CHAPTER X

COMMUNICATION OF MOTION BY LINKWORK. VELOCITY RATIO AND DIRECTIONAL RELATION CONSTANT OR VARYING.

Classification. —Discussion of Various Classes. — Quick Return Motion. —Hooke’s Coupling. — Intermittent Linkwork. —Ratchet Wheels.

176. As has been shown by the general definition (Art. 22), linkwork derives its name from the rigid connecting piece or link. This connecting piece is known by various names under different circumstances, such as connecting-rod, coupling rod, side rod, eccentric rod, etc. The arms are known as cranks when they perform complete revolutions ; and as beams, crank arms, rocker arms, or levers, when they oscillate.
177. Classification of Linkwork. —Linkwork is used,

• I. To transform circular motion into rectilinear reciprocation, or the reverse.
• II. To transform continuous rotation into rotative reciprocation, or the reverse.
  • III. To transmit continuous rotation.

Examples of the first class are seen in slotting and shaping machines, power pumps, and in the usual forms of the steam engine ; of the second class, in steam engine valve motions, where a rocker shaft is employed ; and of the third class, in locomotive side rods.

CLASS I.

Transformation of Circular Motion into Rectilinear Reciprocation, and the Reverse.

178. In Fig. 116, let AP be a crank revolving about the fixed centre A, and connected by a link PQ to a point Q, travelling in a straight line KL whose direction passes through the centre A. Let AP = R, and PQ = l. The length of the path of Q is evidently equal to 2R. When P

is at C or D, the points A, P, Q, will be in one straight line. The points C and D are called $dead\ points$ ; since when P is at either of them, the revolution of AP will cause no motion whatever to be transmitted to the point Q, for that instant. When PQ overlaps AP, as when P is at D, we shall term the point D the inward dead point; and when Q lies at the other extremity of its stroke, so that P is at C, we shall term the point C the outward dead point. In Fig. 116, let fall from P the line PE perpendicular to AQ. Then the distance of Q from A is at any instant, $AQ = QE + AE = \sqrt{l^2 - R^2 \sin^2 \theta} + R \cos \theta$ , the last term of which will be essentially negative when $\theta$ lies between $90^\circ$ and $270^\circ$ .

If PQ were of infinite length, the motion of Q would be equal to that of the point E; but as PQ is of finite length (usually from four to eight times AP), Q is drawn toward A through the distance $PQ - EQ = l - \sqrt{l^2 - R^2 \sin^2 \theta}$ . So

that when AP has moved to its mid-position Ap or Ap’ (or, as it is frequently expressed, AP is on the half-centre), Q will have passed its mid-position M by the distance $qM = AM - Aq = l - \sqrt{l^2 - R^2}$ . Also when Q is at M, P will be at some point S or S’ (Fig. 117), intermediate between

C and p or p’. These points may be readily determined, for in this case AQ = l; hence AQS and AQS’ are equal isosceles triangles, and $\cos \theta = \frac{AD}{AQ} = \frac{R}{2l}$ . The velocity ratio of P and Q varies for each instant, but may be determined at any time by means of the instantaneous centre (Art. 25) or by resolving the velocities, as in Fig. 118.

Let V be the linear velocity of P, and v that of Q. Resolve these along and perpendicular to the link PQ; then, as shown in Art. 24, Pc must be equal to Qf. Draw AN

perpendicular to AQ and intersecting the link (produced) at JSf; draw Ad perpendicular to FQ. Then, from similar triangles, we have

hence

Also

hence

make the link of the same length as the crank arm, as in Fig. 119, the point K (Fig. 116) coincides with A, and the path

of Q is AL = 2R, as before. But Q is drawn toward AP so rapidly on account of the angularity of PQ, that when

AP is perpendicular to AL, Q coincides with A and has completed its stroke. If we produce QP to V, making

PV = PQ = AP (Fig. 120), it is evident that as AP revolves as indicated by the arrow, Q will move from L to A, and V will move from A to W. If now we continue the motion of AP, Q will be driven past A to M, and V will return to A. Thus, the revolution of AP will cause Q to move over the path LM, and V over the path WX. By this means, the arm AP can be made to move the two ends of a link of twice its length through paths at right angles, and each equal in length to 4AP.

180. We have thus far considered the end of the link, Q, to travel in a path of which the direction passes through the axis A, but this path may be a straight line not passing

through A, as in Fig. 121. In this figure, let AP be a crank revolving about the centre A and connected by the link PQ to the point Q travelling in the straight line KL. An arc of a circle struck about centre A, with radius l-R, will cut the line KL at K, the end of the stroke; and the inward dead point D will lie in the straight line KAD. Similarly, the other end of the stroke, L, and the outward dead point C, may be found by striking an arc about centre A, with radius AL = l + R.

The position of the point Q corresponding to any given position of AP may be thus determined:—

Let the perpendicular distance between A and the line KL be AM = e, and let the angle $PAM = \theta$ . Then

$MQ = MS + SQ,$ $MS = R \sin \theta$ ; $SQ = \sqrt{l^2 - Ps^2} = \sqrt{l^2 - (e \pm R \cos \theta)^2}$ .

Hence we have

Also

From these expressions, the distances QL and QK can be determined.

181. By comparing Figs. 116 and 121, it will be seen, that, in the former, the outward and inward dead points, C and D, lie in one straight line; while in the latter, they depart from a straight line by the angle DAV = KAL.

This angle increases as AM is increased, and as the ratio $\frac{R}{l}$

is increased. The practical result is, that, supposing AP to revolve uniformly in the direction of the arrow, the point Q will move over its path from L to K in less time than from K to L, the times being proportional to the arcs DGC and DFC.

182. Eccentric. — A particular form of this class of link motions is the eccentric, shown in Fig. 122. A circular disc called the eccentric, or the eccentric sheave, has its centre at the point P, and is made sufficiently large to embrace the shaft at A, to which it is fastened. The eccentric is enclosed by a strap or band, FG, in which it revolves. This strap is rigidly connected to the rod or bar HN, by which motion is transmitted to the point Q. It will be seen, that, as the eccentric turns about A and slides within the strap, it will communicate exactly the same motion to the point Q as would be given by a crank arm AP and link PQ. In fact,

it is used as a substitute for small cranks on account of the practical difficulties in the formation of the latter. The travel of the point Q will, as in Fig. 116, be equal to 2AP.

The term, throw of an eccentric, is given, by various authorities, either to the arm AP, or to twice that distance; and hence the meaning of the term is often ambiguous.

CLASS II.

Transformation of Continuous Rotation into Rotative Reciprocation, and the Reverse.

183. In Fig. 123, let AP and BQ be arms turning about the fixed centres A and B respectively, and connected by the link PQ. If AP be rotated about A, it will compel BQ to oscillate between the positions Bc and Be, or Bc’ and Be’, according as the arm BQ has been previously placed above or below the centre B.

Let $AP = R,$ $BQ = r$ . $PQ = l$ , $AB = d$ .

To find the dead points: About A as a centre describe circular arcs with radii l + R and l - R. They will cut the circle about centre B, radius r, in the points c, c’, and e, e’,

respectively. These give the outward and inward dead points for R, and hence the limits of the oscillation of r. Drawing the pieces in these extreme positions, it will be seen that we obtain a series of triangles, of which the base is always the line of centres AB (= d), and of which the other two sides are r, and l + R or l - R. We will term these

dead-point triangles. As long as we can construct such triangles with a sensible altitude, it is clear that there can be no dead points for r, and hence the rotation of R will cause r to oscillate. But if with any assumed values of R, l, and r, the triangle will reduce to a straight line, r will have dead points, and we can no longer control the direction of its motion by the single combination shown. Thus, in order that the rotation of an arm R may produce oscillation of an arm r, we must have r greater than R, and also

These conditions are fulfilled by the proportions employed in Fig. 123.

184. Figs. 124, 125, and 126 are inserted to show the effects of shortening the distance between centres, retaining the same lengths of R, I, and r, as in Fig. 123.

In Fig. 124, ^d = ^I + ^r — R, and we have an outward dead point for r, simultaneously with an inward dead point for R. In Fig. 125, ^d = ^Z + i? — ?•, and we have an in ward dead point for r, simultaneously with an outward dead

point for R. These two arrangements are therefore faulty ; since when r is at a dead point, we cannot control the direction of its motion, as it is free to move either up or down

from that point. In Fig. 126, d is made still shorter, and it will be seen that r has dead points, G and G’, when R is at Ag or Ag’ respectively. Hence R cannot perform complete revolutions and drive r.

As R moves in either direction over the arc gEE’g’, r may move in either direction over the path G’e’G’G or GeGG’.

185. Problem. — Given (Fig. 127) the distance AB between two fixed centres A and B, to find the lengths of

arms and links which will cause the following arm BQ to oscillate between the positions BQ and BQ’ when the driving arm AP makes continuous rotations. Assume any convenient length for the arm BQ, and strike an arc with that length as

a radius about B, thus fixing the points Q and Q’. With A as a centre, draw the circular arcs Qq and Q’q’. Then we will have AQ = Aq = l - R; and AQ’ = Aq’ = l + R; $\therefore qq' = 2R$ . Bisect qq’ at a, then aq or aq’ is the length of the arm AP to fulfil the conditions, and Aa is the length of the link PQ = l. If these results do not give convenient lengths for R and l, a longer or a shorter length should be taken for BQ.

186. The velocity ratio in this class of linkwork is to be determined, as in Arts. 24 and 25, by the ratio of the perpendiculars let fall from the fixed centre upon the line of the link. When there are dead points, one or both of the perpendiculars disappear at the instant of passing these points; and this is just as it should be, for no motion is transmitted at that instant.

When the shorter and rotating arm is the driver, its dead points occur, as shown in the figure, at the ends of the oscillations of the following arm, and this arrangement will work satisfactorily when the conditions of Art. 184 are complied with. In case the oscillating arm drive, however, the dead points of the follower must be overcome by the momentum of the rotating pieces, increased if necessary by the addition of a fly-wheel.

CLASS III.

Transmission of Continuous Rotation.

187. Drag Link. — In Figs. 128 and 129, let AP, BQ, be two arms turning about fixed centres A and B, and connected by the link PQ, as before. In order that the continuous rotation of one may produce a continuous rotation of the other, it is necessary that there shall be no dead points. If the link PQ (= l) is made equal to Cc, it is evident that we will have an outward dead point of R at C with an inward dead point of r at c. Or, if l = Ce, we will have simultaneous inward dead points at C and e. Therefore, in order that there may be no dead points, we must make ^I > Cc and < Ce ;

T’ig. 12S

that is to say, l> ^r — B + d,and l<r + R — d; and each of the arms M and ^r must be greater than d.

Fig. 139

In this arrangement, the arms may be equal or not ; but in either case the velocity ratio, being proportional to the per pendiculars from the centres on the link, is constantly varying. The arrangement is termed a drag link. As frequently

constructed, the arms are of the same length, and the centres A and B coincide.

188. Continuous rotation may be transmitted by making the arms H and ^r of Fig. ¹²³ equal, and also ^Z = AB. This

will give us the arrangement shown in Fig. 130. The arms have simultaneous dead points when lying in the direction of the line of centres. Hence if R performs complete revolutions, r may continue to rotate in the same direction, with a constant velocity ratio, or it may move from the dead point in the opposite direction, with a varying velocity ratio ; so that for a given position, AP, of B, ^r may occupy the position BQ or Bq. To insure continuous rotation, then, by this arrangement, it is necessary to provide some means of compelling r to continue its motion past the dead points. This may be accomplished by one of the following arrangements.

189. We may connect the axes by other and similar sys tems, as shown in Figs. 131 and 132.

When two systems of arms and links are employed, as in Fig. 131, they are generally placed with the arms at right angles.

When three are used, as in Fig. 132, the arms are placed at angles of 120°.

190. Another method consists in the use of a third rotat ing arm, connected to the same link, and so placed that the driving arm may lie between the following and the auxiliary

arms. This arm must be equal in length to the other two, and must lie parallel to them in all positions. All three arms may be located on the same line of centres, as in Fig.

133 ; or the driving arm may be to one side of the line of centres of the other two, as in Fig. 134. In the latter case

VT must be part of or be rigidly connected with PQ, so that the points P, V, Q will be the vertices of a rigid triangle.

191. Boehm’s Coupling. —Another method, known as Boehm’s link coupling, is shown in Fig. 135. Two discs placed in parallel planes are fixed to parallel shafts, and connected by two or more links which make an angle with the planes of the discs. The distance between the planes of the discs must be sufficient to enable the links to pass each other,

as shown. The velocity ratio in these three arrangements is constant.

Various Applications of LINKWORK.

192. Bell-Cranks. —A form of linkwork known as bell-cranks is largely used to change the direction of motion. In Fig. 136 let ah be the direction of a reciprocating motion which it is desired to change to the direction cd. In the angle bTd formed by these directions assume any convenient point A ; and from A draw perpendiculars AB and AC on ah and cd respectively. If we construct ^a rigid piece BAG, centred at A, we can by means of it produce the change in direction desired. This piece is termed a bell-crank, and such pieces are largely used in bell-hanging, in the mechanism of organs, etc. As the angular motion of the arms is small, their lengths are sensibly equal to the perpendiculars from A upon the lines of action, and hence the velocity ratio is sensibly constant.

It is clear that we may place the centre A in any one of the four angles about T made by the lines of action. If placed in the angle bTd or aTc, the direction of motion will be as indicated by the arrows; but if we place it in the angles aTd or bTc, the direction of motion along ab being still as indicated by the arrow, that along cd will be reversed.

193. In order that the deviation of the points B and C from the lines of action shall be a minimum, and take place on both sides of the line of action instead of wholly on one side, the length of the arm AB should be equal to the perpendicular distance AB plus one-half the versed sine of the angle through which the arm swings on each side of its mid-position. This is true in all similar cases, and is illustrated by the following example: In a beam engine having a piston stroke of six feet, let the distance between the centres A of the beam gudgeons, and the centre line BP of the cylinder be eight feet. Required, the best length for the beam arm. In Fig. 137 let AB = distance between centres = 8’. Now B is to bisect the length CD, which is the sum of the deviations of the point E on both sides of BP; hence BC = BD. EC must equal the half stroke, or 3’.

In the right triangle ECA, we have

or $(8 + BC)^2 = 9 + (8 - BC)^2$ . Solving this equation for BC, we find $BC = \frac{9}{32}' = \frac{27}{8}'' = 3\frac{3}{8}''$ ,

Therefore the length of the beam arm AE should be 8’ $3\frac{3}{8}$ “, in order that the connection E may deviate the smallest amount from the line of centres BP.

194. To Multiply Oscillations by Linkwork. — In Fig. 138, let AP be an arm rotating about A, and connected by a link PC to an arm BC oscillating about B. For one revolution of AP, BC will move from the position BC to Bc and back again to BC. Now connect C by a link CE to an arm DE oscillating about D. Let AP turn from P to p, and move BC from C to c, then DE will be moved from E to E’ and back again to E. Therefore during a complete revolution of AP, DE will perform two complete cycles of motion ; going from E to E’ and back again to E during each half revolution of AP. If we connect DE to another oscillating arm KH in a similar manner, KH will perform two complete cycles for one of DE, and hence four complete cycles for one revolution of AP.

The length of the arc of oscillation of each arm depends on the length of the preceding arm, and on the versed sine of the angle through which the latter swings on each side of its mid-position.

195. To Protluce a Kapidly Varying Velocity from Uniform Motion. — In Fig. ¹³⁹ let AP be an arm oscillating with uniform velocity about A, and connected by a link PQ to an arm oscillating about ^a centre 5, placed so

that when P is at a dead point, BQ may be perpendicular to PQ. Starting from the dead point P, the uniform motion of AP produces very little motion of BQ at first, but as P moves over the equal arcs Pp, pp’, etc., Q will move through arcs rapidly increasing in length, such as Qq, qq’, etc. That is, the uniform velocity of P produces a rapidly accelerated velocity of Q. When P moves in the other direction, from p’‘’ toward P, the velocity of Q will be rapidly retarded.

196. Slow Advance and Quick Return by Linkwork. — In Fig. 140, let AP be a rotating arm, from whose

Fig. 140

uniform motion we wish to derive the motion of a second arm, such that its period of advance shall be equal, say, to twice its period of return. On a circle about A, lay off the arc PCp = one-third of the circumference; i.e., angle PAp = 120 degrees ; then AP and Ap must be the posi- tions of the driving arm when the following arm is at the ends of its stroke, and the driver is at its dead points.

On PA and Ap produced, lay off PQ = pq = proposed length of link. Then Qq must be the chord of the arc of oscillation of BQ, and the centre B must lie in the perpendicular bisecting this chord. BQ may be of any length to

give the angle of oscillation QBq desired, consistent with the deductions of Art. 184.

197. Hooke’s Coupling or Universal Joint is a contrivance, belonging to the general class of linkwork, for connecting shafts whose axes intersect.

In Fig. 141, A and B are the two shafts having semicircular jaws at their ends. The connecting and rigid cross

QPpq is formed with its four arms at right angles and in the same plane. The centre of the cross is at O, the intersection of the axes. All four arms are of the same length, and turn in bearings at P, p, Q, and q. Let the shaft A be the driver, then the ends of the arms, P and p, move in a circle whose plane is perpendicular to the axis of A; and the ends, Q and q, move in a circle whose plane is perpendicular to the axis of B. We will term these arms the driving and following arms respectively. The planes of the circles evidently intersect in a line through O, perpendicular to the plane of the axes, and the angle between the planes is equal to the angle between the axes $= \beta$ .

198. Let a plane through O, and perpendicular to the driving axis, be taken as the plane of projection. Then, Fig. 142, the circle described with radius OP = Op about as a centre, will represent the path of the points P and p. Let the plane of the axes A and B be perpendicular to the paper, intersecting it in OE. Then COD, perpendicular to OE, will be the intersection of the planes of rotation of the

driving and following arms. Draw the radius OF, making the angle $EOF = \beta =$ the acute angle between the axes; and from F draw a line parallel to COD, and intersecting OE in K. Then

and an ellipse constructed with CD as its major and OK as its semi-minor axis will represent the projection, on a plane perpendicular to the driving axis, of the path of the following arms.

Suppose a driving arm to move from the position OC to OP through the angle $COP = \theta$ . Then a line OQ drawn perpendicular to OP will be the projection of a following arm which has moved from OK, while OP moved through the angle $COP = EOQ = \theta$ . OQ is perpendicular to OP, since the latter lies in the plane of projection; and hence the angle POQ is shown in its true size. The point Q has moved through the actual vertical distance Qn, although the actual

path of Q is a circle of radius equal to OE. Therefore, if through Q we draw a line parallel to OE, and connect OR, then EOB will be the actual angle through which the follow ing arm has moved, while the driving arm has moved through COF = EOQ = e. Let angle EOE = ^.

Hence

(1) $\tan \phi = \tan \theta \cos \beta.$

To obtain the velocity ratio, we must differentiate this expression, whence

(2) $\frac{d\phi}{d\theta} = \frac{\cos^2\phi}{\cos^2\theta}\cos\beta = \cos\beta \frac{1 + \tan^2\theta}{1 + \tan^2\phi} = \frac{a'}{a}.$

Eliminating <^ and in turn from Equation (2) by means of Equation (1), we get

(3) $\frac{a'}{a} = \frac{\cos \beta}{1 - \sin^2 \theta \sin^2 \beta}.$

(4) $\frac{a'}{a} = \frac{1 - \cos^2 \phi \sin^2 \beta}{\cos \beta}.$

Starting with ^a driving arm at 00 and ^a following arm at OK, we measure the angles and rj>from these positions re spectively.

199. The expressions (3) and (4) will have minimum values when sin ^{6 =} 0, and cos (^ ^{= 1} ; in that case — ⁼ cos /?, ^a and ⁶ and <^ both = 0, tt, ² it, etc. That is, the minimum values of the velocity ratio occur when a driving arm is at 00 or OD, and the following arm is at OK or KO produced.

Maximum values occur when $\sin \theta = 1$ ; $\cos \phi = 0$ . In this case $\frac{a'}{a} = \frac{1}{\cos \beta}$ , and $\theta$ and $\phi$ both $= \frac{\pi}{2}, \frac{3\pi}{2}$ , etc. That is, the velocity ratio has its maximum value when the driving arm is at OE or EO produced, and the following arm is at OD or OC.

Hence we see that during each revolution there are two maximum and two minimum values of the velocity ratio, and that it varies between the values $\frac{1}{\cos \beta}$ and $\cos \beta$.

Between the maximum and minimum points there are four points where the ratio is unity.

The variation in the velocity ratio increases as the angle between the axes increases, and this fact must, in general, govern the employment of this joint. If the variation due to the angle between the shafts is not too great for the case under consideration, it may be employed; otherwise some other mode of connection must be used.

200. Double Hooke’s Joint.—The variation in the velocity ratio may be entirely eliminated by the use of two Hooke’s joints, arranged as in Fig. 143.

If we construct the connecting piece with the forks at its ends in the same plane, and place it so that it makes the same angle $(\beta)$ with both shafts A and B, the variation at one end of the connecting piece will be counterbalanced by the variation at its other end; and thus a uniform motion may be transmitted from A to B. For, in Fig. 143, let the plane of the axes be the plane of the paper, then considering

A as the driver, the velocity ratio between A and ab is at its maximum ; i.e.,

Also, considering ab as the driver, the velocity ratio between ab and B is at its minimum ; i.e.,

Multiplying these two equations together, we get

It will be clear upon examination that the variations thus balance each other throughout the revolution, so that — is ^a always unity ; in other words, the velocities of the two principal axes are always equal. If the cross ends of ab are in planes perpendicular to each other, the variations, instead of neutralizing each other, will evidently act together, and make the total variation in velocity of the two principal axes greater than if one joint only were employed.

Intermittent Linkwork.

201. Click and Ratchet. — An example of intermittent linkwork is the click a,nd ratchet-wheel^ a simple form of which is shown in Fig. 144. An arm AB oscillating about A has jointed to it at i? a click or catch BC. Turning about Z) is a ratchet-wheel Cc, having teeth generally of the shape shown. When the arm AB moves as indicated by the arrow, the end C of the click presses against the straight side of ^a tooth,

and thus moves the wheel. Centred at the fixed point ⁶ is a pawl or detent, he, which either rests on the teeth by its weight, or is pressed against them by a spring. This serves, as shown, to hold the wheel, and prevent its backward motion during the back stroke of AB, but offers little or no resist ance during the forward motion of AB. The pawl and tooth act upon each other by sliding contact ; and the direction of the line of action, or of the pressure between them, is a nor mal to the straight face of the tooth and end of the pawl.

Let eg be this normal, and let fall upon it the perpendiculars hg and Do. Then, if the wheel tend to turn backwards, that is,’ in the direction eg, the pawl will tend to turn about h in the same direction, eg, or towards the wheel. That is, the tendency is to force the pawl and the tooth into closer contact, which is as it should be. But if the shape of the face of tooth and pawl is such that the normal occupies some position beyond b, as eh, the tendency is to turn the pawl outward, or to cause it to slide off the tooth. Hence, with a straight pawl or click, as shown in this figure, the normal to the face of the tooth should pass between the centres

of the wheel and of the pawl. And, by similar reasoning, in the cases in which we have a hooked pawl, as in Fig. 148, the normal should pass outside of or beyond the centre of motion of the pawl.

202. Reversible Click. — In feed motions, such as those of shapers and planers, it is frequently desirable to employ a click and ratchet-wheel which will drive in either direction. An arrangement similar to that shown in Fig. 145

is then used. In the position shown, the wheel is being driven in the direction of the arrow. The teeth, as well as the click, are made alike on both sides ; so that when the click is thrown over on the other side of the arm AB, as shown in dotted lines, the wheel will be driven in the contrary direction.

203. It is usually more convenient to have the driving arm and the wheel concentric, as in Fig. 145, rather than as in Fig. 144. It is clear, that, in this case, to have an effec tive arrangement, the driving arm should move through such an angle that the end of the click shall travel through an arc

slightly greater than some multiple of the pitch arc; the excess being simply to insure that the click shall clear the correct number of teeth on its return stroke, and have the smallest possible amount of lost motion. Thus, if the click and arm drive the wheel ahead two teeth at each for ward stroke, the arc of motion of the click should be a trifle over twice the pitch arc, to insure the same amount of motion by each forward stroke.

204. Silent Click. — This contrivance (Fig. 146) avoids the clicking noise and the consequent wear of a common click. BC is the click, which, in this figure, is made to push the teeth.

It is carried by one arm, AB, of a bell-crank lever, which has the same centre of motion, A, as the ratchet-wheel. The other arm of the lever has two studs, E and E’. Between these pins is the driving arm AF, also centred at A, and con nected by ^a link OH to the click. The motion of this arm in the direction of the arrow drives the wheel in the same direction. When the motion of the arm is reversed, it at fi’-st moves back against E’ before it can move the bell-crank lever ; and, during this motion, the link GH lifts the click

clear of the teeth. Then, by pressing against the pin E’, it moves the bell-crank back to the position shown in dotted lines. The driving arm then moves ahead again against the pin E, pulling the click into gear with the teeth, as shown and then, by means of the pin E, drives the wheel ahead, as before.

205. Double-acting Click. —This arrangement, shown in Figs. 147 and 148, may be used when it is desired to drive the wheel ahead during both strokes of the driving arm.

To accomplish this result, the driving arm in Fig. 147 carries two pushing clicks ; and that in Fig. 148, two pulling clicks. The former is the stronger arrangement, and is therefore used wherever great strength is required, as in ships’ windlasses. The centre O being located, the arms GK and GH are made equal, and so placed that in their midpositions they will be perpendicular to the lines of action of the respective clicks. “With these arrangements, two or more

detents or pawls are used, so placed that they will prevent the ratchet-wheel from turning back more than one-third or one-half the pitch.

206. Frictional Catch. — This contrivance is a sort of intermittent linkwork, founded on the dynamical principle, that two surfaces will not slide on each other so long as the angle which the direction of the pressure between them makes with their common normal at the point of contact is less than a certain angle, called the angle of repose. This angle depends on the material of which the surfaces are composed, their condition as to smoothness, and on the lubrication employed. For metallic surfaces, moderately smooth, not lubricated, the sine of this angle is somewhat greater than one-seventh. In Fig. 149, the shaft and rim of the wheel to be acted upon are shown in section. AK is the catch arm, having a rocking motion about the axis A of the wheel; the link by which it is driven is supposed to be jointed to it at K. K’K’’ represents the stroke, or arc of motion, of the point K; so that K’AK’’ is the angular stroke of the catch arm. L is a socket, capable of sliding up and down on the catch arm to a small extent; a shoulder for limiting the extent of that sliding is shown by dotted lines. The socket and the part of the arm on which it slides should be

square, and not round, to prevent the socket from turning. From the side of the socket there projects a pin at D, from which the catch DGH hangs. M is a spring pressing against the forward side of the catch. G and H are two

studs on the catch, which grip and carry forward the rim BBCC of the wheel during the forward stroke, by means of friction, but let it go during the return stroke.

A similar frictional catch, not shown, hanging from a socket on a fixed instead of a movable arm, serves for a

detent, to hold the wheel still during the return stroke of the movable arm.

207. The following is the graphic construction for determining the proper position of the studs G and H:— Multiply the radii of the outer and inner surfaces, BB and GG, of the rim of the wheel, by a coefficient a little less than the sine of the angle of repose, — say one-seventh, — and with the lengths so found, as radii, describe two circular arcs about A; the greater (marked E) lying in the direction of forward motion, and the less (marked F) in the contrary direction. From D, the centre of the pin, draw DE and DF tangent to these arcs. Then G, where DE cuts BB, and H, where DF cuts CC, will be the proper positions for the points of contact of the two studs with the rim of the wheel.

The stiffness of the spring ought to be sufficient to bring the catch quickly into the holding position at the end of each return stroke.

The length of stroke of a frictional catch is arbitrary, and need not be an aliquot part of the circumference of the wheel, as is the case with the click motions described.

208. Another form of frictional catch is shown in Fig. 150.

An arm AB, centred at C, rides on a saddle which slides on the rim NN of the wheel. A piece EE is attached to

one end of the arm, and admits of being pressed firmly against the inside of the rim NN. When the end B is moved as indicated, the rim NN will be firmly grasped or nipped between the saddle and the piece EE, and will be forced to move to the right. When B is pushed back, a stop prevents BCA from turning more than is sufficient to loosen the hold of EE, and the saddle slides freely on the rim. A screw F may be employed to bring up a stop H towards the arm ACB, and so to prevent the arm from twisting into the position which gives rise to the grip of EE. No motion will then be imparted to the wheel, a result which is obtained in any ordinary ratchet-wheel by throwing the click off the teeth.