Dynamical Analysis of Nonlinear Roll Motion and Capsizing of Ships by MMS

To study the capsizing phenomenon caused by the nonlinear roll motion, we consider a more general dynamical system constructed by the non-autonomous delayed differential equation (DDE).

The amplitude equations are derived by the method of multiple scales (MMS) and the stability of the steady-state solutions is analyzed by the Routh-Hurwitz criterion. Based on these amplitude equations, we construct bifurcation diagrams to analyze periodic solutions, Hopf bifurcation, and chaotic phenomena.

We obtain the distinct routes to chaos, i.e., Period-2 window to chaos, classical Period Doubling Bifurcation to chaos, and Period Doubling Bifurcation to chaos of Period-3 periodic solutions of this non-autonomous DDE, so as to analyze the stability of the ship motion and whether there is a risk of capsizing.

The influence of time delay and other parameters on ship motion are discussed, which provides a research basis for selecting appropriate parametric values to avoid capsizing in ship design. Three phenomena related to ship capsizing are captured: chaos, jump phenomenon and multi-stability, which is helpful for the exploration of predicting and avoiding the occurrence of capsizing events. Numerical simulations are performed which are in a good agreement with our theoretical results.