Some Observations on the Techniques for Predicting the Oscillations of Freely-Floating Hulls in a Seaway
Authors: St. Denis Manley 
Published: October 2008

ABSTRACT
The paper is a sequel to an inquiry presented last year into the validity of spectral techniques for describing the seaway (St. Denis, 1973). The problem now addressed is stated as follows: Given a seaway defined by its variance spectrum and a hull whose dynamical properties are nonlinear and frequency dependent, what techniques are available for finding statistical expressions for the oscillatory motions in each of the six degrees of freedom. The nonlinearities appearing in the hydrodynamic coefficients are either the consequence of changes in hull geometry resulting from the oscillatory displacements from the mean position, or they are the manifestation of the nonlinear phenomenon of viscous damping. It is brought out that, although numerical solutions can be obtained in the time domain, no generally applicable nonnumeric technique for solving nonlinear differential equations either deterministic or stochastic is in hand, and that one is forced to have recourse to approximate techniques. Some available such techniques are discussed as to their usefulness in solving the second order set of stochastic differential equations expressing hull response in a random seaway. There appears to be no simple choice.

1. INTRODUCTION
During the past two decades, it has become common practice to describe the seaway through the spectrum of the variance of the surface profile. Some comments on the validity of such a representation have been made in a previous paper (St. Denis, 1973), where it is brought out that the spectral technique can be applied with confidence when the seaway is of mild intensity; that it provides only a first order description of seaways of moderate intensity, but is otherwis'e inadequate for revealing certain important features that these seaways exhibit (launching, cupping, white capping and breaking); and, finally, that it is entirely insufficient to describe the heavy seas of great severity. The purpose of the present paper is to examine the validity of the spectral technique for describing the oscillations to which vehicles and platforms are subject when operating at sea, The scope of the inquiry is limited to those motions that are induced by light seaways which themselves are properly represent able by a variance spectrum. Seaways of moderate and high intensities are nonlinear and are not contained within this restriction. Otherwise stated, the attention is focused on the kinetic behavior in light, i.e., linear, seas of those systems that are characterized by nonlinear transfer functions.
Prediction of the oscillations that a vehicle or platform system experiences in a seaway is a problem in dynamics (the determination of the forces imposed by the seaway on the system) and in kinetics (the determination of the motions resulting from the forces imposed by the seaway). Only in linear system is it feasible to separate the two aspects; in nonlinear systems, the hydrodynamic loading is affected by the oscillatory response and the two are not properly separable; nevertheless, such a separation is made herein for ease of exposition; however, the mutual interaction of loading and response will need be borne in mind.