Filter models for prediction of stochastic ship roll response

• Shaping filter technique is introduced to model the random wave excitation.
• A four-dimensional path integration method is applied to study the stochastic roll response.
• The feasibility of simplifying the dynamic systems is verified by Monte Carlo simulation.
• The influence of external excitation on the statistics of high level response is illustrated.

In this paper, the shaping filter technique is introduced to study the stochastic roll response of a vessel in random beam seas. Specifically, the roll motion is described as a single-degree-of-freedom (SDOF) model in which the stationary random wave excitation term is approximated as the output of a shaping filter, i.e. a filtered white noise process. Therefore, the original SDOF model can be extended into a four-dimensional (4D) and a six-dimensional (6D) dynamic system, respectively, when this second-order nonlinear differential equation couples with the second-order linear filter and the fourth-order linear filter used to model the random wave excitation. Basically, the fourth-order linear filter provides a better approximation, but the subsequent dynamic system would be more complicated to tackle. In this regard, the 4D coupled system can be viewed as a Markov system whose probability properties are governed by the Fokker–Planck (FP) equation. Furthermore, the 4D path integration (PI) method, an efficient approximate technique based on the Markov property of the dynamic system, is applied to solve the 4D FP equation and then the response statistics are derived. In contrast, the statistics of the roll response in the 6D extended system are evaluated directly by a 6D Monte Carlo simulation. The advantages of the 4D PI method as well as the feasibility of simplifying the 6D system by the 4D system in terms of determining the statistics of high-level roll response are demonstrated through various cases corresponding to different sea states. Furthermore, the influence of external excitations on the statistics of high roll response levels is also illustrated by these cases.