Page 283 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

264 SLENDER STRUCTURES AND AXIAL FLOW

u = u,d = 2.7). It is confirmed that the frequencies at which the receptance vanishes
correspond to these clamped-pinned pipe eigenvalues - so that at those frequencies the
pipe oscillates like a clamped-pinned one.
The second point, the migration with u of point P towards the origin, is at
first sight paradoxical, since generally the eigenfrequencies decrease with increasing
u (see Figures 3.27-3-29), implying a softening of the system, while the migration
in Figure 4.36(a,b) indicates hardening! Recalling that the exciter is attached to the
downstream end of the cantilevered pipe, the explanation is once more related to the
characteristics of the clamped-pinned pipe, which is subject to divergence at u,d = 2.7.
Hence, at u,d an infinite force is required to hold the free end in position. At u > u,d,
the tendency to buckle will cause the pipe to press against the support, and hence the
displacement to be out of phase with the applied force; therefore, point P shifts to the
negative %e(cr,, ) axis.+
Bishop & Fawzy tested the theory by conducting forced vibration experiments, using
surgical quality silicone rubber tubes conveying water (cf. Section 3.5.6), excited sinu-
soidally via a carefully designed cross-head mechanism, based on the Scotch-yoke
principle. The force was measured by a force transducer and the displacement by a
displacement transducer, at the same point or elsewhere along the pipe. These experi-
ments illustrate the difficulties in undertaking such experiments, especially near the flutter
boundary.$ Near u,f, since ucf > u,d for a clamped-pinned pipe in all cases when the
system is excited at its lower end, experiments were practically impossible since ‘it was
extremely difficult to arrest the tube, let alone to oscillate it sinusoidally’. Hence, the pipe
was excited at a point x = 0.15L-0.4L. A number of difficulties persisted, however. For
example, for large force amplitudes, the system sometimes behaved as if composed of
two subsystems joined together at the excitation point: a clamped-pinned beam and a
pinned-free one - specifically for the forcing frequency close to that of the lower part
of the pipe; this led to ‘dynamic interference’, beating and so on.
In the end, however, some successful experiments were performed, leading to several
results of the type shown in Figure 4.36(c) - probably the first ever for an active system
so close to the flutter boundary. Here the inverse direct receptance is plotted, so that
at resonance the curve goes through zero. These curves are in terms of raw measure-
ment quantities: a rotameter reading R, related to the flow velocity by U = 1.185R x
10-6/A (mls), where A (m2) is the internal cross-flow area of the pipe; and a frequency
factor, f, equal to 480 times the oscillation frequency in Hz. Unfortunately, the cross-
sectional dimensions of the pipes are not given; hence, u and w cannot be computed, and
these results cannot be compared with the theory. The reason given for not presenting a
comparison with theory is that dissipation, always present in the experiments, has been
ignored in the theory.§
Nevertheless, for the experimental system, Figure 4.36(c) shows that for flutter, 104 <
R,f < 106 and 1340 < fcf < 1390 approximately. This demonstrates that it is feasible

+The value of Ucd in this case is too close to the uc,= = 2.749 for the clamped-free system. Bishop &
Fawzy present another calculation with CY. - 1, = 0.203 and y = 5, for which ucj = 6.07, while UCd = 4.74.
J-.
The receptance curve passes through the ongin for u between 4.6 and 4.8, in agreement with the explanation
given.
*As expressed by the authors with exquisite British understatement: ‘it has to be said that the study of a
resonance test on an active system near an instability boundary is not easy’.
§One may nevertheless suspect, since some comparison, even with this limitation, would have been useful,
that quantitative agreement cannot have been flattering.