A Treatise on the Theory of Screws

192] FREEDOM OF THE THIRD ORDER. 191 191. Reaction of the Constraints. An impulsive wrench on a screw y acts upon a body with freedom of the third order, and the body commences to move by twisting upon a screw 0. It is required to find the screw X, a wrench on which constitutes the initial reaction of the constraints. Let </> denote the impulsive screw which, if the body were free, would correspond to 0 as the instantaneous screw. Then X must lie upon the cylindroid (<£, ??), and may be determined by choosing on (</>, y) a screw reciprocal to any screw of the given screw system. 192. Impulsive Screw is Indeterminate. Being given the instantaneous screw 6 in a system of the third order, the corresponding impulsive screw is indeterminate, because the impulsive wrench may be compounded with any reactions of the constraints. In fact y may be any screw selected from a screw system of the fourth order, which is thus found. Draw the diametral plane conjugate to a line parallel to 0 in the ellipsoid of inertia, and construct the cylindroid which consists of screws belonging to the screw system parallel to this diametral plane. Then any screw which is reciprocal to this cylindroid will be an impulsive screw corresponding to 6 as an instantaneous screw. Thus we see that through any point in space a whole cone of screws can be drawn, an impulsive wrench on any one of which would make the body commence to twist about the same screw. One impulsive couple can always be found which would make the body commence to twist about any given screw of the screw system. For a couple in a plane perpendicular to the nodal line of a cylindroid may be regarded as a wrench upon a screw reciprocal to the cylindroid; and hence a couple in a diametral plane of the ellipsoid of inertia, conjugate to the diameter parallel to the screw 0, will make the body commence to twist about the screw 0. It is somewhat remarkable that a force directed along the nodal line of the cylindroid must make the body commence to twist about precisely the same screw as the couple in a plane perpendicular to the nodal line. If a cylindroid be drawn through two of the principal screws of inertia, then an impulsive wrench on any screw of this cylindroid will make the body commence to twist about a screw on the same cylindroid. For the impulsive wrench may be resolved into wrenches on the two principal screws. Each of these will produce a twisting motion about the same screw, which will, of course, compound into a twisting motion about a screw, on the same cylindroid.