Paidoussis 



V. DYNAMICS OF TOWED FLEXIBLE CYLINDERS 



5. 1 Method of Analysis 



Upon expressing the equation of motion and the boundary con- 

 ditions in dimensionless form, the dynamics of the system may be 

 found to depend on the following dimensionless parameters: 



(i) f , and fg, which were defined in §3; 



(ii) cc^, €c^, c, and c^, where e = L/D; 



(iii) A = s/L; the ratio of tow-rope length to body length; 



(iv) X, - X|/L and X2 ~ ^g/"^ ' w^^®^® ^1 ^^^ ^g ^®^® 

 defined in §3; 



(v) u = (M/EI) ' UL, the dimensionless towing speed. 



It is noted that according to the assumptions made in the theory, 

 m = M, 



We shall not present the analysis here, as it is adequately 

 documented elsewhere [ 26] , [ 27] . Suffice it to say that solutions 

 were obtained of the type 



^=Y(2)e-', 



where Y is a function of x/L, t is a dimensionless time and w 

 is the dimensionless frequency given by 



= (M^^nL^ 



fi being the circular frequency of motion. In general, to will be 

 complex. Clearly, we have an infinite set of frequencies, to,-, as 

 the system has an infinite number of degrees of freedom. If the 

 imaginary components of the frequencies, Im(a)j) , are all positive, 

 then the system will be stable. If, on the other hand, for the jth 

 mode we have Im (wj) < 0, then the system will be unstable in that 

 mode; now if the corresponding real component of the frequency. 

 Re (tOj), is zero this will represent a divergent motion without oscil- 

 lations, which we shall call yawing; if Re (tOj) ^ 0, then the insta- 

 bility will.be oscillatory. 



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