A Treatise on the Theory of Screws

CHAPTER XXIV. THE THEORY OF SCREW-CHAINS*. 338. Introduction. In the previous investigations of this volume the Theory of Screws has been applied to certain problems in the dynamics of one rigid body. I propose to show in the present chapter to what extent the conceptions and methods of the Theory of Screws may be employed to elucidate certain problems in the dynamics of any material system whatever. By such a system I mean an arbitrary arrangement of /z rigid bodies of any form or construction, each body being either entirely free or con- strained in any manner by relations to fixed obstacles or by connexions of any kind with one or more of the remaining — 1 pieces. For convenience we may refer to the various bodies in the system by the respective numerals 1,2,.../,. This numbering may be quite arbitrary, and need imply no reference whatever to the mechanical connexions of the pieces. Die entire set of material parts I call for brevity a mass-chain, and the number of the bodies in a mass-chain may be anything from unity to infinity. I write, as before, of only small movements, but even with this limitation problems of equilibrium, of small oscillations and of impulsive movements are included. By the order of the freedom of the mass-chain, I mean the number of generalized co-ordinates which would bo required to specify a position which that mass-chain was capable of assuming. The order cannot be less than one (if the mass-chain be not absolutely fixed), while if each element of the mass-chain be absolutely free, the order will be as much as 6/z. Starting from any arbitrary position of the mass-chain, let it receive a small displacement. Each element will be displaced from its original position to an adjacent position, compatible of course with the conditions Transactions Royal Irish Acad. Vol. xxvm. p. 99 (1881).