Strumpf 



The criterion given by Eq. (14a) now is applied to the case of a tapered body 

 of revolution with 



-^=43.25 ft; A=49.3sqft; B= 95,500 lb; ly^ = 2. 94 x 10^ slug- f t ^ (23a) 



The bare hull coefficients are 



Z; = -0.71; M; = 0.98; Z' = -0.14; and M' = -0.07; (23b) 



and the mass coefficient m' for the neutrally buoyant case is 1.40. By substitut- 

 ing Eqs. (19) into (14a) and using the given numerical constants in the resulting 

 expression, it is found that the submersible has neutral hydrodynamic stability 

 when the tail appendage has a normal force rate coefficient z^^ = -0.86. Thus, 

 -z^ > 0.86 defines one of the limit curves for acceptable cruising performance. 



Similarly, the criteria given by Eqs. (18) and (22) are applied to the hydro- 

 dynamically stable conditions defined by 



Z; = Z5 = -1.5 and 5 = 0.35 rad (24) 



a 



and Eqs. (23). The cross-hatched areas u^ vs x^ with z^ as parameter shown 

 in Fig. 3 define the region where acceptable cruising performance is possible. 

 The acceptable cruising regions in the u^.x^ plane are characterized by two 

 roughly triangular areas with the same vertex on the x^ = axis and a u^ value 

 there which is always greater than the critical speed* (but approximately equal 

 to it when z^ > 0.03 ft). The upper area is always much larger than the lower, 

 but both angles at the vertex increase with Zq. The base of the upper area is 

 the maximum speed (assumed here to be u^ = 8.0 ft/sec), and the base of the 

 lower area is u^ = 0. For the hydrodynamically stable submersible, acceptable 

 cruising conditions are limited by dynamic instability when z^ is small and by 

 the excessively large elevator angles required for equilibrium when z^ is large. 

 In either case, it is necessary to keep x^ small in order to permit cruising at 

 low speeds. 



The existence of the cruising regions near u^ - x^ = should not be inter- 

 preted as being acceptable hovering conditions. The reason for this is that 

 cruising and hovering conditions refer to two completely different equilibrium 

 states. In the cruising condition no attempt is made to keep the submersible 

 motionless relative to the x^,y^,z^ frame (i.e., x^^ = z^ = 0), whereas this is 

 precisely the situation in hovering. For example, the case where the submers- 

 ible drifts with a current u^ = 1.69 ft/sec (1 knot), so that its longitudinal speed 

 Up = relative to the fluid, is a permissible and acceptable equilibrium cruising 

 condition if z^ = 0,05 ft and x^ = 0.005 ft (see Fig. 3); however, it is not an ad- 

 missible hovering condition because x :j: 0. 



*The critical speed for a given Zq is that for which the denominator of Eq. (16) 

 vanishes. 



286 



