Breaking the symmetry of the parametrically excited pendulum

This paper considers a parametrically excited pendulum whose symmetry is destroyed by a bias term. This study investigates the effect of this symmetry-breaking by comparing the control parameter space of frequency and amplitude of the forcing with its symmetric counterpart. Approximate bifurcation analysis is used to predict the new escape boundary using a harmonic balance scheme. The bifurcations exhibited in the asymmetrical model are echoed in the case of the tilted pendulum, whose drive is not quite vertical. The experimental importance of this alteration is furthermore discussed with a view of predicting the onset of rotating motions. More specifically, an easily viewed drop in amplitude experienced by the asymmetric oscillatory solution may be considered as the trigger of escape and can therefore be regarded as a precursor of imminent danger or operational difficulties. The paper goes on to examine the impact that a variation in the bias term causes in terms of the changes to the region of safe, oscillatory motion. Applications in ship dynamics give a physical significance of this papers findings.