July 6, 1876] 



NATURE 



213 



THE KINEMATICS OF MACHINERY 



THE study of pure mechanism, a branch of kinematics, 

 in general consists of the solution of the following 

 problems : — Given the mode of connection of two or more 

 points or bodies with each other, required their compa- 

 rative motion, and conversely given their comparative 

 motion to find their proper connection. Now the compa- 

 rative motion of two points is determined, as laid down by 

 Willis, when (i) the velocity ratio or the proportion which 

 their velocities bear to each other, and (2) their directional 

 relations, are known ; the latter requiring for its complete 

 determination {a) the angle between the directions com- 

 pared, {b) the angle which the plane containing the two 

 directions makes with a plane hxed in space, and [c) the 

 angle the intersection of the two planes makes with a fixed 

 line on the latter plane. In " Kmematics of Machinery," 

 the English translation by Prof. A, B. W. Kennedy of 

 Prof. Reuleaux's " Theoretische Kinematik," 1 the study is 

 confined within narrower hmits, causing the translator 

 not a little difiSculty, as he expresses in his preface, to 

 translate the word kinematic, carrying as it does a more 

 extended signification here than on the Continent. 

 Starting with the condition that the change of position is 

 definite at each instant, and determined by the form and 

 connection of the fixed and moving parts, Prof. Reuleaux 

 proceeds to investigate the directional relations of the 

 motion and the arrangements of the parts by which the 

 motion is best brought about without any reference to the 

 idea of velocity. 



On turning to the Appendix, pp. 585-589, we find a 

 most interesting historical collection of the definitions of 



a machine, one of which definitions ve remark includes 

 equally an adhesive fly-paper and the red-hot poker of 

 the clown; that given by Reuleaux, p. 35, is more concise 

 and certainly nearer the point than most of these. " A 

 machine is a combination of resistant bodies so arranged 

 that by their means the mechanical forces of nature can 

 be compelled to do work accompanied by certain deter- 

 minate motions." 



Thus the prevention, by the resistance of the different 

 parts, of all motion other than that desired, as well as 

 the conversion into useful work of as much of the energy 

 expended as its efficiency permits, is the function of the 

 machine. " Those parts of a machine transmitting the 

 forces by which the moving points are caused to limit 

 their motions in the definite and required manner, must 

 be bodies of suitable resistant capacity ; the moving parts 

 themselves must belong also to similar bodies." But the 

 determination of the suitable form and sectional area of 

 the resistant parts, though indispensable in the construc- 

 tion of the machine, belongs to another part of the study 

 of machine design, and cannot be included in the kine- 

 matic discussion. 



We now come to the conception of a pair of elements. 



In order that a body B (Fig. i) may prevent all 

 ether motion in the body A than that desired, A being 



' "The Kinematics of Machinery : Outlines of a Theory of M»chines." 

 By K. Reuleaux, Director of and Professor in the Kiiniglichen Gewerbe- 

 Akademie in Berlin, Member of the Konigl. technischen Deputation fur 

 Gewerbe. Translated and edited by Alex. B. W. Kennedy, C.E., Professor 

 of Civil and Mechanical Engineering in University College, Loodon. 

 (London: Macmillan and Co., 1876.) 



assumed to move in a plane, B must be given a form such 

 that it always remains in contact with A in all its required 

 positions ; when that has been done no other motion can 

 be given to A with respect to B. This geometrical form 

 of B is called the envelope of A, and it is plain that the 

 motion of B with respect to A, considered fixed, is the 

 same as that of A with respect to B, and that no other 

 motion of B with respect to A is possible ; that is, A is 

 also the envelope of B. The relation is thus seen to be 

 reciprocal. A combination of this sort is called a pair of 

 elements, and a machine consists solely of such elements, 

 corresponding reciprocally in pairs. " The shaft and the 

 bearing, the screw and the nut are examples of such pairs 

 of elements. We see here that the kinematic elements 

 of a machine are not employed singly, but always in pairs, 

 or, in other words, that the machine cannot so well be 

 said to consist of elements as of pairs of elements. If a 

 kinematic pair of elements be given, a definite motion 

 can be obtained by means of them if one be held fast or 

 fixed in posiiion. The other element is free to be moved, 

 but only in the one particular way allowed by the consti- 

 tution of the pair." 



In order to combine two pairs of elements ab and cd, 

 we must unite each element of one pair with one of the 

 elements of the other pair. If this is effected as in 

 Fig. 2, no new motion is obtained, as also when they are 

 in the same straight line ; but if a and d are united in 

 such a manner as not to be parallel or in one straght 

 line, the motion is entirely altered, and any points between 

 b and c will describe a curve. In either of these two cases 

 having united a to d and b to c, we have only two resistant 

 bodies each limiting and determining the relative motion 



Fro. 2. 



of the other ; thus the two pairs of elements are reduced to 

 one pair, in one case with the same, in the other with a 

 different motion. 



" Accordingly, the reciprocal combination of the ele- 

 ments of two pairs gives us again a pair of elements 

 which may differ from either of the single pairs of which 

 it is composed." Again, a combination of three, four, or 

 more pairs of elements may be made, each element of 

 each pair being combined with one element of another 

 pair, thus forming a linkage returning upon itself, or a so- 

 called closed kinematic chain. Fig. 3 shows this com- 

 bination. As a good example of this, the beam, connecting 

 rod and crank of a beam-engine may be taken ; a and b are 

 the Plummer blocks of the crank shaft and main centre 

 rigidly connected together by the bar a/i, which represents 

 in the engine the rigid connection of these two by the 

 frame, supports, and bed ; b the crank-shaft rigidly con- 

 nected by the crank with the crank-pin c; de the connect- 

 ing rod rigidly connecting the crank pin c with the gudgeon 

 f, and lastly, the beam eh rigioly connecting the gudgeon 

 / with the main centre g. In this closed chain of four 

 pairs of elements the only motions of each part with 

 respect Xo ah regarded as fixed are readily seen. Thus 

 we are led on to the result that " the mechanism is a 

 closed kinematic chain ; the kinematic chain is compound 

 or simple, and consists of kinematic pairs of elements ; 

 these carry the envelopes required for the motion which 

 the bodies in contact must have, and by these all motions 



L 2 



