The system of a vessel and a direct spring operated pressure relief valve is extended
by a downstream pipe. The presented mathematical model consisting of ordinary and
partial differential equations can be transformed to a system of differential difference
equations by means of the travelling wave solutions of the partial differential equations.
This model contains a time delay originated in the pipe length and the speed of sound
in the pipe. The linear stability analysis presents stability charts in a parameter
plane characterising the environment where the valve is built in. The boundary lines
and the corresponding vibration frequencies of the self-excited vibrations are determined
as compact closed form expressions. Several analogies are identified with other physical
systems modelled by delayed oscillators. Both the stability charts and the vibration
frequencies are checked with numerical simulations of the original nonlinear system
consisting of coupled ordinary and partial differential equations. The connection
between the travelling wave solution and the standard quarter standing wave approximation
is explained. A qualitatively relevant phenomenon is identified, namely the appearance
of the quasi-periodic vibration at certain parameter combinations.
