Long-Term Extreme Response and Reliability of a Vessel Rolling in Random Beam Seas

Abstract

In this paper, the long-term extreme response of a vessel rolling in random beam seas and the associated reliability evaluation are addressed. The long-term response analysis is based on the upcrossing rates of the roll motion under different sea states. Generally, for nonlinear roll motion in random seas, the high-level roll response is sensitive and closely related to the nonlinear effects associated with the restoring and damping terms. Therefore, assessing the corresponding statistics of the random roll motion with low probability levels is difficult and time-consuming. In this work, the Markov theory is introduced in order to tackle this problem. Specifically, for the dead ship condition, the random roll excitation moment is approximated as a filtered white noise process by applying a second-order linear filter and an efficient four-dimensional (4D) path integration (PI) technique is applied in order to calculate the response statistics. Furthermore, the reliability evaluation is based on the well-known Poisson estimate as well as on the upcrossing rate calculated by the 4D PI method. The long-term analysis and reliability evaluation of the nonlinear roll motion in random seas, which consider the variation of the sea states could be a valuable reference for ship stability research.

Lyapunov Function based Criteria for Ship Rolling in Random Beam Seas

“…While establishing the mathematical model, the couplings between the roll motion and other modes of ship motion are ignored to make the equation of roll motion simpler [5][6][7][8][9]. This assumption is based on the idea that roll motion has a greater influence on the ship stability in beam seas rather than the other modes of ship motion.…”

“…The single degree of freedom (SDOF) roll motion equation in random beam seas can be represented by the equation (1) [5][6][7][8].…”

“…where |F roll (ω)| represents amplitude of the moment of the roll motion per unit wave height at frequency ω and S ξξ (ω) is the wave energy spectrum [5][6][7][8][9].…”

“…The stochastic roll response and reliability of a ship in random beam seas is studied by using a four-dimensional (4D) path integration (PI) approach in the studies of Chai et al [5][6][7][8] where a 4D Markov dynamic system is established by combining the single-degree-of freedom model used to represent the ship rolling behaviour in random beam seas with a second-order linear filter used to approximate the stationary roll excitation moment.…”

“…In this study, firstly the roll motion in beam seas is represented by the mathematical model derived by Chai et al [5][6][7][8]. Secondly, the methodology of constructing a Lyapunov function of randomly exerted system is outlined.…”