Linearized Potential Flow Theory for ACVs in a Seaway 



;1<Tt( 5aM 1010 + ^ M 0110 ) = ael0t [ mh G (<T ^001 - g ^001 ) + 

 + ( PgI A - m 2 gz B - o Z \) 6 Q01 - pgi 001 Ax A + 



+ Z ^001 +h GV M Xp o ^ 



(7-1) 



These expressions give the restoring forces and moment 

 acting on the craft when it is given oscillatory displacements in sur- 

 ge, heave and pitch of a forced nature. The longitudinal force contains 

 the inertia term only, but the vertical force and pitching moment 

 contain in addition, the contributions due to hydrostatic pressure. 

 The velocity potential of the water does not enter in the forces and 

 moment of this order and we cannot therefore expect to find such 

 effects as the added mass and damping of water which are of a hydro - 

 dynamic nature and of a higher order. 



VII. 3 Higher Order Restorin g Forces and Moment 

 Added Mass and Damping Effects 



The higher order forces and moment can also be written 

 down from (5-6), (5-7) and (5-8) in the same manner as (7-1). These 

 second order forces and moment include the added mass and damping 

 effects of water as the potentials * lftlf) and *oilO enter into ^ aese 

 expressions in addition to the steady potentials and the steady displa- 

 cements as the steady disturbance of the water persists during the 

 oscillations. The pitch and heave stiffness of the air cushion also 

 play their part in determining the forces and moments of this order. 



It is clear that the steady potentials ^iqqq and ^0100 ( which 

 are real) will be in phase with the displacements and will not there- 

 fore yield any damping contribution. The only contribution will be 



from the unsteady potentials *._. _ and * which have a real 



1010, 1 1 n i<rt 



part and an imaginary part, so that when the product with e is 



taken there will be components of force in phase and in quadrature 



with the displacements giving rise to added mass and damping effects. 



It will have to be remembered that we are discussing only 

 hydrodynamic effects here. The damping due to aerodynamic and 

 pneumatic effects in the air cushion will also have to be taken into 

 account. 



159 



