The Nonlinear Ship Rolling And Capsizing Under Irregular Wave Excitation

The stability of a ship under random wave excitation is directly related to thesafety of lives and properties at sea. The stability against capsizing in rough seas isone of the most fundamental requirements considered by naval architects whendesigning a ship. However, an equivalent understanding of ship capsizing in roughwaves has not been fully realized. The obstacles mainly come from two areas: firstly,the nonlinearity of the large amplitude ship rolling; secondly, the randomness of waveexcitation. The fact that many intact vessels were lost in rough seas confirms theshortcomings of the current criteria. Therefore, the problems of ship rolling instochastic waves should be investigated in ship stability analysis, and consequentlynew stability criteria should be established.Considering the nonlinear rolling damping, nonlinear restoring moment andrandom wave excitation, the random nonlinear differential equation of ship rolling isestablished. It seems that time domain method is an appropriate way in handling suchnonlinear problem. What’s more, considering the randomness of wave excitation, shipcapsizing is a random event. The stochastic processes theory can be used forinvestigating this random event. Ship capsizing in rough seas is an irrecoverable event.It is also a typical first passage problem. The first passage theory is suitable forinvestigating the capability of ship anti-capsizing on random waves.The response of a dynamic system, whose random excitation is expressed aswhite noise or filtered white noise process, is a Markov process. Under the theory ofMarkov diffusion process, the probability density of the dynamic system response(such as the transition probability of the ship rolling motion) can be obtained bysolving the Fokker-Planck equation. However, analytical solutions are only possiblefor a very limited class of systems. The path integration (PI) method has been provedto be an effective numerical method to solve the Fokker-Planck equation with highaccuracy.In this paper, the Two-dimensional Fokker-Planck equation in time domain issolved by using the path integration (PI) method and the applicability of the PImethod was verified with the analytical solutions. The transition probability density ofthe ship rolling in white noise was obtained. The distribution of the first passage timewas achieved by first passage theory. The influences of the magnitude of externalexcitation, the rolling damping and the nonlinear restoring moment on the transitionprobability density of the ship rolling and the first passage time were analyzed. Thereal waves were regarded as the coloring of the white noise by a filter which wasintroduced into the system. The Four-dimensional Fokker-Planck equation in timedomain was solved by using the PI method and the transition probability density ofthe ship rolling movement was obtained. The evolution process of ship capsizing probability distribution with the time was achieved by using first passage theory.The differential equation is integrated in time domain by the numerical methodand the roll responses of the ship are simulated. The erosions of the safe basins of atypical ship are investigated and the survivability diagram of the ship is obtained. Thevariations of survival probabilities in different sea states and vessel speeds areanalyzed.In conclusion, the proposed methods provide a reliable and efficient tool forstudying the nonlinear ship rolling and capsizing in random seas. Its application couldprovide valuable reference to the ship design and the safe standards.