Complex near-grazing dynamics in impact oscillators

Impact oscillators frequently appear in various physical and engineering systems with non-smooth characteristics and can exhibit different dynamic behavior from the smooth nonlinear systems, including grazing bifurcation in which an impact with zero velocity occurs. This paper investigates the near-grazing dynamics of the multi-degree of -freedom impact oscillators in the small neighborhood of degenerate grazing points, with a focus on the stability and potential bifurcations of near-grazing period-one impact motions. The high order zero time discontinuity mapping method is applied to perform the prospective analyses of stability and bifurcations. Particularly, this paper shows that the peculiar Neimark-Sacker bifurcations regaining the stability of near-grazing period-one impact motion can be induced by two different ways, either through the interaction between the singular and regular real eigenvalues or via a grazing bifurcation directly. A two degree-of-freedom impact oscillator is taken as an example to present detailed numerical results for the verification of proposed analysis.