Dynamiaa of Submerged Towed Cylinders 



part of an infinite cylinder; boundary layer effects have also been 

 neglected. The virtual mass M(x) = pS(x), and v(x,t) = [ (8/8t) 

 + U(8/8x)][y(x,t)] , which substituted into (4) yield 



Fa = pS[{8/8t) + U(8/8x)]V + pU[(8y/8t) + U(8y/8x)] (dS/dx) . (5) 



The frictional forces, as proposed by Taylor [ 30] , and elabo- 

 rated by Paldoussis [ 24] , [ 25] are taken to be 



^N = ic^(pS/D)u2 sin i and F^ = |c^(pS/D)u2 cos i, 



where i is the instantaneous angle of incidence on the cross-section 

 and is given by i = sin*' (v/U) , and D = D(x) is the diameter. 

 Accordingly, F^ and F^^ are given by 



Fn = icN(pS/D)U[(8y/8t) + U(8y/8x)] and F^ = ic^(pS/D)u2. (6) 



Finally, we note that the bending monnent Is related to the flexural 

 rigidity by 



n = EI(82y/8x2). (7) 



Now, substituting (6) Into (1), neglecting terms of second order 

 of magnitude, and Integrating from x to L, we obtain 



T(x) = T{L) +ic^pU^r [S(x)/D{x)] dx, 



where T(L) Is the Veilue of T at the downstream end. We consider 

 that T(L) Is non-zero and that It arises from possible form drag 

 at the end. We accordingly write 



T(x) = |c2PS{L)U^ + ic^pU^y [S(x)/D(x)] dx. 



(8) 



where Cg is the form-drag coefficient. 



Substituting now (3), (5), (6), (7) and (8) Into (2), making use 

 of (1), 2ind neglecting terms of second order of naagnltude, we obtain 



985 



