Page 148 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

130 SLENDER STRUCTURES AND AXIAL FLOW

Even though each Bi - 6, varies smoothly with ,9, as it crosses ~TC and x, cos(6i - 6,)
and sin(@ - 6,) respectively, change sign - with attendant abrupt changes in the energy
expressions. For example, 02 - 0, > n for low j3, it crosses n at j3 2: 0.24, and then
decreases sharply to - ;IT near PSI; hence, sin(& - Q3) becomes positive for j3 2 0.24
and then increases precipitously near PSI, while cos(& - 03) becomes small. Similarly,
03 - 04 crosses n at j3 2: 0.6 prior to dropping to - :IT at ,9s2 (not shown), and 0, - Qs
does the same at j3 2: 0.83 and Bs3, respectively.
Thus, the answer to the existence of the jumps lies in the modal content of the flutter
mode and the phase differences between its component parts. Moreover, at each jump,
there is a transition zone in which three possible mixes of modes are feasible with different
uCfr one low, one middle (unstable), and the other high (e.g. at j3s2, for 0.65 < j3 -= 0.69
approximately), but as j3 is increased sufficiently, only the one with the higher ucf survives.
As a cautionary note it should be mentioned that, in the foregoing, the travelling
wave component in the mode shape was ignored, whereas in reality (see Section 3.5.6)
8; E 6;(4) generally. Clearly, this also must play a role.

3.5.5 On destabilization by damping

To those with a structural mechanics background the very statement that dissipation, i.e.
energy loss, may make a stable system unstable might appear paradoxical. In gyrody-
namics, however, this effect has been known for a long time (Den Hartog 1956; Crandall
1995a,b) - certainly since Thomson (Lord Kelvin) and Tait demonstrated in 1879 that
damping in a ‘gyroscopic pendulum’ can be destabilizing. A gyroscopic pendulum is an
‘up-standing’, up-turned pendulum to which spin has been added so as to stabilize the
statically unstable system. Stability can nevertheless be destroyed if damping is added,
no matter what the spin-rate (Crandall 1995a).+
The effect is not surprising to fluid mechanicians either. For instance, they know of
Reynolds’ two hypotheses, formulated in 1883, stating that: (a) in some situations the
inviscid fluid may be unstable, while the viscous one is stable, so that the effect of
viscosity is purely stabilizing; (b) in other situations the inviscid fluid may be stable while
the viscous one unstable, indicating that viscosity is destabilizing (Drazin & Reid 1981;
Chapter 4). Examples may be found in shear flow instability (Tritton 1988; section 17.6),
arising in 2-D velocity profiles with a discontinuity (e.g. a jet or a wake) or in profiles
with no point of inflection (e.g. a pipe flow or a boundary layer with a favourable pres-
sure gradient). In the first type of flow, viscosity is primarily stabilizing, preventing the
Kelvin-Helmholtz$ type instability at low Reynolds number (%e). In the second type
of flow this instability does not occur, but viscosity can cause instability of a different
kind. Viscosity now plays a dual role: stabilizing at low Re, but destabilizing at high Re.
In aeronautics the destabilizing effect of damping has been known for a long time, in
relation to aircraft flutter, and has been carefully studied (Broadbent & Williams 1956;
Done 1963; Nissim 1965); also, in satellite dynamics this untoward effect of dissipation is

?Crandall shows that, although ordinary damping is always destabilizing, ‘rotating damping’ is not, thus
explaining how in practice such pendula are stabilized at high spin-rates.
*The Kelvin-Helmholtz instability is the premier example of shear flow instability in profiles with a point of
inflection. It may be demonstrated theoretically by a flow in which the upper half-plane has a uniform velocity
to the right, and the lower half-plane to the left. If waviness develops in the interface, the pressures generated
(via Bernoulli’s equation for inviscid flow) tend to exaggerate the waviness, leading to instability.