An accurate and fast iterative scheme for estimating the ship rolling and capsizing in regular waves

A short number of steep breaking waves hitting a ship from the side may affect its dynamic behaviour and to determine extreme roll angles and, eventually, capsizing. This last phenomenon is a challenging task for naval engineers because it is responsible not only for a lot of material damages, but also for human lives. It is therefore not surprising that there have been published many relevant theoretical and experimental studies on heavy rolling and capsizing but the problem is not yet fully resolved. Generally, the ship rolling in beam regular seas could be described by a second-order non-linear differential equation with the roll angle as dependent variable. Non-linearity comes from the restoring and damping moments, which are usually represented by (odd) polynomials of roll angle or of its time derivative. Numerical integrators incorporated in modern software packages (e.g. ode45 in Matlab) encounter efficiency problems for the unbounded solutions associated with capsizing, the running time being extremely high. The alternative is to write a computer software, appropriate to the topic to be solved. In the paper, the roll equation was solved using a simple, fast and accurate iterative scheme based on Taylor expansion. Compared with Runge-Kutta 4 th order numerical technique, the used scheme demonstrated not only an excellent agreement of the results, but also a significant reduction of the CPU time. The rapidity of the scheme allowed us to conduct a comprehensive investigation on fractal erosion of safe basins and to represent the normalized integrity curves and the amplitude-frequency response curves for different combinations of wave parameters. The restoring and damping coefficients corresponded to a vehicle ferry model, considered to be either intact or damaged, on the one hand, and with or without bilge keels, on the other.