Hidden Dynamics of a Self-excited SD Oscillator

The present study explores the nonlinear dynamics of a self-excited smooth and discontinuous (SD) oscillator with geometric nonlinearity at the switching surface. Using a novel framework called hidden dynamics, introduced by Jeffrey, the research addresses the challenge posed by dry friction oscillators where the static friction coefficient is greater than the kinetic friction coefficient, which is ignored in Filippov’s theory. 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 were determined analytically and validated through numerical simulations. The system's behaviour is analyzed through the examination of bifurcation diagrams and phase portraits, and a comparison is conducted with Filippov’s theory. Some interesting bifurcation phenomena are highlighted, including a novel phenomenon involving the collision and merging of two degenerate boundaries and the bifurcation of a sliding homoclinic orbit to a saddle. Furthermore, the system's response to harmonic excitation is analyzed, wherein the oscillator displays stick-slip limit cycles, pure slip limit cycles and the emergence of chaotic solutions through periodic doubling bifurcation.