Page 206 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

188 SLENDER STRUCTURES AND AXIAL FLOW

where
M
L = N1, = ((M +;)L3N)-'I2 t, p=- M +m'

For e = k, one may consider the articulated system to be a physicalZy discretized
version of the continuous one, with the flexibility of the latter lumped at the mid-point
of each I-length segment and equal to k = EZ/1 - cf. Goldstein (1950; Chapter 11). It
is the transition from the low-N discrete system to the continuous one that is the main
concern of Paldoussis & Deksnis' work.
The dimensionless eigenfrequencies of the articulated system are compared with those
of the continuous one,t first at u = 0, for increasing N. As expected, for N = 2 or 3,
the two sets are appreciably different; with increasing N, however, they converge quite
rapidly. Thus, for N = 10 the lowest five modes in the two sets are within 2%; for N = 20
within 1%, for y = 0; and only slightly less close for y = 10 [see table and figures in
Pai'doussis & Deksnis (1970)l.
Then, the dynamical behaviour of the system with flow is investigated for various N.
Figure 3.79(a,b) gives results for y = 10 and 100. It is seen that for y = 10 stability is lost
by flutter, no matter what N is - although the Argand diagrams show that divergence
is possible at u =. ucf. An interesting observation (cf. Sections 3.5.4 and 3.5.5) is that
for sufficiently low N, no S-shaped jumps are manifested in the curves, Finally, from
the results for N = 8 it is clear that, for sufficiently high N, the stability curve of the
articulated system approaches that of the continuous one; since convergence in the lower
eigenfrequencies is better than in the higher ones, agreement between the N = 8 discrete
and the continuous system is better for lower 0 (cf. Section 3.5.4).
The situation depicted in Figure 3.79(b) for y = 100 is more complex. It is seen that
(i) for N = 2, the system loses stability by flutter only if p < 0.195, and by divergence
for higher f?; (ii) for N = 3 only flutter is possible; (iii) for N = 4 and 8, both divergence
and flutter are possible but ucd > ucf, the difference between the two stability bounds
being much larger for N = 8.
Indeed, observing the trend with increasing N in Figure 3.79(b), it is reasonable to
suppose that u,d -+ 00 as N + 00. This resolves the paradox that, whereas for the artic-
ulated system divergence is possible (and in some cases stability is lost that way), for the
continuous system no divergence can occur. These same results explain the same paradox
as expressed by Benjamin (1961b): that in some cases, divergence is possible with water-
flow but not with air-flow. From Figure 3.79(b) we see that, for N = 2, stability is lost
by divergence when f? = 0.2 or higher and by flutter when f? 2 these two values
off? being typical for water- and air-flow experiments respectively.
The non-occurrence of divergence for N = 3 is explained, phenomenologically at least,
in Figure 3.80. For even values of N, there generally is a mode (typically the first), which
crosses the origin from positive to negative 9m(w), the classical divergence path. In some

+The eigenfrequencies of the continuous system have themselves been obtained from a discretized (Galerkin)
model, unless y = 0 - see Section 3.3.6; however, the discretization in this case is malyrical rather than
physical.