Page 420 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

392 SLENDER STRUCTURES AND AXIAL FLOW

a ‘Takens-Bogdanov’ point, where one of the frequencies in the Hopf-Hopf interaction
becomes zero); the other lines shown in the figure correspond to features of the dynamics
not discussed here. The solution then generally involves two frequencies and the secondary
oscillations are in-plane or out-of-plane. As shown in Figure S.S6(c), heteroclinic cycles
can occur in the narrow range bounded by the solutions for oscillation in one or the other
plane (SW1 and SW2) and the secondary branches marked by and I in the figure.
The same lund of heteroclinic cycles may exist if flutter is rotary rather than planar.
These results offer a reasonable qualitative explanation of Copeland & Moon’s (1992)
experimental observations (Figure 5.49).

5.8.5 Chaos in the articulated system
An extremely complicated bifurcation picture is involved in the nonlinear dynamics of a
horizontal two-segment articulated pipe with an asymmetric spring, so that the two pipe
segments at equilibrium are at an angle I+? to each other, as shown by Champneys (1991,
1993) and as discussed briefly in Section 5.6.2(a). Because the dynamical theory required
to understand the dynamics is beyond the scope of this book, only some selected results
are presented and the reader is encouraged to refer to the primary sources.
A typical sequence of dynamical behaviour with increasing u is given in Table 5.8.
The ‘primary orbit’ in the table is a limit cycle due to a supercritical Hopf bifurcation. It
is followed by a period-doubling bifurcation (at point 4) and gains amplitude by a tower,
as shown in Figure 5.57(a); a tower consists of a number of saddle-node bifurcations (the
first of which occurs at point 5). (Similar towers show how the period can increase with u.)
Eventually, that branch of the curve regains stability at u = 5.5667 via a reverse period-
doubling bifurcation (at point 6) and remains stable thereafter. It is noted in Figure 5.57(a)
that neither the primary nor the period-doubling branches are stable for u > 5.0865, and it
is for that range of u that interesting dynamics occurs. In fact, the bifurcations at points 4
and 8 are the beginnings of a period-doubling cascade leading for u E (5.0865-5.1321) to
‘small-scale’ chaos with periodic windows. This is succeeded for u E (5.1321-5.5667) by
‘large-scale’ chaos involving mixed-mode period-(m, n ) orbits; these have m large-scale
oscillations and n small-scale ones per period, as exemplified in Figure 5.57(b) - see
Glendinning (1994). Eventually, the map shown in Figure 5.57(c) is obtained, with the
regions I-VI as defined in Table 5.8.
The vertical articulated system when I+? = 0 has also been studied by Champneys (1993)
close to the pitchfork-Hopf double-degeneracy point. Once more, a very complex and
interesting bifurcation structure is revealed. An example of a period-( 1,8) figure-of-eight

Table 5.8 The dynamics of the articulated system with @ = 0.6 (Champ-
neys 1991).

Region u Behaviour
I 0-4.7072 Stable stationary point
I1 4.7072-5.041 1 Small-scale primary orbit
111 5.041 1-5.0865 Period-doubling cascade
IV 5.0865-5.1321 Small-scale chaos with periodic windows
V 5.1321-5.5667 Large-scale chaos and period-(1, n) orbits
VI 5.5667-00 Large-scale (1 ,O)-orbit