Page 282 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

PIPES CONVEYING FLUID: LINEAR DYNAMICS I1 263

(2) w3

21
u = 2.0

-8
20
19

Figure 4.36 (a) Variation of the direct receptance all for a theoretical harmonically forced
system with u = 2 and varying values of w (nozzle area parameter a, = A/Aj = 3, j3 = 0.203,
y = 5, zero dissipation, and N = 4). (b) Same as in (a), but u = 2.2, 2.4, 2.6, 2.8 and 3.0.
(c) Inverse direct receptance at 6 = 0.15, close to the instability boundary of the experimental
system (a, = 1.27, ,8 = 0.387, y = 250) for a number of values of the flow parameter, R, and
varying frequency parameter, f, defined in the text (Bishop & Fawzy 1976).

oscillate at its critical frequency, wcf = 15. When w < w,f, energy flows from the pipe to
the driving mechanism, and the displacement leads the excitation - which is not possible
for passive systems. The phase lead continues until w = w,f, when no energy flows to or
from the driving mechanism, since all the energy required to achieve an infinite amplitude
is supplied solely by the fluid. For w > mcf, the pipe is forced to oscillate more rapidly,
and the displacement lags behind the force; hence the receptance curve is now below the
real axis.
A number of other, special and interesting features of these receptance curves are
discussed by Bishop & Fawzy, among them: (i) the vanishing of the receptance at a
finite w; (ii) the migration of the starting point of the receptance curve along the real
axis towards the origin. The first point suggests that the system may have some purely
imaginary antiresonance eigenvalues - which means that these are the resonances of
a system with the excitation point (at the downstream end of the pipe) constrained to
zero, i.e. those of a clamped-pinned pipe. Indeed, it is known that the eigenvalues of
the clamped-pinned system are purely imaginary up to a critical value (in this case