Cruising and Hovering Response of a Tail-Stabilized Submersible 



P ' 

 k-j - - -^ k\ 



W-B 



Ao — A^ 



An — A. 



(4) 



Equation (2a) for the propeller thrust X^ was obtained from momentum 

 theory. Although absolute values of velocity and angular velocity components do 

 not ordinarily appear in motion equations for submerged bodies, they are nec- 

 essary here because in hovering motions there exist physical discontinuities in 

 the derivatives of the forces and moments with respect to these variables when 

 the variable changes sign. 



The hydrodynamic pitch moment arising from the velocity component w of 

 the submersible is separated into two parts in Eq. (Ic). One part, due to the 

 main hull itself, is given by the term 



-Cj, uw = 4 A'^M^ uw , (5a) 



and the other part, due to the appendage, is given by 



-Cg Uw = -^A^M; Uw . (5b) 



This separation is dictated by conditions of symmetry and the need to determine 

 the pitch moment for all angles of attack a of the hovering submersible. Since 

 u = u cos a and w = u sin a, the use of the products uw and Uw in connection 

 with the hull and appendage moments, respectively, is of little consequence in 

 cruising flight where a is small. 



Because the main hull is considered to be symmetric relative to both the 

 yz and xz planes, its pitch moment must be zero when a = + Vjr/2 , k being an 

 integer. The product uw - (U^/2) sin 2a has the required properties of symme- 

 try, and therefore is suitable for estimating the hull moment in hovering flight. 



281 



