A numerical investigation of the global stability of ship roll: invariant manifolds, Melnikov's method, and transient basins

Abstract

A parametrically forced, single-degree-of-freedom equation modelling ship roll is investigated through the numerical study of invariant manifolds, Melnikov's method, and transient basins. The calculation of the manifolds is facilitated through the development of a sophisticated algorithm for approximating the locations of the saddle points of the Poincaré map. For selected fixed values of the restoring-moment and damping parameters (the "base case"), the manifolds of the saddles of the Poincaré map are repeatedly computed for increasingly higher excitation amplitudes until homo clinic , heteroclinic, and mixed manifold intersections are observed. The critical amplitudes at which these tangles first occur are accurately predicted by Melnikov's method, verifying its viability as a tool for analyzing ship roll. Corresponding transient basins indicate that fractally mixed regions of stable and unstable initial conditions appear with the onset of transverse manifold intersections. For parametric forcing, the fractal areas are symmetric about the origin and do not significantly affect the integrity of the safe region near the origin. Test cases involving external or combined external-plus-parametric excitation result in asymmetric transient basins and, following the appearance of manifold tangling, a catastrophic reduction of the safe area. Lastly, Melnikov's method is used to perform a parameter study that indicates the effects of varying the restoring-moment and damping coefficients on the critical excitation level.