Stability and nonlinear self-excited friction-induced vibrations for a minimal model subjected to multiple coalescence patterns

In certain industrial applications with frictional interfaces such as brake systems, the friction-induced vibrations created by coupled modes can lead to a dynamic instability and thus to an important deterioration in the operating condition. As a result, they are considered as a source of critical engineering problem. In addition, the presence of the nonlinearity makes necessary the consideration of the nonlinear dynamic analysis in order to explain clearly the complexity of the contribution of different frequency components due to unstable modes in the self-excited friction-induced vibrations, to get a design as reliable as possible and to avoid catastrophic failure during the operation phase of the mechanism. The present paper is based on previous works of Sinou and Jezequel and extends them to include a developed damped four-degree-of-freedom system with frictional contact and spring cubic nonlinearities. Its essential goals are to analyze numerically the mode-coupling instability of the four-degree-of-freedom system owing to the friction between the surfaces of contact and to predict its nonlinear dynamic behavior. The numerical study of stability for the static solution of the mechanical system is performed by applying the complex eigenvalue analysis of the linearized differential equations of motion and by identifying the Hopf bifurcation points as a function of the coefficient of kinetic friction. Depending on the Runge-Kutta time-step integration scheme and the fast Fourier transforms, quantitative and qualitative nonlinear phenomena related to self-excited friction-induced oscillations and limit cycle evolutions are observed and discussed for various friction coefficients.