Page 433 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 40.5

pipe to perform an almost-periodic or amplitude-modulated motion with small amplitude.
For fi1 < 7 < fi2, however, these motions coexist with larger-amplitude periodic motions
associated with subharmonic resonance which appear as an isolated solution branch. Which
one materializes, in that range, depends on the initial conditions.
Points A-F in Figure 5.63(a) are special in that they are intersections of two bifurcation
curves. Bajaj undertakes the unfolding of these bifurcations by local analysis. An example
is shown in Figure 5.64 for point F. Across Bh2 a supercritical Hopf bifurcation takes place
in the averaged system, leading to the limit cycle in region IIa; the physical system then
performs a subharmonic amplitude-modulated motion. However, beyond this limit cycle,
other fixed points may exist, and hence periodic solutions of the physical system, as
shown in the lower part of this figure.

Figure 5.64 The qualitative types of phase portraits of the averaged system around point F of
Figure 5.63 (Bajaj 1987b).

To summarize, it is shown that, when the mean flow is below u,f, the pipe can only have
periodic solutions which are at half the excitation frequency. Even when the straight position
is stable, there are flow fluctuations for which nonzero stable solutions also exist. Thus, a
large enough disturbance can force the pipe to perform large steady-state periodic motions.
For mean flow above the critical value, the zero solution is always unstable and the
pipe can perform small modulated motions, large periodic motions or large-amplitude
modulated motions, depending on the values of flow rate and excitation frequency. Some
of these motions coexist for the same values of parameters and then the initial conditions
and disturbances determine the motions performed.
A similar study of the articulated system was conducted earlier by Bajaj (1984). Planar
motions of a two-segment cantilevered pipe are considered, and hence the 2-D versions of
equations (5.74) and (5.76) are used. The periodic solutions for the principal parametric