Mathematical model of curvature analysis for conjugate surfaces with generalized motion in three dimensions

Abstract

A general mathematical model is proposed to determine the direct relations between the principal curvatures and directions of two conjugate surfaces in continuous tangency along a line at every instant. Without any assumption regarding the motion of mating surfaces, this model, created on the basis of kinematics, the theory of gearing and differential geometry, is especially suitable for calculating the curvatures of a surface generated on a computer numerically controlled (CNC) machine with surface modifications by non-constant-speed supplemental correcting motions. The model has simplified dramatically the calculation of curvatures by comparison with the differential geometry approach. The principal curvatures of a generated surface and dimensions of contact ellipses can be obtained by the model without the need for complicated equations of the generated surface. For verification of the model, a helical gear set comprising a gear with involuted teeth and a pinion with modified circular arc teeth is investigated. The tooth contact analysis (TCA) technique is also applied to study the shift of bearing contacts and dimensions of contact ellipses under three different kinds of generation setting.