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R. Jurisits, A. Steindl:
"Low order resonance of an elastic fluid-conveying tube";
Talk: 10th International Conference on Flow-Induced Vibration, Dublin; 2012-07-03 - 2012-07-06; in: "Flow-induced Vibration", C. Meskell, G. Bennett (ed.); (2012), ISBN: 978-0-9548583-4-6; 7 - 13.

The two-dimensional motion of a slender elastic
fluid-carrying tube is considered. The tube is clamped
at the upper end, at its lower end a point mass is fixed,
and at some intermediate position two springs support the
tube in transversal direction. Adopting model equations
for the tube-liquid system from previous works, the loss
of stability of the trivial position of the tube is investigated.
While a linear stability analysis indicates several
possibilities of stability losses, special attention is paid
to the Hopf-Hopf (coupled Flutter) instability where the
occurence of a (1 : 3) internal resonance is detected for
distinct parameter values. In a non-linear analysis the
post-bifurcation behaviour is investigated for the case of
the (1 : 3) resonance. For limiting cases, the dynamics
of the tube is shown to be described by Duffing´s
and by Mathieu´s equation, respectively, associated with
(quasi)periodic motions of the tube.

Hopf bifurcation, Resonance, Mode-interaction, Normal Form

http://programme.exordo.com/fiv2012/proceedings.pdf