Impact-induced limit cycles for inelastic impact systems with weak nonlinear damping

This paper investigates the existence and stability of periodic solutions for a class of inelastic impact systems with weak nonlinear damping. The equation governing the system is a second-order differential equation with a small perturbation parameter, which becomes non-smooth when impact conditions are applied. We explore the interplay between the nonlinear damping and impact dynamics, focusing on the preservation of periodic motion in the presence of weak nonlinear damping. The asymptotically stable periodic solution corresponds to a limit cycle on the impact phase plane. To address the challenges posed by the non-smooth nature of the system, we employ generalized polar coordinates to transform the second-order equation into a first-order system, enabling the effective application of the method of averaging. The results shed light on the long-term dynamics and periodic behavior of the system, providing new insights into the effects of weak nonlinear damping in inelastic impact systems.