Ogilvie 



'jk 



where 



= mass of ship, 



I j = moment of inertia in j -th mode, and 



I j ,, = product of inertia. 



The only product of inertia which appears, l^g, vanishes if the ship has fore- 

 and-aft symmetry. The other non-diagonal elements all vanish if the coordi- 

 nate origin coincides with the center of gravity of the ship. 



The last term in (13) arises because we allow the origin to be taken at a 

 point other than the center of gravity of the ship. Such generality is introduced 

 only as a convenience in hydrodynamic studies, where it is often simplest to 

 take the coordinate origin in the free surface directly above the center of grav- 

 ity. Specifically, we assume the center of gravity is located at x' = (0, 0, -x*). 



Equation (13), together with the associated definitions, is essentially 

 Cummins' final form of the equations of motion, although the detailed derivation 

 here differs somewhat from his. Some aspects are worth further note. Let us 

 rewrite (13): 



6 6 6 



k=l k=l k=l 



6 ^t 



k= 1 ''-00 



(r) Kj^(t- T) dr 



= Fj(t) + Gj(t) - /3.mga.(t) 



(13) 



It is clear that ^.^ has the nature of an added mass. Cummins has pointed 

 out that this is a "genuine" added mass, in the sense that it depends only on the 

 body; it is neither frequency nor speed dependent. In heave, pitch, and roll (and 

 their couplings), it is actually one-half of the infinite-fluid added mass of the 

 double body. However, in surge, yaw, and sway (and their couplings), it is the 

 added mass that corresponds to the case of the upper half -body moving oppositel 



34 



