Page 152 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

134 SLENDER STRUCTURES AND AXIAL FLOW

All measurements were straightforward, except perhaps the measurement of EZ for
rubber pipes, required in the determination of the dimensionless flow velocity, u. The
techniques utilized for this are summarized in Appendix D.
Some of the general observations on the dynamical behaviour of this system are worth
giving in some detail; they are similar to those made by other researchers in the experi-
ments to be discussed later.
Small flow velocities increased the damping of the system, and oscillations induced
by light taps close to the free end, which were still of the general shape of the first
cantilever mode, decayed much faster. At somewhat larger flow velocities the system
became overdamped and any displacement of the pipe was followed by a return to rest
without any oscillatory motion. In some cases physical contact of the free end of the
pipe with the hand, momentarily transforming the system to one supported at both ends,
caused the pipe to buckle by bowing out near the middle. When contact was broken
suddenly, the pipe returned rapidly to its position of rest, but when the hand was removed
only slowly, the pipe pressed against and followed the hand with the result that the
timid observer was soon faced with a stream of fluid directed against himself (or nearby
colleagues!) - this, as already remarked, being a demonstration of a negative-stiffness
(divergence) instability.
At still higher flow velocities, light taps resulted in heavily damped oscillation with
a form rather more like that of the second cantilever mode than the first. As the flow
velocity increased further, the system became less heavily damped until at a certain
critical velocity of flow the disturbance produced by lightly tapping the pipe grew into a
self-supporting oscillation. If no outside disturbance was introduced, the system eventually
became unstable spontaneously. This usually occurred at measurably higher flow velocities
than were sufficient for ‘induced’ instability to take place, particularly in the case of
rubber pipes. Further increase of the flow velocity beyond the stability limit resulted in
an increase in both the amplitude and frequency of oscillation.
When decreasing the flow velocity, it was noted that oscillation persisted below the
point where instability, spontaneous or ‘induced’, first occurred. This, and also the fact
that in some cases the onset of instability depended on the amplitude of the applied
disturbance, indicated that the experimental systems behaved in general nonlinearly.
The mode of deformation of the unstable system was recorded with a cinC-camera in
a few selected typical cases and a number of successive frames of the film are shown in
Figures 3.45(a-c). In general, for very small values of ,% the modal form was essentially
that of the first cantilever mode, with a small component of the second. For higher values
of p, the second cantilever mode became more prominent, and for fi > 0.3 approximately
the third mode became apparent [e.g. see frame 8 of Figurc 3.45(c)]. In all cases, the
tangent to the free end of the pipe sloped backwards to the direction of motion of the free
end over the greater part of a cycle of oscillation. This ‘dragging’ motion was predicted
to be necessary for flutter, in conjunction with the energy considerations of Section 3.2.2.
Indeed, all observations described are in agreement with the theoretical predictions
of Sections 3.5.1 and 3.5.2. However, two additional comments should be made. First,
the dynamical behaviour of the system is, to some extent, nonlinear - as noted
above - suggesting that the Hopf bifurcation is subcritical. Second, according to linear
theory, once instability is developed, the amplitude should increase without limit; of
course, once the amplitude becomes large, nonlinear forces come into play, and in this
case evidently their net effect is to limit the amplitude, thereby establishing a limit cycle.