#### Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.23.202101.72–81

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 35Q35, 35Q74, 74H55

#### Investigation of the dynamic stability of bending-torsional deformations of the pipeline

P. A. Velmisov, Yu. A. Tamarova, Yu. V. Pokladova

Ulyanovsk State Technical University (Ulyanovsk, Russian Federation)

Abstract. Nonlinear mathematical models are proposed that describe the dynamics of a pipeline with a fluid flowing in it: a) the model of bending-torsional vibrations with two degrees of freedom; b) the model describing flexural-torsional vibrations taking into account the nonlinearity of the bending moment and centrifugal force; c) the model that takes into account joint longitudinal, bending (transverse) and torsional vibrations. All proposed models are described by nonlinear partial differential equations for unknown strain functions. To describe the dynamics of a pipeline, the nonlinear theory of a rigid deformable body is used, which takes into account the transverse, tangential and longitudinal deformations of the pipeline. The dynamic stability of bending-torsional and longitudinal-flexural-torsional vibrations of the pipeline is investigated. The definitions of the stability of a deformable body adopted in this work correspond to the Lyapunov concept of stability of dynamical systems. The problem of studying dynamic stability, namely, stability according to initial data, is formulated as follows: at what values of the parameters characterizing the gas-body system, small deviations of the body from the equilibrium position at the initial moment of time will correspond to small deviations and at any moment of time. For the proposed models, positive definite functionals of the Lyapunov type are constructed, on the basis of which the dynamic stability of the pipeline is investigated. Sufficient stability conditions are obtained that impose restrictions on the parameters of a mechanical system.

Key Words: differential equations, pipeline, deformation, dynamics, flexural-torsional vibrations, stability, Lyapunov functionals

For citation: P. A. Velmisov, Yu. A. Tamarova, Yu. V. Pokladova. Investigation of the dynamic stability of bending-torsional deformations of the pipeline. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:1(2021), 72–81. DOI: https://doi.org/10.15507/2079-6900.23.202101.72–81

Submitted: 14.12.2020; Revised: 07.01.2021; Accepted: 10.02.2021

Information about the authors:

Petr A. Velmisov, Head of the Department of Higher Mathematics, Ulyanovsk State Technical University (32 Severny Venets St., Ulyanovsk 432027, Russia), Dr. Sci. (Physics and Mathematics), Professor, ORCID: https://orcid.org/0000-0001-7825-7015, velmisov@ulstu.ru

Yuliya A. Tamarova, Postgraduate Student, Department of Higher Mathematics, Ulyanovsk State Technical University (32 Severny Venets St., Ulyanovsk 432027, Russia), ORCID: http://orcid.org/0000-0001-6408-1573, kazakovaua@mail.ru

Yuliya V. Pokladova, Associate Professor of the Department of Higher Mathematics, Ulyanovsk State Technical University (32 Severny Venets St., Ulyanovsk 432027, Russia), Ph. D. (Physics and Mathematics), ORCID: https://orcid.org/0000-0002-9876-4038, pokladovau@inbox.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.