A numerical example for a ship model, with wo = 3.85, (@/wo) = 0.06705, 
B = 0.08, and waves with a Neumann type spectrum and a peak at wo, is shown in Figs. 
10.2 — 10.5. Figure 10.5 shows the overall effect of nonlinear damping, and Fig. 10.4 
shows the effects of nonlinearity on the term of Eq. 10.31, especially the double convolu- 
tion Sg¢g(w) of the spectrum Sgg(w), shown by Eq. 10.32. 
In the same way, the author also calculated the effect of a nonlinear restoring force 
expressed as 
Ip + NO + (Kip + K3*) = M, (10.36) 
d + 2ad + wip + k3h? = mit). (10.37) 
Starting with a 0 approximation go, for k3 = 0, the spectrum of the first order 
approximation @; when k3+ 0, is 
Sob) = Soop (@) — kz + 2RelHggm(@) - 305, 59.9()] 
+ UH gn(00)?|904 Spp() + 60 S51, (@)}. (10.38) 
Here 
oo 
Sy Oe | | Shap Sbop(O2)Sbop(@ —W1-—@2)dw dw». (10.39) 
—co —0O 
10.3.2 General Formulation of the Perturbation Method 
S. H. Crandall’’ formulated a general approach to the case with a nonlinear 
restoring term by the perturbation method, as follows. 
The equation of motion is expressed as 
X + 2aX + wp [X + €9(X)] =f) (10.40) 
where € has a small value. Then X can be expanded in terms of powers of € 
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