i ky(t)x(t —t)dt + | ko(t)x(t —T)dt + | k3(t)x(t —T)dt 
= ic z(t—T). (9.25) 
Actually, however, taking into account the Kramer—Kronig’s theory that connects 
the added mass and the damping for floating and oscillating bodies and also the fact that 
the restoring term a3x(r) is not a function of frequency, the equation of motion is, as 
T. F. Ogilvie” pointed out, in the form of 
{M + m( )} x(t) + | K(t—t) x(t)dt+C x(t) = f(t). (9.26) 
Here m(o) is the added mass at frequency w = ©. 
9.4 NONLINEARITY OF THE BEHAVIOR OF MARINE VEHICLES 
When the waves are moderate and the amplitudes of the motions are mild, the re- 
sponse of marine vehicles is well expressed by linear equations, as was shown in the 
preceding sections. 
The nonlinearity of ocean waves was found to be not as large as was shown in Sec- 
tion 9.2. Even when the wave itself is linear, sometimes the exciting force exerted by the 
wave can be nonlinear with respect to wave height. For example, for drifting vehicles or 
for the slowly varying behavior of moored offshore structures, the exciting force is not 
linear with the wave height because of the effects of the secondary potentials of waves. 
For rolling near the synchronous frequency, the vehicle might oscillate with such a large 
amplitude that the damping and the restoring force are not linear with the velocity or the 
displacement. For oscillatory motions, in many cases viscous resistance that is not linear 
to the velocity of motion sometimes plays a big role in addition to that of wave making 
resistance that is linear, and results as nonlinear damping. 
Usually, however, even in cases of this kind, the effect of nonlinearity is assumed to 
be small or weak in the succeeding discussions, as will be mentioned again in Chapter 11. 
Generally, under certain restrictions (as the sum of the absolute values of all kernel func- 
tions is not infinite), the nonlinear response can be expressed by 
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