A study of the frequency response of impact pairs with application to automotive gear rattle

A torsional lumped parameter multi-degree-of-freedom model of an automotive manual transmission is developed and the corresponding nonlinear equations of motion are derived. The model consists of a set of nonlinear impact pairs coupled by linear springs. Each impact pair is defined by a semi-definite torsional system consisting of two inertia elements joined by a nonlinear clearance type elastic element and a linear energy dissipation element. The method of harmonic balance is used to develop approximate analytical solutions for the impact pair and to study the existence and stability of the solutions. Steady state frequency response predictions, for a single frequency excitation and primary resonance, compare extremely well with analog computer simulations. Potential problems are observed when numerical integration techniques are used to predict the steady state frequency response of the impact pair. The clearance nonlinearity is shown to be significantly different from the classic cubic nonlinearity. The method of harmonic balance is then applied successfully to a multi-degree-of-freedom system composed of three coupled impact pairs. Two special cases are studied in detail and again the frequency response predictions for a single frequency excitation match analog simulations very well. For one of the special cases it is shown that the nonlinear multi-degree-of-freedom system can be studied using a nonlinear single-degree-of-freedom model and for the other special case the nonlinear multi-degree-of-freedom system can be studied using a linear multi-degree-of-freedom model. Finally, a simplified nonlinear model of an automotive transmission is developed. This model is applied to the study of neutral and drive/cost rattle problems and approximate analytical solutions are developed which clearly explain the nonlinear behavior of the system. A set of design guidelines for reduced rattle are then formulated, which are consistent with the experimental observations given in the literature. This study has resolved many modeling issues related to clearance type nonlinear systems as well as geared systems with backlash.