Abstract
The fascinating behavior of tubes conveying fluid has attracted great interest among engineers as well as mathematicians. A nice and rather complete review of the relevant literature is given in [1]. In a fundamental paper Bajaj and Sethna ([2]) studied the Hopf bifurcation (flutter instability) of the downhanging equilibrium position of a cantilever tube. The mathematical interest in this problem stems from the fact that due to the rotational and reflectional symmetry (O(2)-symmetry) each eigenvalue of the linearized problem, associated with the loss of stability, occurs with multiplicity two. The consequence of this double multiplicity is that the system of bifurcation equations (amplitude equations of the critical modes) is rather high dimensional and hence has a richer solution set. However, on the other hand the bifurcation equations must obey the same symmetry properties as the equations of motion. Therefore the number of terms which are consistent with the symmetry requirements is pretty small. Thus the complication resulting from the high dimension is to some extend compensated. In [2] it is shown that depending on the mass ratio β between tube and fluid (see eq. (14) below) two distinct oscillatory motions occur after loss of stability. Both follow from symmetry breaking bifurcations. One is a planar oscillation which breaks the rotational symmetry and the other is a rotating motion which breaks the reflectional symmetry. Both solutions can be easily observed in experiments.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Paidoussis M.P., Flow-induced instabilities of cylindrical structures, Appl. Mech. Rev. 40 (1987) 163–175.
Bajaj A. K., Sethna P. R., Flow Induced Bifurcations to Three-Dimensional Oscillatory Motions in Continuous Tubes, SIAM J. Appl. Math. 44 (1984) 270–286.
Steindl A., Troger H., Bifurcations of the equilibrium of a spherical double pendulum at a multiple eigenvalue, Int. Series of Numerical Math. Vol. 79, Birkhäuser Verlag, Basel 1987, 277–287.
Can J., Applications of Centre Manifold Theory, Appl. Math. Sciences 35, Springer-Verlag, New York, Heidelberg, Berlin 1981.
Lundgren T. S., Sethna P. R., Bajaj A. K., Stability Boundaries for Flow Induced Motions of Tubes with an Inclined Terminal Nozzle, J. Sound and Vibration 64 (1979) 553–571.
Golubitsky M., Schaeffer D., Singularities and Groups in Bifurcation Theory, Appl. Math. Sciences 51, Springer-Verlag, Berlin, Heidelberg, New York 1985.
Oberle H. J., Grimm W., Berger E., BNDSCO — Rechenprogramm zur Lösung beschränkter optimaler Steuerungsprobleme, Benutzeranleitung, TUM-M8509, Math. Institut, Technische Universität München, 1985.
Sugiyama Y., Tanaka Y., Kishi T., Kawagoe H., Effect of a Spring Support on the Stability of Pipes Conveying Fluid, J. Sound and Vibration 100 (1985) 257–270.
Steindl A., Troger H., Bifurcations and oscillations in mechanical systems with symmetries, Proceed. 11th Intern. Conf. on Noni. Oscilations, (M. Farkas, V. Kertesz, G. Stepan eds.) Janos Bolyai Math. Soc, Budapest 1987, 231–238.
Golubitsky M., Stewart I., Symmetry and Stability in Taylor-Couette Flow, SIAM J. Math. Anal. 17 (1986) 249–288.
Lindtner E., Steindl A., Troger H., Stabilitätsverlust der gestreckten Lage eines räumlichen Pendels mit elastischer Endlagerung unter einer Folgelast, ZAMM 67 (1987) T105–T107.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Steindl, A., Troger, H. (1988). Flow Induced Bifurcations to Three-Dimensional Motions of Tubes with an Elastic Support. In: Besseling, J.F., Eckhaus, W. (eds) Trends in Applications of Mathematics to Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73933-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-73933-0_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-73935-4
Online ISBN: 978-3-642-73933-0
eBook Packages: Springer Book Archive