Contact Path Method of Generating Spherical Conjugated Curves as Bevel Gearing Tooth Profiles

Abstract

This paper proposes a general method to generate spherical conjugated curves based on their contact path. The main objective of generating conjugated curves is to use them as gearing tooth profiles. The derivation is based on the prescribed spherical contact path, the equation of meshing, and the rigid body motion between two bodies. This paper demonstrates the method by assigning great circle and small circle contact paths in circular pitch pair transmission to generate conjugated curves. With the great circle contact path, the conjugated curves are exact spherical involute, and with the small circle contact path, the conjugated curves are spherical cycloid. The contact properties of the conjugated curves, such as induced curvature and the sliding ratio, are also provided with mathematical equations. Based on the results, the procedures and the geometries between the planar and spherical contact path methods have corresponding relationships. In contrast to the Camus theorem, the exact spherical involute could be generated from a similar procedure of planar involute cases. The ultimate goal of this research topic is to propose a general approach to generating conjugated profiles in planar, spherical, and spatial cases. The successful derivation of the spherical cases in this paper paves the way for spatial cases in the future.