Page 404 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

SLENDER STRUCTURES AND AXIAL FLOW

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Top view
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(4
Figure 5.49 Transition from equilibrium to chaos for 3-D motions of the system of Figure 5.43(a),
for various end-masses. Top: the ranges of various oscillatory states in terms of increasing ug. S,
stationary pipe; PL, planar oscillation; CW, clockwise rotating motion; CCW, counterclockwise
rotation: PL,CW and PL,CCW, clockwise and counterclockwise rotating planar oscillation; PL,P,
coupled planar and pendular oscillation; N, nutation; PL(R), planar oscillation rotating through a
finite angle; PL, P(R), coupled planar and pendular oscillation rotating through a finite angle; CH,
chaos. Bottom: sketches of various periodic motions. (a) PL; (b) CCW; (c) PL(R); (d) PL,P; (e) N
(Copeland & Moon 1992).

incommensurate frequencies, so that motions evolve on a ‘two-torus, T2’. Then, a third
Hopf bifurcation gives rise to quasiperiodicity involving three frequencies and a ‘three-
torus, T3’. This last torus, however, is nonrobust and it can be destroyed by a certain
type of perturbation, transforming it into a strange attractor. Thus, the appearance of a
third frequency, if it can be captured at all, signals the possible onset of chaos. Therefore,