On the third law of gearing: A study on hypoid gear tooth contact

Abstract

Presented is a procedure to determine the extreme relative curvatures between conjugate gear flanks. This procedure is a modification of the third law of gearing which establishes the extreme relative curvatures between conjugate tooth flanks in terms of normal pressure angle, spiral angle, and hyperboloidal pitch surfaces. The original formulation for the third law of gearing was based on parabolic contact along a line of contact for the limiting condition of relative curvature between two gear flanks. Penetration or existence of negative Gaussian curvature for the relative flank curvature can occur for skew axis gearing with “conjugate” teeth. In such cases, reciprocity between the tooth normal and the twist defined by the gear rotation axes and instantaneous gear ratio is a necessary condition but not a sufficient condition to ensure suitable gear flanks in skew axis gearing. For cases with penetration, a transverse profile amendment is introduced to achieve parabolic contact (zero Gaussian curvature) for the relative curvature between meshing gear tooth flanks. First and second order reciprocity are applied in the transverse direction to determine profile amendment. Ease-off topography and unloaded transmission error (UTE) are used to assess conjugate action between mating gear flanks in direct contact.