Page 188 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

170 SLENDER STRUCTURES AND AXIAL FLOW

compression was provided by an air jet, issuing from a pair of nozzles affixed to the
beam at a slight angle, so as to avoid interaction with it; the compressive reaction force
was towards the clamped end. The air was supplied via pairs of light rubber hoses, one
vertically above and the other below the blade. Despite the obvious difficulties associated
with minimizing the effect of the supply hoses, excellent qualitative and to some extent
quantitative agreement with the theory of Figure 3.70(c) was obtained: for tS > 0.45
divergence was observed, while for tS < 0.45 flutter was observed.
In the case of a cantilevered beam with a tangential end-load at the free end, representing
Beck’s problem (Section 3.2.2), there is no simple way of minimizing the effect of fluid
supply lines. Nevertheless, a successful experiment was conducted by Sugiyama et al.
(1990, 1995) by attaching a solid-fuel rocket to the free end! The aluminium cantilever
(section: 6 x 30 mm, L = 800- 1400 mm) weighed 0.4-0.7 kg. The motor was much more
massive, - 14 kg, and could supply about 390 N force for 4 s. Hence, special techniques
had to be developed for deciding whether a damped or amplified oscillation occurred
from only a few cycles of oscillation in the period over which the rocket supplied full
thrust. Also, not only the mass but the moment of inertia of the motor had to be taken
into account. Agreement of experiment with theory is excellent, provided dissipation is
ignored; once taken into account, viscoelastic damping in the column (a = 5 x low4)
is found to diminish the theoretical critical thrust by a factor of 2 as compared to the
undamped system, thus rendering agreement apparently rather poor. However, once the
criterion ‘for stability in a finite time’ (Leipholz 1970) is used, the two sets of theoret-
ical results come very close to each other, thus leading to very good agreement with
experiment.

3.6.6 Pipes with attached plates

One such system, depicted in Figure 3.71(a), is considered by Herrmann & Nemat-Nasser
(1967) as part of a series of studies on the stability of nonconservative mechanical systems.
It consists of a thin plate or I-section, with two pairs of flexible pipes attached to it and
conveying fluid. This system can execute both flexural transverse motions and torsional
motions [cf. Nemat-Nasser & Henmann’s (1966) work on the same structural system
subjected to a follower load], and it is in the study of the latter that lies the main contri-
bution of this work.
The equation of motion of the system for flexural transverse motion is the same as
before, equation (3.1), except that 2M replaces M, which now is the mass per unit length
for eachpair of pipes. For torsional motions, adapting Benjamin’s statement of Hamilton’s
principle to suit, Henmann & Nemat-Nasser (1967) obtained the following equation of
motion and boundary conditions:

+
+
a29
a49
EC, - [2MU2r2 - GJ]- a29 +MUh2 - (mr2 + iMh2)- a29 = 0; (3.128)
ax4 ax2 axat at2
(3.129)