Dynamias of Submerged Towed Cylinders 



Similar equations were obtained when using the theory of [ 26], 

 i.e. Eqs. (10a), (11), (12) and (13), namely 



j[2+x,(l+f,) +X2(l+f2)] (-<*>') +[-|^CN + f, -f2](coi) + [-| c^p/A]|h 

 + j-4[x,(l+f,) - X2(l+f2)](-<*^') +[2 -^(f, +f2)](a;i) 



+ [\ €(c, + c,) + f , - f, - 1 c,p/A] } $ = 0, (17a) 



j- ["Ix.d+f,) -^X2(i+f2)](-<^')-[i(f,+9]M) -[|c^p/A] JH 



+ j[i +ix,(l+f,) ^{Xz{i+i^]{-'^^)^[{{i,-i^) +^€cj(a)i) 



+ [|c^/A -l(f, +f2)]j$ = 0. (18a) 



For non-trivial solution, the determinant of the coefficients of 

 H and ^ in (17) and (18), or in (17a) and (18a), must vanish, yielding 

 a quartic in co, 



J^ + aJ' + Bco^ + Cw + E = 0. (19) 



6. 2 Calculations Based on the Theory of [ 26] 



The aim here was to compare the dynamical behavior of the 

 rigid body to that of a flexible body; as the rigid body may be regarded 

 as a flexible one of very large flexural rigidity, it would be reasonable 

 to expect correspondence of the dynamical behavior of the rigid body 

 to the 'rigid-body' modes of the flexural one, i.e. the zeroth and 

 first modes. Recalling that the dimensionless flow velocity in the 

 case of a flexible body was defined as u = {M/Kiy^ \31^ , the dynamical 

 behavior of the rigid body should approach that of the flexible one as 

 u — ^ 0. Two sets of calculations were conducted, as described below. 



The four rigid-body frequencies, given by (19), were computed 

 for a number of cases and the values connpared with the existing com- 

 plex frequencies of the flexible body. As an example, let us compare 

 the case corresponding to Fig. 3. The four frequencies are to, = 1,956, 

 co2= -0.761, (Oj^ = ± 0. 582 - 0. 3571. These compare well with the 

 four frequencies associated with the flexible body for u = 0,7, 

 namely CO, = 1.934, oo2=-0.734, (O3 ^ = ± 0. 580 - 0. 350i; the first 



1005 



