Mode coupling instability mitigation in friction systems by means of nonlinear energy sinks: Numerical highlighting and local stability analysis

In this paper, we study the problem of passive control of friction-induced vibrations due to mode coupling instability in braking systems. To achieve that, the well-known two degrees of freedom Hulten's model, which reproduces the typical dynamic behavior of friction systems, is coupled to two ungrounded nonlinear energy sinks (NES). The NES involves an essential cubic restoring force and a linear damping force. First, using numerical simulations it is shown that the suppression or the mitigation of the instability is possible and four steady-state responses are highlighted: complete suppression, mitigation through periodic response, mitigation through strongly modulated response, and no suppression of the mode coupling instability. Then the system is analyzed applying a complexification-averaging method and the resulting slow-flow is finally analyzed using geometric singular perturbation theory. This analysis allows us to explain the observed steady-state response regimes and predict some of them. The boundary values of the friction coefficient for some of the transitions between these regimes are predicted. However, the appearance of a three-dimensional super-slow flow subsystem highlights the limitation of the local linear stability analysis of the slow-flow to predict all these boundaries.