Modeling of spur and helical gear planetary drives with flexible ring gears and planet carriers

A model is presented which enables the simulation of the three-dimensional static and dynamic behavior of planetary/epicyclic spur and helical gears with deformable parts. The contributions of the deflections of the ring gear and the carrier are introduced via substructures derived from 3D finite element models. Based on a modal condensation technique, internal gear elements are defined by connecting the ring-gear substructure and a planet lumped parameter model via elastic foundations which account for tooth contacts. Discrete mesh stiffness and equivalent normal deviations are introduced along the contact lines, and their values are recalculated as the mating flank positions vary, with time. A constraint mode substructuring technique is used to simulate the planet carrier as a superelement which is connected to the planet center. Planetary/epicyclic gear models are completed by assembling lumped parameter sun gear/planet elements along with shaft elements, lumped stiffness, masses and inertias. The corresponding equations of motion are solved by combining a time-step integration scheme and a contact algorithm for all simultaneous meshes. Several quasistatic and dynamic results are given which illustrate the potential of the proposed hybrid model and the interest of taking into account ring gear and carrier deflections.