Page 280 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 26 1

contention that parametric resonance regions calculated with = 0 are representative of
cases with effectively any value of B.

4.5.6 Two-phase and stochastically perturbed flows
As already mentioned, practical interest in this work has been associated with possible
excitation of piping by pump-generated pulsations. A different and interesting applica-
tion was studied by Hara (1977, 1980) in connection with two-phase flow in piping, in
the 'slug-flow' regime, where the flow is essentially composed of alternating liquid and
gaseous slugs.
Clearly, in this case it is not the velocity that is varying with time but the mass per unit
length; moreover, the time variation is more like a square wave than a harmonic function.
This is the phenomenon accidentally discovered in the experiments with piping aspirating
flow, as shown in Figure 4.1 l(b). Hara's experiments, involving a 2.2 m long horizontal
simply-supported pipe were in fact conducted with air-water mixtures simulating true two-
phase flow. Parametric resonances were found for w/w, E 0.65, 0.95 and 1.94, where
w is the frequency associated with 'slug arrival times'; these ratios are remarkably close
to the theoretically expected $, 1 and 2. The strongest excitation in this case was for
w/w, = 1 rather than 2; this is explained as being due to additional two-phase forced
excitation when w = wn .
Finally, the case of a pipe conveying stochastically perturbed flow was studied
by Narayanan (1983), Ariaratnam & Namachchivaya (1986b) and Namachchivaya &
Ariaratnam (1987). By assuming the intensity and correlation time of the stochastic
perturbations to be small (broad-band spectrum), the problem is transformed into a Markov
process, and solutions are obtained by stochastic averaging. It is found that the amount
of damping necessary to ensure stability depends only on those values of the excitation
PSD near twice the eigenfrequencies and near their sums and differences.

4.6 FORCED VIBRATION

There are two aspects of forced vibration of pipes conveying fluid worthy of discussion.
The first is the physical aspect, which sheds further light onto the dynamics of the system,
and the second is related to the analytical techniques which can be used to obtain the
forced response of such systems. These will be dealt with separately in what follows.

4.6.1 The dynamics of forced vibration

Let us consider a pipe subjected to an arbitrary harmonic force field, such that it is
governed by an equation of the form

(4.76)

in which %(v) is given by equation (3.70). By means of Galerkin's method, this may be
discretized into
Mq + C(u)q + K(u) q = F eior, (4.77)