Deproximating Tredgold's Approximation

Abstract

Presented is Tredgold's Approximation for using an equivalent cylindrical gear with spur teeth to “approximate” a bevel gear with straight teeth. This relation is extended to spiral bevel and hypoid gears by utilizing pitch surface curvature in the direction perpendicular to the gear tooth spiral to establish an equivalent gear. Subsequently, the envelope of a planar gear tooth profile in this perpendicular direction is presented. The envelope of the gear tooth profile is used to determine fully conjugate gear teeth profiles for spatial gear elements. This procedure is valid for any tooth profile along with circular and non-circular gears. To validate the methodology, a virtual model of a bevel gear pair (“presented model”) is created and an unloaded tooth contact analysis is performed. The procedure used to perform the unloaded tooth contact analysis and determine the corresponding unloaded transmission error is based on the concept of ease-off topography. An example of a face-milled bevel gear pair (“literature model”) serves as a reference of correctness in determining the ease-off for unloaded contact. To conclude the paper, the obtained ease-off topography and unloaded transmission error of the presented model are calculated and displayed, demonstrating the specification of fully conjugate teeth.