Grazing bifurcations and chatter in a pressure relief valve model

This paper considers a simple mechanical model of a pressure relief valve. For a wide region of parameter values, the valve undergoes self-oscillations that involve impact with the valve seat. These oscillations are born in a Hopf bifurcation that can be either super- or sub-critical. In either case, the onset of more complex oscillations is caused by the occurrence of grazing bifurcations, where the limit cycle first becomes tangent to the discontinuity surface that represents valve contact. The complex dynamics that ensues from such points as the flow speed is decreased has previously been reported via brute-force bifurcation diagrams. Here, the nature of the transitions is further elucidated via the numerical continuation of impacting orbits. In addition, two-parameter continuation results for Hopf and grazing bifurcations as well as the continuation of period-doubling bifurcations of impacting orbits are presented. For yet lower flow speeds, new results reveal chattering motion, that is where there are many impacts in a finite time interval. The geometry of the chattering region is analysed via the computation of several pre-images of the grazing set. It is shown how these pre-images organise the dynamics, in particular by separating initial conditions that lead to complete chatter (an accumulation of impacts) from those which do not.