Asymptotic analysis of self-excited and forced vibrations of a self-regulating pressure control valve

Pressure vibrations in hydraulic systems are a widespread problem and can be caused by external excitation or self-exciting mechanisms. Although vibrations cannot be completely avoided in most cases, at least their frequencies must be known in order to prevent resonant excitation of adjacent components. While external excitation frequencies are known in most cases, the estimation of self-excited vibration amplitudes and frequencies is often difficult. Usually, numerical studies have to be executed in order to elaborate parameter influences, which is computationally expensive. The same holds true for the prediction of forced oscillation amplitudes. This contribution proposes asymptotic approximations of forced and self-excited oscillations in a simple hydraulic circuit consisting of a pump, an ideal consumer and a pressure control valve. Two excitation mechanisms of practical interest, namely pump pulsations (forced vibrations) and valve instability (self-excited vibrations), are analyzed. The system dynamics are described by a singularly perturbed third-order differential equation. By separating slow and fast variables in the system without external excitation, a first-order approximation of the slow manifold is computed. The flow on the slow manifold is approximated by an averaging procedure, whose piecewise defined zero-order solution maps the valve's switching property. A modification of the procedure allows for the asymptotic approximation of the system's forced response to an external excitation. The approximate solutions are validated within a realistic parameter range by comparison with numerical solutions of the full system equations.