235 CHAPTER V. OF AGGREGATE COMBINATIONS IN MECHANISM. SECTION I.-General Explanations. 225. Aggregate Combination Defined.- "Aggregate Combinations" is a term introduced by Professor Willis, to denote those assemblages of pieces in mechanism in which the motion of one follower is the resultant of motions impressed upon it by more than one driver. The number of independently-acting drivers which impress directly a compound motion on one follower cannot be greater than three; because each driver determines the motion of at least one point in the follower; and the determination of the motion of three points in a body determines the motion of the whole body. In most cases which occur in practice, the number of independent drivers which act directly on one follower is two. 226. General Principle of their Action.—The follower which has such a compound motion directly communicated to it by more than one primary piece must necessarily be a secondary piece, as defined in Article 37, page 17; its motion at any instant is the resultant of the motions impressed upon it separately by the pieces which act as its drivers; and the determination of that resultant motion depends upon the principles already explained in Chapter III. of this Division, pages 43 to 75. Several examples of the motion of secondary pieces have been given in the preceding Chapter, in treating of those secondary pieces, such as links and bands, and the sheaves of running blocks, which act as connectors in elementary combinations. 227. Aggregate Combinations terminating in a Primary Piece.— Very often an aggregate combination is of the nature of a train; and although a secondary piece receives in the first instance a compound motion from two or from three primary pieces, that secondary piece communicates motion in the end to a primary piece. In such cases the motion of that last primary follower may be determined, by finding the motions which would be communicated to it through the intermediate secondary piece or pieces by the several primary drivers acting separately, and taking the resultant of those motions. 228. Shifting Trains.-A secondary piece in an aggregate combination has very often a form like that of a primary piece, and is distinguished from a primary piece only by the fact that its bearings, instead of being carried by the fixed frame, are carried by a moving frame; that moving frame being one of the primary pieces from which the secondary piece receives its motion. For example, a wheel may turn about an axis which is carried by an arm that turns about another axis. The compound motions of which such secondary pieces are capable have been treated of in Articles 72 to 79, pages 51 to 62, and Articles 81 to 86, pages 66 to 74. When such a secondary piece is to drive or to be driven by a primary piece, or another secondary piece not carried by the same moving frame, special contrivances, which may be called shifting trains, have to be used in order to keep up the connection between the two pieces during their various changes of relative position. The following are examples: I. When two pieces turning about parallel axes are connected by toothed gearing, and one of them is free to shift its position along its axis relatively to the other, the LONG or BROAD PINION may be used. In fig. 167 A A and B B are a pair of parallel axes; C, a spur-wheel on A A; D, a pinion on B B; and the breadth of the pitch-surface of D is made greater than that of C by a length equal to the distance through which D is capable of being shifted longitudinally. II. When a toothed wheel, C C, fig. 168, gears with a rack, D D, and either the rack is to be capable of turning about an axis, B B, parallel to its pitch-line, or the axis A of the wheel is to be capable of being moved round the axis B B at the end of an arm, FA, the CIRCULAR RACK is to be used, being, as represented in the figure, a solid of revolution generated by the rotation of the trace of the rack-teeth about the axis B B. The pitch-line DD becomes the trace of an imaginary pitch-cylinder generated by its revolution about the axis B B; and the pitch-point E is the point of contact of that cylinder with the pitch-cylinder of the wheel. It is easy to see that by fixing a broad pinion on one part of a shaft, and a circular rack on another, that shaft may receive at the same time two independent motions of rotation about its axis and translation along its axis respectively, from two different spurwheels; the result being a helical motion; and this is one of the simplest of aggregate combinations. III. TRAIN-ARM.-When rotation is to be transmitted from a fixed axis to a shifting axis, or from one shifting axis to another, and the relative motion of the two axes is such that their distance apart, and the angle which their directions make with each other, do not change,-in other words, when one of the two axes revolves round the other as if it were carried by a rotating arm,—the connection between those axes may be kept up by means of one rigid frame, which carries any combination or train of mechanism suitable for transmitting rotation from the one axis to the other: such a frame is called a train-arm. B The general principles of the velocity-ratios which are communicated by means of train-arms will be stated further on; but at present one particular case requires special mention,-it is that in which the train carried by the arm is such that the two axes connected by it are parallel, and the angular velocities of the pieces. which turn about them equal and in the same direction. In fig. 169 the plane of projection is supposed to be normal to the two axes to be connected, A and B the traces of those two axes, and A B their common perpendicular. A moveable frame or train-arm connects the bearings of the axes with each other, so that the distance A B is invariable; A and that frame carries a train of mechanism such as to transmit the angular velocity of the piece which turns about A unchanged in velocity and direction to the piece which turns about B. For example, those pieces may have pairs of parallel and equal cranks linked together by coupling-rods; or they may be equal and similar pulleys connected by a band; or equal and similar toothed wheels, with an intermediate wheel gearing with both. The result is, that while the train-arm turns Fig. 169. into any other position, such as A b, the angular velocities of the pieces which rotate about the axes A and B respectively continue to be equal in magnitude and identical in direction. IV. When rotation is to be transmitted between a pair of axes whose common perpendicular alters in length as well as in direction, a COMPOUND TRAIN-ARM may be used, consisting of two or more train-arms jointed together at intermediate axes. For example, in fig. 170, A and C are the traces of two such axes. B A is the trace of an intermediate axis, connected by means of two train-arms with A and with C respectively, so that the distances A B and B C are invariable; while A B can be turned into any angular position about A, such as A b, and B C into any angular position about B, such as b c. Then the relative position of A and C can be altered either in direction or in distance, so long as their distance apart does not exceed A B+ B C; and the transmission of motion will still be kept up by means of the trains that are carried by the train-arms. Fig. 170. V. When motion is transmitted between two axes by means of a band, the connection may be maintained during changes of the relative position of those axes by means of STRAINING PULLIES and GUIDING PULLIES So arranged as to keep the band tight. 229. Methods of Treating Problems respecting Aggregate Combinations. The methods by which problems respecting aggregate combinations are solved may be distinguished into two classes. I. In one class a piece which may be regarded as a train-arm, or moving frame (and which may be designated by B), has a given motion relatively to the fixed frame, A, of the machine; and at the same time a secondary moving piece, C, has a given motion relatively to B. The resultant of those two given motions is the motion of C relatively to A; and the general rules for finding it in various cases have been stated in Articles 73 to 77, pages 52 to 56, and Articles 81 to 86, pages 66 to 74. II. In the other class of methods the motions of three points in a secondary piece that is free to move in all directions, or, more frequently, the motions of two points in a secondary piece that is guided so as to move in one plane, or about one fixed point, are given; and the motion of the piece as a whole is to be deduced from them. The general rules for doing this have been given in Articles 69 to 71, pages 45 to 51. There is no difference in principle between the kinds of problems that are treated by those two classes of methods respectively; the choice of methods is a matter of convenience only. 230. Aggregate Combinations classed according to their Purposes —Aggregate Velocities—Aggregate Paths.— -The classification of aggregate combinations which will be adopted throughout the rest of this Chapter is that of Mr. Willis, and is founded on the purposes which the combinations are designed to effect. Those purposes are distinguished into (I.) aggregate velocities, and (II.) aggregate paths. I. When an aggregate velocity is the object aimed at, the final piece of the train is usually a primary piece, whose comparative velocity, by the help of an aggregate combination, is made either to have a certain constant value or to vary according to a law which it might be difficult or impossible to realize by means of a train of elementary combinations only. II. When an aggregate path is the object aimed at, a point in a secondary piece is made, by means of an aggregate combination, to move in a path of a figure which may be different from that which a point in a primary piece would describe. * The only paths which points in primary pieces can describe are straight lines, circles, and screw lines; and paths of all other figures must be described by the help of aggregate combinations. Sometimes, indeed, it is found convenient to use aggregate combinations for describing, either exactly or approximately, even those elementary paths themselves-the straight line, the circle, and the screw-line. For example, there is a numerous class of aggregate combinations called parallel motions, whose object is to make a point move sensibly in a straight line. * In other words, paths in which both the curvature and the tortuosity are either none or uniform. The curvature of a path is the reciprocal of the radius of curvature. The tortuosity is the reciprocal of the length, measured along the path, in the course of which the radius of curvature rotates round a tangent to the path as an axis, through the angle which subtends an arc equal to radius. In the case of a helix, or screw-line, let r be the radius of the cylinder on which the screw-line is described, and p the pitch of that = 2 π line; let Pbe the radius of a circle whose circumference is equal to the pitch; let be the obliquity of the screw-line to a plane normal to its axis; let P be its radius of curvature; and let o be the reciprocal of the tortuosity. Then q = r tan ; and according to Article 64, page 41, the radius of curvature is Also, it can be shown that the reciprocal of the tortuosity is |