CHAPTER 14 
CONCLUSIONS FOR PART II 
The so-called “‘spectrum correlation method” discussed in Parts I and II has been 
shown to be a powerful way in the analysis of stochastic processes, but only when the 
processes are linear. Many steps in the manipulations in Parts I and Il are based on the 
assumption of linearity of the process. 
In the analysis of seakeeping data, as is usual in the general engineering field, many 
phenomena can be approximated as linear, and so spectrum or correlation analysis has 
played a significant role in advancing the technique of handling those data. Now, 
however, the nonlinearity of seakeeping data for ocean vehicles and structures has 
gradually been introduced, as mentioned in the Introduction. 
Here in Part III of this lecture, methods for treating the nonlinearity in the stochastic 
process analysis were summarized and reviewed. The conclusions obtained here are as 
follows: 
1. Several works on the effect of the nonlinearity of ocean waves on its spectrum 
were reviewed. It is now clear that, if necessary, we can get the effect of nonlinearity on 
its spectrum as well as on the probability distribution of the maxima and the minima of its 
amplitudes. It is also clear that the nonlinearity of ocean waves is usually quite small. 
2. Even when the nonlinearity of the waves is small, the response of ocean vehicles 
and structures might be nonlinear, because of the very low frequency characteristics of 
their responses that respond rather severely to the higher order nonlinear excitation by 
waves or because of the nonlinearity in their response characteristics. For these cases, 
when the nonlinearity is weak and the response characteristics are expressed by weakly 
nonlinear equations of motions, the equivalent linearization and perturbation methods can 
be applied if the excitation is approximately linear. The perturbation method was applied 
by this author rather early to ship’s oscillation, and now both methods are well formulated ~ 
as shown in Chapter 10. These methods are applicable to obtaining the first approxima- 
tion of weakly nonlinear damping and restoring oscillations under random excitation. To 
proceed to the second or third order approximation is, however, not necessarily easy. 
3. The Voltera or functional expansion is one appropriate way to express the weakly 
nonlinear response to random excitation, and the polyspectral analysis is a reasonable 
way to get the higher order nonlinear frequency characteristics. The procedure for this 
analysis was summarized in Chapter 11. 
4. Recently, J. F. Dalzell has engaged in significant efforts on polyspectral analysis. 
He tried not only bispectrum but even trispectrum analysis in the study of seakeeping in 
irregular waves, with excellent results, although an enormous amount of careful computa- 
tion was necessary which at this stage might not be feasible for practical purposes. 
5. Probability distribution of the maxima and the minima of the oscillatory motions 
can also be obtained from the nonlinear frequency responses. The general procedure for 
obtaining the probability distribution of extremes by calculating the cumulants related to 
the nonlinear frequency response characters and a few other examples are reviewed in 
Chapter 12. 
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