Page 407 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 383

The numerical simulations were pursued to higher ug by means of the eight-dimensional
reduced subsystem. The results for p = 2.30 indicate that the motion remains basically
period-] for E = [ug - (uX)~]/(uLT)~ = 0.1 1, which is not in agreement with experiment.
Those for p = 3.55, however, show a clear path to chaos via the quasiperiodic route,
which agrees with experiment, as illustrated in Figure 5.51. For E = 0.04 the motion
is periodic; for t = 0.0485 it is quasiperiodic with two fundamental frequencies, and
for E = 0.05, shown in Figure 5.51(b), with three frequencies; finally, for E = 0.065
the motion is predominantly quasiperiodic, but with a chaotic component. Again, there-
fore, in view of the results for p = 2.30, only partial agreement with experiment is
obtained.
However, more perplexing is the less than good agreement between the numerical
and the analytical results, which must exist at least in some neighbourhood of t = 0: in
addition to the bistable behaviour (both rotational and planar oscillations) which occurs
only in the former, the variation of frequency with E does not agree. Of course, this has
also perplexed Copeland and this author, but no error has been found, though this remains
a possibility.
Further experiments on the same system were conducted by Muntean & Moon (1 995),
in which the system is additionally excited at the support via a shaker, and the 'end-
mass' may be a little higher up than the end of the pipe. The objective of this work is
to investigate the transition from quasiperiodicity to chaos and this is done by means of
multifractal dimensions, or spectra of fractal dimensions, in a similar manner as in Jensen
et al. (1985) for the forced Rayleigh-BCnard convection experiment. It is shown that the
dynamics of the system can be captured by simple maps, and hence the transition to chaos
displays remarkable universality irrespective of the physical system.

(c) 2-0 motions of a pipe with an end-mass defect

Partly to further explore just how singular the case of A = 0 is, the situation when the
additional mass at the free end is negative (p < 0), i.e. when there exists an end-mass
defect, was investigated by Semler & Paidoussis (1995) and some interesting results were
obtained.
Utilizing equation (5.39) and applying a Galerkin discretization scheme as in
equations (5.115) and (5.116a,b), the inertial term is found to have the form [Si; +

p@i(l)@;(l) + yilk;qkql]q;. Then, making the assumption that p and the inertial
nonlinearities are small, one can write
+
-
16;; + ~@;(l)@j(l) ~rlkjqkqll-' 2 ~ij p@i(lMj(l) - Yilkjqkql?
and so thc nonlinear equation of motion can be recast into first-order form, which may
be integrated via a standard Runge-Kutta scheme.
Typical results are shown in Figure 5.52 for a case with N = 4 in the discretized system
with parameters as for one of the pipes used by Paidoussis & Moon [Section 5.8.11,
namely B = 0.216, y = 26.75, p = -0.3,' and experimentally determined damping
values. The Hopf bifurcation occurs at u = uH = 8.7, so that it is immediately obvious
from the figure that subsequent bifurcations begin to occur at much higher values of u. At
u = 19.82, one of the Floquet multipliers crosses the unit circle at h = +1, indicating a

'This is a relatively large value of I@/, but qualitatively similar results are obtained for smaller, more realistic
values, e.g. = -0.085, although the bifurcations shown in Figure 5.52 then occur at even higher values of u.