Nonlinear rolling of ships in regular beam seas

A second-order approximate solution is obtained for the nonlinear established harmonic roll of a ship in regular beam seas. The perturbation solutions are compared with solutions obtained by numerically solving the nonlinear governing equations. The peak roll angle and corresponding frequency predicted by the second-order expansion are found to be in closer agreement with the numerical simulation than those predicted by the first-order expansion. Increasing the wave slope is found to increase the peak roll angle, increase the resonance width, and bend the response curves to lower frequencies. Decreasing the damping coefficients is found to increase the peak roll angle, decrease the resonance width, and also bend the response curves. Increasing the amplitude of the wave slope beyond a threshold value results in some unstable harmonic responses and a cascade of bifurcations. The roll responses experience either period doubling or period tripling bifurcations leading to chaotic behavior. A Floquet analysis is used to predict the occurrence of these bifurcations. The perturbation expansion predicts fairly well the established harmonic oscillations as well as the start of the period multiplying bifurcations.