g(x] =2°, 7 =€w5. When € >0, we call ita hard spring system, and when € < 0, a soft 
spring system. 
From Eq. 10.12’ 
w2g = w9|1 + €Ex*1/EL"] (10.14) 
When nonlinearity is weak, x can be approximated as Gaussian, and then 
E[x“] = 3(E Le ie (10.15) 
and thus W2q = Wl1 + 3€03]. (10.16) 
When every coefficient is linearized, the response can be obtained. If f(t) is almost 
white in the important range, E [x?] for this case can be computed as 
EL] = 02 -3e0,. (10.17) 
Here oz is the variance for linear oscillation. Eq. 10.17 shows that, whene > 0, E[x?] is 
smaller than by nearly 3€0,,. 
Applying these general results (Eqs. 10.11 and 10.12), L. A. Vassilopoulos”4 
obtained @-, and We, for ship’s rolling, expressed as 
X+ax+Pldixrt+ op xt+ek-r =fir), (10.18) 
as 
eq = a+, = a+ /8/n -B Ox (10.19) 
Wg = w+ [302] -k. (10.20) 
Further, assuming that the damping is small and the spectrum of the wave slopes is 
flat in the important range for rolling, i.e., 0; = @o0; , he derived 
284 
