Page 325 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

306 SLENDER STRUCTURES AND AXIAL FLOW

simply the centre eigenspace. The stable manifold is ignored, and attention is focused
on the centre manifold; then the evolution of the system in this subspace, wherein the
interesting aspects of the dynamics is happening, is examined. In the vicinity of u =
uCr = n, we have h = (0, -, -, -}; re-writing u = u,, + p, where p << 1, one eventually
obtains (Appendix H.2.2):
3
x = c,px - c2dx , p = u - IT, (5.84)
in transformed coordinates, where c1 = 63.015 and c2 = 24.746; this shows that this is
a pitchfork bifurcation. Thus, putting X = 0, it is obvious that there exist fixed points at
and
xst = &[(c1/~2)(p/d)]~/~ is easy to show that they are stable;+ i.e. the new fixed
it
points are sinks (attractors in phase space). Transforming back to the original coordinates,
one finds that these are located at
qst = 1.596(p/d)1/2, (5.85)
i.e. there is a parabolic relationship between qst and p. As may be seen in Figure 5.5,
agreement with numerical results is excellent, almost to p = 0.4, despite having specified
p << 1.
This form of dependence of the post-divergence fixed points on u is also predicted in
another way by Thompson & Lunn (1981), who develop an elegant ‘static elastica’ formu-
lation of the nonlinear deformation of a pipe under equivalent static loading, effectively
the equation of motion with all time-dependent effects ignored. Then, by similarity to the
nonlinear behaviour of struts (columns) subjected to compressive loading (Thompson &
Hunt 1973), they obtain a ‘rising post-buckling path’, the same as shown in Figure 5.5.
The question now is what happens for higher u, in particular for u 3 2n. In this respect,
it is instructive to look at the evolution of the linear system for the specific parameters in
this example. As shown in Figure 5.4, because of the presence of dissipation, the mode
loci evolve similarly to Figure 3.14 rather than to those typified by Figures 3.9-3.1 1.
Dissipation renders restabilization of the first mode followed by a Hamiltonian Hopf
bifurcation (cf. Figure 3.11) impossible, and it also prevents the pitchfork bifurcation
associated with the second mode from happening (cf. Figure 3.9). Thus, at u = 27r it
is the second branch of the first mode that crosses to the unstable part of the complex
frequency or the complex eigenvalue plane, rather than a branch of the second mode. At
that point, one has h = (+, 0, -, -1. However, the new fixed points originating at u = 277
are saddles. The flow on the centre manifold in this case is governed by
X = 31.81~~ 24.99dx3; (5.86)
-
the origin in this case is unstable prior to the bifurcation, and so are the new fixed
points. These fixed points, transformed back to the original coordinates, are also shown
in Figure 5.5, where it is seen that, because the amplitude is smaller, they agree with
numerically computed results even better than those for the stable fixed points.
Furthermore, it is shown that no further bifurcations occur; in particular, the only stable
fixed points, those given by equation (5.85), do not give rise to Hopf or other bifurcations
as u is increased.

tBasically, one perturbs (5.84) such that x = xsI + i, where xsl is given by (5.85) - or takes the Jacobian
at x = xst - and eventually obtains X = -2c1w.f; from the solution i = io exp(-2clyr) it is clear that this
solution branch is stable for y > 0 and unstable for ,u c: 0.