Page 368 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

348 SLENDER STRUCTURES AND AXIAL FLOW

into a centre, when tr([AJ) = 0, where tr stands for trace, which gives

(1 + c)p1 + (1 + b)p2 = 0. (5.130)
Along this line, a heteroclinic loop emerges. This in turn may give rise to the possibility
of heteroclinic tangles and chaos, as discussed in Section 5.8.

5.7.4 Concluding comment

The work presented in this section is representative of a larger set, not fully discussed
for the sake of brevity. The reader is also referred to the next section, primarily devoted
to chaotic dynamics, but also containing work of interest here (e.g. on the dynamics of
pipes fitted with an additional mass at the free end).
It is hoped that the work in Section 5.7 has shown that, similarly to the linear dynamics
of cantilevers conveying fluid (Chapters 3 and 4), the nonlinear dynamics is equally
fascinating. Of special interest are the results in Figures 5.19-5.22, where the regions of
sub- and supercritical Hopf bifurcations are defined, as well as whether motions are three-
dimensional or planar and, in the case of an inclined nozzle, in which plane. Once more,
the special importance of the ‘critical values of p’, associated with S-shaped discontinuities
in the u, versus /3 plot, emerges; thus, it is seen that, for nonlinear dynamics also, these
values of are either separatrices or backbones of peculiar behaviour. Also of importance
are the codimension-2 and -3 bifurcation sets emanating from the vicinity of double
degeneracies, samples of which are given in Figures 5.27-5.29. Of special interest in
this regard is the existence of conditions leading to heteroclinic loops which are often
associated with chaotic behaviour (Section 5.8).
Finally, it is also hoped that the material in Sections 5.5, 5.6 and 5.7 has made
abundantly clear - not only by the substance of the work, but also by the authors’
names - that this system has served both as an example of a physical system on which
the various modem methods of nonlinear dynamics could be demonstrated and as a system
through which these methods could be further developed.

5:8 CHAOTIC DYNAMICS
With the rapidly developing, and deserved, fascination with chaos in engineering systems,
it was inevitable that its possible existence in fluidelastic systems would be explored. As is
well known, however, chaos is usually associated with strong nonlinearities (Moon 1992);
hence, the first set of studies into chaotic dynamics involved modifications to the system
so as to introduce strong nonlinear effects. Three such systems are discussed: (i) the pipe
with loose lateral constraints (Section 5.8.1); (ii) with magnets added; (iii) with an added
mass at the free end. Then, the existence of chaos under more particular conditions, e.g.
near double degeneracies, is discussed in Sections 5.8.4 and 5.8.5.

5.8.1 Loosely constrained pipes

In contrast to other parts of this chapter, the presentation here is chronological as well
as paedagogical in tone. The reason for this is that, in addition to showing how chaos
can arise in loosely constrained pipes conveying fluid, there is another objective also: to