Page 132 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

114 SLENDER STRUCTURES AND AXIAL FLOW

the first-mode shape contains appreciable second-mode content, the second-mode shape
third-mode content, and so on. Nevertheless, the present appellation is clearly a reasonable
one. Another important point is that, similarly to the pipe with supported ends, these are
not stationary, classical modes with fixed nodal points, but contain appreciable travelling
wave components, to be discussed with the experiments in Section 3.5.6.
The critical flow velocity, ucf as a function of #? is shown in Figure 3.30;+ there
exists a similar curve for the corresponding frequency at u = u,f, labelled wcf - see
Figure 3.35. It is clear that, ucf depends strongly on 6. Furthermore, the ucf and wcf
curves contain a set of S-shaped segments. By referring to Figure 3.28 it is recognized that
they are associated with the instability -restabilization-instability sequence discussed in
the foregoing; hence, in Figure 3.30, the negative-slope portions of the curve correspond
to thresholds of restabilization. If an experiment could be conceived in which the material
damping is zero and #? could be varied in very small steps, then around these points there
would be ‘jumps’ in u,f; e.g. for #? in the vicinity of 0.69, from u,.. 2: 11 to ucf 2: 12.8
for a very small increase in #?. The values of #? associated with these S-shaped segments
of the stability curve (at #? 2: 0.30, 0.69, 0.92) will be found to be associated with a large
number of perplexing linear and nonlinear characteristics of the system - in the sense of
acting as separatrices for differing dynamical behaviour. Yet, the origin of their existence
is not fully understood (see Section 3.5.4). As #I -+ 1, more and more S-shaped jumps
are encountered. Mukhin has shown that for #? = 1 no flutter solution may be possible,
i.e. ucf 4 00 (Mukhin 1965; Lottati & Kornecki 1986).

Figure 3.30 The dimensionless critical flow velocity for flutter, urj, of a cantilevered pipe
conveying fluid, as a function of B, for y = 01 = (T = k = 0 (Gregory & Paldoussis 1966a).
Nevertheless, it appears that these jumps are related in some way to mode content.
This is made clear by Figure 3.31 in which, in addition to results obtained by the method

+Numerically, such curves are computed by determining fi for each assumed ucj, rather than vice versa