/5 injo,) 



Understanding and Prediction of Ship Motions 



- <X> '^ - 



cr . S . . ( OJ) + 



o ^o ^o 



3cr' 



0^0 o ^o 



= /3' In J CO) 





where 



H,(a;) 



$(&;) 



I(D(a)) 



q^C-^) 



= Q{HJa;)} , 



I-H ro;) , 



2aa) 



(cd -^ - o)-^) + 4a"'a)'' 



and also from 



2 -^ 



(3.24) 



S(w) = S. .(a)^)-S. .(co^)S. . (CO- oj^- co^)da)^doj^. (3.25) 



By the substitution of Eqs. (3.19) and (3.24) into Eq. (3.10) and from 



S. . (CO) = CO^ S. . (CO) 



X — [A.(aj)l ^ , - CO < aj<03 



•J n 



(3.26) 



2 S . . (oj) dw 



^o^o 



the spectrum of the 1st order approximation 4>^(t) can be calculated. 



As an example, the spectrum of the non-linear rolling of a model with 

 oj^ = 3.85 and a/w^ = y = 0.06705, /3 = 0.08 has been calculated. For the waves, 

 the Neumann shape spectrum which has its peak around ^^ =3.85 was used. In 

 these kinds of computation which include the convolution of a spectrum, the spec- 

 trum should be defined in the range of - » -^ +» of oj, and accordingly the definition 



121 



