of one coordinate system to the other. If the angular motions of 

 the platform are small so that the products of small quantities may 

 be neglected in comparison to the first order terms, all of the 

 nonlinear terms noted above may be neglected. Next, recall that 

 the external forces and moments which form the right hand side of 

 the equations of motion depend on the incident waves and on the 

 motion of the platform. As noted earlier, if the motions of the 

 platform and the waves are small, the motion-dependent forces are 

 decoupled from the wave-dependent forces and the equations of 

 motion may be rewritten in the form of a set of linear differential 

 equations: 



(m..+a..)x.+b..x.+c..x.=F.(t) (1) 



ID ID D ID D ID D D 



Here a. ., b. ., and c. . are coefficients in the expressions for the 



ID ID ID ^ 



motion-dependent forces termed "added mass", "damping" and "restoring 



constant" and F.(t) is the wave-induced time-dependent exciting 



force . 



In the case of a platform containing an appreciable portion 

 of its volume near the water surface, the damping and added mass 

 coefficients are found to be frequency dependent and equation (1) 

 can be considered strictly applicable only for the case of excita- 

 tion due to a system of sinusoidal regular waves. In this case, 

 the system of differential equations is reduced to an equivalent 

 system of algebraic equations in the amplitudes of the motion 

 responses for the given wave frequency. The principle of linear 

 superposition may still be utilized in obtaining the response to 

 random seas by first decomposing the sea into its regular wave 

 components and solving for the response to each. These are then svperiirpDsed 

 to obtain the total response. Such "frequency-domain" analysis 

 procedures are well established in other fields such as control 

 system design. 



In the case of nonlinear motions analysis, the equations of 

 motion must usually be integrated in the time domain using a 

 step-by-step procedure. If a complicated relationship is assumed I 



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