Probabilistic analysis of the instantaneous center of rotation of a floating structure under regular and irregular beam waves

Accurate prediction of roll motion is crucial to the safety of ships and floating structures. The "roll center" mentioned in the literature is ambiguous and various definitions exist. Hence, the instantaneous center of rotation (ICR) is preferred, being an established kinematic concept. Understanding the ICR is important on account of its significant influence on roll damping. A previous study postulated that the ICR moves in a straight line for regular beam waves, based on experimental observations. In this paper, the probability distribution of the ICR coordinates (y1, y2) is derived analytically for regular and irregular beam waves. For regular waves, it is proven a fortiori that the ICR lies on a straight line, and that y1 and y2 follow the Cauchy distribution, for which the mean and variance are undefined. For irregular waves, it is shown that the joint probability density of y1 and y2 is a bivariate Cauchy distribution, thus the iso-density contour lines are concentric ellipses. Regular and irregular wave experiments are performed to investigate realistic conditions. The analytical solutions are found to be quite accurate, notwithstanding some discrepancies arising from nonlinearities and experimental imperfections. The theoretical results presented herein have potential practical applications, for example since the ICR governs damping, the ICR distribution allows damping to be modeled as a stochastic process.