Ship Rolling Capsizing Probability Based On 4-dimension Path Integral Solution Under Random Beams

The stability of the ship is important for sailing safety.The ship must have the ability to resist overturning moment especially while encountering severe sea condition.Although the ship construction rules have described the rules of stability in detail,there are still some factors not considered as rules are made based on hydrostatics and experience.The two main factors are the nonlinearity of rolling and the randomness of the external load.The first one leads to the uselessness of linear methods such as spectral method which makes the problem more complicated.And the last one results in ship capsizing even the sea seems calm.In conclusion,it is necessary to find a suitable method to make new rules of stability.This article analyses the mechanism of ship motion and get the SDOF of the nonlinear differential equation of ship rolling motion by decoupling from the motion of other five degrees of freedom.Meanwhile the nonlinearity of damping resistance and restoring force are taken into consideration.To simulate the real sea condition,the filter is used to transfer the white noise into regarding real waves by simulating the sea spectra.Combining this two equations,the mathematical model can be created.Through the theory of Markov,a system becomes a Markov Process if the input is a white noise.The equation set mentioned above can be transferred into Ito differential formula as in this formula,drift coefficient matrix and diffusion coefficient matrix can be easily obtained to get the Fokker-Planck equation.If the solution of the Fokker-Planck equation can be got,it is possible to get the transition probability density from the beginningto anytime.Then by using the Chapman-Kolmogrove equation,the probability distribution of the ship rolling at anytime can be calculated.However,the analytical result of the high-dimension Fokker-Planck equation is difficult to calculate.It is necessary to use numerical method to solve the problem.In this article the Path Integral Method and Finite Difference Method are introduced and by comparing the results of this two methods,the study wants to prove the reliability of Path Integral Method.The idea of Path Integral Method is to disperse the equation in time and state.As the interval is tiny,the transition probability density can be regard as unchanged.By calculating the transition probability density in tiny interval and using Chapman-Kolmogrov equation,the probability distribution of ship rolling at anytime can be calculated.The Finite Difference get the transition probability density by disperse the Fokker-Planck equation directly.In order to get the variation of the ship capsizing probability with the time,the first pass theory is introduced in this article.If the ship state crossed the safe border,the ship can be considered as overturned which is irreversible.To prove the theories are correct,the analytical result of the Coupling duffuing oscillator is compared with the results of Path Integral Method and Finite Difference Method.The probability distribution and the peak value of the probability is quite different between the ship stimulated by the white noise and the coloured noise.The probability distribution of ship rolling motion under the real sea condition is more consistent with the experiment,which means by using the coloured noise to simulate the sea spectra,the result can be more accurate.The study also finds the ship stability becomes worse if there is an inclination angle at the beginning which is a very important reason causing the ship capsizing.Finally,the article gives an calculation result of a platform.Based on the study,the theory in this article can reflect the change of ship capsizing probability going with the time,which can give guidance to the design of ship safety.