Page 360 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

340 SLENDER STRUCTURES AND AXIAL FLOW

when Hopf bifurcations occur and A{*,) = (+, +, -, -). This probably leads to stable
limit-cycle motion, since there is no other stable equilibrium state. At u = 12.48, it is the
origin {0} that undergoes a Hopf bifurcation. The three fixed points (0) and (411) coalesce
at u = 15.07 = h{o) = (+, +, 0, -)I. A numerical investigation confirms the results
found: limit-cycle oscillations are found to exist before the first Hopf bifurcation occurring
at u = 12.43, showing that these oscillations are due to the subcritical bifurcation of the
(fl} fixed points. For u a little less than 12.43, e.g. at u = 12.35, the orbit can be attracted
either by one of the stable fixed points or by the attracting periodic limit-set.
It is of particular interest that, in this case, post-divergence flutter does materialize,
although not in the manner predicted by linear theory: i.e. it emerges from the new stable
fixed points associated with first-mode destabilization, rather than from the second mode.
From this and other similar calculations, it is clear that the nonlinear dynamics of the
system can be substantially different from linear predictions. Thus, the stability map in
Figure 5.25(a), obtained by linear analysis, can only be relied upon for the jirst loss of
stability: by divergence for -56 < y < 71.9 and by flutter for y > 71.9 approximately
for the particular set of parameters given in the caption; the other predicted instabilities,
beyond the first, do not necessarily materialize.
The region of ‘global oscillations’ in Figure 5.25(a) cannot be obtained by linear or
even local nonlinear analysis, but was found numerically. ‘Global’ is used here to indicate
that the oscillations circumnavigate more than one, in this case three, fixed points. For
u = 7.5, it is seen in Figure 5.25(b) that the origin has become a saddle, but two new stable
equilibria exist. For u = 13.1, however, the dynamics is more complicated, as shown in
Figure 5.25(c). The origin (0) is a saddle, as well as the second pair of fixed points, (f2);
for clarity, not all the stable and unstable manifolds have been drawn in this figure, and the
existence of only one fixed point of the second pair at - 0.1 is revealed by the trajectories
shown. The first pair (fl} at f0.2, is ‘weakly’ attracting. Flows with initial conditions
close to the equilibrium are attracted by one of the fixed points {&l}. However, other
attracting sets also exist: one may observe either limit-cycle oscillations around one of the
equilibria or global limit-cycle oscillations around the five equilibria. Those oscillations
do not come from local bifurcations. For Duffing’s equation, for example, solutions lie
on level curves of the Hamiltonian energy of the system, and these solutions are closed
orbits representing a global stability state (Guckenheimer & Holmes 1983).

(b) 2-0 motions of pipe- spring system; double degeneracy conditions

Here the dynamics of the system is discussed in the vicinity of the double degeneracy,
in this case due to coincidence of a pitchfork and a Hopf bifurcation. Figure 5.26 shows
appropriate combinations of ,4, y and K for which such a double degeneracy is obtained.
Appendix H shows how the nonlinear dynamics in the vicinity of a Hopf or a pitchfork
bifurcation may be analysed by means of centre manifold theory and either normal form
or averaging analysis. Here the same type of analysis is done under conditions of double
degeneracy.
In this case, introducing an appropriate modal matrix [PI and the transformation y =
[PIX, equation (5.117) can be put into ‘standard form’, defined by