CHAOTIC ROLL MOTION AND CAPSIZE OF SHIPS UNDER PERIODIC EXCITATION WITH RANDOM NOISE

A stochastic analysis procedure is developed to examine the properties of chaotic roll motion and the capsize of ships subjected to periodic excitation with a random noise disturbance. To take into account the presence of randomness in the excitation and the response, a generaliz ed Melnikov method is developed to provide an upper bound on the domai n of the potential chaotic roll motion. The associated Fokker-Planck e quation governing the evolution of the probability density function (P DF) of the roll motion is derived and numerically solved by the path i ntegral solution procedure to obtain joint probability density functio ns (JPDFs) in state space. A chaotic response can be found in two regi ons (near the homoclinic and heteroclinic orbits). The global behavior of the roll motion can be depicted by the JPDF. It is found that the presence of noise enlarges the boundary of the chaotic domains and bri dges coexisting attracting basins in the local regimes. The attracting domain of capsize is of the greatest strength. The probability of cap size is considered in this paper as an extreme excursion problem with the time-averaged PDF as an invariant measure. With this measure, the heteroclinic region is identified as an 'unsafe' regime. Numerical res ults indicate that, under the presence of noise, all roll motion traje ctories of a ship that visit the regime near the heteroclinic orbit wi ll eventually lead to capsize.