Page 157 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I I39

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0 0.1 0.2
P
Figure 3.47 Comparison of the experimental values of ucf and wcf for cantilevered metal pipes:
A, measurements; -, theory (Pai’doussis 1963).

short pipes (L/Di 2 36-350 for the rubber pipes) and supercritical Hopf bifurcation for
long pipes (LID; 2 1545-1650 for the metal pipes).
Chronologically, the second set of experiments was conducted by Greenwald &
Dugundji (1967) - see also Section 3.4.4. They conducted experiments with three
elastomer pipes (Do = 3.00-4.75 mm, h = 0.86- 1 SO mm). The pipes were hung vertically
and clamped at their upper end. The authors have made similar general observations
to those discussed in the foregoing. A very nice photograph is shown in Figure 3.48,
corresponding to a pipe with /3 = 0.471, which shows more clearly than Figure 3.45(a-c)
the nonstationarity of the modes and the travelling wave component in the mode shapes.
The measured critical flow velocities are compared with theory in Table 3.3. It is seen
that agreement is at least as good as in Figure 3.46 when viscoelastic damping is taken
into account. In the theory the authors neglected gravity; this is reasonable: from their
data one finds y = 1.68-2.89, which results in theoretical values of ucf higher than those
in Table 3.3 by less than 2%.
A more extensive set of experiments was conducted by Pdidoussis (1970) with vertical
pipes, either hanging or standing. This is the first instance when such pipes were cast
by the researcher, and this allowed the manufacture of truly straight pipes for the first
time, thus facilitating the experiments a great deal - not only for pipes conveying fluid,
but also for experiments with shells and cylinders. The ‘manufacturing techniques’ are
outlined in Appendix D.
The general observations of the dynamical behaviour of hanging pipes are much as
described before and need not be repeated here. However, two additional points are useful