Davis and Oates 



The above equations could be simplified further if the reference forces and 

 moments were to be subtracted from the basic equation. However, hydr ©dynamic 

 forces and moments are more readily derived in terms of their full values rather 

 than in changes from the reference condition. It is more convenient, therefore, 

 to leave the equations in this form. 



Since the full values for the forces and moments have been left in the equa- 

 tions the craft probably will not be in a trimmed condition at the reference con- 

 dition but will stabilize out at some other attitude. 



Some caution is necessary when considering craft centre of gravity height 

 above the sea surface. It is necessary to translate velocities into space axes 

 before integrating to derive position. For example w may be integrated directly 

 to give w the velocity of the craft along the instantaneous, or current direction 

 of the craft Oz axis, but to find the e.g. height, velocities must be converted to 

 space axes, z^ is the parameter that is to be integrated in this case. 



Consider the craft to be moving in the XgZg plane with a constant velocity V 

 directed along OgXg away from a set of space fixed axes O^x^y^Zg which was co- 

 incident with the moving axis system ObxbybZb at time t = . At time t, let 

 OgXg make an ar^le d with o^x^ [Fig. (ii)] 



-*- jc. 



^-^"i^^e* ^-^ 



pixeof Axes. 



% 



Figure (ii) 



The components of 0^ relative to o^ are 



i^ = V cos 6 (28) 



i^ = -V sin ^. (29) 



The acceleration in the z^ direction is obtained by differentiating i^ thus 



z^ = - V sin e - V cos 0^ ; (30) 



however, v = as V is stated to be constant. Therefore 



618 



