Page 405 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 381

unlike the period-doubling route, this route to chaos involves a finite and small number
of bifurcations.
For a system with p = 2.60t at ug = U/(gL)1/2 = 3.048 and in the CCW regime,
the motion is periodic and thus involves but one frequency - as found by delay recon-
structions of the experimental attractor and associated Poincare maps and PSD plots,
similar to those in Appendix I. For ug = 3.080, however, the motion is in the PL,CW
regime, with the pipe oscillating in a plane that is continuously rotating counterclockwise,
and hence is associated with two incommensurate frequencies. As seen in Figure 5.50,
(a) the pseudo-phase-plane plot, (b) the 'closed curve' in the Poincark section and (c) the
power spectrum with a multitude of combination frequency peaks are all indicative of
quasiperiodicity involving two frequencies.

I .
h 1 . 1 .
.. : . ..
0.8 - , *~, : , .',: : ? : _. ' .
..
..
..: :.,'."..;.,. ..*.
.:
.:.. . . .
.. .. .
0.6 0.0 -...;.:. ._/ . .I.. . ,: ;
.. .
..
v1
h h _. .., ..
_.
m
0.4 vl . .. ..
..
0 2 :,. .. ..;, ...
, .'
2 0.2 -0.1 - :._ . r .,.
i.
R . .. ",:'
..
.. . .
.. . :-,:.
0.0 . .. . . ..I ..
-0.2 - ' .. .L..;. ;.,-...+.?,;~:%;-."
':...e&
...__ .
-0.2 D l . 1 . I . I
h -20
!2
a
2
3 -40
sr
-
4
I
g -60
E
e:
-80
0 2 4 6 8 10
(C) Frequency (Hz)
Figure 5.50 Pipe motion in the PL,CCW regime for 3-D motions of the system of Figure 5.43(a),
for p = 2.60, ug = 3.080; scale is arbitrary. (a) Delay reconstruction of attractor, delay = 0.1 17 s (30
sampling periods), correlation dimension d, = 2.105 for 33 0oO data points (b) Poincan5 section, at
x(t) = 0.2; (c) power spectrum (Copeland & Moon 1992).
+End-masses of different aspect ratios and moments of inertia were used, so that p alone is insufficient to
differentiate two experiments. This distinction is not made in this book for simplicity, but it explains why the
values of up in Figures 5.49 and 5.50 do not correspond.