Grazing-induced bifurcations bifurcations in impact oscillators with elastic and rigid constraints

This paper investigates differences between the grazing-induced bifurcations in impact oscillators with one-sided elastic and rigid constraints by a path-following (continuation) method. The grazing bifurcations are computed and classified for both oscillators. Two-parameter smooth (period-doubling, saddle-node) and non-smooth (grazing) bifurcations are analyzed. Frequency response curves including bifurcation points are determined for different values of stiffness ratio and restitution of energy coefficient. As the stiffness ratio increases, the constraint changes from elastic to rigid and the bifurcation structure varies correspondingly. For the first time our numerical results presented in [17] and in the current work show that for the impact oscillators with one-sided elastic constraint, the smooth (period-doubling, saddle-node) bifurcations approach the non-smooth (grazing) bifurcations as the stiffness ratio increases. However, for the impact oscillators with one-sided rigid constraint, there is no smooth bifurcations near the non-smooth (grazing) bifurcation points. Basins of attraction, computed by our newly developed Matlab-based computational suite ABESPOL [5], complement our study.