A two-degree-of-freedom nonlinear numerical model is developed to simulate the heave-excited rolling motion of a vessel. The geometrical concepts of nonlinear dynamics and the methods of Simple Cell Mapping and Simplicial Mapping are applied to achieve a global understanding of the nonlinear behaviour of ship rolling. Parametric resonance, jump and bifurcations of the types of fold, Hopf, period-doubling and pitchfork are observed to occur in the simulated rolling motions. Chaotic motions are also predicted by the numerical model.
Parametric resonance of rolling motions has long been recognized as one of the, most dangerous modes of ship-capsizing. Reviews of both the experimental and analytical studies of ship rolling were recently given by de Kat and Paulling (1989) and Nayfeh (1988). In the analysis of rolling motions, the earlier approach of assuming linear restoring moment and no feedback from roll to heave (or pitch) led to a Mathieu equation for the roll, with the heave (or pitch) treated as a prescribed parametric excitation. Instability regions in the parametric planes were then obtained from the Mathieu equation (Paulling and Rosenberg, 1959; Blocki, 1980). This approach gave qualitative descriptions of roll instabilities but could not explain some of the complicated behaviour associated with large rolling motions. Nayfeh, Mook and Marshall (1973) and Mook, Marshall and Nayfeh (1974) investigated the coupled roll-pitch motions. They demonstrated the existence of a saturation phenomenon involving an energy exchange between the indirectly excited roll and the directly excited pitch motion, when the pitch frequency was twice the roll frequency and the pitch motion was excited near its resonance. Thompson, Rainey and Soliman (1990 1 proposed recently that, the phenomenon of ship instability and the post- instability behaviour could be quantified by applying the new geometrical concepts of nonlinear dynamics (Thompson and Stewart" 1986).