Stochastic nonlinear ship rolling in random beam seas by the path integration method

Nonlinear ship rolling in random seas is a serious threat to ship stability. In this work, the dynamic stability of the vessel in random seas is evaluated by means of probabilistic methods. Specifically, the Markov diffusion theory is applied in order to study the stochastic aspects of the roll motion driven by random wave loads. The roll motion is modeled as a single-degree-of-freedom (SDOF) model in which the nonlinear damping and restoring terms as well as the random wave excitation are all incorporated. The stationary wave excitation moment in the SDOF model is represented as a filtered white noise by employing a second order linear filter. Therefore, a four-dimensional (4D) Markov dynamic system is established by combing the SDOF model with the linear filter model. Because the probabilistic property of the 4D Markov system is governed by the Fokker–Planck (FP) equation, the response statistics of roll motion can be obtained by solving the FP equation via an efficient 4D path integration (PI) method, which is based on the Markov property of the coupled dynamic system. Furthermore, the random wave excitation is approximated by an equivalent Gaussian white noise, and a two-dimensional (2D) PI technique is applied in order to obtain the response statistics of the dynamic system driven by this Gaussian white noise. The rationality and accuracy of applying the equivalent Gaussian white noise to simulate nonlinear ship rolling in random seas is studied. Moreover, the accuracy of the response statistics computed by the 4D and 2D PI techniques is verified by means of the versatile Monte Carlo simulation technique.