Page 346 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

326 SLENDER STRUCTURES AND AXIAL FLOW
u
Region A, 42 Region Bo

40 40 41
+

40
Region Do

LY 90 41
Figure 5.16 Two-parameter unfolding of bifurcations for the symmetric case (p4 = 0) of an
N = 2 vertical articulated cantilever near a point of double degeneracy (Sethna & Shaw 1987).

Analysis of the linear system yields the double-degeneracy conditions, corresponding
to simultaneous Hopf bifurcations in the second and fourth modes, for instance. The
system is transformed to Jordan canonical form, and then centre manifold and normal
form theory are employed to study the dynamics in the neighbourhood of the double
degeneracy. The reduced subsystem on the centre manifold is found to be governed by
the amplitude equations
+
i-1 = r1(& + r: + bri) + O(lr15), i-2 = ~(82 cr: + dr;) + O(lrI5), (5.111)
and similar equations for the phase angles 81 and 82. In (5.111), 61 and 82 are the incre-
ments in the real parts of the eigenvalues (equal to zero at the critical points), which
can be related to the bifurcation parameters and x, associated with u and y, respec-
tively (Appendix F.5). Thus, this is a codimension-two analysis. System (5.1 11) has been
analysed by Guckenheimer & Holmes (1983), and nine topologically different classes of
solutions are found to be possible.
Langthjem found several of these solutions, involving different kinds of periodic and
quasiperiodic motions. Sample results are shown in Figure 5.17: (a,c) in the {q, r2)-
plane and (b,d) in the {&, $1) phase plane, where #1 is the angular deflection of
the first articulation. In Figure 5.17(a) we see a fixed point on the rl-axis, (rI, r2) =
(a, physical system it corresponds to periodic oscillations at frequency
0);
the
in