Page 110 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

92 SLENDER STRUCTURES AND AXIAL FLOW

in Figure 3.9 and u 2: 9.0+ in Figure 3.10), they leave the axis, indicating the onset of
Pafdoussis-type coupled-mode fluttert as defined in Section 3.2.3 and by Figure 3.4(d).
The behaviour of Figure 3.11 (j3 = 0.5) is different. The w = 0 solution for u 2: 8.99
does not correspond to a second divergence, but to restabilization of the system. This lasts
to u 2: 9.3, whereupon coupled-mode flutter occurs via a Hamiltonian Hopf bifurcation,
as defined in Figure 3.4(c).
What is particularly interesting about this predicted coupled-mode flutter is its orig-
ination. As discussed in Section 3.2.1 and as shown by equations (3.5) and (3.6), for
periodic motions there is no energy transfer between the fluid and the pipe. Hence, since
the system is conservative, the question arises as to how the instability can be supported
whilst the total energy of the system remains constant. As pointed out by PaYdoussis &
Issid(1974), the question is not quite like this, since the critical point for the onset of
flutter, unlike for the nonconservative (cantilevered) system, is not a point of neutral
stability; rather, it involves the coincidence of two real frequencies, and hence growing
oscillations of the form q(c, t) = %e[f(c)(a + bt) exp(iwt)], with w real. The source of
energy is of course the flowing fluid, yet how some energy is channelled to generate the
oscillatory state remains the question. A possible answer was provided, via an ingenious
set of arguments, by Done & Simpson (1977) for a pipe with supported ends but with the
downstream end free to slide axially [Figure 3.1(b)].
First, one may consider a two-mode Galerkin approximation of the system, namely

For clamped and pinned ends, b,, = 0 and b,, = -brs; for pinned ends, csr = 0 for all
r # s, while the same applies to clamped ends for r + s odd, which is the case here.
Hence, equation (3.91) may be written as

0 -2/3112~b21] ir+ [A;' +;'cI~
q + [ 2/3'/2~b21 0 A; + u2c22 I q = 0. (3.92)
O
It is of interest to remark that (i) the damping matrix is skew symmetric, which is a
characteristic of the system being gyroscopic conservative, as already remarked, and
(ii) by setting det[K] = 0, [K] being the stiffness matrix, one retrieves the zeros for static
loss of stability Ucd = 7t and u = 237, exactly for simply-supported ends and approximately
for clamped ends (since in this case the matrix is not fully diagonal for N > 2).
Then, solutions of the form q = qo exp(At) are considered, leading to the characteristic
equation
p4h4 + p2h2 + po = 0, (3.93)
with p4 = 1, p2 = [A;' +A; - u2(cll + c22) - 4Bu2b&], po = (A: - u2cll)(h; - u2c22).
The condition of coalescence of two eigenfrequencies corresponds to two equal roots
of (3.93), which occurs if pi - 4p4po = 0. The results for clamped ends are shown in
Figure 3.12, where it is seen that all the critical points of Figures 3.10 and 3.11 are

?This flattering appellation, coined by Done & Simpson (1977), has been retained here for this particular
form of coupled-mode flutter. This phenomenon, however, although analytically intriguing, was shown to be
physically doubtful with he appearance of Holmes' (1977, 1978) work. This would have rendered any claim
to fame by this book's author rather ephemeral, were it not for the fact that, luckily, the physical reality of the
phenomenon is firmly established for another system (Chapter 8)!