Application of Spectral Methods to Non-Linear Dynamic Systems
Authors: Filson John J. , Perez Leonardo Perez 
Published: October 2008

ABSTRACT
The paper describes a technique whereby spectra of irregular seas are applied to dynamic analysis of non-linear offshore systems to produce response and response spectra. It employs the fast fourier transform (FFT) to produce irregular excitation records from a specified sea spectrum. Response is determined in the time domain and response spectra are formed from the response record, again using the FFT. The basic methods used are described and application of the method is discussed. An example application is presented on the dynamic response of a jack-up drilling unit and shows some uses of the results. The intent of the paper is to show how spectral methods can be used to produce analytic results where experimental methods have previously been necessary.

INTRODUCTION
Early treatments of sea forces have utilized deterministic methods whereby the loading was derived from a single period wave. Earliest use dates from Naval Architects in the 19th century employing waves with length equal to that of a ship to determine wave bending. The sea is not of course, composed of single period waves and tends to be quite irregular. Still, there is much practical appeal for the deterministic approach. If by some means one can identify a wave which will indeed place a system in maximum stress, the remaining analysis becomes simple. When a structure becomes sensitive to the period of waves, the deterministic approach can give deceptively erroneous results. Such is often the case with floating structures as well as with highly flexible structures.
Since the introduction of spectral methods to the marine industry some 22 years ago (1), a systematic means of considering wave period and height has been at hand and it has gradually filtered into engineering practice. The irregular sea has been described as a composite of waves with different heights and periods and with random phase relationships to each other. It is possible with statistical methods to describe the sea in terms of a spectrum by considering the distribution of wave energy as a function of wave period. Then, if it is known how a structure in the sea responds to waves of different periods, it is usually possible to form a corresponding response spectrum of the structure in an irregular sea.
The importance of this approach is that once the response spectrum is known, a very useful set of statistical quantities can be simply expressed, (e.g., mean, averages, maximums, etc.), both for the wave and the structure response. With a response spectrum, one can indeed consider the period sensitivity of the structure over all relevant wave periods. Thus, with a relatively standard specification of sea intensity (e.g., "significant height"), one can usually make a meaningful prediction of maximum structure response in a storm of a given duration. Michel (2) has given an excellent description of this approach for linear systems.
One problem in particular has remained in applying spectral methods. 'Being based on the principle of superposition has limited it to linear systems.