Page 142 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

124 SLENDER STRUCTURES AND AXIAL FLOW

sense that it imposes a systematic increase in the mode associated with instability as
j3 is increased. Also, an improvement in the correspondence between jumps and mode
changes is achieved: it works for the first jump in all cases; nevertheless, it fails for the
second jump when y = 10, and for the third jump when y = 100. Clearly something more
profound is involved.
A similar but more mathematical attempt was made more recently by Seyranian (1994),
starting from the same observation that motivated Paldoussis’ (1969) work: the ‘drawing
near’ of two mode loci (e.g. in Figure 3.29 for the second and third modes at u = 8.8125)
with increasing j3, prior to switching of the mode leading to flutter, which often occurs as j3
is varied. Seyranian argues convincingly that this ‘drawing near’ of the loci implies actual
frequency coincidence (a repeated root) at some nearby point in the parameter space - a
‘collision of eigenvalues’ in his terminology - if only an additional parameter (in this
case, other than j3 and u) is varied at the same time. This may well be true, although
Seyranian demonstrates it only for nongyroscopic nonconservative systems (e.g. for an
articulated column with a follower load). As seen in the ‘conventional’ mode-ordering
column of Table 3.2, however, there is not always a mode switch across an S-shaped
jump, nor does mode switching necessarily imply an impending jump (Figure 3.29 vis-u-
vis Table 3.2 being a case in point).
Either of these attempts, even if successful, would have given a mathematical explana-
tion rather than physical insight into the nature of the S-shaped discontinuities. A more
successful interpretation in this respect was provided by Semler et al. (1998), which also
throws some light onto the destabilizing effect of damping.
Semler et al. (1998) consider a double pendulum under zero gravity, subjected to a
follower load, P, as shown in Figure 3.38(a). The two rods are constrained by rotational
springs of equal stiffness, k, and rotational dashpots, c1 and c2. The equations of motion
are rendered dimensionless by introducing t = tdm for the time and the parameters

(3.107)

Stability is lost via a Hopf bifurcation and the critical value of 8 for flutter, Ycr =
f(y1, y2), may be derived in closed form. Figure 3.38(b) shows some results obtained
for fixed y1 while y2 is varied. It is shown, for all y1, that increasing y2 from zero initially
stabilizes the system (i.e. a higher 9 is required to cause flutter), but the trend is eventually
reversed and then y2 becomes ‘destabilizing’.
To understand the mechanism leading to this behaviour, the net energy gained by the
system over a period of not necessarily neutrally stable oscillation, T, is considered,
I’ +
8$ sin xdt - [y~$~ y2x2] dt, (3.108)

where 4 41 and x = 41 - 42 are an alternative set of generalized coordinates. AE has
the same meaning as E - z0 in (C.6). Once the equations of motion are decoupled via
modal analysis techniques (Section 2.1.2), it is possible to consider AE for each of the
two modes separately. One can thus obtain the diagram of Figure 3.39(a). It is seen that
mode 1, the ‘stable mode’ (Le. not the one associated with flutter), becomes more and
more stable as y2 is increased, being associated with progressively more negative AE.
However, mode 2, the flutter mode, becomes less stable with increasing y2; eventually,
for y2 = 0.025 [cf. Figure 3.38(b)] AE > 0 is obtained, and hence amplified oscillations.