Mathematical modelling and manufacturing of the internal toroidal tooth profile

Abstract

This paper investigates the toroidal tooth profile of an epicyclic gear train based on the analysis of the integrated motion from both sun-worm and planet worm-gears and develops a mathematical model based on the coordinate transformation between the coordinate system of the stationary internal gear and that of the planet worm-gear. The model is derived from the geometrical analysis of the meshing system and is used to model the meshing surface which is produced as a result from the enveloping surface. This is further extended to model the internal toroidal tooth profile in the stationary gear, the normal tooth profile and the helix trajectory. Based on these models, the shape of normal tooth section of the stationary toroidal tooth profile is obtained and the three-dimensional modelling of this tooth profile is completed. The paper further examines the case of manufacturing by developing a fly-blade for the machining process and develops a manufacturing procedure which successfully produced a stationary internal toroidal tooth profile manufactured by a fly-blade on a hob machine.