Cruising and Hovering Response of a Tail-Stabilized Submersible 



|Xq| < 0.\0 PT. FOR |Zi,,| AMD |Zb,| < 300" 



.V 8 



rixQ = 20 RAO/SEC 



, PT. 





Fig. 4 - Regions of Current velocity and CG position for stable 

 hovering operations of a tail-fin- stabilized subnnersible 



The sufficient condition for stability analogous to Eq. (38) for positive cur- 

 rents (or u„ < 0) is that 



Z;(M; + m'x,;.) - [m' + (Zq_ -Z;j] (M;^-M;^) > for u^ < 



(39) 



This inequality cannot be satisfied by the tail-stabilized submersible because 

 the tail appendage contribution (Zq^ and M^J to the second product in Eq. (39) 

 causes a destabilizing effect which is greater than that introduced by the body of 

 revolution itself. This places a severe restriction on the hovering capabilities 

 of the tail stabilized vessel in positive (stern-to-bow) ocean currents and ac- 

 counts for the lower limit of stability curve (marked dj in Fig. 4. This lower 

 limit curve is, to a first-order approximation, defined by 



> -6\ Zq 



(zq>0). 



(40) 



where z^ is measured in feet and u^ in feet-per-second. Hence, it is the action 

 of the metacentric pitch moment -WzQ(sin 0) in the vertical plane motions of the 

 vessel which accounts for presence of a small stable hovering region when 

 Ug < 0. In pure horizontal plane motions where there is no stabilizing meta- 

 centric yaw moment present, the tail stabilized submersible is inherently un- 

 stable in hovering equilibrium conditions with u^ < 0. It also is noted that in 

 the numerical applications of Eq. (34), the terms containing x^ are found to have 



291 



