Linearized Potentiel Flow Theory for ACVs in a Seaway 



and this reduces to 



U - A •_• A 



G = x 1 + z k 



for the coplanar motion we are considering. 



We therefore have 



F R = m [x i + ( l - g ) 1] 



and substituting for the displacements, we derive for the components 

 of the rigid body force 



X R = - maZ aelat ( *001 + 6 *101 + ^011 } 

 and (IV. 1) 



Z R = -ma 2 ae 10 (z^ + 3^ + 0^) - mg 



As regards the moment, we have from (5.4) 



M R =JJp X (U -g)tdm 



V _> 



where the absolute velocity U at the point (x, y, z) of the body is 



U = (V + x)^ + y^+z y k 



reducing to 



it = x I + z i 



for coplanar motion and since 



x = x + x 1 cos + (z 1 + h ) sin0 



and 



z = z + z' cos 8 - x' sin0 - h (1 - cos 6) 



G 



we have 



x = x + [(z' + h ) cosfl - x' sin #] 6 - [x' cos + (z* + h„) sin 0] 0* 



207 



