Chapter III: THE TEETH OF GEAR-WHEELS (continued) (5)

"Having completed this ellipse, prolong its major axis, and draw a similar and equal one, with its foci, C, D, upon that prolongation, and tangent to the first one at P; then B D = L P. About B describe an arc with any radius, cutting the first ellipse at Y and the line L at Z; about D describe an arc with radius D Z, cutting the second ellipse in X; draw A Y, B Y, C X, and D X. Then A Y = D X, and B Y = C X, and because the ellipses are alike, the arcs P Y and P X are equal. If then B and D are taken as fixed centres, and the ellipses turn about them as shown by the arrows, X and Y will come together at Z on the line of centres; and the same is true of any points equally distant from P on the two curves. But this is the condition of rolling contact. We see, then, that in order that two ellipses may roll together, and serve as the pitch-lines of wheels, they must be equal and similar, the fixed centres must be at corresponding foci, and the distance between these centres must be equal to the major axis. Were they to be toothless wheels, if would evidently be essential that the outlines should be truly elliptical; but the changes of curvature in the ellipse are gradual, and circular arcs may be drawn so nearly coinciding with it, that when teeth are employed, the errors resulting from the substitution are quite inappreciable. Nevertheless, the rapidity of these changes varies so much in ellipses of different proportions, that we believe it to be practically better to draw the curve accurately first, and to find the radii of the approximating arcs by trial and error, than to trust to any definite rule for determining them; and for this reason we give a second and more convenient method of finding points, in connection with the ellipse whose centre is R, Fig. 193. About the centre describe two circles, as shown, whose diameters are the major and minor axes; draw any radius, as R T, cutting the first circle in T, and the second in S; through T draw a parallel to one axis, through S a parallel to the other, and the intersection, V, will lie on the curve. In the left hand ellipse, the line bisecting the angle A F B is normal to the curve at F, and the perpendicular to it is tangent at the same point, and bisects the angles adjacent to A F B, formed by prolonging A F, B F.

"To mark the pitch line we proceed as follows:--

"In Fig. 194, A A and B B are centre lines passing through the major and minor axes of the ellipse, of which _a_ is the axis or centre, _b_ _c_ is the major and _a_ _e_ half of the minor axis. Draw the rectangle _b_ _f_ _g_ _c_, and then the diagonal line _b_ _e_; at a right angle to _b_ _e_ draw line _f_ _h_ cutting B B at _i_. With radius _a_ _e_ and from _a_ as a centre draw the dotted arc _e_ _j_, giving the point _j_ on the line B B. From centre _k_, which is on line B B, and central between _b_ and _j_, draw the semicircle _b_ _m_ _j_, cutting A A at _l_. Draw the radius of the semicircle _b_ _m_ _j_ cutting _f_ _g_ at _n_. With radius _m_ _n_ mark on A A, at and from _a_ as a centre, the point _o_. With radius _h_ _o_ and from centre _h_ draw the arc _p_ _o_ _q_. With radius _a_ _l_ and from _b_ and _c_ as centres draw arcs cutting _p_ _o_ _q_ at the points _p_ _q_. Draw the lines _h_ _p_ _r_ and _h_ _q_ _s_, and also the lines _p_ _i_ _t_ and _q_ _v_ _w_. From _h_ as centre draw that part of the ellipse lying between _r_ and _s_. With radius _p_ _r_ and from _p_ as a centre draw that part of the ellipse lying between _r_ and _t_. With radius _q_ _s_ and from _q_ draw the ellipse from _s_ to _w_. With radius _i_ _t_ and from _i_ as a centre draw the ellipse from _t_ to _b_. With radius _v_ _w_ and from _v_ as a centre draw the ellipse from _w_ to _c_, and one half the ellipse will be drawn. It will be seen that the whole construction has been performed to find the centres _h_ _p_ _q_ _i_ and _v_, and that while _v_ and _i_ may be used to carry the curve around the other side or half of the ellipse, new centres must be provided for _h_ _p_ and _q_; these new centres correspond in position to _h_ _p_ _q_.

"If it were possible to subdivide the ellipse into equal parts it would be unnecessary to resort to these processes of approximately representing the two curves by arcs of circles; but unless this be done, the spacing of the teeth can only be effected by the laborious process of stepping off the perimeter into such small subdivisions that the chords may be regarded as equal to the arcs, which after all is but an approximation; unless, indeed, we adopt the mechanical expedient of cutting out the ellipse in metal or other substance, measuring and subdividing it with a strip of paper or a steel tape, and wrapping back the divided measure in order to find the points of division on the curve.

"But these circular arcs may be rectified and subdivided with great facility and accuracy by a very simple process, which we take from Prof. Rankine's "Machinery and Mill Work," and is illustrated in Fig. 195. Let O B be tangent at O to the arc O D, of which C is the centre. Draw the chord D O, bisect it in E, and produce it to A, making O A = O E; with centre A and radius A D describe an arc cutting the tangent in B; then O B will be very nearly equal in length to the arc O D, which, however, should not exceed about 60°; if it be 60°, the error is theoretically about 1/900 of the length of the arc, O B being so much too short; but this error varies with the fourth power of the angle subtended by the arc, so that for 30° it is reduced to 1/16 of that amount, that is, to 1/14400. Conversely, let O B be a tangent of given length; make O F = 1/4 O B; then with centre F and radius F B describe an arc cutting the circle O D G (tangent to O B at O) in the point D; then O D will be approximately equal to O B, the error being the same as in the other construction and following the same law.

"The extreme simplicity of these two constructions and the facility with which they may be made with ordinary drawing instruments make them exceedingly convenient, and they should be more widely known than they are. Their application to the present problem is shown in Fig. 196, which represents a quadrant of an ellipse, the approximate arcs C D, D E, E F, F A having been determined by trial and error. In order to space this off, for the positions of the teeth, a tangent is drawn at D, upon which is constructed the rectification of D C, which is D G, and also that of D E in the opposite direction, that is, D H, by the process just explained. Then, drawing the tangent at F, we set off in the same manner F I = F E, and F K = F A, and then measuring H L = I K, we have finally G L, equal to the whole quadrant of the ellipse.

"Let it now be required to lay out 24 teeth upon this ellipse; that is, 6 in each quadrant; and for symmetry's sake we will suppose that the centre of one tooth is to be at A, and that of another at C, Fig. 196. We therefore divide L G into six equal parts at the points 1, 2, 3, &c., which will be the centres of the teeth upon the rectified ellipse. It is practically necessary to make the spaces a little greater than the teeth; but if the greatest attainable exactness in the operation of the wheel is aimed at, it is important to observe that backlash, in elliptical gearing, has an effect quite different from that resulting in the case of circular wheels. When the pitch-curves are circles, they are always in contact; and we may, if we choose, make the tooth only half the breadth of the space, so long as its outline is correct. When the motion of the driver is reversed, the follower will stand still until the backlash is taken up, when the motion will go on with a perfectly constant velocity ratio as before. But in the ease of two elliptical wheels, if the follower stand still while the driver moves, which must happen when the motion is reversed if backlash exists, the pitch-curves are thrown out of contact, and, although the continuity of the motion will not be interrupted, the velocity ratio will be affected. If the motion is never to be reversed, the perfect law of the velocity ratio due to the elliptical pitch-curve may be preserved by reducing the thickness of the tooth, not equally on each side, as is done in circular wheels, but wholly on the side not in action. But if the machine must be capable of acting indifferently in both directions, the reduction must be made on both sides of the tooth: evidently the action will be slightly impaired, for which reason the backlash should be reduced to a minimum. Precisely what _is_ the minimum is not so easy to say, as it evidently depends much upon the excellence of the tools and the skill of the workmen. In many treatises on constructive mechanism it is variously stated that the backlash should be from one-fifteenth to one-eleventh of the pitch, which would seem to be an ample allowance in reasonably good castings not intended to be finished, and quite excessive if the teeth are to be cut; nor is it very obvious that its amount should depend upon the pitch any more than upon the precession of the equinoxes. On paper, at any rate, we may reduce it to zero, and make the teeth and spaces equal in breadth, as shown in the figure, the teeth being indicated by the double lines. Those upon the portion L H are then laid off upon K I, after which these divisions are transferred to curves. And since under that condition the motion of this third line, relatively to each of the others, is the same as though it rolled along each of them separately while they remained fixed, the process of constructing the generated curves becomes comparatively simple. For the describing line, we naturally select a circle, which, in order to fulfil the condition, must be small enough to roll within the pitch ellipse; its diameter is determined by the consideration, that if it be equal to A P, the radius of the arc A F, the flanks of the teeth in that region will be radial. We have, therefore, chosen a circle whose diameter, A B, is three-fourths of A P, as shown, so that the teeth, even at the ends of the wheels, will be broader at the base than on the pitch line. This circle ought strictly to roll upon the true elliptical curve, and assuming as usual the tracing-point upon the circumference, the generated curves would vary slightly from true epicycloids, and no two of those used in the same quadrant of the ellipse would be exactly alike. Were it possible to divide the ellipse accurately, there would be no difficulty in laying out these curves; but having substituted the circular arcs, we must now roll the generating circle upon these as bases, thus forming true epicycloidal teeth, of which those lying upon the same approximating arc will be exactly alike. Should the junction of two of these arcs fall within the breadth of a tooth, as at D, evidently both the face and the flank on one side of that tooth will be different from those on the other side; should the junction coincide with the edge of a tooth, which is very nearly the case at F, then the face on that side will be the epicycloid belonging to one of the arcs, its flank a hypocycloid belonging to the other; and it is possible that either the face or the flank on one side should be generated by the rolling of the describing circle partly on one arc, partly on the one adjacent, which, upon a large scale and where the best results are aimed at, may make a sensible change in the form of the curve.

"The convenience of the constructions given in Fig. 194 is nowhere more apparent than in the drawing of the epicycloids, when, as in the case in hand, the base and generating circles may be of incommensurable diameters; for which reason we have, in Fig. 197, shown its application in connection with the most rapid and accurate mode yet known of describing those curves. Let C be the centre of the base circle; B that of the rolling one; A the point of contact. Divide the semi-circumference of B into six equal parts at 1, 2, 3, &c.; draw the common tangent at A, upon which rectify the arc A2 by process No. 1, then by process No. 2 set out an equal arc A2 on the base circle, and stepping it off three times to the right and left, bisect these spaces, thus making subdivisions on the base circle equal in length to those on the rolling one. Take in succession as radii the chords A1, A2, A3, &c., of the describing circle, and with centres 1, 2, 3, &c., on the base circle, strike arcs either externally or internally, as shown respectively on the right and left; the curve tangent to the external arcs is the epicycloid, that tangent to the internal ones the hypocycloid, forming the face and flank of a tooth for the base circle.

"In the diagram, Fig. 196, we have shown a part of an ellipse whose length is 10 inches and breadth 6, the figure being half size. In order to give an idea of the actual appearance of the combination when complete, we show in Fig. 198 the pair in gear, on a scale of 3 inches to the foot. The excessive eccentricity was selected merely for the purpose of illustration. Fig. 198 will serve also to call attention to another serious circumstance, which is that although the ellipses are alike, the wheels are not; nor can they be made so if there be an even number of teeth, for the obvious reason that a tooth upon one wheel must fit into a space on the other; and since in the first wheel, Fig. 196, we chose to place a tooth at the extremity of each axis, we must in the second one place there a space instead; because at one time the major axes must coincide, at another the minor axis, as in Fig. 191. If then we use even numbers, the distribution and even the forms of the teeth are not the same in the two wheels of the pair. But this complication may be avoided by using an odd number of teeth, since, placing a tooth at one extremity of the major axis, a space will come at the other.

"It is not, however, always necessary to cut teeth all round these wheels, as will be seen by an examination of Fig. 199, C and D being the fixed centres of the two ellipses in contact at P. Now P must be on the line C D, whence, considering the free foci, we see P B is equal to P C, and P A to P D; and the common tangent at P makes equal angles with C P and P A, as is also with P B and P D; therefore, C D being a straight line, A B is also a straight line and equal to C D. If then the wheels be overhung, that is, fixed on the ends of the shafts outside the bearings, leaving the outer faces free, the moving foci may be connected by a rigid link A B, as shown.

"This link will then communicate the same motion that would result from the use of the complete elliptical wheels, and we may therefore dispense with most of the teeth, retaining only those near the extremities of the major axes which are necessary in order to assist and control the motion of the link at and near the dead-points. The arc of the pitch-curves through which the teeth must extend will vary with their eccentricity: but in many cases it would not be greater than that which in the approximation may be struck about one centre, so that, in fact, it would not be necessary to go through the process of rectifying and subdividing the quarter of the ellipse at all, as in this case it can make no possible difference whether the spacing adopted for the teeth to be cut would "come out even" or not if carried around the curve. By this expedient, then, we may save not only the trouble of drawing, but a great deal of labor in making, the teeth round the whole ellipse. We might even omit the intermediate portions of the pitch ellipses themselves; but as they move in rolling contact their retention can do no harm, and in one part of the movement will be beneficial, as they will do part of the work; for if, when turning, as shown by the arrows, we consider the wheel whose axis is D as the driver, it will be noted that its radius of contact, C P, is on the increase; and so long as this is the case the other wheel will be compelled to move by contact of the pitch lines, although the link be omitted. And even if teeth be cut all round the wheels, this link is a comparatively inexpensive and a useful addition to the combination, especially if the eccentricity be considerable. Of course the wheels shown in Fig. 198 might also have been made alike, by placing a tooth at one end of the major axis and a space at the other, as above suggested. In regard to the variation in the velocity ratio, it will be seen, by reference to Fig. 199, that if D be the axis of the driver, the follower will in the position there shown move faster, the ratio of the angular velocities being PD/PB; if the driver turn uniformly the velocity of the follower will diminish, until at the end of half a revolution, the velocity ratio will be PB/PD; in the other half of the revolution these changes will occur in a reverse order. But P D = L B; if then the centres B D are given in position, we know L P, the major axis; and in order to produce any assumed maximum or minimum velocity ratio, we have only to divide L P into segments whose ratio is equal to that assumed value, which will give the foci of the ellipse, whence the minor axis may be found and the curve described. For instance, in Fig. 198 the velocity ratio being nine to one at the maximum, the major axis is divided into two parts, of which one is nine times as long as the other; in Fig. 199 the ratio is as one to three, so that, the major axis being divided into four parts, the distance A C between the foci is equal to two of them, and the distance of either focus from the nearer extremity of the major axis equal to one, and from the more remote extremity equal to three of these parts."

Another example of obtaining a variable motion is given in Fig. 200. The only condition necessary to the construction of wheels of this class is that the sum of the radii of the pitch circles on the line of centres shall equal the distance between the axes of the two wheels. The pitch curves are to be considered the same as pitch circles, "so that," says Willis, "if any given circle or curve be assumed as a describing (or generating) curve, and if it be made to roll on the inside of one of these pitch curves and on the outside of the corresponding portion of the other pitch curve, then the motion communicated by the pressure and sliding contact of one of the curved teeth so traced upon the other will be exactly the same as that effected by the rolling contact (by friction) of the original pitch curves."

It is obvious that on B the corner sections are formed of simple segments of a circle of which the centre is the axis of the shaft, and that the sections between them are simply racks. The corners of A are segments of a circle of which the axis of A is the centre, and the sections between the corners curves meeting the pitch circles of the rack at every point as it passes the line of centres.

Intermittent motion may also be obtained by means of a worm-wheel constructed as in Fig. 201, the worm having its teeth at a right angle to its axis for a distance around the circumference proportioned to the required duration of the period of rest; or the motion may be made variable by giving the worm teeth different degrees of inclination (to the axis), on different portions of the circumference.

In addition to the simple operation of two or more wheels transmitting motion by rotating about their fixed centres and in fixed positions, the following examples of wheel motion may be given.

In Fig. 202 are two gear-wheels, A, which is fast upon its stationary shaft, and B, which is free to rotate upon its shaft, the link C affording journal bearing to the two shafts. Suppose that A has 40 teeth, while B has 20 teeth, and that the link C is rotated once around the axis of A, how many revolutions will B make? By reason of there being twice as many teeth in A as in B the latter will make two rotations, and in addition to this it will, by reason of its connection to the arm C, also make a revolution, these being two distinct motions, one a rotation of B about the axis of A, and the other two rotations of B upon its own axis.

A simple arrangement of gearing for reversing the direction of rotation of a shaft is shown in Fig. 203. I and F are fast and loose pulleys for the shaft D, A and C are gears free to rotate upon D, N is a clutch driven by D; hence if N be moved so as to engage with C the latter will act as a driver to rotate the shaft B, the wheel upon B rotating A in an opposite direction to the rotation of D. But if N be moved to engage with A the latter becomes the driving wheel, and B will be caused to rotate in the opposite direction. Since, however, the engagement of the clutch N with the clutch on the nut of the gear-wheels is accompanied with a violent shock and with noise, a preferable arrangement is shown in Fig. 204, in which the gears are all fast to their shafts, and the driving shaft for C passes through the core or bore of that for A, which is a sleeve, so that when the driving belt acts upon pulley F the shaft B rotates in one direction, while when the belt acts upon E, B rotates in the opposite direction, I being a loose pulley.

If the speed of rotation of B require to be greater in one direction than in the other, then the bevel-wheel on B is made a double one, that is to say, it has two annular toothed surfaces on its radial face, one of larger diameter than the other; A gearing with one of these toothed surfaces, and C with the other. It is obvious that the pinions A C, being of equal diameters, that gearing with the surface or gear of largest diameter will give to B the slowest speed of rotation.

Fig. 205 represents Watt's sun-and-planet motion for converting reciprocating into rotary motion; B D is the working beam of the engine, whose centre of motion is at D. The gear A is so connected to the connecting rod that it cannot rotate, and is kept in gear with the wheel C on the fly-wheel shaft by means of the link shown. The wheel A being prevented from rotation on its axis causes rotary motion to the wheel C, which makes two revolutions for one orbit of A.

An arrangement for the rapid increase of motion by means of gears is shown in Fig. 206, in which A is a stationary gear, B is free to rotate upon its shaft, and being pivoted upon the shaft of A, at D, is capable of rotation around A while remaining in gear with C. Suppose now that the wheel A were absent, then if B were rotated around C with D as a centre of motion, C and its shaft E would make a revolution even though B would have no rotation upon its axis. But A will cause B to rotate upon its axis and thus communicate a second degree of motion to C, with the result that one revolution of B causes two rotations of C.

The relation of motion between B and C is in this case constant (2 to 1), but this relation may be made variable by a construction such as shown in Fig. 207, in which the wheel B is carried in a gear-wheel H, which rides upon the shaft D. Suppose now that H remains stationary while A revolves, then motion will be transmitted through B to C, and this motion will be constant and in proportion to the relative diameters of A and C. But suppose by means of an independent pinion the wheel H be rotated upon its axis, then increased motion will be imparted to C, and the amount of the increase will be determined by the speed of rotation of H, which may be made variable by means of cone pulleys or other suitable mechanical devices.

Fig. 208 represents an arrangement of gearing used upon steam fire-engines and traction engines to enable them to turn easily in a short radius, as in turning corners in narrow streets. The object is to enable the driving wheel on either side of the engine to increase or diminish its rotation to suit the conditions caused by the leading or front pair of steering wheels.

In the figures A is a plate wheel having the lugs L, by means of which it may be rotated by a chain. A is a working fit on the shaft S, and carries three pinions E pivoted upon their axes P. F is a bevel-gear, a working fit on S, while C is a similar gear fast to S. The pinions B, D are to drive gears on the wheels of the engine, the wheels being a working fit on the axle. Let it now be noted that if S be rotated, C and F will rotate in opposite directions and A will remain stationary. But if A be rotated, then all the gears will rotate with it, but E will not rotate upon P unless there be an unequal resistance to the motion of pinions D and B. So soon, however, as there exists an inequality of resistance between D and B then pinions E operate. For example, let B have more resistance than D, and B will rotate more slowly, causing pinion E to rotate and move C faster than is due to the motion of the chain wheel A, thus causing the wheel on one side of the engine to retard and the other to increase its motion, and thus enable the engine to turn easily. From its action this arrangement is termed the equalizing gear.

In Figs. 209 to 214 are shown what are known as mangle-wheels from their having been first used in clothes mangling machines.

The mangle-wheel[10] in its simplest form is a revolving disc of metal with a centre of motion C (Fig. 209). Upon the face of the disc is fixed a projecting annulus _a_ _m_, the outer and inner edges of which are cut into teeth. This annulus is interrupted at _f_, and the teeth are continued round the edges of the interrupted portion so as to form a continued series passing from the outer to the inner edge and back again.

[10] From Willis's "Principles of Mechanism."

A pinion B, whose teeth are of the same pitch as those of the wheel, is fixed to the end of an axis, and this axis is mounted so as to allow of a short travelling motion in the direction B C. This may be effected by supporting this end of it either in a swing-frame moving upon a centre as at D, or in a sliding piece, according to the nature of the train with which it is connected. A short pivot projects from the centre of the pinion, and this rests in and is guided by a groove B S _f_ _t_ _b_ _h_ K, which is cut in the surface of the disc, and made concentric to the pitch circles of the inner and outer rays of teeth, and at a normal distance from them equal to the pitch radius of the pinion.

Now when the pinion revolves it will, if it be on the outside, as in Fig. 209, act upon the spur teeth and turn the wheel in the opposite direction to its own, but when the interrupted portion _f_ of the teeth is thus brought to the pinion the groove will guide the pinion while it passes from the outside to the inside, and thus bring its teeth into action with the annular or internal teeth. The wheel will then receive motion in the same direction as that of the pinion, and this will continue until the gap _f_ is again brought to the pinion, when the latter will be carried outwards and the motion again be reversed. The _velocity ratio_ in either direction will remain constant, but the ratio when the pinion is inside will differ slightly from the ratio when it is outside, because the pitch radius of the annular or internal teeth is necessarily somewhat less than that of the spur teeth. However, the change of direction is not instantaneous, for the form of the groove S _f_ _t_, which connects the inner and outer grooves, is a semicircle, and when the axis of the pinion reaches S the velocity of the mangle-wheel begins to diminish gradually until it is brought to rest at _f_, and is again gradually set in motion from _f_ to _t_, when the constant ratio begins; and this retardation will be increased by increasing the difference between the radius of the inner and outer pitch circles.

The teeth of a mangle-wheel are, however, most commonly formed by pins projecting from the face of the disc as in Fig. 210. In this manner the pitch circles for the inner and outer wheels coincide, and therefore the velocity ratio is the same within and without, also the space through which the pinion moves in shifting is reduced.

This space may be still further reduced by arranging the teeth as in Fig. 211, that is, by placing the spur-wheel within the annular or internal one; but at the same time the difference of the two velocity ratios is increased.

If it be required that the velocity ratio vary, then the pitch lines of the mangle-wheel must no longer be concentric.

Thus in Fig. 212 the groove _k_ _l_ is directed to the centre of the mangle-wheel, and therefore the pinion will proceed during this portion of its path without giving any motion to the wheel, and in the other lines of teeth the pitch radius varies, hence the angular velocity ratio will vary.

In Figs. 209, 210, and 211 the curves of the teeth are readily obtained by employing the same describing circle for the whole of them. But when the form Fig. 212 is adopted, the shape of the teeth requires some consideration.

Every tooth of such a mangle-wheel may be considered as formed of two ordinary teeth set back to back, the pitch line passing through the middle. The outer half, therefore, appropriated to the action of the pinion on the outside of the wheel, resembles that portion of an ordinary spur-wheel tooth that lies beyond its pitch line, and the inner half which receives the inside action of the pinion resembles the half of an annular wheel that lies within the pitch circle. But the consequence of this arrangement is, that in both positions the action of the driving teeth must be confined to the approach of its teeth to the line of centres, and consequently these teeth must be wholly within their pitch line.

To obtain the forms of the teeth, therefore, take any convenient describing circle, and employ it to describe the teeth of the pinion by rolling within its pitch circle, and to describe the teeth of the wheel by rolling within and without its pitch circle, and the pinion will then work truly with the teeth of the wheel in both positions. The tooth at each extremity of the series must be a circular one, whose centre lies on the pitch line and whose diameter is equal to half the pitch.

If the reciprocating piece move in a straight line, as it very often does, then the mangle-_wheel_ is transformed into a _mangle-rack_ (Fig. 213) and its teeth may be simply made cylindrical pins, which those of the mangle-wheel do not admit of on correct principle. B _b_ is the sliding piece, and A the driving pinion, whose axis must have the power of shifting from A to _a_ through a space equal to its own diameter, to allow of the change from one side of the rack to the other at each extremity of the motion. The teeth of the mangle-rack may receive any of the forms which are given to common rack-teeth, if the arrangement be derived from either Fig. 210 or Fig. 211.

But the mangle-rack admits of an arrangement by which the shifting motion of the driving pinion, which is often inconvenient, may be dispensed with.

B _b_ Fig. 214, is the piece which receives the reciprocating motion, and which may be either guided between rollers, as shown, or in any other usual way; A the driving pinion, whose axis of motion is fixed; the mangle rack C _c_ is formed upon a separate plate, and in this example has the teeth upon the inside of the projecting ridge which borders it, and the guide-groove formed within the ring of teeth, similar to Fig. 211.

This rack is connected with the piece B _b_ in such a manner as to allow of a short transverse motion with respect to that piece, by which the pinion, when it arrives at either end of the course, is enabled by shifting the rack to follow the course of the guide-groove, and thus to reverse the motion by acting upon the opposite row of teeth.

The best mode of connecting the rack and its sliding piece is that represented in the figure, and is the same which is adopted in the well-known cylinder printing-engines of Mr. Cowper. Two guide-rods K C, _k_ _c_ are jointed at one end K _k_ to the reciprocating piece B _b_, and at the other end C _c_ to the shifting-rack; these rods are moreover connected by a rod M _m_ which is jointed to each midway between their extremities, so that the angular motion of these guide-rods round their centres K _k_ will be the same; and as the angular motion is small and the rods nearly parallel to the path of the slide, their extremities C _c_ may be supposed to move at a right angle to that path, and consequently the rack which is jointed to those extremities will also move upon B _b_ in a direction at a right angle to its path, which is the thing required, and admits of no other motion with respect to B _b_.

To multiply plane motion the construction shown in Fig. 215 is frequently employed. A and B are two racks, and C is a wheel between them pivoted upon the rod R. A crank shaft or lever D is pivoted at E and also (at P) to R. If D be operated C traverses along A and also rotates upon its axis, thus giving to B a velocity equal to twice that of the lateral motion of C.

The diameter of the wheel is immaterial, for the motion of B will always be twice that of C.

Friction gearing-wheels which communicate motion one to the other by simple contact of their surfaces are termed friction-wheels, or friction-gearing. Thus in Fig. 216 let A and B be two wheels that touch each other at C, each being suspended upon a central shaft; then if either be made to revolve, it will cause the other to revolve also, by the friction of the surfaces meeting at C. The degree of force which will be thus conveyed from one to the other will depend upon the character of the surface and the length of the line of contact at C.

These surfaces should be made as concentric to the axis of the wheel and as flat and smooth as possible in order to obtain a maximum power of transmission. Mr. E. S. Wicklin states that under these conditions and proper forms of construction as much as 300 horse-power may be (and is in some of the Western States) transmitted.

In practice, small wheels of this class are often covered with some softer material, as leather; sometimes one wheel only is so covered, and it is preferred that the covered wheel drive the iron one, because, if a slip takes place and the iron wheel was the driver, it would be apt to wear a concave spot in the wood covered one, and the friction between the two would be so greatly diminished that there would be difficulty in starting them when the damaged spot was on the line of centre.

If, however, the iron wheel ceased motion, the wooden one continuing to revolve, the damage would be spread over that part of the circumference of the wooden one which continued while the iron one was at rest, and if this occurred throughout a whole revolution of the wooden wheel its roundness would not be apt to be impaired, except in so far as differences in the hardness of the wood and similar causes might effect.

"To select the best material for driving pulleys in friction-gearing has required considerable experience; nor is it certain that this object has yet been attained. Few, if any, well-arranged and careful experiments have been made with a view of determining the comparative value of different materials as a frictional medium for driving iron pulleys. The various theories and notions of builders have, however, caused the application to this use of several varieties of wood, and also of leather, india-rubber, and paper; and thus an opportunity has been given to judge of their different degrees of efficiency. The materials most easily obtained, and most used, are the different varieties of wood, and of these several have given good results.

"For driving light machinery, running at high speed, as in sash, door, and blind factories, basswood, the linden of the Southern and Middle States (_Tilia Americana_) has been found to possess good qualities, having considerable durability and being unsurpassed in the smoothness and softness of its movement. Cotton wood (_Populus monilifera_) has been tried for small machinery with results somewhat similar to those of basswood, but is found to be more affected by atmospheric changes. And even white pine makes a driving surface which is, considering the softness of the wood, of astonishing efficiency and durability. But for all heavy work, where from twenty to sixty horse-power is transmitted by a single contact, soft maple (_Acer rubrum_) has, at present, no rival. Driving pulleys of this wood, if correctly proportioned and well built, will run for years with no perceptible wear.

"For very small pulleys, leather is an excellent driver and is very durable; and rubber also possesses great adhesion as a driver; but a surface of soft rubber undoubtedly requires more power than one of a less elastic substance.

"Recently paper has been introduced as a driver for small machinery, and has been applied in some situations where the test was most severe; and the remarkable manner in which it has thus far withstood the severity of these tests appears to point to it as the most efficient material yet tried.

"The proportioning, however, of friction-pulleys to the work required and their substantial and accurate construction are matters of perhaps more importance than the selection of material.

"Friction-wheels must be most accurately and substantially made and kept in perfect line so that the contact between the surfaces may not be diminished. The bodies are usually of iron lagged or covered with wooden segments.

"All large drivers, say from four to ten feet diameter and from twelve to thirty inch face, should have rims of soft maple six or seven inches deep. These should be made up of plank, one and a half or two inches thick, cut into 'cants,' one-sixth, eighth, or tenth of the circle, so as to place the grain of the wood as nearly as practicable in the direction of the circumference. The cants should be closely fitted, and put together with white lead or glue, strongly nailed and bolted. The wooden rim, thus made up to within about three inches of the width required for the finished pulley, is mounted upon one or two heavy iron 'spiders,' with six or eight radial arms. If the pulley is above six feet in diameter, there should be eight arms, and two spiders when the width of face is more than eighteen inches.

"Upon the ends of the arms are flat 'pads,' which should be of just sufficient width to extend across the inner face of the wooden rim, as described; that is, three inches less than the width of the finished pulley. These pads are gained into the inner side of the rim; the gains being cut large enough to admit keys under and beside the pads. When the keys are well driven, strong 'lag' screws are put through the ends of the arm into the rim. This done, an additional 'round' is put upon each side of the rim to cover bolt heads and secure the keys from ever working out. The pulley is now put to its place on the shaft and keyed, the edges trued up, and the face turned off with the utmost exactness.

"For small drivers, the best construction is to make an iron pulley of about eight inches less diameter and three inches less face than the pulley required. Have four lugs, about an inch square, cast across the face of this pulley. Make a wooden rim, four inches deep, with face equal to that of the iron pulley, and the inside diameter equal to the outer diameter of the iron. Drive this rim snugly on over the rim of the iron pulley having cut gains to receive the lugs, together with a hard wood key beside each. Now add a round of cants upon each side, with their inner diameter less than the first, so as to cover the iron rim. If the pulley is designed for heavy work, the wood should be maple, and should be well fastened by lag screws put through the iron rim; but for light work, it may be of basswood or pine, and the lag screws omitted. But in all cases, the wood should be thoroughly seasoned.

"In the early use of friction-gearing, when it was used only as backing gear in saw-mills, and for hoisting in grist-mills, the pulleys were made so as to present the head of the wood to the surface; and we occasionally yet meet with an instance where they are so made. But such pulleys never run so smoothly nor drive so well as those made with the fibre more nearly in a line with the work."[11]

[11] By E. S. Wicklin.

The driving friction may be obtained from contact of the radial surfaces in two ways: thus, Fig. 217 represents three discs, A, B, and C; the edge of A being gripped by and between B and C, which must be held together by a spiral spring S or other equivalent device. These wheels may be made to give a variable speed of rotation by curving the surfaces of the pair B C as in the figure. By means of suitable lever-motion A may be made to advance towards or recede from the centre of B and C, giving to their shaft an increased or diminished speed of revolution.

A similar result may be obtained by the construction shown in Fig. 218, in which D and E are two discs fast upon their respective shafts, and C are discs of leather clamped in E. It is obvious that if D be the driver the speed of revolution of E will be diminished in proportion as it is moved nearer to the centre of D, and also that the direction of revolution of D remaining constant, that of E will be in one direction if on the side B of the centre of D, and in the other direction if it is on the side A of the centre of D, thus affording means of reversing the motion as well as of varying its speed. A similar arrangement is sometimes employed to enable the direction of rotation of the driver shaft to be reversed, or its motion to cease. Thus, in Fig. 219, R is a driving rope driving the discs A, B, and _c_, _d_, _e_, _f_, _g_ are discs of yellow pine clamped between the flanges _h_ _i_; when these five discs are forced (by lifting shaft H), against the face of a motion occurs in one direction, while if forced against B the direction of motion of H is reversed.

For many purposes, such as hoisting, for example, where considerable power requires to be transmitted, the form of friction wheels shown in Fig. 220 is employed, the object being to increase the line of contact between wheels of a given width of face. In this case the strain due to the length of the line of contact partly counteracts itself, thus relieving to that extent the journals from friction. Thus in Fig. 221 is shown a single wedge and groove of a pair of wheels. The surface pressure on each side will be at a right angle to the face, or in the direction described by the arrows A and B. The surface contact acts to thrust the bearings of the two shafts apart. The effective length of surface acting to thrust the bearings apart being denoted by the dotted line C. The relative efficiency of this class of wheel, however, is not to be measured by the length of the line C, as compared to that of the two contacting sides of the groove, because it is increased from the wedge shape of the groove, and furthermore, no matter how solid the wheels may be, there will be some elasticity which will operate to increase the driving power due to the contact. It is to preserve the wedge principle that the wedges are made flat at the top, so that they shall not bottom in the grooves even after considerable wear has taken place. The object of employing this class of gear is to avoid noise and jar and to insure a uniform motion. The motion at the line of contact of such wheels is not a rolling, but, in part, a sliding one, which may readily be perceived from a consideration of the following. The circumference of the top of each wedge is greater than that of the bottom, and, in the case of the groove, the circumference of the top is greater than that of the bottom; and since the top or largest circumference of one contacts with the smallest circumference of the other, it follows that the difference between the two represents the amount of sliding motion that occurs in each revolution. Suppose, for example, we take two of such wheels 10 inches in diameter, having wedges and grooves 1/4 inch high and deep respectively; then the top of the groove will travel 31.416 inches in a revolution, and it will contact with the bottom of the wedge which travels (on account of its lesser diameter) 29.845 inches per revolution.

Fig. 222 shows the construction for a pair of bevel wheels on the same principle.

A form of friction-gearing in which the journals are relieved of the strain due to the pressure of contact, and in which slip is impossible, is shown in Fig. 223. It consists of projections on one wheel and corresponding depressions or cavities on the other. These projections and cavities are at opposite angles on each half of each wheel, so as to avoid the end pressure on the journals which would otherwise ensue. Their shapes may be formed at will, providing that the tops of the projections are narrower than their bases, which is necessary to enable the projections to enter and leave the cavities. In this class of positive gear great truth or exactness is possible, because both the projections and cavities may be turned in a lathe. Fig. 224 represents a similar kind of gear with the projections running lengthways of the cylinder approaching more nearly in its action to toothed gearing, and in this case the curves for the teeth and groves should be formed by the rules already laid down for toothed gearing. The action of this latter class may be made very smooth, because a continuous contact on the line of centres may be maintained by reason of the longitudinal curve of the teeth.

Cams may be employed to impart either a uniform, an irregular, or an intermittent motion, the principles involved in their construction being as follows:--Let it be required to construct a cam that being revolved at a uniform velocity shall impart a uniform reciprocating motion. First draw an inner circle O, Fig. 225, whose radius must equal the radius of the shaft that is to drive it, plus the depth of the cam at its shallowest part, plus the radius of the roller the cam is to actuate. Then from the same centre draw an outer circle S, the radius between these two circles being equal to the amount the cam is to move the roller. Draw a line O P, and divide it into any convenient numbers of divisions (five being shown in the figure), and through these points draw circles. Divide the outer circle S into twice as many equal divisions as the line O P is divided into (as from 1 to 10 in the figure), and where these lines pass through the circles will be points through which the pitch line of the cam may be drawn.

Thus where circle 1 meets line 1, or at point A, is one point in the pitch line of the cam; where circle 2 meets line 2, or at B, is another point in the pitch line of the cam, and so on until we reach the point E, where circle 5 meets line 5. From this point we simply repeat the process, the point E where line 6 cuts circle 4, being a point on the pitch line, and so on throughout the whole 10 divisions, and through the points so obtained we draw the pitch line.