Instability of Ship Roll with Nonlinear Heave-Roll Coupling

Study on the instability regions of ship rolling with nonlinear coupling of heave and roll was presented. The Hill's equation, which describes not only the linear but also the quadratic coupling term, was developed from the heave roll coupled governing equations of a vessel in head sea to simulate the parametrically excited system of the roll motion coupled with heave. And the Mathieu's equation, including only the linear coupling term, was also developed. Using the method of strained parameters for the two types of the parametrically excited systems including nonlinear and/or linear coupling term, the stability regions were derived. The results were verified by numerical analysis using Runge Kutta method with the parameters on the margins or in out of the instability regions. The periodic steady state solutions, the convergent solutions, the divergent solutions and the chaotic solutions for the ship roll response were reported. It is shown by the theoretical and numerical investigation that the nonlinear coupling term is a very important factor to ship motion, and the methods of strained parameters is effective in the analysis of stability of ship rolling.