Page 83 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

66 SLENDER STRUCTURES AND AXIAL FLOW

A strange characteristic of this system is that, at high flow velocities but before the onset
of flutter, supporting the downstream end of the cantilever by one's finger or a pencil
causes it to become unstable by divergence (Benjamin 1961b; Gregory & Pdidoussis
1966b). So, here is a case where added support causes instability! If one tries to remove
the finger or pencil slowly, the pipe follows! This shows clearly and physically that the
divergence is a negative stiffness instability. This also gives rise to an interesting paradox,
discovered by Thompson (1982b) and elucidated in terms of the strange black box of
Figure 3.3(a,b). As more weight is placed on the scale, the scale goes up.+ What could
be in the box is shown in Figure 3.3(c). The phenomenon is nonlinear and its discussion
properly belongs to Chapter 5 (Section 5.6.1); it has nevertheless been outlined here to
whet the appetite, so to speak, for the many interesting aspects of the nonlinear behaviour
Black A-
of this system.

r box Black 4 +"

Figure 3.3 Illustration of the negative stiffness mechanism of a buckled pipe conveying fluid,
analysed by Thompson (1982b).

The stability of this system was linked to the classical nonconservative problem of a
column subjected to a tangential follower-type load at the free end,: known as Beck's
problem or Nicolai's paradox, by Nemat-Nasser ef af . (1966), Henmann (1967) and
Herrmann & Nemat-Nasser (1967). Beck's problem may be summarized as follows
(Bolotin 1963; Ziegler 1968). As already suggested in Section 3.2.1, the stability of
a conservative system may be assessed statically, i.e. by ignoring the time-dependent
forces; e.g. in the case of a column with supported ends or of a cantilevered one with
a compressive load of fixed orientation. The same may be attempted - as first done by
Nicolai in 1928 - for a cantilevered column with a follower load, i.e. a compressive
load with fixed orientation relafive to the column, notably a load always tangential to
the free end (as in Figure 2.2). The paradoxical result is then obtained that the system

~
'A second, but dynamically trivial paradox is that the black box of Figure 3.3 is in fact white!
*By the analogy between equations (3.1) and (3.3) it may easily be shown that the equivalent of (3.5) is
AW = -Ps$ [(aw/ar) (aw/ax)]k dt # 0, since neither (aw/at)L nor (awlax), are zero for all t E [O. TI.