The nonlinear ship rolling and safe basin erosion in stochastic beam seas

Abstract

A short number of steep breaking waves hitting a ship from the side may affect its dynamic behaviour and determine extreme roll angles and, eventually, capsizing. Even the capsize is a rare event, its consequences for the ship and crew are often fatal. Generally, the ship rolling in regular or stochastic beam 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 polynomials of roll angle or of its time derivative. In the paper, we used such a model equation to estimate the ship rolling and capsizing in a stochastic beam sea. The sea action was simulated by a harmonic function with random frequency and phase. The roll equation was solved using a simple, fast and accurate iterative scheme based on Taylor expansion which proved to be very competitive in terms of accuracy with more elaborate methods and which allowed a substantial reduction of the CPU time. Due to these properties of the scheme, we were able to conduct an extensive investigation on fractal erosion of safe basins and to represent the boundaries between capsizing and non-capsizing regions in wave frequency – wave amplitude plane and normalized integrity curves for different combinations of random wave parameters. The restoring and damping coefficients corresponded to a vehicle ferry model, considered to be either with or without bilge keels.