Page 264 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 245

which substituted into equation (4.70) leads once again to a matrix equation equivalent
to equation (4.73) but with a vector {. . . u4, u2, bo, b2, 64. . . .}T, and thence to a vanishing
determinant which yields the boundaries of secondary resonances.
It has been found that, typically, the resonance boundaries associated with the first
mode of pinned-pinned and clamped-clamped pipes can be determined with adequate
precision by using the k = 1 approximation for the principal primary region and the k = 2
approximation for the main secondary one, and truncating the Galerkin series at N = 2.
For cantilevered pipes, on the other hand, it was found necessary to use typically k = 3
and k = 2, respectively, and N = 5. Generally, the higher the flow velocity, or p, or the
mode number, the higher N and k have to be. Most of the calculations presented here
have been performed with k = 3 or 2 and N = 5.
Typical results for a clamped-clamped pipe are shown in Figure 4.23, showing the
effect of uo and damping (a) on the principal resonance (w 2 2wl) and the main secondary
or ‘fundamental’ resonance (w 2 wl), where w1 is the first-mode eigenfrequency or the
real part thereof for any given UO. Resonance oscillations exist within the quasi-triangular
regions, while the system is in its trivial equilibrium state outside. It is noted that, as the
flow velocity is increased, the regions of parametric resonance are displaced downwards,
which reflects the decrease in the first-mode eigenfrequency 01 with increasing UO. Had
w/w1 been plotted in the figure instead of w/wol - where 001 is the zero-flow-velocity
value of w1 - then all the curves would have been centred about w/w1 = 2 and 1. For
a = 0, the parametric resonance regions exist even for p = 0, but at that point have zero
width; for 0 > 0, there is a minimum p for each case, below which no resonance is
possible.
It is also noticed that the resonance regions become broader with increasing flow, and
that the effect of damping is correspondingly attenuated. Thus, at uo = 3, the primary
resonance region is very little affected by fairly large dissipation (a = OS), and the
system is subject to parametric resonance even when p < 0.1.
Figure 4.24 shows the effect of ,6 on parametric resonances. It is seen that with
increasing B the resonance regions become broader; it is perhaps worth mentioning that
with the equation of motion used by Chen the width of the unstable regions appears to be
independent of B. The unstable regions are also displaced downwards with increasing p,
which reflects the lowering of the natural frequencies as ,6 is increased for this particular
flow velocity; this is because w1 for B1l2 = 0.8 is lower than for = 0.2’ at uo = 2
(to the scale of this figure the difference is more clearly seen in the principal resonance
regions).
It is noticed that the lower boundary of the principal primary resonance associated
with pl/* = 0.8 in Figure 4.24 is not straight; this is a characteristic of cases where the
resonance boundary concerned is close to another resonance region (which is not shown
here for the sake of simplicity). This also applies to the primary region for uo = 3 in
Figure 4.23.
As seen in the foregoing, the dynamics of pipes with supported ends subjected to
pulsating flow is, after all, not too dissimilar to that of columns with a harmonically
perturbed conservative end-load. Nevertheless, a significant difference is that gyroscopic
effects, operative in the case of pipes, and hence B, affect the location and particularly
the extent of the resonance regions in the {p, wbplane.

+Some of Paidoussis & Issid’s (1974) calculations were conducted with specific values of rather than
,S, for direct comparison with Chen’s (1971b).