KINEMATICS OF MACHINERY. Me chanics may be defined as the study of motion and of the circumstances which influence or affect it. The study of motion alone is a branch of geometry in which the element of time en ters. Ampere, 'Philosophic des sciences' (1843), gave to this geometry of motion the name kinematics; Reuleaux, 'Theoretische Kinematil0 (1875) and others before him called it phoro nomics but this term has not been accepted in mechanics. Kinematics of machinery — also called kinematics of mechanism, theory of mechanism, applied or technical kinematics — is that phase of the subject which is useful in engineering. It deals with the motions of a complex deformable system subjected to such constraints as will make the motion unique or determinate. According to Koenigs, 'Introduc tion a une theorie nouvelle des micanismes' (1905), it is the study of the constraints in machines, a machine being an assemblage of resistant bodies (rigid, flexible or fluid) under mutual constraint upon which force may act. All the methods necessary for the mathemat ical, as distinguished from the graphical, solu tion of the most complicated problems were known long before kinematics existed as a separate science. The first attempt to separate it from mechanics in general was made by Monge in 1794; his program was elaborated and published in 1808 by Lanz and Betancourt, 'Essai sur la composition des machines.' Their system aroused interest, but whatever benefit to the science may have accrued through this stimulus was certainly outweighed by the harm that arose through emphasizing the idea that it was more the province of kinematics to classify and describe machines than to devise simple methods of finding velocities and accel erations. This point of view was strengthened by Willis, who published his famous Principles of Mechanism' in 1841. To judge from the 20-page preface to his second edition (1870) Willis regarded classification as of first and last importance. Nevertheless he gave proper kinematic descriptions of an enormous number of mechanisms, was the first to investigate velocity ratios—the ratio of the velocity of the driver to that of the follower — and oc casionally improved methods of designing and calculating. It is not easy to-day to understand how Willis exerted so deep an influence for a generation. When one considers the brilliant work done in matheMatical kinematics by Euler, Lagrange, Laplace, Poisson, Poinsot and Cori olis, it is not unfair to say that Willis was as antiquated in his own time as he is now. The next noteworthy book in the development of kinematics is Reuleaux, 'Theoretische Kine matik: Grundziige einer Theorie des Maschin enwesens' (1875), English edition by Kennedy in 1876. To Reuleaux classification is still of fundamental importance. Willis classifies ma chines according to the way they transmit mo tion, for example, by rolling contact, sliding contact, wrapping connectors (belts and chains), linkwork and ratchets, each being sub divided according to whether the velocity ratio or the direction of the velocity is constant or variable. Reuleaux on the other hand examines
the elements of which machines consist and finds that they always occur in pairs, a number of pairs being connected to form a chain. In kinematic pairs there may be relative sliding (e.g., crosshead and guide), turning (hinge or pin Joint) or sliding and turning (screw and nut). If the contact is between surfaces, as in these examples, the pair is called lower; if there is line contact, as in gear wheels, the pair is higher. Reuleaux showed that many differ ent mechanisms may be made from the same kinematic chain by holding different elements or links fixed. Neither Willis nor Reuleaux had an efficient method of finding velocities in a mechanism; neither attempted to find accel erations and neither gave sufficient emphasis to the fundamental problem of kinematics, namely, to devise general methods for the determination of velocities and accelerations. The most im portant treatise on kinematics from this point of view is Burmester's der Kine matik' (1888) ; this is decidedly superior to anything written before that time and has not yet been surpassed.
Fundamental Three theorems suffice as a basis for the analysis of velocity and acceleration in mechanisms. On account of their importance a rigorous mathematical proof will he given. It is customary to base their derivation on the postulate of superposi tion which states, briefly, that kinematic vectors may be superposed according to the parallelo gram law ; this method is not convincing when applied to accelerations and sometimes leads to serious mistakes.
In Fig. 1 it is required to find the velocity of any point P of a rigid body moving in a plane, the velocity of 0 and the angular velocity of the body being given. The body has three degrees of constraint specified by the co-ordi nates f, n, 8, i.e., given values of them will fix its position. If a constraint is removed by permitting the corresponding co-ordinate to vary, the body has one degree of freedom. In the following equations one dot over a letter will denote the first time-derivative. From Fig. 1 x--=f r rose, y r sin 8; hence sincer is constant and 8=u .
x=--f —rn sin 0, y---= r.) cos 8 ...(1) where (f, n) and (x, y) respectively the velocities of 0 and P with reference to the fixed axes xy and are thus the so-called ab solute as distinguished from the relative veloc ities. is that part of the velocity of P which is due to the rotation of P about 0; it is normal to OP and is the relative velocity of P with respect to 0. Equations (1), giving the components of the relative and absolute ve locities of 0 and P, state that P has two super posed velocities: the absolute velocity (f, n) of O and the linear velocity r due to the rotation of P about 0. Hence