Dynamics of Submerged Towed Cylindevs 



A new theory was also presented (for both flexible and rigid 

 cylinders) which, it is believed, represents the physical system 

 more closely. The main difference in the results obtained by the 

 old and new theories are associated with the behavior of the rigid- 

 body modes of the system; specifically, the new theory predicts the 

 system to be more stable In its first (oscillatory) mode and less stable 

 In Its zeroth (yawing) mode than does the old theory. 



The new theory Is In general qualitative agreement with ex- 

 periment. Quantitative agreement cannot be assessed definitively 

 until a means Is found for accurately determining the values of some 

 of the dlmenslonless system parameters , particularly f| and £2' 

 Nevertheless, It Is possible to make Intelligent estimates of these 

 parameters based on experience from other experiments [ 25] . On 

 that basis quantitative agreement between theory and experiment, for 

 one particular experiment (Table 1), is seen to be fair, although 

 clearly leaving a good deal to be desired. 



In all the above discussion, as In [ 26] , the observed criss- 

 crossing Instability was identified with the theoretically predicted 

 first-mode oscillatory Instability , despite the fact that In most cases 

 theory predicts that the system Is also subject to yawing Instability 

 over the same range of towing speeds. This Is supported by the 

 observed frequency characteristics of the oscillation and the ob- 

 served effect of varying A, for instance, being essentially as 

 theoretically Indicated for the behavior of the first mode. It has 

 thus been presumed that oscillatory Instability Is the prevalent form 

 of Instability, There Is, however, an alternative Interpretation of 

 the observed behavior, namely that criss-crossing oscillation Is a 

 nonlinear manifestation of yawing. This may be postulated, but can- 

 not be proven by the present linear theory. 



In fact a number of questions remain. More careful and ex- 

 tensive experiments, including experiments with rigid cylinders, and 

 more extensive theoretical calculations are necessary to resolve 

 these questions. 



We next consider briefly the mechanism underlying the onset 

 of Instabilities, to the extent of identifying the physical forces at 

 work. 



We first consider the mechanism Involved In yawing. The 

 first thing to recognize Is that yawing must Involve angular motion as 

 opposed to pure translation. This Is evident upon considering the 

 cylinder momentarily displaced parallel to the x-axls; In this case 

 the forces acting on the cylinder are exactly as in the equilibrium 

 configuration, except that the tow-rope exerts a restoring force on 

 the body. We next imagine the cylinder momentarily displaced such 

 that the y-dlsplacement of the nose Is positive and that of the tall 

 negative. Then, considering boundary conditions (11) and (12), we 

 note that the Invlscid hydrodynamic force at the nose is f ,MUl9y/9x) 



lOii 



