Page 114 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

96 SLENDER STRUCTURES AND AXIAL FLOW

another kind arises, at a slightly lower critical flow velocity, in which the two branches
of the same mode are involved rather than two different modes. We shall continue calling
this a coupled-mode flutter since, strictly speaking, the two branches on the 9m(w)-axis
should be considered as being associated with different modes, from the left-hand (not
shown) and right-hand sides of the complex o-plane - see Figure 2.10(a).
The Done & Simpson argumentation for coupled-mode flutter may be extended to
dissipative systems by supposing that, at the threshold of flutter, a sustained correction
in the contraction c may be effected by the discharging axial momentum flux, so as to
maintain a constant-amplitude motion. Thus, effectively, a sustained rate of work occurs
through axial motion, whereas the dissipation occurs through lateral motion; note also
that AW = 0 in equation (3.95) in the undamped system applies to lateral motions.
It is important to stress, yet again, that both the restabilization of the system after
divergence (e.g. in Figure 3.11) and the coupled-mode flutter are due to the gyroscopic
nature of the system, i.e. to the Coriolis terms in the equation of motion. As pointed
out by Shieh (1971) and Huseyin & Plaut (1974), purely conservative systems cannot
be restabilized after divergence ‘on their own’, but gyroscopic forces can restabilize an
otherwise conservative system, a fact known since Thomson & Tait’s (1879) work. The
possibility of coupled-mode flutter is a much newer ‘discovery’ which may be attributed
to Shieh, who illustrated its existence with an example from gyrodynamics involving a
shaft under an axial compression P, rotating with angular velocity R. The equations of
motion are
Ely”” + Py” + M(y - 2ni - LPy) = 0,
(3.97)
EZZ’”’ + Pz” + M(i + 2ny - D2Z) = 0,

in which y and z are mutually perpendicular deflections in a plane normal to the long
axis; these equations clearly bear close similarity to that of the problem at hand - cf.
equation (3.1).
Huseyin & Plaut (1974) discuss the dynamics of gyroscopic conservative systems in
general, as well as the rotating shaft and pipe systems as examples. The latter will be
discussed here briefly, partly (i) to introduce the concept of the ‘corresponding nongyro-
scopic system’ and (ii) to demonstrate the use of the so-called ‘characteristic curves’.
Huseyin & Plaut considered a two-degree-of-freedom discretization of the horizontal
system, i.e. of equation (3.1), by using the beam eigenfunctions as suitable comparison
functions. In the case of a clamped-pinned system, the results are shown in Figure 3.15
for three values of B;+ also plotted are the results for B = 0, which is the corresponding
nongyroscopic system, representing a column subjected to a load 9 = u2. The results are
plotted in the form of characteristic curves, i.e. curves of loading versus 02, namely u2
versus w2. Clearly, only u2 > 0 is meaningful, but the extension of the curves to u2 < 0
helps to show that the curves (full lines) are conic sections. In (a) it is seen that the system
is initially stable (0; > 0, o; > 0), but for u2/n2 = 2.05 (at point A) corresponding to
u,d = 4.49 [cf. equation (3.9Oc)], the first-mode locus crosses to the w2 0 half-plane,
indicating divergence in the first mode. The system remains unstable with increasing u2,

+These curves are not identical to Huseyin & Plaut’s (1974), which are quantitatively in error (Plaut 1995);
thus, the values of for each of the three distinct types of behaviour are incorrect, and so is the value of u2/n2
for point B; otherwise, the results are qualitatively similar to those in Figure 3.15.