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.
The work of the second author (Sections 1, 2, and 4) was supported by the Russian Science Foundation under grant no. 19-71-30012, https://rscf.ru/en/project/19-71-30012/, and performed at the Steklov Mathematical Institute of Russian Academy of Sciences.
Citation:
K. E. Abdul'manov, V. V. Vedeneev, “Linear and Nonlinear Development of Bending Perturbations in a Fluid-Conveying Pipe with Variable Elastic Properties”, Modern Methods of Mechanics, Collected papers. On the occasion of the 90th birthday of Academician Andrei Gennad'evich Kulikovskii, Trudy Mat. Inst. Steklova, 322, Steklov Math. Inst., Moscow, 2023, 10–23; Proc. Steklov Inst. Math., 322 (2023), 4–17
\Bibitem{AbdVed23}
\by K.~E.~Abdul'manov, V.~V.~Vedeneev
\paper Linear and Nonlinear Development of Bending Perturbations in a Fluid-Conveying Pipe with Variable Elastic Properties
\inbook Modern Methods of Mechanics
\bookinfo Collected papers. On the occasion of the 90th birthday of Academician Andrei Gennad'evich Kulikovskii
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 322
\pages 10--23
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4344}
\crossref{https://doi.org/10.4213/tm4344}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 322
\pages 4--17
\crossref{https://doi.org/10.1134/S0081543823040028}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85180194602}