762 MECHANICS [APPLIED MECHANICS. ties of which turn the gudgeons at the ends of the arms of a rec tangular cross, having its centre at 0. This cross is the link ; the connected points are the centres of the bearings F 1; F 2 . At each instant each of those points moves at right angles to the central plane of its shaft and fork ; therefore the line of intersection of the central planes of the two forks at any instant is the instan taneous axis of the cross, and the velocity ratio of the points Fj, F. 2 (which, as the forks are equal, is also the angular velocity ratio of the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity ratio is that of equality, for each successive quarter-turn is made by both shafts in the same time ; but its actual value fluctuates between the limits when F! is the plane COS0 and = cos0 when F 2 is in that plane (39). Its value at intermediate instants is given by the following equa tions : let fa, < 2 be the angles respectively made by the central planes of the forks and shafts with the plane OCjCg at a given instant ; then cos 6 = tan fa tan fa (40). ttj dfa tan < 2 + cot 77. Intermittent Link-work Click and Hatchet. A click acting upon a ratchet-wheel or rack, which it pushes or pulls through a certain arc at eich forward stroke and leaves at rest at each back ward stroke, is an example of intermittent linkwork. During the forward stroke the action of the click is governed by the principles of linkwork ; during the backward stroke that action ceases. A catch or 2>aU, turning on a fixed axis, prevents the ratchet-wheel or rack from reversing its motion. Division 5. Trains of Mechanism. 78. General Principles. A train of mechanism consists of a series of pieces each of which is follower to that which drives it and driver to that which follows it. The comparative motion of the first driver and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the directional relations of the several elementary combinations of which the train consists. 79. Trains of Wheelwork. Let A lt A 2 , A 3 , &c. , A m _i, A m denote a series of axes, and a^ a 2 , o 3 , &c., a m -i, a m their angular velocities. Let the axis A l carry a wheel of Nj teeth, driving a wheel of ?i. 2 teeth on the axis A. 2) which carries also a wheel of N 2 teeth, driving a wheel of n 3 teeth on the axis A 3 , and so on ; the numbers of teeth in drivers being denoted by Ws, and in followers by ris, and the axes to which the wheels are fixed being denoted by numbers. Then the resulting velocity ratio is denoted by <^ = <^ . 3. &c _ a, = N! . N 2 . . &c. . . . N TO -i , c^ % o 2 a m _i ?i 2 . 3 . . &c. . . . n m that is to say, the velocity ratio of the last and first axes is the ratio of the product of the numbers of teeth in the drivers to the product of the numbers of teeth in the followers. Supposing all the wheels to be in outside gearing, then, as each elementary combination reverses the direction of rotation, and as the number of elementary combinations m - 1 is one less than the number of axes m, it is evident that if m is odd the direction of rotation is preserved, and if even reversed. It is often a question of importance to determine the number of teeth in a train of wheels best suited for giving a determinate velocity ratio to two axes. It was shown by Young that, to do this with the least total number of teeth, the velocity ratio of each ele mentary combination should approximate as nearly as possible to 3 59. This would in many cases give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6 ought to be the limit of the velocity ratio of an elementary combination in wheelwork. The smallest number of teeth in a pinion for epicy- cloidal teeth ought to be twelve (see sect. 59), but it is better, for smoothness of motion, not to go below fifteen; and for involute teeth the smallest number is about twenty-four. Let p be the velocity ratio required, reduced to its least terms, and let B be greater than C. If ^ is not greater than 6, and C lies
between the prescribed minimum number of teeth (which may be of wheels will answer the be the numbers required. they are, if possible, to be (or if they are too small, of teeth. Should B or C, and prime, then, instead of the exact ratio -- , some ratio approximating to that ratio, and called {) and its double 2t, then one pair purpose, and B and C will themselves Should B and C be inconveniently large, resolved into factors, and those factors multiples of them) used for the number or both, be at once inconveniently large capable of resolution into convenient factors, is to be found by the method of continued fractions. T> Should p be greater than 6, the best number of elementary com binations m - 1 will lie betw r eeu log B - log C - and log 6 log B - log C Ios3 Then, if possible, B and C themselves are to be resolved each into m - 1 factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6^ ; or if B and C contain inconveniently-large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substi tuted for ~- as before. U So far as the resultant velocity ratio is concerned, the order of the drivers N" and of the followers n is immaterial ; but to secure equable wear of the teeth, as explained insect. 54, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and n shall be either 1, or as small as possible. 80. Double Hooke s Coupling. -It has been shown in section 76 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cos0 and . Hence one or both J cos 6 of the shafts must have a vibratory aud unsteady motion, injurious to the mechanism and framework. To obviate this evil a short in termediate shaft is introduced, making equal angles with the first and last shaft, coupled with each of them by a Hooke s joint, and having its own two forks in the same plane. Let a lt a 2 , ct 3 be the angular velocities of the first, intermediate, and last shaft in this train of two Hooke s couplings. Then, from the principles of sect. 76 it is evident that at each instant -- = - 2 , and consequently that a l a 3 03 = 0,! ; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant. 81. Converging and Diverging Trains of Mechanism. Two or more trains of mechanism may converge into one, as when the two pistons of a pair of steam-engines, each through its own connecting- rod, act upon one crank-shaft. One train of mechanism may diverge into two or more, as when a single shaft, driven by a prime mover, carries several pulleys, each of which drives a different machine. The principles of comparative motion in such converging and diverging trains are the same as in simple trains. Division 6. Aggregate Combinations. 82. General Principles. Willis has designated as "aggregate combinations" those assemblages of pieces of mechanism in which the motion of one follower is the resultant of component motions impressed on it by more than one driver. Two classes of aggregate combinations may be distinguished which, though not different in their actual nature, differ in the data which they present to the designer, and in the method of solution to be followed in questions respecting them. Class I. comprises those cases in which a piece A is not carried directly by the frame C, but by another piece B, relatively to which the motion of A is given, the motion of the piece B relatively to the frame C being also given. Then the motion of A relatively to the frame C is the resultant of the motion of A relatively to B and of B relatively to C ; and that resultant is to be found by the prin ciples already explained in division 3 of this chapter, sects. 34 to 41. Class II. comprises those cases in which the motions of three points in one follower are determined by their connexions with two or with three different drivers, so that the motion of the follower, as a whole, is to be determined by the principles of 71, 78, pp. 690, 692. This classification is founded on the kinds of problems arising from the combinations. Willis adopts another classification, founded on the objects of the combinations, which objects he divides into two classes, viz., (1) to produce aggregate velocity, or a velocity which is the resultant of two or more components in the same path, and (2) to produce an aggregate path, that is, to make a given point in a rigid body move in an assigned path by communicating certain motions to other points in that body. It is seldom that one of these effects is produced without at the
same time producing the other ; but the classification of Willis