Page 84 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 67

apparently never loses stability! The resolution of the paradox is that the system never
loses stability statically. The critical compressive load was determined by Beck in 1952
(Bolotin 1963) by solving the full equation of motion, equation (3.3). It is given by
9, = eL2/EI = 20.05,* at which point coupled-modejutter arises, otherwise known as
a Hamiltonian Hopf bifurcation, in contrast to the cantilevered pipe, which loses stability
by single-mode flutter via an ordinary Hopf bifurcation - see Section 3.2.3.
The fact that the cantilever conveying fluid is not only a nonconservative problem
similar to Beck’s (a circulatory dynamical system in Ziegler’s classification), but is also
subject to gyroscopic forcesS helps explain the fascination it has exerted, and does so still,
on applied mechanicians and mathematicians for the last 30 years. An additional point
fort of this system is that it can readily be realized and studied experimentally, unlike
the original Beck’s problem which requires a rocket-engine mounted to the free end of
a beam column, or something similar - not an easy task! Indeed, it was implied in a
lecture (Paidoussis 1986a) that such a task was much too hard to contemplate, which a
team of Japanese researchers promptly disproved (Sugiyama et al. 1990), by doing the
difficult experiment with a solid-fuel rocket, demonstrating the occurrence of flutter and
obtaining good agreement with theory - see also Section 3.6.5.
Finally, a few words on the case when the flow is from the free end towards the clamped
one: by reinterpreting (3.11) for U < 0 it would appear that the system is unstable by
flutter for small U (indeed for infinitesimally small U if dissipation is ignored!) and is
then stabilized for larger IUI, as first pointed out by PaYdoussis & Luu (1985) - the
inverse behaviour to that described heretofore. More will be said about this in Chapter 4
(Section 4.3), but in what follows we return to the system with U > 0.

3.2.3 On the various bifurcations

A general discussion of the evolution of the eigenvalues and the corresponding eigenfre-
quencies leading to some of the standard bifurcations or linear instabilities was given in
Section 2.3. This is reinforced and expanded here for the phenomena of interest in this
chapter.
The Argand diagrams for divergence via a pitchfork bifurcations are shown in
Figure 2.10(a). If the system is conservative (zero dissipation), the diagram for the
eigenfrequencies is modified. The eigenfrequencies are wholly real for u < u,, and
then become wholly imaginary (a conjugate pair), as shown in Figure 3.4(a); hence
w = 0 for u = u,. The corresponding eigenvalues are wholly imaginary for u < u,, and
then for u > u, become wholly real; the eigenvalue Argand diagram for each of the
cases in Figure 3.4 is obtained via a 90” counterclockwise rotation of the corresponding
eigenfrequency diagram. For the pipe and the column with simply-supported ends, u, = 75
and 9, = n2, respectively - see equations (3.7) and (3.9).

‘This value of PC is about eight times higher than the Euler buckling load for fixed-orientation compression
of the cantilevered column, gC = 4.’ (Ziegler 1968).
tunlike the system with supported ends, if this system is discretized, the Coriolis-related matrix is not
skew-symmetric; it can of course be decomposed into symmetric and skew-symmetric parts.
%rktly speaking, the type of bifurcation involved is defined by the nonlinear terms in the equation of
motion. In this case, the flow-related nonlinearities in the stiffness term are cubic and similar to those in a
softening cubic spring. This is what gives rise to two stable static equilibria for u > uc - cf. equation (2.165)
and the discussion following it in Section 2.3.