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By a combination of averaging techniques with the theory of Markov processes, an approximate theory is developed for the rolling motion of a ship in beam waves. A simple expression is obtained for the distribution of the roll angle, and is tested by a comparison with a set of digital simulation estimates due to Dalzell. Good agreement is obtained over a realistic range of damping values.

A theoretical model is proposed for a moored vessel, with nonlinear restoring forces, responding to random waves. The model takes into account the nonlinear nature of the relationship between wave elevations and wave forces. Fokker-Planck equations are derived for the distribution of the amplitude response of the vessel. By simplifying these equations, with the introduction of a white noise approximation for the force process, an expression is obtained for the stationary amplitude distribution. The accuracy of this approximate result is assessed by a comparison with corresponding digital simulation estimates.

A common method of assessing the damping present in ship rolling motion is to perform a free-decay experiment, in which, in the absence of waves, the ship is given an initial roll amplitude and then released. By processing the resulting decaying, oscillatory trace it is possible to estimate quantitatively the degree of damping, even when this is nonlinear. In this paper an approach to the estimation of nonlinear damping is proposed which involves the use of a cubic spline interpolation of the peak amplitudes, followed by a parametric identification procedure. This method is first applied to some simulation data and then to some real free-decay data, obtained from a scale model of a ship, with various initial conditions. The experimental data are analyzed using two alternative parametric forms, that is, linear-plus-quadratic and linear-plus-cubic damping. A principal advantage of the proposed technique is that, because nonlinearities in the restoring moment are properly accounted for, it is not limited to small angles of roll. Thus it can be applied to free-decay data relating to very large initial roll amplitudes.

The results of an experimental study of a ship rolling in random beam waves at zero speed are presented. The experiments were conducted in a large wave tank using a 1:20 scale model of the fisheries protection vessel Sulisker. By digitally processing the roll response measurements, obtained over long periods of time, estimates of the probability distribution of the roll peak amplitudes were obtained and compared with some corresponding theoretical predictions. The theory was found to give good agreement with the experimental findings for four different wave elevation spectra. In particular, the experimentally observed deviation from the Rayleigh distribution at high peak amplitudes was correctly predicted by the theory. This deviation is largely due to the pronounced nonlinear character of the roll damping.

A method of estimating the parameters in a nonlinear, single-degree-of-freedom model of ship roll motion is described. It is shown that the "linear-in-the-parameters" nature of the model allows formulation of a relatively simple, and computationally very efficient, estimation procedure, based on a recursive, linear least-squares algorithm. The method is applied to the case of forced roll, where the roll moment excitation can be accurately determined. The general nature of the estimation scheme allows the combining of data from a sequence of experiments with sinusoidal forcing of varying frequency, and also allows the use of data arising from a nonsinusoidal forcing moment. In the latter case all the required parameters in the ship motion model can be estimated from a single test. The technique is illustrated by applying it to both digitally simulated data (where the parameters are known, a priori) and to real data, obtained from a model ship. In the latter case the parameter values obtained by the proposed method are shown to agree well with estimates obtained by alternative, independent methods.

An equation of motion for uncoupled ship rolling motion in irregular seas is considered in which parametric excitation, arising from the time dependence of the restoring moment, is included. It is shown that, when the damping is light, the method of stochastic averaging can be applied to yield an approximate Markov model for the rolling motion. Appropriate Fokker-Planck equations are given from which some simple expressions for the stationary response distribution are derived. For the case of linear damping it is found that the motion is unstable if the damping factor is sufficiently small. The inclusion of a quadratic damping term in the equation of motion is shown to limit the unstable motion predicted by the linear theory. Explicit expressions for the distribution of the roll response are derived for the case of combined linear and quadratic damping. The case of purely parametric excitation is considered in some detail.