Study On The Parametrically Excited Nonlinear And Random Roll Of Ships In Longitudinal Waves

Parametric rolling in the longitudinal waves is an important mode leading to ship capsizing and relates to many disasters. This paper studies on the prediction method of parametric excitation in regular and random longitudinal waves, and investigates the conditions on which the parametric rolling occurs as well as the dynamic characteristics of large amplitude roll motions.The parametrically excited rolling equation of ships in longitudinal waves is set up with considering the influence of metacentric height varying with time. By assuming quasi-static equilibrium of heave and pitch motions, the method for the calculation of metacentric height fluctuations is derived for ships in regular and random waves.The method of multiple scales is used to obtain the approximate analytical solutions of the regular case, and to determine the stable and instable space of the trivial solutions from which we can determine the conditions of parametric rolling. The parametric roll motions in regular and random waves are numerically simulated by using Runge-Kutta algorithm for different navigation speeds, and the dynamic characteristics of large amplitude roll motions are investigated. The method of stochastic averaging and FPK equation are applied to yield the probability density function of stochastic roll amplitude process. The probability distribution function and mean value are also derived by integration the roll amplitude probability density function.The numerical and analytical results show that parametric rolling exists both in regular and random high waves. When the ship-wave encounter frequency is near the twice of roll natural frequency, the principal parametric resonance occurs, and the large amplitude rolling arises. The roll motion decays otherwise. The stochastic parametric rolling amplitude changes abruptly and is more difficult to predict than in regular waves because of the frequency characteristics of random waves. Also the jump phenomenon is found in the parametric rolling. The jump of roll amplitude will result in the lost of motion stability and the increase of ship capsize probability.