Transient Nonlinear Rolling: The Domains of Attraction

The rolling motion of a ship exhibits well known nonlinear features already for moderate sea excitations. However, the theoretical analysis carried out for confused seas is in general limited to linear model predictions, missing in this way many important features. These can be determinant in the context of ship safety. For example, the motion instability related to jump phenomena between different steady-state oscillations or the onset of subharmonic oscillations, are not considered. In this paper, the rolling motion has been described by a differential equation nonlinear both in damping and restoring. An approximate analytical solution valid in the regions of nonlinear resonance relevant to ship stability is obtained by means of a perturbation method. The transient nonlinear rolling oscillations due to a deterministic excitation are studied in detail. Such an approach is relevant insofar as the behavior of a system under stochastic excitation can be seen as a continuously changing initial conditions. The importance of initial conditions on the subsequent motion is illustrated through the construction of the domains of attraction. These are the regions of the initial conditions leading to a certain steady-state solution rather than another, when more than one are coexisting. In this way, the probability of the different oscillatory states can be roughly estimated. The employed procedure allows a better understanding of the time evolution of the oscillation and allows a time saving with respect to the application of the Monte Carlo method.