A Treatise on the Theory of Screws

132] FREEDOM OF THE SECOND ORDER. HO 132. Reaction of Constraints. We shall now consider the following problem:—A body which is free to twist about all the screws of a cylindroid G receives an impulsive wrench on a certain screw p. It is required to find the screw X, a wrench on which constitutes the impulsive reaction of the constraints. Let 0 represent the cylindroid which, if the body were perfectly free, would form the locus of those screws, impulsive wrenches on which correspond to all the screws of G as instantaneous screws. Since a wrench on r/, and one on X, make the body twist about some screw on C, it follows that the cylindroid (??, X) must have a screw p in common with C. The wrench on X might be resolved into two, one on y, and the other on p, and the latter might be again resolved into two wrenches on any two screws of G'. It therefore follows that X must belong to the screw system of the third order, which may be defined by ?/, and by any two screws from G". Take any three screws reciprocal to this system, and any two screws on G. We have then five screws to which X is reciprocal, and it is therefore geometrically determined (§ 26). When X is found, the cylindroid (,/, X) can be drawn, and thus p is deter- mined. The position of p on G' will point out the screw on G, about which the body will commence to twist, while the position of p on (,/, X), and the known intensity of the wrench on rj, will determine the intensity of the 'wrench on X.