Thus, for small angular motions, the angular velocity com- 

 ponents are approximately equal to the time derivatives of the 

 Euler angles. The equations of motion, (7) may be written in a 

 more compact m.atrix form in this case: 



{F} = [m]{x} (10a) 



where F - vector of all six eternal forces and moments. 



[m] =6x6 matrix containing mass, moments of inertia and 



products of inertia of the platform, 

 {x} = vector containing three translational displacements 



of the platform center of gravity and three rotations 

 about the coordinate axes. 



If the vector {F} may be expressed in terms of linear functions 

 of all relevant motion variables, equations (10a) form the basis 

 for a linearized motion response computation. 



In general, however, the force system contains nonlinear terms 

 in such quantities as the motions (displacements, velocities and 

 accelerations) and incident wave height. The latter may be of 

 importance in computing the large amplitude motions in high waves. 

 In this case, it is convenient to introduce a new variable, 



(V) = iff) 



which may be combined with equations (7) and (9) to yield the 

 complete nonlinear motion equations as 



{dV} ^ [ni]-l{F} 

 dx 



<al> - <^> 



{d^} = [i]-l {{M} - {(u) X {lo)}} 

 dt 



dt 



(11) 



Equations (11) form a system of twelve first-order differential 

 equations in state variables (v) , {xl , {03} and {8} which express the 



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