Modal Analysis of Planetary Gearsets

This contribution deals with modal analysis - the computation of eigenfrequencies and eigenmodes - of a multibody systems, which is based on the linearization of the equation of motions at a loaded state. Special emphasis is taken on transmission systems of automotive applications, especially the modal analysis of planetary gearsets (PGS). In general, the multibody system representing the PGS consists of rigid and/or flexible bodies. These bodies are interconnected by nonlinear elements (joints), which represent forces and moments from radial and axial bearings and from gear interactions. In order to obtain the standard quadratic eigenvalue problem which has to be solved for eigenvalues and mode shapes, a linearization by a first order Taylor approximation is used. The linearization includes the FEM based body mass, damping and stiffness matrices in case of an elastic body. Additional terms may result from linearization of the inertia forces and the joint forces. The linearization of the joint forces and moments yields contributions to the linearized stiffness and damping matrices. Most parts of the linearization of the equations of motion are computable analytically, especially the linearization of the inertia forces. For joint types, where no explicit and a rather simple functional relationship is given, as for instance inner and outer gearing in the PGS, the partial derivatives of the joint forces with regard to node positions and velocities are approximated by the method of finite differences. A standard finite difference method for computing the joint stiffness matrix is rather time consuming. Therefore, an approach will be presented, which halves the number of gear joint evaluations when taking the geometrical properties of the joint into account. Furthermore, the desired properties of the stiffness matrix as symmetry and positive definiteness are enforced. They allow the selection of the most suitable eigenvalue solver for this class of matrices which further reduces the overall computation time compared to solving a general eigenvalue problem. The approach is applied to a typical engineering task of a PGS, which consists of the central sun gear, the ring gear (fixed in the gearbox housing) and three planets. Gear meshes are taken into account by contact models resolving all relevant details of the multi-flank-contact. The effect of the helix angle on the eigenfrequency distribution is investigated in detail.