Page 281 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

262 SLENDER STRUCTURES AND AXIAL FLOW

where the matrices C and K are functions of u. The steady response of the system is
written as q = Qexp(iwt) and hence equation (4.77) leads to
[K(u) + iwC(u) - w2M]Q = S(u, iw)Q = F. (4.78)

Hence, we may define the direct receptance (Bishop & Johnson 1960; Bishop & Fawzy
1976) at any generalized coordinate qj as the generalized displacement at that coordi-
nate due to a generalized force of unit amplitude and frequency w applied at the same
coordinate; then, application of Cramer’s rule to equation (4.78) shows that the direct
receptance ajj is given by
2N-2

(4.79)

e= 1
where the A, are the 2N complex eigenvalues of S associated with resonances of the
system. The A, are eigenvalues of S when coordinate qj is locked and are associated with
antiresonances. The treatment and the results to be presented are taken from Bishop &
Fawzy (1976), in terms of plots of receptance and its inverse, the inverse receptance,? a
form of mechanical impedance involving displacement rather than velocity. The motiva-
tion behind this study is to gain understanding useful in the dynamical testing of aircraft
near the flutter boundary.
Typical results for a vertical cantilevered pipe fitted with an end-nozzle are shown in
Figure 4.36, for the direct receptance at the free end, all, which relates the response
at 6 = 1 to the excitation at the same point. The system is discretized by a Galerkin
scheme with N = 4, and so four modes are involved; the critical flow velocity for flutter
is ucf = 2.749.
The four circular loops in Figure 4.36(a) correspond to the four degrees of freedom
of the discretized system, which are traced by the solution as the forcing frequency w is
increased from zero (point P). The real parts of the eigenvalues of the system, (-2.28 f
5.46i), (-1.41 f 19.19i), (-1.77 f 58.22i), (-1.74 f 117.5%) are close to the minima
of Sim(a,). Every point of a receptance diagram represents the sum of the responses in
all the modes. This sum may be such that the curve intersects the positive real axis, as
in Figure 4.36(a,b); at the frequency corresponding to such an intersection, no work is
done by the driving force. Intersection with the negative real axis is also possible, again
signifying no work done by the driving force, but for a different reason: this intersection
occurs only for u > u,f, at w = wcf - see Figure 4.36(b), where the intersection for
u = 3.0 occurs at w = 15, beyond the confines of the figure.
It is also noted, in Figure 4.36(b), that as u -+ ucf the diameter of the first loop of
the receptance curve diminishes, while that of the second one, associated with the flutter
mode tends to infinity; thus, as w is increased, the receptance shoots off to infinity through
the first quadrant, which seems reasonable on physical grounds. At uCfr the pipe tends to

+Bishop developed the concept of receptance into a powerful tool for the analysis of all conceivable aspects
of vibration of mechanical systems (Bishop & Johnson 1960). An anecdote making the rounds, in the U.K. at
least, in the early 1960s is that one day the following sign was affixed (by a frustrated student, probably) on
the door to this office: No admittance without receptance!