Muvthy 



s < z iooi =*< *&* '"* fa [" kz ' + i(kx " >]& dx ' di 



and 



■o 

 *«M 1001 =* ( i^J e^'jLp [- kz' + i(kx + 7)] 



(8-3) 



(8-4) 



It will be noted that these forces and moment are purely of a 

 hydrostatic nature. 



VIII. 3 Free Oscillation 



When the ACV responds freely in waves taking up displace- 

 ments appropriate to the excitation, restoring forces are developed 

 on account of the displacements. It is assumed, of course, that there 

 are no external forces acting on the craft apart from the constant 

 thrust. It has been shown earlier in Section 6 that the small displa- 

 cement in pitch will not provide a restoring force in the longitudinal 

 or vertical directions although the thrust line is displaced in space. 

 There will also be no restoring moment available in pitch on this 

 account as the contribution is of a higher order. As the ACV is in 

 equilibrium under the action of the restoring forces due to the dis- 

 placements caused by the exciting forces, we may set 



Restoring 



Force 

 Moment 



+ Exciting 



Force 



Moment 



= 



i. e. 



8a 



+ Be 



X 

 Z 



M 



X 



z 



M 



+ 



X 



Pa) Z 



(M 



(x 



I M 



+ higher order restoring forces + 



•+• higher order exciting forces = 



It is clear from the above system of equations that the order 

 of the displacement caused by the incident wave has to be the same 

 as the order « of the waves themselves. 



Using the expressions derived in Section 7 for the restoring 

 forces and moment, we can now write the following set of equations 



162 



