Cruising and Hovering Response of a Tail-Stabilized Subnnersible 



The solution of Eq. (29) is of the form 



cr.t o-.t cr.t , (31) 



u- = XujC ' , w - Zw.e ^ , 9 - 2i9.e ' with a. 4 cr. , 



and the Ui,w^,(9j are constants depending upon the initial conditions. This sys- 

 tem yields an indicial equation of the form 



d^cr." + dicr.3 + dja.^ + djcr. + d^ - . (32) 



where 



do = Ai(B2C3-B3C2) - CiAjBj 



di = AiCBjCis-BijCj- 83017 + 3,703) - CiA3B,7-Aig(B3C2-B2C3) 



dj - Ai6(B,7C3-B3C,7-B,5C2+B2Ci5) - Aj(Bi5Ci7-B„C,s-B2C,2) <33) 



'^a - AjgCBjCij - B15C17 + BJ7C15) + Bj7A,Ci2 



•^4 ~ A 16^ 17^ 12 



The submersible is dynamically stable under the prescribed hovering conditions 

 if the real parts of all four a. are negative. The Routh stability criteria give 

 the necessary and sufficient conditions for this to be true. These are that 



d, / d,d \ 



(a) d. > and (b) d^ < -^ K - -^J . (34) 



The other bases for defining acceptable hovering conditions are that lu^l <u„ 

 (or |nx^ I < Hx^), and that each equilibrium vertical thrust force be less in mag- 

 nitude than z , or from Eq. (28) 



'-bjuj + [(b,2 + 4a,A7)u,2]'''- 



(a) 



and (35) 



wIxgI 



2x. 



< Z. • (b) 



These hovering criteria now are applied to the submersible with the con- 

 stants given by Eq. (23) and 



Z^ = 300 lb; x^ ^ 16.0 ft; a^ = 2.46 slug-ft; b^ = -6.40 slugs. (36) 



The hydrodynamic data include 



289 



