Skip to main content
SpringerLink
Log in

Stability for a composite of Timoshenko laminated beams

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

The present model describes the vibrations of a composite beam consisting of two interconnected beams. One component is a material that exhibits only structural dissipation, while the other is a material subjected to additional frictional dissipation. It is well known that the solutions for laminated Timoshenko beams with one component converge to the equilibrium point at rates that depend on the wave propagation velocity. Here, we show that the solution of the complete system decays with exponential or rational rates when the components share a transmission condition. Proofs of the main results are provided using the multiplier method in the frequency domain. Technically, the main ingredients are inequalities of observability type that serve as “propagators” of asymptotic properties from one component to another.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this published article.

References

  1. M. S. Alves, C. A. Raposo, J. E. Munõz Rivera, M. Sepulveda and O. V. Villagran, Uniform Stabilization for the Transmission Problem of the Timoshenko System with Memory, J. Math. Anal. Appl., 369 (2010) 323–345.

  2. T. A. Apalara, Uniform stability of a laminated beam with structural damping and second sound, Z. Angew. Math. Phys., 41 (2017) 1–16.

    MathSciNet  Google Scholar 

  3. A. Borichev, Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347(2) (2009) 455–478.

    Article  MathSciNet  Google Scholar 

  4. S. Cox, E. Zuazua, The rate at which energy decays in a damped string, Commun. PDE, 19(1–2) (1994) 213–243.

    Article  MathSciNet  Google Scholar 

  5. S. W. Hansen and R. Spies, Structural damping in a laminated beam duo to interfacial slip, J. Sound Vib., 204 (1997) 183–202.

    Article  Google Scholar 

  6. Z. Liu and S. Zheng, Semigroups Associated to Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, 398, 1999.

  7. A. Lo and N.-e. Tatar, Stabilization of laminated beams with interfacial slip, Electron. J. Differ. Equ., 129 (2015) 1–14.

  8. M. I. Mustafa, On the stabilization of viscoelastic laminated beams with interfacial slip, Z. Angew. Math. Phys., 33 (2018) 1–14.

    MathSciNet  Google Scholar 

  9. M. I. Mustafa, Boundary control of laminated beams with interfacial slip, J. Math. Phys., 59 (2018) 1–9.

    Article  MathSciNet  Google Scholar 

  10. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, New York, 1983.

    Google Scholar 

  11. J. Pruss, On the spectrum of \(C_0\)-semigroups, Trans. Amer. Math. Soc., 284 (1984) 847–857.

    MathSciNet  Google Scholar 

  12. C. A. Raposo, W. D. Bastos and M. L. Santos, A transmission problem for the Timoshenko system, Comput. Appl. Math., 26(2) (2007) 215–234.

    Article  MathSciNet  Google Scholar 

  13. C. A. Raposo, Exponential stability for a structure with interfacial slip and frictional damping, Appl. Math. Lett., 53 (2016) 85–91.

    Article  MathSciNet  Google Scholar 

  14. N.-e. Tatar, Stabilization of a laminated beam with interfacial slip by boundary controls, Bound. Value Probl., 169 (2015) 1–11.

  15. S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag., 41 (1921) 744–746.

    Article  Google Scholar 

  16. J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44 (2005) 1575–1597.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Federal University of Viçosa, Viçosa, MG, 36570-000, Brazil

    M. S. Alves

  2. Department of Mathematics, Universidade Estadual de Londrina, Londrina, PR, 86.057-970, Brazil

    R. N. Monteiro

Authors
  1. M. S. Alves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. N. Monteiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. N. Monteiro.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alves, M.S., Monteiro, R.N. Stability for a composite of Timoshenko laminated beams. J. Evol. Equ. 23, 63 (2023). https://doi.org/10.1007/s00028-023-00911-0

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00028-023-00911-0

Keywords

Mathematics Subject Classification

Search

Navigation