APPLIED MECHANICS.] MECHANICS 7G3 them to depends upon which of those two effects, even supposing occur together, is the practical object of the mechanism. 83. Reduplication of Cords Differential Windlass Blocks, Sheaves, and Tackle. The axis C (tig. 26) carries a larger barrel AE and a smaller barrel DB, rotating as one piece with the angular velocity otj in the direction AE. The pulley or sheave FG has a weight W hung to its centre. A cord lias one end made fast to and wrapped round the barrel AE ; it passes from A under the sheave FG, and has the other end wrapped round and made fast to the barrel BD. Required the relation between the velocity of trans lation r. 2 of W and the angular velocity c^ of the differential barrel. In this case v^ is an aggregate velocity, pro duced by the joint action of the two drivers AE and BD, transmitted by wrapping connectors to FG, and combined by that sheave so as to act on the follower W, whose motion is the same with that of the centre of FG. The velocity of the point F is ctj.AC, upward Pi motion being considered positive. The velocity of the point G is - ctj.CB, downu ard motion being negative. Hence (p. 690, 71, 72) the instantaneous axis of the sheave FG is in the diameter FG, at the distance FG 2 AC-BC AC + BC from the centre towards G ; the angular velocity of the sheave is AC + BC a 2 - ai . FG and, consequently, the velocity of its centre is FG AC - BC _ ni (AC-BC) r -~ a -- 2 AC + BC ~ ~2 or the mean between the velocities of the two vertical jiarts of the cord. If the cord be fixed to the frame-work at the point B, instead of being wound on a barrel, the velocity of W is half that of AF. A case containing several sheaves is called a block. A fall-block is attached to a fixed point ; a running-block is movable to and from a fall-block, with which it is connected by two or more plies of a rope. The whole combination constitutes a tackle or purchase. The two plies of a rope at opposite sides of a sheave in the fall- block have equal and opposite velocities. The two plies at opposite sides of a sheave in the running-block have velocities (as in the case of the sheave FG) differing equally in opposite directions from the velocity of the running-block. One end of the rope is fastened either to the fall-block or the running-block The other, or free end, is called the full. Let i be the velocity of the fall, v. 2 that of the running-block ; and let it be required to find their ratio ; and let velocities towards the fall- block be positive, and from it negative. CASE 1. If the fastened end of the rope be attached to the fall- block its velocity is 0, and this also is the velocity of the first ply. The rope passes under a sheave in the running-block, so that the velocity of the second ply is 2v. 2 . It then passes over a sheave in the fall-block ; the velocity of the third ply is - 2i 2 ; then under a sheave in the running-block ; the velocity of the fourth ply is 4i, ; and so on, the general law being this : let n be an even number, then the velocity of the n th ply = ?ii>., ,, (?i + l) th ply= -nv., . . . (43). = t- 1; if the fall be the (?i + l) th ply" CASE 2. If the fastened end of the rope be attached to the run ning-block, the velocity of the first ply is v. 2 ; of the second, -i; 2 ; of the third, 3r. 2 ; of the fourth, - Bv. 2 ; and, generally, if n be an odd number, velocity of the ?i th ply = ?it? 2 j ,, " (?i + l) th ply= -nv. 2 > . . . (44). = 1 !, if the fall be the (?i + l) th ply ] Generally, (45), where n is the number of plus of rope by which the running-block hangs. 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 of 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. For details and technical terms see SHIPBUILDING. 84. Differential Screw. On the same axis let there be two screws of the respective pitches p } and^.,, made in one piece, and rotatin" with the angular velocity a. Let this piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut A. The velocity of advance of B relatively to C is (according to sect. 41) oft, and of A relatively to B (according to sect. 67)-<yv hence the velocity of A relatively to C is o-(Pi -!>) (46), being the same with the velocity of advance of a screw of the pitch Pi -p. 2 . Tim combination, called Hunter s or the differential screw, combines the strength of a large thread with the slowness of motion due to a sraall one. J35. Epicyclic Trains. The term epicyclic train is used by Willis to denote a train of wheels carried by an arm, and having certaia rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels or assist in driving them. The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in sects. 37 to 40. 86. Link Motion. Let S (fig. 27) be the shaft of a steam-engine, its axis, E/ iliefonvard eccentric, suitably placed for moving the slide-valve when the shaft rotates forwards, F its centre, OF its crank-arm, C/ its rod, E& the backward eccentric, suitably placed for moving the slide-valve when the shaft rotates backwards, B its centre, OB its crank-arm, C& its rod. L is a long narrow box called the link, jointed at T/ and T& to the eccentric rods ; R is the valve- rod which works the slide-valve, jointed to P, a slider, which, by moving either L or R, or both, can be adjusted to any required posi- Fig. 27. tion in the link. When P is at T/ the valve is said to be in full forward gearing, being acted upon by / alone. When P is at Tj the valve is said to be in full backward gearing, being acted upon by E;, alone. When P is placed in an intermediate position, the valve has an aggregate motion due to the joint action of E/ and 5. The most exact mode of determining that motion is to make a skeleton drawing of the apparatus in various positions ; but an approximation to the motion of the valve may be obtained by join ing FB, and taking Q so that T/P : T 6 P : : FQ : BQ ; then the valve will move nearly as if it were worked by one eccentric, having its centre at Q. 87. Parallel Motions (exact}. A parallel motion is a combination of turning pieces in mechanism designed to guide the motion of a reciprocating piece either exactly or approximately in a straight line, so as to avoid the friction which arises from the use of straight guides for that purpose. Fig. 28 represents an exact parallel motion, first proposed, C ( it is believed, by Mr Scott Russell. The arm CD turns on the axis C, and is jointed at D 1o the middle of the bar ADB, whose length is double of that of CD, and one of whose ends B is jointed to a slider, sliding in straight guides along the line CB. Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis of the bar ADB ; and the direction of motion of A is at every instant perpendicular to EA, that is, along the straight line ACrr. While the stroke of A is AC, extending to equal distances on either side of C, and equal to twice the chord of the arc Dd, tl stroke of B is only equal to twice the sagitta ; and thus A is guided through a comparatively long stroke by the sliding < through a comparatively short stroke, and by rotatory motions a the jomts C, D, B. (For details, see STEAM-ENGINE) Fig. 28. , , . , 88. Parallel Motion Watt s Apirroximatc (see ng. 29). Let , ct be a pair of levers connected by a link Ft, and osci out the. axes Cc between the positions marked 1 and 3. Let 1 middle positions of the levers CT 2 , ct. 2 be parallel to each other is required to find a point P in the link It such that its muU CT
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