dead-points of the path of the connected point of the turning piece, where the line of connexion is continuous with or coincides with the crank-arm.
Let S be the length of stroke of the reciprocating piece, L the length of the line of connexion, and R the crank-arm of the continuously turning piece. Then, if the two ends of the stroke be in one straight line with the axis of the crank,
and if these ends be not in one straight line with that axis, then S, L − R, and L + R, are the three sides of a triangle, having the angle opposite S at that axis; so that, if θ be the supplement of the arc between the dead-points,
| S2 = 2 (L2 + R2) − 2 (L2 − R2) cos θ, cos θ = | 2L2 + 2R2 − S2 | . | |
| 2 (L2 − R2) |
 |
| Fig. 111. |
§ 66. Coupling of Intersecting Axes—Hooke’s Universal Joint.—Intersecting axes are coupled by a contrivance of Hooke’s, known as the “universal joint,” which belongs to the class of linkwork (see fig. 111). Let O be the point of intersection of the axes OC1, OC2, and θ their angle of inclination to each other. The pair of shafts C1, C2 terminate in a pair of forks F1, F2 in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at O. This cross is the link; the connected points are the centres of the bearings F1, F2. 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 instantaneous axis of the cross, and the velocity ratio of the points F1, F2 (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:—
| α2α1 = 1cos θ when F1 is the plane of OC1C2 | ||
| and | α2α1 = cos θ when F2 is in that plane. |
Its value at intermediate instants is given by the following equations: let φ1, φ2 be the angles respectively made by the central planes of the forks and shafts with the plane OC1C2 at a given instant; then
| cos θ | = tan φ1 tan φ2, | |
| α2α1 = −dφ2dφ1 | = tan φ1 + cot φ1tan φ2 + cot φ2. |
§ 67. Intermittent Linkwork—Click and Ratchet.—A click acting
upon a ratchet-wheel or rack, which it pushes or pulls through a
certain arc at each forward stroke and leaves at rest at each backward
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 pall, turning on a fixed axis, prevents the ratchet-wheel or
rack from reversing its motion.
Division 5.—Trains of Mechanism.
§ 68. 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.
§ 69. Trains of Wheelwork.—Let A1, A2, A3, &c., Am−1, Am denote a series of axes, and α1, α2, α3, &c., αm−1, αm their angular velocities. Let the axis A1 carry a wheel of N1 teeth, driving a wheel of n2 teeth on the axis A2, which carries also a wheel of N2 teeth, driving a wheel of n3 teeth on the axis A3, and so on; the numbers of teeth in drivers being denoted by N′s, and in followers by n’s, and the axes to which the wheels are fixed being denoted by numbers. Then the resulting velocity ratio is denoted by
| αm | = | α2 | · | α3 | · &c. . . . | αm | = | N1 · N2 . . . &c. . . . Nm−1 | ; |
| α1 | α1 | α2 | αm−1 | n2 · n3 . . . &c. . . . nm |
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 elementary 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 wheel-work. The smallest number of teeth in a pinion for epicycloidal teeth ought to be twelve (see § 49)—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 B/C be the velocity ratio required, reduced to its least terms, and let B be greater than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of teeth (which may be called t) and its double 2t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are, if possible, to be resolved into factors, and those factors (or if they are too small, multiples of them) used for the number of teeth. Should B or C, or both, be at once inconveniently large and prime, then, instead of the exact ratio B/C some ratio approximating to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.
Should B/C be greater than 6, the best number of elementary combinations m − 1 will lie between
1log B − log Clog 6 and 1log B − log Clog 6.
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 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.
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 in § 44, 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.
§ 70. Double Hooke’s Coupling.—It has been shown in § 66 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cos θ and 1/cos θ. Hence one or both of the shafts must have a vibratory and unsteady motion, injurious to the mechanism and framework. To obviate this evil a short intermediate 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 α1, α2, α3 be the angular velocities of the first, intermediate, and last shaft in this train of two Hooke’s couplings. Then, from the principles of § 60 it is evident that at each instant α2/α1 = α2/α3, and consequently that α3 = α1; 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.
§ 71. 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.
§ 72. General Principles.—Willis 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 principles already explained in Division 3 of this Chapter §§ 27-32.
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.
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
