Computing method for induced curvature parameters based on normal vector of instantaneous contact line and its application to Hindley worm pair

Yaping Zhao,Yimin Zhang

doi:10.1177/1687814017721884

Abstract

The main objective of this article is to propose a practical methodology to compute the induced curvature parameters for a line-conjugated tooth surface couple based on the normal vector of its instantaneous contact line. The computing formulae in different form are thus suggested for the preceding normal vector in order for convenient use. Based on this, the formulae for the induced curvature parameters are accordingly given out. Compared to the previous work, the deducing process is more rigorous and more laconic, and the obtained results are more practical. Concurrently, different deducing techniques are provided, and this is beneficial for the development of the meshing theory for gear drives. The theory and method presented are applied to the modified Hindley worm drive, and its basic and important equations are attained. The concrete numerical example is a type I drive, whose conjugate zone is divided into two parts. In one part, the value of the induced principal curvature is much smaller than that in the other one. The biggest induced principal curvature comes into view about at the inlet portion of the worm, and certainly the highest contact stress between the teeth will happen consistently at that position.

Highlights

As for a line-conjugate tooth surface couple, the induced normal curvature means the difference between the normal curvatures of its two tooth surfaces along a common tangential direction at an instantaneous meshing point on an instantaneous contact line
The second surface, that is, the enveloped surface, in a line-conjugate surface couple essentially is the envelope to a one-parameter surface family, which is formed by the first surface, that is, the enveloping surface, under the given relative motion
On the worm gear tooth flank, the lines AB and A0B0 stand for the inlet end of the worm and the line EF is the conjugate line of the meshing limit line, which splits up the whole conjugate zone A0B0BA into two subconjugated zones

Summary

Introduction

As for a line-conjugate tooth surface couple, the induced normal curvature means the difference between the normal curvatures of its two tooth surfaces along a common tangential direction at an instantaneous meshing point on an instantaneous contact line. The induced normal curvature can completely be determined by the geometry of the enveloping surface and the relative motion of the surface couple. The curvature parameters of the enveloped surface, even its equation, are not required to be known when computing the induced normal curvature. This is the reason why a word ‘‘induced’’ is put into the phrase ‘‘induced normal curvature.’’

Objectives

Discussion

Conclusion