Page 434 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

406 SLENDER STRUCTURES AND AXIAL FLOW

resonance, in the vicinity of the Hopf bifurcation in steady flow (at u,f), are determined
by the method of alternate problems (Appendix F.6.3) and their stability assessed from the
Floquet exponents of the associated variational equations. The results, in this case also,
are discussed in terms of (i) the mean-flow velocity perturbation Q = u0 - u,f, (ii) the
detuning parameter 3, and (iii) the harmonic flow-perturbation amplitude p, where u =
u0 + cos 2wt.
The results are presented in diagrams of (i) p versus 5 for a given q, (ii) amplitude A
versus q, and (iii) A versus 6, for a = K = 1, y = 0.25, and fi = f and 5 x lop2, some
of which are given here in Figure 5.65.
In the A versus q diagrams of Figure 5.65(a-c) the Hopf bifurcation in steady flow
(for B = :) is supercritical. For a large negative 3, e.g. 6 = -0.31 as in (a), there is

-0.2-0.1 0 0.1 -0.3-0.2-0.1 0 0.1 0.2 0 0.1 0.2 0.3
(a) D 17 17

2r

0 0.3 0.6 0 0.3 0.6 0.9

'IA

-4.0 -2.0 0 2.0 4.0 -2.0 0 2.0 4.0 -2.0 0 2.0
(g) 17 (h) 17 6) 17
Figure 5.65 Response diagrams for the parametrically excited articulated system. Top: A versus
q diagrams for p = and (a) 0 = -0.31, (b) 0 = -0.1, (c) 0 = 0.4. Middle: A versus 5 diagrams
for /I = 2 and (d) r] = -0.075, (e) q = 0.188, (f) r] = 0.30. Bottom: A versus r] diagrams for
3
p = 5 x lop2 and (g) 0 = -0.3, (h) 0 = 0, (i) 0 = 1.7 (Bajaj 1984).
3