Page 366 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

346 SLENDER STRUCTURES AND AXIAL FLOW

AA
++ I
---------- -------.

Figure 5.28 Amplitudes of solutions, A, versus u - u, for 3-D motions of the cantilevered
pipe-spring system, with K slightly higher than K,: (a) for a system with a double set of zero
roots, which starts from a trivial stable state (TS), then develops buckling (SB), and then planar
oscillations about the buckled state (SW3) and about the origin (SW2); (b) for a system with a
double set of one zero root and one purely imaginary pair, which starts from the trivial state (TS),
develops buckling (SB), and then planar oscillations perpendicular to the buckling plane (SW4).
Solution branches drawn as dashed lines are unstable (Steindl & Troger 1995).

under its own weight; as the flow velocity is increased, the system is restabilized
via a reverse pitchfork bifurcation, and then loses stability by flutter via a Hopf
bifurcation at higher flow. For the special sets of parameters shown in Figure 5.26 for
K = 0, these two bifurcations become coincident. The system is studied analytically
and numerically, starting again with equation (5.42) and a two-degree-of-freedom
discretization thereof. The system is first projected on the centre manifold and then the
unfolding parameters PI, ,LL~ and p3 are computed, as shown explicitly in Appendix F.5,
equations (F.58)-(F.62). For B = 0.2, for which u, = 2.246, yc = -46.001, computing
the derivatives in (F.61) numerically and eliminating p3, the following relationships are
obtained: