Page 371 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 35 1

probability of finding the vibrating system there is high. Departures from this double-
masted, suspension-bridge shape indicate departures from regularity in the motion: a
Gaussian-like distribution corresponds to random or random-like (chaotic) motion. The
autocorrelation of a periodic signal is periodic with time, whereas an aperiodic signal has a
‘damped-response versus time’ characteristic, showing loss of memory after a few cycles
of motion - which is characteristic of random or more generally stochastic signals.
Typical results with pipe 9 (Table 5.3), water flow and the polycarbonate impact
bars at wb/L = 0.055 are presented in Figures 5.31 and 5.32: power spectra (PS).
phase plots, probability density functions (PDF), autocorrelations and PoincarC maps.
In Figure 5.3 l(a), the limit-cycle amplitude is sufficiently large to allow impacting,
preferentially on one of the two bars. Therefore, the motion at this stage is asymmetric,
biased towards one of the bars.+ The profusion of harmonics of the main oscillation
frequency (f 2 2.6Hz) is due to the impacting. The double-masted shape of the PDF,
the essentially constant-amplitude autocorrelation and the single-loop phase-plane plot
all indicate periodic (period-1) motion. In Figure 5.31(b) it is seen that the motion is
still periodic, but the strong subharmonic at if in the PS plot and the double-loop
phase-plane plot indicate period-2 motion; physically, a typical sequence of motions is
this: the pipe impacts on the bar, then executes a complete ‘free’ cycle of oscillation,
before impacting again. The corresponding Poincark map, Figure 5.32(a) shows two fuzzy
‘points’, indicating that a small chaotic component may be in existence already, but
indicating a predominantly period-2 motion.
The motion is considerably more erratic for U = 7.48 m/s (not shown), but still periodic
(period-2): there is a slight reduction of the autocorrelation with time and the trough of
the PDF is considerably more filled out, with a clear double peak on the right side,
corresponding to the period-2 trajectory. This perhaps is the limit of periodic or almost
periodic motion. In Figure 5.31(c), the two subharmonics of the main frequency are at $
and $, signifying a period-3 motion (found in a few other instances also); however, the
low-frequency content of the signal is wide-banded and erratic and hence the oscillation
should be considered chaotic. Significantly, the PDF has become less concave in the
centre region, almost convex, and the autocorrelation decays fairly rapidly with time, with
beating (more easily visible if displayed over a longer time period). Finally, the motion
is quite chaotic in Figure 5.31(d), as shown by the PS, the PDF and the autocorrelation
equally. The phase portraits in Figure 5.31(a,b) correspond to the pipe impacting on one
motion-constraint bar, whereas in Figure 5.3 l(c,d) it is impacting on both.
The PoincarC maps of Figure 5.32 correspond to (a) conditions just after the period-
doubling bifurcation, with two attractors, as already remarked; (b) more erratic motion,
with a suggestion of more complex attractors; (c) where the motion is more wide-band
chaotic. In the last case, although the PoincarC map does not display the artistic merit of
that in (b) and even less of the ‘fleur de PoincarC’ and some other remarkable examples
(Moon 1992), still it does seem to have some structure. In this connection, the point should
be made that these PoincarC maps represent two-dimensional sections of a multidimen-
sional attractor, which may indeed have a great deal of structure that cannot be discerned
in the planar sections taken; the construction of double-Poincar6 maps (Moon 1992) might
have been more successful in rendering any such structure more conspicuous. It is also

+This occurs soon after - in terms of increasing u - impacting begins to take place