Stability of nonlinear oscillatory systems with application to ship dynamics

Abstract

A procedure to generate an approximate bifurcation diagram for a single-degree-of-freedom system in a selected parameter space is developed. The procedure is based on the application of Floquet analysis to determine the stability of second-order perturbation approximations of the solutions of the system in the neighborhoods of specific resonances. As a control parameter is varied, a combination of elementary concepts of bifurcation theory and the proposed method are used to detect the first bifurcation from the periodic solutions and hence infer the qualitative changes that the system experiences. Codimension-one bifurcations are investigated in a two-dimensional parameter space composed of the amplitude and frequency of the excitation. The behavior of a softening Duffing oscillator is analyzed under external and parametric excitation. The dynamics of a ship rolling in waves is also considered and three types of excitations are treated: external, parametric, and a combination of both.

Analog- and digital-computer simulations are used to verify the accuracy of the analytical predictions. It is found that the predictions based on the first bifurcation of the analytical solution give a good estimate of the actual behavior of the system. The stability regions of the solutions near each of the resonances display a self-similar structure in the parameter space. The physical implications of these bifurcation patterns are important for the prediction of the capsizing of ships. The dangerous regions of the parameter space where capsizing might occur are identified for a given system.

Capsizing is found to occur via two distinct scenarios: one evolving from a large oscillation through a disappearance of a chaotic attractor (crises) and a second, potentially more dangerous, developing from a small oscillation through a sudden tangent instability. These scenarios agree with previous experimental studies.