A Treatise on the Theory of Screws

392 THE THEORY OF SCREWS. [355, Of two reciprocal screw-chain systems, each expresses the collection of wrench-chains of which each one will equilibrate when applied to a mass-chain only free to twist about all the chains of the other system. This is, perhaps, one of the most comprehensive theorems on Equilibrium which could be enunciated. 356. Impulsive screw-chains and instantaneous screw-chains. Up to the present we have been occupied with considerations involving kinematics and statics: we now show how the principles of kinetics can be illustrated by the theory sketched in this chapter. We shall suppose, as before, that the mechanical arrangement which we call the mass-chain consists of y, elements, and that those elements are so connected together that the mass-chain has n degrees of freedom. We shall also suppose that the mass-chain is acted upon by a wrench about any screw- chain whatever. The first step to be taken is to show that the given wrench-chain may be replaced by another more conveniently circumstanced, fake any n chains of the given system, and Qy — n chains of the reciprocal system, then the given wrench-chain can be generally decomposed into components on the n + (ßy-n) chains here specified. The latter, being all capable of neutralization by the reaction of the constraints, may be omitted, while the former n wrench-chains admit of being compounded into a single wrench-chain. We hence have the following important proposition:— Whatever be the forces which act on a mass-chain, their effect is in general equivalent to that of a wrench on a screw-chain which belongs to the system of screw-chains expressing the freedom of the mass-chain. The application of this theorem is found in the fact that, while we still retain the most perfect generality, it is only necessary, either for twists or wrenches, to consider the system, defined by n chains, about which the mass- chain can be twisted. Let us consider the mass-chain at rest in a specified position, and suppose it leceives the impulsive action of any set of forces, it is required to determine the instantaneous motion which the system will acquire. The first operation is to combine all the forces into a wrench-chain, and then to transform that wrench-chain, in the manner just explained, into an equivalent wrench- chain on one of the screws of the system. Let 0 be the screw-chain of the system so found. In consequence of this impulsive action the mass-chain, previously supposed to be at rest, will commence to move; that motion can, however, be nothing else than an instantaneous twist velocity about a screw- cham a. We thus have an impulsive screw-chain 0 corresponding to an instantaneous screw-chain a. In the same way we shall have the impulsive screw-chains </>, f, &c., correlated to the instantaneous chains, ß, y, &c.