Nonlinear rolling of biased ships in regular beam waves

Abstract

The method of multiple scales is used to determine a second-order approximate solution for the nonlinear harmonic response of biased ships in regular beam waves. A Floquet analysis is used to predict the stability of steady-state harmonic responses. The perturbation solutions are compared with solutions obtained by numerical integration of the nonlinear governing roll equation. The results show that the first-order perturbation expansion may be inadequate for predicting the peak roll angle and its corresponding frequency. On the other hand, the peak established roll angle and corresponding frequency predicted by the second-order expansion are found to be in good agreement with the numerical simulation. Moreover, the perturbation expansion predicts fairly well the start of period multiplying bifurcations that lead to chaos. Biased ships are found to be more susceptible to period multiplying bifurcations and chaos than unbiased ships.