A TRUSTWORTHY COMPUTATION OF THE TRANSITORY AND STATIONARY RESPONSES FOR THE SHIP ROLLING BY A MODIFIED MULTIPLE TIME SCALE METHOD

Abstract

An important step in any analysis concerning the ship behaviour in waves is a reasonable accurate assessment of high roll amplitudes. The most commonly used way to estimate the roll oscillations is to solve a nonlinear differential equation with the roll angle as unknown. The nonlinearities refer to different ways of approximating the damping and restoring moments. The lack of exact analytical solutions and the inability of a numerical solution to highlight the influence of various parameters on the ship motion, leads to the need to use approximate analytical methods. Operating the benefits of a modified multiple time scale method recently proposed in the literature, we obtained the governing equations for the transition towards the stationary solutions, the first – order approximations for these solutions and the frequency – amplitude relationship. The accuracy of the analytical results derived in the paper was demonstrated by comparing them with the numerical results (considered to be exact). The agreement between the two was excellent for relatively small forcing amplitude and satisfactory for high level of them on almost the entire range of used values for excitation frequency, including the primary resonance domain.