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JOURNALS // Trudy Matematicheskogo Instituta imeni V.A. Steklova // Archive

Trudy Mat. Inst. Steklova, 2023 Volume 322, Pages 10–23 (Mi tm4344)

Linear and Nonlinear Development of Bending Perturbations in a Fluid-Conveying Pipe with Variable Elastic Properties

K. E. Abdul'manova, V. V. Vedeneevb

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991 Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: We consider bending vibrations of a fluid-conveying pipe resting on an elastic foundation with nonuniform elasticity coefficient. Previously A. G. Kulikovskii showed analytically that the elasticity parameters can be distributed in such a way that at every point the system is either locally stable or convectively unstable. In this case, despite the absence of local absolute instability, there exists a global growing mode whose formation is associated with the points of internal reflection of waves. In the present paper, we perform a numerical simulation of the development of the initial perturbation in such a system. In the linear formulation we demonstrate how the perturbation is transformed into a growing eigenmode after a series of reflections and passages through a region of local instability. In the nonlinear formulation, where the nonlinear tension of the pipe is taken into account within the von Kármán model, we show that the perturbation growth is limited; in this case the vibrations acquire a quasi-chaotic character but do not leave the region bounded by the internal reflection points determined by the linearized problem.

Keywords: absolute/convective instability, global instability, internal reflection, development of perturbations, hydroelasticity.

UDC: 534.11

Received: January 29, 2023
Revised: February 28, 2023
Accepted: May 20, 2023

DOI: 10.4213/tm4344


 English version:
Proceedings of the Steklov Institute of Mathematics, 2023, 322, 4–17

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© Steklov Math. Inst. of RAS, 2024