Ship Stability in a Seastate by the State-Space Fokker–Planck Method

Abstract

The development of a new methodology for the modeling of the nonlinear responses and stability of ships in stochastic steep waves is presented. The radiation and diffraction hydrodynamic forces are cast in a state-space form by using a panel method followed by the implementation of the estimation of signal parameters via rotational invariance techniques algorithm. Strong free-surface nonlinearities present in the hydrostatic restoring and Froude-Krylov exciting forces are modeled by the fluid impulse theory. The ambient seastate is represented by a state-space diffusion modeled after a given wave spectrum. This enables the study of the nonlinear seakeeping and stability of ships in a seastate by invoking the methods of stochastic calculus. A state-space stochastic differential equation is derived for the states governing the vessel nonlinear responses, and a linear Fokker-Planck partial differential equation is obtained for the joint probability density function of the vessel motions. 1. Introduction Ships advancing in steep random waves may experience large responses and loss of stability. The need exists for the development of rational stability criteria that will enable the derivation of the safe operation envelope of vessels operating in severe seastates. The past several decades have witnessed significant progress toward the development of potential and viscous flow computational methods for the treatment of the seakeeping hydrodynamics of ships. These methods combined with experiments have been the primary tools for the treatment of ship stability in realistic environments (Umeda et al. 2008). A comprehensive list of articles is contained in the previous reference. Yet, we still lack a general-purpose theoretical framework that may be applied for the assessment of the stability properties of ships. The study of ship stability in a seastate requires the explicit account of nonlinearities of potential and viscous origin, the coupling of all modes of motion, and the development of statistical models for the nonlinear ship responses and their extreme statistics in a stochastic seastate. The study of the stability of ships by the methods described previously is typically carried out by Monte-Carlo simulations for the determination of the safe operating envelope. This approach may, however, be very time consuming when the treatment of the six-degree-of-freedom seakeeping problem is necessary and in conditions where nonlinear coupling between the vessel responses lead to loss of stability (e.g. parametric rolling, broaching).