Page 175 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 157

Finally, I/L was varied systematically and the critical flow velocities for flutter or
divergence was obtained and plotted versus l/L, as illustrated in Figure 3.60, where they
are compared with theoretical results (apparently with dissipative forces ignored). It is
seen that theory and experiment are in excellent agreement.
It is of interest that if the system lost stability by divergence, then, provided 1/L was
close to 1,/L [Figure 3.58(b)], flutter about the buckled state was observed to occur. On
the other hand, if stability was lost by flutter, limit-cycle oscillation persisted at higher
flows, ‘and the tube does not buckle’; but it is not clear whether any asymmetry in the
motion takes place which might be taken as evidence of a coexisting divergence.

3.6.2 Cantilevered pipes with additional spring supports
As we have seen in the foregoing, cantilevered pipes lose stability by flutter, whereas
pipes supported at both ends do so by divergence. It was of interest, therefore, to study
‘intermediate’ support conditions, as initially done by Chen (1971a) [and later, appar-
ently independently, by Becker (1979)], who examined the dynamics of the system of
Figure 3.61(a). Physically, one would expect that for a very weak spring-constant K, the
system would behave essentially as a cantilevered pipe; for sufficiently large K, however,
the system would approach a clamped-pinned one. This, in fact, is what is obtained.
The dynamics of the system (neglecting gravity, dissipative effects, etc.) is governed
by equation (3.1), or in dimensionless form by (3.76), and the same boundary conditions,
except that the fourth, related to the shear at the downstream end, EZ(a3w/ax3) = 0, is

........
.......... .................................................... ......... --. ... _______............. .____... .....
...
.....
ir

(a)

. L .
(C)
Figure 3.61 Various types of additional spring supports for cantilevered pipes conveying fluid:
(a) translational spring at the downstream end, x = L; (b) translational spring at x = 1 < L;
(c) translational and rotational springs at x = L.