9.3 RESPONSE OF THE BEHAVIOR OF A MARINE VEHICLE ON WAVES 
When the amplitude of oscillation in six degrees of freedom is small, the behavior 
of a marine vehicle in waves can be expressed approximately by linear equations. 
Here, for simplicity, if we assume that a floating body in a regular wave train 
f(t) = Z(w)e!' is subjected to a force bC(r) that is linear with wave height C(t), the 
equation of motion is well approximated°? by 
Q4X(t) + arx(t) + a3x(t) = E(t). (9.11) 
The response x(t) will be in the form X(w)e, 
where 
X(w)el" = X(w)lel@t#9x@)+0.0)]. (9.12) 
O¢(w), 0,(@) are the phase relations of the exciting force to the wave height, and the 
response to the exciting force, respectively. 
Generally, for a body floating on the surface of water, the coefficients a), a,b of 
the inertia term a@;x(r), the damping term, and the compulsory term, respectively, are 
generally functions of the frequency and so we find a; = Aj(@),a2 = A2(w), and 
a3 = A3(W),b = B@). 
The equation of motion, Eq. 9.11, will be 
—w7A;(w)X(w)e! + jwAx(w)X(w)e! + A3(w)X(w)e!! = Bw)Z(w)e!. (9.13) 
Therefore, the frequency response function H,¢(w) of the behavior X(w) to wave height 
Z(@) is 
nexX(@) i B@) 
ES) 7G) | a AOL) eet 
From this, expressing H,-(w) justas H(w), 
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