The complicated behaviours of a novel model of smooth and discontinuous dynamics with quasi-zero-stiffness property

In this paper, we propose a spring-rods-mass system being of smooth and discontinuous property which can be designed as a vibration isolator due to its SQZS (stable-quasi-zero-stiffness). Both smooth and discontinuous bifurcation sets are obtained including pitchfork-like bifurcation and heteroclinic-like bifurcation, based upon which the complicated nonlinear dynamics is discovered, including multiple equilibrium states and discontinuous singularities depending on a geometrical parameter \(\beta \) $\beta$ and stiffness ratio \(\alpha \) $\alpha$ . The SQZS condition is presented and low frequency isolation performance is verified by analyzing amplitude and transmissibility response using average method, and high order SQZS property is also discovered performing even lower isolation frequency. Furthermore, asymptotically stable type and static-asymptotically stable type of the focus-like singularity on the manifold is distinguished by the convergence property of impact time \(\{\tau _n\}_{n=1}^{\infty }\) ${\{{\tau }_n\}}_{n=1}^{\infty }$ . And chaotic thresholds under the perturbation of both viscous damping and external harmonic forcing are detected by employing Melnikov function, while the coexistence of some periodic orbits is also discovered by numerical simulation.