Page 80 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I 63

similarly for the pipe problem, as U is increased. Hence, it is obvious that the frequen-
cies of the system must decrease with increasing U. At u,, the lowest (fundamental)
frequency vanishes as the stiffness in that mode vanishes. In the linear sense, the original
straight configuration becomes unstable, and all adjacent deformed states in that mode
become possible equilibria. In the nonlinear sense, a pitchfork bifurcation takes place,
the original equilibrium is unstable and two stable equilibrium states, one on either side,
emerge - defined by the nonlinear forces acting on the system, as will be demonstrated
in Chapter 5.
However, the analogy of the pipe with supported ends to the column with the same
boundary conditions should not be carried too far, because the latter problem is purely
conservative, while the former is gyroscopic conservative. As will be shown later, despite
the fact that the gyroscopic (Coriolis) forces do no work in the course of free oscillations,
they do exert important influence on the overall dynamical behaviour.’
Finally, it should be mentioned that, according to linear theory, there should be no differ-
ence in the dynamics of systems (a) and (b) of Figure 3.1. In physical terms, however, it
is obvious that buckling implies lateral deflection of the pipe. In system (b), once u 2 u,,
the pipe may develop large static deflection since it is axially unrestrained. In system (a),
on the other hand, where axial sliding of the lower end is prevented, lateral deflection is
associated with axial extension of the pipe; this implies stretching and hence the gener-
ation of a deflection-related axial tension, a nonlinear effect. In practice, this means that
the zero-frequency state is never achieved, as will be discussed further in Section 3.4.

3.2.2 Cantilevered pipes

As will be shown, a cantilevered pipe conveying fluid is a nonconservative system, which,
for sufficiently high flow velocity, loses stability by flutter of the single-mode type, i.e.
via a Hopf bihrcation - see also Section 3.2.3.
The stability of cantilevered pipes conveying fluid [see Figure 3.l(c)] was first studied
by Bourrihres (1939), who examined the problem of general motions of an infinitely
flexible and inextensible string, and the special case where the string is circulating (travel-
ling) between two fixed supports; he then tackled the problem of one such string within
another, which could have flexural rigidity - this of course being equivalent to the case of
a pipe conveying fluid. He obtained the general nonlinear equations of motion, but did not
develop them fully. Then, he linearized them and proceeded to study such diverse aspects
as the difference between spontaneous and perturbation-induced instabilities (cf. Gregory
& Paidoussis 1966b), and the wave propagation characteristics; he also attempted to
predict the period of self-excited motions, and studied several other aspects of the problem,
as well as conducting experiments. On the other hand, he could not calculate the critical
flow velocity, which, unlike the case of a pipe with supported ends, requires the use of
computersi - of course, then unavailable. Bourrikes’ was a truly admirable effort, and
it is a pity that it was lost to posterity, until recently (Section 3.1). His work did not have

+In this respect, as civil servants the world over discovered long ago (and as viewers of BBC’s Yes Minisrer
have witnessed to their delight), it is not necessary to do actual work in order to exert influence; see also Lynn
& Jay (1989).
*Although Padoussis (1963). in order to check computer calculations - computers then being a relatively
new device - did do a hand calculation, thereby demonstrating its feasibility.