Limit cycle oscillation and dynamical scenarios in piecewise-smooth nonlinear systems with two-sided constraints

Piecewise-smooth models are widely explored to investigate occasional contact or impact systems, such as gear systems in mechanical engineering and folding wings in aircraft structures. However, the complicated nonlinear responses of non-smooth features may result in unexpected negative consequences. In this paper, a general vibro-impact single-degree-of-freedom system with symmetric and asymmetric constraints is identified and studied. It is first classified into several categories based on the relative placement of equilibrium points and switching manifolds. The geometric structures of the system's phase space are analyzed to derive the analytical formulations of its backbone curves. The numerical findings show that the theoretical expression of the backbone curves can precisely estimate the amplitude of the limit cycles with high energy. Analytical and numerical approaches are used to study grazing bifurcations, phase angle jumps, and energy changes in switching manifolds. Finally, the dynamical responses of these systems under sinusoidal period external stimulation with varying frequencies and amplitudes are explored. The results reveal that the occurrence of subharmonic resonance at low frequencies is determined by the magnitude of external stimulation, particularly when a grazing bifurcation threshold is exceeded. Notably, this criterion is not present under free-play systems. With increasing external excitation frequencies, the response amplitude first rises, then falls, and the system's energy rapidly diminishes due to superharmonic resonance, potentially resulting in transient chaos in free-play systems.