Strumpf 



(a) STERN STABIUHATION 



(b^ BOW AND STCRN STABI HZ /\T tON 



L t/2 —4-. l/z 



H 



Uc < o 

 STABLE 



ii,'^ 



Hur. >0 86 



u« > o 



STABLE 



-Z^^>0.86 



Fig. 5 - Comparison of hydrodynamic stability of vessel 

 with (a) stern fins only and (b) bow and stern fins 



a very small contribution when |xq| < 0.10 ft. Hence, the only restriction on Xq 

 shown in Fig. 4 is that given by Eq. (35b). 



One way in which the hovering stability problem can be eliminated is de- 

 picted in Fig. 5. Bow fins are added to the stern-finned submersible to obtain a 

 vessel with fore-and-aft symmetry which is hydr ©dynamically stable in both di- 

 rections of motion. To demonstrate this, representations of the hydrodynamic 

 rate coefficients analogous to Eq. (19) first are written for the bow and stern fin 

 case, i.e.. 



■h "b 



K - M' - 2" (z' - z; ) 



K-- z;, -|(z;,-^^^> ^^ = ^%"i(^w, + z;^) 



(41) 



when the subscripts b and s represent bow and stern fin contributions, respec- 

 tively. Now assume that both fins are equivalent hydrodynamically, i.e., that 



b s 1 



(42) 



and use this in Eq. (41) to get 



Zw ~ Z^ + 2Z^ M^ - M^ 



Z' = Z' 



K -K,'\ K, . 



(43) 



292 



