Some Remarks on the Excitation Threshold of Parametric Rolling in Non-Linear Modelling

It is known that the parametric excited rolling in following seas can lead to dramatic roll amplitudes and eventually capsizing. On the other hand, the parametric rolling in head waves is a less studied phenomenon: in this case, relative ship velocity is high and consequently the roll damping increases for the effect of forward speed. In this paper, the phenomenon of parametric rolling has been investigated from a mathematical point of view. The KBM perturbation technique has been applied to the nonlinear Mathieu equation describing parametric rolling. The advantages of analytical solutions via perturbation techniques are: “easy” determination of amplitudes of periodic motion and possibility to study the threshold for the onset of parametric rolling with respect to some coefficients. It has been shown that numerical simulation and analytical investigation of simplified equation of motion provide solutions that are in good agreement with model testing.