offset of the platform in the downv/ave direction. This effect 

 is clearly seen in Figure 11 which gives the results of a nonlinear 

 time-domain integration of the equation of surge motion. The 

 effect is found to be closely related to the wave steepness. The 

 mean force is a nonlinear function of the wave height, but in 

 this case, unlike the wave reflection force discussed earlier, 

 the force depends on viscous drag. 



VII. APPENDIX 



The rigid-body equations of motion . The two coordinate systems, 

 one fixed in space and one fixed in the platform are illustrated 

 in Figure 1. The objective of the motions analysis, simply 

 stated, is to describe the time varying position of one system 

 with respect to the other. The translatory position of the origin 

 of the platform coordinates is given by the three coordinates 

 (x, , x„, X-.) measured in the directions of the three space axes. 

 The angular motion is expressed in terms of the three components 

 of the platforms angular velocity (oj., , ll)„ , w-.) resolved in the 

 •directions of the axes fixed in the platform. The details of the 

 derivation of the equations of motion of a rigid body having six 

 degrees of freedom are given in standard textbooks on dynamics and 

 need not be repeated here. In vector form the equations may 

 be written as follows: 



[m]{|^} = {F} 



[I]{^} + {o)} X {103} = {M} (7) 



Here {f}, {M} = vectors of external forces and moments respectively. 



{f} is expressed in the space coordinate system and 



{m} in body coordinates, 

 [m] =3x3 diagonal matrix in which the three nonzero 



terms are the platform mass. 

 [I] =3x3 matrix containing the moments and products 



of inertia of the platform. 



134 



