A hybrid 3D finite element/lumped parameter model for quasi-static and dynamic analyses of planetary/epicyclic gear sets

A model is presented which enables the simulation of the three-dimensional dynamic behaviour of planetary/epicyclic spur and helical gears. Deformable ring-gears are introduced by using either beam elements for simple structures, or 3D brick elements for complex geometries. Based on a modal condensation technique, internal gear elements can be defined by connecting the ring-gear sub-structure and a planet lumped parameter model via elastic foundations which account for tooth contacts. Discrete mesh stiffnesses and equivalent normal deviations are introduced along the contact lines, and their values are re-calculated as the mating flank positions vary with time. Planetary/epicyclic gear models are completed by assembling lumped parameter sun-gear/planet elements along with shaft elements, lumped stiffnesses, 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 quasi-static and dynamic results are given which illustrate the potential of the proposed hybrid model and the interest of taking into account ring-gear deflections.