Summary
This work explores discontinuities in dynamical systems, emphasizing the pivotal concept of the switching surface, where continuity is lost. This phenomenon holds particular significance in mechanical systems, contributing to stick-slip vibrations, impact, or a combination of these phenomena. Achieving uniqueness in the presence of such discontinuities poses challenges. Rigorous ideas from piecewise-smooth dynamics and singular perturbation theory provide valuable perspectives on the problem. Filippov employed the convex combination of neighbouring vector fields, defining Filippov sliding dynamics (linear switching). Jeffrey extended this framework and incorporated a nonlinear term into the Filippov convex model. This term, referred to as the hidden term, vanishes outside the switching surface, marking the introduction of hidden dynamics (nonlinear switching). We employ hidden dynamics to investigate a self-excited smooth discontinuous (SD) oscillator with geometric nonlinearity at the switching surface. This exploration highlights a gap in Filippov’s theory regarding dry friction oscillators, where the static friction coefficient is often larger than the kinetic coefficient. By modelling the belt friction in the SD oscillator as Coulomb friction, we investigate the consequences of the discontinuity in the friction model. The sliding regions are calculated from the theory and verified using numerical simulations. Bifurcations within the system are discussed, and a comparative analysis of the system’s dynamics under Filippov and hidden dynamics theories is presented. Additionally, we analyze the system’s response to harmonic excitation, providing valuable insights into its behaviour under external forces.
