International Journal of Differential Equations

Unboundedness of Solutions of Timoshenko Beam Equations with Damping and Forcing Terms

Kusuo Kobayashi and Norio Yoshida

Full-text: Open access

Abstract

Timoshenko beam equations with external damping and internal damping terms and forcing terms are investigated, and boundary conditions (end conditions) to be considered are hinged ends (pinned ends), hinged-sliding ends, and sliding ends. Unboundedness of solutions of boundary value problems for Timoshenko beam equations is studied, and it is shown that the magnitude of the displacement of the beam grows up to ∞ as t under some assumptions on the forcing term. Our approach is to reduce the multidimensional problems to one-dimensional problems for fourth-order ordinary differential inequalities.

Article information

Source
Int. J. Differ. Equ., Volume 2013, Special Issue (2013), Article ID 435456, 6 pages.

Dates
Received: 15 January 2013
Accepted: 21 February 2013
First available in Project Euclid: 24 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485226923

Digital Object Identifier
doi:10.1155/2013/435456

Mathematical Reviews number (MathSciNet)
MR3038082

Zentralblatt MATH identifier
1270.35298

Citation

Kobayashi, Kusuo; Yoshida, Norio. Unboundedness of Solutions of Timoshenko Beam Equations with Damping and Forcing Terms. Int. J. Differ. Equ. 2013, Special Issue (2013), Article ID 435456, 6 pages. doi:10.1155/2013/435456. https://projecteuclid.org/euclid.ijde/1485226923


Export citation

References

  • S. Timoshenko, D. H. Young, and W. Weaver Jr., Vibration Problems in Engineering, John Wiley & Sons, New York, NY, USA, 4th edition, 1974.
  • T. M. Wang and J. E. Stephens, “Natural frequencies of Timoshenko beams on pasternak foundations,” Journal of Sound and Vibration, vol. 51, no. 2, pp. 149–155, 1977.
  • E. Feireisl and L. Herrmann, “Oscillations of a non-linearly damped extensible beam,” Applications of Mathematics, vol. 37, no. 6, pp. 469–478, 1992.MR1185802
  • L. Herrmann, “Vibration of the Euler-Bernoulli beam with allowance for dampings,” in Proceedings of the World Congress on Engineering, vol. 2, London, UK, 2008.
  • M. Kopáčková, “On periodic solution of a nonlinear beam equation,” Aplikace Matematiky, vol. 28, no. 2, pp. 108–115, 1983.MR695184
  • T. Kusano and N. Yoshida, “Forced oscillations of Timoshenko beams,” Quarterly of Applied Mathematics, vol. 43, no. 2, pp. 167–177, 1985.MR793524
  • N. Yoshida, “Forced oscillations of extensible beams,” SIAM Journal on Mathematical Analysis, vol. 16, no. 2, pp. 211–220, 1985.MR777463
  • N. Yoshida, “Forced oscillations of nonlinear extensible beams,” in Proceedings of the 10th International Conference on Nonlinear Oscillations (Varna, 1984), pp. 814–817, Sofia, Bulgaria, 1985.
  • N. Yoshida, “On the zeros of solutions of beam equations,” Annali di Matematica Pura ed Applicata, vol. 151, pp. 389–398, 1988.MR964521
  • N. Yoshida, “Vibrations of Timoshenko beams with damping and forcingterms,” in Proceedings of the IEEE International Conference on Industrial Engineering and Engineering Management of (IEEM '12), 2012.
  • J. M. Ball, “Stability theory for an extensible beam,” Journal of Differential Equations, vol. 14, pp. 399–418, 1973.MR0331921
  • W. E. Fitzgibbon, “Global existence and boundedness of solutions to the extensible beam equation,” SIAM Journal on Mathematical Analysis, vol. 13, no. 5, pp. 739–745, 1982.MR668317
  • T. Narazaki, “On the time global solutions of perturbed beam equations,” Proceedings of the Faculty of Science of Tokai University, vol. 16, pp. 51–71, 1981.MR632661