Paidoussis 



to the new theory occurs over a much more limited range of system 

 parameters, while yawing is more prevalent. 



Comparing Fig. 1 6 to Fig, 14 we note the following essential 

 differences: (i) yawing, being independent of c-p according to the 

 new theory, is represented by a single line; (ii) according to the 

 new theory oscillations persist to progressively lower values of fg 

 as ccj is reduced, while the opposite trend was predicted by the 

 old theory; (iii) according to the new theory, for f , = 1 , there 

 are large regions in the (ec,^, i^ parameter space where yawing 

 occurs alone, but not where oscillations occur alone; on the other 

 hand, according to the old theory the opposite is true. However, 

 this last point applies only for f| = 1, It may be seen that for f , = 0,8 

 and 0, 7, the results of the new theory become much more like those 

 of Fig, 14 in this respect. 



We note that the onset of yawing is independent of f, as well 

 as c., so that the line shown in Fig, 16 applies to all cases examined 

 therein. Once again considering term E of Eq, (19), which in this 

 case is given by E = (Cjp/2A)(2 ^c,^- 2f g) . we see that f , , c,, Cg, Cy 

 and A are all parameters that cannot affect the onset of yawing. 



We next compare Fig. 17 to Fig. 15. The results are quite 

 similar, except that (when f | = 1) oscillatory instability occurs over 

 a more limited range according to the new theory than predicted by 

 the old theory. However, the results of the new theory for f, =0.8 

 when compared with those of the old one for f| = 1 are quite similar. 

 The results for f, =0,7, not shown in Fig, 17, are of interest in 

 that oscillatory instability, in that case, occurs practically over the 

 whole plane, i.e. for f2> 0.013 for A =0,1 and for f2> 0,008 for 

 A = 0.2, 



VII. CONCLUSION 



In this paper we have reviewed an existing theory for the dy- 

 namics of flexible cylindrical bodies towed underwater, and developed 

 a parallel theory for rigid cylinders. It was shown that, whereas the 

 dynamical problem in the case of rigid cylinders is independent of 

 towing speed, in the case of flexible cylinders the dynamical behavior 

 (and stability) of the system is highly dependent upon towing speed. 

 It was found that, in general, flexible towed cylinders are subject 

 to both flexural and 'rigid-body' instabilities, the latter occurring 

 at relatively low towing speeds. It was also established that at low 

 towing speeds, the dynamical behavior of the flexible cylinders in 

 their two lowest modes (the so-called zeroth and first) correspond to 

 that of rigid cylinders, which of course have but two degrees of free- 

 dom. Thus the study of the dynamics of towed flexible cylinders 

 yields sufficient information to establish the dynamical behavior of 

 the corresponding rigid bodies. 



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