Skip to main content
SpringerLink
Log in

The optimal decay rates for viscoelastic Timoshenko type system in the light of the second spectrum of frequency

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

The stabilization properties of dissipative Timoshenko systems have been attracted the attention and efforts of researchers over the years. In the past 20 years, the studies in this scenario distinguished primarily by the nature of the coupling and the type or strength of damping. Particularly, under the premise that the Timoshenko beam model is a two-by-two system of hyperbolic equations, a large number of papers have been devoted to the study of the so-called partially damped Timoshenko systems by assuming damping effects acting only on the angle rotation or vertical displacement (Almeida Júnior et al. in Math Methods Appl Sci 36:1965–1976, 2013; in Z Angew Math Phys 65:1233–1249, 2014; Alves et al. in SIAM J Math Anal 51(6):4520–4543, 2019; Ammar-Khodja et al. in J Differ Equ 194:82–115, 2003; Muñoz Rivera and Racke in Discrete Contin Dyn Syst Ser B 9:1625–1639, 2003; J Math Anal Appl 341:1068–1083, 2008; Santos et al. in J Differ Equ 253(9):2715–2733, 2012). In these cases, the desired exponential decay property of the energy solutions is achieved when the non-physical equal wave speed assumption plays the role to stabilization according since the pioneering Soufyane’s paper (C R Acad Sci Paris 328(8):731–734, 1999). Recent results due to Almeida Júnior et al. (Z Angew Math Phys 68(145):1–31, 2017; Z Angew Math Mech 98(8):1320–1333, 2018; IMA J Appl Math 84(4):763–796, 2019; Acta Mech 231:3565–3581, 2020) show that the second vibration mode or simply second spectrum of frequency and it’s damaging consequences appears as a lost element in analysis of stabilization and now it’s more clear that the damping importance into stabilization scenario of Timoshenko type systems. This paper considers a one-dimensional viscoelastic Timoshenko type system in the light of the second spectrum of frequency where the equal wave speed assumption is not needed for getting the exponential decay property. Precisely, we consider the so-called truncated version for the Timoshenko system according studies due to Elishakoff (Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010; ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11, 2015; Int J Solids Struct 109:143–151, 2017; J Sound Vib 435:409–430, 2017; Int J Eng Sci 116:58–73, 2017; Acta Mech 229:1649–1686, 2018; Z Angew Math Mech 98(8):1334–1368, 2018; Math Mech Solids, 2019) and we added a viscoelastic damping acting on shear force. We firstly prove the global well-posedness of the system by Faedo–Galerkin approximation. By assuming minimal conditions on the relaxation function, we establish an optimal explicit and energy decay rate for which exponential and polynomial rates are special cases. This result is new and substantially improves earlier results in the literature where the equal wave speeds plays the role for getting the stability properties. It is likely to open more research areas to Timoshenko system and probably others.

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
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Timoshenko [63] introduced Eqs. (1.3)–(1.4), which take into account the shear deformation and rotary inertia. According to Elishakoff et al. [22, 24], Timoshenko had two predecessors, namely Bresse [12] and Rayleigh [54]. However, Timoshenko did not reference Bresse, though he sometimes referenced Rayleigh. Moreover, Ehrenfest’s name did not appear in his papers dated 1920 and 1921. Koiter [35] did not know these facts when he wrote: “What is generally known as Timoshenko beam theory is a good example of a basic principle in the history of science: a theory which bears someone’s name is most likely due to someone else.” Elishakoff [27] unequivocally proves that the modern theory with the shear coefficient was introduced by Timoshenko and Ehrenfest. It is therefore fair that the theory should be called the Timoshenko–Ehrenfest theory.

References

  1. Almeida Júnior, D.S., Santos, M.L., Muñoz Rivera, J.E.: Stability to weakly dissipative Timoshenko systems. Math. Methods Appl. Sci. 36, 1965–1976 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Almeida Júnior, D.S., Santos, M.L., Muñoz Rivera, J.E.: Stability to 1-D thermoelastic Timoshenko beam acting on shear force. Z. Angew. Math. Phys. 65, 1233–1249 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Almeida Júnior, D.S., Ramos, A.J.A.: On the nature of dissipative Timoshenko systems at light of the second spectrum. Z. Angew. Math. Phys. 68(145), 1–31 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Almeida Júnior, D.S., Ramos, A.J.A., Santos, M.L., Miranda, L.G.R.: Asymptotic behavior of weakly dissipative Bresse–Timoshenko system on influence of the second spectrum of frequency. Z. Angew. Math. Mech. 98(8), 1320–1333 (2018)

    Article  MathSciNet  Google Scholar 

  5. Almeida Júnior, D.S., Elishakoff, I., Ramos, A.J.A., Miranda, L.G.R.: The hypothesis of equal wave speeds for stabilization of Bresse–Timoshenko system is not necessary anymore: the time delay cases. IMA J. Appl. Math. 84(4), 763–796 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Almeida Júnior, D.S., Ramos, A.J.A., Soufyane, A., Cardoso, M.L., Santos, M.L.: Issues related to the second spectrum, Ostrogradsky’s energy and stabilization of Timoshenko–Ehrenfest type systems. Acta Mech. 231, 3565–3581 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Alves, M.O., Gomes Tavares, E.H., Jorge Silva, M.A., Rodrigues, J.H.: On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J. Math. Anal. 51(6), 4520–4543 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ammar-Khodja, F., Benabdallah, A., Muñoz Rivera, J.E., Racke, R.: Energy decay for Timoshenko system of memory type. J. Differ. Equ. 194, 82–115 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)

    Book  Google Scholar 

  10. Bhaskar, A.: Elastic waves in Timoshenko beams: the “lost’’ and “found’’ of an eigenmode. Proc. R. Soc. 465, 239–255 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bhashyam, G.R., Prathap, G.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 76(3), 407–420 (1981)

    Article  Google Scholar 

  12. Bresse, M.: Cours de méchanique appliqée. Mallet-Bachelier, Paris (1859)

    Google Scholar 

  13. Cazzani, A., Stochino, F., Turco, E.: On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Z. Angew. Math. Phys. 67(2), Article 24 (2016)

  14. Cazzani, A., Stochino, F., Turco, E.: On the whole spectrum of Timoshenko beams. Part II: further applications. Z. Angew. Math. Phys. 67(2), Article 25 (2016)

  15. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. Z. Angew. Math. Phys. 96(10), 1220–1244 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Coucha, A., Ouchenane, D., Zennir, K., Feng, B.: Global well-posedness and exponential stability results of a class of Bresse–Timoshenko-type systems with distributed delay term. Math. Methods Appl. Sci. 1–26 (2020)

  17. Dell’oro, F., Pata, V.: On the stability of Timoshenko systems with Gurtin–Pipkin thermal law. J. Differ. Equ. 257(2), 523–548 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Eden A., Foias, C., Nicolaenko B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations. RAM: Research in Applied Mathematics, vol. 37, Masson, Paris; Wiley, Chichester (1994)

  19. Elishakoff, I.: An equation both more consistent and simpler than the Bresse–Timoshenko equation. In: Advances in Mathematical Modelling and Experimental Methods for Materials and Structures, Solid Mechanics and Its Applications, pp. 249–254. Springer, Berlin (2010)

  20. Elishakoff, I., Kaplunov, J., Nolde, E.: Celebrating the Centenary of Timoshenko’s study of effects of shear deformation and rotary inertia. ASME Am. Soc. Mech. Eng. Appl. Mech. Rev. 67(6), 1–11 (2015)

    Google Scholar 

  21. Elishakoff, I., Hache, F., Challamel, N.: Critical contrasting of three versions of vibrating Bresse–Timoshenko beam with a crack. Int. J. Solids Struct. 109, 143–151 (2017)

    Article  Google Scholar 

  22. Elishakoff, I., Hache, F., Challamel, N.: Variational derivation of governing differential equations for truncated version of Bresse–Timoshenko beams. J. Sound Vib. 435, 409–430 (2017)

    Article  Google Scholar 

  23. Elishakoff, I., Hache, F., Challamel, N.: Vibrations of asymptotically and variationally based Uflyand–Mindlin plate models. Int. J. Eng. Sci. 116, 58–73 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Elishakoff, I., Hache, F., Challamel, N.: Comparison of refined beam theories for parametric instability. AIAA Technical notes 56(1) (2018)

  25. Elishakoff, I., Tonzani, G.M., Marzani, A.: Effect of boundary conditions in three alternative models of Timoshenko–Ehrenfest beams on Winkler elastic foundation. Acta Mech. 229, 1649–1686 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Elishakoff, I., Tonzani, G.M., Zaza, N., Marzani, A.: Contrasting three alternative versions of Timoshenko–Ehrenfest theory for beam on Winkler elastic foundation—simply supported beam. Z. Angew. Math. Mech. 98(8), 1334–1368 (2018)

    Article  MathSciNet  Google Scholar 

  27. Elishakoff, I.: Who developed the so-called Timoshenko beam theory? Math. Mech. Solids (2019)

  28. Fatori, L.H., Muñoz Rivera, J.E., Monteiro, R.N.: Energy decay to Timoshenko’s system with thermoelasticity of type III. Asympt. Anal. 86, 227–247 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Feng, B., Almeida Júnior, D.S., Dos Santos, M.J., Miranda, L.G.R.: A new scenario for stability of nonlinear Bresse–Timoshenko type systems with time dependent delay. Z. Angew. Math. Mech. 100, 1–17 (2020)

    Article  MathSciNet  Google Scholar 

  30. Feng, B., Jorge Silva, M.A., Caixeta, A.H.: Long-time behavior for a class of semi-linear viscoelastic Kirchhoff beams/plates. Appl. Math. Optim. 82, 657–686 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Guesmia, A., Messaoudi, S.A.: On the control of a viscoelastic damped Timoshenko-type systems. Appl. Math. Comput. 206(2), 589–597 (2008)

    MathSciNet  MATH  Google Scholar 

  32. Graff, K.F.: Wave Motion in Elastic Solids. Dover Publication, New York (1991)

    MATH  Google Scholar 

  33. Hassan, J.H., Messaoudi, S.A., Zahri, M.: Existence and new general decay results for a viscoelastic Timoshenko system. Z. Anal. Anwendungen 39(2), 185–222 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Jin, K.P., Liang, J., Xiao, T.J.: Coupled second order evolution equations with fading memory: optimal energy decay rate. J. Differ. Equ. 257, 1501–1528 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Koiter, W.T.: Timoshenko beam theory is not always more accurate than elementary theory. J. Appl. Mech. 44, 357–358 (1977)

    Article  Google Scholar 

  36. Levinson, M., Cooke, D.W.: On the two frequency spectra of Timoshenko beams. J. Sound Vib. 84(3), 319–326 (1982)

    Article  MATH  Google Scholar 

  37. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  38. Manevich, A., Kolakowski, Z.: Free and forced oscillations of Timoshenko beam made of viscoelastic material. J. Theor. Appl. Mech. 49(1), 3–16 (2011)

    Google Scholar 

  39. Meleshko, V.V., Bondarenko, A.A., Dovgiy, S.A., Trofimchuk, A.N., van Heijst, G.J.F.: Elastic waveguides: history and the state of the art. I. J. Math. Sci. 162(1), 99–120 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Meleshko, V.V., Bondarenko, A.N., Trofimchuk, A.N., Abasov, R.Z.: Elastic waveguides: history and the state of the art. II. J. Math. Sci. 167(2), 197–216 (2010)

    Article  MATH  Google Scholar 

  41. Messaoudi, S.A., Said-Houari, B.: Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl. 348, 298–307 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Messaoudi, S.A., Mustafa, M.I.: A stability result in a memory-type Timoshenko system. Dyn. Syst. Appl. 18, 457–468 (2009)

    MathSciNet  MATH  Google Scholar 

  43. Messaoudi, S.A., Hassan, J.H.: General and optimal decay in a memory-type Timoshenko system. J. Integral Equ. Appl. 30(1), 117–145 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mustafa, M.I.: General decay result for nonlinear viscoelastic equations. J. Math. Anal. Appl. 457, 134–152 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  45. Muñoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. Ser. B 9, 1625–1639 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Muñoz Rivera, J.E., Racke, R.: Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 341, 1068–1083 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Nesterenko, V.V.: A theory for transverse vibrations of the Timoshenko beam. J. Appl. Math. Mech. 57, 669–677 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  48. Prathap, G.: The two frequency spectra of Timoshenko beams: a reassessment. J. Sound Vib. 90, 443–445 (1983)

    Article  Google Scholar 

  49. Prüss, J.: Evolutionary Integral Equations and Applications. Monographs in Mathematics, vol. 87. Birkhäuser Verlag, Basel (1993)

  50. Ramos, A.J.A., Almeida Júnior, D.S., Freitas, M.M., Dos Santos, M.J.: A new exponential decay result for 1d porous dissipation elasticity from second spectrum viewpoint. Appl. Math. Lett. 101, 106061 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  51. Ramos, A.J.A., Almeida Júnior, D.S., Miranda, L.G.R.: An inverse inequality for a Bresse–Timoshenko system without second spectrum of frequency. Arch. Math. 115, 361–481 (2020)

    MATH  Google Scholar 

  52. Ramos, A.J.A., Aouadi, M., Almeida Júnior, D.S., Freitas, M.M., Araújo, M.L.: A new stabilization scenario for Timoshenko systems with thermo-diffusion effects in second spectrum perspective. Arch. Math. 116, 1–17 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  53. Raposo, C.A., Ferreira, J., Santos, M.L., Castro, N.N.O.: Exponential stability for the Timoshenko beam with two weak damping. Appl. Math. Lett. 18, 535–541 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  54. Rayleigh, J.W.S.: Theory of Sound. Macmillan Publications Co. Inc., London (1877)

    MATH  Google Scholar 

  55. Santos, M.L., Almeida Júnior, D.S., Muñoz Rivera, J.E.: The stability number of the Timoshenko system with second sound. J. Differ. Equ. 253(9), 2715–2733 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  56. Simon, J.: Compact sets in the space \(L^p(0, T; B)\). Ann. Mat. Pura Appl. 146, 65–96 (1986)

    Article  MATH  Google Scholar 

  57. Smith, R.W.M.: Graphical representation of Timoshenko beam modes for clamped-clamped boundary conditions at high frequency: beyond transverse deflection. Wave Motion 45, 785–794 (2008)

    Article  MATH  Google Scholar 

  58. Soufyane, A.: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Paris 328(8), 731–734 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  59. Soufyane, A., Wehbe, A.: Uniform stabilization for the Timoshenko beam by a locally distributed damping. Electron. J. Differ. Equ. 29, 1–14 (2003)

    MathSciNet  Google Scholar 

  60. Stephen, N.G.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 80, 578–582 (1982)

    Article  Google Scholar 

  61. Stephen, N.G.: The second frequency spectrum of Timoshenko beams theory—further assessment. J. Sound Vib. 192, 372–389 (2006)

    Article  MATH  Google Scholar 

  62. Traill-Nash, R.W., Collar, A.R.: The effects of shear flexibility and rotatory inertia on the bending vibrations of beams. Q. J. Mech. Appl. Math. 6, 186–222 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  63. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41(245), 744–746 (1921)

    Article  Google Scholar 

Download references

Acknowledgements

The authors express sincere thanks to the referee for his/her constructive comments and suggestions that helped to improve this paper.

Funding

D. S. Almeida Júnior thanks the CNPq for financial support through the project: “Impact of the second spectrum of frequency on the stabilization of partially dissipative Timoshenko type systems,” Grant 314273/2020-4. B. Feng was supported by the National Natural Science Foundation of China—Grant 11701465. A. Soufyane has been supported by University of Sharjah through Grant 1802144069.

Author information

Authors and Affiliations

  1. Federal University of Pará, Augusto Corrêa Street, 01, Belém, Pará, 66075-110, Brazil

    D. S. Almeida Júnior

  2. Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    B. Feng

  3. Département de Mathématiques et Informatique, Faculté Polydisciplinaire de Safi, Université Cadi Ayyad, Marrakesh, Morocco

    M. Afilal

  4. Department of Mathematics, College of Sciences, University of Sharjah, P.O. Box 27272, Shariqah, UAE

    A. Soufyane

Authors
  1. D. S. Almeida Júnior
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Afilal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Soufyane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. S. Almeida Júnior.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Almeida Júnior, D.S., Feng, B., Afilal, M. et al. The optimal decay rates for viscoelastic Timoshenko type system in the light of the second spectrum of frequency. Z. Angew. Math. Phys. 72, 147 (2021). https://doi.org/10.1007/s00033-021-01574-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01574-y

Keywords

Mathematics Subject Classification

Search

Navigation