Higher Path-Curvature Analysis in Plane Kinematics

Abstract

Classical kinematics furnishes us with powerful techniques for the determination of the motion characteristics of a rigid body, or “plane,” or link. The present analysis is concerned more with the motion characteristics of the path generated by a point on a link, rather than with the link as a whole. For four infinitesimal displacements, it is shown that the locus of all points on a link with a prescribed ratio of path-evolute curvature to path curvature is a higher algebraic curve, the “quartic of derivative curvature.” For five infinitesimal displacements, five points on a moving link can in general be found having five-point contact with an arbitrary, prescribed curve. For circular motion, these results reduce to the classical theories of Burmester and Mueller. Equations have been derived for the first and second rates of change of path curvature in terms of the evolutes to the path. The results are applicable to the kinematic analysis and synthesis of mechanisms and are illustrated, specifically, for the generation of involute, parabolic, and elliptic arcs.