Numerical Method for Calculating the Pipe Spatial Vibrations

  • Natalia YaparovaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)


The paper is devoted to the problem of calculation of the spatial vibrations of elementary pipeline sections that appear under shock pulse. The equation of the motion and the boundary conditions describing the pipe deflections are presented as the forth-order partial differential equation with the Dirichlet and Neumann boundary conditions. To solve the problem, the numerical method based on finite-difference equations and a regularization technique is proposed. In order to evaluate the efficiency of the proposed method, the computational experiments were carried out. The results demonstrated sufficient accuracy of numerical solutions and confirm the sensitivity of method to changes in system.


Finite-difference method Spatial vibrations of pipe Regularization Computational scheme 



The work was supported by the Ministry of Education and Science of the Russian Federation within the framework of the basic part of the State task “Development, research and implementation of data processing algorithms for dynamic measurements of spatially distributed objects”, Terms of Reference 8.9692.2017/8.9 from 17.02.2017.


  1. 1.
    Timoshenko, S., Young, D.H.: Vibration Problems in Engineering. Van Nostrand, New Jersey (1955)Google Scholar
  2. 2.
    Pfeiffer, P.: Vibrations, of Elastic Bodies. The All-the-Union Scientific and Technical Publishing, Moscow (1934). (Russian; original edition: Handbuch Der Physik, band VI. Mechanik Der Elastischen Korper. Redigiert von R. Grammel. Berlin (1928))Google Scholar
  3. 3.
    Paidoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 2. Elsevier Academic Press, London (2003)Google Scholar
  4. 4.
    Nesterov, S.V., Akulenko, L.D., Korovina, L.I.: Transverse oscillations of a pipeline with a uniformly moving fluid. Rep. Acad. Sci. Mech. 427(6), 781–784 (2009)zbMATHGoogle Scholar
  5. 5.
    Kutin, J., Bajsic, I.: An analytical estimation of the Coriolis Meter’s characteristics based on modal superposition. Flow Meas. Instrum. 12, 345–351 (2002)CrossRefGoogle Scholar
  6. 6.
    Khakimov, A.G., Shakiryanov, M.M.: Spatial oscillations of the pipeline under the influence of variable internal pressure. Bull. UFA State Aviat. Tech. Univ. 14(37), 30–35 (2014)Google Scholar
  7. 7.
    Mironov, M.A., Pyatakov, P.A., Andreev, A.A.: Forced flexural vibrations of a tube with a liquid flow. Acoust. J. 56(5), 684–692 (2010)Google Scholar
  8. 8.
    Baderko, E.A.: A method of potential theory in boundary value problems for \(2m\)-parabolic equations in a semibounded domain with a nonsmooth lateral boundary. Differ. Equ. 24(1), 1–5 (1988)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Prokudina, L.A.: Nonlinear evolution of perturbations in a thin fluid layer during wave formation. J. Exp. Theor. Phys. 118(3), 480–488 (2014)CrossRefGoogle Scholar
  10. 10.
    Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Inverse Problems of Mathematical Physics. Walter de Gruyter, Germany (2007)CrossRefGoogle Scholar
  11. 11.
    Chernyatin, V.A.: A Substantiation of the Fourier Method in the Mixed Problem for Partial Differential Equations. Moscow State University, Moscow (1991)zbMATHGoogle Scholar
  12. 12.
    Kabanikhin, S.I.: Inverse and Ill-Posed Problems: Theory and Applications. Walter de Gruyter, Germany (2011)CrossRefGoogle Scholar
  13. 13.
    Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, London (1995)CrossRefGoogle Scholar
  14. 14.
    Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)CrossRefGoogle Scholar
  15. 15.
    Yaparova, N.: Numerical method for solving an inverse boundary problem with unknown initial conditions for parabolic PDE using discrete regularization. In: Dimov, I., et al. (eds.) NAA 2016. LNCS, vol. 10187, pp. 752–759. Springer, Heidelberg (2017). Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computational Mathematics and High Performance ComputingSouth Ural State University (National Research University)ChelyabinskRussia

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