Stochastic dynamic analysis and reliability evaluation of the roll motion for ships in random seas

Abstract

Large amplitude roll motion in realistic seas is a serious threat to ship stability because it can lead to damage or even capsizing of the vessel. For the excessive roll motion in random seas, the nonlinear effects and dynamics associated with damping and restoring terms should be considered. Currently, the criteria of the International Maritime Organization (IMO) for evaluation of the intact stability are based on both hydrostatics and dynamics (IMO, 2008). However, due to the stochastic nature of the ocean environment and the randomness of the roll response, the assessment of extreme rolling should inevitably be based on dynamic considerations and probabilistic methods. With the awareness of the deficiencies of the current criteria for intact stability evaluation, the IMO is currently developing the next generation (also second generation) of such criteria with a certain consideration of the physics associated with the dynamics of nonlinear roll motion and the randomness of the ocean environment and roll response. In this work, probabilistic methods are proposed for studying the nonlinear behavior of the random roll motion as well as for evaluating the dynamic stability of the vessel in random seas. The problems of dead ship condition in random beam seas and parametric roll in random head seas are studied. For the first problem, the Markov theory is applied in order to study the stochastic nonlinear roll response. Specifically, the rolling behavior is described by a single-degree-of-freedom (SDOF) model which incorporates the nonlinearities associated with the damping and restoring terms as well as the randomness of the wave excitation. The linear filter technique is employed to approximate the random external excitation and then the SDOF model is extended into a four-dimensional (4D) Markov system whose probabilistic properties are governed by the Fokker-Planck (FP) equation. Based on the Markov property of the coupled dynamic system, the 4D path integration (PI) method is introduced in order to solve the high-dimensional FP equation. The numerical robustness and high efficiency of the 4D PI method are evaluated by comparing with the results from Monte Carlo simulation (MCS). Based on the PI method and the MCS method, the feasibility of applying the secondorder linear filter and the Gaussian white noise to approximate the random external excitation are studied. The 4D dynamic system is shown to be an appropriate model for studying the stochastic roll response. With the assistance of the 4D PI method, the influence of ship parameters and wind excitation on the stochastic roll response are investigated. Furthermore, the reliability evaluation associated with high response levels is studied and long-term extreme response is predicted by combing the 4D PI method with the metocean description. In the second part of the thesis, the Grim effective wave model is introduced in order to approximate the variation of the restoring moment in random head seas. Based on the Grim effective wave approximation, the mathematical model for describing the rolling behavior is established. The linear filter technique and an efficient MCS method are applied to predict the extreme roll response of a vessel sailing in random head seas. It is demonstrated that the efficient MCS method is able to provide satisfactory estimation of the extreme roll response with a dramatic reduction of computation time, which is important for the subsequent long-term statistical evaluation. The probabilistic methods mentioned above and the results and conclusions obtained in this work hopefully can provide a useful reference for the second generation IMO intact stability criteria which are currently being developed as well as for stochastic dynamic analysis of nonlinear systems.