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A physically based averaging procedure is applied to a stochastic nonlinear singledegree-of-freedom equation for ship rolling, leading to a one-dimensional continuous Markov model for the energy envelope of the roll motion. It is shown that this model enables various statistics of the roll response to be estimated, including its stationary distribution and the mean time for the energy to reach a critical level. Moreover, it is demonstrated that the Markov model can be used as the basis of a new stochastic identi cation technique for estimating the spectrum of the excitation, and the nonlinear damping moment, from measurements of the roll response alone.

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