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.
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
\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}
Linking options:
https://www.mathnet.ru/eng/vsgtu1474
https://www.mathnet.ru/eng/vsgtu/v220/i2/p249
This publication is cited in the following 5 articles:
Elvin Azizbayov, Yaşar Mehraliyev, “On an inverse boundary-value problem for the pseudohyperbolic equation with nonclassical boundary conditions”, Hacettepe Journal of Mathematics and Statistics, 54:1 (2025), 142
A. V. Bogatov, “Zadacha s integralnym usloviem dlya odnomernogo giperbolicheskogo uravneniya”, Vestn. SamU. Estestvennonauchn. ser., 24:4 (2018), 7–12
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
A. B. Beilin, “O zadache upravleniya smescheniem odnogo iz kontsov tonkogo sterzhnya”, Vestn. SamU. Estestvennonauchn. ser., 2017, no. 3, 12–17
A. B. Beylin, L. S. Pulkina, “A PROBLEM ON VIBRATION OF A BAR WITH UNKNOWN BOUNDARY CONDITION ON A PART OF THE BOUNDARY”, Vestnik of Samara University. Natural Science Series, 23:2 (2017), 7
Citation in format AMSBIB
Contact us: