Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 934714, 32 pages
Review Article

Modeling of Ship Roll Dynamics and Its Coupling with Heave and Pitch

Department of Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA

Received 14 May 2009; Accepted 11 June 2009

Academic Editor: Jose Balthazar

Copyright © 2010 R. A. Ibrahim and I. M. Grace. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In order to study the dynamic behavior of ships navigating in severe environmental conditions it is imperative to develop their governing equations of motion taking into account the inherent nonlinearity of large-amplitude ship motion. The purpose of this paper is to present the coupled nonlinear equations of motion in heave, roll, and pitch based on physical grounds. The ingredients of the formulation are comprised of three main components. These are the inertia forces and moments, restoring forces and moments, and damping forces and moments with an emphasis to the roll damping moment. In the formulation of the restoring forces and moments, the influence of large-amplitude ship motions will be considered together with ocean wave loads. The special cases of coupled roll-pitch and purely roll equations of motion are obtained from the general formulation. The paper includes an assessment of roll stochastic stability and probabilistic approaches used to estimate the probability of capsizing and parameter identification.