Therefore, the third order nonlinear frequency response function is obtained as 
SYXXX(W 1),@2,(@3) —_ S¥Xxx(@1, @2, 3) (11.86) 
OSxx(W1) Sxx(@ 2)Sxx(@3) Sys xxy(W1,@2, 03) 
H3(@1,@2,@3) = 
6. Application of the trispectrum to ship’s rolling by Dalzell. 
J. F. Dalzell!’ discussed the characters of higher order nonlinear responses in Vol- 
tera expansion, especially the interferences between different order nonlinear responses, 
based on E. Bedrosian and S. O. Rice® and also tried to solve the problems encountered 
on their numerical computations. Besides Dalzell’s trial°>-** on the application of the 
bispectrum analysis to the added resistance of ships advancing in waves (already referred 
to in Section 11.4), he also tried!” an application of trispectrum analysis to the problem of 
nonlinear ship’s rolling, expressing*! the nonlinear roll damping by cube of roll angular 
velocity. 
Dalzell expressed the equation of motion as 
3 
[AilPo}+ B Yoh + cro} = X(2). (11.87) 
j=1 
S 
Then, using the so-called incommensurate frequency technique,” the relations of 
Aj, Bj,C;j = 1 — 3) and frequency response functions H;(w), H2(@,,@2), 
H3(@1,@2,@3) were derived. 
Setting appropriate values for A, to C3, and using the simulated series of wave 
heights X(r) that have Pierson—Moskowitz type spectra, Dalzell synthesized the roll 
response Y(t) as shown in Fig. 11.12, using impulse responses h)(T), h2(t,T2) and 
h3(T), 72,73) obtained through H\(w), H2(@1,@2) and H3(w1,@2,w3) expressed by Aj, B;, 
and Cj(j=1 — 3). 
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