Bifurcation structures and bursting dynamics in a two degrees of freedom quasi-zero stiffness system with elastic constrain

In this manuscript, bifurcation structures and bursting patterns involving grazing phenomena are discussed based on a nonsmooth quasi-zero stiffness vibration isolation system. Within the selected parameter zone, analysis about stability and bifurcations of equilibria belonging to two sub-systems in phase space show that, equilibria of both sub-systems lose their stability through supercritical Hopf bifurcations. Numerical continuation performed on two sub-systems indicates that, the sub-system S− presents bi-stability consisting of small amplitude cycle branch circling around the S-shape equilibrium curve and large amplitude cycle branch, while the sub-system S+ presents mono-stability. Meanwhile, both cycle branches are terminated by fold bifurcations of cycle. Furthermore, as the quasi-static external force changes, periodic motions governed by the small amplitude branch of S− and cycle branch of S+ may touch the nonsmooth boundary and enter the other side of the boundary, which indicates the grazing bifurcations. By computing the bifurcation diagrams of the overall system, it is shown that grazing bifurcations result in complex distribution of cycle attractors as well as the occurrence of chaos. Based on these, multiple patterns of periodic and aperiodic bursting and the corresponding generation mechanisms are obtained. Besides, it is found that increasing the damping in coupling parts can improve the isolation efficiency.