Understanding and Prediction of Ship Motions 



(p[(t) being the modification term and is 





(3.7) 



Of course the convergence of this approximation should be certified strictly at 

 first; however, here the convergence is assumed as far as the linear damping 

 a/wj is rather large and the non-linear damping /3 is pretty small. 



The autocorrelation function of this 1st approximation 4>i(t) is then, 



= E[4>^(t+T) - cp[(t + T)] [cpjt)-4'[(t)] 



E[0„(t + r) cPJt)] - E[0^(t + T) 0;ct)] 



E[cP[(t + T) cPJt)] + E[cP[(t + r) cp[(t)] 



= E[0^(t + T) 0^(t)] - 2RE[0^(t + T) 0;(T)] + E[0;(t + T) <p[(t)] . (3.8) 



Accordingly the spectrum s^ ^ (co) of the 1st approximation, 4>y(t), is 



+ ^ ^"'"" E[<^;(t + T) •0;(t)] dr. 



'' - CO 



The substitution of Eq. (3.7) into this equation yields 



= S^^^^(a)) --^2^J J hg(t-a) e-J-- e[0„( t + r) -c^^Cm) I -^^Cm) I] d/xdr 



-co ^00 -.CO 



'^'' / / ^ 



*^ ~ CD *'-Q0 »^-03 



(3.9) 



,(t + r-M) hg(t- I.) 



/CO ^co 



IVT^ ei-(--) dMJ 

 - m '^ - CO 



117 



(3.10) 

 (Cont.) 



