Bifurcation Analysis of a Simplified Model of a Pressure Relief Valve Attached to a Pipe

A two-parameter bifurcation analysis is performed on a mathematical model that represents the motion of a pressure relief valve connected to a reservoir via a pipe. The system comprises a dimensionless system of five differential equations representing the valve position and velocity, the reservoir pressure, and the velocity and pressure components of a quarter-wave within the pipe. Two key dimensionless parameters represent the mass flow rate into the reservoir and the length of the pipe. It is found that there are two independent forms of Hopf bifurcation: a so-called valve-only instability, which involves the valve alone and occurs when the flow rate is too low, and a coupled acoustic resonance, which involves quarter-waves within the pipe, that occurs when the pipe is too long. Secondary bifurcations are also traced including curves of grazing bifurcations where the valve body first impacts with its seat. Using a mixture of simulation and numerical continuation, it is found that these instabilities interact in a complex bifurcation diagram that involves Hopf-Hopf interaction, bistability through catastrophic grazing bifurcations, and, for long enough pipes, subcritical torus bifurcations. A particularly important discovery is a range of parameters for which the valve-only instability is subcritical, so that large amplitude impacting chaotic motion coexists with the stable equilibrium operation.