Page 403 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: NONLINEAR AND CHAOTIC DYNAMICS 379

detected, but also jump phenomena, quasiperiodic and chaotic oscillations. This conclusion
is reinforced by the findings of Sections 5.8.3(b) and (c).

fbl3-0 motions of a pipe with an added end-mass
As discussed in Section 5.8.3(a), there is a natural tendency for motions to be three-
dimensional once they become chaotic, even if at the onset of flutter they are planar.
Copeland (1992) and Copeland & Moon (1992), whose work is discussed here, observed
that for larger end-masses this tendency to three-dimensionality exists even before the
onset of chaos. Hence, they studied 3-D motions of the system from the outset, both
experimentally and theoretically.
In the experiments, the apparatus is very similar to that of Figure 5.43. The pipes used
are also similar to those in Section 5.8.3(a), but very long and slender (L - lm, L/Di 2
125, B = 0.219, y = 292), while the end-masses are much larger (.nil = 83.8-816.9 g); for
the largest, p = 3.81.
The experimental critical flow velocities for the onset of flutter have already been
discussed [Figure 5.44(b)]. For higher flows, there exist a series of increasingly compli-
cated periodic and quasiperiodic motions, eventually leading to chaos; their sequence
and range are shown in Figure 5.49 (top), with the motions sketched below - definitely
among the most captivating of experimental results with pipes conveying fluid.
As seen in Figure 5.49, rotational motions do not arise for p = 0 and 0.367, the
smallest experimental value of p. However, they are increasingly evident for higher p. For
p = 3.55, the response is predominantly circular. It is seen that, in addition to planar and
rotational motions, there are three periodic states of greater complexity: rotating planar,
planar and pendular, and nutating oscillations. As evidence of circular symmetry, clock-
wise and counterclockwise motions may both occur; likewise, the planar oscillations are
not biased towards particular vertical planes.
There are three kinds of rotating planar motion. The rotation is either backwards and
forwards through a finite angle [PL(R)], as shown in Figure 5.49(c), or more commonly
continuous rotation in either the clockwise (PL,CW) or counterclockwise (PL,CCW)
sense. Generally, the period of rotation is ten or more times the period of planar oscillation.
For p = 1.24, there exists a state of motion that appears to be coupled planar oscillation
with the pendular mode (PL,P) [Figure 5.49(d)]. The period of pendular oscillation is
approximately four times the period of planar oscillation. Finally, the motion described
as nutating [Figure 5.49(e)}, for its resemblance to the nutation of a spinning rigid body
with axial symmetry, is perhaps the most interesting; it occurs for p = 3.81,3.55 and
2.30. The motion can be characterized in terms of how many small nutations (the small
loops) are made in a single precession (the motion about the vertical axis) and in terms of
the relative amplitude of the nutation, Rl/RZ. The number of nutations per precession is
generally an irrational number between 4 and 12. The loops are not stationary, but occur
at different points for each cycle of precession, suggesting a nonresonant response.
With decreasing flow, the sequence and type of oscillatory states are generally different;
e.g. for p = 0.746, chaos is succeeded by PL, P(R) and PL oscillations, before the pipe
regains static equilibrium.
In at least some of the cases, a clear quasiperiodic route to chaos is followed, as put
forward by Ruelle, Takens and Newhouse (Newhouse et al. 1978; Berg6 et al. 1984;
Moon 1992), observed for example in Taylor-Couette flow. In this scenario, a secondary
Hopf bifurcation transforms periodic motions into quasiperiodic ones, involving two