Strumpf 

 Z; =-1.5; Z;=-2.21; Z' =m; =-0.75; M' = -0.38, (a) 



a ^ a a ^ 



(37) 



X.' = -0.07; X' = Z! = -1.273; M.' = -0.054, X' = -0.15. (b) 



The values of Z^ and x^ were given by SP, a^ and b^ were determined using 

 propeller design charts (Ref. 7) for a three-bladed propeller with diameter of 

 4.0 ft and pitch-to-diameter ratio of 0.50, the coeffecients of Eq. (37) were ob- 

 tained using Eq. (19) and the previously given data (23b), the "added mass and 

 inertia" coefficients were estimated on the basis of Lamb's "accession to iner- 

 tia" coefficients for prolate spheroids, and the estimate of x^ is an average 

 value based on the fact that the submersible will operate over a range of speed 

 from to approximately 5 knots. The Schoenherr friction drag coefficients as 

 well as estimates of form drag were made for the main hull, tail appendages, 

 rescue skirt, thruster ducts and gear, and other exposed equipment to obtain the 

 given value of x^. However, a rough estimate of x^ is justified because it has 

 been found (1) that reasonably large changes in its value have little influence on 

 the measures of performance treated herein. 



The cross-hatched area of Fig, 4 shows the region in the u^.Zq plane of 

 acceptable hovering performance defined by the criteria of stability, maximum 

 propeller speed, and maximum vertical thrust forces. The region is closed at 

 the top by the criterion u^ < 6.95 ft/sec which was obtained by arbitrarily as- 

 suming a maximum propeller speed n^^ of 20 rad/sec in Eq. (35a). The notation 

 in Fig. 4 that |xq| < 0.10 ft for Iz^ | and Iz^, | < 300 lb is the result of using 

 the given data in Eq. (35b). 



The other hovering limit curves in Fig. 4 are the result of applying the 

 stability criteria. It is found that d^ and dj are positive for all u^ , but d^ > 

 only if Zq > 0. This latter condition accounts for the z^ = limit line in Fig. 4. 

 When Up > 0, Zq > 0, and |xq| < 0.10 ft, d^ and d^ are positive and condition 

 (34b) is satisfied if 



+ 



Z;(M;-m'x^) - (m' + Z;) M; > 0, for Up>0 



or equivalently, if 



(38a) 



Z;(M;-m'x^) - [m' + (Z' +Z; )] (M; +M; ) > 0, for u^ > , ^^Sb) 



n ^ a h a 



where the sign of each quantity is shown above its symbol. For the hydrodynam- 

 ically stable submersible considered, the first product of Eq. (38) is positive 

 (Stabilizing) and, although the second product gives a negative contribution (de- 

 stabilizing), its effect is diminished by the stabilizing influence of the tail ap- 

 pendage contributions Zq^ and M^^. Hence, Eq. (38) is satisfied, and the neu- 

 trally buoyant, tail stabilized submersible with a small margin of hydrodynamic 

 stability can hover acceptably in large negative (bow-to-stern) ocean currents if 

 z^. > and |x„| is smaller than 0.10 ft. 



290 



