Analysis of externally excited nonlinear Mathieu equation model for floating bodies

Abstract

Parametric resonance is a dynamic instability that causes exponential growth in the amplitude of an oscillating system. This study performs a nondimensional analysis on a nonlinear Mathieu-type equation model for floating bodies excited by waves, developed to capture parametric resonance in both the heave and pitch degrees of freedom. A non-cylindrical axisymmetric spar-buoy is used as a test case. The wave excitation forces are pre-calculated for various heave and pitch positions and interpolated with a third-order polynomial. The model includes nonlinear hydrostatic restoring forces and incorporates position-dependency of the wave excitation forces. Through the nondimensional analysis, the model is simplified. Simulations show parametric resonance when the wave frequency is twice the natural frequency of the structure. The behaviour of the oscillations is investigated by a frequency sweep. The results compare favourably to those from a benchmark model with nonlinear Froude-Krylov forces, but achieving a 10000-fold speed increase, with the computation time going from at least three hours, to less than one second. On top of this increased computational efficiency, the presented model facilitates analytical approaches, such as perturbation analysis or harmonic balance.