A new closed-form calculation of envelope surface for modeling face gears

Abstract

Envelope approaches have been widely applied in mechanical engineering to build mathematical and 3D models, which are fundamental elements for the design and manufacturing. It is difficult to obtain both models of face gears since their tooth surfaces are derived from a complex double envelope generation process: the face gear generation with a shaper, and the shaper generation with a rack cutter. According to this process, the conventional approaches calculate the face gear tooth surface from the well-known nonlinear equation of meshing. To avoid this problem, a completely new calculation method of envelope surface is established according to the geometry characteristic of the shaper tooth surface, so the mathematical model of the face gear tooth surface can be directly represented as a closed-form (explicit) rather than implicit one. This method is available to the face gear drives with general spur pinions or the involute helical pinion. Furthermore, an efficient approach is developed to automatically generate the 3D models of face gears. The presented work has two main contributions: (1) the closed-form envelope approach is beneficial to general envelope calculations; (2) The efficient modeling method can also be applied to other types of gear.