Ocean Systems Engineering, Vol. 1, No. 3 (2011) 249-261 249 Nonlinear ship rolling motion subjected to noise excitation Arada Jamnongpipatkul, Zhiyong Su * and Jeffrey M Falzarano Department of Civil Engineering, Texas A&M University, College Station, Texas, USA (Received April 6, 2011, Accepted September 23, 2011) Abstract. The stochastic nonlinear dynamic behavior and probability density function of ship rolling are studied using the nonlinear dynamical systems approach and probability theory. The probability density function of the rolling response is evaluated through solving the Fokker Planck Equation using the path integral method based on a Gauss-Legendre interpolation scheme. The time-dependent probability of ship rolling restricted to within the safe domain is provided and capsizing is investigated from the probability point of view. The random differential equation of ships’ rolling motion is established considering the nonlinear damping, nonlinear restoring moment, white noise and colored noise wave excitation. Keywords: nonlinear ship rolling; noise excitation; path integration method; Fokker Planck Equation. 1. Introduction Safety against capsizing in heavy seas is one of the major concerns of ship operators and designers. Although the data of ship capsizing is scattered, it was often reported in the media. Existing criteria consider only the ship’s static stability, which is based only on the ship’s nonlinear restoring moment curve. The criteria do not correspond to the complex nature of the capsize phenomenon and the large number of possible scenarios. Dynamical behavior of ships have been of interest to many researchers and engineers, particularly in the stability of roll motion. Prior to capsizing, ships will undergo severe roll motion. The eventual motion may even be chaotic. Identifying chaotic motion and the critical conditions are important for both predicting ships’ capsizing and studying the capsize mechanism. Ship rolling even in a regular sea can exhibit complicated behavior, leading to instability and eventually capsizing. Much work has been done on the analysis of vessels subjected to a periodic excitation in a simplified sea state in order for us to understand the mechanism of ship roll motion under the influence of nonlinear stiffness and nonlinear damping. Falzarano et al. (1992) analyzed the global stability of a ship in regular waves by the use of the Melnikov method and lobe dynamics to define the critical parameters for the onset of chaos that might lead to capsizing and explained the unexpected capsizing in both the homoclinic and heteroclinic regions. Ship rolling in the homoclinic region is the case of ship oscillation around the loll angle while ship rolling in the heteroclinic region represents large amplitude roll motion, which results in ship rolling between *Corresponding author, Research Scientist, E-mail: chotiros@arlut.utexas.edu DOI: http://dx.doi.org/10.12989/ose.2011.1.3.249