Steady-state dynamics and discontinuity-induced sliding bifurcation of a multi-module piecewise-smooth vibration-driven system with dry friction

Abstract

The vibration-driven locomotion system subject to the Coulomb dry friction is a typical multi-degree-of-freedom piecewise-smooth dynamical system with rigid-body displacement. Rather than ignoring the sticking behaviors caused by dry friction in previous studies, this paper focuses on the steady-state dynamics and the discontinuity-induced sliding bifurcations of a non-smooth three-module vibration-driven system. It is shown that the presence of sticking partially invalidates the averaging method and prevents an accurate prediction of the locomotion performance. As a result, numerical methods are employed in this research, which not only gives the distribution pattern of the average steady-state velocity with respect to the actuation phase differences but also elucidates its variation trend with respect to the actuation frequency and friction anisotropism, revealing the significance of the resonance effect and phase coordination for locomotion performance enhancement. Subsequently, this paper focuses on the stick–slip dynamics associated with the non-smooth boundaries induced by dry friction. Through comprehensive numerical calculations, we identify four different types of stick–slip motions and their distributed zones, whose boundaries are essentially the sliding bifurcation curves. Analytical expressions of these bifurcation curves are derived by means of the least-square regression, and they together constitute a bifurcation diagram for a given actuation and environment conditions. Each time a sliding bifurcation curve is traversed, the stick–slip trajectories experience qualitative changes. We also systematically reveal the evolution of the bifurcation diagram with the actuation frequency and friction anisotropism.