h,(T\,T2 . . . Tn) is called the 7» th order impulse response function and 
H,(@1,@2 . . . @,) the n th order frequency response function. The expansion of x(f) in 
Eq. 9.29 is called the functional expression or the Voltera—Wiener expression. The treat- 
ment of nonlinear processes from this stand point and the application of polyspectra in 
the analysis of the nonlinear response system is dealt with in Chapter 11. 
When the nonlinearities are expressed explicitly in the form of the equation of 
motion as 
Mx + B(x) + K(x) = F(), 
with the nonlinearity in the damping and restoring terms, there have been a few efforts to 
overcome the difficulties. In Chapter 10, these efforts will be summarized. 
Hasselman’! formulated the equations of motion for a quadratic nonlinear ship’s 
motion, expressing the wave field by surface displacement €( x,t) = ¢, and its normal 
surface velocity (df/dt) (x,t) = €2, where xX = (x;,x2) is the horizontal Cartesian coordi- 
nate vector. He derived a generalized nonlinear equation expressed in functional form 
and made it clear that, through cross—spectrum and cross—bispectrum analysis, it is possi- 
ble to get the linear and quadratic frequency responses from the data obtained from 
irregular waves on the ocean. For his formulation, the frequency components that are the 
sum and difference of two component frequencies appear to be important. His expression 
is, however, in generalized form and is not necessarily adequate for practical applications. 
Almost the same content will be explained later in scalar form in Section 11.3, for a gen- 
eral dynamic system with quadratic nonlinear characteristics. 
279 
