Page 129 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 111

regard to the F given in the figures. For instance, for r = 5, u,d/n = 1.23 should have
been obtained (cf. Section 3.4.2) and not 1.62.+ However, since it is the qualitative nature
of the frequency-versus-u variation that is perplexing, item (d) will be ignored here. On
reflection, neither (a) nor (b) can provide a convincing explanation, but they do point out
how demanding these deceptively simple experiments can be. Item (c), however, provides
a likely explanation. As will be shown in Chapter 6, initially curved pipes in fact do not
diverge. Thus, for semicircular pipes the reduction in frequency with flow is minimal; in
that sense, strictly according to this hypothesis, this represents an intermediate system,
behaving as a straight pipe for low u and as a curved one for higher u. Another outlook on
this is provided by nonlinear theory. As discussed in Chapter 5, the pitchfork bifurcation
is structurally unstable (in the mathematical sense), and the displacement-versus-u curve
evolves more smoothly$ in the presence of a small, or not-so-small, initial asymmetry.
This corresponds physically to a gradual exaggeration of the asymmetry as u is increased
(as observed), in contrast to the explosive divergence of the imperfection-free system.
Furthermore, since neither the initial (u = 0) nor the ‘final’ state (for u larger than the
theoretical u,d) is associated with w = 0, the frequency in-between tends to bridge these
two states without passing through zero.
Experiments were also conducted on clamped-clamped pipes by Jendrzejczyk & Chen
(1985), with no sliding permitted. They found that divergence does not occur for the
reasons already given; indeed the r.m.s. vibration amplitude was found to decrease as the
theoretical critical &d is exceeded, which was attributed to deflection-induced tensioning.
A final comment is that in all these experimental studies there has been no reported
observation of post-divergence coupled-mode flutter. Although this does not prove that
it cannot exist - especially noting that the violence of the onset of divergence makes
experimentation, at more than twice the critical flow rate, problematical - it would tend
to support Holmes’ finding, via nonlinear analysis, that pipes with supported ends cannot
flutter. as discussed in Chapter 5.

3.5 CANTILEVERED PIPES

3.5.1 Main theoretical results
The essential dynamics of cantilevered pipes conveying fluid has already been outlined in
Sections 3.1 and 3.2. Referring to the dimensionless equation of motion, equation (3.70),
it is noted that, for cantilevered systems, F = l7 = 0 always; furthermore, since the case
of time-varying flow and elastic foundations will not be considered till later, ii = k = 0
as well. Hence, the only parameters that remain to be considered for the results to be
presented in this section are the damping parameters a! and o, the mass parameter ,B. and
the gravity parameter y.
The simplest system is considered first, in which a! = o = y = 0 additionally, Le. a
horizontal system with the dissipation ignored, which thus depends only on /3. In this
case, solutions are possible via the First Method of Section 3.3.6(a) and involve no
approximations (due to Galerkin truncation, for instance). Typical results are shown in
Figures 3.27 and 3.28 for = 0.2 and 0.295, respectively. It is seen that for small ii (u < 4

‘The effect of gravity was neglected by the authors (y = 0), but in fact it is very small
*It is ‘unfolded’, in nonlinear terminology.