Mathematical modelling and sliding characteristics analysis on the spherical movable teeth for the toroidal drive

Toroidal drives combine a planetary gear drive with a worm gear drive. Furthermore, in toroidal drives, the planetary gear teeth are movable. The motion of spherical movable teeth for planetary gears affects the friction, wear and transmission efficiency of the meshing surface, but this effect has not been thoroughly studied to date. This study involved a kinematic analysis of planetary gear teeth. Frenet-fram was first introduced into the toroidal drive to describe the motion of spherical movable teeth. A contact analysis of the spherical movable teeth was carried out, and a mathematical model of rolling-sliding motion between the spherical movable teeth and the central worm spiral groove was established via the conversion mechanism method. The instantaneous velocity formula and relative sliding velocity formula at the meshing points of movable teeth were derived. The influence of the system parameters on the sliding velocity was analysed, and several useful conclusions were drawn. These results can provide a theoretical foundation for subsequent research on the friction loss and transmission efficiency of toroidal drives.