Paul ling 



Motion- Dependent Hydrodynamic Forces 



We now consider a velocity vector, U , whose components 

 are the velocities of the center of gravity of the structure in oxyz 

 and are given by Xj , Xg, Xj in Eq. (6). We may similarly define 

 an angular velocity vector, J2 , whose components are the three 

 rotations about oxyz, and were denoted X4, Xg, x^ in (6). The 

 resultant velocity of a point P(x,y,z) is given by 



"u = IJ +sf xT, 



where r is the radius vector drawn from the origin of oxyz to the 

 point P. Similarly, the resultant acceleration is given by 



a"p ="u + if xT . 



(Note that this neglects the radial or "centripetal" acceleration which 

 Is of the order of the square of the (small) angular velocity.) Point 

 P may be thought of as lying on the x-axis of a member of the 

 structure^ We therefore may obtain the velocity and acceleration 

 vectors u^,^ and a^jj appearing in Eq. (7) by a trans formation^s_imilar 

 to (12). The a-. , however, relate the nnember coordinates oxyz 

 to the space coordinates oxyz in this case. Also note that the 

 velocities and accelerations are now unknown quantities. 



Applying these transformations, we obtain expression^ similar 

 tp (12) where the u; , u: are replaced by the components of up and 

 ap . The elementary forces are given by an expression which is the 

 negative of (13) but containing the same jx's and \'s. These are then 

 transformed back to the oxyz-directions by the inverse of the above 

 transformation. Integration of these elementary forces and their 

 moments over the length of the member is somewhat simpler now 

 since the velocities and accelerations vary in a somewhat simpler 

 manner than in the case of the wave-induced velocities and accelera- 

 tions. 



The Added Mass and Drag Coefficients 



The idealization made in arriving at the subdivision of forces 

 illustrated In Eq. (7) Is a computational expedient at best. In reality, 

 the total force experienced by a member of the structure will be the 

 resultant of distributed normal pressure forces and tangential shear- 

 ing forces arising from viscosity. The subdivision Into a Froude- 

 Krylov force, a force proportional to relative fluid velocity, and a 

 force proportional to relative fluid acceleration, Is done partly on an 

 Intuitive basis and partly on the basis of our knowledge of simpler 

 problems. Such a subdivision has the great advantage of leading to 

 solutions In which there Is a linear relationship between platform 

 motion and the exciting wave motion. Let us review some of the 



1094 



