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Distribution of eigenfrequencies in long one-dimensional systems

Published online by Cambridge University Press:  28 April 2025

Vasily Vedeneev Open the ORCID record for Vasily Vedeneev [Opens in a new window]  and
Anastasia Podoprosvetova Open the ORCID record for Anastasia Podoprosvetova [Opens in a new window]
Vasily Vedeneev*
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia
Anastasia Podoprosvetova
Affiliation:
Institute of Mechanics, Lomonosov Moscow State University, Moscow 119192, Russia
*
Corresponding author: Vasily Vedeneev, [email protected]
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Abstract

It is known that the complex eigenfrequencies of one-dimensional systems of large but finite extent are concentrated near the asymptotic curve determined by the dispersion relation of an infinite system. The global instability caused by uppermost pieces of this curve was studied in various problems, including hydrodynamic stability and fluid–structure interaction problems. In this study, we generalise the equation for the asymptotic curve to arbitrary frequencies. We analyse stable local topology of the curve and prove that it can be a regular point, branching point or dead-end point of the curve. We give a classification of unstable local tolopogies, and show how they break up due to small changes of the problem parameters. The results are demonstrated on three examples: supersonic panel flutter, flutter of soft fluid-conveying pipe, and the instability of rotating flow in a pipe. We show how the elongation of the system yields the attraction of the eigenfrequencies to the asymptotic curve, and how each locally stable curve topology is reflected on the interaction of eigenfrequencies.

JFM classification

Instability: Absolute/convective instability Aerodynamics: Flow-structure interactions Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1010 , 10 May 2025 , A9
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Supplementary material: File

Vedeneev and Podoprosvetova supplementary material movie 1

Eigenfrequency loci of elastic panel in a supersonic flow for changing panel length L.
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File 696 KB
Supplementary material: File

Vedeneev and Podoprosvetova supplementary material movie 2

Eigenfrequency loci of an elastic tube conveying fluid for changing tube length L.
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File 1.8 MB
Supplementary material: File

Vedeneev and Podoprosvetova supplementary material movie 3

Eigenfrequency loci of a rotating pipe flow for changing pipe length L.
Download Vedeneev and Podoprosvetova supplementary material movie 3(File)
File 615.3 KB