Page 272 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 253

Following the trend just described in reverse, it is clear that one might expect the
simple and combination parametric resonance regions to go on increasing in extent with
increasing ug beyond ucf. Nevertheless, the results in Figure 4.29(b) for u > ucf are both
startling and interesting: the ‘combination resonance region’ corresponding to quasiperi-
odic motions has increased quite dramatically,+ virtually covering all the previously stable
area; nevertheless, it is of interest that a small region remains where the system is stable
in pulsating flow, whereas in the absence of pulsation it would not be!

4.5.3 Experiments

Experiments were conducted with an apparatus and elastomer pipes similar to those in
Paldoussis’ (1970) steady-flow experiments, described in Section 3.5.6. However, the
apparatus was modified to enable the addition of a harmonic perturbation component to
the mean flow via a ‘plunger pump’ driven by a variable-stroke reciprocating mechanism,
connected to a variable-speed drive, as shown in Figure 4.30. Thus both the amplitude
and the frequency of the imposed harmonic perturbation could be varied; the frequency
range was 1 to 16 Hz. Flexible bellows were inserted to isolate, as much as possible, the
test pipe from vibration arising from the reciprocating mechanism and drive.
The flow velocity was measured just upstream of the elastic pipe by a hot-film
anemometer. Traces of the periodically perturbed flow showed that the plunger pump
gave almost truly sinusoidal perturbations to the flow, so that the flow velocity
could be represented by U = UO( 1 + p cos at). Both p and the mean flow velocity,
UO, were determined by the hot-film anemometer. Experiments were performed with
clamped-clamped and cantilevered pipes. The lower clamped end in the former case was
such as to permit axial sliding. The apparatus and the experiments are described in greater
detail in Paidoussis & Issid (1976).
In general, the dynamical behaviour of clamped-clamped pipes is similar to that of
columns subjected to periodic end-loading. The dominant resonances are associated with
the first mode. The secondary parametric resonance was difficult to pin-point, particularly
for small p and for U far removed from ucd; the main reason being that there was always
a small-amplitude vibration of the pipe at the pulsation frequency, transmitted either
mechanically or through the fluid, which proved impossible to eliminate completely. A
stable region usually separated the secondary from the primary resonance, except for
high p and UO close to Ucd, where it was observed that the frequency of pipe oscillation
changed directly from a to +a. When the pulsation frequency was increased beyond the
first-mode primary region, resonances associated with the second mode were observed.
In some cases combination resonances or mixed resonance regions were encountered. In
general, the observations are in qualitative agreement with theory.
A quantitative comparison between theory and experiment is made in Figure 4.31. It is
evident that if the theoretical curves were shifted downwards, agreement with experiment
would improve substantially; this would indicate that the theoretical frequencies may be
incorrect and leads one to suspect that the lower sliding clamped support was not perfect;
in fact, it was slightly loose to permit unimpeded axial movement. It is also noted that

‘The notation of ‘combination resonance’ for ti > tic! is inappropriate. ‘Quasiperiodic’ is much better to
denote the presence of two incommensurate frequencies in the response.