On the Three Laws of Gearing

Abstract

Three laws of gearing are presented in terms of a three link 1-dof spatial direct contact mechanism. The first law of gearing defines the instantaneous relationship between an infinitesimal displacement of an output body to an infinitesimal angular displacement of an input body for a specified tooth contact normal. A system of cylindroidal coordinates are introduced to facilitate a universal methodology to parameterize the kinematic geometry of generalized motion transmission between skew axes. The second law of gearing establishes a relation between the instantaneous gear ratio and the apparent radii of the hyperboloidal pitch surface in contact as parameterized using a system of cylindroidal coordinates. The third law of gearing establishes an instantaneous relationship for the relative curvature of two conjugate surfaces in direct contact and shows that this relation is independent of the tooth profile geometry. These three laws of gearing along with the system of cylindroidal coordinates establish, in part, a generalized geometric theory comparable to the existing theory for planar kinematics.