Grazing bifurcations and chaos in the dynamics of a hydraulic damper with relief vales

This paper numerically investigates the closed loop dynamics of a lumped mass attached to a linear spring and a nonlinear hydraulic damper with relief valves. The damper in question is designed to have piecewise linear characteristics. The inclusion of the nonlinear damper into the closed loop system leads to regions of periodic motion separated by regions of nonperiodic dynamics as the forcing frequency changes. Analysis of the system uses discontinuous nonsmooth numerical continuation techniques to follow the stable and unstable solutions, along with bifurcation diagrams to aid the understanding of the nonperiodic dynamics. The impacting of the relief valves leads to grazing bifurcation analysis using discontinuity mappings, which explains some "corners" observed in the bifurcation diagram and predicts the onset of chaotic dynamics. Other nonsmooth events such as the grazing of an invariant torus are suggested by numerical simulations.