Page 257 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

238 SLENDER STRUCTURES AND AXIAL FLOW

theory the system loses stability by divergence, at Ucd 2: 3.16 - vide also Paldoussis &
Laithier (1976). On the other hand, according to Timoshenko refined-flow (TRF) theory the
system is shown to lose stability by fluttert at a higher value of u (u,f = 3.95, wcf = 8.85
for y = 10;~~. = 3.63, w,.. = 8.40 for y = 0).
It is recalled that according to Euler-Bernoulli beam theory (and a simple plug-flow
model) a cantilevered pipe conveying fluid can only lose stability by flutter. It is only
in an earlier version of this work, in which the plug-flow model was used (Paldoussis
& Laithier 1976), and here according to TPF theory, that loss of stability by divergence
is predicted. On the other hand, once a more appropriate model for the fluid mechanics
is used, flutter is predicted once again. Now, it cannot be said that the present TRF
theory never predicts divergence for short cantilevered pipes, but simply that in some of
the cases where TPF theory predicted divergence the present theory predicts flutter. In
this connection, it is recalled that when the cantilevered pipe system is subjected to a
second conservative force - other than the flexural restoring force - it sometimes loses
stability by divergence. Examples are (i) the pipe-plate system of Section 3.6.6, subjected
to warping as well as torsion, and (ii) the articulated pipe system of Section 3.8, subjected
to gravity. Hence, there may be areas in the parameter space of the present system, also,
according to TRF theory as well.
where stability may be lost by divergence ~

4.4.8 Comparison with experiment

The theory is compared with experimental results for cantilevered pipes, obtained by
Laithier (1979). The pipes were made of silicone rubber, 15.60mm in outside diameter
and 6.35 mm in inside diameter. The fluid conveyed was water.
The pipes were specially moulded, with the upper end cast onto a special adaptor
(Appendix D.2). The adaptor could be screwed directly to the piping supplying steady
water flow. Special care was taken in designing the adaptor to ensure that (a) the upper
support approaches the clamped condition as closely as possible, and (b) the entrance
of the fluid to the supported part of the pipe is effected without disturbance (which in
short pipes could have an important effect on their dynamical behaviour). The measured
Young’s modulus for these pipes was E = 1.49 x lo6 N/m2, Poisson ratio u = 0.45, and
the hysteretic damping coefficient p = 0.02. Utilization of equations (4.37) and (4.32) in
this case gives A = 0.538~~. In the experiments, A was varied by progressively reducing
the length of the pipe (by carefully cutting pieces off the free end), thus reducing E; L
was varied between 140 and 51 mm in one case, and 73 and 27 mm in another.
The flow velocity was measured by standard means. Oscillation was sensed by a fibre-
optic sensor, measuring the lateral displacement close to the supported end of the pipe; the
frequency of oscillation was measured from oscillation time-traces, recorded on a storage
oscilloscope.
The critical flow velocities for flutter, uCj, according to the three theories are compared
with the experimental data in Figure 4.22(a) and the corresponding critical frequencies,
w,j, in Figure 4.22(b); it is important to mention that the experimental values of ucf were
measured at just the onset of instability and are not the limit-cycle values (which in this
case are quite different), so that they should correspond better to those predicted by linear

?Surprisingly, this is the behaviour predicted by the Euler-Bernoulli theory, but at a very different critical
flow velocity, ucf = 8.7, and in the second mode.