Theory and Model for Active Design of Spiral Bevel Gears

Abstract

Starting from the fundamental principles of differential geometry, this paper systematically establishes the mathematical foundation and model representation of spiral bevel gear tooth surfaces. Following a logical chain of “geometric foundations—parametrization principle—normal theory—integrability conditions—metric and curvature—closed-form equations—conjugate meshing”, all core contents of spiral bevel gear tooth surface modeling are unified within an axiomatic framework. It is proved that the essence of a spiral bevel gear tooth surface is a smooth immersion from a two-dimensional parameter manifold into three-dimensional Euclidean space; the normal direction angle functions are the angular coordinate control functions of the Gauss map defined on this manifold; and the geometric essence of conjugate meshing is the constraint condition of relative velocity in the tangent space. This theoretical system not only provides a unified mathematical basis for precise modeling, active design, and modification optimization of spiral bevel gears, but also opens avenues for research on non-circular bevel gears, novel tooth profiles, and intelligent tooth surfaces.