Technical note on ship rolling associated to high degree polynomial restoring moment using the Melnikov method

Abstract

Closed solutions of one-dimensional unperturbed Ordinary Differential Equations (near Hamiltonian systems) are obtained when the restoring force is represented by a quintic polynomial (with either three or five zeros). This is of great practical interest for the study of rolling ship motion with both configurations of vanishing stability and lolling effects. A numerical extension to higher degree of the odd polynomial restoring moment (with three zeros) is considered as well. The standard Melnikov method is used when the forcing is a harmonic excitation. This provides a critical amplitude of the forcing above which the dynamical system should be unstable (capsizing or chaotic motion). Comparisons between different forms of the restoring moment are made and the Melnikov criterion (critical forcing) is analysed in terms of frequency of excitation. It is shown that the Melnikov method predicts that an infinite critical excitation amplitude is required to destabilize systems at some frequencies. This is confirmed by analysing the erosion of basins of attraction.