Stability Analysis of the Helmholtz Oscillator with Time Varying Mass.pdfVIP

STABILITY ANALYSIS OF THE HELMHOLTZ OSCILLATOR WITH TIME VARYING MASS. RICHARD A. MORRISON Abstract. We study the effects of periodically time varying mass on the stability of the Helmholtz oscillator, which, when linearised, takes the form of Ince’s equation and exhibits parametric resonance. The resonance 7 regions in the parameter space are mapped and we use the Melnikov function to demonstrate that the 0 parametric instability does not affect the qualitative dynamics of the nonlinear system. 0 2 1. INTRODUCTION n a The recent activity in the application of Nonlinear Dynamics to problems in Naval Architecture is exemplified J by the papers in [13]. One particular success has been a more rational understanding of beam sea capsize 1 and the development of a robust mathematical toolkit for the study of this problem. 2 Following the work of Wright and Marshfield, [20] it has become common to study the roll dynamics using ] a one degree of freedom nonlinear oscillator taking the form S ¨ ˙ D (1) θ + g (θ) + GZ (θ) = F (t). ˙ . Here θ is the roll angle relative to the wave normal, g (θ) is the, often linear–viscous, damping function, and h t GZ (θ) (so denoted as it is usually some polynomial fit to the GZ curve) encapsulates the restoring moment. a The forcing, F (t), is often taken to be a simple sinusoidal function, see for example [17], although stochastic m models of forcing are also studied, e.g. [3]. [ Observing that capsize is equivalent to the roll angle exceeding the angle of v