This paper investigates the nonlinear interaction behaviour of the heave-excited rolling of ships_ The coupled heave-roll system is shown to have two coupling terms due to parametric excitation: A linear term, usually included in most roll stability analyses, which causes Mathieu's type of instability, and a quadratic term which is often ignored. The significance of including the quadratic coupling effect of heave is studied. The analytical condi tions of heave-excited ship rolling are derived using the harmonic balance method. It is shown that, from the viewpoint of dynamic stability analysis, the quadratic coupling term is not a high order term and should be included in the governing equation of motion. The parametrically excited heave-roll behaviour should be modelled by a Hill's equation instead of a Mathieu's equation. Comparisons of the instability regions predicted by these two models are made. The important factors affecting the instability regions are also discussed.


The dynamic behaviour of ships in extreme waves is highly nonlinear and very complex. In recent literature, several research programs based on experimental investigations of ship capsizing have been reported, e.g., de Kat and Paulling (1989) and Grochowalski (1989). The former also discussed numerical simulations of the large amplitude motions of ships; several modes of capsize were suggested. One of the most dangerous capsize modes discussed was the parametrically excited large roll response. Even without direct roll excitation, rolling motions can be excited parametrically by heave or pitch of the ship through the coupling effect between these motions. Parametric excitations are dangerous to ship stability because they introduce a timevarying component to the stiffness, or restoring moment, term of the governing dynamic equation. Sudden loss of ship stability can occur when the restoring moment is reduced significantly.

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