A further subdivision of the purposes of aggregate combinations leads to the following classification : AGGREGATE VELOCITIES. Production of Uniform Velocity-Ratios (as in Willis's Class A). AGGREGATE PATHS. Description of Curved Paths, (Ellipses, Epicycloids, &c.) 231. Converging Aggregate Combinations. This term may be applied to denote those trains in which the drivers in an aggregate combination are themselves the followers in aggregate combinations. By means of trains of that kind, any number of component motions may be combined. Suppose, for example, that a piece, A, is driven jointly by B and C, and that B is driven jointly by D and E, and C by F and G; then the motion of A is the resultant of four component motions, due respectively to the actions of D, E, F, and G. B G K Fig. 171. E SECTION II.-Production of Uniform 232. Differential Pulley and Windlass.In this combination, two pulleys, B and C (fig. 171), of different radii, rotate as one piece about a fixed axis, A. An endless chain, BDE CLK H, passes over both pulleys. The rims of the pulleys are shaped so as to hold the chain, and prevent it from slipping. The lines in the figure represent the pitch-lines of the pulleys and the centre line of the chain respectively. As to the relation between those lines and the actual figures of the pieces, see Articles 166, 176, pages 180, 190. One of the bights or loops in which the chain hangs, D E, passes under and supports the running block F. The other loop or bight, H K L, hangs freely; and very often the combination is driven by hauling upon the part H K; which therefore may be called the hauling part. It is evident that the velocity of the hauling part is equal to that of the pitch-circle B. Sometimes the compound pulley is driven by other means; as by a second endless chain acting on a sprocketwheel. In order that the velocity-ratio may be exactly uniform, the radius of the sheave F should be an exact mean between the radii of B and C; but it is not necessary to follow this rule strictly in practice. In stating the velocity-ratio, however, it will be assumed that the rule has been observed. Let the velocities of the pitch-circles of B and C be denoted by B and C respectively. Then the proportion of those velocities to each other is Let F denote the velocity of the running block. Then, if C were a fixed point, and consequently CE a "standing part" of the chain, the value of F would be B, and the direction of its motion would be upward (agreeably to the principles of Article 201, page 215). Also, if B were a fixed point, and B D a standing part, the value of F would be C; the negative sign being used to denote downward motion. The actual value of F is the resultant of those two components; that is to say, whence we have the comparative motion of the larger pitch-circle B, and the running block F, expressed by the following velocityratio: The velocity of the running block is the same with that of the with the same angular velocity with the actual differential or compound pulley. To calculate the length of chain required for a differential pulley, take the following sum: half the circumference of A + half the circumference of B + half the circumference of F + twice the greatest distance of F from A+ the least length of the loop HK L. This last quantity is fixed according to convenience. R The differential windlass or differential barrel (fig. 172) is identical in principle with the differential pulley; the difference in con Fig. 172. A struction being, that in the differential windlass the running block hangs in the bight of a rope whose two parts are wound round, and have their ends respectively made fast to, two barrels of different radii, which rotate as one piece about the axis A. The differential windlass is little used in practice, because of the great length of rope which it requires. That length is expressed by the following sum:-Twice the least distance of the running block from A+ half circumference of running B block + x total distance through which F F is lifted; and the last term is often an inconveniently great quantity. 233. Compound Screws. (A. M., 505.)—A compound screw consists of two screws cut upon the same spindle, and each having a nut fitted upon it. The screw turns: one of the nuts is usually fixed, so that the screw in turning in that nut is made to advance; the other nut slides, but does not turn; and the sliding motion of the second nut relatively to the first nut is the resultant of the advance of the screw relatively to the first nut, and of a motion equal and opposite to the advance of the screw relatively to the second nut; that is to say, the second nut moves relatively to the first nut as if it were acted upon by a single screw of a pitch equal to the difference between the pitches of the two screw-threads that are cut on the spindle; supposing those threads to wind the same way. But if the threads are contrary-handed, for the difference of their pitches is to be substituted the sum. Fig. 173 represents a differential screw: that is, a compound screw in which the threads wind the same way. N1 and N, are the two nuts; S1 S1, the longer-pitched thread; S S2, the shorter-pitched thread: in the figure both those threads are left-handed. At each turn of the screw the nut N2 advances relatively to N, through a distance equal to the difference of the pitches. The use of the 1 differential screw is to combine the slowness of advance due to a fine pitch with the strength of thread which can be obtained by means of a coarse pitch only. Fig. 174 represents a compound screw in which the two threads are contrary-handed; and the effect of each turn of the screw is to alter the distance between the nuts N, and No by an amount equal to the sum of the pitches of the threads, which are usually equal to each other. This combination is used to tighten the couplings of railway carriages. C B 234. Epicyclic Trains with Uniform Action.-An epicyclic train for producing an uniform aggregate velocity-ratio consists essentially of four parts, whose general arrangement may be held to be represented by the diagram in Fig. 175-viz., the primary wheels B and C, turning about the same axis, O, with different uniform velocities; the train-arm A, being a moveable frame, turning with an uniform velocity about the same axis; and the shifting train of secondary pieces, carried by the train-arm A, and transmitting an uniform velocity-ratio from B to C, in the manner of an ordinary train. The shifting train may consist of any kind of mechanism belonging to Class A; such as circular toothed wheels, whether spur, bevel, or skew-bevel; screw-gearing; circular pulleys and bands; links with equal parallel cranks; and double universal joints. Fig. 175. The comparative motions of the three primary pieces, A, B, and C, are determined in the following manner :-Let a, b, and c represent numbers proportional to the respective angular velocities of those pieces; it being understood that rotations in one direction are to be considered as positive, and those in the contrary direction as negative. First, suppose that B is fixed relatively to A; that is to say, that it simply turns along with A, having the same angular velocity; or, in symbols, that b a; then it is evident that C must turn along with A also, with the same angular velocity; that is to say, on this supposition, we have c = a. = Next, let B have a different angular velocity from A; then ba will represent the angular velocity of B relatively to A. Determine, from the construction of the shifting train, the ratio of the velocity of C to that of B, as if the train-arm A were fixed; and denote that ratio by n; taking care to mark the value of n as positive or negative, according as the rotations of B and C are in similar or contrary directions. That ratio will also be the ratio which the angular velocity of C relatively to A bears to the angu lar velocity of B relatively to A. when the train-arm A is in motion; that is to say, in symbols, and this is the general equation of the action of an epicyclic train. Two particular cases may be distinguished, according as the wheel C or the train-arm A is the follower in the combination. CASE I-The wheel B and the train-arm A are driven by means of diverging trains, with angular velocities proportional to given numbers, b and a; then the proportionate angular velocity of C is given by the following formula: c = n (b − a) + a = n b + (1 − n) a................................ (2.) CASE II The primary wheels B and C are driven by means of diverging trains with angular velocities proportional to given numbers, b and e; then the proportionate angular velocity of the train-arm a is given by the following formula : 1 1 - N n ..(3.) In some examples of both cases one of the primary wheels is fixed. Let B be that wheel; then b = 0; and we have One of the uses of epicyclic trains is to obtain with precision velocity-ratios in toothed wheel-work which are expressed by numbers whose factors are too large to be suitable for the teeth of wheels. For example, may be such a ratio; and it may be such that each of those parts is expressed by numbers whose factors are not too large; and then, by using a train-arm with the velocity |