Micro-Slip as a Loss of Determinacy in Dry-Friction Oscillators

Abstract

Dry-friction contacts in mechanical oscillators can be modeled using nonsmooth differential equations, and recent advances in dynamical theory are providing new insights into the stability and uniqueness of such oscillators. A classic model is that of spring-coupled masses undergoing stick-slip motion on a rough surface. Here, we present a phenomenon in which multiple masses transition from stick to slip almost simultaneously, but suffer a brief loss of determinacy in the process. The system evolution becomes many-valued, but quickly collapses back down to an infinitesimal set of outcomes, a sort of “micro-indeterminacy”. Though fleeting, the loss of determinacy means masses may each undergo different microscopic sequences of slipping events, before all masses ultimately slip. The microscopic loss of determinacy is visible in local changes in friction forces, and in creating a bistability of global stick-slip oscillations. If friction forces are coupled between the oscillators then the effect is more severe, as solutions are compressed instead onto two (or more) macroscopically different outcomes.

Hidden Dynamics of a Self-excited SD Oscillator

“…Stick-slip vibrations have been studied extensively for their bifurcations and chaos by using Filippov sliding dynamics [8][9][10][11][12]. A dry-friction oscillator as a paradigm for the role of nonlinear and hysteretic terms in discontinuity is studied in [13], and the microscopic loss of determinacy are studied in [14]. Much of the literature concerns on linear self-excited oscillators although relatively few sutudies have been of interest on nonlinear self-excited oscillators by introducing the cubic term in stiffness and damping coefficients [15,16].…”