Dynamics of Submerged Towed Cylinders 



Fig. 2 Diagram of a towed flexible, slender cylinder 

 with streannllned "nose" and "tail" 



the bending moment and shear force are zero, and (b) at the upstream 

 end, X = 0, the bending moment is zero, but the shear force is equal 

 to the normal component of the tow-rope pull. It is obvious, however, 

 that the very form of Eq. (9) will depend on the shape of the nose and 

 the tail. As we are only looking for the general characteristics of the 

 dynamical problem, this is not convenient. We shall instead proceed 

 as follows: (i) we shall use Eq, (10) which satisfactorily applies 

 over the uniform, cylindrical part of the body; (il) the forces acting 

 on the non-cylindrical ends will be lumped and incorporated in the 

 boundary conditions. For this process to be meaningful we must have 

 i. « L and i- "^"^ ^» where i. and i_ are the lengths of the nose 

 and tall, respectively; yet i| and ^2 ^.re considered to remain great 

 enough to permit the use of slender body approximations [ 23] , [ 26] • 

 Since i| and i2 ^^® small, compared to L, we may further simplify 

 the problem by considering y and the lateral velocity v to be approxi- 

 mately constant over 0^ x ^ i| and L - ig — ^ — L, and by neglecting 

 the skin frlctlonal forces over the same Intervals, Hence, Integrating 

 Eq, (2) and using (4), and Incorporating the forces arising from the 

 tow-rope pull, P, we obtain 



eind 



iu^'-'^Li^-'^^^y^'^'^-i. 



|4 dx = 0; 



at 



the parameters f , and fg. which are equal to unity according to 

 slender-body Invlscld-flow theory, were Introduced to account for 

 the theoretical lateral force at the nose and the tall, respectively, 

 not being fully realized because of (a)-the lateral flow not being truly 



987 



