being so long as to make the point A describe a very flat circular arc, deviating very little from a straight line. Choose a convenient point b for the attachment of the bridle to the bar AB, and lay off pb = p'b' = PB, so as to find the extreme and middle positions of that point. Next find the centre J) of a circular arc passing through 6, jB, and b' ; then D will be the axis of motion of the bridle Db. The error of this parallel motion is less, as 6 is nearer the middle of pa. 2G8. Kobert's Approximate Straight-Line Motion. — Fig. 189 illustrates Robert's parallel motion. Two equal arms AC and BD are jointed to fixed centres at one end, connected at the other end to the ends of the base of a rigid isosceles triangle CPD. In this triangle, CP = DP = AC = BD, and CD = lAB. It is evident that in the mid position shown, the point P is in the straight line AB ; also, that it will lie in this line when PD coincides with BD at one AGGREGATE COMBINATIONS. 285 end of the stroke, and when PC coincides with AG at the other end of the stroke. Between these positions, however, P deviates slightly from AB. 264. Tchebiclieff's Approximate Straight - Ivine Motion. — Another close approximation to a straight-line motion is that due to Prof. Tchebicheff of St. Petersburg, and illustrated by Fig. 190. The arms are of the following proportions : Let AB = 4, then AG = BD = 5, and GD = 2. I^ig. ISO The path of the guided point P, midway between C and Z), will then closely approximate to a straight line parallel to AB. It maj' be easily proved that the distance of P from AB is the same at the ends of the stroke, where P is in the perpendiculars to AB through A and B, and in the mid position being that shown in the figure. In intermediate positions P deviates slightly from a straight line. Both this and the preceding motion give a closer approximation than can be obtained by Watt's motion. 265. A Trammel is a device for drawing ellipses. It consists (Fig. 191) of a bar, PCD, carrying a pencil at P, and fitted with pins, or pieces mounted on pins, which slide in grooves, as shown in the figure. The grooves are usually at right angles with each other, and the cross-shaped piece 286 ELEMENTARY MECHANISM. in which they are formed is fastened in place on the paper. Let PD = a = the semi-major axis of the ellipse to be drawn, PC = 6 = the semi-minor axis, PM = x, and PN = y. Then we have r=r=r = - = sin PDM = sin 6 ; PD a PG b~ cos CPJSf = cos (f>. $ + $ = sin^<^ + cosV = 1, which is the equation of an ellipse. By varying the lengths PC and PD, ellipses of different sizes and eccentricities can be drawn. B 266. Oval Cliuck. — If in Fig. 192 we keep the bar CPD stationary, and turn the grooved piece and paper, an ellipse will be described upon the paper by the point P as AGGREGATE COMBINATIONS. 287 before. This fact is taken advantage of in the so-called "oval" chuck for turning ellipses, and of whicli Fig. 192 illustrates the principle. In this figure P is the cutting tool, C the centre of the mandrel of the lathe, and D the centre of a circular piece whicli is fixed to the headstock of the lathe. One part of the chuck is fixed to the mandrel, and has cut in it a diametral slot represented by aCh. A second part of the chuck, being that which carries the piece to be turned, has two lugs which project through the slot aCh and form part of two straight pieces, represented by ad and be, which slide on the circular piece previously referred to. The result is, that, as the mandrel revolves, the piece being turned, or the work, receives a combination of this motion of rota- tion and a reciprocating motion in the slot, by which the distance of the centre of the work from the tool is varied in the manner necessary to form an ellipse. Draw De par- allel to Ga, and CO perpendicular to Ga. Then when the work has been turned about G through the angle a'Ga, it has also been moved through G the distance OD. We now see that the triangle GOD of Fig. i92 corresponds to GOD 288 ELEMENTARY MECHANISM. of Fig. 191, and drawing PJlf perpendicular to De, we have, as before, PM -—- = sin PDM = sin ^ ; •^ = cos PDM = cos <^ ; PLf or is for the instant the centre of the ellipse. Evidently since P, G, and D are fixed, the position of this centre is con- stantly changing, lying always at the junction of a perpen- dicular to aCb through C, and a parallel to aCb through D. PEOBLEMS. 1. An "engine makes 600 strokes per minute. Fly-wheel is on the crank shaft. Find the linear and angular velocity of a point in the fly-wheel 3 feet from the centre of the shaft. Ans. a = 1884.96; V = 5654.88 feet per minute. 2. The speed of the periphery of a wheel 8 feet in diameter is 4,000 feet per minute. Find the linear velocity of a point 3i feet from the centre. 3. A point in a fly-wheel, 4 feet from the centre of the wheel, moves through 2,500 feet per minute. The stroke of the engine being 2 feet, find the mean piston speed. Ans. V = 39Y.89 feet per minute. 4. A locomotive moving at the rate of 35 miles per hour, has driving wheels 63 inches in diameter and cylinders 24 inches stroke. Find the linear and angular velocities of the crank-pins relatively to the frame of the engine. 5. Two shafts are centred 4 feet apart. Find the diameters of wheels to work by rolling contact, so that the driving shaft will make 5 revolutions while the following shaft makes 7 revolutions. Ans. Driver, 28 inches; follower, 20 inches. 6. The distance between the centres of two shafts = 54 inches. The driving-shaft makes 80 revolutions per minute. The follower is to make 100 revolutions per minute. Find the diameter of wheels for rolling contact. 7. A shaft making 120 revolutions per minute is to drive by spur gearing a second shaft 28 inches from it at a speed of 300 revolutions per minute.' Find diameters of pitch circles. 290 ELEMENTARY MECHANISM. 8. Velocity ratio to be transmitted = f = — . Diameter of the a driver is 15 inclies. Find tlie diameter of tlie follower, and the dis- tance between parallel axes. (Direct contact.) Ans. Diameter, 20 inches; distance, 17i inclies. 9. A wheel 32 inches in diameter is fixed on a shaft making 325 revolutions in 5 minutes. This wheel and shaft are to drive a second wheel by rolling contact, so that the latter will make 52 revolutions per minute. Find the size of the second wheel, and the distance between the centres of the wheels. 10. Given two intersecting axes at right angles, velocity ratio — = -. Show how to find the pitch cones graphically. a 3 11. The angle between two intersecting axes is 75°. Show how to find graphically the sizes and positions of conical frusta which will transmit a velocity ratio — =; — . a 80 12. P = circular pitch, iV = number of teeth. D = pitch diameter, M = diametral pitch. (1). Given P= 2i inches, iV = 40. Find D. (2). Given P — ll Inches, N = 75. Find D. (3). Given P - | inch, D — 12 inches. Find N. (4). Given D = 24 inches, Sf = 50. Find P. (5). Given 8 pitch wheel, N = 40. Find D. (6). Given 3 pitch wheel, N = 60. Find D. (7). Given 4 pitch wheel, D = 20 inches. Find JV. (8). Given 2 pitch wheel, JD = 35 inches. Find N. (9). Given D = 15 inches, N = 75. Find M. (10). Given D = 27 inches, N = 81. Find M. 13. Two axes 27 inches apart are to be connected by two 3 pitch wheels. Velocity ratio J. Find diameters of pitch circles and num- bers of teeth. Ans. Numbers of teeth 63 and 45. 14. Construct a cam on a base circle of 3 inches diameter, to revolve once per minute, and give to a bar, whose line of motion passes through the centre of motion of the cam, a stroke of 2 inches. Tlie bar rises during 25 seconds with a uniform velocity; remains at rest 20 seconds; and descends during the remainder of the revolutiori with a uniformly accelerated velocity, PROBLEMS. 291 16. Draw a cam, wWch, by oscillating through an angle of 60°, shall give a uniform ascending and descending motion to a bar whose line of motion passes 4 inches to right of the centre of the cam. Stroke of the bar, 3 inches. 16. Design a cam on a base circle of 3 inches diameter, to raise a point whose line of motion passes one inch to the right of the centre of motion of the cam, by a uniform step-by-step motion, during | of a revolution of the cam, and allow it to descend with uniform velocity during the remaining i of the revolution. 1 7. In Fig. 107 given AP = 2 feet, AB = \ foot, BC - Zk feet, CQ = 6 feet. AB is vertical, and aQ is horizontal. P revolves in the direction of arrow, making one revolution per minute. (1). Find length of stroke of Q l (2). Find time of forward stroke in seconds [ By computation. (3). Find time of backward stroke in seconds J (4). Find position of P when Q is at the middle of ' forward stroke (5). Find position of P when Q is at the middle of bacljward stroke Graphically. 18. Having a crank 2 feet long, and a connecting-rod 8 feet long, find the angle of the crank with line of centres when the piston is at the middle of its stroke. Ans. ± 82° 49' 9". 1 9. Having a crank 1 foot long, and a connecting-rod 5 feet long, revolutions per minute 120. Find piston velocity in feet per minute, when the crank makes an angle of 45° with the line of centres. Ans. 609.22 feet per minute. 30. Having an engine of 5 feet stroke and a 10-foot connecting- rod, find distance of the piston from the end of stroke when the crank has made \ of a revolution. Ans. 2 feet 2.19 inches. 21. Having an engine of 3 feet stroke, connecting-rod lOi feet long, find what angles the crank makes with line of centres when the velocity of the piston equals that of the crank. Ans. Sin-i . I. 22. Having a beam engine of 10 feet stroke, 13 feet between the centres of beam and cylinder. Find the best length for the beam arm. 292 ELEMENTARY MECHANISM. 23. Show graphically how to construct a quick return motion by jointed links, such that pe'-iod of advance _ 3 period of return 2 24. Design a quick return motion, such that the period of return = ^ of the period of advance. 25. Design a cam on a base circle of 2 inches diameter, to give to a point whose line of motion passes i inch to the right of the centre of motion of the cam, the same motion as piston in problem 20. 26. Connect two parallel shafts by a crossed belt, so that — =z-, and a 4 find the length of the belt by exact calculation. 27. Two shafts are to be connected by an open belt, distance be- ffi' 2 tween axes — 10 feet and — — -■ Find diameters of pulleys and the d o length of the belt. 28. A pulley {A) on a driving-shaft drives pulley (B) by a crossed belt. A spur gear (C) on shaft with {B) drives pinion (D). Pulley (jE"), on the shaft with (Z>), drives pulley (F) by an open belt. Given ^ = 20 inches diameter, 40 revolutions per minute. Given B = 15 inches diameter. Given C = 90 teeth, B = 15 teeth. Given E = 30 inches diameter, F = 10 inches diameter. Find number of revolutions per minute of F, and direction of rotation relatively to A. 29. An engine of 3 feet stroke, piston speed of 360 feet per min- ute, has a main driving pulley 8 feet in diameter, from which is driven a pulley 4 feet in diameter. A pump having a plunger displacement of 2 gallons, is to be driven from the shaft carrying the 4-feet pulley, and is to pump 5,000 gallons per hour. Find arrangement of the con- necting train of mechanism, by belts or gearing. 30. A lathe has a set of change wheels whose pitch diameters are 2 inches, 3 inches, 5 inches, 6 inches, 7i inches, and 9 inches respect- ively. Leading screw has 4 threads to the inch, and is right-handed. Distance between the centres of mandrel and leading screw is 16 inches. Select and arrange wheels to cut a left-handed screw of 6 threads to the inch. PROBLEMS. 293 31. A lathe has 4 threads per inch on a right-handed leading screw. Find the sizes of least number of change wheels to cut right-handed threads of 5, 6, 8, 9, and 10 to the inch. Smallest wheel to have 20 teeth. Arrange table for change wheels for the various cuts. 32. Find trains for an 8-day clock, 16 turns of weight cord on barrel. The escape wheel has 30 teeth. Number of teeth on wheels not to exceed .... 96 Number of teeth on pinions not less than .... 8 Required hour, minute, and seconds hands. 33. Find the trains for a 32-day clock, the barrel to carry 24 coils of the weight cord ; pinions to have not less than 8, and the wheels not over 108 teeth; swing wheel (escajTe wheel) to have 60 teeth, and the pendulum to make 120 vibrations per minute. Required hour, minute, and seconds hands. 34. Find trains for a 12-day clock; 18 turns of weight cord on bar- rel. Escape wheel revolves twice per minute. Pendulum makes 120 beats per minute. Least number of teeth for pinions = 9. Greatest number for wheels — 108. Required hour, minute, and seconds hands. 35. Find trains for an 8-day clock. Pendulum makes 150 vibra- tions per minute. Swing wheel has 25 teeth. Dead beat escapement. Least number of teeth for pinions = 10. Greatest number for wheels — 108. 12 coils of weight cord on barrel. 36. Find numbers of teeth for a train to give approximately E = 194 with an error of less than 1 ; maxiinum number of teeth for wheel = 90 ; minimum number for pinion = 12. 37. Design a, drill press (Fig. 174), pitch of screw to be finch; drill to make 60 revolutions per minute; driving axis to make 40 revo- lutions per minute. Drill to descend ^^ inch per revolution. 38. In Fig. 176, C has 121 teeth, and is fixed, D has 120 teeth, d has 119 teeth, E = 120 teeth. Find how many revolutions of arm A will cause E to revolve once. 39. In Fig. 176, C is a fixed wheel, and has 20 teeth, D = 36 teeth, d: = 24 teeth, ^ = 32 teeth. Find velocity ratio = ^. 294 ELEMENTARY MECHANISM. 40. In Fig. 179, C has 30 teeth, and M has 40 teeth. Keqvured n velocity ratio — -. OD 41. In Fig. 179. If C makes 3 turns while OB makes 5 turns, find number of teeth for E and C. 42. In Fig. 181. Let the arm BC be suppressed, and let P' be guided in a circle drawn on ^P as a diameter. Prove that C will move in a straight line perpendicular to A P. 43. Find dimensions of a trammel to describe an ellipse of which the major axis is 2\ times as fong as the minor axis. INDEX. PAGE Action, angle and arc of ..... 86 Action, line of 26 Action of bevel and spnrwheels, rela- tive 156 Addendum 85 Addendum circle 85 Aggregate combinations 263 Aggregate velocities by link work . . 263 Analogy between coues and hyperbo- loids 42 Anchor escapement .... . . 188 Angle of action, approach and recess, 86 Angular velocity 4 Annular wheels 47,101,138 Approach, angle and arc of .... 86 Approximate forms of teeth . . . 113 Approximate numbers for trains . . 260 Approximate straight-line motion, Roberts' 284 Approximate straight-line motion, Tchebicheff's 285 Approximate straight-line motion, Watt's 278 Arc of action, approach and recess 86 Arc of action and pitch, relation between 95 Arcs, rectification of circular ... 74 Axes, least number of 250 Automatic drill-feed 267 Bauds 15 Backlash 86 Bar and cam, slotted 183 Bar and pin, slotted 182 Bell-crank 209 PAGE Belts 228 Belts, length of 236 Belts, shifting 231 Belts, twisted 232 Bevel gearing 47, 151 Bevel and spur wheels, relative action of 156 Bevel wheels 151 Bevel wheels, skew . 157 Boehm's coupling 208 Cams . . 165 Cam and slotted bar 183 Cam curve, construction of . . . 166 Cam for complete revolutions . . . 169 Catch, frictional 223 Centre, instantaneous . . . 18 Chuck, oval ... 286 Circle, addendum ... ... 85 " of the gorge 36 *' pitch 45 •' size of describing 94 Circular arcs, rectification of ... 74 " pitch 80 Clearance 86 Click and ratchet 219 " double-acting 222 " reversible 220 " silent 221 Clock, eight-day 256 " month 257 " trains, examples of 251 Clockwork 245 Cog, hunting 260 Combination, elementary 7 295 296 INDEX. PAGE Component motions H CorapoBitioD of motions 11 Condition of constant velocity ratio . 24 *' of rolling contact .... 24 Cones and hyperboloids, analogy be- tween 42 Cones, rolling 29 Conjugate curves 73 Connectors, wrapping 15, 227 Contact between teeth, frequency of . 259 " condition of rolling .... 24 " motions, directional relation in 24 " motions, velocity i-atio in . . 21 " rolling 14, 27 " sliding 14, 66 Coupling, Boehm's 208 " or universal joint, Hooke's, 214 " Oldham's 185 Crank 193 Cranks, bell 209 Crank and pin, slotted 179 Crank substitute, Watt's 272 Crown-wheel escapement ..... 187 Curves, conjugate 73 Curvilinear motion 2 Cycle . . . = , 6 Cycloid, construction of 79 Cylinders, rolling 27 Dead-beat escapement 190 Derived tooth outline, pin gearing . 133 Describing circle, size of 94 Diameter of pin, limiting 136 " pins of sensible .... 133 " table of pitch 81 Diametral pitch 82 Differential pulley, Weston's . . . 265 Directional relation S Directional i-elation in contact mo- tions 24 Directional relation in linkwork . . 20 Directional relation in trains . . . 243 Directional relation in wrapping con- nectors 21 Double-acting click 222 Drag link 204 Drill feed, automatic 267 Driver 7 Driver, least follower for given , . 144 PAGE Eccentric 199 " rod 193 Eight-day clock 256 Elementary combination 7 Ellipses, compulsory rotation of roll- ing 60 Ellipses, lobed wheels from .... 57 " rolling 55 Endless eciew 174 Epicyclic trains 268 " " velocity ratio in . . 269 Epicycloid, construction of .... 76 " and hypocycloid .... 67 " and pin 69 " and radial line .... 68 Escapement, anchor ....... 188 " crown-wheel .... 187 " dead-beat 190 Exact straight-line motion, Feaucel- lier's 275 Exact straight-line motion, Russell's, 282 Faces and flanks, teeth with hoth . . 02 Face gearing 49, 159 Faces or flanks only, teeth with . , 87 Face of tooth 85 Feed, automatic drill 267 Ferguson's paradox 271 Flank of tooth 85 Follower 7 " for given driver, least . . . 144 Frequency of contact between teeth . 259 Friction gearing 44 Frictional catch 223 Gearing 45 " bevel 47,151 " classification of 45 " face 49,159 " friction 44 ** pin 131 " screw 49,172 " skew 48,157 " spur 46, 85 " twisted 50, 149 Gears, mitre 48 Gorge, circle of the 36 Grant's odontograph 124 Graphic representation of motion . . 10 Grasshoppe;* motion, 283 INDEX. 297 PAQE Guide pulleys 233 Hooke'B coupling or universal joint . 314: Hour-glasa worm 177 Hunting cog 260 Hyperbolold of revolution .... 34 Hyperboloids, rolling 36 Hyperboloids, velocity ratio of roll- ing 37 Hyperboloid and coneB, analogy be- tween 42 Hypocycloid and epicycloid .... 67 " couetructioa of . ... 78 Idle wheel 244 Inside screw 173 InstaDtaneous centre 18 Interchangeable lobed wheels ... 59 *' spur wheels ... 97 Intermittent motion 2, 62 Involutes 70 Involute, construction of 79 " teeth, interference of . . . 109 " teeth, peculiar properties of ■. Ill •* system of teeth 104 Length of belts 236 Line of action 26 '* pitch 45 •' epicycloid and radial .... 68 Linear .velocity . 3 Links 15,193 Liukwork 193 '* aggregate velocities by . . 263 " directional relation in . . 20 " multiplication of oscillation by 211 " quick return motion by . . 213 *' rapidly varying velocity by, 212 ** circular motion into recti- linear reciprocation and the reverse by .... 194 " continuous rotation into ro- tative reciprocation and the reverse by . . . 200 " transmission of continuous rotation by . . . - 204 " velocity ratio in .... 16 Lobed wheels 56 PAGE Low-numbered pinions 140 Logarithmic spiral .... . . 52 *' '• construction of . . 53 " " lobed wheels from, 56 Machine, definition of 1 Machines, parts of 6 Mangle rack 65 " wheels 63 Mechanism, definition of pure ... 1 " train of . . .' . . . 7, 240 Mitre gears 48 Month clock 257 Motion and rest 2 Motions, composition of 11 Motion, curvilinear 2 " directional relation in contact, 24 " graphic representation of . . 10 " grasshopper 283 " intermittent 2, 62 " kinds of 2 " modes of transmission of, 7, 16 " oscillating 2 " parallel 275 " parallelogram of 11 " Peaucellier's exact straight- line 275 " periodic 5 " polygon of 12 " quick return by linkwork . . 213 " reciprocating 2 *' rectilinear 2 *' resolution of . .... 13 " resultant 11 " Roberts' approximate straight- line 284 " Russell's exact straight-line . 282 " screw for variable .... 178 " Tchebicheff's approximate straight-line 285 " velocity ratio in contact . . 21 *' vibi-atiug 2 " Watt's approximate straight- line 278 " Watt's sun and planet . . .272 " Whltworth's quick return . 181 Non-circular wheels 150 Notation . 247 Numbers for trains, approximate . . 260 298 INDEX. PAGE Odontograph, G-rant's 124 " improved Willis's . . 122 " Robinson's templet . 126 " Willis's 119 Oldham's coupling 185 Oscillating motion 2 Oscillations by linkwork, multiplica- tions of 211 Outside screw 173 Oval chuck 286 Paradox, Fet-gusott's 271 Parallel motions 275 Parallelogram of motions 11 Parts of machines 6 Path 2 Peaucellier's exact straight-line mo- tion 275 Period 6 Periodic motion 5 Pin and epicycloid 69 Pin and slotted bar 182 Pin and slotted crank 179 Pin gearing , ... 131 Pin gearing, limiting diameter of pins 136 Pin gearing, pins of sensible diame- ter 133 Pinions 47 " low-numbered 140 " two-leaved 142 Pilch and arcs of action, relation be- tween 95 Pitch circle 45 " circular 80 •' diameters, table of 81 " diametral 82 « line 45 " of screw . . . ^ 172 " surface 45 '* point 85 Polygon of motions 12 Problems 289 Properties of involute teeth, pecu- liar .... Ill Pulley, convexity of 232 " guide 233 " stepped 229 " tightening 231 " Weston's differential . . . 265 PAGE Quick return motion by linkwork . . 213 " " « Whitworth's . 18 Rack . . - 47 Rack and wheel 98, 110, 137 Ratchet, click and 219 Ratio, velocity 8 Recess, angle and arc of ...'.. 86 Reciprocating motion ...... 2 Rectification of circular arcs ... 74, Rectilinear motion 2 Relation, directional 8 Representation of motion, graphic . 10 Resolution of motion 13 Rest and motion 2 Resultant motion 11 Reversible cUck 220 Revolution 2 " cam for complete .... 169 " hyperboloid of 34 Roberts' approximate straight-line motion 284 Robinson's templet odontograph . . 126 Rolling cones 29 " contact 14 <» " condition of .... 24 " cylinders 27 " ellipses 55 " ellipses, compulsory rotation of 60 " hyperboloids 36 hyperboloids, velocity ratio of, 37 Rotation 2 " eynchronal 28 Russell's exact straight-line motion . 282 Screw 172 " endless 174 '• for variable motion . . . .178 " gearing 49, 172 " inside 173 '* outside 173 " pitch of 172 Shifting belts 231 Similarity in all modes of trans- mission 26 Size of describing circle .... 94 Skew bevel wheels .... . . 1J7 Skew gearing 48 Sliding contact 14 INDEX. 299 PAGE Sliding in contact motions, percent- age of 40 Sliding in contact motions, velocity of, 24 Slotted bar and cam 183 " bar and pin 182 " crank and pin 179 Spiral, logarithmic 52 Spiral, logarithmic, lohed wheels from . . 56 Spur and bevel wheels, relative action of 156 Spur gearing 46, 85 Stepped pulley . . 229 " wheel 50 Straight-line motion, Peaucellier'e exact 275 Straight-line motion, Roberts' approx- imate 284 Straight-line motion, Russell's exact . 282 Straight-line motion, Tchebicheff's approximate 285 Straight-line motion, Watt's approxi- mate 278 Surface, pitch . . 45 Sun and planet motion, "Watt's . . . 272 Synchronal rotations 28 Tappets 165 Tchebicheff's approximate straight- line motion 285 Teeth, approximate forms of . . . 113 '* customary dimensions of . . 102 *' definitions of parts of . . 85 " frequency of contact between, 259 " interference of involute . . . 109 " peculiar properties of involute, 111 " unsymmetncal 148 " with both faces and flanks . . 92 " with faces or flanks only . . 87 Templet odontograph, Robinson's . 126 Tightening pulleys 231 Tooth outline in pin gearing, derived, 133 *' " in bevel gearing ■ • . 153 Trains, approximate numbers for . . 260 •' directional relation in . . . 243 ** epicyclic 268 " examples of clock .... 251 " of mechanism, definition . , 7 '* for rope-making machinery .273 " method of desiguing . . .249 PAGE Trains, value of ....#>>• 240 " velocity ratio in epicyclic . • 269 Trammel 285 Transmission of motion, modes of, 7, 16 Trauemission, similarity in all modes of 26 Tvedgold's method for bevel wheels . 154 Twisted belt 232 " gearing 50, 149 Two-leaved pinion 142 Uniform periodic motion 6 " velocity 3 Unsymmetrical teeth 148 Value of a train 240 Variable velocity 4 Velocity, angular 4 " constant or uniform ... 3 " linear . . 3 " of sliding in contact motions, 24 " ratio . . 8 " *' condition of constant . 24 *' " in contact motions . . 21 *• " in epicyclic trains . . 269 •' " in linkwork ... 16 " " in wrapping connect- ors 20 *• " of rolling hyperbo- loids 37 " " variable 4 Verge . . 187 Vibrating motion 2 Watt's approximate straight-line mo- tion .278 Watt's crank substitute, or sun and planet motion 272 Weston's differential pulley .... 265 Wheel and rack 98, 110, 137 Wheels, annular .... 47, 101, 138 " bevel 151 " idle 244 " interchangeable 97 « " lobed ... 59 " lobed 56 '* mangle 63 " non-circular 150 " skew bevel 157 " stepped 50 300 INDEX. 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Law and Practice of United States Naval Courts-Martial. Adopted as a Text-Book at the U. S. Naval Academy. 8vo, law-sheep 3 00 HASKINS, C. H.— The Galvanometer and its Uses. A Manual for Electricians and Students. Second edition. 13mo, mo- rocco 1 50 HAUPT, Brig.-Gen. HERMAN.— Military Bridges. For the Passage of Infantry, Artillery, and Baggage-Trains ; with sug- geslions of many new expedients and constructions for crossing streams and chasms. Including also designs for Trestle and Trass Bridges for Military Railroads, adapted specially to the wants of the Service of the United States. Illustrated by 69 lithographic engravings. 8vo, cloth 6 50 HEAD, Capt. GEORGE E.— A New System of Fortifications. Illustrated. 4to, paper 50 HEAVY ARTILLERY TACTICS : 1863. Instructions for Heavy Artillery ; prepared by a Board