LENGTH OF STROKE-HOOKE'S JOINT. 461 tinuously rotating piece, the ends of the stroke of the reciprocating point correspond with the dead points of the continuously revolving piece (Article 486). Let S be the length of stroke of the reciprocating piece, L the length of the line of connection, 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 their 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 be the supplement of the arc between the dead points, S= 2 (L2+ R)-2 (L-R) cos ;) 2 L2+2 R2 S2 cos = 2(L'-R) (2.) 491. Hooke's Universal Joint (fig. 220) is a contrivance for coupling shafts whose axes intersect each other in a point. Let O be the point of intersection Fig. 220. F/ F of the axes O C1, O C, and i their angle of inclination to each other. c The pair of shafts C1, C, terminate in a pair of forks F, F, 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 F, F. 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, F, (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, Its value at intermediate instants, as well as the relation between the positions of the shafts, are given by the following equations:Let 1,, be the angles respectively made by the central planes of the forks and shafts with the plane of the two axes at a given instant; then 492. The Double Hooke's Joint (fig. 221) is used to obviate the vibratory and unsteady motion caused by the fluctuation of the velocity-ratio indicated in the equations of Article 491. Between the two shafts to be connected, C1, Cs there is introduced a short intermediate shaft C, making equal angles with C, and C,, connected with each of them by a Hooke's joint, and Fig. 221. having both its own forks in the same plane. Let i be the angle of inclination of C, and C, and also that of C, and C,. Let 1, 2, 3, be the angles made at a given instant by the planes of the forks of the three shafts with the plane of their axes, and let a1, a, as, be their angular velocities. Then whence tan tan . = cos i = tan 1tan ; tan 1; and σ =αr; so that the angular velocities of the first and third shafts are equal to each other at every instant. 493. A Click, being a reciprocating bar, 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. SECTION 5.-Reduplication of Cords. 494. Definitions.-The combination of pieces connected by the several plies of a cord or rope consists of a pair of cases or frames called blocks, each containing one or more pulleys called sheaves. One of the blocks called the fall-block, B1, is fixed; the other, or running-block, B., is moveable to or from the fall-block, with which it is connected by means of a rope of which one end is attached either to the fall-block or to the running-block, while the 495. The Velocity-Ratio chiefly considered in a tackle is that between the velocities of the running-block, u, and of the tackle-fall, v. That ratio is given by equation 6 of Article 402 (which see), viz :— Fig. 222. Fig. 223. v = nu;...... where n is the number of plies of rope by which the running-block is connected with the fall-block. Thus, in fig. 222, n = 7; and in fig. 223, n = 6. 496. The Velocity of Any Ply of the rope is found in the following manner : I. For a ply on the side of the fall-block next the tackle-fall, such as 2, 4, 6, fig. 222, and 3, 5, fig. 223, it is to be considered what would be the velocity of that ply if it were itself the tacklefall. Let that velocity be denoted by v', and let n' be the number of plies between the ply in question and the point of attachment by which the first ply (marked 1 in the figures) is fixed to one or other block. Then v' = n' u....... ..... II. For a ply on the side of the fall-block farthest from the tackle-fall, the velocity is equal and contrary to that of the next succeeding ply, with which it is directly connected over one of the sheaves of the fall-block. III. If the first ply, as in fig. 223, is attached to the fall-block, its velocity is nothing; if to the running-block, its velocity is equal to that of the block. 497. White's Tackle.—The sheaves in a block are usually made all of the same diameter, and turn on a fixed pin; and they have, consequently, different angular velocities. But by making the diameter cf each sheave proportional to the velocity, relatively to the block, of the ply of rope which it is to carry, the angular velocities of the sheaves in one block may be rendered equal, so that the sheaves may be made all in one piece, and may have journals turning in fixed bearings. This is called White's Tackle, from the inventor, and is represented in figs. 222 and 223. SECTION 6.-Hydraulic Connection. 498. The General Principle of the communication of motion between two pistons by means of an intervening fluid of constant density has already been stated in Article 411, viz., that the velocities of the pistons are inversely as their areas, measured on planes normal to their directions of motion. Should the density of the fluid vary, the problem is no longer one of pure mechanism; because in that case, besides the communication of motion from one piston to the other, there is an additional motion of one or other, or both pistons, due to the change of volume of the fluid. 499. Valves are used to regulate the communication of motion through a fluid, by opening and shutting passages through which the fluid flows; for example, a cylinder may be provided with valves which shall cause the fluid to flow in through one passage, and out through another. Of this use of valves, two cases may be distinguished. I. When the piston moves the fluid, the valves may be what is called self-acting; that is, moved by the fluid. If there be two passages into the cylinder, one provided with a valve opening inwards, and the other with a valve opening outwards; then during the outward stroke of the piston the former valve is opened and the latter shut by the inward pressure of the fluid, which flows in through the former passage; and during the inward stroke of the piston, the former valve is shut and the latter opened by the outward pressure of the fluid, which flows out through the latter passage. This combination of cylinder, piston, and valves, constitutes a pump. II. When the fluid moves the piston, the valves must be opened and shut by mechanism, or by hand. In this case the cylinder is a working cylinder. 500. In the Hydraulic Press, the rapid motion of a small piston in a pump causes the slow motion of a large piston in a working cylinder. The pump draws water from a reservoir, and forces it into the working cylinder; during the outward stroke of the pump piston, the piston of the working cylinder stands still; during the inward stroke of the pump piston, the piston of the working HYDRAULIC HOIST-TRAINS. 465 cylinder moves outward with a velocity as much less than that of the pump piston as its area is greater. When the piston of the working cylinder has finished its outward stroke, which may be of any length, it is permitted to be moved inwards again by opening a valve by hand and allowing the water to escape. 501. In the Hydraulic Hoist, the slow inward motion of a large piston drives water from a large cylinder into a smaller cylinder, and causes a more rapid outward motion of the piston of the smaller cylinder. When the latter piston is to be moved inward, a valve between the two cylinders is closed, and the valve of an outlet from the smaller cylinder opened, by hand, so as to allow the water to escape from the smaller cylinder. The larger cylinder is filled and its piston moved outward, when required, by means of a pump, in a manner resembling the action of a hydraulic press. (See also p. 642.)* SECTION 7.-Trains of Mechanism. 502. Trains of Elementary Combinations have been defined in Article 435, and illustrated in the case of wheelwork, in Article 449, and in the case of a double Hooke's joint, in Article 492. The general principle of their action is that the comparative motion of the first driver and last follower is expressed by a ratio, which is found by multiplying together the several velocity-ratios of the series of elementary combinations of which the train consists, each with the sign denoting the directional relation. 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. *See Hydraulic Power and Hydraulic Machinery, Robinson, 2nd edition, and A Manual of the Steam Engine and other Prime Movers, Rankine. |