Paidoussis 



Re(w)= O 



Fig. 12 



The dimensionless complex frequencies of the zeroth and 

 first modes of a flexible cylinder with f, =0.7, c, =0, 



f2 = 1 - cg = 0. 7, A = 1 , ec^ = ec^ = 1 , -X, = Xo = 0.01 ( ), 



Also shown ( ), portions of the zeroth and nrst modes 



with €c,^ = €c^ = 0. 5. (New theory). 



system. The observed behavior of flexible cylinders with increasing 

 towing speed [ 26] can be summarized as follows: (a) at low towing 

 speeds a 'criss-crossing' oscillation developed in which the cylinder 

 inclination was of opposite sign to that of the tow-rope; (b) at 

 slightly higher towing speed, sometimes a narrow region of stability, 

 or a region of stationary buckling, was observed; (c) at higher 

 towing speeds, second-mode, and at yet higher towing speeds, 

 third-mode flexural oscillation developed. The above are typical 

 observations provided that the tail is not blunt and the tow-rope not 

 too short; if they are, then the system remains stable for apparently 

 all towing speeds. 



We first note that, in terms of qualitative agreement, the 

 results depicted in Figs. 12 and 13, for instance, agree with the 

 experimental observations. Thus, at very low towing speeds the 

 system is subject to first-mode oscillatory instability and yawing, 

 the former ceasing at slightly higher towing speeds, while yawing 

 persists (presumably corresponding to the observed buckling). At 

 yet higher towing speeds, the second mode loses stability, followed 

 by the third mode at even higher towing speeds. 



1000 



