Page 225 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 207

Figure 4.5 Argand diagram of the complex eigenfrequencies of a conical-conical cantilever
conveying fluid and immersed in quiescent water, with and without dissipation in the pipe material
taken into account (t= 22. S= 0.5, Be= 0.03, B, = 0.016, y= 16.47, y, = -0.08. ye = 1.7, c,, = 0):
-
~ _ . pd = ud = 0 (undamped); -, p,, = 0.20, ud = 0.04 (damped) (Hannoyer & Pai'doussis
1979a).

the reduced flexural rigidity of conical-conical pipes and the diminished gravity effect
(p > pe in the case presented). Second, there are two flutter instabilities close to each
other (in terms of ul). Comparing the undamped and damped systems, there is little
evident similarity in the root loci. The differences are more apparent than real, however.
Although different modes become unstable in the two cases, the critical flow velocities
are not too different. It is recalled that this being a nonconservative system. dissipation
can actually destabilize it.
Figure 4.6(a) shows that, for tubular cantilevers of constant cone angle Be (and similarly
for PI), varying E by cutting pieces off the free end entails variations in Be (and similarly
in B,) - see equations (4.24). Figure 4.6(b) shows the effect of the slenderness ratio
E = L/D,(O) on the critical flow velocity u,, for a conical-conical pipe with constant j3,
and De. (It is noted that as 6 changes, the corresponding ae and a, also change.) It is
seen that with increasing slenderness the system loses stability at a lower flow velocity.
This contrasts with the case of uniform pipes where u,, is almost independent of e. Of
course, the more slender the system, the lower is the diinensional critical flow velocity,
in any case (vide definition of u,: since u, cx U,(O)L, as L increases, U,(O) decreases for
a constant u,); but in conical systems this effect is greatly amplified. Finally, the effect
of the surrounding fluid is seen to be the same as for cylindrical-conical pipes.