Page 30 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

CONCEPTS, DEFINITIONS AND METHODS 13
This transcedental equation yields an infinite set of eigenvalues, the first three of which are

AIL = 1.875 10, A2L = 4.69409, A3L = 7.85476; (2.25)

the corresponding natural- or eigenfrequencies are

(2.26)

The modal shapes or eigenfunctions are

Cbr(x> = cosh Arx - cos A,x - a, (sinh Arx - sin h,x) , (2.27)
where
sinh ArL - sin ArL
a, = (2.28)
cosh A,L + cos h,L’
Before proceeding further, an iniportarzt note should be made. It is customary in vibra-
tion theory and in classical mathematics to define the eigenvalue as being essentially the
square or, as in equation (2.26), the square-root of the frequency, except possibly for a
dimensional factor as in (2.26); the main point is that a positive eigenvalue here is associ-
ated with a positive eigenfequency. In dynamics and stability theory, however, solutions
are expressed as being proportional to exp(iRt) or exp(Ar), so that f2 and A are 90”
out of phase; a positive eigenvalue in this case would represent divergent motion, i.e. an
unstable system! This can lead to confusion, no doubt. However, these different meanings
and notations are so deeply embedded in these fields [cf. equations (2.26) and (2.36)] that,
in the author’s opinion, trying to unify the notation and meanings would create even more
confusion. Instead, the context and occasional reminders will be preferable, to make the
reader aware of which of the two notations for eigenvalue is being used.
When a concentrated mass Me is added at the free end of the pipe,’ the equation of
motion is the same, but the boundary conditions are

(2.29)
hence there is a shear force at the free end, associated with the inertia of the supplemental
mass. Of course, for a simple problem like this, it is possible to proceed in the normal
way and determine the eigenvalues and eigenfunctions of the modified problem. It will
nevertheless be found convenient to transform such systems into discrete ones by the
Galerkin method. To this end, for the problem at hand, the end-shear is transferred from
the boundary conditions into the equation of motion, which may be re-written as

(2.30)

‘The main purpose here is purely tutorial; nevertheless, the dynamics of a pipe conveying fluid with an
added mass at x = L is considered in Chapter 5 (Section 5.8.3), and it is shown to add a lot of zest to the
dynamics of the system.