A mathematical model is derived that
describes the dynamics of a single stage relief valve
embedded within a simple hydraulic circuit. The aim
is to capture the mechanisms of instability of such
valves, taking into account both fluid compressibility
and the chattering behaviour that can occur when
the valve poppet impacts with its seat. The initial
Hopf bifurcation causing oscillation is found to be either
super- or sub-critical in different parameter regions.
For flow speeds beyond the bifurcation, the
valve starts to chatter, a motion that survives for a
wide range of parameters, and can be either periodic
or chaotic. This behaviour is explained using recent
theory of nonsmooth dynamical systems, in particular
an analysis of the grazing bifurcations that occur
at the onset of impacting behaviour.