Page 139 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 121

For a typical pipe used in the experiments (Pai’doussis 1969, 1970) and Q = Ion, one
finds (T 2 lo-’ or less. The effect of this is clearly very small as compared to, say,
hysteretic damping with p - 6(10-2), since in the equation of motion ,u is multiplied
by Nevertheless, the effect of an artificially large (T on stability as investigated by
Pai’doussis (1963) and Gregory & Pai’doussis (1966b) is of interest; in these calculations
the whole of the observed damping in the first and second mode of one of the pipes used
in the experiments is assumed to be entirely due to (T (which, of course, cannot be so),
yielding (T = 0.23 and (T = 1.42, respectively.* As seen in Figure 3.35, viscous damping
with (T = 1.42 destabilizes the system only for B > 0.55. With (T = 0.23 this occurs for
#I > 0.60, while for 0.3 < /3 < 0.6 the critical flow velocity is less than 1% higher than
for the undamped case. The critical frequency, wCf, is reduced in almost all cases.
The effect of very large values of (T is examined by Lottati & Kornecki (1986). Such
large cr would arise if the pipe were immersed in water or a more viscous fluid (but in
that case m, the pipe mass, must be presumed to include the fluid added mass). As shown
in Figure 3.36, cr is stabilizing for /3 5 0.5 approximately, as in the foregoing, but for
B = 0.8 it is destabilizing. with an interesting ‘negative jump’ in the curve.

14 - -I

12 -

-
2 IO p =os

1
6k&
p =O.l

4
0 2 4 6 8 IO
U
Figure 3.36 The effect of large values of viscous damping, 0, on the critical flow velocity for
flutter, ucf. of a cantilevered pipe for various ,!l (Lottati & Kornecki 1986).

Another interesting dynamical feature of nonconservative systems is related to the non-
smooth variation in the critical load as damping is varied from vanishingly small to zero,
as first discussed in general terms by Bolotin (1963). This has been studicd cxtcnsively for
two-degree-of-freedom articulated columns [looking like the pipe system of Figure 3.1 (d)
but without flow] subjected to compressive follower loads (Hsmann & Bungay 1964;
Henmann & Jong 1965, 1966).S Such systems lose stability either by divergence or by

‘See Sections 3.3.5 and 3.3.6.
tThese values correspond to p = 0.065 and are computed via m = Afp and u = Azp, respectively.
“See also Section 2.1.5.