Page 333 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

3 14 SLENDER STRUCTURES AND AXIAL FLOW

more strictly it is a motion on a torus, rather than a limit cycle which would imply a
single frequency.
Thaefore, coupzed-mode flutter does exist, even in the nonlinear context. However,
it is pathologically nonrobust: the slightest amount of damping destroys it utterly. On
reflection, this is as unremarkable as ‘finding’ that periodic solutions can exist for a
conservative system, but not when damping is included; what is remarkable, nevertheless,
is that such periodic solutions are academically possible, even after divergence.
(d) Concluding remarks

The same conclusion as Holmes’ with regard to the nonexistence of coupled-mode flutter
for dissipative systems was reached by Ch’ng (1978) and Ch’ng & Dowel1 (1979), who
utilize the same equation as Holmes, equation (5.82), discretize it and then integrate it
numerically. [It is of interest that an error in an earlier attempt by Ch’ng (1977) led
to the opposite conclusion. Holmes (1977) also admits that, in an earlier version of his
work, a mistake led him too to the opposite conclusion. All this shows how sensitive this
type of analysis can be.] On the other hand, Lunn’s (1982) work shows that sustained
oscillation, i.e. flutter, about the unstable initial equilibrium is possible, theoretically at
least, provided that there exists no dissipation whatsoever. This, of course, is impossible
in any real physical system.
In fact, as discussed in Section 3.4.4, no experimental evidence exists that pipes
supported at both ends do flutter, whether axial sliding at the supports is permitted or
not; in the former case violent divergence (buckling) develops and the w = 0 condition
is obtained, while in the latter case this is not so. The main point here is that, for realistic
systems, predictions of linear theory, beyond the onset of the first instability (divergence),
do not materialize. This is not general, and in fact Holmes (1977) discusses another case,
involving panel flutter, where post-divergence flutter does indeed materialize. This is also
known to occur in cylindrical structures subjected to external axial flow (Chapter 8).

5.5.3 Pipes with an axially sliding downstream end
When a pipe has a laterally supported but axially free-to-slide downstream end, its equa-
tions of motion are essentially those of a cantilevered pipe (see end of Appendix G.2):
the centreline may be taken to be inextensible, and the nonlinearities are mainly due
to curvature effects, while the mean deformation-induced tension is zero. The nonlinear
dynamics of such a system has been studied analytically, numerically and experimentally
by Yoshizawa et al. (1985, 1986) up to and beyond the point of divergence.
The system considered is a clamped-pinned pipe, supplied by a constant-head tank,
while the flow velocity is generally deformation-dependent. The equations derived are
similar to Rousselet & Herrmann’s [Section 5.2.8(b)]: (i) a ‘flow equation’ similar to equa-
tion (5.53), with a friction parameter a; (ii) an equation for the pipe motion involving both
axial and transverse displacements, u and w, and the angle of deformation, 8 - interrelated
via sin B = aw/as, cos B = 1 + (&/as) as per equations (5.4) for an inextensible pipe.
The eigenfunctions of the subsystem 8”” - y[(l - c)~” - 41 + u2f + ij = 0 are deter-
mined and then the deflection of the pipe is approximated by a one-mode Galerkin scheme,
~(4, t) = 41 (t)q1 (t). Analytical solutions are obtained with this approximation, adequate
for relatively modest deflections, as well as more accurate solutions for the post-divergence