Strumpf 

 (B) Hovering 



When acceptable hovering conditions are possible, the vessel has inherent 

 dynamic stability relative to an equilibrium state with ^o^ - '^o^ = 0, e - 0^, 

 and where lu^l < u^, Iz^J < z^ and IZb^l < z^, z^ being the maximum vertical 

 force obtainable with either the bow or stern thruster. 



The inherent dynamic stability of the submersible in the hovering equilib- 

 rium state 



K = '^r. = 61^ = (9^ = w^ = and u„ = -u, (25) 



c e 



has been investigated (1) under the conditions that 



5 = 0, Z3 = Zj , Zb = Zb , n^ = n^ . and W - B = . (26) 



Letting 



u=Ug + u, w=w, 6-6, etc., (27) 



and substituting Eqs. (25), (26), and (27) into the equations of motion give 



2a [-(Sign Ug) (a) 



2s = 2^ = -Zb (b) 



(28) 



for the equilibrium equations, and 



AjU + Aj^ + A,gU = (a) 



62^ + 63^ + Bj5^ + Bj^w = (b) 



CiG + Cj^ + C^ + Ci5^ + Cj2^ + C17W = (c) (29) 



^o = ^ («*) 



for the linearized perturbation equations, where 



A16 = 2|uJ - b^ln^J 



Bis = B^u^ + BjluJ; B,^ = BgluJ (30) 



C,5 = C,u^ + CjluJ; Ci, = CjuJ + C,u^. 



and the other constants are defined in Eq. (4). 



288 



