KINEMATICS OF MACHINERY. A BRIEF TREATISE ON CONSTRAINED MOTIONS OF MACHINE ELEMENTS. BT JOHN H. BARR, M.S., M.M.E. Consulting Engineer, Union Typewriter Company; Formerly Professor of Machine Design, Sibley College, Cornell University; Member of the American Society of Mechanical Engineers. REVISED BY EDGAR H. WOOD, M.M.E. Professor of Mechanics of Engineering, Sibley College, Cornell University. SECOND TOTAL ISSUE', EIGHT THOUSAND, , , > NEW YORK: JOHN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1911 Copyright, 1899, 1 91 it by JOHN H. BARR. THE SCIENTIFIC PRESS ROBERT DRUMMONO AND COMPANY BROOKLYN, N. Y. KINEMATICS OF MACHINERY. such changes take place are very different in the two cases, the ratio of these velocities remains the same. 7. Path, A point in changing its position traces a line called its path. The statements in the preceding articles on the motions and velocities of bodies apply equally to every point in a moving body, whether the path of the point be rectilinear or otherwise. This is consistent with the definitions of motion and velocity; for these definitions state that motion is measured by space traversed (not restricted to space in a right line), and that velocity is the rate of motion. The path of a point may be of any form whatever, in a plane or in space; it may be a straight or curved line of finite length, along which the point moves from end to end, reversing its direc- tion of motion at either end, so that it passes any particular posi- tion first in one direction and then in the opposite direction ; it may be a closed curve so that, unless the curve crosses itself, suc- cessive passings of any position are always in the same direction ; or it may be an infinite straight or curved line, the point never twice occupying the same position, except in the special case in which the curved path crosses itself. There are many cases, however, in which the path is definite and limited in both form and extent, and nearly all motions of mechanisms are of this class. 8. Cycle ; Period ; Phase. In most mechanisms the members go through a series of relative motions, at the end of which they occupy the same relative positions as at the beginning. The completion of such a series of relative motions, with the re- turn of the members to the relative positions which they had at first, constitutes a cycle. In the ordinary steam-engine, for example, the cycle corresponds to one revolution of the crank, whatever the time occupied by the revolution. In a common type of gas-engine, the cycle corresponds to two complete revolutions of the crank, for the four strokes of the piston during these two revolution are: a suction stroke; a com- pression stroke ; a working stroke (impulse) ; and an exhaust stroke. The valve-gear, in this case, is so arranged that valve, piston, etc., CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 7 only return to their initial relative positions after the completion of four strokes of the piston, or two revolutions of the crank. The time elapsing during a cycle is called the period. The simultaneous positions occupied by the members, at any instant during the cycle, constitute a phase. 9. Continuous, Reciprocating, and Intermittent Motion. If the direction of motion does not reverse, the motion is sometimes said to be continuous (using the word somewhat differently ifcan-ifi the strict mathematical sense, in which all motion is continuous). Motion is said to be reciprocating if its direction reverses. Motion is called intermittent when it is interrupted by intervals of rest. Motion in a closed path may be continuous, reciprocating, or intermittent; and it may vary as to velocity in any manner what- soever. Motion in a path of finite extent, not forming a closed figure, must be reciprocating, and may or may not be intermittent. 10. Plane Motion ; Rotation, Translation. Of the great num- ber of motions available in machinery, a very large proportion are included in three classes of comparatively simple nature, viz. : Plane Motion, Helical Motion, and Spherical Motion. Plane Motion is by far the most common, and it is the simplest class as well. If any plane section of a body moves in its own plane, all points in this section move in this plane, and all points outside of this sec- tion move in planes parallel to the given section. Such a motion constitutes a plane motion. Any point in a body having plane motion may trace any path in its plane; but all points similarly located in the other parallel planes, that is all points lying in a common perpendicular to the different planes of motion, have paths of identically the same form. Thus, in Figs. 1 or 2, if the section shown shaded always moves in its own plane, the successive positions of the perpendicular through any point as p must always be parallel, and therefore all points in this perpendicular move in equal paths. The property of plane motions, just discussed, greatly simplifies the treatment of these motions, as the motion of one point (or of a set of points) in any section represents the motion of all similar KINEMATICS OF MACHINERY. points in other sections ; or the motion of a single section (a plane figure) in its own plane represents the motion of the entire body. This can be extended even farther, for the motion of a point not in the particular plane represented can be replaced by that of its corresponding point on that plane (its projection on the plane), and thus the motion of a single plane figure represents all the Fig. I ! A Fig. 2 motions of all the points in the body. For example, the motions of the points p, s, and q in Figs. 1 or 2, are in equal paths, and the motion of any one of these points may be taken to represent that of any other. The motion of an engine crank and of the eccentric can be, and often are, conveniently shown together, as if actually in one plane. In case of all other than plane motions, however, it is necessary to show the various positions of the members by two or more pro- jections, or by some equivalent system, if it is desired to completely represent the motion. Plane Motion is either a Rotation, a Translation, or a motion which can be reduced to a combination of these. The reduction of the general motion to a combination of rotation and translation is not always to be desired, however, and such motion will often be treated as a class of itself, without relation to the simpler and more special classes to which it can be reduced. If a body moves, as in Fig. 1, so that all points travel in paral- lel planes and at constant distances from a fixed right line, it has a plane motion of rotation. Examples: pulleys, cranks, levers, etc. It is not necessary that the motion be continuous ; the rota- tion may be continuous, reciprocating, or intermittent. CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 9 If a body moves, as in Fig. 2, so that all points move with equal velocities in equal paths, the motion is a translation. If these paths are parallel right lines, the motion is a rectilinear translation. Examples : the carriage of a lathe, piston or cross-head of an engine, platen of a planer, etc. If, however, the paths of all the different points are equal curves, the motion is a curvilinear translation* Example : the side rods of a locomotive. Eectilinear translation is always to be understood when the word translation is used without qualification. A rectilinear translation may be treated as the special case of rotation in which the distance to the axis is infinity, or as rotation in a circle of infinite radius. It has been shown that the plane motion of a body is completely represented by the motion of any section taken in a plane of motion, or by the change of position of a plane figure. Two points suffice to locate a figure in a plane, and hence the plane motion of a body is determined by the motion (successive positions) of any two of its points not in the same perpendicular to the plane of motion. In the general case of motion in space, the motion is determined only by the motions of at least three points, not in one right line. For if the motions in space of two points are known, the body may, in the general case, have a motion of rotation about the line connecting these two points; but the motion of a third point, outside of this line, deter- mines the motion of the body completely. In general, the motion of a body in a plane may be reduced to an equivalent rotation and a trans- lation. Thus, Fig. 3, the motion of the body A, which is complete- ly determined by the motion of two points such as a and #, or by the motion of the line connecting these points, corresponds to a change of position from A to A'. This change of position can be conceived as made up of a transla- tion, a-b to a'-V, and a rotation, about I' through the angle 10 KINEMATICS OF MACHINERY. a' V a". Or, the rotation can be conceived to take place first, fol- lowed by the translation. As this motion is a perfectly general case of plane motion, the same reasoning applies to all such cases, no matter how large or how small the motion may be. 11. Helical Motion. If all the points in a body have a motion of rotation about an axis, combined with a translation parallel to that axis, the motion is a Helical Motion (see Fig. 4). In nearly all cases the helical motions met with in machines are regular hel- ical motions, in which there is a constant relation between the rotation and the translation; that is, the ratio between the transla- |A ft* or Fi 9 . 4 Fig. 5 tion component and the angular component is constant. The pitch of the helix is the translation along the axis corresponding to one complete rotation, and in a regular helical motion the pitch is con- stant. 12. Spherical Motion. If the motion of a body is such that all points in the body remain at constant distances from a fixed point (see Fig. 5), the motion is spherical. All points in the body move in the surfaces of spheres, having the fixed point for a com- mon centre. 13. Relation between Plane, Helical, and Spherical Motions. If the translation component (pitch) in a helical motion be in- creased till it equals infinity, the motion reduces to a plane trans- lation. On the other hapd, if the translation component be re- CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 11 duced to zero, the motion reduces to plane rotation. It is thus seen that both of the limits of helical motion are plane motions, and that plane motion of rotation or of translation may be treated as special cases of helical motion. If the distance from the fixed point to the moving body in a spherical motion be increased to infinity, the surfaces of the spheres in which the points of the body move are reduced to planes, and we thus see that plane motion may be treated as a special case of spherical motion. The much greater frequency of plane motion and its simplicity makes its consideration, in practical cases, as a Fig. 6 Fig. 7 Fig. 8 Fig. 9 special form of these more complex motions undesirable, though this view of the case is not without interest. Motions more complicated than the classes just mentioned are sometimes met with in machinery, and some of these will be dis- cussed in subsequent articles; but they are comparatively so 12 KINEMATICS OF MACHINERY. few, and are so varied in character, that a classification of them is not practicable. Figs. 6, 7, 8, and 9 show practical examples of plane rotation, plane translation, helical motion (regular) and spherical motion respectively. 14. Graphic Representation of Velocity. The direction and velocity of a motion may be represented by a right line, the direc- tion of which indicates the direction of the motion, while its length represents the velocity to some convenient scale. If, for instance, it is desired to represent the velocities : 20 feet per second, 35 feet per second, 55 feet per second, and 40 feet per second, by lines on a drawing, or diagram, a scale can be adopted which will give convenient lengths (say 10 feet per second to the inch); and, to this scale, these velocities will be represented by lines 2 inches, 3.5 inches, 5.5 inches, and 4 inches long, respec- tively. In a similar way, velocities in other units, as feet per pa- . 10 Fig. II minute, miles per hour, etc., can be indicated to suitable scales. The velocity of the point p (Fig. 10) whicli has a motion of 300 feet per minute in a rectilinear path, is represented to a scale of 200 feet per minute to the inch, by the line p-v , 1.5 inches long. If the motion is in any other path than a right line, the velocity at any position may still be represented by a straight line tangent 'to the path at that position, for the direction of the curved path at any point coincides with the tangent at that point. In Fig. 11, the velocity of the point p, moving in the curved path at the rate of 45 fee'.; per second, is represented to a scale of 40 feet per second to the inch by the line p-v, 1.125 inches long, lying along the tangent to the path and through the given posi- tion of p. This graphic representation of velocity is of the greatest importance, as it makes many solutions possible on the drawing- board without the use of calculations; giving the results CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 13 required directly in connection with the regular process of designing, and permitting the easy determinations of results that could only be arrived at otherwise by tedious algebraic methods. As to the accuracy of these graphic methods, it may be said that they are as close as can be used in a drawing itself; so, for the ordi- nary purposes of designing, they are all that can be desired in this respect. Furthermore, the graphic method has the advantage of showing a number of connected quantities in their true relation, appealing to the mind through the eye much more effectively than do numerical quantities. A limited experience with such problems as follow in this work will impress upon one the value of this method. 15. Newton's Laws of Motion. Starting with the statement of Newton's Laws, which enunciate fundamental relations between, force and motion, and with the familiar Parallelogram of Forces, we can readily develop the theory of the very important subject of Resolution and Composition of Motions. NEWTON'S LAWS. I. Any material point acted upon by no force, or by a system of balanced forces, maintains its condition as to rest or motion; if at rest it remains at rest; if in motion it .moves uniformly in a right line. II. Any material point acted upon by a single force, or by a system of unbalanced forces, has an acceleration of motion pro- portional to, and in the direction of the force, or the resultant of the system of forces. III. Action and reaction are equal, opposite, and simultaneous. 16. Parallelogram of Forces. The resultant of two or more forces applied at a point of a body is the single force which, if ap- plied at the same point, will have the same effect on the body, as to rest or motion, as the given forces themselves. These forces which act together are called components of the single force, which is equivalent to their combined action. Forces may be represented graphically, in a similar manner to that already explained in connection with the representation of 14 KINEMATICS OF MACHINERY. velocities; the direction of the line indicating the direction of the force, and the length of the line representing the magnitude of the force. If two forces, acting on a point, are represented in this way, the resultant of these forces is similarly represented by the diagonal of the parallelogram formed on the components as sides. For the proof of this, see Mechanics of Engineering, by Professor I. P. Church, page 4. This proposition can be extended to cover the case of any num- ber of forces acting at a point; for the resultant of any two of such a system of forces can be found, then the resultant of this first re- sultant (which exactly replaces the two original forces), and an- other of the forces can next be found, the resultant of this last resultant and another component can then be found, and so on till all of the original forces have been combined. The last resultant is the resultant of the system. By the reverse of the process just outlined, a single force can be replaced by two or more components. The process of finding the resultant of several forces is called the Composition of Forces ; the reverse process of finding the com- ponents of a force is called the Resolution of Forces. P Fig. 12 ^- -V> 2 ^g. 13 17. Resolution and Composition of Motions and Velocities. While a point may be acted on by any number of forces simulta- neously, it can have but one motion at any time. This motion may, however, he considered as the resultant of two or more component motions, in the same way that any force may be considered as the resultant of two or more forces. According to Newton's second law, if a point p (Fig. 12), which is initially at rest, is acted on by a single force F u the point will move in the direction of the force. The velocity at any instant may be represented to scale by Vi. If (Fig. 13) a second force F 2 acts simultaneously on p, the motion will be in the direction of the CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 15 resultant, F r , of F l and F 2 . This motion may then be considered as the resultant of two component motions in the directions of the respective component forces. Similarly, the velocity of p may be considered as the resultant of the velocities of the two component motions. Evidently this resultant velocity is the diagonal of the parallelogram of which the component velocities are adjacent sides. From the preceding discussion, the following proposition can be drawn: PARALLELOGRAM OF VELOCITIES. If two component velocities of a point be represented, to scale, by the adjacent sides of a parallelogram, the diagonal of the paral- lelogram will represent the resultant velocities to the same scale. Conversely, a velocity represented by a line, to scale, may be resolved into any pair of component velocities, which are repre- sented to the same scale by the sides of a parallelogram of which the first line is the diagonal. As in the case of forces, the reduction of more than two com- ponent velocities to one resultant can be effected by an extension of the above principles. This method can be applied to any number of velocities, whether in one plane or otherwise. In Fig. 14 the resultant of v^ and v 2 =v a ', the resultant of v a and v 3 = vy, the resultant of Vb and v^ = v r , = the resultant of the system. In Fig. 15 the resultant of v 1 and v 2 = y a ; resultant of v a and v 3 v r . If the single velocity v r (Figs. 13, 14, or 15) is given, it can be replaced by the velocities of which it is the resultant; for they, combined, are its equivalent. Determining the resultant of a system of velocities is called Composition of Velocities; finding the components of given veloc- ities is called Resolution of Velocities. A system of velocities can have but one resultant; but a given -velocity can have an infinite number of sets of components. The 16 KINEMATICS OF MACHINERY. velocity v (Fig. 16) may have for components v, and #,; v' and v a ', or any number of sets of components; or the resolution is in- definite, because an infinite number of parallelograms can be drawn with the line v for a diagonal. Fig. 14 Fig. 15 If we know: (a) the direction of both components; () the mag- nitude of both ; or (c) the magnitude and direction of one, there is a definite resolution [case (b) admits of a double solution]. For illustration of these three cases see Figs. 17, 18, and 19, respec- tively. Fig. 16 -m. Fig. 17 Case (a). The given velocity p-v, Fig. 17, is to be resolved into components in the directions p-m and p-n. From the point v draw line v-Vi parallel to p-n, and cutting p-m in Vi ; also from point v draw line v-v*, parallel to p-m, cutting p-n in v 2 ; p-Vi and p-v 2 are adjacent sides of a parallelogram meeting in p, and p-v is the diagonal of this parallelogram through this same point p, hence the velocities represented by p-Vi and p-v 2 are the components of p-v in the given directions, p-m and p-n. It is evi- CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 17 dent that no other parallelogram can be formed on p-v as a diag- onal with its sides in these given directions. Case (b) : The given velocity p-v, Fig. 18, is to be resolved into two components of the magnitudes indicated by m and w, direc- tions to be determined. With radius m and centre p draw the arc m,-m/, and with same radius and centre at v, draw the arc m a -rw 3 '; also with radius n and centre at v draw the arc n^nf , and with same radius and centre at Fig. 18 P> draw the arc P Fig. 19 this gives four intersections. Connect these intersections with p by the lines p-v iy p-v^ p-vj, and p-vj. By drawing lines from v to each of these intersections of the arcs, it is seen that two parallelograms are formed (p-v^v-v^ and p-v/- v-v 2 f ), each having the given velocity p-v for a diagonal, with sides (p-Vi and p-Vt, and p-Vi and p-v 2 ' ', respectively) equal to the re- quired components ; hence there are two solutions to this case, both satisfying the condition that the velocity p-v be resolved into two components of values m and n. Case (c) Fig. 19: The given velocity, p-v, is to be resolved into the component p-v l9 known as to magnitude and direction, and another component, entirely unknown. Draw a line from v to v l9 also draw a line from v parallel to p-v lt then draw a line from p parallel to v-v l9 cutting the line last drawn in v a . p-v^ is the required component; for the given com- ponent p-v l and this line last found form adjacent sides of a parallelogram with p-v, the given velocity, as a diagonal. 18 KINEMATICS OF MACHINERY. It will be seen from the preceding discussion, that in the reso- lution of a velocity into two components, or the composition of two velocities into one resultant, that there are six elements involved, viz. : the directions and magnitudes of three velocities, and that if four of these elements are known the other two may be determined, (except for the double solution of case 6, in which two values satis- fying the conditions are obtained). The first case (a) is by far the most common in practical examples. 18. Angular Velocity. When a point is revolving about some axis, permanently or temporarily, it is frequently convenient to express its rate of motion in angular rather than in linear measure. This rate of motion may be expressed in any system of time and angular units, as revolutions per minute or per second, degrees per second, radians per minute, etc. In many practical problems the rate of angular motion of a member is most conveniently stated in terms of revolutions per unit of time ; but in analytical expressions the arc passed over by a point is often more readily measured in other units. The radian is an arc of a length equal to the radius r ; hence there are %n 6.283 radians to a circumference; or a radian is T equivalent to 360 -=- 6.283 = 57.3, nearly. If a revolving point makes n revolutions about its axis per unit of time the space passed over in time unity, or its linear velocity, is v = %7irn\ and the angle traversed in the same time will be 2 nrn GO = %7rn radians. r If the body A (Fig. 20) is revolving about the axis through O (which is perpendicular to the plane of the paper), at the rate of n revolutions per unit of time, the point p, at a distance r from the axis, has a linear velocity of %nrn\ another point at p', at a dis- tance r' from the axis, has at the same time a linear velocity 1 'Znr'n ; and any two points at different distances from the axis have different linear velocities at any instant. But all points in the same rigid body when revolving about an axis must describe equal angles in the same time, and the angle (or arc) is being de- scribed at a rate, expressed in radians by %nn. This expression CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 19 for the rate of angular motion is what is called the Angular Veloc- ity of the body; and it is necessarily the same at any instant for all points in the same rigid body. It will be noticed that the only varia- ble in the expression for angular velocity as just derived is n, the num- ber of revolutions per unit of time. Comparing the expression for angular velocity with that for the linear velocity of a revolving body, it is seen that it corresponds with the linear velocity of a point in the body at a distance from the axis equal to unity : from which we deduce the statement : The angular velocity of a body is numerically equal to the linear velocity of a point in the same body at unit distant from the axis. The relation between the linear and the angular velocity of a point which is most frequently used and one that should be firmly fixed in the memory, is Fig. 20 . Linear velocity Angular velocity = J - Radius v or oo = . r If a point revolves about a fixed centre with a linear velocity of 60 feet per second (720 inches per second), and with a constant radius of 18 inches (1.5 feet), its angular velocity is or GO = = 40 (radians per second), 720 GO = -r-Q- = 40 (radians per second). The space-units which measure the radius and the linear velocity must be the same, and the angular velocity is then ex- pressed in radians per second, or per minute, according to whether the linear velocity time-units are seconds or minutes. Angular velocity may be constant, or it may vary, uniformly or otherwise. If the radius remains constant, as in a body rotating about an axis to which it is rigidly connected, the angular velocity 20 KINEMATICS OF MACHINERY. must vary just as the linear velocity of any one point varies, as is seen from the above relation; or it varies directly as n. 19. Instantaneous Motion, Instant Centre, Instant Axis. The instantaneous motion of a point is its motion at any point in its path. It was shown in Art. 14 that the direction of this instan- taneous motion is along a tangent to the path at the position of the point. Thus, in Fig. 21, if a point is moving in the path m-n, the di- * rection and velocity of its instan- taneous motion when it occupies the position p, may be repre- sented by the line p v, tangent to m n at p. This is equally true whether the point is mov- ing in the path t-t', a-b, a'-b', a"-6", or in any path whatever which is tangent to p-v at p. The instantaneous motion of a Fi a . 21 N' point is therefore independent of the form of its path. Motion in any of the possible paths is equivalent, for the instant, to rotation about c, c', or c", or about any point in the line N-N', drawn through p perpendicular to p-v, for the path of a point having such rotation would be tangent to the other paths at p. An Instantaneous Centre (called more briefly an Instant Centre) af any plane motion of a body is a point about which the body may be considered as rotating at any instant relative to another body in the same plane. In Fig. 22 let p-v and p'-v' represent the velocities of the instantaneous motions of any two points, p and p' in the rigid body A, moving in the plane of the paper, and let p n and p f -n r be perpendicular to p-v and p'-v f at p and p' respect- ively. Then the instantaneous motion of p is equivalent to rota- tion about some point in p-n as a centre. Likewise the motion of p' is equivalent to rotation about some point in p'-n'. Since A is a rigid body, p and p' can have no motion relative to each other, and the centre of rotation of p must also be the centre of rotation of p'. The point 0, at the intersection of p-n and p'-n' y is the only point that meets this requirement. It was shown CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 21 in Art. 10 that the plane motion of a body is determined by the motion of any two of its points not in the same perpendicular to the plane of motion. Therefore the instantaneous motion of A is equivalent to a rotation about as a centre. In other words, is an instant centre of the motion of A. This motion is assumed to be relative to the paper or any reference body in the plane of the paper. If the material of A and the reference body, B, are assumed to be extended to include 0, Fig. 23, a pin could be put through 0, materially connecting A and B and the instantaneous motion of A with reference to B would :not be interfered with. The instant centre of the relative motion of two bodies is a point Fig. 22 Fig. 23 at which they have no relative motion; it is the only point common to the two bodies for the instant. It will be noted that the points p and p' may be moving in any paths whatever so long as these paths are tangent to p-v and p'v' at p and p' respectively. Unless these paths are circular arcs having a common centre at 0, the rotation of A about does not continue for any finite time, which is implied when is called an instant centre. If the paths of p and p' are such circular arcs, is a permanent centre as well as an instant centre. In general, it is only necessary to know the direction of motion of two points in a body having plane motion, in order to deter- mine the location of the instant centre. When the points are moving in parallel paths special treatment is necessary. If the motions are at right angles to the line joining the two points, 22 KINEMATICS OF MACHINERY. as in Fig. 24, the instant centre lies somewhere on this line. The linear velocities of points in a rigid body being propor- tional to their distances from the centre of rotation, the loca- tion of the instant centre, 0, may be found from the proportion p-v:p'-v' : : 0-p : 0-p f , by the construction indicated. When, as in Fig. 25, the common direction of motion of the two points is not at right angles to the line joining them, the perpendiculars p-n and p'-n f are parallel, and their intersection, 0, is at infinity. The instantaneous motion of the body is a rotation about a centre at infinity, or it is translation. In this case all points of the body have equal linear velocities. It is more exact to refer to rotation or revolution about an n at oo Fig. 24 Fig. 25 axis than about a centre. In the case of plane motion the axis of rotation is always perpendicular to the plane of motion, and pierces every section of the body parallel to the plane of motion at its centre of rotation (Fig. 1). Since the motion of each section completely represents the motion of the whole body, it is customary in dealing with plane motions to refer to instant centres instead of the corresponding instant axes. It was shown in Art. 10 that the motion of a body in space is determined by the motion of any three of its points not in the same right line. Such motion is at any instant equivalent to rotation about an axis combined with translation parallel to that axis. That is, it is a form of helical motion, as denned in Art. 11. In Fig. 26, the instantaneous velocities of any three points of a rigid body having any motion whatever in space are CONCEPTIONS OP MOTION. NATURE OF A MACHINE. 23 represented by p-v, p'-v', and p ff -v ff * In order that the instan- taneous motion of the body may be equivalent to a rotation about some such axis as XX', combined with a translation parallel to that axis, the components of pv, p'v f , and p"v" perpendicular to X-X' must represent a plane rotation about X-X', and those parallel to X-X f must all be equal. That the axis X-X' can be located, and that the three instantaneous velocities can be resolved into such components is shown as fol- lows: From p in Fig. 26 (a), the lines p -v , p -v ', and p ~ v o" are drawn respectively parallel and equal to p-v, p'-v', and p"-v" Fig. 26 in Fig. 26. The three points V Q , V Q ', and v " determine a plane. The line p ~ s o is drawn perpendicular to this plane, piercing it at s Q . The projections of PQV O , PQVQ, and PQV O " on p ~ s o are all equal to P O -SQ. The projections of the same lines on the plane perpendicular to > ~ s o are respectively s -v , s -t> ', and s o~ v o"' These are all perpendicular to p ~~ s o'- I* 1 Fig. 26 the components of p-v, p'-v', and p"-v" taken parallel to p ~ 5 o are p s, p's', and p"s". These are all equal to p s and represent * When these velocities are assumed it is necessary that the velocity of each point be so taken that its component along the line joining the point to each of the other points shall be equal to the component of the velocity of that point along the same line. Otherwise the motion would change the distance between the points, which is not consistent with the conception of a rigid body. 24 KINEMATICS OF MACHINERY. a translation of the body parallel to p -s . The components p-r, p'-r', and p"-r" are respectively parallel and equal to s -^ , s o~ v o'> s o~V- They represent plane motion of the body, since all are parallel to the plane perpendicular to > ~ s o- Every plane motion of a body has been shown to be equivalent to an instan- taneous rotation about an axis perpendicular to the planes of motion. The axis, X X', of the rotation represented by p r, p'-r f , and p"-r" must therefore be parallel to p -s Q . It pierces every plane section through the body perpendicular to p -s Q in the instant centre of the motion of that section. The motions ordinarily used in machinery may be considered as special cases of space motion. When the axis of rotation is fixed and there is a constant relation between the rotation and the translation, uniform helical motion results. When the trans- lation is reduced to zero and the axis of rotation passes through a fixed point, the motion is spherical. Both these motions may be further reduced to plane motion as explained in Art. 13. 20. Free and Constrained Motion. It follows from the state- ments of Art. 15 that if a point is to move in any prescribed path, the resultant of all forces acting upon the point in any of its posi- tions must lie in a tangent to the path at the position of the point.* If the path be other than a straight line, this involves a constant change in the direction of the resultant force, caused either by a change in direction or magnitude (or both) of at least one of the components of this resultant. This is exactly what takes place in every such case; but the method of this readjustment of the resultant force affords the basis of a very important division of motions into two classes, viz.: Free and Constrained Motions. A body which has no material connection with other bodies is called a free body; the planets are examples of this class of bodies. A planet revolves around the sun in a path or orbit determined by the resultant of all forces acting upon it; every disturbing action or force alters its path. The motion of a body which has a material connection with another body, permitting motion relative to that body only in * In this and the following discussion the point or body must be considered as initially at rest in each of the various positions it occupies in tracing its path. CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 25 certain restricted paths, is said to be constrained. The crank-pin of an engine has constrained motion. In this case, if motion takes place under the action of any force it must be in a fixed path, and no force, whatever its direction, short of one that will break or injure the machine, can cause motion in any other path. The primary actuating force in the case of the crank-pin is the pressure or pull exerted upon the pin along the direction of the connecting-rod (neglecting frictional influence). This primary force does not, except at two instants in each revolution of the crank, act tangentially to the path of the body acted upon ; there- fore we must look for some other force which combined with this primary force gives a resultant acting tangentially to the path of the crank-pin. While the study of such actions is not strictly within the province of the present treatise, it is important to clearly fix the nature of constrained motion, and for this purpose the distribution of force acting through the connecting-rod upon the crank-pin of the ordinary reciprocating steam-engine will now be briefly considered. Fig. 27 indicates the mechanism of the engine (without valve-gear). Figs. 28, 29, 30, 31, 32, 33, 34 show the connecting-rod and crank in different positions, or phases. The full lines indicate the directions of motion, ana UKJ aasii-and-dot lines indicate forces acting. Fig. 28 In Fig. 28 the connecting-rod is at right angles to the crank, and therefore its centre line coincides with the tangent to the circle in which the crank-pin must move. As the force P, acting on the pin, is in the direction of the centre line of the rod, this force alone would produce motion in the prescribed path, and no other force need be considered as acting to produce such motion at this par- ticular phase. The rod is under compression. In Fig. 29 the condition is similar, except that the connecting- rod is now under tension instead of compression, and the action on 26 KINEMATICS OF MACHINERY. the pin is a pull instead of a thrust, but, as before, the force acts tangentially to the path. In Fig. 30, however, the force P' exerted by the connecting-rod on the pin (thrust) is not in the direction of the tangent to the path, and hence it alone cannot produce motion in the required p' .. Fig. 29 direction. If, however, a force P", be introduced in the direction of the centre line of the crank, of such magnitude that it, com- bined with P 1 ', will have a resultant P in the line of the tangent to- the path of the pin, the conditions necessary to produce motion in the required direction will be present ; and unless such a com- ponent of force is acting in conjunction with P', the required motion cannot take place. If the crank-pin were a free body this- force would be an external force, but it will be seen that it would be very difficult to apply such an external force in the right direc- tion and of the proper magnitude, for these requirements con- stantly change. In case of constrained motion the material con- nection (the crank in this case) supplies this force by its own resistance to a change of form. The primary acting force, alone, would impart motion in the direction of its own line of action, but this motion could not take place without changing the form of the crank, and the crank offers resistance to this change, by just the necessary amount for constrainment. As action and reaction are always equal, the force exerted on the crank to change its form is met by a corresponding counter-action, or reaction, just sufficient to give the required constraining force, and to cause motion in the circle of which the crank is the radius. This external force tend- ing to change the form of the crank calls out within the material an internal molecular action known as stress, and this action is just equal to the external force. In this particular phase both con- necting-rod and crank are subjected to compression. CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 27 In Fig. 31 the condition is similar to that of the preceding, ex- cept that the connecting-rod is under tension, the action on the pin Fig. 31 Fig. 32 is a pull, and the resistance of the crank is necessarily reversed, the stress now being a tension. This results of course in subjecting the crank to tension also, and as it is of a material that will resist this action, the motion in the required path is secured by the combined action of the force exerted upon the pin through the connecting-rod and of the secondary force called out in the crank. Figs. 32 and 33 show phases in which the stresses in the connecting- rod and crank are not similar. When the crank-pin is at one of the "dead centres" A or B y as in Fig. 34, it will be noticed that the force exerted by the con- Fig. 33 Fig. 34 necting-rod is at right angles to the direction of the pin's motion, and hence no force combined with it can give a resultant in the direction of the tangent to the path ; the whole effect of this force, P', is now to compress or extend the crank (change its form), and none of it is available in moving the crank-pin. If it were not for the resistance of the crank at this time the pin would be impelled in the direction of P', at right angles to its proper path, but the resistance of the crank just balances the force received from the rod, and, according to Newton's laws, the pin is, at the instant, under a system of balanced forces, and, if in motion, it continues to move in a tangent to its path, unaffected by these forces except as they 28 KINEMATICS OF MACHINERY. influence friction. Of course this is only an instantaneous con- dition, and therefore the pin does not move through any finite dis- tance under such a balanced system of forces. Strictly speaking, the condition last considered is not equivalent to the action of no force at all, although the forces are balanced, for the pressure of the pin and of the shaft against the bearings results in a frictional resistance tending to retard the motion. The action of the fly-wheel also modifies the motion of the engine, reducing the fluctuation of velocity that would be experienced under the great variation of the resultant force throughout the revolution; but neither of these cases need be treated in connec- tion with the present discussion, which is simply intended to ex- emplify the nature of constrained motion. The distinguishing characteristic of a constrained motion is that, in a body having such motion, all points in the body have definite paths in which they move, if motion takes place under the action of any force whatever. The stresses produced in the restraining connections supply the components of force necessary to combine with the primary force, or forces, to give a resultant in the direc- tion of the prescribed path. If these connections are strong enough to resist the maximum stress to which they are thus subjected, no farther attention is required to secure the proper adjustment of the resultant force to the prescribed path. The provision of the necessary strength is in the province of another branch of me- chanics, and it may be assumed in the present work that such strength is provided. It will be seen that absolute constrainment is not possible by the ordinary methods employed in machinery construction ; because all materials are somewhat deformed under stress. Practical constrainment may always be secured ; that is, the depart- ure from the desired motion can be reduced to any required limit. The nature of the constrainment depends upon the form of the constraining members. This is illustrated in Figs. 6, 7, 8 and 9, in which the nature of the relative motion of the parts is plainly determined by the form of the contact surfaces. The degree of constrainment is determined by the dimensions and material of the constraining members. CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 29 r All motions used in machinery are either completely or partially Constrained. 21. Mechanics is the science which treats of the relative motions and of the forces acting between bodies, solid, liquid, or gaseous. " The laws or first principles of mechanics are the same for all bodies, celestial or terrestrial, natural or artificial." (Rankine.) 22. Mechanics of Machinery treats of the applications of those principles of pure mechanics involved in the design, construction, and operation of machinery. Every problem of mechanics arising in connection with ma- chinery is subject to the laws of pure mechanics, and we could con- ceive of its solution by the general methods of the larger science; but the operation would often be needlessly difficult, if not practi- cally impossible, and more convenient special treatment has been developed for the limited class of phenomena connected with prob- lems of mechanism. It has been seen that constrained motions are much more easily treated than are free motions; and all problems of motions of machines are included under constrained, or partially constrained, motions. It is mainly the distinction between free and constrained motions, in fact, that separates the Mechanics of Machinery from the more general science of Pure Mechanics. 23. A Mechanism, or train of mechanism, is a combination of resistant bodies for transmitting or modifying motion, so arranged that, in operation, the motion of any member involves definite, relative, constrained motion of the other members. 24. A Machine consists of one or more mechanisms for modify- ing energy derived from natural sources and adapting it to the per- formance of useful work. A machine may consist of a single mechanism according to this definition; but it has seemed best to make the following distinction between a mechanism and a ma- chine: the primary function of the former is to modify motion; while that of the latter is to modify energy, and, of course, inci- dentally motion. The term "mechanism" becomes more general, and it includes the elements of a large class of instruments or appa- ratus, such as clockwork, engineers' instruments, models, and also most forms of governors, as well as some larger constructions, the function of which is essentially the modification of motion, and 30 KINEMATICS OF MACHINERY. which only do work incidentally, such as the overcoming of their own frictional resistance. There is a real and vital distinction be- tween machines and such apparatus; but so far as a study of their motions is concerned, no such distinction need usually be made. From the above definitions of mechanisms and machines, we may derive the following: A Machine is a combination of resistant bodies for modifying energy and doing work, the members of which are so arranged that, in operation, the motion of any member involves definite, relative, constrained motion of the others. The essential characteristics of a machine are: (a) A combination of bodies. (b) The members are resistant. (c) Modification of energy (force and motion) and the perform- ance of work. (d) The motions of the members are constrained. (a) A machine must consist of a combination of bodies. The lever does not, by itself, constitute a machine, nor even a mechanism, and it only becomes such when combined with the proper fulcrum or bearing. Without this complementary member, properly placed and sustained, a definite, constrained motion is impossible. The fulcrum is just as important an element in the make-up of the machine as is the lever itself. The screw is of no use in modi- fying motion or energy unless it is fitted with the proper envelope, usually called a nut. So with the wheel and axle. It makes no difference whether made from a single piece of material or built up from several pieces of stock, the wheel and axle is essentially one piece when completed, as there is no relative motion between the various parts, and it can only be of use in connection with appro- priate supports or bearings. And so on with all other examples; the simplest machine must have at least two members, between which relative motion is possible. (b) The members of a machine are generally rigid, but not neces- sarily so. Flexible belts, straps, chains, etc., confined fluids (liquid CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 31 or gaseous), and springs, often form important parts of machines. The flexible bands can only transmit force when subjected to ten- sion; the confined fluids transmit force only under compression; springs may act under tension, compression, torsion, or flexure. These bodies are not rigid, in the usual sense of the word, but they are resistant under the particular action for which they are adapted; hence they can be used in special applications to great advantage. In fact their value in such applications is due to the absence of the property commonly designated as rigidity. No material is absolutely rigid, and what is commonly and con- veniently called a rigid body is one in which the distortions under load are so small as to be negligible for many purposes. The action of springs, when carefully analyzed, is found to be identical in quality with that of the so-called rigid bodies. The characteristic of springs is the magnitude of the distortions. Every solid body possesses the property of yielding under a load to. a greater or less degree, following the same general law as springs, within the safe working limit at least. The difference is one of degree only, but in this difference of degree lies the special fitness of springs for certain parts of machines. (c) The machine is used to modify energy and do work. It is interposed between some source of energy and the work to be done, and it adapts this energy, as supplied by or derived from natural sources, to the required work. The conception of a machine involves the conception of some source of energy, an effect to be produced, and a train of mecha- nism suitably arranged to receive, modify and apply the energy derived from this source to the desired end. The nature of the source of energy and of the work to be done determine the character of the machine, and the forms of the members for receiving the energy, transmitting, modifying, and applying it. The primary natural source of energy may be the muscular effort of animals, wind, water, heat (acting through sucL vehicles as steam, air, or other gases), etc. The secondary sources may be pulleys, gears, shafts, etc., deriving their energy, directly or indirectly, from some of the primary sources. The prime movers windmills, water-wheels, heat-engines, etc. are driven 32 KINEMATICS OF MACHINERY. from primary sources, while machinery of transmission machine tools, dynamos, etc. are actuated from secondary sources. In a machine-shop, for example, the source of energy of the tools is the line-shaft, or the counter-shaft, according as the latter is, or is not, treated as a part of the tool; it evidently makes no difference how this shaft is driven, so far as the study of the indi- vidual tool is concerned. The source of energy being a rotating- shaft, the member of the machine receiving the energy must be a pulley, gear, sheave or other form capable of connection with such rotating-shaft. Energy may be transmitted by compressed air or by water under pressure ; then the receiving member may be a Dis- ton, reciprocating in a suitable cylinder, or a wheel with appropri- ate vanes or blades attached. In a similar way the desired result determines the motions and forms of the members producing it. When metal is to be planed a reciprocating motion is usually imparted to the member to which the piece operated upon is attached, or to the cutting tool. Thus, different classes of work, such as grinding grains or min- erals, pumping water or other fluids, compressing air or other gases, weaving or spinning, cutting woods, stones, or metals, the transportation of materials, etc., each require an appropriate modification of the energy imparted to the receiving member of the machine. In general, any of the sources of energy may be applied to pro- duce any mechanical effect by means of proper trains of mechan- ism; and this gives rise to a very great number of possible ma- chines. The working members of machines have been classified by Willis as: (a) Parts receiving the energy. (b) Parts transmitting and modifying the energy. (c) Parts performing the required work. To these might be added : (d) Auxiliary parts, as regulators, etc. (e) Frames for restraining the motions and sustaining the machines. Various classifications of the parts of machines have been made CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 33 by different writers, but that of Willis has perhaps been most gen- erally accepted. From a kinematic standpoint, such classifications are of doubtful value, and Reuleaux's masterly treatment of the subject indicates that all such divisions are artificial and arbitrary. This will be more fully discussed under Inversion of Mechanisms. The working, or moving, members of a machine may be levers, arms, beams, cranks, cams, wheels with treads, blades, vanes, or buckets, with teeth or with flat or grooved rims, etc. , screws and nuts, rods, shafts, links, and other rigid members; as well as belts, bands, ropes, chains (flexible members); and occasionally confined fluids, as water, oil, air, etc. Many modifications of these are used, and an indefinite variety of forms result, yet, kine- matically, when reduced to the simplest forms, the variety of mech- anisms is much less than would at first appear. The frames which support the working parts and determine their motions are almost as varied in form and materials as the moving members themselves, but are capable of similar simple treatment. In fact, as will appear later, the frame may be treated as exactly equivalent to any other member of the machine, and so far as relative motion of the members is concerned, it mat- ters not which particular piece is made " stationary." The leading distinction between a machine and a "structure" (such as a bridge) is that the former serves to modify and transmit energy, or force and motion; while the latter modifies and trans- mits force only. Some parts of machines, as the fixed frames, are properly structures, while as a whole the construction is a machine. (d) The relative motions of the members of a machine are constrained, or restricted to certain definite predetermined paths, in which they must move, if they move at all relative to each, other. The nature of constrained motion has been considered in Art. 20. The leading characteristic of a mechanism or a machine is the constrainment of its motion. A structure does not permit relative motion of its members, or at most it only allows the very limited incidental motions, due to deformation of its members under loads, the effect of changes of temperature, etc. Occasionally what are 34 KINEMATICS OF MACHINERY. usually classed as structures, or parts of structures, do have pre- scribed motions, as the draw of a bridge, or a turn-table. These are not properly machines, but they do come under the preceding definition of a mechanism. All artificial combinations of bodies, to which may be given the general name of constructions, and which have constrained motion, may be classed as mechanisms; and if intended to be employed in the performance of useful work, as machines. The constrainment in mechanisms is sometimes 'partial, or in- complete. Thus in the case of a crane, in which the load is sus- pended by a chain or cable, the slightest horizontal force will sway the load-hook, and therefore change its path from the right line perpendicular to the earth's surface in which it normally moves. This does not affect the useful operation of the crane, as the hook is constrained against all undesirable motion, and for practical pur- poses the action is just as good as if the constrainment were com- plete. Often, in fact, this degree of freedom of motion is de- sirable. Again, consider the familiar fly-ball (conical-pendulum) gov- ernor (Fig. 9) as used on many classes of steam-engines. The balls of the governor are constrained to the extent that their cen- tres always lie in the surface of a certain sphere, that is, they have spherical motion ; but they may move in any path whatever (within the limits of action), lying in the prescribed spheres. The path in which they actually travel depends upon the relations between their mass, angular velocity, and radius relative to the axis about which they revolve at the instant under consideration, and their motion can be determined only in connection with these forces. In all but a few such cases as the latter we may study con- strained motion quite apart from the forces involved in the opera- tion of the mechanism. 25. Machine Design; Kinematics. The design of a new ma- chine or the analysis of an existing machine divides itself natu- rally into two quite distinct processes, as will appear upon brief reflection. In every machine energy is supplied from some source and so modified as to produce some useful effect. A train of CONCEPTIONS OF MOTION. NATURE OF A MACHINE. 35 mechanism is employed to secure the required transfer or modifi- cation, and this intermediate mechanism must be adapted, first, to secure the motion demanded to produce the desired result ; and second, to transmit the necessary force without breakage or undue distortion of the members of the machine. It will readily appear that the motion system can often be planned or studied without ^considering the magnitude of the forces transmitted. As an illus- tration, consider the lever of Fig. 35, in which the distance from the fulcrum, 0, to A is, say 1 foot, and distance from to B is 2 feet. Now if B be moved through a dis- tance of 2", A will move through 1". Suppose the resistance to act at A, and the force which over- comes it at B. The resistance at A Fig. 35 may be 1 pound, 1 ton, or of any -other amount whatsoever, then the force required at B to overcome it will be of a corresponding magnitude; but in any case the ratio of the motions or the velocity ratio of A to B will be determined by the length of the lever-arms, independent of the actual forces in- volved. Furthermore, if B moves 2" in one second, A will move 1 inch in one second; if B moves 6" in one second, A will move 3" in one second ; or if B makes 4 strokes per second, A will also make 4 strokes per second; but the (total) path described by A, or distance moved through, will be but half the path of B. So for any motion whatever of B, A will have a definite, corresponding motion, and the ratio of the motions will remain invariably the s.ime, whatever the actual motion and the forces acting may be. It appears then that the ratio of the motions which A and B have, relative to the fixed member, is determined purely by geometrical considerations, and may be studied without taking into account anything else. The lever must not only give a required motion to one point for a given motion of another, but it must transmit a certain force; .and having satisfied the motion requirement, it is necessary to give the lever, the pivot, and all parts subjected to load, sufficient strength to safely carry the loads. This second operation requires & knowledge and application of the physical properties of the mate- 36 KINEMATICS OF MACHINERY. rials used, and of other laws of mechanics than those relating to simple motion. A similar discussion would apply to all ordinary mechanisms, but the foregoing is perhaps sufficient for the present purpose, viz., showing how the design of a machine may be taken up as two distinct processes, one of which can be completed, subject to certain modifications, before the other is considered. Frequently the actual motions, velocities, and accelerations, as well as the external loading, are involved in the second process; for the weight of a part and the changes in direction and velocity of its motion produce stress in the restraining members. An example is the stress due to "centrifugal force." In the complete design of a machine, there are many other considerations affecting the durability, freedom from frictional and other losses, etc., that are no less important than the preceding, and all of these must be carefully weighed in their proper place; but these consid- erations are not within the province of the present work. The two grand divisions of the Mechanics of Machinery out- lined above are called: (I) The Geometry of Machinery, Pure Mechanism, or Kine- matics ; (II) Constructive Mechanism, or Machine Design. In beginning the study of machinery, it is both logical and convenient to take up the above divisions, in the order given. The first division, Geometry of Machinery, Kinematics, or Machine Motions, will be the leading subject of the present work. While, as in the illustration of the lever given above, the con- sideration of the forces acting is not, generally, involved in the study of machine motions, there are important classes of mechan- isms the motions of which cannot be treated without taking cog- nizance of certain forces. Examples of these are the centrifugal governors, already mentioned, so commonly used on steam-engines and other motors, in which centrifugal force and the force exerted by springs, or by gravity, can not be separated from a treatment of the motions; also escapements such as are used in clocks and watches. CHAPTER II. GENERAL METHODS OF TRANSMITTING MOTION IN MACHINE?. 26. Transmission through Space without Material Connec- tion. In most mechanisms motion is transmitted and controlled through actual contact of members of the mechanism; but there are certain exceptions as, for example: electric motors, escape- ments of clocks, governors, etc. The armature of the motor is caused to rotate by electromagnetic forces acting across an open space; the pendulum or balance-wheel of the clock- work is driven by the intermittent action of gravity or a spring, though the resultant motion is affected by the length of the pen- dulum or proportions of the wheel, independently of the intermit- tent connection with the escape- wheel ; and the motion of the governor-balls is determined by the combined effect of centrifugal force and of gravity or of springs. In these instances, the motion of the members is not fully constrained, and the motions of such mechanisms cannot be treated by purely geometric methods, inde- pendently of the forces involved in the actual operation. With the exceptions of these and similar mechanisms, motion is only transmitted by direct contact of one material body with another. We are at present only concerned with such cases as the latter. 27. Transmission by Actual Contact. These cases may be conveniently treated under two divisions; and the second division may be subdivided into two classes. In every mechanism we have one member, frequently called a link, whatever its form driving another link; the former is called the driver, and the latter the follower. The driver may have a surface which bears directly upon the follower, or there may be an intermediate piece serving to transmit the force and motion. This intermediate connector may be a rigid 37 38 KINEMATICS OF MACHINERY. bar or block ; it may be a flexible band (as a belt, cord, or chain) ; or it may be a confined fluid column. These various modes of connection give rise to the following classification : Direct Contact. . * Modes of Transmission of Motion. Intermediate Connectors " Rigid Links. Flexible | Bands ' etc ' Connectors Fluid Oonnectors . 28. Higher and Lower Pairing. Figs. 36 and 37 represent Fig. 36 Fig. 37 examples of direct contact transmission, in which A will be con- sidered as the driver and B as the follower. The contact in these examples is confined to a line (or a point), instead of being dis- tributed over a finite surface. . Such contact line or point con- tact constitutes what is termed higher pairing ; while contact over a finite surface is called lower pairing. Higher pairing usu- ally involves greater wear at the contact surfaces, and is generally to be avoided if it is possible to do so. The contact between the teeth of gears and that between most cams and their fol- lowers is necessarily higher pairing. However, there are many cases where it is perfectly practicable to introduce modifications in the construction which distribute the contact over a surface, with- out sacrifice of the kinematic relations. While this does not change the relative motion of driver and follower, it is practically advan- tageous in reducing the intensity of contact pressure, and conse- quently the wear of parts. It is usually desirable to substitute lower pairs for higher pairs where practicable. In certain cases, TRANSMITTING MOTION IN MACHINES 39 where pure surface contact is not possible, a modification, which does not eliminate line contact, may be advantageously employed. Figs. 38 and 39 show cases of transmission from the driver A to follower B by direct contact, higher pairing being used. In Fig. 38 Fig. 39 these cases (Figs. 38 and 39), the kinematic action is the same as would result from contact between the point p of driver and the dotted "pitch-line" of the follower, as indicated by Figs. 40 and Fig. 40 Fig. 41 41. These latter figures do not represent practical mechanisms, for of course it is necessary to have the contact parts of sen- sible size. Figs. 42 and 43 show similar arrangements, each having a suit- able block interposed between the driver and follower. These in- termediate pieces do not change the transmission of motion in any degree, but it will be noticed that the driver now acts upon the block, and the block upon the follower, eliminating line contact entirely without sacrifice of the desired motion, and a better prac- tical mechanism is thus obtained. The mechanisms of Figs. 38 and 39 are respectively identical, kinematically, with those of Figs. 42 and 43. An intermediate connector, or a new link, has been KINEMATICS OF MACHINERY. introduced, and, in a sense, the mechanism comes under the second division in the above classification; but this intermediate con- nector, C, does not alter the transmission of motion. As we are not concerned in the least with the motion of this block itself, it Fig. 42 Fig. 43 ^-_| -' may be neglected in the kinematic analysis, and such substituted mechanisms will be treated as direct contact under Division I. If desired, A could be treated as the driver of (7; and C (which is the follower with regard to A) as the driver of B. Except in cases where the contact surfaces of both links are either planes, surfaces of revolution, or regular helical surfaces, such substitution cannot be made; for these are the only contact surfaces possible in lower pairing. In gear-teeth, for example, it is not possible to avoid higher pairing. There are other cases, as in cams, where it is practically of advantage even though line contact is not thus eliminated to introduce an intermediate piece, replac- ing one kind of line contact by another kind. Thus, in Fig. 44, the cam could act directly upon the end of the rod B; but the friction would be excessive and the action would not be smooth, especially if the form of the cam departs much from that ,of a surface of revolution whose geometrical axis coincides with the axis of rotation. When a roller, C, of suitable size, is attached to Fi 9- 44 the end of the rod much smoother action is obtained. The roll does not rub on the cam, as in the TRANSMITTING MOTION IN MACHINES. 41 direct sliding contact of the follower upon the driver, and the sliding action is transferred to the pin which carries the roll, where surface contact is procured. In this case, as in those of Figs. 42 and 43, the intermediate connector can be neglected kinematically. The motion transmitted to the follower corresponds to the contact of the centre of the pin, p, with a hypothetical surface called the pitch surface of the cam indicated by the dotted line. The rela- tion of this pitch surface to the actual working surface, and the derivation of the latter from the former, will be treated later under the head of Cams. In the following discussions of the angular velocity ratio of driver to follower, in direct-contact mechanisms, the auxiliary con- nector the block, cam-roll, etc. will be neglected, as it has been seen that its own motion is immaterial, and that it does not affect the velocity ratio of driver to follower. It is interesting, in connection with the preceding discussion, to note that it is sometimes advantageous to employ line or point contact when the case will, kinematically, permit surface contact. The familiar roller-bearings and ball-bearings are examples. In these, friction and wear are reduced by the substitution of line or point contact for the ordinary surface-bearing, because by this sub- stitution the grinding effect of sliding is replaced by a rolling of each member upon those with which it comes into contact. 29. Direct-contact Transmission. The most general case of direct contact is between two surfaces such as are shown in Figs, Fig. 45 Fig. 46 36, 37, and 45. The surfaces may both be convex, or one may be concave, as in Fig. 45 ; but in the latter event the radius of curva- ture of the concave surface must always be at least as great as that of any portion of the other member that can come in contact with 42 KINEMATICS OF MACHINERY. it ; otherwise a certain part of the concave surface will not come into contact, as in Fig. 46, between e and /, and the action will be discontinuous or irregular. Except for this limitation, the surfaces may be of any form; but the present discussion will be confined to those cases in which all the elements of the contact surfaces and both axes of rotation are parallel to each other. Members having such contact surfaces can have only plane motion relative to each other. There are special cases not coming under this class which will be treated later in the work. In treating these plane motions the simplification referred to in Art. 10 can be applied, that is, the representation of these surfaces and their motions by their projection on a plane parallel to the plane of motion. Motion can be transmitted by direct contact only by normal pressure between the surfaces. The action between the two parts in contact may have the nature of rolling, sliding, or mixed rolling and sliding. The last condition is the most general. The precise nature of these actions and the method of determining them will form the subject of a later section. Referring to Fig. 47, it is evident that all points in A must rotate about 0, and, likewise, all points in B must rotate about 0' '. Consequently the velocity of any point in either is represented by a line through that point perpen- dicular to the radius connecting it with its centre, or 0', as the case may be. The point of contact between the two members is at P, which may be regarded as the coin- cident position of two points, one of which, called P a , is a point in A, while the other, called Pb, is a point in B. Then Pm and Pn represent the veloc- ities of P a and Pb, respectively. The pressure between A and B is transmitted in the direction of the common normal to the two surfaces at the point of contact, and whatever the actual velocities of P a and P&, the components of Fig. 47 TRANSMITTING MOTION IN MACHINES. 43 these two velocities along the line of this normal (NN f ) must be equal when they are in contact (as Ps). If the normal component of the velocity of Pb were greater than the normal component of the velocity of P a , B would quit contact with A . On the other hand, if the normal component of the velocity of P a is greater than that of Pb, A would enter the space occupied by B, and this is incon- sistent with our conception of a rigid body. As we are concerned only with the velocity ratio of the two members,, and as this ratio is independent of the actual velocities, the velocity either angular or linear of one member may be assumed if not known, and this affords a means of determining the velocity of the other member at that instant. The angular velocity ratio of the driver to the follower is, in the general case, varying continually. Simple methods of determining this angular velocity ratio at any phase of the motion may be used, and a close analogy exists in these methods as applied to the three different classes of transmission. Each class will be discussed by itself, and the gen- eral relation will be deduced afterward. T' Fig. 48 Fig. 49 If in Figs. 48 and 49, oo^ = the angular velocity of A, be known, for the phase under consideration, the linear velocity of P a can be found from the relation : linear vel. ang. vel. = =-. : radius. or Pm OP' Represent the linear velocity of P a by Pw, and resolve it into its components along and perpendicular to the common normal 44 KINEMATICS OF MACHINERY. Thus the normal component Ps, and the tangential component Pt a are obtained. The direction of the motion of P& is known (perpen- dicular toPO'), and the normal component of its velocity must equal that of P a , or it is Ps, hence the actual velocity of P&, Pn, can be found (Art. 17, Case ()); and its tangential component, Pt bf may be found from this if desired. Having found in this way the linear velocity of P 6 , its angular velocity, &> 2 , may be obtained by dividing this quantity by the radius PO' ; and the angular velocity ratio of A to B, for this phase of the motion = % becomes known. fi, A similar method could be employed in determining this ratio for any number of phases, and thus the motion of the follower, cor- responding to the motion of the driver, whether uniform or other- wise, could be derived. The following demonstration establishes relations of the angular velocity ratio of driver and follower, in direct-contact mechanisms, which are much more expeditious and convenient in drawing-board practice. In Figs. 48 and 49, let Pm and Pn represent the linear veloci- ties of P and P 6 , respectively. Drop perpendiculars Of and O'g from and 0' upon the normal NN'. Ps is the common normal component of Pm and Pn. Pms and 6/P/are similar triangles; also Pns and O'Pg are similar triangles. co l = angular velocity of P a about = - = --., . (1) a = angular velocity of B about 0' = = --. . (2) - Of Ps ~ - Of Prolong the normal and line of centres till they intersect at /; then lOf and 10' g are similar triangles and - = =. (4) 10 Of co, TRANSMITTING MOTION IN MACHINES. 45 It follows from the above discussion that: In direct-contact mechanisms the angular velocities of the members are, at any phase, inversely as the perpendiculars let fall from their fixed centres upcn the line of the common normal of the two curves ; or inversely as the segments into which the line of centres is divided ~by this normal. > 30. Link-connectors. A relation very similar to that just de- rived can be deduced for the angular velocities of a driver and fol- lower connected by a rigid link. In this case we are not concerned with the motion of the inter- mediate connector itself. Figs. 50 and 51 show two arms, OA and O'B, free to turn about the fixed centres and O r , and connected by the link AB. The velocity of the point A is rep- F| 9- 50 resented by Am, perpendicular to OA. The velocity of B is shown by Bn, perpendicular to O'B', and its magnitude is determined by the fact that its component in the direction of AB must equal the component of Am in this same line ; for if these components are not equal, the distance between A and B must change, which is incon- sistent with the conception of a rigid body; hence, if Am is assumed, En is thereby determined. 46 KINEMATICS OF MACHINERY. Let G?, = angular velocity of A about = -j, . . (1) Tt Let a?, = angular velocity of B about 0' = --=. . (2) Drop perpendiculars 0/and O'g upon AB-, then triangles OAf and .4ms are similar; also 0' Bg and Bnr are similar. From #4/" and Ams, ~ = - =GJ i ( 3 ) From O'Bg and Bnr= = ^ ... (4) Produce -4.5 and 00' (if necessary) to intersect in /; then 10' __ 0^ _ , /O " 0/ << From this reasoning the following statement is drawn : In two arms connected ~by an intermediate link the angular ve- locities of the arms are to each other inversely as the perpendiculars let fall from the fixed centres upon the line of the link ; or inversely as the segments into which the line of centres is cut by the line of the link (both of these lines produced, if necessary). These relations may be seen from direct inspection, by assuming the system to be replaced by the two effective arms, Of and O'g, connected by the link fg. This new system would evidently be equivalent to the original system, for this particular position (but for no other) ; and as the arms Of and O'g are perpendicular to the link, the linear velocities of / and g are equal ; hence the angular velocities of the arms are inversely as the radii, or = -^. This would apply to any phase; but the substituted arms, Of and O'g, would not be of the same lengths for different phases. TRANSMITTING MOTION IN MACHINES. 47 The relation deduced above may be arrived at also by means of the method of instantaneous axes. In Figs. 52 and 53, co, and a? t have the same signification as before, and co = angular velocity of the connecting-link about its instant centre. A and B are two points in the connector AB ; therefore the motions of these two points completely determine the motion of this link. The velocity of A is Am, perpendicular to OA ; and the velocity of B is Bn, perpendicular to O'B. Therefore Q, at the intersection of OA and 0' B, is the instant centre for the link AB in the phase shown (see Art. 19). As the angular velocity equals the linear velocity divided by the radius : Am Bn GO = -7r-r = QA (7) Drop perpendiculars Qlc, Of, and O'g from Q, 0, and #', respectively, upon the line of the link AB. OAf and QAk are similar triangles; also O'Bg and QBTc are similar. . i 4m QA_QA_Qk ca ' OA Am ~ OA ~ Of . . (8) (*>__ Bn_ __ _ 5; == QB X ~Bn ~ ~QB " Qk' KINEMATICS OF MACHINERY. From equations (8) and (9), = ^ = 77T- (W) CO This last demonstration thus gives a result identical with equa- tion (6). ^31. Wrapping-connectors. This term includes belts, bands, ropes, chains, and all flexible members used to connect a driver and follower, and transmitting force only under tension. The working surfaces may be of any convex cylindrical form ; but concave forms are excluded, as the wrapper would not follow the depressions of such a form, and if used the action would not be smooth and con- tinuous. The term cylindrical as used above applies strictly in case of flat bands. In case of round cords, ropes, etc., the contact surface is usually grooved to correspond x more or less closely to the form of the wrapper, but the motion is in this case equivalent to that which would be obtained by the neutral axis of the connector, wrapping upon an ideal pitch line of the member upon which it is carried (see Fig. 54). The mathematical (and kinematic) wrapper, or the pitch line, is the line xxx. In case of flat bands, also, the true pitch surface, and line of action, are at a distance from the physical face of the rigid mem- ber or carrier, equal to about one-half the thickness of the band. For the present purpose the connector will be treated as of no sensible thickness, and the surfaces shown in Figs. 55 to 58 are to- be taken as the true pitch surface. The effect of thickness of con- nector will be discussed in a later chapter. The band is flexible, but is supposed to be practically inexten- sible; and as it is subjected only to tension, the distance between any two points of the band cannot change. This implies that, whatever actual velocities two such points may have, the components- - 54 TRANSMITTING MOTION IN MACHINES. 49 of these velocities in the direction of the connector must be the same at any instant. In Figs. 55 and 56, a is the driver and I is the follower. Either Fig. 55 Fig. 56 of the tangent points of the band and carrier (A or B) is a coinci- dent point of the band and of the member which it meets at that point; GL>J and &?., are the angular velocities of the points A and B respectively, and Am and Bn are the corresponding linear veloc- ities. Am Bn Since A and B are two points in the inextensible band, their components of velocity in the direction of the connector are equal, or As = Br. Drop perpendiculars from and 0' upon the line of the connector ; then OAf and Ams are similar triangles ; also 0' Bg and Bnr are similar. co l = angular velocity of A about = 7 -. = -j^, . (1) (JA Uj ? s = angular velocity of B about 0' = - = r*, . (2) oo, ~0f */ w = r). (3) 50 KINEMATICS OF MACHINERY. Prolong the line of centres, 00', and the line of the band, AB t to meet in / ; then lOf and 10 'g are similar triangles. 10' _0'g _ a?, ''7o^~~0f~^ *' From equations (3) and (4) we can formulate the statement: In wrapping -connectors, the angular velocity of the driver is to that of the follower inversely as the perpendiculars let fall from the fixed centres upon the line of the connector ; or inversely as the seg- ments into which the line of centres is cut by the line of the connector (both produced if necessary). This relation may be shown by the instantaneous centre method also (Figs. 55 and 56). A, as a point in the driver, has a linear velocity Am and GO I = Am -f- OA. B, as a point in the follower, has a linear velocity Bn, and o? 3 = Bn -f- O'B. The acting part of the connector, AB, has an angular velocity about Q of Am Bn = QA '- = OB' Let fall Qk perpendicular to AB ; then OAfand. QAk are similar; also O'Bg and QBk are similar. , _ Am QA QA_Qk GO OA x Am ~ OA ~ Of 9 { } GO Bn O'B O'B O'g _ _ v _ - _ i fi i GO, ~ QB ' Bn QB~ Qk' Multiply (5) by (6) : C* v O'g __ O'g _ 10' Of X Qk - Of '- 10' This result accords with that of equation (4). 32. Similarity of Expressions for Angular Velocity Eatio in the Three Modes of Transmission. By substituting the term line of action for "line of the normal," "line of the link," and "line of wrapping-connector," in the three cases of direct contact, link-con- TRANSMITTING MOTION IN MACHINES. 51 nector, and wrapping-connector, respectively; the following state- ment will apply to all of these modes of transmitting motion : The angular velocities of the members are inversely as the per- pendiculars let fall from their fixed centres upon the line of action ; or inversely as the segments into which the line of action cuts the line of centres. There are special cases in which the preceding theorems are not available, because the expressions become indeterminate; though these cases can be reconciled to the general statement. For ex- ample : see the direct-contact mechanism of Fig. 57, or the link- work of Fig. 58. In these mechanisms the centre about which B Fig, 57 Fig. 58 rotates is at QO , hence the perpendicular from it upon the normal = QO (also the segment from its centre to the line of action = 00); and by the theorem of Art. 29; = n = o> This is consistent, as the angular velocity of the follower B, is o; hence that of the driver (A) is infinitely greater than that of the follower; but the result does not enable us to derive the linear velocity of the follower, for this equals the angular velocity multiplied by the radius, = o X oo , an indeterminate expression. The linear velocity of the fol- lower is easily found by other means, however, as its normal com- ponent must equal that of the linear velocity of the driver, and the direction of the follower's motion is known. From this data, the linear velocity of the follower is derived (see Art. 17). A similar course of reasoning applies to the mechanism shown in Fig. 58. 52 KINEMATICS OF MACHINERY. In the linkwork shown by Fig. 59 the follower is not under control of the driver at the particular phase there represented. As Fig. 60 A reaches the position shown (at either dead centre), it has no component of motion in the line of the link AB, hence there is no component compelling motion of B. As A passes this position, B might be moved in either of the two directions indicated by Bn or Bri. If the shaft about which B rotates is provided with a fly- wheel, or similar device, its direction of motion may be maintained, and as soon as the dead centre is passed, A again exerts an influence over its motion. In the case of Fig. 60 B comes to rest as A passes the dead centre, Bn being zero at this phase. 33. Directional Relation. It will be seen by reference to Figs. 48, 50, and 55 that the driver and follower both rotate in the same direction; that is, loth members are turning in the right-handed, clockwise, or negative direction as angles are usually reckoned ; or both rotate in the reverse direction.* It will also be noticed that in the cases shown in Figs. 48, 50, and 55, the two fixed cen- tres lie on the same side of the line of action. In Figs. 49, 51, and 56, on the other hand, the fixed centres lie on opposite sides of the line of action, and the two members ro- tate in opposite directions; if one member has right-handed rota- tion, the other has left-handed rotation, and vice versa. From this observation of all of these general cases, the following statement in * The follower is converted into the driver when such reversal takes place in Figs. 48 and 49, but not necessarily in Fig. 50. That this conversion does not take place in all direct contact and wrapping-connector mechanisms is evi- dent from Figs. 39 and 204-209, in which either member may be the driver. TRANSMITTING MOTION IN MACHINES. 53 regard to the Directional Relation is quite evident: In any of the three ordinary modes of transmission of motion the directions of rota- tion of the driver and follower are the same if the fixed centres ofboih lie on the same side of the line of action ; and the directions are -opposite if these centres lie on opposite sides of the line of action. 34. Condition of Constant Angular Velocity Ratio. It has been shown that with any of the three common methods of transmitting motion the angular velocities of the members are inversely as the segments into which the line of centres is cut by the line of action. Thus in any figure from 48 to 56 (except Fig. 54) GO I : G0 a : : 10' : 10. If the angular velocity ratio is constant,, - 1 -=-^ = a constant, and as 00' the distance between the fixed centres is a constant, /must be a fixed point in this line (or its extension) in order that ihe above condition be realized. Therefore it may be stated that : The Condition of Constant Angular Velocity Ratio is that the line of action must always cut the line of centres (produced if necessary] in a fixed point. This condition is fulfilled by an infinite number of pairs of curves which may be used as the outlines of the acting faces of di- rect-contact members. It will appear later that the proposition just stated is of fundamental importance in the theory of teeth of gears. The condition of constant velocity ratio is fulfilled in the case of wrapping-connectors when the driver and follower have faces which are right cylinders with the axes of the cylinders as the axes of revolution ; for example, in the case of ordinary pulleys with crossed or open belts. Constant velocity ratio is secured with link transmission when the driving and the driven arms are equal (Fig. 59), and the length of the connecting-link is equal to the distance between the fixed centres and parallel to the line of centres, as in the parallel rods of locomotives. 35. Nature of Rolling and Sliding. When two pieces act to- gether by direct contact they may roll upon each other, they may slide upon each other, or they may move relative to each other with a combined rolling and sliding action. Fig. 61 shows two such members, in which p is the contact 54 KINEMATICS OF MACHINERY. point in the phase shown. If r and s are any two points which meet as the action continues (becoming coincident contact points), the arcs pr and ps must be equal if the action is pure rolling. If for any increment of motion the corresponding arcs of action of the two curves are not equal, there must be some sliding between them. In pure rolling action no one point of either body comes in contact with two successive points of the other. If a point of one of the bodies comes in contact with all suc- cessive points of the acting surface of the other (within the limits of its path), the action is purely sliding; for example, the piston in the cylinder of an engine. In some cases, as in many cams and all gear-teeth, the action is mixed sliding and rolling. The sliding action must occur along^ the common tangent at the point of contact of the two surfaces. 36. Rate of Sliding and Condition of Pure Rolling. It has- been shown that in direct-contact mechanisms the normal compo- nents of the velocities of the points of contact must be equal. The tangential components may have any values, either in the same or in opposite directions. When the tangential components of the velocities of the contact points are in the same direction and equal there is no sliding, and the two velocities are identical as corresponding components are equal. The rate of sliding is the difference of the tangential components if they are in the same direction, or their sum if they are in opposite directions ; or : The rate of sliding is the algebraic difference of the tangential components of the velocities of the points of contact. TRANSMITTING MOTION IN MACHINES. 55 In direct-contact mechanisms the normal components of the velocities of the points of contact are always equal, and the tangen- tial components are also equal when the action is pure rolling. Figs. 62 and 63 illustrate this condition, and the two velocities, Pm N" Fig. 63 and Pn, are identical. But P, as a point in A, moves at right angles to OP', and, as a point in B, it moves at right angles to O'P; therefore, when Pm coincides with Pn, OP and O'P are both perpendiculars to the same line at the same point and they must therefore lie in one right line. In order that this may occur, P must lie in the line of centres; or: The condition of pure roll- ing is that the point of contact shall always lie in the line of centres. Any pair of direct-contact pieces bounded by curves which satisfy the condition just stated act upon each other with a pure rolling action; and any departure of the contact point from the line of centres is accompanied by sliding action. There are many sets of curves which may be employed to thus transmit motion by direct contact and with pure rolling action, among which may be mentioned : tangent circles (or circular arcs) rotating about their centres; pairs of equal ellipses each rotating about one of its foci with a distance between the fixed centres equal to the common major axis; and pairs of similar logarithmic spirals rotating about their foci. As the common normal to two direct-contact members passes through the point of contact, and as this point always lies in the line of centres if the action is pure rolling, the common normal 56 KINEMATICS OF MACHINERY. cuts the line of centres in the contact point when pure rolling occurs. The angular velocity ratio is inversely as the segments into which the line of centres is divided by the normal; or inversely as the perpendiculars let fall from the fixed centres upon the normal (see p. 45, Art. 29). In pure rolling these segments are the contact radii themselves (OP and O'P of Figs. 62 and 63), and therefore in such cases the angular velocities are inversely as the contact radii. Drop perpendiculars, Ok and O'l, from the fixed centres (Figs. 62 and 63), upon the common tangent, TT', and it will be seen that the triangles OPk and O'Pl are similar. .'.7:7- =-7777 == -~ OK Or 0)2 the angular velocity ratio of the members. We may then use, if convenient, the following relation: The angular velocity ratio, in direct-contact mechanisms having pure rolling action, is inversely as the perpendiculars from the fixed centres w the common tangent. In the circular-arc forms (Figs. 64 and 65) the perpendiculars from the centres to the tangent are the contact radii; thus the well-known relation for tangential wheels of circular section, that the angular velocities are inversely as the radii of the circles, is seen to agree with the more general relations deduced in this article. 37. Constant-velocity Ratio and Pure Rolling Combined.' As stated in the preceding two articles, there are many pairs of curves which will satisfy either the condition of constant-velocity TRANSMITTING MOTION IN MACHINES. 57 ratio, or of pure rolling. There is but one class of curves, how- ever, viz., circular arcs rotating about their centres, which can have at the same time loth constant-velocity ratio and pure roll- ing (see Figs. 64 and 65). For constant- velocity ratio, the normal must cut the line of centres in a fixed point; for pure rolling, the contact point (through which the normal passes) must lie in the line of centres. If both of these requirements are met at the same time the contact point must be a fixed point in the line of centres; hence the contact radii must be constant; and therefore the out- lines of the members are circular arcs. 38. Positive Driving. Circular-arc members (right cylinders), as shown in Figs. 64 and 65, do not transmit motion positively. Actual physical bodies of the corresponding forms can transmit motion from one to the other only through frictional action. In the absence of friction, with such forms, no motion could be trans- mitted against any resistance; with friction a limited resistance can be overcome. There is no assurance that more or less slipping may not occur, and if this does take place the velocity ratio be- comes both variable and uncertain.* In such forms as those shown in Figs. 62 and 63, on the other hand, motion of the driver involves a positive and definite motion of the follower. It is now in order to determine the conditions necessary to insure positive or compulsory driving. It is some- times stated that positive driving is only produced when the contact radius of the driver increases as the action proceeds ; thus in Fig. 61 A can only drive B positively when Op is greater than any pre- ceding contact radius, as Or, and less than any succeeding radius as Or'. While this is the case with many forms, it is not a general * The tangential Component of the velocity of either the driving or driven point represents its rate of sliding along the tangent. If the tangential com- ponents of the velocities of both these points are equal and in the same direc- tion there is no sliding between them. With perfectly smooth surfaces, one of the members could not move the other against the smallest resistance. In the practical cases where motion is transmitted by frictional action, the effectiveness increases as the departure from ideal smooth cylindrical surfaces becomes greater. Perfect cylinders, if such were possible, would not be of the slightest use in such causes. 58 KINEMATICS OF MACHINERY. requirement for positive driving. Figs. 66 and 67 show mechan* isms in which A is the driver and B is the follower. It will be seen that A can rotate indefinitely, causing continuous rotation of B (though not with a uniform velocity ratio between A and B), and Fig. 66 at the end of each rotation the two members will return to the same- relative positions they had at the start. It is evident then that the contact radius of the driver cannot increase indefinitely. It will be seen, also, that B may be the driver, and that a similar remark, will apply in this case.* Eeferring to Figs. 64, 65, and 68, it is seen that the motion, Pm, of the contact point of the driver lies in the direction of the common tangent, TT f \ hence the normal component of this motion. is zero. It is only the normal com- ponent of the driving point's motion which tends to impart positive mo- tion to the follower ; and in the cases of Figs. 64, 65, and 68, where the motion of this point is wholly tan- gential and the normal component is zero, there is no tendency to pro- duce positive driving. In other words, positive driving is assured only when the driving contact point has a component of motion in * The presence of the intermediate roll or block is not essential, and the above statement would be equally true if A were simply provided with a pia engaging the follower. Fig. 68 TRANSMITTING MOTION IN MACHINES. 59 the direction of the normal, and as the contact point moves per- pendicularly to the contact radius, there can be no such normal component of motion when this radius is perpendicular to the com- mon tangent ; or, what is the same thing, when this radius coin- cides with the common normal. Positive driving cannot occur then if the common normal passes through the centre about which the driver rotates. The contact radius may coincide with the tan- gent, and in fact this is a very favorable position, as the motion of the contact point is then entirely in the direction of the normal, and there is no tendency to slide. If the common normal passes through the fixed centre about which the follower rotates the driver cannot impart positive motion to the follower; for in this position the normal component of the motion of the driven point is per- pendicular to the path, and any motion in this direction is prohib- ited by the nature of constrained motion. A force directed toward the centre does not tend to produce rotation, but only to exert pressure against the bearings. It is thus seen that positive driving cannot occur if the common normal passes through either of the fixed centres. If the normal passes through the centre of the driver only, the driver can move, but motion is not transmitted to the follower. If the normal passes through the centre of the follower, the driver is locked, for its motion can have no normal component; but the follower may still move if under other influences, such as the action of a fly-wheel. If the common normal passes through loth fixed centres, as in Figs. 64 and 65, the motions of both contact points are tangential and wholly independent, except for the frictional action. In conclusion the following statement may be formulated : The condition of positive driving is that the common normal shall not pass through the fixed centre of either the driver or the follower. 39. Inversion of Mechanisms. It was explained in Art. 5 that one body may have at any time distinct motions relative to different bodies, and that it is sometimes convenient to refer the motion of a member to some other part than the fixed frame. Take, for example, the crank and connecting-rod mechanism of the ordinary reciprocating-engine, as shown in Figs. 27 and 69. There art 60 KINEMATICS OF MACHINERY. four members of this mechanism : the crank, connecting-rod, cross- head and piston (the last two are kinematically one piece), and the frame (including the cylinder).* In Fig. 69 these parts are designated by the letters a, b, c, and d respectively, and the shading of d is used to indicate that it is the stationary member. The crank, a, rotates about the centre O ad relative to the frame, d, and this centre, O ad , is the instant centre (also a permanent centre) for the motion of a relative to d. The frame, d, also rotates rela- tive to the crank, a, about this same centre; for if we imagine the crank to be the fixed mem- ber, as in Fig. 70, d actually does rotate about this centre as the mechanism operates; but the change in the relative positions of the members is simply that due to the mode of constrainment, which- ever member is fixed, and we have not changed the form of the mechanism in any way; hence the relative motions of the parts are the same under both conditions. The members in Fig. 70 are identical with those of Fig. 69, and their connections with each other are the same as before. If the member #, the original connecting-rod, be made the fixed mem- ber, as in Fig. 71, a and d are both moving members; but they still rotate, relatively , about O ad . This mechanism, as shown in Fig. 71, is that of the oscillating steam-engine, in which a corre- sponds to the crank, b to the frame, c to the cylinder, and d to the piston-rod and piston. Fig. 72 represents another condition of this same mechanism, in which c, the original crosshead, is the fixed member. Under any of these four conditions the relative * The notation used in the following discussion is this : small letters, a, b, e, etc., are used to designate the different members ; is used for all centres (instant or permanent), and the subscripts of indicate the members which rotate relatively about it. Thus O ac indicates that the member a rotates rela- tive to c (or c relative to a) about the point designated as O a > TRANSMITTING MOTION IN MACHINES. 61 motion of a to d (or of d to a) is a simple rotation about the centre O ad . In a similar way, the relative motion of a and ~b is a rotation about the centre O ab \ that of Z> and c is a rotation (or oscillation) Fig. 71 Fig. 72 about Obc ; that of c and ^ is a translation parallel to the centre line of d, or a rotation about a centre, O c to d. Each of these motions is equivalent to rotation or oscillation about a centre (per- manent or instant). Four of these are permanent centres in the mechanisms referred to above, viz. : O ab , 6c , O cd , and# ad .* In Fig. 73, OK rotates relative to d in an arc of finite radius. In Fig. 69, the motion of O bc relative to d is equivalent to rotation about the centre 0^, in an arc of infinite radius. If it were pos- sible to supply a link of infinite radius connecting the point O cd of d with the point O bc , the mechanism of Fig. 69 would be similar in character to that of Fig. 73; or the former may be considered as a limiting form of the latter. The practical mechanism of Fig. 69 is the exact kinematic equivalent of such an imaginary mechanism. The relative motions of the opposite links, a and c, and b and d, of Figs. 69 and 73, are not so evident as are the motions of the adjacent members; but the instant centres for these motions are readily located from the principles of Art. 19. In the first place it is to be noted that the instant centre for two members is a common or coincident point of both, for it is a point with regard to which neither of them has any motion. If this point lies * In the mechanism of Fig. 69 the centre Ocd is at infinity. While it is not an actual physical pin like the other permanent centres, still it is properly a permanent centre rather than an instant centre, because it is equivalent to a fixed centre of a link of infinite length. TRANSMITTING MOTION IN MACHINES. 63 outside of either actual bod}*, this body may be imagined as extended to include this centre, for a body may have rigid con- nection with any point relative to which it has no motion, as stated in Art. 5. It is to be borne in mind, then, that O ab is a point in both a and b\ O ac is a point in both a and c, etc. This enables us to locate the instant centres for the opposite links. In Fig. 73 O o6 , as a point in a rotating relative to d, must move in a line perpendicular to the line joining the points O ad -0 ab ; hence its motion is equiva- lent to a rotation about some point in this line or its extension (see Art. 19). Likewise, the motion of Ob c relative to d is equiv- alent to a rotation about some point in the line O cd -0 bc or its extension. But O a b and O bc are two points in the link 6, and, as such, must have the same motion, i.e., rotation about the point O bd at the intersection of the lines O ad -0 ab and O cd -0 bd . The motion of these two points determines the motion of the link. Therefore O bd is the instant centre for the motion of b relative to d. In a similar way, all points of b rotate relative to a about the centre O a b, and all points of d rotate relative to a about O ad . The points O bc and O cd are points in b and d, respectively; hence they move, relatively to a, perpendicularly to the lines O bc -0 ab and O cd -0 ad respectively, and the intersection of these lines, or O acj is their common centre of rotation relative to a. But O bc and O cd are two points in the link c; therefore the point O ac is the instant centre for the motion of c relative to a. We have thus located all of the instant centres for the mechanism of Fig. 73. Four of the centres for the mechanism of Fig. 69 have been located, viz.: the permanent centres O ad , O ab , O bc , and O cd (the last at infinity). The centres for the two pairs of opposite links, b and d, and a and c, are yet to be found. The former, O bd is readily found; for O ab (a point in 6) moves perpendicularly to the centre line of a, and O bc (also a point in b) moves perpendicu- larly to NN; therefore the intersection of these lines, or M , is the required instant centre for the motion of 6 relative to d. The reasoning by which the centre for the relative mofion of a and c is found is somewhat more involved. The point O bc is a point common to b and c. AH points in 6 rotate relative to a about the 64 KINEMATICS OF MACHINERY. centre ab ; therefore O bc as a point in b rotates about this point, or it moves, relative to a, perpendicularly to the centre line of b (the line O bc -0 ab ). The point O cd (at infinity) is a point common to c and d. As a point in d it rotates about O ad relative to a, moving perpendicularly to a vertical line through O ad , or to N' N'. One point of c (O bc ) moves perpendicularly to O bc -0 ab , and another point of c (Oca) moves perpendicularly to N' N' \ hence the instant centre for the motion of c relative to a is at the intersection of these two lines, or at the point 0^. By considering c, instead of a, as the stationary member, the same result may be reached rather more easily. It is possible to locate all of the instant centres of a mechanism of four members by the principles already given; but with higher numbers of members this cannot always be done, and a very im- portant theorem given by Professor Kennedy affords a ready solu- tion in these more difficult cases. This theorem is often advan- tageous even in four-link mechanisms, and by its aid the rather tedious reasoning employed above in finding O ac for the mechanism of Fig. 69 can be avoided. The statement of this theorem, as given by Professor Kennedy, is : " If any three bodies a, 1), and c have plane motion, their vir- tual \instant\ centres O ab , O bc , and O ac are three points upon one straight line." This theorem applies to any three bodies having plane motion, whether they be members of the same mechanism or not. Two of the three centres being known, or assumed, the following demon- stration proves that no point lying outside of the line connecting the known centres can be the required third centre; hence this third centre must lie in the line connecting the other two, as stated in the theroem. In Fig. 74, let a, b, and c be any three bodies moving in a plane, members of a single mechanism as indicated by the heavy lines, or entirely independent bodies. Suppose that a rotates relative to b about the centre O ab , and that c rotates relative to b, about the centre O bc . Then aB is a point common to a and b, and O bc is a point common to b and c. As- sume that such a point as 0' is the instant centre for the relative TRANSMITTING MOTION IN MACHINES. 65 motion of a and c; then this point is a common point of a and c. All points in a must rotate relative to b about O a &, and all points in c must rotate relative to b about Obc- As a point in a the motion of 0' relative to b must be in a direction perpendicular to Oab-0 f , while as a point in c its motion relative to b must be in a direction perpendicular to 0& c 0'. A point can have but one motion relative to a given body at any time, and therefore these perpendiculars must coincide. This is possible only when O a b-0' and 0& c -0' are in one straight line, viz.: the line joining the given centre O a & and 0& c . It is thus seen that the point 0' cannot be the centre required, unless it does lie in such line. This theorem does not locate the centre O ac definitely; for it may be any place along the line of O a b-0b c , between these centres, or beyond either of them. This is as it should be, for in the arrangement of Fig. 74 there is no prescribed connection between a and c, and their relative motion is therefore not definitely con- strained.* If a fourth member, d, which will constrain the motion of a relative to c such as a link connecting the free ends of a and c be introduced, another combination of three members (as a, c, and d) may be taken, in which O ad and O cd are known, and, by the theorem, the other centre for this combination (O ac ) will lie in the line of the known centres. In this constrained four-link * It is to be noticed that the theorem discussed above has reference to three members, and that these three members involve three instant centres ; any member has a centre with reference to each of the other members. In a combination of three bodies every letter which stands for one body is used twice as a subscript to 0. If two of the three centres are given their symbols will have one common letter in their subscripts, and the third (required) centre will have for a subscript the two odd letters. Thus if O a b and Obc are the given centres, O ac is the third. This is a convenient aid in applying the above theorem to a mechanism. 66 KINEMATICS OF MACHINERY. mechanism there are two lines, each of which contains O ac (viz.: O a b-0b c , and O ad -0 cd )', hence O ac is at their intersection, or its position is definitely determined (see Fig. 73). . By referring to Figs. 69 and 73 it will be seen that the loca- tions of the centres, as already determined, agree with the state- ment of the theorem. Thus, as to the links a, b, and c, the O e f at oo at intersection Fig. 75 centres O a &, 0& c , and O ac lie in one line; also, as to a, c, and d, O ac , O ad , and O cd lie in one line, and O ac lies at the intersection of these two lines. As an illustration of the application of the above theorem to more than four members, the mechanism of a common type of crank shaper, indicated in Fig. 75 by what is called a skeleton drawing, may be taken. TRANSMITTING MOTION IN MACHINES. 67 In this machine there are six members: the cranK, a; the sliding block, 6; the vibrator, c; the link, d; the ram, e\ and the frame, /, to which the members a and d are pivoted, and which is provided with guides for the motion of e. This mechanism has 15 instant centres,* of which 7 are also permanent centres. Of the permanent centres, O af , O a &, O cd , O ce , and O df are located at the points of connection of adjacent members, while 0& c and O e f are at infinity. The other centres are found by the use of .Kennedy's theorem. f The following scheme suggests the solution of this problem : Centre Required. Lies-at Intersection of the Lines. Ocf Oae Oac Obf Oae Oad Obd Obe Ocd Odf and Oce Oef Ocd Oce " Odf Oef OabObc " Ocf Oaf OabOaf " Obe Ocf Oac Oce " Oaf Oef Oaf Odf " Oac Ocd OabOad " Ocd Obe Obf Oef " Obe Oce The method of instant centres will be frequently used in the iater part of this work, especially in treating linkwork ; but it may be well to give an illustration of its use at the present place. It is to be remembered that the linear velocity of a point which is mov- ing relative to any body is proportional to its distance from the centre about which it rotates relative to that body. In Fig. 69, for example, the body a rotates relative to the stationary member, * In a mechanism of n members, there are instant centres, 2 some of which are also permanent centres. f A diagram such as is shown in connection with Fig. 75 is convenient in this work. The unshaded spaces indicate the centres to be located, and the memory is aided by checking off, in the proper place, each centre as it is found. 68 KINEMATICS OF MACHINERY. d (the frame), about O ad . If the linear velocity, v t , of O ab is known, the linear velocity, t> 9 , of the crosshead (piston) is readily found by the principles of instant centres. The point 6c is com- mon to the connecting-rod, #, and the crosshead, c\ while the point. O ab is common to the crank, a, and the connecting-rod, Z. The instant centre of b relative to d is M ; then, as all points of b must have the same angular velocity about M their linear velocities are proportional to their distances from this centre ; hence v 9 : v l : : O b(I -O bc : O bd -0 ab . If v 1 be laid off from O ab toward O bd , along the line connecting these points, and then a line, mn, parallel to the connecting-rod, be drawn till it cuts the normal NN, the length on this normal from O bc to n equals v a , from the above proportion. As the motion of c relative to d is a translation, all points of c have the same velocity relative to d ; hence v^ is the velocity of the crosshead, or piston,, relative to the frame, or cylinder. In an engine the crank rotates about the shaft with a velocity which is usually taken as uniform ; while the velocity of the cross- head (or piston) is variable. The velocity of the piston can be found for any phase by laying off the crank-pin velocity along the extension of the crank, drawing a line (as mn, Fig. 69) parallel to- the connecting-rod till it cuts the normal (NN) through the cross* head pin. A modification of the preceding construction is often eveiL more convenient. Lay off the line N'-N' (Fig. 69) through the centre of the shaft, O ad , perpendicular to the line of the piston travel. The connecting-rod (extended if necessary) cuts N'-N' in the point O ac , then, since v 2 :v l :: O bd -0 bc : O bd -0 ab and, from similar triangles, O bd -0 bc :0 bd -0 ab ::0 ad -0 ac :0 ad -0 ab , it fol- lows that v 2 :v 1 : :0 ad -0 ac ' 'O ad -0 a b. It. appears from this proportion that when the length of the crank, O ad -0 ab , is taken to represent the uniform crank-pin velocity, the cross-head velocity is represented by the distance, O a< j-0 ac , the intercept on the perpendicular, N'-N f , between TRANSMITTING MOTION IN MACHINES. 69 the shaft centre and the line of the connecting-rod, the latter extended if necessary. This relation is also evident from the consideration that the instant centre, O ac , as a common point of a and c, has the same velocity and direction of motion in c as in a. As a point of c the linear velocity of O ac is v 2 , since all points of c have the same velocity of translation. This velocity is found from the motion of O ac as a point in a by the proportion v 2 :v l : :0 ad -0 ac :0 ad -0 a b. In general, when the instant centres, O a &, O ac , and 0& c , for the plane motion of any two bodies, a and 6, relative to each other and to a third (reference) member, c, are located, the linear velocity of any point in 6 corresponding to a given linear velocity Fig. 75a of any point in a can be found graphically. In Fig. 75a, let v t represent the given linear velocity of any point, P, in a, and let the corresponding velocity, v 2 , of any point, Q, in 6, be required. The linear velocity, v 1 ', of O a b as a point of a is found from the proportion v' : v l : : O ac -0 a b O ac -P, by the construction shown. Using O a b as a point of 6, the corresponding linear velocity of Q is found, by a similar construction, from the proportion, v 2 :v f ::O bc -Q:Ob c -O ab . The ratio of the angular velocities of any two bodies, a and b, having plane motion relative to a third body, c, may also be determined when the three instant centres are located. Let oj v and w 2 be the respective angular velocities of a and b relative to c in Fig. 75a. Since O ab as a common point of- a and 6, has a 70 KINEMATICS OF MACHINERY. linear velocity i/, the angular velocities of a and 6 relative to c are equal to this linear velocity divided by the respective instant radii. That is, w l =v' -r-0 ab -0 ac and aj 2 =v' + ab -0 bc . Hence (1)^:0)2- -Oab-Obc'-Oab-Oac- When this proportion is used in the case of any mechanism the resulting value of the angular velocity ratio is identical with that obtained by the methods of Arts. 29-32. 41. Velocity Diagrams. It has been shown in the preceding article how the method of instant centres can be used to determine the linear velocity of one point from the known velocity of another point. It is often desirable to represent, graphically, the velocities of a point at various phases of a mechanism, and this is done con- veniently by velocity diagrams. Fig. 76 shows the mechanism of Fig. 76 (a) the reciprocating engine in outline. C is the crank-pin, c is the crosshead-pin, Q is the centre of the shaft. The crosshead moves from to 9 and back again to during one complete rotation of the crank. The simultaneous positions of crosshead-pin and crank- pin are indicated respectively by 0, 1, 2, 3, etc., and 0', V, 2', 3', etc. As shown in the preceding article, if the linear velocity of the crank-pin is represented by the length of the crank, r, the velocity of the crosshead for any phase is represented by the segment, s, of s TRANSMITTING MOTION IN MACHINES. 71 the line N-N, which lies between Q and the line of the connecting rod, C-c. If the segment, s, is found for each of the crosshead positions from to 9, the corresponding lengths of s may be erected as ordinates to 0-9 at the corresponding crosshead positions. A curve passing through the upper ends of these ordinates gives a velocity diagram of the point c with the path, 0-9, as a base. This diagram is called a Velocity-Space Diagram. If a sufficient number of ordinates have been determined 'the diagram gives quite accurately the velocity of c for intermediate positions. Fig. 77 shows a method of constructing a velocity diagram upon Fig. 77 a curved path as a base. The driving arm, or crank, a, imparts, by its rotation, a reciprocating motion to the arm c in the arc 1-9. The point Ob c occupies the positions 1, 2, 3, etc., when the point Oab, is at the corresponding points 1' ', 2', 3', etc. If 2'-2/ is laid off equal to the linear velocity of O a & upon the extension of 72 KINEMATICS OF MACHINERY. the line of a, and 2/-2J is drawn parallel to 6, the segment of the extension of c cut off by this parallel equals the linear velocity of the point O bc . This is proven by reference to the instant centre of b and d, O bd ; for the linear velocity of O ab relative to d is to the velocity of O bc as O bd -0 a b is to O bd -0b c (the linear velocities of two points in 6 relative to d are proportional to their radii from O bd ). But 2'-2/ (the velocity of O ab ) is to 2-2 t as O bd -0 ab is to O bd -0 bc , and therefore 2-2 t is the velocity of O bc . By a similar construction for other phases, the corresponding velocities of the point O bc may be obtained. If these velocities of the driven point are laid off as radial ordinates at the corresponding points in its path, the curve 1^-2^-3^ etc., may be drawn, and it is the velocity diagram of O bc on its path as a base. This is called a Radial Velocity Diagram. If the motion of the driving point, O ab , is uniform, its velocity diagram is a circle concentric with its path, as drawn in Fig. 77, but the method applies equally well if the driving point has a variable velocity. A velocity diagram with rectangular co-ordi- nates may be constructed from the one just determined by rectify- ing the path of O bc , 1-2, etc., and erecting, at the various points, parallel ordinates of lengths found as above. This derived velocity diagram is shown in Fig. 77a, but it is seldom necessary to con- struct it. If on various positions of the crank (Fig. 76) the corresponding velocities of the follower, c, are laid off radially from Q, as Q-l", Q-2", etc., and a curve is then drawn through I", 2", 3", etc., a Polar Velocity Diagram of the motion of c is obtained. This is sometimes preferred to the rectangular diagram on the path of the follower. It is often desirable to show the relation between velocity and time. For this purpose a diagram may be constructed (Fig. 76a) in which ordinates represent velocity and abscissas represent time. This is called a Velocity Time Diagram. In the illustrations of Arts. 39, 40, and 41, linkwork mech- anisms have been taken, as the methods developed in these arti- cles are especially useful in the treatment of this class; but the deductions are also applicable to other mechanisms. TRANSMITTING MOTION IN MACHINES. 73 42. Acceleration Diagrams. In any velocity-space diagram the subnormal to the curve at any point is proportional to the corresponding acceleration. When different scales are used for velocity (ordinates) and for space (abscissas), as is usually the case, still another scale must be used for acceleration. Let OPQ, Fig. 78, be any velocity-space curve in which I" of ordinate represents n times as many velocity units (ft. per sec.) as 1" of abscissa represents space units (ft.). Let PM, PT, and PN be the respective ordinate, tangent and Fig. 78a normal to the curve at any point, P, and let 6 be the angle between the tangent, PT, and the base line, TN. ds v = =nPM = velocity represented by the ordinate PM, when the length PM is measured by the space scale; dv p=-r = corresponding acceleration; at PM MN dv = = ~PM 7ivdt' Hence the subnormal, MN, is proportional to the acceleration and may be used as an ordinate, at M, of an Acceleration Space Diagram (see also Fig. 76). When so used 1" of ordinate repre- sents n 2 times as many acceleration units (ft. per sec. 2 ) as 1" of abscissa represents space units (ft.). Since 1" of velocity 74 KINEMATICS OF MACHINERY. ordinate represents n times as many velocity units as 1" of abscissa represents sp^ace units, 1" of accleration ordinate repre- n 2 seats =n times as many acceleration units as V of velocity n ordinate represents velocity units. The engine mechanism of Fig. 76 is drawn to a scale of J size.* The ordinates of the velocity curve represent the velocity of the cross-head to a scale on which the length of the crank on the drawing measures the linear velocity of the crank-pin. Taking this velocity as 9 ft. per sec. and the actual length of the crank as 9" (represented on the drawing by 9"Xi = lJ"); the velocity scale is 1 J" =9 ft. per sec. or V =8 ft. per sec. The space scale is 1"=8", or 1"=J ft. .-. w=8-f-=12. The acceleration scale is l"=n 2 Xf =144X J=96 ft. per sec. 2 In any velocity-time diagram the slope of the tangent to the curve at any point is proportional to the acceleration. Fig. 78a shows a method of constructing an Acceleration-Time Diagram. PM and QH are two ordinates of any velocity-time curve OPQ, at any convenient distance apart, TK is tangent to OPQ at P, and makes an angle, 6, with the base line, TH . QH is extended to cut TK at K, and PG is drawn parallel to TH. From M take ML=KG as an ordinate of the acceleration curve, and determine other ordinates in the same way, the distance between the two ordinates used being equal to PG in each case. In Fig. 78a, p = - = tan = -. Therefore KG measured in at r(j velocity units is the acceleration in corresponding acceleration units during an amount of time represented by the length of PG. When PG = sec., p =mKG. On the corresponding acceleration m scale I" represents m times as many acceleration units (ft. per sec. 2 ) as 1" on the velocity scale represents velocity units (ft. per sec.). Using the same data as in the preceding example, i.e. : velocity crank-pin = 9 ft. per sec., and length of crank = 9" = f foot, the crank rotates = 1.91 times per sec. The time of 1 rev. t X2;r * The figure is reduced to about size in reproduction. TRANSMITTING MOTION IN MACHINES. 75 is - =.52 sec. This time is divided into eighteen equal parts i .y i in constructing the diagram in Figs. 76 and 76a, and two of these parts are used as the distance between ordinates in constructing the acceleration-time diagram. These two parts represent 52x2 1 sec. = .058 sec. m = - =17.3. The acceleration scale is 18 I" =8X 17.3 =138 fi. per sec. 2 \a the preceding discussion the foot-second system of units was used throughout. Any other system of units may be used in a similar manner and the corresponding scales determined by the same methods. It may be noted that the acceleration is indeterminate, graphic- ally, on the velocity-space diagram, where the curve crosses the axis of Jf. It can be found for several ordinates near that point and extended to the end position without much error. On the time- velocity diagram, however, it is wholly determinate. Both methods are open to the objection that considerable error is necessarily in- troduced in drawing tangents to curves which are not very well defined themselves. If the weight of the moving body is known the force required to accelerate or retard it at any position can be found from the acceleration curve. If F~be this force, W the weight of the body, and p the acceleration, F = - p. The acceleration can be read off from the acceleration scale at any point and the force corresponding W may be found simply by multiplying the acceleration by . Or y a force scale may be constructed, as can readily be seen. 43. Centrodes and Axodes, The instant centre for two bodies having plane motion may also be a permanent centre, in which case it remains a fixed point in both bodies ; but in the general case the instant centre does not occupy the same position in either body for any two successive relative positions of these bodies, and the locus of the instant centre upon each of the bodies is called a Cen- trode. The instant centre is a point common to the two bodies for the instant, and therefore the two coincident points of the bodies which lie at this centre have for the instant no relative motion ; but 76 KINEMATICS OF MACHINERY. any other two points (one in each of these bodies) do move rela- tively. The pair of centrodes traced on the bodies by the motions of the instant centre are tangent to each other at the instant centre; and as these contact points of the centrodes have no relative motion, the pair of centrodes roll on each other with a pure rolling action. Points in the pair of centrodes which previously coincided in the instant centre are now in common with other points of the two bodies rotating relatively about the present instant centre; and a similar remark applies to a pair of such points which may coincide in the instant centre at any succeeding phase. Any plane motion between two bodies, whatever the mechan- ism adopted for producing this motion, is exactly equivalent to that resulting from the rolling upon each other of two members whose contact lines conform to the centrodes for this motion. The nature of this action may be made clearer by reference to Fig. 79. Let a and b be two points in the body -4, moving rela- Fig. 79 tive to the fixed body M so as to occupy in succession the positions a-b, a'-b', a"-b", etc. From the middle of a-a' and b-b' erect per- pendiculars to these lines, intersecting in ; then the motion from a-b to a'-b' is equivalent to a rotation about the point as a centre through the angle aOa' = bOb' = (p. The motion a'-b' to a"-b" is likewise equivalent to a rotation about 0' through the angle 0', etc. 0, 0', etc., are temporary or shifting centres, and they may be connected by the polygon 0-0'- 0", etc., which lies in the station- ary body M. If the line O-O/ be laid off on the moving body A of TRANSMITTING MOTION IN MACHINES. 77 a length equal to 0-0' and making the angle with the latter, it is evident that when a-b moves to a'-b' ', 0-0 '/ will fall along 0-0' and 0' will coincide with 0'. From 0' lay off an angle with O'-O" equal to 0'; extend 0-0' to # through this last angle; and let the angle gO'O" = /?'. This extension divides 0' into /?' and a' v and a' = 0' P + 0/P ; . . (2) .'. OP + 0*P = OP + O'P = O'P + 0/P = OiP + OS P. . (3) From either the first and second, or the third and fourth mem- bers of (3) we get: 0>P = O'P, (4) from which it is seen that the arcs PB' and PA are equal. In a similar way it can be shown that OiM' = 0'N\ 9 and that the arcs AM' and B'N' are equal; therefore the arc PM' = AP - AM' is equal to the arc PN' = B'P - B'N'. This demonstration is general and will apply to any pair of points which can meet as contact points. If the points P and M lie on opposite sides of AB, and P and N lie on opposite sides of A'B', the values of PM and PN become PB + BM, and PA' + A'Nj respectively, but the equality of the arcs is maintained. The driving will be positive in the direction indicated, until the phase shown in Fig. 83 is reached, when the normal passes through both fixed centres, and the driver might continue to rotate without imparting further motion to the follower. To secure con- 82 KINEMATICS OF MACHINERY. tinuous driving for the half revolution succeeding this phase it must be provided for otherwise than by the simple contact of the two ellipses. It has been shown that the free foci O l and O/, are always at a distance apart equal to the major axis, A-B, and these foci could therefore be connected by a link. This system of link- work alone would transmit motion exactly equivalent to that of Fig. 83 the rolling ellipses; but in an actual mechanism the two pieces would have to be at the ends of the shafts between which motion is to be transmitted, or the link would interfere with the shafts.* Another obstacle to such a link connection, as a substitute for the rolling ellipses, is that at the phase shown in Fig. 83 (and at 180 from this position) the linkwork would reach a " dead-centre " position, when it would not be effective in transmitting motion. Teeth may be placed at the ends of the elliptical members (as indicated in Fig. 83), which would engage near the dead-centre phases, and thus carry the follower past this critical position. If such teeth were placed around the entire halves of the ellipses which are in contact after the direct-contact driving ceases to be operative, the link could be omitted, and the necessity of placing the ellipses at the ends of the shafts thus avoided. Where the action is to continue through half a revolution, or more, such teeth are usually placed entirely around the peripheries of the ellipses, and the result is a pair of elliptical gears such as is shown in Fig. 84. The method of forming such teeth, to secure the exact equivalent of the rolling ellipses, will be discussed in a later chapter. With the transmission through such elliptical members * It will be noted that the pair of rolling ellipses correspond to the centrodes of such a system of links as that just suggested. PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 83 as have just been discussed, the angular velocity ratio is inversely as the contact radii at any phase. If the driver has a uniform Fig. 84 A angular velocity, the angular velocity of the follower is a maximum in the phase shown in Fig. 83, when - = -^p = T^- When the driver has made a half revolution from this position, the angular velocity of the follower is a minimum, and l = 7 ^. . These G? a OA 84: KINEMATICS OF MACHINERY. extreme ratios are reciprocals of each other. Of course the driver and follower both complete the half rotations from these two positions (where the contact radii coincide with the major axes) in equal times. If it is required to connect two shafts by rolling ellipses either the maximum or the minimum angular velocity of the follower may be taken at will, but one of these being deter- mined the other is fixed the driver being supposed to have a constant angular velocity. Suppose it is required to construct a pair of rolling ellipses such that the maximum value of - = --. Divide the distance between oo, I centres 00' (Fig. 83) into such segments that OP : O'P :: 2 : 1. Lay off" PA and PB' each equal to 00' '; then lay off PO, and JB'O/ equal to PO'. PA and PB' are the major axes of the re- quired ellipses, whose foci are and 0, , and 0' and O/, respec- tively; from these data the curves can be constructed. Sectors of ellipses can be used for transmitting a reciprocating motion from the driver to the follower. In this case the angle through which one of the members turns, and both the maximum and minimum angular velocity ratios, can be assumed; but the angle through which the other member rotates is not then subject to control, for the two sectors are necessarily alike. Thus (Fig. 85) Fig. 86 the centres are at and 0', and it is required to construct a pair of elliptical sectors such that an angular motion, <*, of the driver will transmit motion to the follower pivoted at 0', and GO I -f- G? a is to have for extreme values O'P -4- OP, and O'P' ~ OP'. PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 85 Draw a Hue from making the angle a with OP, and on this line lay off OM = OP'. P and M are then points in the ellipse which rotates about one of its foci at 0. The distance from /' to the free focus of this ellipse, O l , equals the major axis minus OP', or O t P = 00' OP O'P. With this length as a radius and P as a centre draw an arc, ee. Also, the distance from O t to M, or O^M, = 00' OM= O'P'. With this length as a radius, and a centre at M, draw an arc ff. The intersection of the two arcs ee and ff is 0,. The foci being located and the major axis known, the ellipse can be drawn. The elliptical arc, PN 9 of the follower is equal to that of the driver. The constructions just outlined apply either for actual rolling elliptical members, or for finding the " pitch curves " for toothed gears, or segmental gears. Elliptical gears have been applied in many cases where a " quick- return " action is required, as to shaping-machines, in order to give a quick return motion to the tool with a slower stroke during the cutting. They have also been used to actuate the slide-valve in a steam-stamp used for crushing rock, where it is desirable to admit the steam above the piston throughout nearly the entire downward stroke in order to cause a more effective blow; while on the upward stroke economy demands that only sufficient steam be used to return the stamp-shaft. 47. Rolling Logarithmic Spirals. One of the properties of the logarithmic spiral is that the tangent to the curve makes a con- stant angle with the radius vector at all points. Owing to this property, the curve is also called the equiangular spiral. The polar equation of this curve is = log b r, in which b is the base of the system of logarithms. The angle made with the tangent by the radii vectores is different for different values of 5, but it is constant for any one system of logarithms.* * See Fig. 86, 9 = log^r. Let m = modulus of the system of logarithms, dr , rdB rmdr ,'. dQ = m ; but tan d> = -7 = 5- dr = m = the modulus of the sys- r dr r iem of logarithms; .*. = tan -1 m = a constant. 86 KINEMATICS OF MACHINERY. If two similar logarithmic spirals are placed tangent to each other, as in Fig. 87 or 88, the tangents to the two coincident con- tact points lie in the same line; and as the angles made by these tangents with their radii vectores are equal, these radii lie in a straight line. This holds for all tangent positions of the curves; hence if the curves turn about fixed centres at their foci, the con- tact point always lies in the line of centres, thus meeting the re- quirement for pure rolling. The sum of the contact radii if the foci are on opposite sides of the contact point, and their difference if the foci are on the same side of this point, is a constant and is equal to the distance be- tween the fixed centres. Thus, in Fig. 87, OP + O'P = 00'; o Fig. 88 and, if r and s are two points which may become coincident con- tact points, Or + O's = 00'. Also, in Fig. 88, O'P -OP =00'; and, if r and s are two points which may become coincident con- tact points, O's -Or= 00'. In Fig. 87: OP + O'P = Or + O's, .'. Or - OP = O'P - O's. . (1) In Fig. 88: O'P - OP = O's ~r Or, .'. Or - OP = O's - O'P. . (2) Equations (1) and (2) show that in either external or internal contact the difference between two contact radii of one of the PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 87 spirals equals the difference between the corresponding contact radii of the other spiral. It can be shown that any two arcs , of similar logarithmic spirals are equal in length when the difference of the radii to the extremities of these arcs is the same. Hence in Figs. 87 and 88, Pr = Ps, as it should for pure rolling.* A single pair of logarithmic spirals cannot transmit motion continuously in one direction, but they may be used for a reciprocat- ing transmission with pure rolling. The angular motion of the driver and both extreme angular velocity ratios may be assumed, in which case the angle through which the follower moves can not be con- trolled. Thus, in Fig. 87, the driver may rotate about through the angle POr = a, and the angular velocity ratio varies from O'P -5- OP to O's -~ Or. These conditions determine the points P and r in the spiral which has its focus at 0. The focus 0' ', the point P, and the length of a second radius vector, O's =00' Or, are also fixed for the second spiral; but as this must be similar to the first spiral, the angle PO's cannot be assigned in advance. It is possible to fix the angles of motion of both driver and follower, but with these conditions only one angular velocity ratio can be taken arbitrarily. 48. General Case of Rolling Curves. A general method will now be given for constructing a pair of curves which will roll upon each other in turning about two fixed centres. By this method the angular velocity ratios at the beginning and end of any angular motion of one member may be assigned ; but the cor- responding angular motion of the other member cannot be pre- determined. Or, if one of the curves is prescribed, a curve can be found which will roll upon it. The method gives only approxi- * See Fig. 86. B = Iog 6 r; m = modulus, (ds)* = (rdQ)' -f (dr}*\ but .-. ds = V(m* -f l)dr, .'. 8 = v(m* + 1) dr = v(m + l)(r, - r,); hence /r, the length of tlie arc * included between two radii vectores of the same differ- ence in length is constant. 88 KINEMATICS OF MACHINERY. mate results inasmuch as it does not absolutely insure theoretically perfect rolling between the points located; but the approximation can be carried to any required limit by locating a sufficient number of points. Suppose the distance between the fixed centres, and 0', Fig. 89, to be given, and that it is required to construct a pair of rolling curves such that the angular ve- locity ratio of B to A shall be OP-f- O'P, OP^O'P,, OP t -t-0'P t , OP 3 -^0'P 3 , etc., when the lines PO, m v O, m 2 0, w,0, etc. respectively, lie in the line of cen- tres; these last lines being drawn to cor- respond with required angular motions of A. The first pair of radii are OP for A, and O'P for B. With as a centre and OP, as a radius, describe the arc P l m l , cutting the line m v O, then draw With 0' as a centre and O'P, as a radius, now take a radius equal to Pm 1; with P as a centre, and cut the arc P^ at n x ; and connect this point n x with P t . It is evident that m t and n can meet in the line of centres when A has turned through the angle mf)P and B has turned through the angle n^O'P^. Next draw an arc through P 2 , from centre 0, cutting the line m 2 in ra 2 , and connect m t and m 2 . Also draw the arc P 2 n 2 with 0' as a centre andO'P 2 as a radius; now with a radius equal to m l m 2 , and with n 1 as a centre, cut this last arc at w 2 ; then draw the line nji^. Proceed in a similar way with the points P 3 , P 4 , etc., locating the points m s , w 4 , etc., of A ; and w 3 , n t , etc., of B. It will be seen that the polygons P-m l -m t . . . m t , and P-n r n^ . . . n t may act together with a rough rolling action, and that two curves can be passed through P-w^-ra, . . . m 6 , and P~n^-n 2 . . . n 6 , the action of which will closely approx- imate pure rolling if the points located are sufficiently close together; that is, if the arcs approximate the chords. Evidently, a line from P to draw the arc P PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 89 if the outline of A had been given, the curve of B could have been derived by laying off the lengths Om lf Om 2 , etc., from upon 00', and then proceeding as before in the location of the points of the outline B. Fig. 90 shows the derivation of a curve B to roll upon the straight line which rotates about as a centre aud constitutes the acting line of A. The construction will be obvious from the pre- ceding explanation in connection with Fig. 89. This method cannot usually be applied where complete rotation of both of the members is required; for, as appears from the con- structions given, the angular motion of the follower for a given Fig. 90 motion of the driver cannot be controlled ; hence it is not certain, in the general case, that a complete rotation of one member will correspond to a complete rotation of the other. But with con- tinuous action in one direction, when one member has turned through 360 the other must have turned through an angle of 360, or else some exact multiple or exact divisor of 360. This require- ment does not apply to rolling circles, but it holds for all other pairs of rolling curves. 49. Lobed Wheels. It has been seen that a pair of equal ellipses can rotate continuously with rolling contact, and that the angular velocity ratio passes through one maximum and one minimum value for each revolution. It is sometimes desirable to have several maxima and minima values of this ratio to a single revolution, and 90 KINEMATICS OF MACHINERY. a class of rolling mechanisms called Lobed Wheels may then be used. Fig. 91 shows a pair of these wheels, each having three lobes. The outlines are all logarithmic spirals. If it be desired to have an unequal number of lobes on the two Fig. 91 Fig. 92 wheels these spirals cannot be used; but curves which are derived from ellipses permit this condition. Fig. 92 shows a set of three such wheels in series which roll perfectly; there is a one-lobed wheel acting on a two-lobed wheel,, and this latter rolls with a three-lobed wheel. These figures are drawn from MacCord's Kinematics, to which the reader is referred for a full treatment of Lobed Wheels. In all of these wheels, as in the rolling ellipses, there are periods during which the driving is not positive; but these outlines can be used as the pitch curves for toothed wheels, and teeth can be formed upon these curves which will transmit a positive motion, exactly equivalent to that of the pure rolling of such curves. Ia these derived toothed wheels there is sliding between the teeth themselves, but no sliding (if the teeth are properly formed) be- tween the pitch lines. 50. Rolling Surfaces. In the preceding pages plane curves which roll upon each other while rotating about fixed centres have been considered. It was shown in Art. 10 that the plane motion of any body can be represented completely by the motion of a plane figure ; thus these plane rolling curves may represent corre- sponding bodies which rotate about axes through the fixed centres- PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 91 and perpendicular to the plane of motion. When two or more such bodies can be represented by figures lying in the same plane, it is evident that the axes of all of these bodies must be parallel. The actual contact surfaces of such bodies are generated by a line which travels along the curved outline, always remaining parallel to the axes; hence these surfaces are cylindrical. The actual bodies corresponding to Figs. 64 and 65 are figures of revolution or right cylinders (see Fig. 93); while the bodies corresponding to Figs. 82 to 90 are cylinders only in the general sense. Certain other forms may roll together in rotating about fixed axes which are not parallel, when the motion of each member about its axis is still plane, but the planes of motion of the different members do not coincide. If the two axes intersect, tangent cones, or frusta (as in Fig. 95), having a common contact element and a common apex at the intersection of the axes, may act together with pure rolling. These cones are not necessarily right cones, but the use of cones of other than circular transverse sections is so rare that only right cones will be treated in this work. If the two axes are not in one plane (i.e., if they are neither parallel nor intersecting) they may still be connected by two mem- bers which will roll upon each other, with contact along a common rectilinear element. Fig. 101 shows the general form of a pair of such members; they are called Hyperboloids of Revolution. The general method of generating these latter figures and the nature of the action will form the subject of a later article, in which it will be shown that there is, in a sense, a certain departure from pure rolling in the action; however, this does not prohibit the use of these forms as pitch surfaces for toothed gears, owing to the pecul- iar character of the sliding component. 51. Rolling Cylinders. In rolling right cylinders the angular velocities are inversely as the radii; or - Let d be the dis- GO, r g tance between the fixed axes. In external contact, r 1 -j- r * = d ; and in internal contact r l r a = d, (in this expression r l is taken as the radius of the larger cylinder, inside of which the smaller one 92 KINEMATICS OF MACHINERY. rolls). It is frequently required to find the diameters or radii of tangent cylinders which will connect two shafts and transmit motion (when there is no slipping) with a given angular velocity ratio. This ratio is the same as the ratio of the revolutions made in a given time by the two cylinders, and in practical problems it is usually stated in these terms. Thus, one shaft is to make n t revo- lutions imparting n t revolutions to the other shaft, per unit of fi>7 Al A* time; then - = *=-?. In many cases the required radii, r, and r a , can be found by inspection, or by mental calculation; but it may be convenient to use the following expressions if n l and w, are high numbers with no common divisor. For Cylinders in External Contact : r, + r 2 = a, .'. TI = d r 2 , and r, = d n. GO. r t GO GO r , .*. r. = r. - = (d r.) *; <, r,' '&?, * V Similarly : r, = r, ^ = (d - r,) -^ ; For Cylinders in Internal Contact : r l r a = d (r, being the radius of the large cylinder). .'. r, = d -f r,, and r, = r l d. PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 93 Similarly : or or r. d. . (4) (< \ , G?, ^,7 W, 1 M = d *; ..r.= ! d',r 2 = l - 09j <,' ' (,- <, n,- n, The directions of the rotations of the two members are oppo- site when they are in external contact, and the same when one is tangent to the concave surface of the other, as previously pointed out. 52. Rolling Right Cones. Two right cylinders, combined with two right cones, are shown in Fig. 94. Each cylinder has one base in common with that of one of the cones, hence the axis of this cylinder and cone must coincide. The bases of the two cones (and of the corresponding cylinders) need not be equal, but the Fig. 93 Fig. 94 Fig. 95 slant height of both cones is the same. The bases of the two cones have a common tangent, in their plane (perpendicular to the paper), passing through M. Now imagine the two axes, A A and BB, to rotate in their common plane, about this tangent to the bases through M as an axis (or hinge), till the apex a meets the apex b at Q, as in Fig. 95; when the two cones become tangent along the element QM. It will be seen that the two base circles 94 KINEMATICS OF MACHINERY. still have a common tangent through M and they can roll upon each other in the new position, the two contact points having equal velocities along their common tangent, as in the original position. Any other corresponding transverse sections of the cones, equidis- tant from Q along the elements, as m-m' and m-m" will also roll together; or the two cones roll upon each other in a similar manner to the original rolling of the cylinders. If it is required to connect two given intersecting shafts by rolling cones, so that their rotations per unit of time shall be in the ratio of n l to n %9 it is only necessary to construct two tangent right cones with these shafts for axes, and with a common contact element lying in such a position between the axes that any pair of transverse sections which roll together shall have radii in LL_B^ the inverse ratio of the required angular Fig. 96 motions. If A- A' and B-B', Fig. 96, are the given axes, the position of the contact element may be found by lay- ing off from Q, on these shafts, the distances Qa and Qb, directly propor- tional to the required numbers of rotations of these shafts ; thus Qa : Qb :: n l : n v On Qa and Qb form a parallelogram, and the diagonal of this parallelogram, Qc, or its extension, is the required common contact element. This can be proved as follows : from c drop perpendiculars ce and cf upon the axes A- A' and B-B'; the angle cbf= eac = a (sides parallel) ; ce = ca sin at, cf = cb sin a. .'. ce : cf :: ca : cb :: MM' : MM", hence the cones with MM' and MM" as the diameters of the bases, will roll together with the required angular velocity. The frusta used for this transmission may be taken from any part of the two cones, giving bases greater or less than those indicated, if more convenient. The parallelogram might have been drawn in any of the four PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 95 angles made by the intersection of A- A' and B-B' \ thus if the angle B'QA had been selected, the diagonal Qc' would have been located for the contact element, and two such frusta as those shown with M^M' and M^M" as bases would give the required angular velocity ratio. Either of the other two angles, A'QB' or A'QB, might have been taken if desired. It will be noticed that the cones first found are not similar to those obtained in the second construction; but the pairs constructed in both of the acute angles are similar, as are the pairs in both of the obtuse angles. If the driving-shaft A- A' rotates as indicated by the arrows, it will be seen that the first construction (in the acute angle) imparts rotation to B-B' in one direction ; while the second construction (in the obtuse angle) causes B-B' to rotate in the opposite direc- tion. The choice of angle for the location of the contact element is governed by the required directions of the rotations, and the locations of the actual shafts. It is evident that one of the ma- terial shafts, but not both of them, can pass through Q. Fig. 97 shows a shaft A-A', from which four shafts (making equal angles with A-A') are driven. One of the followers on either side of A-A' is rotated in one direction ; while the other followers (one on each side of the driver) rotate in the opposite direction. It may happen, as in Fig. 98, that one wheel cuts through the 96 KINEMATICS OF MACHINERY. axis of the other wheel. The shaft can then be led off only ii: the direction indicated by the full lines ; for if it were to be carried through Q, in the direction of the dotted lines, the shaft and wheel would interfere. This condition can only occur when the contact radius is located in the obtuse angles. The acute-angle con- struction is to be preferred as avoiding this difficulty in all cases, and also because it gives smaller wheels ; but there are conditions as to location of shafts and required directional relation of rotation which may make the other construction desir- able or necessary. The conditions of the problem may be such that the contact element is perpendicular to one axis, when the cone on this axis is of the special form (a flat disk) shown in Fig. 99. With somewhat different conditions, one of the rolling surfaces may be the concave surface of a cone, as shown in Fig. 100. Fig. 98 a Q Fig. 99 Fig. 100 In a great majority of the cases requiring the construction of rolling cones on intersecting axes, these axes are at right angles to each other. With this condition the pairs of cones formed in any of the four angles (for a given angular velocity ratio) have similar inclinations. The location of the contact radius in one of these angles, and the selection of the particular angle in which it lies, are determined by the general relations previously treated in this article. PURE ROLLING IN DIRECT-CONTACT MECHANISMS. 97 53. Rolling Hyperboloids. If one right line revolves about another right line not in the same plane, and all points in these lines remain at constant distances apart, the revolving line gen- erates a surface called the hyperboloid of revolution. This is a warped surface, the elements of which are straight lines corre- sponding to the successive positions of the generating line. A meridian plane through this figure cuts the surface in an hyper- bola, and it is evident that this hyperbola would generate a sur- B Fig. 101 Fig. 101a face, in revolving about the axis, identical with that generated by the straight line; hence the name given to these figures. Fig. 101 represents a pair of these hyperboloids of revolution tangent to each other along a common element mm. If the axes are fixed in the positions corresponding to this tangency, it is evident that the two surfaces will remain tangent as the two figures rotate about their axes; for each is symmetrical about its axis Hyperboloids of revolution can be placed tangent along an element only when the radii of the ft gorge circles " are propor- tional to the tangents of the angles between the contact element 98 KINEMATICS OF MACHINERY. and the respective axes; i.e., when P^ :P i B l :: tan a : tan/J. This is shown in Fig. 10 la, where A A and BB are the two axes and mm is the common element. A l B l is perpendicular to both axes, and P l A l and PB are the respective radii of the gorge circles. These radii are normal to the hyperboloids and intersect the common element, mm, which is therefore perpendicular to A 1 B 1 at Pi, and parallel to a plane through A A perpendicular to AJ$i. B'B' and m'm' are the projections of BB and mm on this plane , and a and /? are equal to the angles between mm and the respective axes. The lines cd and ce, perpendicular to mm, and intersecting A A and BB aid and e, respectively, are normals to the hyperboloids at c, on the line of tangency. Therefore they lie in one right line, de, the projection of which on the plane of AA and B'B' is de', perpendicular to m'm f at c'. P^c" is the pro- jection of P^c on the plane of BB and A^B^ It is evident that tan a c'd cd cc f P i A l tan/? = cV = ce = c"e = PA' All points in the hyperboloid which rotates about A A, Fig. 101, must move in planes perpendicular to AA; likewise, all points in the other hyperboloid move in planes perpendicular to BB, and as the two axes are not parallel, two contact points can not have identical motions. Thus if V a is the velocity of a contact point in the former figure, Vb is the simultaneous velocity of the corresponding point in the latter figure when they roll together. These two velocities must have equal components per- pendicular to the contact element, but their components along this common line will not coincide. This is the characteristic of the action of these bodies referred to in Art. 50, and, as stated there, it does not affect the angular velocity ratio of the two members, for this relative sliding along the common element can not transmit motion, nor can it affect the component of V a and Vb perpendicular to the common element. The angular velocities of the hyperboloids. (Fig. 101) when they roll together are, repectively, cu 1 =V a -^P i A l and a> 2 = Vb + FRICTIONAL GEARING. 99 P^j. V a = V + cosa, and F 6 = F-^cos /?. It has been shown that P^A l : P l B l :: tan a : tan /?. Therefore ait V a P l B l _ V tan cos /? _ sin 3 a> 2 P i _A l Vb V tan a cos a sin a To construct a pair of rolling hyperboloids to transmit motion between two shafts with a given angular velocity ratio : project these shafts on a plane parallel to both of them, Fig. 101: lay off Pa and Pb on A A and BB proportional to the required revolutions; construct the parallelogram P-a-c-b, and draw PC; this locates the projection of the contact element. At any point c on PC erect a perpendicular, cutting AA and BB in d and e respectively. Divide the perpendicular distance (A^B^ between A A and BB, at P lt in the ratio of the segments cd and ce; thenP 1 A l and P^ will be the radii of the gorge circles of the required hyperboloids. 54. Frictional Gearing. It has been shown that two axes, whether parallel, intersecting, or neither parallel nor intersecting, may be provided with contact members the surfaces of which will roll upon each other. In many mechanisms it is necessary to maintain, exactly, a prescribed relation between the motions of the members throughout the entire cycle of operations. In other instances this is not essential, a reasonable departure from the precise relative motions contemplated being permissible. Thus in cutting a screw-thread in a lathe, it is essential that the relation between the rotation of the spindle and the translation of the tool shall be strictly constant, and the positive mechanism (gears and the lead screw) insure this uniformity of action. But in plane turning the feed may vary somewhat without serious results, and the belt-driven rod-feed, depending upon friction, is often used, thus saving unnecessary wear of the screw. It sometimes happens, as in machinery subject to severe shock, that a positive transmis- sion is not desired; and in many cases this is not an absolute necessity. When a limited variation of the motion transmitted may be permitted, and the two shafts to be connected are at a considerable distance apart, belting or rope transmission is most often employed. Occasionally, because the distance between the shafts is too small to employ these methods of transmission 100 KINEMATICS OF MACHINERY. advantageously, or for other reasons, the substitution of contact members rolling upon each other is convenient. In all such trans- missions having circular transverse sections the action is purely frictional throughout the revolution, and these mechanisms are classed as Frictional Gearing. If the sections are non-circular the action may still be pure rolling, as shown in the preceding chapter; but the driving can- not be positive during the entire rotation; for a critical phase is reached at which the action is only frictional, and beyond this phase driving does not occur, even by friction, unless other ex- pedients (as teeth) are introduced (see Fig. 83). It is evident, then, that frictional gears must have circular transverse sections in order to transmit continuous rotation. The force that can be transmitted through frictional gearing depends upon the physical character of the surfaces in contact and on the normal pressure between the two surfaces. Some slipping or "creeping" almost inevitably occurs; its magnitude depending upon the character of the surfaces, the normal pressure between them and the resistance to be overcome. In certain applications this liability to slip is desirable rather than otherwise. For example, in hoisting, where it is not essen- tial that the load raised shall move through precisely the same distance for each increment of motion of the driver. If any ob- struction to motion of the load be met, the slip prevents the sud- den strain (shock), that would be thrown upon the entire train of mechanism if this elasticity (using the word in a somewhat popular sense) were absent. If a car, or " skip," in being hoisted from a mine leaves the track, meets an obstruction, or is overwound, the yielding through the slipping of friction gears (or of belts) lessens the danger of breakage over that encountered with a positive connec- tion. Furthermore, these friction mechanisms are much simpler in design and construction, and quieter in running than toothed gears; and, owing to such considerations, the employment of fric- tional gears, or "frictions," as they are frequently called for brev- ity, is not uncommon, under proper circumstances. FRICTION AL GEAKlffC.' " '' Frictional gearing is important in itself, and the study of it also affords a good basis for investigation of toothed gearing. Kinematically, any of the figures of revolution which will roll together, as pairs of right cylinders, right cones, or hyperboloids of revolution, might be used as friction gears; but, practically, rolling cylinders (Fig. 93), and the disk and plate (" brush-wheel") (Fig. 102), are by far the most common as the basis of such gearing. Rolling cones are also used, but less frequently. Two cylinders (Figs. 64, 65, and 93) may be used to transmit motion and energy, up to the limits fixed by the friction at the contact element. Supposing no slip to occur, any two contact points have the same linear velocity, and the angular velocities of the two members, A and B, are inversely as their radii. If it is required to impart to a shaft a given number of revolu- tions per unit of time, from a shaft of given rotative speed, the distance between centres being also determined; the required radii can be found by the expressions of Art. 51. For example, d = 48", n l = 210 rev. per min.; n y = 270 rev. per min. The solution of the kinematic part of this problem is extremely simple. 55, Grooved Frictions. The consideration of the force that can be transmitted by friction-gears involves the normal pressure and the coefficient of friction between the contact surfaces. This consideration often modifies the forms of the members, without altering the kine- matic action; and in many cases it may be advan- tageous to use certain derived forms, known as grooved frictions or " V " frictions, in pluce of the fundamental rolling cylinders. Fig. 103 shows a pair of these derived forms in contact. It will be seen that the original, or ideal, rolling cylinders are replaced by rolls with circumferen- tial grooves, the sections of which (in planes pass- " Fig. 103 ing through the axis) are triangular, or more usually, trapezoidal. W2 KHJ-EM-AT1CS OF MACHINERY. The actual contact surfaces are frusta of cones of equal slant and on parallel axes. In order to discuss the action of these grooved rolls, and to understand clearly their advantage over the simple rolling cylin- ders, it will be necessary to treat briefly the action of the forces in- volved in frictional transmission. If two bodies are in contact, with a force F acting in the direc- tion of their common normal, there is a resistance to the sliding of one body upon the other, and this resistance, called friction, is what makes frictional transmission possible. The resistance is a function of this normal pressure and of the physical character of the sur- faces. If the surfaces are very smooth, the resistance under any normal pressure becomes comparatively small. If rough, the pro- jecting particles of one member interlock with those of the other and the friction increases. As absolutely perfect surfaces are not attainable, absolute freedom from friction (absence of this resist- ance to sliding) is impossible ; and the greater the departure from ideal perfection of surface (smoothness), the greater is the friction between any given pair of bodies. The friction varies inversely as the smoothness, and this varies both with the nature of the materials in contact and with the degree of "finish." In every case the friction is greater than zero; and the ratio of this resist- ance,/, to the normal force, F, is called the coefficient of friction, /*. This coefficient can only be derived from experiment, directly or indirectly. Let the normal pressure between the surfaces of the two cylin- ders (Fig. 93) be represented by F. According to Newton's third law, action and reaction are equal and opposite; hence, the pressure of A towards B is met by an equal and opposite pressure of B towards A. These pressures can only be brought to bear upon the contact surfaces through the bearings of the wheels (neglecting weight), and the action and reaction at the bearings are equal ; therefore a pressure F must be exerted by the bearings upon the axle supported by them. In other words, the pressure between the bearings and journals equals the pressure between the contact sur- faces of the two wheels. As the bearings themselves, however per- FR1CT10NAL GEARING. 103 fectly formed and lubricated, are not frictionless, the normal force, F, necessary to transmit energy from A to 5, involves a frictional action at the bearings, resulting in a wasteful resistance to be overcome, and also, incidentally, in wear of these parts. It is therefore desirable to reduce the pressure at the bearings as much as possible; but the friction at the contact surfaces must be sufficient for driving, and the normal pressure at these surfaces is one of the elements which determine this friction. It is in order, then, to investigate the relation between the bearing pressure and the normal pressure at the contact surfaces, and to see if the former can be reduced without undue sacrifice of the latter. With simple cylindrical rolls (Fig. 93) the total bearing pressure for each wheel equals the normal pressure, F, at the con- tact element. In the case of "V" frictions, however, the normal pressure between the contact surfaces may be much greater than the bearing pressure. This can be shown in connection with Fig. 104, in which the wedge of A is inserted in the corresponding groove of B. The common normals to the contact faces of A and B through the centres of the faces are Pn and Pn' 9 and the normal forces between these faces may be taken as acting in the lines of these normals (such normal forces are really the resultants of systems of parallel forces, uniformly distributed over these faces). The force F, acting in the centre line of A and B as indicated, passes through P, and it can be re- solved into components along Pn and Pn' by the parallelogram of forces. These components are represented by F l and F^. The effect of the initial force, F, is equivalent to the combined effect of its components, and it may be replaced by them; therefore, the effect of F is equivalent to the normal actions, Fig. 104 F, sin d + *V sin 0' = F ; and F l cos 6 = F,' cos 6'. If 6 = 6' (the usual condition), Fj=F/, and the total normal action, F!+F/ = 2Fi=F -4- sin0. It is seen, from this last expression, that 104 KINEMATICS OF MACHINERY. the normal pressure increases, for a given value of F, as 6 becomes smaller. When 6 = 90, the total normal pressure equals F, as it should ; for in this case the groove and wedge have disappeared and the contact surfaces are flat and perpendicular to the line of F. For any value of less than 90, the total normal pressure is greater than F. The action between the grooved faces of the "V" frictions is exactly like that of this wedge. The normal pressures between the sides of the acting ridges and the grooves correspond to the normal pressures in the wedge, and the initial force F equals the radial force exerted between the bearings and journals of the wheels. It follows from this discussion that any necessary normal pressure at the acting contact surfaces of the "V" frictions can be maintained by a bearing pressure less than the normal pressure. Hence, the friction loss in the bearings is decreased by the substi- tution of the ' ' V " frictions for the fundamental rolling cylinders ; or, to state it somewhat differently, for a given pressure at the bear- ings, a greater resistance can be overcome at the rim by ' 'V " frictions than by cylindrical rolls. As there is a practical limit to the press- ure that can be safely carried at the bearings, and as excess of bearing pressure means waste through friction, the importance of the wedge-like action in frictional transmission is apparent. The angle between the sides of the grooves (2(9) is usually from 40 to 50. Assuming 40 as this angle, V = 20% and sin = 0.342. F Then 2 F l - = 2.93 F ; or the total resulting normal pressure ,o4/4 is nearly three times the force at the bearing, and the coefficient of friction and bearing pressure remaining the same, nearly three times as great a resistance can be overcome with these grooved rolls as with corresponding true cylindrical rolls. Grooved frictions are frequently so mounted that the distance between the shafts can be changed somewhat. This makes it possible to throw the wheels out of gear, so that the follower can be stopped without checking the driver. It also permits control- ling the bearing pressure, so that it need not be any greater than is required to prevent serious slipping at the driving surfaces. FRICTION AL GEARING. 105 This adjustment also affords a ready means of taking up the wear of the "V" or of the bearings, so that good contact (without which driving is impossible) is maintained. When in gear (close contact) there is, as usually constructed, a small clearance at the bottoms of the grooves, as indicated in Fig. 104. If this were not provided, the edges of the rings might "bottom; " that is, the contact might be entirely or mainly at the bottoms of the grooves, instead of at the inclined sides. Such a condition would defeat the object of the grooves, and to render it impossible, even after considerable wear at the sides, this clearance is provided. The depth of the grooves of either wheel is the difference between the radii to the tops and bottoms of the grooves or rings. This distance minus the clearance at the bottoms may be called the ivorlcing depth, and the faces of the " Vs " above the clearance may be called the working surfaces. The nominal radius, or pitch radius, of a grooved friction may be taken as the mean radius of the working surface, and the hypothetical cylinder corre- sponding to this radius will then be the pitch cylinder, or pitch surface. , The angular velocity of two V friction wheels, when in full con- tact and working properly, may be taken, for most practical pur- poses, as that corresponding to the rolling together of the pitch cylinders, or, inversely, as the pitch radii. The relative sliding or creeping of the wheels along the common tangent to the pitch sur- faces may usually be neglected in well-constructed frictions; for these wheels are only employed where some variations in the angu- lar velocity ratio is admissible. Assuming that no sliding of this character takes place that is, that the angular velocities of the two wheels are inversely as their pitch radii there is nevertheless some relative motion between the two surfaces when they are in contact, causing a grinding action. The nature of this action may be seen in connection with Fig. 105, in which and 0' are the fixed centres of A and B, p is the contact point at the pitch circles, and s and t are two coincident points in the working surfaces, one on each side of p. The linear velocity of p, pv is assumed to be the 106 KINEMATICS OF MACHINERY. same for the coincident points of both wheels which lie at p. It is evident that all points in either wheel which lie outside of its pitch circle have linear velocities greater than pv, and all points of either wheel lying inside of its pitch circle have linear velocities less than pv\ but those points of the working surface of one wheel which are inside of the pitch curve come in contact with points of the other wheel which are outside of its pitch circle; consequently, if the points at the pitch circles have equal linear velocities, all contact points not in these circles have different velocities, and there must be some relative motion or sliding between any pair of such points. Thus, in Fig. 105, the linear velocity of s, as a point in A, is sv' ; and, as a point in B, the velocity is sv" ; therefore the rate of sliding of these points equals sv" sv'. Simi- larly, the rate of sliding at t is ,s ^'' X A tvf tv". An expression for the ^X^X greatest sliding is derived below. Let ^.^'x R and r be the two pitch radii, N ~^^ ' N B and n be the numbers of revolutions per unit of time of the correspond- Fig. 105 i n g w i iee is^ an d ]i the working depth & of the grooves. Then the velocity of XL a point in the pitch circle of either wheel is ZnRN = ^nrn . rn. An extreme outer point of the working surface of the first wheel has a radius R -+- %h, and it comes in contact with a point of the other wheel having a radius r \h ; hence the sliding at these points per unit of time equals snce = rn. By taking extreme contact points on the other side of the pitch circles, having radii R %h and r -j- JA, the same result can be reached by a similar process. The grinding action just noted tends to wear the working faces, even if no slipping occurs at the pitch circles. Such action does not take place in simple cylinder friction- rolls, but it cannot be avoided if the grooves have sensible depth. FR1CTIONAL GEARING. 107 The rate of this sliding action is directly proportional to h ; there- fore the working depth should be as small as practicable. This dimension is limited in practice, without sacrifice of the necessary total contact surface, by using several grooves, side by side, as indi- cated in Fig. 103, instead of fewer and deeper ones. 56. Brush-wheels. Fig. 102 shows a mechanism sometimes used where it is desired to vary the angular velocity of a shaft which is driven by another shaft of constant angular velocity. Suppose the plate on the shaft AA to be the driver, and the disk, or " brush wheel," on BB to be the fol- lower. A long keyway, or spline (or its equivalent), permits the disk to be placed at different positions along a line parallel to a diameter of the plate, as indicated by the dotted locations. The disk is imagined to be of no sensible thickness ; hence it touches the plate at a single point, p. This point, p, is at a distance from BB equal to the radius of the disk, r' ; and at a distance from the axis A A equal to r, which may vary from zero to R (plus or minus). Assuming no slipping at p 9 the angular velocity ratio of AA to BB is r' -r- r (inversely as the radii). When the plane of the disk is in the axis A A, the velocity of p is zero, hence the follower is at rest. If the disk is carried beyond this position (to the opposite side of AA), the direction of the rotation of the follower is reversed. This mechanism is not well adapted for heavy forces; but is very con- venient in many cases for light work, as in feed mechanism and for similar purposes requiring considerable change in the rotative speed of a follower, or reversal of direction of rotation. The disk must have sensible thickness in practical applications, and this gives rise to a grinding action somewhat similar to that mentioned with "V" frictions. If the disk is a cylinder, with contact between one of its elements and a radius of the plate, it is evident that all points in Fig. 102 I A 108 KINEMATICS OF MACHINERY. this cylindrical element must have the same linear velocity (being points in a body at the same distance from the axis); while the cor- responding contact points in the radius of the plate have different linear velocities (being at different distance from the axis of this plate). The disk should therefore be as thin as practicable, and Its edge is sometimes rounded to approximate the point contact of the ideal disk. The plate should usually be the driver ; for if this is not the case, when the disk is in contact with the centre of the plate, the latter is at rest, and the edge of the disk is compelled to slip on the contact surface. The working disk is often made of leather, wood, or other yield- ing material, held between metal washers of slightly smaller diam- eter. This construction increases the adhesion, and makes it easier to maintain the required normal pressure at the contact point as slight wear takes place. In other friction mechanisms one of the members is frequently made of a non-metallic substance for a similar reason, and this member should usually be the driver; for if any slip occurs, by reason of the resistance being greater than the friction can overcome, the tendency is to wear off the edge of the rotating driver evenly, and to wear a depression, or notch, in the stationary follower. If the driver is made of the softer mate- rial the more irregular and objectionable wear of the follower is thereby reduced. In the brush-wheel mechanism it is not so easy to support the soft face on the driver (the plate), and there is not the same reason for doing so; because, even if the wear were all concentrated on this face, it would not be worn off evenly all over, for the follower only covers a small portion of its working surface in any position. When the follower (disk) is at the centre of the plate there is a tendency to wear a small flat place on the edge of the former. This may be avoided in many cases by cutting a slight depression at the centre of the plate, so that contact does not take place in this position of the disk. 57. Cone Friction. Intersecting axes are sometimes connected by rolling conical friction wheels similar to the arrangements indi- cated in Figs. 96 to 100; but these are not so satisfactory as the FRICTIONAL GEARING. 109 frictions on parallel axes, as it is more difficult to adjust the posi- tions of the shafts to maintain the required normal pressure. If the force to be transmitted between intersecting axes is consider- able, it may be better to use positive connections, as bevel-gears, to connect the intersecting shafts, and to introduce the friction ele- ment, if necessary, by means of a supplementary shaft parallel to one of these. Two cones, as shown in Fig. 106, are sometimes used to connect parallel shafts, where changes in the angular velocity of the fol- lower are required. These two cones are similar in inclination, and placed with the adjacent elements parallel, but not touching. An intermediate disk, C (or its equivalent), capable of being moved along the lengths of the F ' 9- l06 cones, is in contact with both of them. Assuming no slipping at either contact, the linear velocity of the edge of this disk will be that of the part of the driver with which it is in contact, and this same linear velocity will be imparted to the follower ; hence the linear velocities of the contact points of the two cones will be equal, and the angular velocities will be inversely as the contact radii of the cones at these points. If the disk is placed nearer the large base of the driver it acts on a smaller section of the follower, and the angular velocity of the latter is correspondingly increased. This device, or modifications of it, is now on the market, for use as a countershaft. Provision is made for maintaining proper con- tact between the disk and the cones. In this case, as in that of the brush-wheel, the disk must have appreciable thickness; hence its contact element engages with points on the cones which must have somewhat different linear velocities, and a corresponding grinding action occurs. Similar remarks as to the means of reducing the practical effect of this action apply to both cases. CHAPTER IV. OUTLINES OF GEAR-TEETH. SYSTEMS OF TOOTH-GEARING. 58. Pitch Surfaces. It has been shown that many pairs of bodies (as cylinders, cones, etc.) may transmit motion from one to the other with pure rolling, while these bodies rotate about axes fixed in the proper relative positions; but that the action of the driver upon the follower is not continuously positive. The application of these rolling bodies as frictional gearing has already been treated. There are many cases, however, where it is desirable to secure a motion equivalent to one of these rolling actions, but where it is absolutely essential that no practical vari- ation from this prescribed motion shall occur. This requirement is frequently met by using the surfaces of the appropriate rolling members as bases, and attaching interlocking teeth to them to prevent slipping. These rolling surfaces, when so used, are called pitch surfaces ; and sections of them perpendicular to their axes are called pitch lines, or pitch curves. Toothed gearing may he classified according to the pitch sur- faces, relation of the axes, and character of the tooth elements as follows : * Class of Gearing. Relative Position of Axes. Pitch Surfaces. Elements of Teeth. 1 Spur Parallel Cylinders Rectilinear 2 Bevel Intersecting Cones a 3 Skew In different planes Hyperboloids " 4 Twisted Any Either Helical 5 Screw In different planes Cylinders {( 6 Face Any None Circular * MacCord's Kinematics. 110 OUTLINES OF GEAR-TEETH. Ill The action of these various classes will be treated in detail in later articles. The two tangent circles of Fig. 107, representing rolling cylin- ders, may have their circumferences divided up into arcs of equal Fig. 107 length, p = Pa = ab = be, etc., = Pa' = a'b' = b'c', etc. This length of arc, p, must be a common divisor of both circumferences, and the numbers of divisions on the two circles are proportional to their circumferences, diameters, or radii. Let the radii be repre- sented by r and r', and the numbers of divisions of the respective circles be called t and t' '; then as p p t * As t and t' are directly proportional to r and r' , it follows that the angular velocity ratio is inversely as the number of the divisions of the two circumferences. It is to be noticed that a and a', b and b', c and c', etc., are pairs of points which become coincident con- tact points as the circles roll together. Now if we bisect the arcs Pa, ab, Pa', a'b', etc., and place pro- jections and corresponding notches on the alternate subdivisions, as indicated by the shaded outlines of Fig. 107, it will be seen that the wheels resemble, somewhat, the familiar toothed gears. The part of the tooth outside of the pitch circles is called the adden- dum or point; the portion inside of the pitch circle, between the spaces, is called the root. The acting surface of the point, or adden- dum, is called the face, and the acting surface of the root is called 112 KINEMATICS OF MACHINERY. the flank. By the formation of such teeth the pitch circles have lost their physical identity, but they are, nevertheless, important kinematically as the basis of the toothed wheels. The distances Pa, ab, Pa' , etc., from any point on one tooth to the corresponding point of the next tooth of the same wheel, measured on the pitch curve, is called the circumferential pitch, circular pitch, or simply the pitch. It is evident that the pitch must be the same for both wheels. If these -wheels are "meshed" (that is, placed with a tooth of one in a space of the other, and with the pitch curves tangent), as shown in Fig. 107, it is apparent that the rotation of one of them will cause the other one to rotate, and that the transmission is now positive. As this rotation goes on, the successive pitch points of the teeth of the two wheels come into contact on the line of centres, and the mean angular velocity ratio for complete rotations, or for angular motions of the wheels measured by their pitch arcs, is identical with that due to the pure rolling of the pitch circles. This might be sufficient for some purposes ; but we have, as yet, no assurance that this angular velocity ratio is strictly constant throughout the angular movements corresponding to the pitch angles. That is, the mean angular velocity ratio during such an angular motion agrees with that of the rolling circles ; but at any phase intermediate between contact at two pitch points the angular velocity ratio may be either greater or less than this mean. It is imperative in many cases, and desirable for smoothness of action and quiet running in nearly all cases, that the angular velocity ratio be constant for all phases, 59. Conjugate Gear-teeth. The condition of constant angular velocity ratio in direct contact is that the common normal to the acting faces, through the point of contact, shall always cut the line of centres in a fixed point; hence the desired constancy of this ratio in such wheels as those of Fig. 107 demands that the common normal shall always pass through the point marked P. If the teeth are of such form that this condition is met, the motion trans- mitted is exactly equivalent to the rolling of the pitch circles,. OUTLINES OF GEAR-TEETH. 113 otherwise there is some departure from the , required relative motion. In general, the form of the teeth of one wheel may be taken quite arbitrarily, and an outline can be found for the teeth of the other whe"el which will give the required angular velocity ratio at all phases; but this statement is subject to practical limitations. A pair of teeth which work together properly are called conjugate teeth. A practical mechanical method of finding a conjugate tooth outline, when both pitch curves and the form of the tooth to be mated are known, will be explained before treating the formation of teeth geometrically. This method is applicable when it is required to construct a wheel to mesh with an existing gear, whether the teeth of the latter have lost their original form through wear or not ; and whether the pitch curves are circles or not. Cut out two segments of wood, .1 and B (Fig. 108), correspond- ing to the two pitch curves, and mount them on centres properly located. Upon the segment A, repre- senting the existing gear, attach, in proper position, a sheet metal templet corresponding in form to one of its teeth, and have this slightly raised above the surface of the wooden seg- ment by inserting a piece of thick paper or cardboard between them, so that a $ ./^ F; piece of drawing-paper attached to the segment B can pass under the templet. Now roll the segments, without slipping, and trace the outline of the templet on the paper attached to B in several positions quite close together; a curve tangent to all of these tracings of the tem- plet is the required tooth outline for B. A thin strip of metal be- tween the edges of the two segments, one end of which is attached as indicated to each of the segments, will prevent slipping during the operation. It is evident that as B is rolled back and forth upon A the outline just derived on B will always be tangent to the 114 KINEMATICS OF MACHINERY. tooth of A y and if .B is provided with a tooth of this form such a tooth in acting upon the given tooth of A will transmit motion identical with that due to the rolling of the pitch curves. The method just explained is convenient for use in the shop, and it suggests a corresponding process for the drafting-room. Draw the given tooth and its pitch curve upon a piece of heavy paper, and then draw the pitch curve of the other member upon tracing-paper, thin celluloid, or other transparent material. Place this last drawing above the other, with proper tangency of the pitch curves, and trace the outline of the given tooth upon the tracing-paper; roll the curves through a small arc, being careful to avoid slipping, and trace the tooth outline in its new position; re- peat this operation until the entire arc of action of the teeth has been covered, and then draw on the tracing-paper a curve tangent to all of the tracings of the given tooth. This tangent curve is the required tooth outline. From what has preceded', it will be seen that two cylinders may be provided with teeth such that the positive motion transmitted from one to the other will be identical with that of the two cylin- ders when rolling upon each other without sliding. This applies to cylinders other than those of circular cross-section ; for the methods of finding a conjugate tooth, as given above, apply to any pair of rolling curves, such as rolling ellipses, logarithmic spirals, etc. 60. General Method of Describing Tooth Ontlines. The general method of describing gear-tooth outlines by means of an auxiliary rolling curve, or generator, will be developed in this article. Suppose A and B (Fig. 109) to be any two rolling plane figures upon the outlines of which a pair of gear-teeth are to be described. As the pitch lines are rolling curves their point of contact is always on the line of centres. In the phase shown by the full lines, the angular velocity ratio of A to B is O'P -f- OP\ in the phase indicated by the broken lines, this ratio O'P' -f- OP' '; or for any phase of these rolling curves, the angular velocities of the members are inversely as the contact radii. If a pair of teeth give OUTLINES Of GEAR-TEETH. 115 Fig. 109 a motion identical with that due to rolling of the pitch curves, it is evident that the common normal to the two teeth in contact must always pass through the point on the line of centres at which the pitch curves are tangent to each other; for these teeth are exam- ples of direct contact members, in which the angular velocities are inversely as the segments into which the line of the normal cuts the line of centres. If such a figure as G be rolled upon the convex side of the pitch curve of A, the point g of the figure G will trace the curve ga on the plane of A. Likewise, by the rolling of G on the con- cave pitch curve of B, the point g will generate the curve gb on the plane of B. The curve G is the generating line of the teeth outlines, and it may be any line capable of rolling on the convex side of A and the concave side of B. The point, g, in this generating line is the describing point of the teeth. Now suppose the pitch curves and the generating line to be in the positions shown by the broken lines, with the generating point at P' 9 the common point of tan- gency of the three lines. If A is turned to the right, as indicated by the phase shown in full lines, B will turn to the left in rolling upon it, and G can be rolled upon the pitch curves so that it remains tangent to both of them at their contact point in 00'. When the pitch lines have reached such a position as is shown by the full lines, G will lie in the position shown by the full line, and the original contact points of A, B, and G will be at a, b, and g, respectively. The arcs Pa. Pb, and Pg must be equal, as the action has been pure rolling. During this rotation the point g describes a curve upon the surface of A (this surface being supposed to rotate with A about 0) such as ag, as noted above; g has, in' a similar way, 116 KINEMA TICS OF MA CHINEE Y. generated a curve bg on the surface of B (rotating about 0'), and, at the instant under consideration, g is the contact point common to ag and bg. Now as G is rolling upon the pitch curves of A and B, and is in contact with them at P, P must be the instant centre of G relative to both A and B' y therefore the point g (a point in G) is, at the instant, rotating about P, and its motion must be in the line gv, perpendicular to gP. As the point g is generat- ing the curves ag, and bg, the common tangent of these curves must coincide with the line of motion of g (gv), and gP, perpendic- ular to gv, is, therefore, the common normal to ag and bg. The curves ag and bg are described upon the surfaces of A and B 9 respectively; and it is evident that teeth upon these members, having the outlines ag and bg, will transmit a motion exactly corresponding to that of the rolling pitch lines; because their common normal passes through the point in the line of centres at which these rolling pitch curves are tangent to each other. The reasoning of the foregoing discussion is perfectly general. It applies to any phase, if the condition that the three curves roll together with a common contact point is met at every instant of the action; hence the curves derived by this construction fully satisfy the kinematic requirements of tooth outlines. The discussion immediately following will be confined to wheels having circles (or circular arcs) for pitch lines. 61. Usual Systems of Gearing. There are a great many curves that can be used as generating lines for the outlines of gear-teeth, but only two are commonly used, viz., circles and right lines. The curve traced by a point in a circle as it rolls upon the con- vex side of another circle is called an epicycloid; if it rolls upon the concave side of another circle, the curve traced is &liypocydoid\ and if it rolls along a straight line a cycloid is described. When a right line rolls upon a circle any point in this line traces a curve called an involute. The common systems of gearing in which the teeth are generated by circular or rectilinear describing lines are called, respectively, the Epicycloidal System and the Involute System. OUTLINES OF GEAR-TEETH. 117 62. Epicycloidal Gearing. Fig. 110, A and B are two pitch circles, with centres at and 0', and tangent at the point P. The generator or describing circle, G, has its centre at o, on the line t of centres 00' . If these circles \ all turn about their respective centres (rolling , upon each other), the paths of these points, a, 6, and a, which originally coincided at P, will be along the arcs Pa, P6, and Pg. Since there is roll- / ing contact Pa, P6, and Pg are i pjg. no all of equal length. During this motion the point g will generate an epicycloid by rolling on the outside of A, and a hypocycloid by rolling on the inside of B. At any instant these two curves will be in contact at g, in the circumference of the describing circle. As the instant centre of the generator, relative to either of the pitch circles, is always at P, g moves perpendicular to Pa, and Pg is normal to both curves at their point of contact. This normal always passes through P, hence the angular velocity ratio is constant. The curves just discussed are suitable for the outlines of gear- teeth, and if the driver, A, has teeth with epicycloidal faces, and the follower, B, has teeth with hypocycloidal -flanks, generated by the same circle, G, the action would begin at the pitch point, P and continue through a period depending upon the length of the teeth. It is evident that a generator G' could be made to describe flanks for A and faces for B, as shown by the curves a'g', and b'g', respectively, which would satisfy the conditions of constant velocity ratio, and that the action of this pair of curves is entirely independent of the first pair ; hence G and G' may be any two circles. At the sides of the figure are shown complete teeth of A and B, the outlines of which correspond to the curves traced by the 118 KINEMATICS OF MACHINERY. describing circles G and G' '. The faces of A and the flanks of B are the epicycloid and hypocycloid generated by (7, and are identical in form with the curves ag and bg, respectively. The faces of B and. flanks of A are of the forms generated by G', as shown by Vg* and a'g', respectively. The teeth are symmetrical; therefore either side may be the acting side, and either wheel may drive. If the common pitch, p, is an exact divisor of both circumfer- ences; if the lengths of the teeth are such that at least one pair shall always be in contact; and if the spaces are deep enough to allow the points to clear in passing the centre line, these wheels will meet all essential requirements. 63. Action of Epicycloidal Gear Teeth. In Fig. Ill the tooth outlines, at the left are just coming into contact, and those at the right are just quitting, contact. The angle aOa' through which gear A turns while one of its teeth is in contact with a tooth of B, is called Fig. in ' the angle of action of A. The angle aOP, passed through while the contact point is approaching the pitch point is the angle of approach of A, and angle POa' passed through while it is receding from the pitch point is the angle of recess. The angles of action, approach, and recess of B are bO'b' ', bO'P, and PO'b' ', respectively. The path of the point of contact during approach is along the arc gP of the describing circle G, and during recess it is along the arc Pg' of the describing circle G'. It has been shown (Art. 62) that the arcs Pa, Pb, and Pg are of equal length. The arcs Pa', Pb' , and Pg' are also of equal length. Therefore the arcs of action, aPa' and bPb' ', subtended by the respective angles of action are of equal length, and this; length is equal to that of the path of the point of contact, gPg' '. When the point of contact is at P , the teeth have pure roll- ing action; at all other times the action is mixed sliding and OUTLINES OF GEAR-TEETH. 119 rolling (Art. 36) . The rate of sliding is greatest when the point of contact is farthest from the pitch point, and it decreases to zero at the pitch point. Since this sliding causes friction it is desirable to reduce it to a minimum. Decreasing the length of teeth lessens the angle of action and the length of the path of contact, and therefore reduces the sliding. For con- tinuous action, one pair of teeth must come into contact before the preceding pair quits contact ; therefore, the angle of action cannot be less than the angle subtended by the pitch arc; or the arc of action aPa' (or bPb') must at least equal the dis- tance between similar points (on the pitch line) of two adjacent teeth of either wheel. This condition fixes the minimum length of the teeth. If the given pitch of the two wheels (Fig. Ill) is aa r W, this determines the minimum arc of action. This arc may be distributed in any way between the approach and recess arcs, though these are commonly nearly equal. Lay off Pa = Pb = Pg, and Pa' = Pb' = Pg r equal to the desired arcs of approach and recess, respectively; then g and g' are the extreme points in the faces of B and A, respectively ; or circles drawn with the radii O'g and Og' are the boundaries of the teeth of the two wheels. The strength of the teeth depends upon their thickness, and the pitch is ordi- narily twice the thickness of the teeth at the pitch circle, or slightly greater to allow clearance at the sides, which is called "backlash;" thus the pitch is a function of the force to be transmitted. As has been shown, the arc of action must at least equal the pitch; it is often made great enough to insure that two teeth shall always be in contact ; or that as one pair is in contact at the centre Ihie, the preceding pair shall be quitting contact, and the succeeding pair shall be beginning contact. This requires an arc of action tqual to twice the pitch arc, and correspondingly longer teeth, for i given pitch. The force acting between the teeth is transmitted in the direc- tion of the common normal (neglecting the effect of friction), or in a line through P and the contact point of the teeth. This contact point always lies in the describing circle G during approach, and 120 KINEMATICS OF MACHINERY. in G f during recess ; hence it appears that the force transmitted is more oblique as the contact point is removed from P. The effect of this obliquity is to increase the pressure between the teeth and at the bearings, with a corresponding increase in the energy wasted through friction. The friction due to the sliding action of the teeth tends to increase the obliquity of the pressure between the teeth during approach and to decrease it during recess by the amount of the angle of friction. Consequently the action during recess is smoother than it is during approach. For this reason gears are sometimes made in which the action is confined to the angle of recess, in which case the driving gear has faces only, and the driven gear has flanks only. Wheels of smaller pitch have shorter teeth, other things being equal, and their action is smoother under the ordinary conditions because the contact point is always nearer the line of centres, where the rate of sliding of the teeth upon each other is less. It will be seen that, for a given pitch, the length of teeth re- quired for a given arc of action is less as the describing circles used are larger in diameter. 64. Determination of Describing Circles. During contact the faces of the teeth of A act only upon the flanks of the teeth of B', similarly, the faces of B act only on the flanks of A', hence the form of the faces of one wheel does not affect that of its own flanks nor of the faces of the mating wheel. There is no necessary fixed relation between the two describing circles G and G' . If the describing circle has a diameter equal to the radius ( the diameter) of the pitch circle within which it rolls in tracing a hypo- cycloid, this special hypocycloid is a right line passing through the centre of the latter circle, or a diameter of it. Hence if the describ- ing circles, G and G' (Fig. 110), have diameters equal to the radii of B and J, respectively, both wheels will have radial flanks ; but these will operate properly in conjunction with the corresponding epicycloidal faces. The faces would not, in this case have the forms shown *n Fig. 110, as the faces of one wheel and the flanks of the other one must be derived from equal describing circles. The OUTLINES OF GEAR-TEETH. 121 radial flank forms are simple in construction and describing circles are som itimes used for a pair of gears which will give such teeth. If the describing circle has a diameter less than the radius of the pitch circle within which it rolls in tracing the hypocycloid, the flanks lie outside of radii through the pitch point ; while if the diameter of the describing circle is greater than the radius of this pitch circle, the hypocycloidal flanks lie inside of the radii to the pitch points. The first of the forms gives spreading flanks which are much stronger than the converging or undercut flanks of the latter form. The radial flank is intermediate between these forms in strength. Except in small gears (frequently called pinions) for light work, undercut flanks are seldom used; the radial flank usu- ally being the weakest form allowed. While it is desirable for strength of the teeth to have spreading flanks, and therefore to use a small describing circle, large describing circles give teeth which act upon each other with less obliquity. In exceptional cases, when a single pair of gears are to work together, it may be good practice to choose the largest pair of de- scribing circles which will give the necessary strength of flanks, and flanks of a comparatively weak form may be used by giving a small excess to the pitch (thickness of teeth). In such cases of single pairs of gears, for reasons already given, radial flanks will sometimes be used for both wheels. In making a set of patterns (or cutters for cut gears), however, it is desirable on the score of economy to provide for the working of any wheel of the set with any other wheel of the same pitch. If this is possible the set is said to be interchangeable. Suppose that in the two gears, A and B (Fig. 110), the faces of the former and the flanks of the latter are gener- ated by a describing circle G, and that the faces of B and the flanks of A are generated by another circle G'. It has been shown that these two wheels will work together. A third wheel, C, of the same pitch, can not work properly with both A and B ; for if the faces of C are generated by G, and its flanks are generated by G', it may engage with #; but it can not act correctly with A, for the faces of A and the flanks of C are not generated by the same circle ; neither 122 KINEMATICS OF MACHINERY. are the flanks of A and the faces of (7, and the conditions of con- stant velocity ratio are not met by this construction. If G = G f , C won Id work correctly with either A or B, or with any other wheel of the same pitch, the faces and flanks of which are epicycloids and hypocycloids generated on its pitch line })jG= G'. We may then state that: The conditions necessary in an Interchangeable Set of Gears are that all of the ivheels of the set shall have the same pitch, and that the teeth of all of them shall have faces and flanks generated by equal describing circles. It is common to assume that the smallest wheel that will prob- ably be required will be a pinion of either 12 or 15 teeth, and to take a describing circle which will give radial flanks to such a pinion; that is, a describing circle with a diameter half that of the pitch circle of this smallest pinion. If t is the number of teeth in the smallest pinion, its pitch-circle radius, or the diameter of the describing circle, = . 65. Annular Wheels. Fig. 65 shows two rolling circles, one of which is tangent to the concave side of the other. The correspond- ing rolling cylinders may be used as pitch surfaces of gears. The larger of these is called an annular gear. Fig. 112 The method of generating the teeth of such gears is indicated in Fig. 112, and it is similar to that explained for external gears> OUTLINES OF GEAR-TEETH. 123 except that the faces of B and flanks of A (generated by G) are both epicycloids, and the faces of A and the flanks of B (generated by G') are loth hypocycloids. 66. Rack and Pinion. If one pitch line is a right line (a circle of infinite radius), as shown in Fig. 113, teeth may be formed by a method similar to that given for the more general case of spur gearing. Such .a gear is called a rack, and the wheel which meshes with it is usually called a pinion. The faces and flanks of the rack are loth cycloids ; and they are alike in an interchangeable set of gears, where but one describing circle is used. In such a set, any wheel will engage properly with the rack. The construc- tion of teeth for a rack and pinion is shown at the left of Fig. 1 13 ; and at the right, the complete teeth are shown in the acting positions. Of course the rack is necessarily of limited length, . Fig. 113 and the motion transmitted between a rack and pinion must be reciprocating. 67. Pin Gearing. If the describing circle equals one of the pitch circles, the hypocycloid in this pitch circle becomes a mere point; and this point acting on an epicycloid generated on the other wheel by this same describing circle will transmit a motion identical with the rolling of the two pitch circles. Fig. 114 shows such a point in B acting on the epicycloidal faces of A. In an actual gear a pin of sensible diameter must be used, and Fig. 114 shows such a pin, and dotted line curves parallel to the 124 KINEMATICS OF MACHINERY. Fig. 114 original epicycloid of A and at a distance from this epicycloid equal to the radius of the pin. This pin and the dotted outline will transmit the same motion as that due to the point and epicycloid. With the point and the epicycloid the angle of action is entirely on one side of the line of centres, and the pin gear should always be the follower, in order that the action shall take place during recess rather than ap- proach. With a pin of sensible diam- eter the action begins at a distance, practically equal to the radius of the pin, before the line of centres is reached, and there is consequently also an angle of approach. The derived curve of the driver gives shorter teeth than the full epicycloids, and the height of the driver's teeth, above the pitch line, is therefore diminished, thus decreasing the angle of recess. These gears were formerly much used, when teeth were commonly made of wood, as the pin I'orm is easily constructed; but this class of gearing is now used but little, except for light gearing, such as clockwork, etc. 68. Involute Teeth. When a right line rolls on the circum- ference of a circle any point of the line traces an involute of the circle. It is a property of this curve that the normal at any point is tangent to the base circle. In Fig. 115, if the right line EE' , tangent to the base circles aa and 66, has rolling contact with these circles as they rotate about the fixed centres and 0', re- spectively, any point g of EE' traces an involute of each base circle.* In every phase of this operation these two involutes are tangent to each other at the position of g, and the line EE' is * The right line EE may be considered as the tangent portion of a flexible band which wraps upon one base circle and unwraps from the other, as they rotate. The point g in this band generates the two involutes upon the rotat- ing planes of the respective circles. OUTLINES OF GEAR-TEETH. 125 normal to both curves at this point. This common normal always cuts the line of centres, 00', in a fixed point, P. If these involute curves are used as the out- lines of teeth for two gears A and B, turning about the fixed centres and 0' respectively, a constant angular velocity ratio, equivalent to pure roll- ing contact between two pitch circles tangent to each other at P, will be A maintained as long as the involutes are in contact. When A turns in a clockwise direction, the first contact between the tooth curves occurs when Fjg< the tracing point is at E, and contact continues as g moves along the line EE' until E' is reached. For continuous action with teeth of involute outline the pitch angles must not exceed the angles through which the respective gears have turned during this time. The line EE' is the locus of the point of contact. The angle between EE' and the common tangent to the pitch circle is the angle of obliquity. Standard gear cutters are so made that the sine of the angle of obliquity is 0.25, which corresponds to an angle of 14. It is an important property of involute tooth outlines that the distance between the centres of rotation may be changed without affecting the velocity ratio. Whatever the distance between the centres of the base circles of the two involutes, the common normal to both curves, in any position of tangency, is always tangent to both base circles, and divides the line of centres into segments which are proportional to the radii of the respective base circles. Since the angular velocity ratio is in- versely proportional to the ratio of these segments, it is independent of the distance between centres.* This property is peculiar to * It will be noted that the mathematical pitch circles vary in diameter with such adjustment of the centre distance, and the pitch circles of involute gears have not the same physical significance as in epicycloidal gears. 126 KINEMATICS OF MACHINERY. the involute system, and is exceedingly valuable, especially in in gears connecting roll -trains, in change gears, etc., where exact spacing of the centres can not be maintained. When a pair of rolls connected by involute gears becomes worn, or when adjust- ment is necessary for passing material of different thickness between them, the centre distance may be changed considerably (if sufficient initial backlash has been provided between the teeth) without affecting the angular velocity ratio. The limits of allow- able adjustment of the centre distance are reached when it becomes so great that as one pair of teeth are engaging the pre- ceeding pair are quitting contact, and when it is so small that the backlash is reduced to zero, on account of the greater thick- ness of the teeth inside the original pitch line. The angles through which the gears may be turned while a a pair of involute tooth outlines are in contact depend on the angle of obliquity. When the angle of obliquity is zero, the base circles coincide with the pitch circles, and the involutes are both entirely outside the pitch circles, and can not come into contact except when passing the pitch point. The angle of action is zero. This represents a special (though impossible) case of epicycloidal gearing in which the diameter of the describ- ing circles is increased to infinity. As the angle of obliquity is increased the angle of action also increases. The length of teeth necessary for this action is indicated by the circles through E and E f with centres at 0' and respectively- Since the distance between either of these addendum circles and the corresponding pitch circle always exceeds that between the pitch circle and the base circle of the mating gear, it is neces- sary to extend the tooth spaces inside the base circles to accommo- date the ends of the teeth. These extensions of the tooth out- lines are usually radial lines tangent to the involute curves at the base line and to fillets at the roots of the teeth. Involute teeth are sometimes called teeth of single curvature, as there ia not a reversal of curvature at the pitch line. OUTLINES OF GEAR-TEETH. 127 The tooth outlines of an involute rack are composed of straight lines perpendicular to the locus of the point of contact. Fig. lloa shows an involute rack and pinion in mesh. 69. Interference in Involute Teeth. The length of standard gear teeth is determined by their pitch, rather than by the con- siderations stated in the preceding article. When one gear has a small number of teeth of standard proportions the ends of the teeth of the mating gear extend beyond the point of tangency of the common normal and the base circle. This is shown in Fig. 116, which illustrates a pair of teeth in contact at the point of tangency between the base circle of the smaller gear B and the common normal. The part of the tooth outline of B inside the base circle is a radial line. It is evident that any further turning of A toward the right will result in contact with the radial part of the outline of B, and the angular velocity ratio will not be constant. This contact of the teeth inside the base circle is called interference. To avoid interference the flanks of the teeth of B may be hollowed out or the points of the teeth of A may be cut away. The latter is the usual remedy. 128 KINEMATICS OF MACHINERY. If the portion of the face which comes into contact with the radial flank of the mating tooth is given the form of an epicycloidal arc generated on the pitch circle by a describing circle of half the pitch diameter of the mating wheel, the action will be correct, for the radial flank is equivalent to a hypocycloidal flank formed by this same describing circle. Fig. 116 This is strictly correct only if the given centre distance is maintained. The least number of teeth that an involute gear of 14^ obliq- uity may have and mesh without interference with an equal gear is 23; the least number in a gear that will mesh with a rack without interference is 32. 70. Comparison of the Systems Since involute teeth trans- mit constant angular velocity ratio when the centre distance varies, exact setting is not so necessary, and wear of the bearings does not disturb the action as it does in the epi- cycloidal system. OUTLINES OF GEAR-TEETH. 129 The line of action is always in the same direction, and the force acting between the teeth is nearly constant in the involute system; while the acting force is variable both in direction and magnitude in the epicycloidal system. The former teeth wear more evenly as a consequence. The mean thrust on the bearings is slightly greater, but more uniform, with in- volute teeth. All involute teeth have the same generator; hence the gears are interchangeable if of the same pitch. Epicycloidal teeth are better for low-numbered pinions, but otherwise have no great advantage and many disadvantages. They are, how- ever, quite commonly used for gears having cast teeth; while the involute system has largely supplanted the epicycloidal system for cut gears. 71. Clearance and Backlash. The term backlash has already been explained as the clearance at the sjdes of the teeth; it is equal to the width of a space minus the thickness of a tooth, both measured on the pitch line. The backlash provides for any irregularity in the form or spac- ing of the teeth. It may be very small in accurate cut gears; but must be larger in cast gears. The spaces are always made deeper than is required to allow the points of the teeth to pass; this allowance is called bottom clear- ance, or simply clearance ; it also provides a lodging-place for a moderate quantity of dirt or other foreign substance which may get between the teeth. 72. Pitch of Gear Teeth. The action of the teeth is smooth- est when the contact point is near the line of centres; hence a large number of small teeth gives more uniform action than fewer and larger teeth. The teeth must be thick enough to sustain the load, however, and the pitch is determined by this consideration. The formula W^spfy, known as the Lewis formula, is in general use for determining the load that may be carried by the teeth of gears. In this formula ir = the total 130 KINEMATICS OF MACHINERY. load transmitted by the teeth, in pounds; s = the safe working stress allowed in the teeth, in pounds per square inch; p = the circular pitch, and/= the width of face, both in inches; ?/= a factor depending on the form of the teeth. For standard epicycloidal and 14 involute teeth, y== 0.124 ; where n = ihe number of teeth in the gear. The pitch of such teeth necessary to sustain a working load, W lt per inch of width of face, for a gear of given diameter, D, as determined from the above formula is: . . . (1) The relation between the circular pitch, the number of teeth, and the pitch diameter of a gear is expressed by the equation, pn=nD, from which p = r:D + n, D = pn+-K, and n = 7zD + p. When either the pitch or the diameter is taken of a convenient size, the other must be expressed as a decimal value. Thus the pitch diameter of a gear having 36 teeth of 1J" circular pitch 36 X 1 5 is -=17.19", while if the diameter is taken as 18" the 7C pitch= =1.571". It is much more convenient to express ob the tooth spacing in terms of the number of teeth per inch of diameter of the gear. This ratio is called the diametral pitch, for which the symbol p' is used. The relation between the dia- metral pitch, the number of teeth, and the pitch diameter of a gear is expressed by the equation p f =n -f- D, from which D=n + p', and n=Dp'. For a gear 18" in diameter and having 36 teeth, p' -36-7-18=2. The relation between diametral and circular pitch is found by combining the equations for the value of D. D= .'. p = 7r-t-p f , and p' = x-t-p. OUTLINES OF GEAR-TEETH. 131 Expressed in terms of the diametral pitch equation (1) becomes p^-l (0.194 + ^0.037-2.15^'). i ' 73. Proportions of Gear Teeth. The dimensions of all other parts of gear teeth are usually expressed as functions of Fig. 117 the pitch. Fig. 117 will serve to explain the terms and symbols used for these parts. p = circular pitch (measured on the pitch line). t= thickness of tooth (measured on the pitch line). s^ width of space (measured on the pitch line). a = addendum = length of tooth outside the pitch line. d= working depth of tooth = 2a. c clearance. h = whole depth of tooth = 2a + c. b = backlash = s t. r= radius of fillet at root of tooth. For gears having cast teeth the following are usual values: ; c=0.05p; a=0.33p; d=0. 132 KINEMATICS OF MACHINERY. The rims of such gears should be made about as thick as the base of the tooth not including the fillet. The hubs are usually about twice the diameter of the shaft. When the arms are of elliptical cross-section, the width of arm at the inside of the rim should be about 2J times the circular pitch. Such arms are tapered from J" to f " per foot on each side, and the thickness is made equal to J the width. The standard proportions of cut gear teeth are expressed in terms of the diametral pitch as follows: a= !"-?-//; 6 = 0; c = 0.157" -s-p'; h=2.W7"+p'. The radius of the fillet at the roots of cut teeth is J the width of the space between the teeth measured on the addendum circle. The outside diameter = D + 2a = (n + 2) -:-/>'. The above proportions are used for cut gears of the epicy- cloidal and 14^ involute systems. Where greater strength is desired without a corresponding increase in the pitch, involute teeth havjng an angle of obliquity of 20 are sometimes used. These teeth are called " Stub Teeth " because they are made shorter than teeth of standard proportions. In addition to being much stronger, these teeth have less sliding and no interference. On account of the greater obliquity of action the normal pressure between the teeth and the thrust on the bearings are greater for a given load than with standard teeth. The pitch of stub teeth is usually expressed as a com- bination of two standard diametral pitches. Thus, a 4/5 pitch tooth has a thickness on the pitch line equal to that of a standard 4 pitch tooth, while the addendum is equal to that of a standard 5 pitch tooth. Other pitches are 5/7, 6/8, 7/9, 8/10, 9/11, 10/12, and 12/14. The depth of the space inside the pitch line is made 1J times the ad- dendum. OUTLINES OF GEAR-TEETH. 133 74. TJnsymmetrical Teeth. If gears are always to turn in one direction, the opposite sides of the teeth may have different out- lines. Fig. 118 shows such teeth. The working sides may belong to any system ; the backs being so formed that they will not interfere, simply. Involutes, with an angle of the normal greater than would be practicable for working faces of teeth, are suitable for the backs. Stronger teeth, for any pitch, are obtained by this construction. Gear-teeth are seldom made un- \^ F '9- ll8 symmetrical. Such forms will generally be more expensive, and the necessary strength may be obtained by increasing the pitch. If the force transmitted is excessive, and driving is always in one direction, teeth of this form may be used to avoid excessive pitch. 75. Stepped and Twisted Gearing. The action of gear-teeth is smooth- est when the contact point is at the line of centres; for in this phase there is pure rolling between the teeth, and, in the epicycloid system, the obliquity is also zero at this in- stant. It is desirable to have the teeth as short as the required arc of action permits; but, as has been shown, this is governed by the pitch, which is a function of the force transmitted. 1 1 is therefore unsafe to reduce the pitch beyond a certain limit in a given case ; but it is possible by " stepped " teeth to retain the required pitch, and still have a pair of teeth always in contact near the line of centres. Suppose a spur gear 134 KINEMATICS OF MACHINERY. to be cut by a series of equidistant planes, perpendicular to the axes ; then let the slices into which the gear is divided be placed as in Fig. 119. If there are N of these slices, each maybe^th the pitch ahead (or behind) the adjacent one. In this arrangement, the maximum distance of the nearest contact point from the line of centres is the corresponding distance with ordinary spur-gears of the same pitch and length of teeth, divided by N. The thickness of the teeth has not been reduced by this modi- fication, hence the strength has not been sacrificed. Large gears are sometimes constructed on this principle, with two sets of teeth, stepped one-half the pitch. As the number of slices into which the gear of Fig. 119 is cut increases (their thickness decreasing correspondingly) the teeth approach those of Fig. 120, which represents the limiting form of the stepped wheels. That is, when the number of slices becomes infinite, the stepped elements become spirals. Gears with teeth of this kind are called twisted gears. It is to be noticed that the action of these twisted teeth is similar to that of the corresponding spur-gears, and they must not be confused with screw -gears which they resemble in form, but which are not constructed for parallel axes. The distinction will be considered more fully in a later article. With twisted gears there is a component of the pressure transmitted which tends to slide the wheel along the axis, or to crowd the shaft to which the wheel is attached against the bearing. This thrust against the bearing can be taken up by a collar, and axial motion thus prevented, but such an expedient results in an undesirable frictional loss, with risk of heating, etc. By twisting the teeth on the opposite sides of the central section in opposite directions, as shown in Fig. 120 (a), the axial efforts due to these two halves Fig. I20(a) IB balance each other, and there is no such thrust imparted to the shaft. In actual gears of this form, the two halves may be cast in one piece, if the teeth are not to be OUTLINES OF GEAR-TEETH. 135 machined ; but if cut gears are used, the two halves are made as two separate gears of opposite inclinations (the elements of one half are right-handed helices, and of the other are left-handed helices), and these two gears may be attached firmly to the shaft, side by side, thus constituting practically one wheel. It will be seen that these twisted gears always have one contact point on the line of centres, if the twist within the width (or the half width, with the double form of Fig. 120 (a)) of the gear is at least equal to the pitch, and the action at this one point is pure rolling. Now if the addenda of these teeth are relieved as indi- cated in Fig. 121 by the dotted lines, the faces will not act upon the flanks of the mating gear till the pitch point of any section comes into contact; that is, till the line of centres is reached, when the action is pure rolling. If the mating gear is similarly treated, the two gears only touch at a point, and this point is always in the line of centres. Thus contact for any tooth begins when the foremost section reaches the line of centres ; it travels along the pitch element of the tooth ending as the last section passes the line of centres, the most advanced sec- tion of the next pair of teeth taking up the action in turn. This concentrates the force transmitted at a single point, theoretically, which may result in too intense a pressure in heavy work ; but it has the effect of producing pure rolling between the teeth in con- tact at all times. This is probably the only example of combined pure rolling, constant angular velocity ratio, and positive driving. 76. Non-circular Gears. In Arts. 46, 47, 48, and 49 the ac- tion of rolling ellipses, rolling logarithmic spirals, general rolling curves, and lobed wheels was briefly explained. It has been seen that cylinders (in the general sense) corresponding to these curves may roll together, and it has been stated that such surfaces may be used as pitch surfaces for non-circular gears. The general method of describing gear-teeth, as given in Art 13o KINEMATICS OF MACHINERY. 59, may be applied in designing teeth for such gears, but a con- venient approximate method will be indicated, using the rolling ellipses for illustration. Circular arcs can be drawn which closely approximate the ellipse at any point, and the methods for circular pitch line gears can then be used for the teeth. Or accurate elliptical curves may be drawn ; then lay off the pitch upon them and apply the method given in Arts. 59 or 62, in generating the teeth on these arcs. Of course, with a given describing curve, the teeth on portions of the pitch line which have different curva- ture will not have the same form. This approximate method can be applied to other rolling curves as well as to the ellipse, and it is thus possible to form teeth for any of the non-circular forms (including the lobed wheels of article 49) which will transmit motion similar to that due to the rolling of the pitch curves. The general method of Art. 60 may be used if preferred. 77. Approximate Methods of Constructing Profiles. The exact construction of the tooth profiles is somewhat tedious, and in many practical applications simpler approximate outlines may be substi- tuted. Gears with cast teeth, especially if the pitch is small, de- part somewhat from the ideal form, however carefully the patterns may be made ; and, therefore, some one of the approximate methods is generally used for laying out the patterns. In making cutters for cut gears, the exact method is usually employed.* The arcs of the curves (epicycloids and hypocycloids, or invo- lutes) used in gear-teeth are so short that circular arcs can be found which very closely approximate these curves ; and most of the approximate constructions are circular-arc methods. A method given in Unwinds Machine Design has the merit of requiring no tables or special instruments, and it will be described first. In Fig. 122, A and B are the two pitch circles, and G and * A little book published by the Pratt & Whitney Co. gives a description of the machine used by this company for accurately making these cutters auto- matically. This treatise was written by Professor McCord, and is reproduced in his Kinematics. OUTLINES OF GEAR-TEETH. 137 G' are the describing circles. Ph is the height of the addendum of B t and Pe represents two thirds this height. If a circle is drawn through e with a centre at the centre of the pitch circle B, it cuts the describing circle G in g\ and if the arc Pb, on the pitch circle B, is laid off equal to the arc Pg on the generating circle G, b and g are two points in the tooth outline, as in the exact con- struction. Draw gP, the normal to the tooth profile at g. Now find by trial a circular arc with a centre on Pg, or its extension beyond P, which will pass through g and b. This arc passes through two points of the exact tooth outline, and its tangent at g also corre- sponds in direction with that of the true epicycloid, as the normal at this point is Pg for both the exact and the approximate curves. If the arc Pa is laid off on the pitch circle A, equal to the arc Pg, another circular arc, with a centre on the normal Pg, will pass through a and g, and it will approximate the flank of A. By a similar method, Pe' is laid off equal to two thirds the height of the addendum of A, and a circular arc with a centre on Pg', and passing through a' and g', is an approximation to the exact face profile of A. The approximate outline for the flank of B is an arc passing through b' and g', with a centre on g'P, or its extension. The point e need not necessarily be two thirds of Ph ; but this fraction gives a good distribution of the error. A method due to Professor Willis has been more widely used, perhaps, than any other. It may be briefly explained as follows: 138 KINEMATICS OF MACHINERY. Lay off from P (Fig. 123) the arc Pa = Pa' one-half the pitch, and draw radii Oa and Oa'\ then draw the lines mm and m'm' through a and a', making an angle with Oa and Oa' respect- ively. At a point c (on mm) take a centre, and draw a circular arc Pf through P; also with a centre at c' (on m'm') draw the arc Pf through P. The angle and the centres c and c' may be so chosen that these arcs will have radii of curvature equal to the mean radii of curvature of the proper epicycloidal and hypocycloidal faces and flanks of the tooth. The angle was found by the originator of this method to give the best results when taken at 75; a . . c and a' . . c' are given by the following formulas, in which p = pitch, and n == number of teeth in the wheel: a . . c f or faces = "ff . -10) > a' . . c' for flanks = -^-f -j. An instrument known as 2 \n I'ZJ Willis's Odontograph facilitates these operations. The form of this instrument is indicated by the dotted lines of Fig. 123. It is graduated along the edge m'm' each way from the point ', and a table which accompanies the instrument gives the positions of the centres c and c' in terms of these graduations for wheels of given numbers of teeth and pitch. OUTLINES OF GEAR-TEETH. 139 Mr. George B. Grant has improved upon this odontograph by tabulating the radii for faces and flanks, and also tabulating the radial distances of the centres c and c' from the pitch circle. Mr. Grant has compiled a similar set of tables, called the Grant Odonto- graph, which are used in precisely the same way; but the values are different, giving somewhat different tooth profiles from those of the Willis system. The Willis method gives circular arc tooth outlines which are correct for one point, and which have radii equal to the mean radius of curvature of the exact curve. The approximate faces derived by this system lie entirely within the true epicycloids. Mr. Grant's system gives arcs which pass through three points of the exact profiles of the faces, and thus more closely approximate the correct curves. The application of Grant's Cycloidal Odontograph, or, as its author calls it, from the method of deriving it, the Three-Point Odontograph, is shown in Fig. 124. The accompanying table is Fig. 124 given, together with his table for constructing approximate involute teeth. This matter is taken from Odontics, or the Teeth of Gears, by George B. Grant, and is reproduced by permission of the author. To use the table in drawing approximate epicycloidal teeth pro- ceed as follows: Draw the pitch line and set off the pitch, dividing the latter properly for thickness of tooth and space. The table gives values both in terms of One Diametral Pitch (equal 3.14" 140 KINEMATICS OF MACHINERY. circular pitch), and of One Inch Circular Pitch. Use the part of the table corresponding to the system of pitch employed. THREE-POINT ODONTOGRAPH. STANDARD CYCLOIDAL TEETH. INTERCHANGEABLE SERIES. (From Geo. B. Grant's " Odontics.") For One Diametral Pitch. For One Inch Circular Pitch. Number of Teeth. For any other pitch divide by that pitch. For any other pitch multiply by that pitch. Faces. Flanks. Faces. Flanks. Exact. Intervals. Bad. Dis. Rad. Dis. Rad. Dis. Rad. Dis. 10 10 1.99 .02 - 8.00 4.00 .62 -01 -2.55 1.27 11 11 2.00 .04 -11.05 6.50 .63 .01 -3.34 2.07 12 12 2.01 .06 oo CO .64 .02 GO CO is* 13-14 2.04 .07 15.10 9.43 .65 .02 4.80 3.00 15| 15-16 2.10 .09 7.86 3.46 .67 .03 2.50 1.10 m 17-18 2.14 .11 6.13 2.20 .68 .04 1.95 0.70 20 19-21 2.20 .13 5.12 1.57 .70 .04 1.63 0.50 23 22-24 2.26 .15 4.50 1.13 .72 .05 1.43 0.36 27 25-29 2.33 .16 4.10 0.96 .74 .05 1.30 0.29 33 30-36 2.40 .19 3.80 0.72 .76 .06 1.20 0.23 42 37-48 2.48 .22 3.52 0.63 .79 .07 1.12 0.20 58 49-72 2.60 .25 3.33 0.54 .83 .08 1.06 0.17 97 73-144 2.83 .28 3.14 0.44 .90 .09 1.00 0.14 290 145-300 2.92 .31 3.00 0.38 .93 .10 0.95 0.12 00 Rack 2.96 .34 2.96 0.34 .94 .11 0.94 0.11 The example of Fig. 124 is a wheel of 20 teeth, 2 diametral pitch, hence the pitch circle is 10 inches diameter (this figure is not reproduced full size). Opposite 20 teeth in the table we find .13 in the column of distances for faces ("dis.") ; divide this by the diametral pitch (2), giving .06" as the distance of the circle of face centres from the pitch circle. Lay this distance off inside the pitch circle, and draw a circle through this point, concentric with the pitch circle. In a similar way the distance for flanks (1.57) is divided by 2, giving .78", which is laid off outside the pitch circle, and a circle is drawn through this point. All tooth faces are to be OUTLINES OF GEAR-TEETH. 141 drawn with circular arcs having centres on the first of these lines of centres, and the flanks are drawn by arcs having centres on the last found line of centres. The tabular radius of faces for 20 teeth is given as 2.20, and dividing this by the diametral pitch, we get 1.10" as the radius for the faces of the 2-pitch wheel. With this radius and centres on the face centre line, draw arcs through the proper points in 'the pitch circle, of course having the concave sides of the arcs toward the body of the teeth. In a similar way, the tabular radius im flanks (5.12) is divided by the diametral pitch, giving 2.56" as the corrected radius. With centres on the flank centre line, draw arcs with this radius meeting the face arcs already drawn at the pitch point, and with the concave sides towards the spaces. Terminate the tooth profiles by the addendum and root circles, determined as in Art. 73, and put in fillets at the bottoms of the spaces. If the circular pitch is used the construction is similar, using the appropriate portion of the table, but multiplying the tabular values by the circular pitch in inches instead of dividing. As explained above, this table is calculated for arcs which pass through three points in the true curve. It is recommended that the student construct tooth profiles on a large scale by the exact method, and then draw the approximate profiles (superimposed), for comparison. Grant's involute Odontograph given on page 142 is used as fol- lows : Lay off the pitch circle, addendum, root and clearance lines, as in the preceding case. " Draw the base line one sixtieth of the pitch diameter inside the pitch line. Take the tabular face radius on the dividers, after multiplying or dividing it as required by the table, and draw in all the faces from the pitch line to the addendum line from centres on the base line. Set the dividers to the tabular flank radius (corrected), and draw in all the flanks from the pitch line to the base line. Draw straight radial flanks from the base line to the root line, and round them into the clearance line. '' [Grant's Teeth of Gears, p. 30.] 142 KINEMATICS OF MACHINERY. INVOLUTE ODONTOGRAPH. STANDARD INTERCHANGEABLE TOOTH, CENTRES ON THE BASE LINE. Teeth. Divide by the Diametral Pitch. Multiply by the Circular Pitch. Teeth. Divide by the Diametral Pitch. Multiply by the Circular Pitch. Face Radius. Flank Radius. Face Radius. Flank Radius. Face R dins. Flank Radius. Face Radius. Flank Radius. 10 2.28 .69 .73 .22 i 28 3.92 2.59 1.25 0.82 11 2.40 .83 .76 .27 29 3.99 2.67 1.27 85 12 2.51 .96 .80 .31 30 4.06 2.76 1.29 0.88 13 2.62 1.09 .83 .34 31 4.13 2.85 1.31 0.91 14 2.72 1.22 .87 .39 32 4.20 2.93 1.34 0.93 15 2.82 1.34 .90 .43 33 4.27 3.01 1.36 0.96 16 2.92 1.46 .93 .47 34 4.33 3.09 1.38 0.99 ir 3.02 1.58 .96 .50 35 4.39 3.16 1.39 1.01 18 3.12 1.69 .99 .54 36 4.45 3.23 1.41 1.03 19 3.22 1.79 1.03 .57 37-40 4.20 1.34 20 3.33 1.89 1.06 .60 41-45 4.63 1.48 21 3.41 1.98 1.09 .63 46-51 5.06 1.61 22 3.49 2.06 1.11 .66 52-60 5.74 1.83 23 3.57 2.15 1.13 .69 61-70 6.52 2.07 24 3.64 2.24 1.16 .71 71-90 7.72 2.46 25 3.71 2.33 1.18 .74 [ 91-120 9.78 3.11 26 3.78 2.42 1.20 .77 121-180 13.38 4.26 27 3.85 2.50 1.23 .80 181-360 21.62 6.88 Grant's special directions for drawing the teeth of the involute rack are substantially as follows: To draw the teeth for the involute rack, draw lines at 75 with the pitch line of the rack; the outer quarter of the tooth length (one half the addendum) is to be rounded off by an arc with a radius equal to 240" divided by the diametral pitch, or .67" multiplied by the circular pitch. This is to avoid interference. 78. Bevel-gears. - It was shown (see Art. 52) that a pair of cones can "be placed on intersecting axes in such a manner that they will transmit motion with a given angular velocity ratio if they roll to- gether without slipping. Such rolling cones may be used for pitch surfaces of bevel-gears just as rolling cylinders are used for the pitch surfaces of spur-gears, and teeth can be formed on these conical pitch surfaces which will transmit a positive motion equiva- lent to that of the rolling cones. OUTLINES OF GEAR-TEETH. 143 In treating of spur-gearing plane sections at right angles to the axes were used to represent the gear, and the tooth outlines were considered to be developed by the rolling of one plane curve (the describing line) upon another plane curve (the pitch line). The real teeth are, of course, solids bounded by ruled surfaces, all transverse sections of which are exact counterparts of the plane curves discussed. That is, the actual teeth are not lines generated by a point in the describing curve as it rolls upon the pitch line, but they are really surfaces generated by an element of the describing cylinder as it rolls upon the pitch cylinder. Spur-gears coming under the head of plane motions permit representation by plane sections, as explained in Art. 10. This simple treatment cannot be applied to bevel-gears, for although each separate gear has a plane motion (rotation) about its axis, taking two cones rolling together, the relative motion is a spherical motion (see Art. 12). To construct bevel gear-teeth two projections are required. Just as an element of one cylinder in rolling upon another cylin- Fig. 125. tier generates a tooth surface, so an element of one cone in rolling upon another cone sweeps up a surface which can be used as the basis of a bevel-gear tooth. Fig. 125 shows a pitch cone A with the generating cone G (of equal slant height) in contact with it 144 KINEMATICS OF MACHINERY. along a common element PQ. All points in the bases of both cones are at the same distance from the common apex, hence these bases are small circles of a sphere which has a radius equal to the common slant height. If the generating cone be rolled upon the pitch cone, a point in the base of Gr, as g, will describe a curve on the surface of the sphere, relative to the base of the pitch cone. This curve is analogous to the epicycloid, the derivation of which was treated in Art. 62, and the curve now under consideration may be called a spherical epicycloid. In a similar manner another generating cone G' can roll inside the pitch cone, a point g' in its base tracing a spherical hypocycloid on the surface of the sphere. Points on other transverse sections of these generating cones would trace similar curves on spheres of different radii. A line passing through the centre of the sphere (the common apex of the cones) and moving along the spherical epicycloids and hypocycloids described as above, would give surfaces portions of which could be used as tooth boundaries. In other words, the elements of the describing cones which pass through g and g' would sweep up these surfaces. If two pitch cones of equal slant height have teeth generated in the manner just outlined, they will work together properly, trans- mitting a positive motion equivalent to the rolling of the pitch surfaces, pro- vided the pitch of the teeth agree, and that the faces of each wheel are de- scribed by the same generating cone /] which describes the flanks of the other wheel. Fig. 126 indicates two pitch cones A and B, with axes QO and QO', respect- ively, and a common element of contact PQ. The describing cones are G and Fig ' 126 - ', with axes Qo and Qo'; and if all. four axes lie in one plane (a meridian plane of the sphere), they OUTLINES OF GEAR-TEETH. 145 can roll together about fix.ec! axes, always having a common con- tact element in PQ. As the rolling proceeds, such an element of G asgQ sweeps up the faces for B and the flanks for A; and, at the same time, the element g' Q of G' sweeps up the faces of A and the flanks of B. The faces of B and the flanks of A always have gQ for a common element,, and a plane through the three points PgQ > the common normal plane at the element of contact of these surfaces, always passes through the contact element of the pitch cones PQ. Likewise, the normal plane to the faces of A and the flanks of B, Pg'Q, always passes through PQ', hence teeth bounded by these swept-up surfaces will transmit motion with a constant angular velocity ratio. The analogy between this case and that of the spur-gear teeth, treated in Art. 62 (Fig. 110), is so close that further discussion is hardly necessary. The describing surfaces are not necessarily right cones of circu- lar cross-section, though these are the figures which correspond to the epicycloidal class of spur-gears, and are the only forms com- monly employed. Any cone with an apex at the common apex of the pitch cones, and tangent to them along their common element might be used, as it would satisfy the kinematic requirements. Involute teeth for bevel gears maybe generated in a manner anal- ogous to that used in Art. (58 for the generation of spur gear teeth. It is difficult to construct spherical epicycloids and hypocycloids, and to represent the teeth of bevel gears on paper, arid in practice a method know as Tredgold's approximation is always employed. 79. Tredgold's Approximate Method of Drawing Bevel-gear Teeth. Fig. 127 shows the projection of two cones (with bases PM and PN) on a plane parallel to both axes QO and QO'. The line 00' is drawn perpendicular to the contact element PQ, then OM is drawn perpendicular to MQ and O'N is drawn perpendicular to NQ. A cone can be constructed on the axis OQ with OP and OM as elements, and another on O'Q with O'P and O'N as elements. These cones are called normal cones to A and J5, respectively, as any element of one of these cones is perpendicular to an element of the pitch cone 146 KINEMATICS OF MACHINERY. having the same axis and the same -base. The surfaces of these noimal cones approximate the spherical surface for a short space either side of the pitch circles, and the conical surfaces have the practical advantage that they can be developed upon a plane for the construction of tooth profiles. Tredgold's approximate method consists in describing tooth outlines on these developed surfaces of the normal cones, and then wrapping these surfaces back to their original positions. The development of the normal cone surfaces is indicated in Fig. 127 by PM'O, and PN'O'. Upon the de- Fig. 127. veloped bases of these cones (PM' and PN f ) as pitch lines, tooth outlines can be drawn by any of the methods used for spur-gears, just as if these were the pitch lines of such gears; and when these surfaces are rolled back into the normal cones the ends of the teeth are given by the profiles constructed in this way. A straight line passing through Q and following such a profile would sweep up tooth surfaces, all elements of which are right lines converging at Q. Within the limits of practice, such teeth, if properly con- structed, agree quite closely with the exact forms. The application of this method is shown in detail in Fig. 128 for the teeth of a single wheel. The pitch cone is shown in side elevation by MQM, and in plan by the circle M^Mi. The side ele- vation of the normal cone is projected in MOM, and the develop- OUTLINES OF GEAR-TEETH. 147 ment of the pitch circle is show by MM'. The pitch, which must be an aliquot part of the pitch circle M l M l ( = MM times n), is laid off on MM', and the addendum and root circles (AA f and RR', respectively) are drawn to give the proper length of teeth. The tooth outline is now constructed on MM' as for a spur-gear. It is evident that all of the pitch points will fall upon the line MM, in the side elevation, when the normal cone surface is re- Fig. 128. turned to its original position; that the outer ends of the teeth will fall upon A A; and that the bottoms will fall upon RR. In the other projection, the pitch points will all lie on the circle MiMi; the tops will fall on the circle through A^A^ and the bottoms will be in the circle RiR^ In this last projection (plan) the teeth will all appear the same, and they will have their true thickness at all parts; but the height (AR] will be shortened to AiR,. Divide up the circle J/, Mi into the proper number of divisions for teeth and 148 KINEMATICS OF MACHINERY. spaces, and draw radii through the middle of the tooth divisions. Lay off half the thickness of the tooth at the pitch line (as obtained from the construction on developed pitch line MM') each side of these middle radii upon the circle M^Mj then lay off half the thickness of the top in a similar way on the circle A^A^ and half the thickness at the bottoms on the circle R 1 R 1 . The half thick- ness at positions intermediate between the pitch circle and the ad- dendum or root circles can also be laid off on the corresponding circles in the plan, taking as many such thicknesses as the desired accuracy requires. The method of finding these intermediate points is indicated in Fig. 128. Through the points thus found, the curves A 1 M 1 R 1 can be drawn, giving the plan of the large ends of the teeth. To complete the plan of the wheel we may proceed as follows: Draw radii from A v M l9 R^, etc., to Q^ then lay off Aa, on the side elevation, equal to the desired length of the tooth, or the face of the gear, and draw amr parallel to AMR; from a, m,and r, points lying in elements from Q to A, M 9 and R, respectively, carry lines across parallel to QQ t ; then circles with Q lf as a centre and tangent to these several parallels are the projections in the plan of the addendum, pitch, and root circles for the small end of the teeth. As all elements of the teeth converge in plan at Q lf the intersections of radii through A l9 M l9 and 7, with these circles last drawn locate points a l9 m l9 and r l in the plan of the small ends of the teeth. Curves through these intersections, with the portions of the radial elements intercepted between them and the outer curves, will complete this projection of the teeth. Returning to the side elevation, the lines aa, mm, and rr, are the projections of the smaller addendum, pitch, and root circles. To complete the side elevation of the teeth project across (parallel to Q^) from the various points on A^A^ to AA\ from M } M^ to MM; from R^R^ to RR, etc., and draw curves through the intersections thus made. This will give the side elevation of the large ends of the teeth by passing curves through the corresponding intersections. In a similar way the side elevation of the smaller ends is obtained, and elements through Aa, Mm, Rr, etc., completes this view. It is OUTLINES OF GEAR-TEETH. 149 evident that each of the teeth appears in this projection with its own distinctive form. The ring, or rim, which supports the teeth usually has a thick- ness equal to the roots of the teeth at the large ends, and this rim, with the hub, arms, etc., can now be drawn. 80. Peculiar Smoothness in Operation of Bevel-gearing. Among spur-gears of an interchangeable system, those with the larger pitch circles will drive more smoothly, other conditions being the same. By referring to the bevel-gear of Fig. 128 it will be seen that there are 16 teeth on a pitch circle of radius QiMi (diameter = MM}} but these teeth have profiles similar to those of a spur- gear with a pitch radius OM, equal to the slant height of the normal cone, and therefore the action of the bevel-gear would cor- respond to that of this larger spur-gear instead of to a spur-gear of diameter MM. The actual pitch diameter of the bevel-gear is to that of the equivalent spur-^ear (so far as smoothness of running is con- cerned) as sin AOQ : 1 ; thus if the pair of bevel-gears on shafts at right angles are equal, angle AOQ = 45, when sin AOQ : 1 : : i\/2 : 1 : : .707 : 1. 81. Non- inter changeability of Bevel-gears. Bevel-gears are almost always made to work together in pairs, and it is not there- fore of great importance to adopt a standard describing circle for all pairs of the same pitch. If two intersecting axes approach each other at a fixed angle, there is but one bevel-gear which will work properly with any other gear;* for a change in the angular velocity ratio involves a change in the direction of the contact ele- ment (Ac of Fig. 96), and hence a change in both pitch cones. A given bevel-gear could work with more than one other wheel if the inclination of the axes varied correspondingly, but this is a con- * Mr. Hugo Bilgram has produced sets of bevel-gears, by his gear-shaper, in which several different sizes of gears tvork correctly with a single gear, and all the axes make the same angle with the common driver. However, the pitch cones all of these gears do not have a common apex, although the teeth elements all converge to a common, point. These gears are 'not, properly; of the common type of bevel-gears. 150 KINEMATICS OF MACHINERY. dition seldom met, and so these gears may generally be designed to work in pairs without regard to other gears. The describing circles (if the epicycloidal system is used) may be so taken that both wheels will have radial flanks, which gives a simple form of teeth to construct, though this is not always de- sirable. For convenience of manufacture it is desirable to have a uniform system, usually; and when bevel-gears are cut with rotary milling cutters of the common type, standard cutters are used for various gears of the same pitch. This will be discussed in a later article. In a great majority of cases requiring bevel-gears the axes are at right angles to each other, and " stock-gears " for such cases can frequently be obtained from gear-makers, if the proportions are not unusual. These stock-gears are generally much less expensive than gears made to order; but special gears are almost always required when the angle of the axes is other than 90. When the two gears are equal (angular velocity ratio 1:1), the gears are- called Mitre Gears. 82. Helical Gears. If two cylinders are tangent to each other,, they will have line contact when the axes are parallel, and point contact when the axes are not parallel. In the latter case the two elements (one of each cylinder) through the point of contact determine a plane tangent to both cylinders. The radii of both, cylinders at the point of contact are perpendicular to this plane. When the cylinders rotate on their axes, each of the points in con- tact moves in the tangent plane in a direction at right angles to the axis of rotation. This is shown in Fig. 129, which represents a plan view of two cylinders, A and 5, in contact at P ; a ... a and b ... b being the axes of the respective cylinders. The angle between the axes of the cylinders is 6. The linear velocities of the points of A and B in contact at P are assumed to be v, and v-t respectively. v t is represented by PZ, and v 2 by Pm. These velocities have a common component, v, represented by Pn, drawn through P perpendicular to Im. The angle between v and v^ is a, that between v and i\ is /?. The components of v t and v 2 , in OUTLINES OF GEAR-TEETH. 151 a direction perpendicular to v are Pt and Pi' respectively. The algebraic difference of these components represents sliding at the points of contact, while the common component rep- resents rolling contact in a direction perpendicular to the sliding. It appears that the action of the tangent cylinders may be considered as a com- bination of sliding in a direc- tion, it', determined by the re- lative magnitudes of the linear velocities of the contact points, and rolling in a direction, Pn, Fifl> I29 perpendicular to the sliding. Considering the rolling and the sliding separately, it is evident that, with a constant angular velocity ratio between the cylinders corresponding to the assumed linear velocities, the rolling action will result in rolling between two helices, each of which is called a normal helix, crossing the elements of the respective cylinders at angles of 90 a and 90 /?, while the sliding is between helices crossing the elements at angles of a and /? respectively. If these cylinders are used as the pitch surfaces of gears to have an angular velocity ratio corresponding to the assumed values of v l and v 2 the teeth of these gears must be so formed as to permit the sliding action, and to transmit a velocity ratio corresponding to the rolling action. If the teeth are of uniform cross-section, and the pitch elements are helices making angles a and /? with the elements of the respective pitch cylinders, the sliding may take place. The form of tooth outlines necessary to transmit constant velocity ratio equivalent to pure rolling of the normal helices is determined by the curvature of these helices, being the same as that for spur gears, the pitch lines of which have radii equal to the radii of curvature of the respective helices. 152 KINEMATICS OF MACHINERY. Such gears are called helical gears. They are included (together with worm gears, which are treated in the next chapter) under the general head of screw gears in the classification on page 110. Helical gears are also commonly known as spiral gears. They closely resemble in form the twisted gears described in Article 75, but their action is entirely different. It is to be noted that the axes of twisted spur gears are always parallel, while helical gears may be designed for any angle between the axes, although they are usually at right angles. The tooth action of helical gears is widely different from that of twisted gears. The former have point contact, and the latter line contact, while the screw-like action of helical gears results in a large amount of sliding in the direction of the common tangent to the tooth elements. This sliding action is entirely absent in the case of twisted gears. In twisted gears the angular velocity ratio is inversely proportional to the radii; while in helical gears it depends not only on the radii, but also on the relative values of the angles a and /?. 83. The Pitch of Helical Gear Teeth. Fig. 130 represents the pitch surface of a helical gear. The pitch elements of the teeth are shown crossing the cylin- der elements at an angle (f>, called the angle of cut of the teeth. The pitch p of these teeth is the dis- tance between corresponding pitch elements of adjacent teeth measured along a normal helix. The dis- tance between the same elements measured on a transverse section of the pitch cylinder is p -=- cos (/>. If there are n teeth, xd=nXp * cos (p \ Fig. 130 or d=n-r- cos 6. Since = p' the diametral pitch correspond- P P ing to p, the equation may be written ' cos (j>, or n d OUTLINES OF GEAR-TEETH. 153 Xp' cos (/>. The corresponding values for spur gears are d = n + p', and n = dXp'. 84. The Velocity Ratio of Helical Gears. If d t and d 2 are the respective diameters of A and B in Fig. 129 the corresponding angular velocities are (tj l = v l ^-^ and w 2 = v 2 +-^. The angular velocity ratio is ~ = ~ L j-- Since i\ cos a = v 2 cos/? this equa- W 2 V 2 dl a). d 2 cos /? tion may be written = 3 . It is evident from the rela- a> 2 d l cos a tion between the diametral pitch, the angle of cut and the num- ber of teeth determined in the preceding article, that the number of teeth of A is n^ = d^p' cos a and the number of teeth of B is n 2 =d 2 p f cos /?. Substituting in the above equation, - 1 = , (o 2 n^ from which it appears that the angular velocity ratio of a pair of helical gears is equal to the inverse ratio of the number of teeth of the respective gears. This relation is the same as for spur and bevel gears, and all other classes of gears. 85. Outlines of Helical Gear Teeth. It was stated in Art. 82 that, for constant angular velocity ratio, the tooth outlines of helical gears must have the same shape as those of spur gears having radii equal to the radii of curvature of the normal helices. The radius of curvature of a helix crossing the elements of a cylinder at any angle is equal to that at the end of the minor axis of the ellipse in which a plane making the same angle with the axis of the cylinder cuts the surface. In Fig. 131, d is the diameter of the pitch cylinder of a helical gear; sos is a tooth element; non is a normal helix; is the angle of cut. The major axis of the ellipse cut from the cylinder by a plane making the angle with a transverse section of the cylinder is d~cos . The radius of curvature at the end of the minor axis is (- 7) ^-= - \2 cos $/ 2 2 cos 2 (/> Hence the tooth outlines of the helical gear should be the same 154 KINEMATICS OF MACHINERY. as those of a spur gear the diameter of which is d-r-cos 2 . The number of teeth of a helical gear is n =d p' cos <, that of a spur d . ^ !_ n gear of diameter d + cos 2 is ri = , from cos* 9* n cos a which it appears that the teeth of a helical gear having n teeth Fig. 131 should have the same outlines as those of a spur gear having n-7-cos 3 2. QhjE.J!b. - ^i = ^ gk~n 2 ~co l ' gk gb oa' ' n> 2 ob Also xy=2c=d l + d 2 = n l + p' cosa+n 2 + p'cosp. Since /?=#-, this equation may be written n x -r- cos a -f n 2 -f- (cos a) = 2p'c. Fig. 132 A similar diagram (Fig. 133) may be constructed when only the centre distance, c; the shaft angle, 6; the velocity ratio, co l -=- oj 2 and the diametral pitch, p', are given. From this diagram the values of n t and n 2 , d^ and d 2J and a and /? may be determined. Draw ox and oi/ making the angle 6 at o. Lay off oa and 06 on ox and oi/, respectively, so that oa :ob 1:0^ : a) 2 . Draw 06 and 6e parallel to oy and ox respectively, intersecting at e. Draw oe. Draw the line XT/, equal in length to 2c, inter- secting oe at #, so that the distance og is a maximum. Draw #/i and gk perpendicular to ox and oy respectively. Multiply the lengths of gh and gk by the diametral pitch, p' t to obtain approxi- mate values of n v and w 2 . For the actual values of n t and n 2 take the largest whole numbers, not greater than the approximate 156 KINEMATICS OF MACHINERY. values, wJiich will satisfy the equation n 2 + n l = a) l + a) 2 . Compute the corresponding values of n^p' and n 2 +p', and take h'g' and k'g' equal to these values and parallel to hg and kg respectively, locating g f on oe. Through g' draw x'y' equal in length to xy, terminating in ox and oy at x' and y' respectively. Then angle x'g'h' = a, and angle y'g'k f = p; distance g'x' = d v and g'y' = d 2 . Since it is not possible to obtain exact values by construction, these values should be considered as approximate. The exact Fig. 133 value of a is obtained by trial from the equation ^-r-co 4- cos (8 a) = 2p'c, using the approximate value first and changing it slightly until an exact equality results. The corre- sponding exact value of /?=# a; those of d t and d 2 are obtained from the equations d l = n l -i-p f cos a, and d 2 = n 2 -p' cos /?. When the shafts are at right angles, as is usually the case, = 90, and cos (0 a) = sin a. Substituting this value, the above equation becomes n^cos a + n 2 -v-sin a = 2p'c. This may be written n tan a + n 2 =2p'c sin a. OUTLINES OF GEAR-TEETH. 157 It will frequently be found that some of the values determined by this construction are not suitable for actual gears. The next less number of teeth in each gear that will give the required velocity ratio may then be tried. If this does not give satis- factory values, either the pitch or the distance between centres must be changed., 87. Cast Gears. Gear-teeth are either cut in a machine or are cast. For the rougher classes of work, it is common practice to use gears with cast teeth; but cut gears are now used almost exclusively for the better grades of work. When gears are cast, it is impoitant to form the patterns very carefully, and especially to space the teeth accurately. With the utmost care, however, it is impossible to get very smooth and accurately spaced teeth, so clearance between the sides of the teeth, or backlash, must be provided. With small gears the enlargement of the mould, due to " rapping " the pattern, more than com- pensates for the shrinkage, and unless this is looked after in the pattern shop and foundry, the teeth may be too thick when cast. A convenient device for forming the teeth of the pattern is shown in Fig. 134. A block of hardwood (preferably of a color quite distinct from the wood of the pattern) is shaped as shown, so that the sections at amr and AMR correspond to the two ends of the teeth (for a spur- gear these sections are, of course, p- tg I34 alike). The middle portion is cut out, so that the distance L equals the length of a tooth; that is, the part removed corresponds to the form of a tooth. The stock for the teeth of the pattern is gotten out in lengths equal to L, and large enough in cross-sections to make a tooth. The block of Fig. 134 is screwed in the vise, and it may have two pointed brads projecting upward through the bottom. Then a piece of the prepared stock is forced down into the space 158 KINEMATICS OF MACHINERY. in this " form/' and is then planed up with " hollow and round " planes. By working down to the form it is quite easy to produce a large number of teeth very uniform in shape. Where many large cast gears are made, a gear moulding machine is sometimes used, as it produces accurate work and reduces the cost of patterns. A stake, or arbor, is set upright at the centre of the mould, which may be swept up in loam to approximately the outside form of the wheel. The pattern for the teeth, simply a block with a few teeth attached which corresponds to a segment of the entire rim, is fastened to the stake by an arm. This arm holds the segmental pattern at the proper distance from the axis of the wheel, and the arm and pattern can be turned about this axis. The pattern can also be withdrawn towards the centre or upward. In moulding the gear, this segment is placed in position and a few teeth are moulded by filling in about the pattern with the sand. The pattern is then drawn, rotated- about the axis through a small angle (one or two teeth less than the number in the pattern), and a few more teeth are moulded. In this way the entire rim is moulded by sections. An index plate, or ring, is used to insure accurate spacing. After completion of the rim the pattern (or the cores) for the arms, hub, etc., is used to complete the mould. 88. Methods of Cutting Gear-teeth. \7hen a gear having cut teeth is to be made a gear-blank is prepared which is identical with the finished gear in every respect, except that it has no teeth. The teeth are then formed in this blank by cutting out the spaces between them. This may be done either by planing or milling. In planing machines the tool has a reciprocating motion, and cuts during the stroke in one direction only; in milling machines rotary cutters are used ; and the cutting is continuous during the process of forming a space. In Fig. 135, a illustrates a tool used for planing gear-teeth, and b shows a standard milling-cutter. The cutting edges of both of these tools are formed to the exact shape of the space between two OUTLINES OF GEAR-TEETH. 159 teeth of the gear to be cut. Another class of tools have cutting edges formed to the shape of the tooth outline of another gear of the same pitch as the one to be cut. An important difference between these two classes of tools is that while the range of the number of teeth in a gear that may be cut with a single cutter of the first class is very narrow, any gear from the smallest pinion to a rack may be cut with a single cutter of the second type. On the other hand, it should be noted that the former class of tools may be used in the ordinary milling and shaping machines, without Fifl< 135t any special attachments other than an index-head for rotating the blank into the correct positions for cutting the teeth, while the latter class is used, in general, only in machines designed especially for cutting gears. To reduce the wear of the finishing tool, it is customary (except in fine pitch gears) to roughly form the teeth by "gashing " the blank before the final cutting operation. When the teeth of a gear have been thus roughed out, they may be finished by planing with a tool ground to a sharp point which cuts the tooth surfaces element by element. In the following articles the use of these methods in Cutting spur, bevel, twisted and helical gears will be briefly outlined. A detailed description of these operations and of the machines which perform them may be found in a book entitled " Gear Cutting Machinery " by Mr. R. E. Flanders. 89. Planing Spur-gears. When the teeth of a spur-gear are to be cut with a tool such as that shown in Fig. 135a, the gear-blank is mounted between the centres of an indexing mechan- ism, placed on the table of a planing or shaping machine. The stroke of the tool is parallel to the axis of the gear. Between 160 KINEMATICS OF MACHINERY. strokes the tool is fed radially toward the axis of the blank until the required depth of space is obtained. The tool is then with- drawn, and the indexing mechanism used to turn the blank on its axis into the proper position for cutting the next space. This process is repeated until all the teeth are completed. This is the simplest method of cutting gear teeth. It may be used for any system of tooth outlines. It is especially adapted to cutting the teeth of annular gears, and gears of large diameter. It is also useful when a gear is needed and no standard milling-cutter is available. A hardened steel pinion (properly " backed off " to give cutting clearance) having a reciprocating motion parallel to its axis may be used as a cutter for spur-gears. The gear-blank is mounted with its axis parallel to that of the cutter. The cutter and the blank are connected by a train of mechanism so that as the cutter is rotated very slightly on its axis between strokes, the blank also turns on its axis with a velocity ratio corresponding to pure rolling contact between the pitch-cylinders. The cutter thus meshes with the gear it is cutting as shown in Fig. 136. At the beginning of the operation of cutting a gear by this method the cutter is fed toward the blank until the distance between the axes corresponds to tangency of the pitch-cylinders of the cutter and the blank. The rotation on the axes is then begun. All the teeth are completed when one rotation of the blank on its axis has been made. A single cutter serves to cut all gears of a given pitch. A tool having a cutting edge shaped like a single tooth of a rack of the same pitch as the gear to be cut may also be used in a somewhat similar manner. In this case the slight rotations of the Fig. 136. OUTLINES OF GEAR-TEETH. 161 blank on its axis are accompanied by a motion of the tool in a direction at right angles to the stroke, the relative motion corre- sponding to pure rolling between the pitch surface of the gear being cut and that of the imaginary rack of which the cutter is a tooth. Fig. 137 shows the position of the tool relative to the space being cut at several stages during the operation. By indexing the blank after every stroke of the cutter, equal cuts are taken on all of the teeth. The rolling action between the pitch surface of the gear and that of the imaginary rack is independent Fifl. 137. of the indexing, and occurs each time the blank has been indexed through a complete revolution. Spur-gear teeth which are too large to be cut with formed tools may be finished with a planing tool having a sharp point, after the spaces have been roughly cut out by other means. In this opera- tion the feeding of the tool toward the axis of the blank is accom- panied by a lateral movement at right angles to the feed, causing the cutting point to trace the outline of the tooth. In this way the teeth are formed element by element. The lateral movement is usually produced by a templet formed to the exact shape of the tooth outline. After one side of one tooth has been completed the blank is indexed and the operation repeated on the next tooth. By providing a second tool to work simultaneously on the other side of the teeth, the necessity for turning the blank over after all the teeth have been finished, on one side b avoided. 162 KINEMATICS OF MACHINERY. 90. Milling Spur-gears. When a standard milling-cutter, such as is shown in Fig. 1356, is to be used to cut spur-gear teeth, the blank is mounted with its axis at right angles to the axis of rotation of the cutter. The blank is fed toward the cutter and the whole space between two teeth is cut out by one passage of the cutter across the face of the blank. The blank is then turned into position for cutting the next space, and the operation repeated until all the teeth are completed. Standard cutters are made in sets suitable for cutting all gears of a given pitch from a 12-tooth pinion to a rack. The following table shows the cutters in one set for epicycloidal and involute teeth, as manufactured by the Brown & Sharpe Mfg. Co. EPICYCLOIDAL SYSTEM INVOLUTE SYSTEM 24 Cutters in each Set. 12 Cutters in each Set. Cutter. Teeth Cutter. Teeth. Cutter. Teeth. A cuts 12 M cuts 27 to 29 1 cuts 135 to rack B 13 N 30 to 33 2 55 to 134 C 14 O 34 to 37 3 35 to 54 D 15 P 38 to 42 4 26 to 34 E 16 Q 43 to 49 5 21 to 25 F 17 R 50 to 59 6 17 to 20 G 18 s 60 to 74 7 14 to 16 H 19 T 75 to 99 8 12 to 13 I 20 U 100 to 149 J 21 to 22 V 150 to 249 K 23 to 24 w 250 or more L 25 to 26 X rack For absolute accuracy a different cutter would be required for each number of teeth of each pitch. Practically this is not necessary, as the change in the form of the teeth for a small change in number is slight, except in case of gears having few teeth. The form of teeth changes more rapidly for epicycloidal than for involute teeth. For this reason a larger number of cut- ters is required in an epicycloidal set. When a large number of duplicate spur-gears are to be pro- duced, the teeth are usually milled with a hob similar to that OUTLINES OF GEAR-TEETH. 163 shown in Fig. 154a. When the helix angle of the thread of the hob is i = angular velocities of wheel and woim, respectively; 9 = helix angle, or inclination of threads to transverse section of the worm. OL-L. T-^f ~~ This relation usually fixes T and t. Tp = nD] D=Tp + x; or, p = nD+T. The strength of the gear depends upon p } and hence p should be fixed and D made to agree, if the conditions will permit. If, however, A is fixed, Z> is limited, for \ (D + d) = A ; but D and d may have any value consistent with this requirement. The value of p gives the num- ber of threads to the inch on the worm, and hence p, d and t give the helix angle ; for tan (j> = pt+r.d. If the teeth of the wheel are ordinary screw-threads (all trans- verse sections of the wheel being identical in form) upon a cylindrical pitch surface, this pitch cylinder and that of the worm are tangent at a single point, and the teeth have point contact only. That is, the worm always engages with points on the central transverse section of the wheel. The worm-wheel may be made of the form shown in Fig. 154, when it is called a close-fitting wheel. The teeth of this wheel may be drawn by passing a series of planes through the worm, parallel to the f axis and to each other and per- pendicular to the axis of the wheel. Each of these sections of the worm will be a rack section, but they are not all alike. Then make the corresponding sections of the wheel those appropriate for wheels to work with such racks.* This process is tedious, and is seldom required in practice, as by the method of cutting the wheels it is not necessary to lay out the teeth. If a cast worm and wheel are to be made, it is of course necessary to lay out the teeth. * See Unwin's Machine Design, Part I, Art. 234, for a full description of this method of drawing worm-wheel teeth. 184 KINEMATICS OF MACHINERY. 101. Hobbing Worm-wheels. A worm-wheel maybe accurately cut by the following process : Turn up the blank to correspond to the outside of the teeth (Fig. 154)*. Next cut a screw of tool steel to the exact form of the worm, then make a milling-cutter of this tool steel worm by cutting flutes across the threads, and " backing Fig. 154. off " the teeth thus formed for clearance. This is called a " hob " (see Fig. 154a), and it is hardened, tempered, and then used as a milling-cutter. The hob and worm-wheel blank are mounted in the gear-cutting machine, with their axes at right angles but necessarily somewhat farther apart than the desired distance * Figs. 154, 154a, and 1546 are taken from Brown & Sharpe's Treatise on Gears. CAMS Ai\D OTHER DIRECT-CONTACT MECHANISMS. 185 between the axes of the worm and wheel. They are then rotated about their axes with the velocity ratio that the worm and wheel are to have, and the blank is fed toward the hob very slowly until the distance between the axes is the same as the desired distance between the axes of worm and wheel. The wheel is sometimes caused to rotate simply by the driving action of the hob, the teeth of the wheel having been roughly cut or "gashed" with an ordinary milling cutter ; but better results are attained when it is driven positively from the cutter-spindle, with the required velocity ratio, through a suitable train of gearing. It will be seen that the teeth thus formed on the wheel will work correctly with a worm which is an exact reproduction of the hob, except that the cutting-teeth are omitted. The worm and hob may be cut like any screw in a lathe, with a tool which will give the desired form of threads. Fig. 1546 shows a method of cutting an approximate close- fitting worm-wheel with an ordinary gear-cutter, the diameter and section of which corresponds to the worm. If the cutter, as shown in this figure, is fed diagonally across the wheel-blank, a straight (point contact) wheel will be produced. If the cutter is fed radially inward, toward the axis of the wheel, a "drop-cut" worm-wheel is produced. Such a drop-cut wheel resembles a bobbed wheel in form ; but the method does not give a truly close-fitting wheel, such as is obtained by the hobbing process. CHAPTEE VI. LINKWORK. 102. General Scope of Linkwork. The simplest form of a con- strained link-mechanism consists of four links, each pivoted at two points to adjacent links. A link with hut two pivots, and joined to two adjacent members, is called a simple link. If a link has more than two such pivots and is joined directly by them to more than two separate members it is called a compound link. A complete linkwork "chain," as link-mechanisms are some- times called, cannot have less than four links ; for if three links are connected in a closed chain they form a triangle, which is a rigid construction not permitting relative motion between the members. If more than four simple links are connected in a closed chain,, forming a jointed polygon of more than four sides, a given motion of one member does not compel the others to move in a definite manner. Link-mechanisms of more than four members are used ;, but, in these cases, one or more of the members must be a compound link. Linkwork can be used to convert : (a) Continuous rotation into reciprocation (rectilinear or circu- lar) or the reverse. (b) Reciprocation into reciprocation with a constant or a vari- able angular velocity ratio. (c) Continuous rotation into continuous rotation, with a COD- stant or a variable velocity ratio. One or more of the links in a linkwork chain may be replaced by a sliding block or similar piece. Certain of these modified chains are of very great importance in practical machine con- struction. 186 LIXKWORK. 187 103. The Four-link Chain. The general form of the four-link chain is shown by Figs. 50 to 53 and other figures already given. It is now in order to examine the influence of the proportions of the members of the four-link chain upon the motion transmitted. The following notation will be used : The driver will be desig- nated as a ; the follower, b ; the connector, c\ and the stationary link, or frame, d. Fig. 155 shows a mechanism in which circular reciprocation of a produces circular reciprocation of b. The phase shown by the light lines a', b', c' , is a limiting phase, and a can move no farther to the left (left-hand rotation).* The driver (Fig. 155) can move through the arc a'-a-a"-a'" 9 causing the follower to move from V through b to 5", and then return over this path to b'". Within these limits circular recipro- cation can produce circular reciprocation, and either member might be the driver; but, practically, the action is not smooth when the follower is near a dead-point; and if b is the follower, the range of action should be somewhat less than the maximum given. In this figure it will be seen that b-\-c d b. In Fig. 156 the follower reaches an inner dead-point when it is in the position b' ', and a can rotate no farther to the right than the position a'. In this case the driver can vibrate through the angle a f aa"-a rf ', causing the follower to reciprocate from V through b to b" and back to b'". It is evident that d < than a -f- c b; and the driver cannot make a complete rotation unless d is equal to, or greater than, a -\- c b \ or a -\- c < d -f b. It will be noticed that the follower might pass (but cannot be positively driven by a) beyond the position #', in either Fig. 155 or Fig. 156; if this should occur in any way, the motion transmitted would be completely changed. To sum up these two cases, we find that the driver cannot make a complete revolution unless these conditions are present: c a _> d b\ and c + a <^d -\- b. li c a = d b, the driver and follower have simultaneously inner and outer dead-points, respectively, as shown by Fig. 157 (in the phase #', Z>'), and the c a = d : ? \^\d[ * \ J^^\ysN\N^sXX motion of the follower may be towards either b" or b'", as the driver passes this position. If c + a = d + b (Fig. 158) the driver reaches the outer dead-point as the follower reaches the inner dead-point (#' and b', respectively); and the follower may either return to b" or pass on to b'". If c a > d b, and LINKWORK. 189 c-f a < d+6, as in Fig. 159, the motion of the follower is fully constrained, and the driver can make a complete rotation. / g\\\Nv^NN^^^J^\N^N^^cs^^\^^^^^^^^^c J Fig. 159. 104. Continuous Rotation of both Driver and Follower. (See Fig. 160.) If a single four-link chain is used to transmit positive continuous rotation to a follower from a rotating driver there must be no dead-points ; for if the driver, a, reaches a dead-point, b will come to rest; and if b reaches a dead-point, its motion will not be fully constrained, and it will generally lock the driver, preventing complete rotation of the latter. With the proportions of Fig. 160, neither a nor b reaches a dead-point, and either of these members may be used as a driver, compelling the other to rotate continuously, but the velocity ratio will be variable. If rotation of both a and b is to be continuous, they will have simultaneous dead-points if either reaches a dead-point. If c = mn = d-}-b a,a will have an outer dead-point, and b will have an inner dead-point at the same instant. If c = mn' = a -\- b d, a and b will both have inner dead-points at one phase. Hence, for continuous positive rotation of both a and b the following conditions must be fulfilled : c > d + b , and c < a + b d\ hence, d -\-b a < a -\-b d .'. d < a.* The mechanism of Fig. 160 is called a "drag-link"; and it is sometimes used to connect the two arms of a centre-crank or double- throw crank. In this case a and b are equal, and d equals zero in the proper adjustment, that is, the fixed axes of a and b coincide. * If dead-points are permissible (as in the parallel rods of locomotives), other provision is made for insuring that the dead-points shall be passed; theu d may be, and usually is, greater than a. 190 KINEMATICS OF MACHINERY. As long as this condition is maintained, the link forms the equiva- lent of a rigid connection, and as the mechanism is reduced to a three-link chain, the motion transmitted is exactly similar to that of a solid crank. If either axis is shifted, through improper align- ment, springing of the shaft supports, or wear, the motion is trans- mitted from one section of the shaft to the other with a slight variation in their angular velocity ratio during the revolution, and the wrenching action on the shaft is much less than it would be with the usual form of rigid crank-shaft. If a and b are equal the angular velocity ratio is constant when d equals zero, or when d =; c\ for with these proportions the two perpendiculars from the fixed centres to the line of the link (c) are always equal for any phase. The former condition (d = 0) is that of the drag-link as applied to engine-cranks in proper alignment. The second condition (d = c) is one met in the locomotive side-rod connection; but in this case the driver and follower have simulta- neous dead-points, and special means must be resorted to for com- plete constrainment of the follower. The essentials of the locomotive side-rod connection are shown in Fig. 59, in which and 0' correspond to the centres of the connected wheels, A' B' is the side rod (the dotted circles represent the paths of the pins by which the side rod is pivoted to the wheels); OA' and O'E' (radii of the pin circles) are the driver and follower between which it is desired to transmit rotation with a constant velocity ratio; and 00' (the frame) is the fourth link. The full lines of Fig. 59 show a phase at which the driver and follower both lie on the line of centres. As the driver passes its dead-point po- sition, the follower might move in either of the directions indicated by the arrows at B. Means of overcoming this defect in the con- strainment will be shown later. In the mechanism under consideration it is necessary that the four links shall form a parallelogram in all phases; that is, in Fig. 59, A'B' must equal 00' \ and OA' must equal O'B'. When this condition is fulfilled the angular velocity ratio must always be unity, for the perpendiculars from the fixed centres (0, 0') to the con- LINKWORK. 191 nector (A'B') are equal in any phase (see Art. 30). In order to insure continuous rotation of the follower when the dead-points are passed, the simple mechanism of Fig. 59 must be supplemented. The method used on locomotives is shown in Fig. 161. Each axle has two driving-wheels secured to it; the two wheels on either side being coupled by a side rod. The pins on the two 1 1 I 1 1 1 1 1 1 1 I 1 S" \ 1 1 1 __ c V c Fig. 162. wheels of each axle are placed so that they are not in line, but one of these pins is ahead of the other, as shown in Fig. 161 by the angle 0,00,= 0,'0'C,'. This angle is commonly 90, so that when the system is at a dead centre phase on one side, the complementary system on the other side is in the best phase for transmission of motion. Other possible arrangements to secure complete constrainment are shown in Fig. 162. In this case three equal cranks, not neces- sarily having their centres in one straight line, are connected by a rigid member (a compound link) which has a bearing for the pin of each. These bearings must be spaced to agree with the spacing of the fixed centres, and the cranks are always parallel to one an- other. The middle crank (shown with the arrow) should be the driver. 105. Combined Linkwork and Sliding-block Mechanisms. The preceding articles of this chapter have been devoted to the four- link chain, and it was seen that by the mechanism of Fig. 159 a 192 KINEMATICS OF MACHINERY. circular reciprocation of the follower may be imparted by the re- ciprocation or rotation of the driver. Fig. 163 shows a mechanism in which one member of the four-link chain is replaced by a curved block b f sliding in a corresponding circular arc groove in an exten- sion of the fixed link d. It is evident that the motion transmitted to this block is exactly equivalent to the motion which it would re- ceive if it were connected to d by the dotted link b'. Whatever the length of V, it may be replaced by a block and groove, the centre line of which corresponds to the path of the moving end of the link b', without altering the character of the motion. Any other link might be replaced in a similar way by the equivalent slot and block. When any one of the links is very long this substitution of a sliding-block for a link may be convenient, the radius of curva- ture of the slot being equal to the length of the link replaced. If the link is of infinite length the centre line of the slot becomes a straight line, and the motion of the block is then a rectilinear translation. The mechanisms shown in Figs. 69, 70, 71, and 72 represent this modified form of the four-link chain, or what Reuleaux has called the "slider-crank chain." This mechanism is so prominent in practical machine construction that it will be treated in detail. 106. Crank and Connecting-rod. The connection between the piston and crank of the ordinary direct-acting engine, as shown in Fig. 27, is one of the most important examples of the modified four-link chain. In this case the motion of the piston and cross- head relative to the frame is equivalent to that of a link of infinite length. The piston and cross-head, being rigidly connected, are kinematically one piece; though we may be only concerned with LINEWOEK. 193 the motion of the piston, it is often convenient to speak of the mo- tion of the cross-head, which is identical with that of the piston. With the usual arrangement the path of the cross-head is in a line which passes through the crank-centre, and this line will be spoken of as the centre line. This is not a necessary condition, and it is sometimes departed from with a result that will be dis- cussed in a later article. Unless otherwise stated it is to be under- stood that this ordinary arrangement is meant. In the steam-engine the reciprocating piston is the driver and the crank is the follower. In case of an air- compressor or power- pump, the reverse is the case; but the relative motion of the mem- bers is not affected by this relation, as it depends simply upon the proportions of the mechanism. The crank usually has a uniform rotation, or approximately such, and this condition will be assumed in the following discus- sion. It is evident that the cross-head (piston) must come to rest as the crank-pin passes the line of centres (dead-centre positions). Its motion is accelerated as it leaves either extreme position, attain- ing a maximum velocity near the middle of the stroke, followed by retardation (negative acceleration) through the latter part of the stroke. The motion of the reciprocating parts is approxi- mately harmonic, departing from true harmonic motion more as the ratio of connecting-rod length to crank length becomes smaller. The position of the crosshead, c (Fig. 164), for any crank posi- tion C, is obtained graphically by taking a radius equal to the length of the connecting-rod Cc, and, with a centre at the given crank position, cutting the path of the cross-head by an arc of this 194 KINEMATICS OF MACHINERY. radius. In a similar way the crank position corresponding to any cross-head position is found by taking a centre at the given cross- head position and cutting the crank-circle with an arc of the above radius. This last process gives two intersections, one above and one below the line of centres, as it should; for the cross-head passes the same point in its path during the forward and return strokes, both of which are accomplished during a single revolution of the crank. If a series of equidistant cross-head positions are taken, it is evident that the corresponding crank-pin positions will not be equidistant, and vice versa. That is, equal increments of cross- head (piston) motion do not impart equal increments of motion to the crank. In drafting-room practice these graphic methods of finding simultaneous crank and cross-head positions are usually most con- venient; but sometimes it is desirable to use analytical expressions for the relations between the crank and connecting-rod. The more important kinematic relations will be derived, trigo- metrically, using the following notation (see Fig. 164) : Centre of crank-circle = Q. Centre of crank-pin = C. Centre of cross-head pin = c. Length of connecting-rod = I. Length of crank = r. Ratio of connecting-rod to crank (? r) = n. Crank dead-centres = A and B. Corresponding cross-head positions (ends of stroke) = a and b. Mid-stroke cross-head position = q. Mid-crank positions (quarters) = M and M*. Simultaneous cross-head position = m. Crank angle, ahead of A, = B. Corresponding angle of connecting-rod with line of centres = 0. Drop a perpendicular, Ck, from upon the centre line; then Ck = r sin 8 = 1 sin ; QJk = r cos ; clc = V? - (CKf = V? - r* sin a 6. LINKWORK. 195 For any crank angle, 0, the distance from c to Q = ck -{- Qk = Vl* r* sin 2 8 + r cos 0. When C is at the quarter (M or M f ), c is at in, a distance from its mid-position = mq = Qq Qm = I Vl* r 3 = nr r Vn* 1 = r (n V n* 1), a quantity which increases as w de- creases, and equals zero when n infinity. It is seen from the above expression that when the crank has rotated through 90 from A, the cross-head has moved through more than half its stroke; while for the next 90 crank rotation the cross-head moves through less than half its stroke. It follows that, with uniform rotation of the crank, the half-stroke, aq, is made in less time than the half-stroke, qb, this variation decreasing as the connecting-rod length increases. The influence of this angularity of the rod on steam distribution will be seen to be important, when the subject of valve motions is studied. In the illustrations of velocity diagrams (see Art. 41 and Figs. 69 and 76) it was shown that the ratio of the linear velocities of the crank-pin and cross-head is equal to the ratio between the length of the crank and that segment of a perpendicular to the line of centres through the shaft which lies between the centre line and -the line of the connecting-rod, the latter prolonged if necessary. Thus, in Fig. 164, if QC represents the velocity of the crank-pin, to some scale, s = Qt is the velocity of the piston (to the same scale) when the crank is at C. If the crank-pin velocity can not be represented conveniently to this scale, lay oif Qv, along the line of the crank, to represent its velocity, and draw vt r parallel to cC; then Qt f = s f is the required velocity of the piston to the scale assumed ; and the value thus obtained for the piston velocity can be used, as in Fig. 76, for constructing the velocity diagram. Another method of determining the ordinates of the velocity diagram is shown in Fig. 165. With this method, Cv is laid off on the extension of the line of the crank to represent the crank-pin velocity to a convenient scale ; an ordinate is erected at c, and this is cut by drawing the line w' parallel to the connecting-rod; then 1S6 KINEMATICS OF MACHINERY. Fig. 165. cv f gives the velocity of the cross-head for this phase (Art. 40). The linear velocities of Cto c are in the ratio of OC to Oc ; as C and c are two points in the connecting- rod, which at the instant has a motion equivalent to a rotation about the instant centre 0; hence the velocity of C is to that of c as OC is to Oc. The construction of the complete diagram will readily be seen from the figure. The method is the same as that used for the four-link chain in Fig. 77. From Fig. 164 it will be seen that the velocity of the piston is equal to that of the crank-pin when s = r. There are two positions of the piston in each stroke where this condition is fulfilled. The first of these positions is shown in Fig. 166, where cC, produced, passes through M\ hence s = QM = r. This equality of velocities can be seen directly by locating the instant centre 0; for OCc- is similar to QCM at this phase; hence OC '= Oc, and the linear velocity of c = linear velocity of C. The same relation can be shown by resolution of the velocities. k R When C (Fig. 167) coincides with M, s = QO= QM, and the crank-pin and the piston have the same velocity. Between these two positions of equal crank-pin and piston velocity, the piston moves faster than the crank-pin ; for s is greater than r. (See Fig. 169.) The second of these positions of equal velocity is always LINKWORK. 197 at which the crank is perpendicular to the line of centres, and is independent of the ratio of connecting-rod to crank; provided this ratio is greater than unity. The first position is a function of this ratio; and the crank-angle corresponding to this phase is found as follows (Fig. 166) : Let fall Qe perpendicular to CM, then as QMC is isosceles, QMe and QCe are equal triangles, and the angle eQO = eQM = 0, .-. MQC = 20. .-. = 90 - 20. Ck = 1 sin = r sin = r sin (9020) = r cos 2 0, .-. - sin = n sin = cos 20; =1 2 sin 2 0, . '. 2 sin 9 + n sm = 1; dividing by 2 and completing the square : A} 7? ' fl =i + jg- and sin = J- 8 + . j The double sign of the radical may be dropped, for if the minus sign be taken, with any value of n greater than 1, we would get a value for sin numerically greater than 1, which is impossible. Taking the plus sign: sin = i i i/(8 + n 9 ) - n \ As sin 6 = n sin 0, sn = This form is convenient for graphical solution as follows (Fig. 168) : Lay off distance AB = n to a scale of r = 1, and erect a perpen- dicular BC = 2.828 to the same scale. Connect A and C, then the hypothenuse AC = 4 / (2.828) 3 + n\ With A as a centre and AB (= n) as a radius, describe arc BD, cutting AC in D. DC= i/(2.828)' + ra a - n. DG X - = sin 6>; or DC X = r sin = tffc (Fig. 166.) 198 KINEMATICS OF MACHINERY. Between the two positions of the crank at which the crank-pin- velocity equals the velocity of the reciprocating parts, the velocity of the latter is greater than that of the crank, as noted above. These reciprocating parts (piston, piston-rod and cross-head) have very nearly the maximum velocity at the position where the connecting rod and crank form a right angle at (Fig. 169). The true phase for the maximum velocity of the piston is a little later than the above position; but it is difficult to locate this exact position, and with the proportions of crank and connecting-rod used in ordinal y engines (I -~ r = n = from 4 to 6 usually), this error is of no prac- tical account, and the approximation is much more conveniently used. To find the crank position (Fig. 169) at which the crank is perpendicular to connecting-rod, erect Ac' perpendicular to the line of centres at A, equal to the length of the rod, and connect c' with Q. The intersection of c'Q with the crank-circle locates the required position of the crank, C ; for Ac' = Cc\ AQ = CQ\ and in the two triangles AQc' and CQc, the angle A QC= is common. "When two sides and the corresponding angle of two tri- angles are equal the triangles are equal; therefore, as QAc' is a right angle by construction, cCQ is also a right angle, and C is the posi- tion of the crank-pin required. The phase at which the piston has its maximum velocity is of importance in certain problems relating to the mechanics of the steam-engine, for it is the phase at which the acceleration of the reciprocating parts is zero. In high-speed engines the acceleration of the reciprocating parts has a very im- portant bearing upon pressures transmitted from the piston to the crank. L1NKWORK. 199 107. The Eccentric. The eccentric is a modified crank, and all that has been said in the preceding article applies to the eccentric and rod. If the crank-pin be grad- ually enlarged, its throw remaining unchanged, the motion transmitted to a given connecting-rod is un- altered. Fig. 170 shows such a crank-pin enlarged in diameter \ until it includes the shaft, and it gives the familiar eccentric and rod. The throw of the eccentric is the radius of the equivalent crank, QC\ or it equals the distance from the centre of the eccentric to the centre of the shaft about which it turns. The enlargement of the pin increases the friction, although it has no kinematic effect. The eccentric is a useful expedient when a crank of small throw is required which cannot be conveniently located at the end of the shaft, for under such conditions the ordi- nary connecting-rod would "interfere" with the shaft unless a double-throw crank were used, and this latter form would weaken the shaft by cutting into it, besides being a more expensive con- struction. For these reasons the eccentric is very commonly em- ployed for operating the valves of engines, imparting a reciprocat- ing, and nearly harmonic, motion to them. 108, Connecting-rod of Infinite Length. It has been seen that the stroke of the cross-head (Fig. 164) equals the diameter of the crank-pin circle, = 2r ; and that the obliquity of the connecting- rod distorts the cross-head motion from a true harmonic motion, causing the half-stroke farthest from the shaft (at the head end of the cylinder) to be made in less time than is taken by the half- stroke nearest the shaft (the crank end). It was shown in Art. 106 that the displacement of the piston from mid-stroke, when the crank is at either "quarter/ 7 or 6 = 90 (measured, in Fig. 164, by qm) is less as the connecting-rod is made longer, relative to the crank; or as I ~- r = n becomes greater. If the rod were of infinite length, the cross-head would be at the middle of its stroke when the crank is at the quarter (d = 90); 200 KINEMATICS OF MACHINERY. r for it was shown that mq = I Vl* r* ; hence, mq = I I = o, when the length of the rod is infinity. It is, of course, impossible to have a rod of infinite length ; but it was shown in Art. 105 that the cross-head and guides give the equivalent of an infinite length of link as to one member of the four- link chain; and the slotted rod and block of Fig. 171 may be in- troduced as an equivalent to an infinite connecting-rod. That is, this mechanism is the equivalent of the four-link chain with two links of infinite length. With the mechanism of Fig. 171 the crank is acted upon by the slotted rod through the block. The component of the motion of the crank-pin, which is normal to the acting faces of the yoke, equals the motion of the rod. This normal component is seen to equal the motion of the crank-pin multiplied by cos ; and as = 90 0, cos = sin 0, hence the velocity of a piston attached to the slotted rod is equal to v sin 0, when v is the velocity of the crank. The piston velocity is a maximum when sin is a maximum (assum- ing the crank to rotate uniformly) ; or when = 90. At this phase the piston and crank have the same velocity, since sin = 1. This agrees with the statement in Art. 106 that the velocity of the piston always equals that of the crank when = 90. With the finite rod there is another crank position, for a smaller value of 0, at which this equality also exists, and between these two crank posi- tions the piston velocity is greater than that of the crank. As the rod is increased in length, these two positions for equality approach each other, the first one more nearly corresponding to = 90. With the infinite rod the two phases for equality coincide, and the phase for maximum velocity, which in the general case lies between them, also falls at = 90, as seen above. These conditions will be found to harmonize with the general relations deduced above. Fig. 171 indicates the application of this mechanism, as it is some- LINKWORK. 201 times made to steam fire-engines and other steam-pumps. P indi- cates the pump-cylinder and S the steam-cylinder. The crank- shaft carries the fly-wheel. The practical effect of this "rod of infinite length," or the Scotch yoke, as it is frequently called, is to make a more compact mechanism than would be obtained with a finite rod of ordinary length ; for the jdistance between the " glands " of the stuffing- boxes on the two cylinders needs be only equal to the stroke plus the outside width of the slotted yoke, with a small allowance each side for clearance. The slid ing-block is not an essential, kinematically, as the crank-pin could act directly on the faces of the slot ; but, as shown in Art. 28, it is generally desirable, when the conditions will per- mit, to use surface contact instead of line contact, thus distribut- ing the pressure transmitted over a larger area. The sliding of the block in the slotted member produces friction and resultant wear, which is not so easily overcome as in a pin con- nection ; and the ordinary form of connecting-rod is therefore pre- ferred as an engine connection when the utmost compactness is not a leading consideration. 109. Connecting-rod of Length Equal to Crank. If the connect- ing-rod is of a length equal to the throw of the crank, as in Fig. 172, these two members always form an isosceles triangle, with the inter- cept on the centre line between the cross-head and shaft as a base. The distance Qa = r + I = 2r, and 6, the end of the stroke next to the shaft, coincides with Q. In this arrange- d ment, c would be drawn from a to Q during a crank movement AM and the displacement from the centre of stroke, due to angularity of the rod, = A Q = r. If the cross-head comes to rest at Q when C reaches M, with any farther motion of the crank the con- 202 KINEMATICS OF MACHINERY. necting-rod would simply rotate around Q with the crank. If the cross-head continues to move in the line aQ, produced beyond ft the crank movement MB would drive the cross-head to d, a distance from Q = 2r, and the total stroke of the cross-head would = 4r. The inertia of the cross-head as it approaches Q would tend to pro- duce this effect ; but such a motion can be made positive by the mechanism shown by the extension of cC to c'. In this form the rigid rod cc r ( = 2r) is pivoted at its centre to the crank-pin, and cross-heads at the ends of the rod move in guides at right angles to each other which intersect in Q. When C is at A, c is at a, and c' is at Q. As Amoves to M, c moves to Q, and c' moves to a f ; then, as the motion of C continues to the position B, c passes Q, moving to d, and c' returns to Q. As (7 passes B and moves to M', c' passes from Q to d', and c returns to Q. During the completion of the crank revolution, C moves from M' to A , c moves from Q to , and c' returns to Q, completing the cycle. At any phase the distance of c' from Q corresponds to Qt, of Fig. 164, and hence is proportional to the velocity of c\ likewise, Qc is proportional to the velocity of c' at any phase. In this mech- anism there is a transverse stress, as well as tension or compression on the rod cc'. 110. Path of Cross-head Passing Outside of Shaft-centre. If the line of cross-head motion, gh (Fig. 173), does not pass through Q, the motion is modified as follows : To find the ends of stroke a and b : first, take a radius = I -f- y i with a centre at Q, and cut gli in a ; second take a radius = I r, with the same centre, and cut gh in b ; the required points are a LINK WORK. 203 and b, and the corresponding crank positions are QA and QB. The stroke from a to b is made while the crank moves through the arc AMB ; and the return stroke takes place as C moves through the arc BM'A. If the crank motion is uniform, the forward and return strokes are made in unequal times, and this mechanism gives one form of "quick-return motion." If it is required to de- sign such a quick-return motion, the relative times of forward and return strokes being given: draw the crank circle and divide its circumference into two arcs having the required ratio, AMP., BM'A. Extend the radii through these points of division A and B, in directions QA and BQ\ then lay off from Q on the exten- sion of QA, I 4- r, locating a ; and on BQ lay off I r from Q, giving 6 ; a and 6 are the ends of the stroke, and ab is the line of cross-head motion. In the Westinghouse engine the above construction is applied ; that is, the line of piston travel passes to one side of the shaft- centre. Two cylinders are placed side by side, with connecting-rods acting on cranks which are opposite each other (180 apart). This engine is single-acting, steam acting on each piston only during its downward stroke ; therefore, by giving the quick return to the up- ward stroke, one piston makes its exhaust-stroke and takes steam again before the other piston has quite completed its "working" stroke ; thus, there is no period at which the rotative effort is abso- lutely zero. Furthermore, the greatest angularity of the connect- ing-rod occurs on the exhaust-stroke, and for a given length of con- necting-rod, the maximum obliquity of action is reduced for the stroke during which steam-pressure is acting on the piston. Or, to state the case somewhat differently, the length of the rod can be re- duced lor a given maximum obliquity during the period of heavy pressure, thus permitting a more compact construction. 111. Motion of a Point in the Connecting-rod between the Cross-head and Crank. In certain valve-mechanisms, motion is taken from some point in a connecting-rod (or eccentric rod) other than either of the pin-centres previously considered. Let P (Fig. 174) be such a point, the motion of which it is desired to find. 204 KINEMATICS OF MACHINERY. Find the instant centre for any chosen phase of the rod Cc. All points of the rod, at the instant, rotate about with the same angular velocity, and with linear velocities proportional to their radii. Hence, the linear velocity of P is to that of C as OP is to OC. The direction of the motion of P is perpen- dicular to OP, as indicated by Pv 3 . A similar method can be used if the point P lies beyond either the crank or cross-head in an extension of the connecting-rod. This problem can be solved by the resolution and composition of relative velocities also, but not so readily. 112. Inversion of Crank and Connecting-rod Chain. It was shown in Art. 39 that a kinematic chain may have the appearance of entirely different mechanisms when different members of it are held stationary. Thus, Figs. 69, 70, 71, and 72 show the four possible inversions of the crank and connecting-rod chain. The case of Fig. 69 has been treated in preceding articles of this chapter. Fig. 70 represents the condition when the former crank is made the fixed member; this case is next in practical importance to the ordinary crank and connecting-rod mechanism. This form may be used to secure a variable angular velocity of a continuously rotating follower from a uniformly rotating driver. It somewhat resembles the drag-link in its action. In conjunction with another linkage this mechanism is frequently used to produce a slow advance and a quick return of the cutter-bar of a shaping-machine. The condition shown in Fig. 71 is, as already pointed out, the mechanism of the oscillating steam-engine. The case of Fig. 72 has comparatively little practical application. Any of these can be readily analyzed by the instant centre method. The form in which a is fixed (Fig. 70), will be treated in some detail, on account of its extended practical use ; the others will not be taken up as special forms. Fig. 175 shows a crank a which rotates about and is pivoted to a sliding block by the pin P. This block fits a slot in the arm LINKWORK. 205 b, which rotates about 0'. The stationary member d supports the fixed centres and 0'. The point P rotates in the circle AB ; hence, its motion at any instant is perpendicular to the radius PO (the centre line of the crank a). The velocity, which is usually uniform but not necessarily so, is designated by v\. We may con- sider the point P to act upon the centre line (pitch line) of the slotted member, as the block does not affect the kinematic problem. The point in a which lies at P has the velocity Pv^ and the point in the slotted bar 5, which is also at P for the instant, has the velocity Pv y As these two velocities have equal components along the common normal to the contact surfaces, the normal component of Pv l = Pv.>. As a point in a, P is fixed at the distance OP from the centre 0. As a point in , it travels back and forth along the pitch line of the slot, its distance from 0', or its effective radius, varying from O'A to 0' B as the driver moves from A to B. During the next half-revolution of the driver (B to A) the effective radius of the follower decreases from O'B to O'A, thus completing the cycle. Only the component of the motion of the driving-point which is normal to O'P can impart rotation to the follower. The velocity of this component is represented by v 2 v t cos (in which expression < is equal to the angle OPO f ) ; because Vi and v 2 are perpendicular to 206 KINEMATICS OF MACHINERY. OP and O'P, respectively. If a circle be drawn on 00' as a diam- eter, the intercept, Pn, of the centre line of the follower (extended through 0' if necessary), which lies between P and this circle is to v 9 as the constant radius of the driver is to v r Or, : v t : i\ : : PO : Pn\ for, as OnO' is an angle subtended in a semicircle, On is perpendicular to O'P, hence Pn = PO cos 0. The velocity of the driver may be represented to a scale which will make it equal to PO) when Pn becomes the velocity v v If this velocity scale is not convenient, v l may be laid off from P towards 0, as Pk, and a line kl drawn perpendicular to O'P will give PI = # a , to this latter scale. 113. Quick-return Motions. If (Fig. 175) a sliding block, c y travels in the path ef, which passes through 0', and is connected to a point C in an extension of the slotted follower by the rod Cc 9 it will reach one end of its stroke when the driving-point P is at E, and this block c reaches the other end of its stroke when P is at F. While P is moving through the arc FEE, c moves from e tof; while P moves through the arc EAF, c makes its return stroke from f to e. Now if the driver rotates uniformly the times of these forward and return strokes are in the ratio of the arcs FEE to EAF. This is, in principle, the Whitworth quick-return mechanism, as it is frequently applied to shapers. The slow stroke is used for the cutting stroke of the tool, while the return stroke is made more rapidly, thus economizing time and increasing the capacity of the machine, In designing such a mechanism the circle in which P rotates may be drawn with as a centre; then divide its circumference by E and F into two arcs having the ratio desired for the times of the forward and return strokes. Draw a line through EF, ex- tended to one side, and the path of c lies in this line. Drop a perpendicular from upon EF and its foot will locate 0', the fixed centre for the slotted arm. Take C at a distance from 0', which will give the required length of stroke, and choose a suitable length for the connecting-rod Cc, In practice C is a pin which can be set at different distances LINKWORK. 207 along a radius to 0', so that the length of stroke of c can be varied to suit the work. The pin C might be placed on the same side of 0' as the slot ; but it is usually more convenient to locate it as in Fig. 175. The velocity of C is to v. A as O'C is to O'P, since these are the Velocities of two points in one piece which rotates about 0' . The motion of C is perpendicular to O'C, as shown by Cv v To find its velocity lay off 27, (found as above) and draw the line v^0 f v 9 cutting the perpendicular to O'C at v 9 , and giving Cv t as the velocity sought. To find the velocity of c, erect at c a per- pendicular to its path ; lay off Cv 3 ' = Cv 3 on the extension of O'C, and draw a line v/v/ parallel to the rod Cc ; v is an ordinate of the velocity diagram of c. The student should complete this diagram for both strokes, by the method indicated. When the location of the instant centre O ac can be determined, the linear velocity of c corresponding to the given linear velocity of P may be determined directly by the general method outlined at the end of Art. 40. The practical construction of the Whitworth quick-return motion is shown in Fig. 176, in which the letters correspond to those of Fig. 175. The pin P is attached to a gear which rotates about 0, Fig. 176. Fig. 177. the centre of a large fixed stud. The centre 0' is a pin secured in the fixed stud, and the slotted member rotates about this centre 0'. The pin C can be clamped at different points along its slot to secure corresponding lengths of stroke of c. 208 KINEMATICS OF MACHINERY. Another quick-return mechanism, also much used for shapers, is indicated by Fig. 177. The slotted bar is pivoted to the frame at 0', and is driven by the crank pin P, which rotates about ay in the preceding case. The slotted bar vibrates between the positions O'e and O'f, reaching an end of its stroke when its centre line ip tangent to the crank-pin circle; or when the crank is at either E or F. It will be seen that the driver passes over the arc FEE for the forward stroke, and through the arc EAFior the return stroke. The former arc is greater than the latter; hence the times of tlie strokes are in the ratio of these arcs, if the driver rotates uni- formly. The normal component only (v 2 ) of the crank-pin velocity (;,} transmits motion to the follower; and v a = v l cos 0. in which is the angle OPO'. If a semicircle be drawn on 00' as a diameter, cutting O'P at n, Pn = OP cos ; hence v l : v y :: OP : Pn; or if OP represents the velocity of the crank-pin, Pn represents the velocity of the driven point of the slotted arm to tne same scale. The upper end of the slotted arm drives the cutter-bar of the shaper as indicated, through a pin, C, which is between two parallel projections attached to the cutter-bar. The velocity of C is i\ , perpendicular to O'C, and v a : v 2 : : O'C : O'P. To find this veloc- ity draw a line through O'v 2 extended till it cuts Cv 3 in v s . The- motion of the tool, v t , is the horizontal component of # 8 . It dif- fers little from v a ; but can be easily found by the graphical con- struction shown. The fundamental portion of this mechanism is a modified form of the one used in the Whitworth motion ; the only difference be- ing that 0', in this case, lies outside of the orank-pin circle; while in the other case it lies inside this circle. This difference in the- proportions causes the slotted bar to vibrate through a definite angle in one case while it rotates continuously in the other case. The methods of connection with the ram of the shaper are quite: different in these two cases, as is also the means of changing the length of the stroke. In the second form this change is made by changing the length of the driving crank-arm, means being L1NKWORK. 209 provided for moving the pin nearer to, or farther from, its cen- tre, 0. The adjustment can usually be made without stopping the machine. With the Whitworth device, the relative time of forward and return strokes is not varied by changing the length of stroke. With the second mechanism the ratio between the times of the forward and return strokes is greatest with long strokes. The angle through which the driver passes for the forward stroke is 180 + Q, where is the angle of vibration of the slotted bar; and during the return stroke the driver passes through 180 - 6. The sine of -J 6 = OP -f- 00', and as 00' is a constant, varies with changes of OP. To design this machine, decide upon the ratio of the times to be occupied in the forward and return strokes for some particular length of stroke. Draw the crank-circle for this particular stroke (Fig. 177) and divide it into the arcs FBEaud. EAF, having this ratio. Draw tangents to this circle at B and F, and their intersec- tion locates 0'. The velocity diagram is readily constructed for both strokes by finding the velocity = ?' 4 for various positions of the ram, by the method given. This diagram should be drawn as an exercise. The crank and connecting-rod when arranged so that the centre line passes outside of the crank-circle centre (as discussed in Art. 110), may be used for a quick return. Elliptical gears (see Art. 46) are also used for quick-return mechanisms. 114. Bell-cranks. Fig. 178 shows the method of designing a bent lever, or bell-crank, to transmit motion from line OA to line OB, with linear velocity ratio = m -f- n. Lay off Oa = m on B, and Ob = n on OA ; complete the parallelogram Obqa by drawing aa and bb parallel to OA and OB,, respectively, and inter- secting at q. Through and q draw a line. Any centre, as Q, on this line may be taken as the bell-crank centre. From Q, drop perpen- diculars QP and Qp on OA and OB', these are centre lines of the 210 KINEMATICS OF MACHINERY. arms of the bell-crank. The angular motion of the arms should be the same on each side of QP and Qp. It will be seen that any motion, either side of the central posi- tion, will shorten the effective arms. To reduce the deviation of the connected points to a minimum, the lengths R and r should be greater than QP and Qp, respectively, by one-half the versed sine of the angle described each side of the central position multiplied by the respective radii. 115. The Beam. Fig. 179 indicates the beam of an engine ; CO is the line of piston motion; QF= distance of beam centre from this line, = d ; AB stroke ; A E = J- stroke = s ; DF should = EF, to minimize the angularity of the connecting-rod. The beam length to fulfil this requirement is found thus: QD = QA = d ~ EF\ but QA t = , d--^* ->-' Fig. 179. ~AE~\ EF?-\-s\ .'. d* + 2d .EF + EF* =d* - 2d . EF + ~EF* -f s\ .-. 4d . EF=s\ .-. EF = DF = *" -r- 4=d ; .-. the length of the beam = d + DF = d + TJ- 116. Rapid Change in Angular Velocity of the Follower. A means of imparting rapidly changing angular velocity to a follower by the use of the linkwork is shown by Fig. 180. As C reaches the vari- ous positions in its path marked 0, 1, 2, 3, etc., c lies at 0', 1', 2', 3', etc., respectively, in its path. The prin- ciple of this arrangement is of fre- quent use. It is applied, as indicated in the figure, in the " wrist-plate " of the Corliss valve-gear, to give a rapid opening of the valves (arid quick closing of the exhaust- valves) with a small motion when they are closed. In this applica- tion the rotating (vibrating) valve is attached to the arm 0' c, and $r LINKWORK. 211 the mechanism is so proportioned that the required port opening is given quickly to a valve at one end of the cylinder, while the valve- arm at the other end moves but little during this period. In general, the motion of the follower c is small compared to a given motion of the driver G as the angle 0' c C approaches a right angle and the angle OCc approaches or 180. On the other hand, the relative motion of c to C is great as the angle 0' c C ap- proaches or 180 and the angle OCc approaches 90. 117. Straight-line Motion. A large number of linkages have been devised to make a point move in a straight line independently of any planed guides. The term Parallel Motions is usually given to such mechanisms, but straight-line motions is a more appropriate term. Fig. 181 shows what is known as Watt's par- allel motion. R and r are arms cen- tred at Q and q ; Aa is a link con- necting the free ends of R and r, and P is a point in Aa which traces an approximately straight line, within certain limits of the motion. If R moves from its central posi- tion, A is drawn to the right, while the accompanying motion of r carries a to the left. The path of P is a function of both of these motions and the result is that P, if properly located in Aa, moves very nearly in a straight line, pro- vided the angular motion of R and r does not exceed about 20. The complete path of P is the " figure 8 " shaped curve Pmn. If R = r, AP = aP. In general, the segments AP and aP are inversely as the length of the adjacent arms. Watt used this mechanism to guide the piston-rod in place of the slides now generally employed ; but the principal application of " parallel motions " at present is on steam-engine indicator pencil motions. The Richards indicator, the earliest of the modern type, has the Watt mechanism. The Tabor indicator has a motion in which' a curved guide is 212 KINEMATICS OF MACHINERY. m used ; it is, therefore, of a different type from the pure linkwork mechanisms usually classed as par- allel motions. Fig. 182 indicates this pencil movement. It is desired that the pencil point P shall move in a right line, mm. It is evident that the curved guide nn can be given such a form that this will occur, and this curve can be found by moving P along mm, tracing the curve nn by the point p. Hav- ing found nn, a circular arc may be found which agrees closely with it, within the range of motion ; and if the centre of this arc be at d, a link dp can be substituted for the curved guide nn. An arrangement similar to this substitution is used on the Thomp- son indicator. If a moved in a straight line, instead of in the arc, yy ; if p were at the centre of Pa ; and if dp = Pp = pa, the mechanism would be the same as that shown in Fig. 172, and the result would be an exact straight-line motion ; requiring a straight guide, however, for the point a. The Crosby indicator has a pencil mechanism similar to that of Fig. 183. If P be moved in a straight line mm, p (a point in the link be) traces a curve ; the bridle link dp is one that will give an arc most nearly approaching this curve. Fig 184. Peaucellier's straight-line motion is exact, and it is a pure link- work. It is shown in Fig. 184. Two equal links a and b have a fixed centre, Q. The links d, e, f, g are equal ; and c, with a fixed centre at q, equals the distance Qq. P is constrained to move in the straight line mm. LINKWORK. 213 118. Pantographs. There are various linkages in which if one point ia made to travel in any path, some other point will be con- strained to describe a similar path, enlarged or reduced. Fig. 185 shows one such arrangement in which Pa = bQ; and aQ = Pb. These links form a parallelogram which has a fixed centre at Q. A bar cd is attached to aQ and Pb parallel to Pa, and the point p, in cd, which lies on the line connecting P and Q, will move in a path similar to that traced by P. Suppose P to move to P', then p moves to p', and from similar tri- angles, QP:Qp::Qa: Qc; also QP' -and Qc = Qc' Q ^ Fig. 185 Qp'::Qa':Qc'',l>utQa=Qa', QP : Qp :: QP' : Qp', hence the distance of p from Q is proportional to the distance of P from Q. As p always lies in the line QP (because QaP and Qcp are similar triangles), the angular motion of p about Q is equal to the angular motion of P about Q. Any path of P is determined by its angular motion about Q and its radius vector to Q as a pole; as the angular motion of P and of p about Q are seen to be equal for any motion of either of these points, and as the radius vector of p bears a constant ratio to that of Pj the path of p is similar to that of P. A form of pantograph, called the "lazy-tongs," is shown in Fig. 186. It is frequently used to reduce the piston motion of an Fig. 186. Fig. 187. engine, in using the indicator. P is attached to the cross-head, and the indicator cord is attached at p. The practical objection to this contrivance is the great number of joints, and consequent liability to lost motion from wear. 214 KINEMATICS OF MACHINERY. Fig. 187 shows another pantograph for the same use. P is at- tached to the cross-head, and the cord is attached at p as before. With either of these arrangements the point p must lie in the line connecting P and Q, and the cord must be led off parallel to the cross-head motion. Watt combined the pantograph with his straight-line motion so* that the piston-rod, air-pump rod, and feed-pump rod were all guided in straight lines by means of one combination of links. 119. Hooke's Coupling, or the universal joint, is used for con- necting two shafts which intersect. It is equivalent to what Reu- leaux calls the four-link conic chain that is, to a four-link chain, in which the pivots are not parallel as in the ordinary case already treated, but their axes lie in radii of a sphere. Every point moves in the surface of a sphere, instead of in a plane. In its typical form (Fig. 188), each shaft has a forked end, and the two forks are united by an equal armed cross aft, cd, or its equivalent. The Fig. 188. Fig 189. shafts Mm and Nn and the arms of the cross (the axes of the pivots) intersect in a common point 0. If only one half of each fork be considered, as mb of Mm and nd of Nn, and these are assumed to be connected by the spherical link bd equal to the fixed -distance between the two adjacent points of the cross, a four-Jink conic chain is produced in which the axes of all the turning pairs inter- sect in 0. With this arrangement the fork could be omitted, and LINK WORK. 215 we would have the kinematic equivalent of the original mechan- ism. The driver Mm and the follower Nn make complete revolutions in the same time; but the velocity ratio is not constant throughout the revolution. If a plane of projection be taken perpendicular to the axis of Mm, the path of a and b will be the circle ABCD in Fig. 189. If the angle between th'e shafts is ft, the path of c and d will be a circle which is projected on the ellipse AB'CD', in which OB' = OD' = OB cos ft = OA cos ft. If one of the arms of the driver is at A an arm of the follower will be at B' '; and if the driver-arm moves through the angle 6 to P the following arm will move to Q-, OQ will be perpendicular to OP* hence B'OQ = B. But B'OQ is the projection of the real angle described by the follower. Qn is the real component of the follow- er's motion in the direction parallel to AC, which line is the inter- section of the planes of the driver's and follower's paths. The true angle 0, described by the follower, while the driver describes the angle 6, can be found thus: draw QR parallel to OB so that Rm = Qn, then OR equals the radius of the follower, and BOR = = the true angle in plane AB' CD' which is projected as B' OQ = 0. Now tan = Rm -f- Om, and tan 6 = Qn ~- On ; but Qn = Rm, tan 6 _0m _ OB _ 1 "' tan0 "~ ~0n ~ ~OB' ~~ cos~fi' A tan = cos /3 tan B (1) The angular velocity ratio of follower to driver is therefore found as follows by differentiation of Eq. (1), remembering that ft is a constant in this equation: a' __ d(f) __ cos ft sec* B _ cos ft sec* B a ~~ ~dB = aec 1 ~ 1 + tan* 0> ' ' ' V 2 ) 216 KINEMATICS OF MACHINERY. Eliminating by means of (1) cos ft a' cos sec 2 cos 9 9 a 1 + cos' /ftan 2 cos 2 6 -f sin 2 6 cos a ft COS 2 cos /? cos ft (3) cos 2 + sin 2 (1 - sin a ft) 1 - sin 2 6 sin* ft' By a similar process could be eliminated, giving of_ _ 1 cos' sin 2 /? a cos ft It is seen from (3) that #'-*- or is a minimum when sin 0=0, or when = 0, TT, etc., which corresponds to a value of = 0, TT, etc. The same thing is seen from (4), which gives a minimum value of a' ~ a when cos = 1, or = 0, TT, etc. Also, a' - a is a maximum when sin 01, or cos 0=0, corresponding to = 90, JTT, etc. ; = 90, $7r, etc. To summarize the foregoing, the follower has a minimum angu- lar velocity, if the driver has a uniform velocity, when the driving- arm is at A or C and the following arm is at B f or D'. The fol- lower has a maximum angular velocity when the driving-arm is at B or D and the following arm is at A or C. By using a double joint the variation of angular velocity be- tween driver and follower can be entirely avoided. To do this an intermediate shaft is placed between the two main shafts, making the same angle, ft, with each. The two forks of this intermediate shaft must be parallel. If the first shaft rotates uniformly, the angular velocity of the intermediate shaft will vary according to the law deduced above. This variation is exactly the same as if the last shaft rotated uniformly, driving the intermediate shaft ; therefore, as uniform motion of either the first or the last shaft imparts the same variable motion to the intermediate shaft, uniform motion of either of these shafts will impart (through the intermediate shaft) uniform motion to the other. This is the combination used in the feed-rod of the Brown & Sharpe milling machines and elsewhere. LIXKWORK. 217 120. Ratchets. The ratchet-wheel and pawl (Fig. 190) resemble both the direct-contact motions and linkwork. The driving-pawl CP acts by direct contact ; but dur- , IL . m > ing driving the action is similar to that of a four-link chain, consisting of QC, qP, PC, and the fixed link Qq. Such mechanisms are sometimes termed intermittdfti linkwork The two centres Q and q may co- incide, the pawl-lever vibrating about the axis of the wheel. In this case there is no relative motion between the members during the forward (working) stroke. The supplementary pawl,c/?, has a fixed centre, c, and its object is to prevent the backward motion of the wheel when CP is not driving. If pn is the common normal to the end of cp and the tooth with which it engages, there is no danger of the pawl becoming dis- engaged under the reaction of the tooth upon it ; for the centres c and q are on the opposite sides of the line of action, and the ten- dency is for the wheel to run backward (right-handed rotation) and for the pawl to turn with a left-handed rotation, which only forces them together. If the direction of the common normal is pn' , the centres both lie on the same side of the line of action, when the tendency is for both pawl and wheel to rotate in the right-handed direction, and the pawl would be forced out of contact, unless held by friction. In a similar way the normal Pm of the driving-pawl CP and the tooth on which it acts should pass between C and Q- The pawl cp only prevents backward motion of the wheel after Ihe wheel has moved back far enough to come in contact with the pawl. The amount of backward motion possible may vary from zero to the pitch of the teeth. This action could be limited by making the teeth small ; but this would weaken the teeth, and the expedient is sometimes adopted of placing several pawls side by side on the pin, c, the pawls being of different lengths. With this arrangement the maximum backward motion may be reduced to the pitch divided by the number of pawls provided. 218 KINEMATICS OF MACHINERY. Sometimes, for feed-motions, etc., the pawl and wheel are made as shown in Fig. 191. This pawl can be reversed for driving in the opposite direction. Fig. 192 shows a double-acting ratchet by which both strokes of the lever drive the wheel. The locking-pawl may be omitted in this case. Fig. 192 Fig. 191. Frictional pawls (Fig. 193) are sometimes used, in which case the wheel is made without teeth. The pawl grips the wheel by fric- tion during one stroke and releases it on the return stroke. These have the advantage of being noiseless, and the angular motion of the wheel for each stroke is not restricted to some multiple of the Fig. 193. Fig. 194. arc between two teeth, but the driving is not positive. Another frictional pawl with a fixed centre at c can be used to prevent "overhauling" of the wheel. The letters of Fig. 193 correspond with those of Fig. 190. LINKWORK. 219 In the form of frictional ratchet shown by Fig. 194, the wheel is surrounded by a ring, which can be vibrated about the axis of the wheel. One of these members (either) has teeth of the form shown; and in the depressions formed by the teeth, rolls, or balls, are placed. Motion of the driver in one direction causes these rolls to bind the follower, while they release it on the return. Positive " silent" ratchets have been made with a special device for holding the pawl clear of the teeth on the return stroke. The forms of ratchets shown by Figs. 190 to 195, and numerous modifications of them, are suitable for many cases requiring the conversion of a reciprocating action into an intermittent rotation. They are especially convenient in feed-mechanisms when the vibra-, tions of the driver are not too rapid. At high speeds the shock between the pawl and tooth, as the driving begins, may be objec- tionable, and the inertia of the wheel is liable to make it move farther than desired, or to cause " overtravel." This last tendency prevents the employment of the ordinary ratchet when, as in revo- lution registers or continuous counters, a definite motion of the follower must be insured. A device for such purposes is shown by Fig. 195. Fig. 195. The lever to which the pawl is attached has a tooth or beak so formed and placed that overtravel is impossible. When the pawl first acts on a pin, another pin passes close to the point of this beak; the beak then follows in behind this pin, crossing the path of pin-motions, and thus limiting the motion of the next pin. The outline ab should be a circular arc with Q as a centre, so that 220 KINEMATICS OF MACHINERY. the pin which it stops will rest against it during the return stroke of the driver. Another device much used for counters is shown by Fig. 196. The "star" wheel is driven through half of its pitch arc by the action of the projection b upon the tooth a during one stroke of the driver, and V acts upon the opposite tooth a' during the return stroke, thus moving the wheel an equal distance in the same direction. It will be seen that the motion of the wheel for a double stroke of the driver is equal to the angle between two teeth, and if the wheel has ten teeth, it will make a complete rotation for ten double strokes of the driver. 121. Escapements. The mechanism of Fig. 196 resembles the escapements used to control the motion of a train of clockwork, and it might, with slight modification, be used for such a purpose. If the wheel A is acted upon by a spring or weight which tends to rotate it continuously in the left-hand direction, this wheel would tend to produce reciprocation of the piece B. If B is a pendulum, it has a normal period of vibration corresponding to its length, and if the pendulum is so heavy that the rota- tive effort of A cannot alter this period, the pendulum in swinging will control the motion of the wheel. The tendency of the wheel to produce vibration of the pendulum may be made sufficient to overcome the frictional resistance which acts to stop the pendulum, and thus the ampli- tude of the vibrations is maintained. Other outlines of teeth for the wheel and pendu- lum are better, practically, and one common form is shown in Fig. 197. The teeth of the piece which vibrates with the pendu- lum are called pallets. Many modifications of the escapement have been devised to meet special requirements. In watches and other portable time- pieces a balance-wheel is used instead of the pendulum to regu- late the period of the vibrating member, but all are similar in their general action. CHAPTER VII. WRAPPING-CONNECTORS. BELTS, ROPES, AND CHAINS. 122. Belts, Hopes, Chains, etc. Flexible members are frequently used for transmission of motion between two pieces provided with properly formed surfaces upon which the connector wraps or un- wraps as the action takes place. The connector may be a flat belt or band, a rope, or a chain composed of jointed members each one of which :s itself rigid. The great majority of the practical applications in which bands are used for transmitting motion are those in which the velocity ratio is constant. Figs. 198 and 199 show pairs of wheels of T M Fig. 198. T circular section connected by bands. These evidently fulfil the condition of constant velocity ratio, for the segments (QF and qF) into which the line of the band cuts the line of centres (or its ex- tension) are constant; also, the perpendiculars (R and r) let fall from the fixed centres upon the line of the band are constant (see Art. 31). In case exact motion through only part of a revolution (or at most through a limited number of revolutions) is to be trans- mitted, the ends of the bands may be fastened to the wheels. The action, with this condition, is positive, provided the direction of the 221 222 KINEMATICS OF MACHINERY. motion is such that the band is always kept in tension. Thus in Figs. 55 and 56, the piece which rotates about must be the driver, while the one rotating about 0' is the follower, for transmission of motion in the directions indicated. The motions of two pieces con- nected in this way are necessarily of a reciprocating character, for when the band is all unwrapped from the follower the mechanism comes to rest, and any farther motion most be in the reverse direc- tion. Such motion can only be secured when the former follower becomes the driver. An example similar to this case is seen in a hoisting-drum which pulls a car up an incline. While hoisting, the drum is the driver relative to the car; but in lowering, the action of gravity on the car causes it to turn the drum backward. In most common applications of flexible connectors the ends of the band are joined together and not fastened to the wheels, and the motion is continuous; this is commonly called an endless band. In these cases the motion is not positive, as the bands may slip (except when chains are used), but usually very exact motion is not essential where these devices are employed. It follows from the demonstration of Art. 31, referred to above, that if wheels of circular transverse sections are connected by a flex- ible band their angular velocities are inversely as their radii. The effective radii are greater than the radii of the wheels by about one half the thickness of the band; but generally the correc- tion for thickness of the band need not be made with thin flat belts. The exact effective diameter is the length of band that will just encircle the wheel divided by TT. When a round cord or rope or a chain is used this affords a convenient way to get the effective or pitch diameter. Wheels for such ropes or cords have grooves cut in the rims to keep the band on the " sheave." For hemp or cotton rope transmissions the grooves are given such a form that the rope is wedged into them slightly, thus increasing the tractive force. With wire rope this wedging is inadmissible, as it would injure the rope, and the bottom of the groove has a somewhat larger radius than that of the rope. The bottom of the groove in wire- rope sheaves WRAPPING- CONNECTORS. 223 is usually lined with rubber, leather, wood, or some such material, to increase the adhesion and save wear of the cable. Fig. 200 shows the section of the rim of a sheave as commonly designed for hemp or other fibrous ropes. Fig. 201 shows a section of rim employed with wire ropes. If sup- porting sheaves or tighteners are required in a hemp-rope transmission the groove is made similar to that shown for wire rope but without the soft lining; for as these sheaves are not intended to transmit ' 9 ' 200- Fig. 201. power, the increased adhesion due to the wedging of the rope is not required, and unnecessary wear of the rope is avoided by making the groove larger. With chain-bands the wheels, called " sprocket " wheels, have projections fitting the links of the chain (more or less closely) to prevent slipping. With flat belts the pulleys have flat or nearly flat faces. The forms of sprocket-wheels and of the faces of pulleys for flat belts will be treated in later articles. 123. Shifting Belts. If a pressure is brought to bear upon the advancing side of a belt (Fig. 202) the belt is deflected in the direc- tion of this force. As the belt passes upon the pulley, each successive portion of it passes upon a part of the pulley farther from the side from which the shifting J force acts, and the belt takes up a new position, as shown by the dotted lines. A pressure upon the receding side of the belt does not have this effect, -[ | i unless the force is great enough to overcome the T | * adhesion of the belt and pull it over bodily. It must be remembered, however, that the receding side of ^ the belt relative to one pulley is the advancing side Fig. 202. relative to the other pulley. 124. Crowning Pulleys. If a flat belt is placed upon a cone (Fig. 203) the edge nearest the base of the cone is stretched more than the other parts, and the belt tends to take the position shown by the dotted line. The effect of this is to shift the belt towards 224 KINEMATICS OF MACHINERY. Fig. 203. the base of the cone, as the advancing portion of the belt runs on nearer to the base. If a similar cone is so placed that its base coincides with that of the first one, when the centre line of the belt has mounted to the common base it will remain in that position, as any displacement from such position -o would bring about the condition tending to return it. Pulleys are, therefore, usually made " crowning" to keep the belt on the centre. If the pulley is crowned about -J inch for each foot in width, the belt will ordinarily evince no tendency to run off, provi- ded the axes of the connecting shafts are parallel. If the shafts are out of alignment, the belt tends to run toward the edges at which the belt is tightest, unless the shafts are very much " out." It is frequently desirable to stop the driven shaft without stop- ping the driver, and a common method of doing this is by means of " tight-and-loose " pulleys. Two pulleys are placed side by side on the driven shaft, one of which is fastened to the shaft, while the other is free to rotate relative to this shaft, but is prevented by collars from moving axially. The hub of the tight pulley usually serves as one of these collars, and the rims should not quite touch. A pulley is secured on the driving-shaft, opposite the tight-and-loose pulley, having a width (or face) equal to the combined width of both of the latter. A belt of about the width of either of the single pulleys connects one of them and the wide-faced driving pulley. When this belt is on the tight pulley, the follower is driven; but if it is shifted to the loose pulley the follower will stop, although the belt continues to run. The belt is easily shifted by applying a lateral pressure to the advancing edge, as explained in Art. 123. It is usual with tight-and-loose pulleys to make them both crown- ing, so that the belt will remain upon either when it is shifted; but to facilitate shifting the wide driving pulley is generally made with a straight face (cylindrical surface). 126. Length Of Belt The length of belt is usually determined WRAPPING- CONNECTORS. 225 by direct measurement if the pulleys are constructed and mounted, or by measuring a drawing if the work is not built and erected. This length may be calculated for either an open or a crossed belt (Figs. 198 and 199, respectively). This calculation is seldom of practical value simply for the determination of the length, but it plays an important part in the correct design of "stepped-cone pulleys," such as are used on the countershafts and spindles of lathes and other machines for securing changes of speed. The importance of this calculation will appear from the discussion of the next article. The open band of Fig. 198 causes the follower to rotate in the same direction as the driver, while the crossed band (Fig. 199) gives the follower a rotation in an opposite direction. This will be seen to agree with the general statement, of Art. 33; for with the open belt both fixed centres are on the same side of the line of action (the driving side of the belt) ; while, with the crossed belt these centres are on opposite 'sides of the line of action. Owing to the rubbing of the sides of the belt where they cross, the open band is used when it is feasible. The crossed band has the advan- tage of a larger arc of contact, which has an important effect on the adhesion, especially on the smaller pulley; but with wide, stiff belts, particularly when the distance between centres is small, the warp- ing of the belt may largely destroy this advantage. It is evident that the length of belt is different in the two cases, other conditions being the same. The following are the algebraic expressions for the length of belts: The angle MQT = mqt = SqQ = 0. For crossed belts (Fig. 199), sin = , and Tt = qS = V7F (R n _ pen belts, sin = ^ The length of the crossed belt n _ __ For open belts, sin = , and Tt = qS = V d* (R r)*. *= L = 2 v'^-p). ... (1) 226 KINEMATICS OF MACHINERY. The length of the open belt -r)2sm->--. - . . (2) It follows from (1 ) that a crossed belt which is of proper length for any pair of pulleys, R and r, will be of correct length for any other pair of pulleys, R' and r' (on the same shafts) if R -{- r = R' -f r', that is, if the sum of the radii is constant; for (R -f r) is the only variable quantity. It will be seen, however, in (2) that if R' -f r' = R .+ r ; 72' r' cannot equal 7? r, unless R' = R and r' = r. An open belt of the correct length for two pulleys, R and r, on fixed shafts would not, therefore, be of exactly the right length for another pair of pulleys, R' and r', on these same shafts, if R' -f- r' = R + r, unless the two larger pulleys are equal, and the two smaller pulleys are also equal. Such a belt might be made to run if the distance between shafts were quite great and the change in sizes of pulleys were small ; but it would not be equally tight on the different sets. 126. Stepped Cones. It is often important to change the speed of a machine which is driven from a shaft having uniform speed. Cones, as shown in Fig. 204, might be placed upon the counter- shaft and on the spindle of the machine. If a crossed belt is used, it would be equally tight at all corresponding positions on these cones, but an open belt would not be; and in order to have it so, "swelled " cones, as shown (exaggerated) in Fig. 205, would be re- quired. Such conical drums have the advantage of permitting every possible variation in speed within limits; but the belt tends to mount towards the large ends of both, which increases the strain upon the belts and the pressure upon the bearings. The stepped cones, Fig. 206, are more compact than conical drums, and they avoid the objection just mentioned. It follows from the preceding discussion that for a crossed belt the sum of WRAPPING-CONNECTORS. 227 the radii of any mating pair of steps should be a constant. But the sum of the radii of the intermediate pairs of steps should be greater than the sum for the outside steps when using an open belt. Rankine's Machinery and Millwork gives a method of deter- mining the swell of the cones (Fig. 205) from which the radii of the intermediate steps of a stepped cone can be derived. A much Fig. 204. Fig. 205. Fig. 206. more convenient approximate graphical method is described by Mr. . A. Smith, in the Trans, of the A. S. M. E., Vol. X, page 269. Lay off Qq (Fig. 207) equal to the distance between shafts; draw the circles with radii R and r, equal to the radii of the known pul- leys; at C, half way between q and Q, erect the perpendicular CG = .SHQq, and with G as a centre, draw the arc mm tangent to iT. The belt line of any other pair of steps should be tangent to mm. R' and r r are radii of two such steps; and the velocity ratio when using these ^teps will be R' -r- r' = FQ -f- Fq. Let Qq = d\ let Fq = x; and call the desired ratio a. Now = a. x = a- r value of x\ draw FT' tangent to mm* respective centres, and tangent to FT', give the required wheels with Lay off Fq equal to this Circles with Q and q as the 228 KINEMATICS OF MACHINERY. radii R' and r f . This method, as here outlined, only applies when the belt angle, $ of Fig. 207, is less than about 18. Tl The original paper, referred to above, gives a modified method for use when watches to secure a more uniform driving action to the mechanism as the spring runs down. The spring is placed in the cylindrical piece, called the barrel, and as it uncoils the small chain is wound upon the barrel and unwound from the conical piece, called a "fusee." It will be seen that as the spring runs down the pull on the connector diminishes, but the "leverage" of the connector upon the follower is increased corre- spondingly, and, therefore, the driving effort transmitted to the mechanism may be kept quite uniform. CHAPTER VIII. TRAINS OF MECHANISM. 132. Substitution of a Train for a Simple Mechanism. It is kinematically possible to transmit motion between two parallel shafts with any required angular velocity ratio by either a single pair of gears or of pulleys; but there are practical conditions which often make it desirable to effect the required transmission of motion by a series of mechanisms, or a compound mechanism, instead of by a single pair of gears, of pulleys, etc. Such an arrange- ment constitutes a train of mechanism. The train may contain pulleys with belts, ropes, chains, gears, screws, and linkwork, any or all; and it may be used to transmit motion between other members than parallel shafts. If two shafts are to be connected by gears, and the required velocity ratio is high, the difference in the size of the gears may be inconveniently great if a single pair is used. That is, the large wheel may occupy too much room, or be difficult to swing, or the small gear may have so few teeth that it would be objectionable. For example : suppose the velocity ratio is 25 to 1, and that strength requires wheels of 2 (diametral) pitch. Then if the pinion be given only 12 teeth, it will be 6 inches in diameter, and the large wheel will be 25 X 6 150 inches in diameter ( = 12 feet). Now, suppose that an intermediate shaft be introduced. This intermediate shaft can be connected to the slower of the original shafts by using a pair of gears which will cause it to rotate 5 times to 1 rotation of this primary shaft, and it can be connected to the faster of the original shafts by a pair of gears which will give it 1 rotation to 5 of the latter shaft; then as each 233 234 KINEMATICS OF MACHINERY. revolution of the first shaft corresponds to 5 revolutions of this intermediate shaft, and as each of its revolutions corresponds to 5 revolutions of the last main shaft, it is evident that the velocit 7 ratio between the first and last shaft is 5 X 5 to 1, equal 25 to 1 - as required. The velocity ratio of first shaft to intermediate and of interme- diate to last shaft are not necessarily equal. They may be any- thing whatever if the product of the separate angular velocity ratios equals the required ratio between the first and the last shaft. Furthermore, the three axes need not lie in one plane; that is, the centres need not be in one straight line. It is thus seen that the use of a train in place of a simple mechanism permits considerable flexibility in the arrangement; this will be clearly seen from an examination of various actual trains. In a similar manner to that of the preceding illustration, an in- termediate shaft may be used in a belt transmission when the velocity ratio is high. Such an arrangement is frequently seen when a slow-speed engine drives a dynamo. The engine is belted to a " jack-shaft/' which in turn drives the dynamo. This may be desirable either to avoid an excessively large pulley or to avoid an extremely wide angle between the sides of the belt. The effect of a large belt angle is to reduce the arc of contact on the smaller pulley; this reduces the adhesion of the belt and increases liability of slip of the belt. Other considerations than a high velocity ratio may make it desirable to substitute a train for a simple mechanism; for instance, to secure a required directional relation, for compactness, etc. A familiar example of such a train is seen in the back-gear mechanism (Fig. 217), as used on lathes and other machine tools. A shaft which carries intermediate gears of a train may itself drive some member which requires a motion different from that of the last member. Thus, in clockwork, the gear on the shaft to which the minute-hand is fixed drives the hour-hand through a reducing pair of gears, and it may also drive a second hand at a higher rate. TRAINS OF MECHANISM. 235 133. Value of a Train. Suppose four axes, I, II, III, and IV (Fig. 214) to be arranged as shown and connected by toothed gears of which the circles , b, c, etc., are the pitch lines. The wheel a meshes with b; c meshes with d, and e meshes with f. Both of the wheels b and c are secured to the shaft II; hence they must rotate as one piece, having the same angular velocity at any in- stant. Likewise, d and e are both secured so shaft III, and they have the same angular velocity. Let the angular velocities of the shafts I, II, III, and IV be repre- sented by a l9 a^ <* 3 , and a^ respectively. Two gears which mesh together must have the same pitch ; hence the numbers of teeth are proportional to the circumferences, to the diameters, or to the radii. But their angular velocities are inversely as the radii, and therefore inversely as the numbers of teeth on the wheel. It follows that if a, b, c, etc., are the numbers of teeth on the wheels designated by these letters, that. 214. a, = ' X (Xa . Give the teeth the involute rack-and-pinion outline at middle section, and mark contact points of teeth. 46. [Art. 104.] If, in the four-link chain of Fig. 159, a = 3"; 6 = 4"; d=10"; find the limits between which the length of c must lie in order to permit continuous rotation of a. 47. [Art. 104.] Taking same data as Prob. 46, is it possible to give c such a length that a drag-link mechanism results; that is, so that both a and 6 shall rotate continuously? Test this by finding the limiting values of c for the drag-link chain, with given values of a, b, and d. 48. [Art. 104.] If, in the drag-link chain of Fig. 160, a = 7"; 6 = 6"; d=3"; find the limiting values of c which will permit continuous rotation of both a and 6. PROBLEMS AND EXERCISES. 255 49. [Art. 106.] (See Fig. 164.) An engine with a stroke of 18" has a connecting-rod 45" long. (a) Calculate the distance of the piston (or cross-head) from the end of stroke (a-c) when the crank angle (6 measured from A) is 60. (6) Calculate the distance from the other end of the stroke when 6 = 120. (c) Calculate the distance from cross-head to middle of the stroke (q ra), when 6 = 90, or 270. 50. [Art. 106.] With data as in Prob. 50, except that connecting-rod is 54" long, calculate (a), (6), (c). 51. [Art. 106.] Data as in Prob. 49. Calculate crank angles at which velocity of cross-head (piston) equals velocity of crank-pin. Also find ratio of piston velocity to crank-pin velocity when crank and connecting-rod form a right angle at C. 52. [Art. 110.] (Fig. 173.) The perpendicular distance from Q to the line of stroke, h g, is 2"; radius of a crank = 3"; crank makes 20 rev. per minute; connecting-rod C c = 9". Find length of stroke of c; and con- struct velocity diagram of c for forward and return strokes on a b as a 53. [Art. 113.] (Fig. 175.) Design a Whitworth quick-return mechanism such that length of stroke c shall be 10"; ratio of times of forward and re- turn strokes = 2 :1; rqadius of driving-crank (OP) = 4"; length of connecting- rod =12". Construct velocity diagram of c for both strokes. 54. [Art. 115.] (Fig. 179.) The stroke of a beam-engine is 4 feet; dis- tance from line of piston motion to beam centre (d) = 5 feet. Find proper length of beam for minimum obliquity of connecting-rod. 55. [Art. 126.] A countershaft runs at 100 rev. per minute. This countershaft is to drive a spindle through stepped cones and an open belt at 150, 100, or 75 rev. per minute. Largest step on countershaft = 14" diam. Distance between centres = 7 feet. Find, graphically, the diameters of all the steps. Check the accuracy of the method by calculating the lengths of belts for each of the three pairs of steps. 56. [Art. 133.] (See Fig. 215.) The diameter of a = 24"; 6 = 40"; c=36"; d=54"; e has 15 teeth; and / has 48 teeth. Find velocity ratio and the directional relation between a and/. 57. [Art. 133.] (Fig. 214.) Data: a has 60 teeth; 6 has 16 teeth; diam. of c = 24"; diam. of d = 8"; e makes 75 rev. per min. and/ 250 rev. per min. How many rev. per min. does a make; and what is the directional relation between a and /? 58. [Art. 133.] (Fig. 216.) The number of teeth on a, 6, c, d, e, and / are, respectively, 15, 45, 23, 35, 1, and 50. Determine velocity ratio be- tween axes I and IV. 59. [Art. 135.] (Fig. 217.) The cone-pulley is driven by an equal cone on a countershaft which makes 90 rev. per min. The steps have diame- 256 KINEMATICS OF MACHINERY. ters of 12", 9f," and 7". The gear a is keyed to the cone-pulley, and it has 28 teeth; gears 6 and c are fast to the shaft B, and have, respectively, 100 and 24 teeth; d is keyed to the spindle and has 88 teeth. Calculate the various possible speeds of the spindle. 60. [Art, 137.] The lathe has a lead screw with 4 threads per inch. The change-gears include wheels with the following numbers of teeth: 24, 30, 36, 42, 48, 48, 54, 60, 66, 69, 72, 78, 84. The " stud " makes the same number of revolutions as the spindle in a given time. With the 24-gear on the stud what gears should be used on the screw to cut 9, 10, 11, 11$ and 12 threads, respectively? What arrangement would be used to cut 4 threads per inch? What for 2 threads? 61. [Art. 137.J Same data as Prob. 60. Arrange table showing what gears to use on the stud and screw to cut threads from 2 per inch up to 14 per inch. INDEX. A PAGE Absolute motion 3 Acceleration 1 Acceleration diagrams 73 Angular velocity. . ... 18 '*. ratio 50,69 " , constant ! 53,56 Angularity of connecting rod 195 Annular wheels 122 Approximate tooth profiles 136 Axis, instant '. 20 Axodes... 75 B Back-gears 238 Backlash and clearance, gears 129 Bands 221 Beam motion , 210 Bell-cranks 209 Belt, length of 224 Belts 221 Belt-tighteners 230 Bent levers 209 Bevel-gears 142 " '.*;., non-interchangeability of 149 " " , smoothness in operation of 149 Brush-wheels 107 Burmester's method for open belts 228 C Cams 169 Cast gears 157 Centre, instant 20, 60, 62, 66 257 258 INDEX. PAGE Centrodes 75 Chain, four-link 187 Chain wheels 230 Change gears 243 Circular pitch 129, 131 Classes of gearing 110, 167 Clearance and backlash, gears 129 Close-fitting worm-wheel 183 Common methods of transmitting motion 37 Comparison of systems of gearing 128 Composition and resolution of motion 14 Condition of constant angular velocity ratio 53 positive driving 57 pure rolling 54 Cone frictions 108 Cone pulleys 226, 239 Cones, rolling 91, 93 Conjugate teeth 112 Connectors, link 37 , wrapping 48, 221 Constant angular velocity ratio 53 Constant velocity ratio and pure rolling 56 Constrained motion 24 Contact transmission, direct 37 Continuous motion 7, 189 Corliss wrist-plate motion 210 Crank and connecting-rod 192 Crossed belts 221, 225 Crowning pulleys 223 Curvilinear translation 9 Cut gears 158 Cutters, gear 162 Cycle, definition < 6 Cylinder cams 178 Cylinders, rolling 91 Cycloids 116 D Dead points, or centres 187 Describing circles in gears 120 Diagrams, acceleration 73 ' ' , velocity 70 Diametral pitch 130 INDEX. 259 PAGE Dimensions of gear-teeth 131 Direct-contact transmission 37, 41 Directional relation in trains 238, 247 Distance of centres in involute gears 125 Drag-link 189 Driving, positive 57 E Eccentric 199 Ellipses, rolling 55, 80 Epicyclic trains 245 Epicycloid 116 Epicycloidal system of gears 116 teeth 117, 118 Escapements 220 F Forces, parallelogram of 13 Four-link chain 187 Free motion 24 Frictional gearing 99 G Gear-cutters 162 Gearing, tooth 110 , frictional 99 Gear moulding machines 158 Gear-planers 159, 164 Clears, bevel 142 ' ' , cast 157 " , cut 158, 159, 161, 163 ' ' , helical 1 50 ' ' , non-circular 135 11 , spiral 152 ' ' , stub tooth 132 " , worm 180 Gear teeth, methods of cutting 158 , proportions of 131 Generating circles, epicycloidal gears 120 Grant's odontographs 139 Graphic representation of motion 12 Grooved friction-wheels 101 Guide-pulleys 230 260 INDEX. H PAGES Helical gears 150 " " , graphical method for 154 Helical motion 7, 10 Higher pairing 38 Hobbing worm-wheels 184 Hooke's coupling 214 Hyperboloids, rolling 91, 97 Hypocycloid 116 I Idler gear 240- Indicator pencil mechanisms 211 Instant axis. 2O Instant centre 20, 60, 62, 66 Instant centre theorem 64 Interchangeable set of gears 122. Interference in involute gears 127 Intermediate connectors 37 Intermittent motion 7 Inversion of mechanism 59, 204 Involute gearing 116, 124 Involute teeth, interference of ,. 127 K Kennedy's theorem 64 Kinematics, definition 34 L Lazy-tongs 213 Length of belts 224 Length of connecting-rod 195 Length of teeth 120 link.?. 37 Link-connectors 45 LInkwork 186 Lobcd w-ieels 89 Logarithmic spirals, rolling 55, 85 Lower pairing. 38- IXDEX. 261 M PAGE Machine, definition 29 Machine design, definition, 34 Mechanics, definition 29 Mechanism, definition 29 ' ' , inversion of 59 1 ' , trains of 233 Methods of transmitting motion 38 Milling-cutters, standard 158, 162 " bevel-gears 164 1 ' spur-gears 161 Mitre gears 153 Motion, absolute 3 ' ' , continuous, reciprocating and intermittent 7 ' ' , definition 1 ' ' , free and constrained 24 ' ' , graphic representation of 12 " , helical 10 ' ' , instantaneous. 20 ' ' , Newton's laws of 13 ' ' , plane . 7 " , relative 3, 61 ' ' , resolution and composition 14 ' ' , spherical 10 N Newton's laws of motion 13 Non-circular gears 135 Non-interchangeability of bevel-gears 149 O Obliquity of connecting-rod 195 Open belts 221, 225 Oscillating-engine mechanism 204 Outlines of conjugate gear-teeth 113 Outlines of gear-teeth, general method 114 helical gear-teeth 153 P Pairing, higher and lower ; 38 Pantographs . 213 262 INDEX. _ ,, , PAGE Parallel motions 211 Parallel rods, locomotive 190 Parallelogram of forces 13 ' ' motions 15 Path, definition 6 Pencil motions, indicator 211 Period, definition 5 Phase, definition 5 Piston, velocity ratio to crank-pin 196 Pitch of gear-teeth 129, 152 surfaces 110, 135, 142, 151 Planing gear-teeth 159 ; 165 Positive driving in direct contact 57 Positive return cams 174 Problems and exercises 249 Proportions of gear-teeth 131 Q Quick-return motions 202, 204, 206 Quarter-turn belts 229 R Rack and pinion 123 Rapid change in angular motion of link 210 Ratchets 217 Rate of sliding in direct contact 54 Ratio, velocity f . 5 Reciprocating motion 7 Rectilinear translation 9 Relation of direction of rotation 52 Relative motion 3, 61 Resolution and composition of motion 14 Reverted train 248 Rolling circles 79 " cones 91, 93 ' ' curves 78, 87 ' ' cylinders 91 " ellipses 55, 80 " hyperboloids 91, 97 ' ' logarithmic spirals 55, 85 ' ' pure, condition of 54 ' ' surfaces . . 90 INDEX. 263 PAGE Rolling and sliding 53 Rope transmission 221 Rotation 7 S Scotch yoke , 201 Screw 179 Screw-cutting train 242 Shaper quick-return motion 206 Sheaves for ropes 223 Shifting belts 223 Side rods, locomotive 190 Slider-crank mechanism 60, 192 Sliding, rate of 54 Sliding and rolling 53 Slip in frictional gearing 106 Spherical motion 7, 10 Spiral gears 152 Sprocket-wheels 231 Stepped cones 226 Stepped gearing- 133 Straight-line motions 211 Strength of gear-teeth 129 Stub teeth 132 Systems of gearing, usual 116 T Teeth, conjugate 112 ' ' , epicycloidal 117 ' ' , involute ; 124 " , stepped 133 " , stub 132 " , twisted 133 ' ' , unsymmetrical 133 ' ' of bevel gears 145 ' ' of gears, proportions of 131 Tight-and-loose pulleys 224 Tooth-gearing 110 Tooth outlines, general methods 114 Train, screw-cutting 242 Trains, epicyclic 245 Trains of mechanism. . . 233 264 INDEX. PAGE Translation cams 177 Translation, rectilinear and curvilinear 7, 9 Transmission by actual contact 37 without material connection 37 TredgolcTs approximate method for bevel-gear teeth 145 Tumbling gears 241 Twisted gearing 133 U Universal joint 214 Unsymmetrical teeth 133 Unwin, approximate method for gear-teeth 136 V ' 'V" friction-gears 101 Value of a train of mechanism 235 Varying angular velocity, wrapping connectors 232 Velocity, angular , 18 " , ' " , determined by instant centres 69 ' ' diagrams 70, 71 ' ' , linear 1 ' ' , ' ' , determined by instant centres 68 11 ratio 5 " " , helical gears 153 1 ' , uniform and variable 2 W Wheels, brush 107 " ,lobed 89 Whitworth's quick-return mechanism 206 Willis' odontograph 137 Worm and wheel 181 Worm gearing 180 Wrapping connectors .... 48, 221 Wrist-plate motion 210