PART I 
TREATMENT OF NONLINEARITIES 
CHAPTER 9 
INTRODUCTION 
9.1 INTRODUCTION FOR PART III 
In the preceding sections, Part I and Part II, the processes were assumed to be al- 
most linear, and in so far as the process is linear, the periodogram analysis shown in Part I 
and the model fitting method in Part Il are effective methods of analysis. For example, a 
Fourier—Stieltjes expression for the process assumed to exist in Part I is originally a linear 
expression, and the AR, MA, or ARMA models treated in Part II are based on the idea of 
decomposing the processes into independent or orthogonal processes, i.e., assuming their 
linearity. However, if the process is nonlinear, these methods can no longer be applied 
directly. We need some special considerations for their treatment. 
In most engineering fields, many phenomena can be approximated as linear. How- 
ever, no phenomena are purely linear but include some elements of nonlinearity. Today in 
nonlinear phenomena such as the effect of viscosity, the secondary potential forces that 
include the interaction of two frequency components of excitations have come to be con- 
sidered important in treating the behavior of ocean vehicles and structures, although 
useful information has been derived even under the limitations of linear approximation. 
Here in Part II, the nonlinearity of waves themselves that are the source of excita- 
tion to systems of ocean vehicles and structures is first investigated. According to a few 
works already published, their nonlinearity is usually not large. 
The approximation methods that have been used to treat nonlinear response pro- 
cesses, such as the linearization method and the perturbation method, are summarized. As 
more advanced approximations, the Voltera or functional expansion of the nonlinear pro- 
cess and the application of polyspectra are then introduced. 
As an extension of these approximation method, a slightly different aspect of the 
analysis of the probability characteristics of the process, the probability distribution func- 
tion of the extremes, is summarized. As extensive work has been published recently on 
the analysis of nonlinearity response including these probability characteristics by J. F. 
Daizell,!*!3.55-8 only the derivation of general characteristics of several functions and 
the results of these applications are reviewed. 
Next in Part III, the treatment of stochastic processes as Markov processes is intro- 
duced, and then the application of the Fokker—Planck equation, also recently introduced 
by J. B. Roberts>*! in the analysis of seakeeping data, is reviewed. 
Finally, as a slightly different approach, or as the extension of the model fitting tech- 
niques discussed in Part II, a few examples of this extension to the nonlinear process are 
briefly reviewed. 
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