Cruising and Hovering Response of a Tail-Stabilized Subnnersible 



which means that the appendages have the following net effects on the rate 

 coefficients: 



K^ -- 2z;^; z;_^ = m;^ = o; and M^^ = \ z;^ (44) 



Use of these relations in Eqs. (38) and (39) with Xq = shows that the stability 

 criteria for u^ > and u^ < reduce identically to the same one, namely 



"^'^'^ - ("^' + 2;,) m;^ > 0- (45) 



or 



K\ - (y z;, ^ 2m;Jz;^ . [z;^m;^ - (m' .z;j] m;^ > o . (46) 



Substitution of the bare hull hydrodynamic coefficients given by Eq. (23b) and 

 the mass coefficient m' = 1.40 into Eq. (46) and solving for z' shows that if 



-Z;^ > 0.86 



the bow- and stern-finned vessel will be hydrodynamically stable in both posi- 

 tive and negative ocean currents. It is noted that the value of zi for neutral 

 hydrodynamic stability of the submersible with only a stern appendage also is 

 -0.86. This type of result occurs generally because a stern appendage alone, 

 regardless of size, cannot stabilize the vessel when u^ is negative, and the bow 

 appendage alone, regardless of size, cannot stabilize it when u^ is positive (6). 



V. UMTT MANEUVER RESPONSE CHARACTERISTICS 



The adequacy of stern stabilization alone in a submersible which must both 

 cruise and hover in ocean currents is investigated further by resort to limit 

 maneuver response calculations for the tail-stabilized submersible. Although 

 Eqs. (1) through (4) were used in digital machine computations of one cruising 

 maneuver and three hovering maneuvers (1), for brevity only one type of the 

 latter class is considered herein. Similarly, the parametric studies covered in 

 Ref. 1 were extensive, but the present discussions deal only with those aspects 

 relating to the use of tail stabilization. 



(A) Cruising Limit Maneuvers 



Figure 6 is an example of computed response in a cruising pitch overshoot 

 maneuver (Run 102). The vessel is initially in straight, level, equilibrium flight 

 with Up = 5.07 ft/sec. The water current velocity components u^ = w^ = o, 

 Z^ = 2M8 = -0.50, Z4^ = -1.5, and Xq = Zq = 0.01 ft. The other data (where ap- 

 plicable) are the same as given previously in Eqs. (23), (36), and (37). The ele- 

 vator then is deflected at the maximum rate '^^ = 0.20 rad/sec until it reaches 

 the maximum angle S^ = 0.35 rad and held there until the time tj when the 



293 



