Page 343 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 323

global branches of some of the orbits, and towers, which are sequences of period-doubling
and saddle-node bifurcations. Champneys goes on to show the existence of chaotic regions
in this and in a subsequent paper (Champneys 1993) where the system asymmetry was
removed - as discussed in Section 5.8.5.

fb) 3-0 motions
A very sophisticated analysis of three-dimensional motions of the N = 2 system - effect-
ively generalizing to 3-D the foregoing analysis, but utilizing entirely different
techniques - has been conducted by Bajaj & Sethna (1982a,b). The equations of motion
used are equations (5.74)-(5.77); hence the flow velocity is motion-independent. The
springs are considered to be so designed as to allow both planar motions in two directions
and rotational motion with zero torsional stiffness - a challenging design problem if
attempted experimentally.
The particular problem investigated is the loss of stability by flutter. Because of the
rotational symmetry of the system, a double pair of complex eigenvalues crosses simul-
taneously the imaginary axis from negative to positive. The nonlinear phenomena in
this case are more complicated than those associated with simple Hopf bifurcations
(Figure 2.1 1); e.g. supercritical bifurcations do not necessarily imply a stable system
in this case (Iooss & Joseph 1980).
After considering the linear dynamics, the problem is transformed into Jordan canonical
form. Then, periodic solutions of the nonlinear equations are analysed by the method
of alternate problems (Hale 1969; Bajaj 1981, 1982), which is similar in spirit to the
Lyapunov-Schmidt method (Appendix F, Sections F.6.2 and F.6.3). Two independent sets
of periodic solutions are found to exist: clockwise or counterclockwise rotary motions
about the x-axis and planar transverse motions. Their stability is determined by the Floquet
exponents of the corresponding variational equations, leading finally to the following set
of interesting results:

(i) both supercritical and subcritical solutions of both the rotary and planar kinds are
generally possible for 0 < 3 < 3 (0 < #I < 1) and for given ranges of a, K and
y, as defined by (5.68); as already mentioned, these are associated with double
pairs of eigenvalues crossing the imaginary axis;
(ii) if both planar and rotary motions are supercritical, then the one with the larger
amplitude is stable and the other unstable, whereas normally one would expect
all supercritical solutions to be stable;
(iii) for a given p, if either of the solutions (rotary or planar) is subcritical, both
solutions are unstable.

Typical results are shown in Figure 5.15, where l/lp201'/~ is a measure of the ampli-
tude. In Figure 5.15(a) it is seen that both rotary and planar supercritical solutions exist.
For 8 < 0.5 1, the latter being larger, the planar oscillations are stable and hence should
physically materialize; for B > 0.51, it is the rotary motions that are stable. When gravity
is present, as in Figure 5.15(b), the situation is more complex. Thus, for p < 0.45, we
once again have stable planar motions, while for 0.45 < B < 1.19 we have stable rotary
motions. For p > 1.19, however, there also exist subcritical solutions, not present in
Figure 5.15(a); since one of the solutions is subcritical, both are unstable. This does
not imply that there are no stable periodic solutions for > 1.19; it merely reflects the