Dynamics of Submerged Towed Cylinders 



II. THE EQUATION OF SMALL LATERAL MOTIONS OF A 

 FLEXIBLE SLENDER BODY IN AXIAL FLOW 



•We shall derive the equation of small lateral motions of a 

 slender body of revolution of the type shown in Fig, 1(a); the body 

 is supposed to be supported somehow so that it is not washed away 

 downstream. The fluid is incompressible and of density p; it is 

 flowing with velocity U parallel to the x-axis, which coincides with 

 the undisturbed longitudinal axis of symnnetry of the body. The body 

 is of mass p6r unit length m(x) , cross-sectional area S(x) , and 

 flexural rigidity EI(x). 



Fig, 1(a) Diagram of a flexible, slender body of 

 revolution in axial flow 



We consider small motions y(x,t) and assume that y, 

 8y/9x, 9 y/9x to be all small, so that no separation occurs in cross - 

 flow. Moreover, we assume that dS/dx is small everywhere, 

 except perhaps at the ends of the body, so that no separation occurs 

 in the axial flow (except perhaps at the rear end) , and so that 

 slender-body theory may be used. Also d(EI)/dx is assumed to be 

 small, which, together with the restrictions on the displacement 

 function, allows us to use the simple Euler-beam approximation to 

 describe the flexural forces. The body is further assumed to be 

 of null buoyancy and uniform density, so that no constraining force 

 in the y-dlrection nor a moment is necessary to keep it lying along 

 the x-axis, at least at zero flow velocity. Furthermiore , the motions 

 are considered to take place within the (x,y) -plane, which for the 

 sake of simplicity is assumed to be horizontal. Finally, we neglect 

 Internal dissipation in the material of the body. 



We now consider an element 5x of the body. The forces and 

 mioments acting on it are shown in Fig. 1(b). Q is the transverse 

 shear force, ''71 is the bending moment, T is the axial tension, 

 Fjj and F^ are the normal and longitudinal components of frictional 

 forces per unit length, and F^ is the lateral inviscid force per unit 

 length. 



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