Nonlinear Ship Rolling in the Presence of Narrow Band Excitation

The rolling motion of a ship in the presence of a narrow band stochastic beam sea is studied by means of the Multiple Scales perturbation method. The analytical solutions indicate that the roll variance can be a multivalued function. A numerical time domain simulation confirms these indications and gives an insight into the dependence of the solution behaviour on initial conditions of the motion as the bandwidth of the process diminishes. Finally, the envelope probability density is obtained in analytical form. The consequences of the multivaluedness on the statistical modelling of nonlinear rolling are discussed