Coexistence of Stable Solutions in the Nonlinear Ship Motion

The dynamic stability and complicated motions of a vessel in regular sea are investigated when the frequency in the pitch is nearly twice the frequency in the roll. We use the Method of Multiple Scales to determine four first-order nonlinear ordinary differential equations that govern the modulation of the amplitudes and phases when the forcing frequency is near either the pitch or the roll natural frequency. These equations are used to determine the steady-state solutions and their stability. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found in the presence of quadratic nonlinearities. The linear solution is symmetric. The pitch mode has multi-valued solutions and the roll mode has a single-valued solution for the case of primary resonance of the pitch mode. Both pitch and roll modes have multivalued solutions in small intervals for the case of primary resonance of the roll mode.