Analytical and numerical solutions of a non-linear ship rolling motion

The method of multiple time scales is applied to construct a second-order uniform expansion of the non-linear rolling response of a ship in regular beam seas. The analysis took into consideration linear, quadratic, cubic, quintus, and seven terms in the polynomial expansion of the relative roll angle. The damping moment consists of the linear term associated with radiation and viscous damping and a cubic term due to frictional resistance and eddies behind bilge keels and hard bilge corners. The analytical solution is conducted to construct the steady-state amplitude and phase of harmonic oscillations. The stability of the proposed solution is determined applying Liapunov's first method. Effects of the different parameters on system behavior are investigated numerically. Results are presented graphically and discussed. (C) 2002 Elsevier Science Inc. All rights reserved.