Explicit solution to the 3-D contact analysis problem of non-conjugate gear teeth in parallel-axis and cross-axis helical configurations

Abstract

Modern gear tooth contact models are implicit multiple equation systems which rely on numerical solution techniques. Often, they require initial guess values. If they are not accurate enough, problems with solution convergence may appear and demand significant computational effort to be solved. In this paper a new model is developed using a modified version of the fundamental law of gearing in three dimensions. Gear surfaces were parametrized using two parameters which are radius from center of gear to the point of contact and its projection on the center line. Due to such specific parametrization, it was possible to derive a set of modified vector equations for surface coincidence and tangency which could be reduced to 2 scalar explicit equations with 2 unknowns, and which provides effectiveness in finding the solution of tooth contact compared to other models. The profile tangency conditions are expressed through the following aspects: cross product of surface normal that is equal to zero, which is more closely linked to the physical meaning of the tangency condition, and the fact that difference of vectors from centers of gears to point of contact at the meshing position is equal to distance between centers of gears.