Page 117 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 99

drop and pa is the atmospheric pressure, both of which do not enter the equation of
motion (Section 3.3.4).
It is clear that the pressure term acts in the same way as the MU2 term, and hence it
is not surprising that, given a sufficiently high level of pressurization, divergence may be
induced by pressure alone -just as it may do by compression alone, i.e. for T < 0 and
sufficiently large. Physically, one may think of the pressurization as being produced by
floating pistons acting on both sides of a segment of the pipe, as shown in Figure 3.16(b).
An easy experiment to demonstrate pressure-induced divergence consists in joining two
rigid pipes with a straight rubber hose and then connecting the other ends of the rigid
pipes to the same regulated pressure supply. As the pressure is increased, eventually the
rubber hose buckles. The same effect may be obtained if, instead of a rubber pipe, bellows
are used [Figure 3.16(c)].
The effect of pressurization may appear to be obvious and hence trivial. Nevertheless,
consider the following two systems: (i) a pipe with an axially sliding end under pressur-
ization p and tension T, with zero flow [i.e. as in Figure 3.16(a) but with axial sliding
permitted and U = 0; 7; being provided by a weight acting through pulleys], and (ii) a
closed tube pressurized to p. In both cases, the equation of motion is

+
a4w a2w - a2w a2 w
+
EI - FA - T - (m + M)- = 0. (3.99)
-
ax4 ax2 ax2 at2
In case (i), p and T are independent of each other, and 7 may possibly be zero. In
case (ii), however, in the linear limit, PA = T and the net effect of pressurization is nil.
This, nevertheless, has not stopped an intrepid would-be inventor from obtaining a patent
for stiffening hollow rotors against whirling by 'pressurization-induced tensioning', as
shown in Figure 3.17(a,b), while conveniently forgetting about the destabilizing effect
of pressurization illustrated in Figure 3.17(c). Thus, the inventor took into account the

w w

............. ...................................... .._.
............ .................. ..............
.....
.%. !
DsI Y rm'

Figure 3.17 (a) The fallacious patent for delaying the onset of whirling through pressuriza-
tion-induced tensioning; (b) the stabilizing effect of pressurization-induced tension, T; (c) the
destabilizing effect of pressurization, 7.