Page 244 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 225

Galerkin procedure yields

(4.44)

where k = 1,2. ..., 00, and

(4.45)

The constants 1;;. i = 1, 2. ..., 11, are defined as follows:

The evaluation of these integrals in terms of the Timoshenko beam eigenfunctions is
discussed in Appendix E.2.
The solution as expressed by equations (4.43) is then truncated at n = N, and equations
(4.44) yield a vanishing determinant of order 2N. This is solved to give the eigenfrequen-
cies w of the system, for different values of the dimensionless flow velocity u and of the
other system parameters, p, A, y, etc.

4.4.3 The inviscid fluid-dynamic force

Here the inviscid fluid-dynamic force, fi, will be derived, first according to the plug-flow
approximation and then in a more refined manner.
(a) The inviscid fluid-dynamic force for plug flow

This approximation, which applies to large length-to-diameter ratios, small displacements
and, as we shall see, long wavelengths of deformation of the pipe as compared to its
diameter, is what has been used in all of the foregoing. Thus, by using d’Alembert’s
principle, the force fi is equal to the mass of the fluid per unit length multiplied by the