Page 392 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SLENDER STRUCTURES AND AXIAL FLOW

if excited sufficiently so that it crosses the vertical (unstable) equilibrium position, may
develop chaotic oscillation, as a result of ‘hesitation’ as to which of the two potential wells
it will next gravitate towards. Conceptually, the same may be achieved if the excitation
is provided via flow-induced flutter.
The equation of motion is given by the linear equation (3.70), with k = r = I7 = U = 0
and a!, 0 # 0, as modified by (3.74) to account for the end-nozzle, and the presence of
t)
the magnetic forces represented by ~lr(1, + K~v~(I, t). In the experiments, ~1 and ~2
are determined by measuring the first buckled natural frequency in the post-buckling state
of the system, and the location of the statically buckled pipe end. The equations are
first discretized by Galerkin’s method and then studied by simulation via a fourth-order
Runge-Kutta integration scheme, both for the autonomous and the forced system. Useful
experimental data, however, could only be obtained for the forced system; flutter of the
buckled autonomous system could not be achieved because of experimental limitations
(Tang 1997).
Numerical phase-plane plots for an autonomous system similar to the experimental
one but with zero damping (a! = 0 = 0), discretized to fourth order (N = 4), show (a) a
stable limit cycle for u = 3.30, just beyond the flutter threshold, (b) period-4 motion for
u = 3.89, and (c) chaotic oscillation for u = 3.96 and 4.29. Interestingly, for a model
with N = 2, periodic rather than chaotic oscillation is displayed for u = 4.29. For N = 4
it is shown that the dynamics evolves about both potential wells: a small orbit about one
of them, followed by a larger orbit leading to the other one.
Additional work on the effect of damping shows that, with increased damping in the
pipe material (a), the chaotic attractor is progressively weakened, so that eventually, for
a! - 6( lop2), the oscillation becomes periodic.
Typical numerical results for the case of forced oscillation of the system are shown
in Figure 5.42 for N = 2 (cf. the early work in Section 5.8.1) and a low value of u. It
is shown that with increasing Fo and constant w, there is an alternation of periodic and
chaotic regions, which is mapped in (e) for varying w. A similar map for N = 1 (not
shown) is completely different from that of Figure 5.42(e), but another for N = 3 is not
too radically different from that for N = 2, showing the beginnings of convergence.
The phase-plane plots of Figure 5.42(a-d) show clearly that for low f = FoL2/EI the
vibration is in the vicinity of the potential we11 in which the system is buckled. For higher
f, it is about both wells, and the motion is essentially as follows: one or more orbits
around one of the potential wells, followed by a trajectory over to the other potential
well, and so on.
The theoretical (N = 3) thresholds for chaos with forced vibration are reasonably close
to the experimental ones for a case with u = 0.35. However, this does not represent the
best test for the theory since, at such low u, the main effect of flow is to contribute some
additional damping vis-&vis u = 0. Of more interest would have been an experiment
at u close to the flutter boundary; however, this was precluded by the apparatus used
(Tang 1997).

5.8.3 Pipe with added mass at the free end
As discussed in Section 3.6.3, the linear dynamics of a cantilevered pipe conveying fluid
is modified in interesting ways by the addition of a point mass, notably at the free end.
The nonlinear dynamics is equally interesting, as will be seen in what follows.