A Treatise on the Theory of Screws

CHAPTER X . FREEDOM OF THE FIRST ORDER. 108. Introduction. In the present chapter we shall apply the principles developed in the preceding chapters to study the Statics and Dynamics of a rigid body which has freedom of the first order. Ensuing chapters will be similarly devoted to the other orders of freedom. We shall in each chapter first ascertain what can be learned as to the kinematics of a rigid body, so far as small displacements are concerned, from merely knowing the order of the freedom which is permitted by the constraints. This will conduct us to a knowledge of the special screw system which defines the freedom enjoyed by the body. We shall then be enabled to determine the reciprocal screw system, which involves the theory of equilibrium. The next group of questions will be those which relate to the effect of an impulse upon a quiescent rigid body, free to twist about any screw of the screw system. Finally, we shall discuss the small oscillations of a rigid body in the vicinity of a position of stable equilibrium, under the influence of a given system of forces, the movements of the body being limited as before to the screws of the screw system. 109. Screw System of the First Order. A body which has freedom of the first order can execute no movement which is not a twist about one definite screw. The position of a body so circumstanced is to be specified by a single datum, viz., the amplitude of the twist about the given screw, by which the body can be brought from a standard position to any other position which it is capable of attaining. As examples of a body which has freedom of the first order, we may refer to the case of a body free to rotate about a fixed axis, but not to slide along it, or of a body free to slide along a fixed axis, but not to rotate around it. In the former case the screw system consists of one screw, whose pitch is zero; in the latter case the screw system consists of one screw, whose pitch is infinite.