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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 2, Pages 249–258 (Mi vsgtu1474)  

This article is cited in 3 scientific papers (total in 3 papers)

Differential Equations and Mathematical Physics

A problem on longitudinal vibration of a bar with elastic fixing

A. B. Beylin

Samara State Technical University, Samara, 443100, Russian Federation

Abstract: In this paper, we study longitudinal vibration in a thick short bar fixed by point forces and springs. For mathematical model we consider a boundary value problem with dynamical boundary conditions for a forth order partial differential equation. The choice of this model depends on a necessity to take into account the result of a transverse strain. It was shown by Rayleigh that neglect of a transverse strain leads to an error. This is confirmed by modern nonlocal theory of vibration. We prove existence of orthogonal with load eigenfunctions and derive representation of them. Established properties of eigenfunctions make possible using the separation of variables method and finding a unique solution of the problem.

Keywords: dynamic boundary conditions, longitudinal vibration, loaded orthogonality, Rayleigh's model

DOI: https://doi.org/10.14498/vsgtu1474

Full text: PDF file (807 kB) (published under the terms of the Creative Commons Attribution 4.0 International License)
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Bibliographic databases:

UDC: 517.956.3
MSC: 35L35, 35Q74
Original article submitted 10/II/2016
revision submitted – 18/V/2016

Citation: A. B. Beylin, “A problem on longitudinal vibration of a bar with elastic fixing”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:2 (2016), 249–258

Citation in format AMSBIB
\Bibitem{Bey16}
\by A.~B.~Beylin
\paper A problem on longitudinal vibration of a bar with elastic fixing
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2016
\vol 20
\issue 2
\pages 249--258
\mathnet{http://mi.mathnet.ru/vsgtu1474}
\crossref{https://doi.org/10.14498/vsgtu1474}
\zmath{https://zbmath.org/?q=an:06964485}
\elib{https://elibrary.ru/item.asp?id=27126227}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. B. Beilin, L. S. Pulkina, “Zadacha o kolebaniyakh sterzhnya s neizvestnym usloviem ego zakrepleniya na chasti granitsy”, Vestn. SamU. Estestvennonauchn. ser., 2017, no. 2, 7–14  mathnet  elib
    2. A. B. Beilin, “O zadache upravleniya smescheniem odnogo iz kontsov tonkogo sterzhnya”, Vestn. SamU. Estestvennonauchn. ser., 2017, no. 3, 12–17  mathnet  crossref  elib
    3. A. V. Bogatov, “Zadacha s integralnym usloviem dlya odnomernogo giperbolicheskogo uravneniya”, Vestn. SamU. Estestvennonauchn. ser., 24:4 (2018), 7–12  mathnet  crossref  elib
  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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