Strumpf 



•30 



. TEST DATA (REF. 3") 

 FROM EO.l»3> 



Fig. 2 - Comparison with test results to 

 determine validity of theoretical repre- 

 sentation of normal force in motion 

 equations 



IV. CONDITIONS FOR CRUISING, HOVERING, AND 

 DYNAMIC STABILITY 



(A) Cruising 



When acceptable cruising conditions are possible, the vessel has inherent 

 dynamic stability relative to a level, straight course equilibrium state (u^.^^.s^), 

 where < u„ < u and |s„| < s„, u„ and s„ being the maximum attainable speed 

 and elevator angle, respectively. An inherently stable vessel can recover from 

 very high degrees of control saturation, whereas the inherently unstable vessel 

 cannot (4). In a proportional control, the degree of control saturation may be 

 defined as the ratio of the elevator angle s (or elevator angular velocity i> ) that 

 would be attained if there were no limit, to its limiting value s^ (or bJ). 



Using small perturbation theory and Eqs. (1), it can be shown (5) that under 

 constant X^, and zero z^ and M^, conditions, a submersible is dynamically stable 

 in cruising flight if 



z;m' - (m' + z;)M; > o 



(14a) 



and 



X' 



z;w'(^6-''G^e) - m;(w'-b')^, 



+ (w'-B')(ZsM;-z;M5)Se > o. (I4b) 



where 



W ^ —7- Au 2 and B' 



p/2 ^ 



p/2 



Au. 



Inequality (14a) is the hydrodynamic stability criterion, and (14b) reduces to 



284 



