Page 112 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

94 SLENDER STRUCTURES AND AXIAL FLOW

q2] and amplitude C:

Clearly, at no time in the course of the oscillation can the contraction become instanta-
neously zero. A similar argument may be made in the case of PaYdoussis flutter; in this
case, q20/q10 is not purely imaginary but complex, and the phase angle is not neatly in
but an angle 4. Nevertheless, the same conclusion may be reached with regard to the
overall contraction never becoming zero during oscillation.
The implication of this is that the momentum flux of the fluid issuing from the sliding
end of the pipe does work on the system in achieving a certain oscillation, MU2 acting
as a compressive load P as discussed in Section 3.1 and acting over a distance equal to
the mean contraction, C. No net work is required thereafter to maintain the oscillation,
but there is an oscillatory flow of energy because of the axial motion of the downstream
end of the pipe, which nevertheless is zero over a cycle of oscillation. This energy
may be thought of as being carried in the form of travelling waves, as will be seen in
Figure 3.13, with a node moving down to the pipe exit in half a cycle of oscillation. It
is in this ingenious way, thanks to Done & Simpson, that the paradox of oscillation with
no net energy expenditure may be explained!

11
0 -3-

,
3 5
-9-
8
d 8
-.
4

8 2

2 4

Figure 3.13 Variation of modal forms of the fundamental mode of a simply-supported
pipe of vanishing flexural rigidity during a period of oscillation: (a) u = 0; (b) u/u, = 0.25;
(c) u/u, = 0.75; the fractions denote fractions of the period (Chen & Rosenberg 1971).