Parametric oscillations in planetary wheel gearboxes

Abstract

A mathematical model of parametric oscillations in the planetary wheel gearboxes (PWG) of the trolley buses has been developed. The model takes into account the cyclic deformations of a thin-walled rim and teeth in the meshing of the epicycle with the satellites. The differential equation of the oscillations of the epicycle is reduced to the form of the Mathieu-Hill equations with variable periodic coefficients. On the basis of the Ains-Strett diagram, regions of instability of parametric oscillations of the "criteria element" – a thin-walled epicycle for spur and helical gears are determined, taking into account the influence of gaps in the gears. The ranges of values of the critical velocities of the trolley buses corresponding to the manifestation of the first three parametric resonances of the PWG epicycle in the structure of the traction bridges of the 1st and 3rd generations of well-known manufacturing companies are established. The criterion of the vibration stability of the epicycle is substantiated, the calculated values of the probabilities of parametric resonances are established. Theoretically, the positive effect of helical engagement in PWG has been confirmed, which provides a minimum depth of hardness pulsation, a narrowing of the instability region of parametric oscillations of the epicycle, reduces the probability of nucleation and development of fatigue cracks.