An algebraic methodology to identify the principal screws and pitches of screw systems

Abstract

This article investigates the principal screws and their pitches of screw systems from the viewpoint of line geometry. It starts from the fact that any n-independent screws pertaining to a specified n-screw system can be utilized to determine the system, and the pitch of a unit screw is the half of the reciprocal product with itself. The reciprocal product matrix of a screw matrix is therefore defined to determine the principal pitches and the directions of the principal screw axes of the system. With matrix diagonalization procedure, one can immediately obtain the principal pitches and the principal screws of a screw system. The whole procedure is straightforward and easily understood by an engineer with primary knowledge of linear algebra. Besides, the computational effort is greatly reduced and the efficiency is better compared with other methods. As a matter of fact, the pitch of any screw of a screw system can be represented by a homogeneous quadratic form of the linear combination coefficients. The homogeneous quadratic equation of the pitch of a screw system represents a conic curve in a two-system and an ellipsoid or a hyperboloid in a three-system. Therefore, this article also provides such visualizations of the principal axes in two- and three-dimensional spaces at last.