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Issues related to the second spectrum, Ostrogradsky’s energy and the stabilization of Timoshenko–Ehrenfest-type systems

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Abstract

In this paper, we discuss the stabilization properties of a beam model on a Winkler foundation by using Timoshenko–Ehrenfest-type systems, taking into account the influence of the so-called second spectrum. We consider the well-known classical version of the Timoshenko–Ehrenfest beam model as well as the truncated (or simplified) version of the same beam model according to the approach given by Elishakoff (in: Banks-Sills (ed.), Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications. Springer, Berlin, pp 249–254, 2010). The main novelty of our approach is the concept of applying Ostrogradsky’s energy to both beam models to highlight the physics issues arising in the frequency spectra. Our ideas are an attempt to fill the gap regarding the consequences of the second spectrum in the stabilization scenario for dissipative Timoshenko systems that are partially damped.

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Notes

  1. Elishakoff et al. [27] documents (strongly based on the papers and autobiographical book by Timoshenko) that Stephen Timoshenko was one of the developers of this theory. Specifically, he had a co-author, the physicist Paul Ehrenfest (1880–1933), who collaborated with him on his 1921 and 1922 papers. See also Challamel and Elishakoff [13] for a historical presentation of the beam and plate models in elasticity theory as well as studies of predecessors such as Bresse (i.e., his studies in the nineteenth century), who rigorously derived the set of equations for the curved beam shear in dynamics, which was later (1913; 1916; 1920; 1921; 1922) generalized by Timoshenko and Paul Ehrenfest.

  2. Timoshenko [63] introduced Eqs. (1.3)–(1.4), which take into account the shear deformation and rotary inertia. According to Elishakoff et al. [23, 25], Timoshenko had two predecessors, namely Bresse [12] and Rayleigh [50]. 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 [45] 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.

  3. Emil Winkler (1835–1888) was the German civil engineer and professor responsible for formulating and solving the problem of an elastic beam on a deformable foundation, which today is known as the Winkler foundation (Fig. 3). It is a beam model on an elastic foundation that assumes a linear force–deflection relationship.

References

  1. Abbas, B.A.H., Thomas, J.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 51(1), 123–137 (1977)

    Google Scholar 

  2. Abramovich, H., Elishakoff, I.: Application of the Krein’s method for determination of natural frequencies of periodically supported beam based on simplified Bresse–Timoshenko equations. Acta Mech. 66(1–4), 39–59 (1987)

    MATH  Google Scholar 

  3. 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)

    MathSciNet  MATH  Google Scholar 

  4. 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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. 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)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  8. Ammar-Khodja, F., Benabdallah, A., Muñoz Rivera, J.E., Racke, R.: Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194(1), 82–115 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Anderson, R.A.: Flexural vibration in uniform beams according to the Timoshenko theory. J. Appl. Mech. 20, 504–510 (1953)

    MATH  Google Scholar 

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

    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)

    Google Scholar 

  12. Bresse, M.: Cours de méchanique appliqué, pp. 122–128. Mallet-Bachelier, Paris (1859)

    Google Scholar 

  13. Challamel, N., Elishakoff, I.: A brief history of first-order shear-deformable beam and plate models. Mech. Res. Commun. 102, 103389 (2019)

    Google Scholar 

  14. Chervyakov, A.M., Nesterenko, V.V.: Is it possible to assign physical meaning to field theory with higher derivatives? Phys. Rev. D. 48, 5811–5817 (1993)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Elishakoff, I., Livshits, D.: Some closed-form solutions in random vibration of Timoshenko beams. In: Petyt, M., Wolfe, H.F. (eds.) Proceedings of the 2nd International Conference on Recent Advances in Structural Dynamics, pp. 639–648. Institute of Sound and Vibration Research, University of Southampton, Southampton (1984)

    Google Scholar 

  17. Elishakoff, I., Lubliner, E.: Random Vibration of a Structure via Classical and Non-classical Theories. Proceedings of the IUTAM Symposium on Probabilistic Mechanics of Structures. Springer, Stockholm (1985)

    Google Scholar 

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

    Google Scholar 

  19. Elishakoff, I., Clement, S.: A consistent set of nonlocal Bresse–Timoshenko equations for nanobeams with surface effects. J. Appl. Mech. 80(6), 061001 (2013)

    Google Scholar 

  20. Elishakoff, I., Kaplunov, J., Nolde, E.: Celebrating the centenary of Timoshenko’s study of effects of shear deformation and rotary inertia. Appl. Mech. Rev. 67(6), 060802 (2015)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. 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)

    Google Scholar 

  23. 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 (2018)

    Google Scholar 

  24. 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)

    MathSciNet  MATH  Google Scholar 

  25. Elishakoff, I., Hache, F., Challamel, N.: Comparison of refined beam theories for parametric instability. AIAA Tech. Notes 56(1), 438 (2018)

    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)

    MathSciNet  Google Scholar 

  27. Elishakoff, I.: Who developed the so-called Timoshenko beam theory? Math. Mech. Solids 25(1), 97–116 (2019)

    MathSciNet  Google Scholar 

  28. Elishakoff, I.: Handbook on Timoshenko–Ehrenfest Beam and Uflyand–Mindlin Plate Theories. World Scientific, Singapore (2020)

    Google Scholar 

  29. Feng, B., Almeida Júnior, D.S., Dos Santos, M.J., Rosário Miranda, L.G.: A new scenario for stability of nonlinear Bresse–Timoshenko type systems with time dependent delay frequency. Z. Angew. Math. Mech. 100(2), e201900160 (2020)

    Google Scholar 

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

    MATH  Google Scholar 

  31. Hache, F., Elishakoff, I., Challamel, N.: Critical comparison of exact solutions in random vibration of beams using three versions of Bresse–Timoshenko theory. Probab. Eng. Mech. 53, 95–108 (2018)

    Google Scholar 

  32. Han, S.M., Benaroya, H., Timothy, W.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 225(5), 935–988 (1999)

    MATH  Google Scholar 

  33. Huang, F.L.: Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1, 43–56 (1985)

    MathSciNet  MATH  Google Scholar 

  34. Kim, J.U., Renardy, Y.: Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25(6), 1417–1429 (1987)

    MathSciNet  MATH  Google Scholar 

  35. Lanczos, C.: The Variational Principles of Mechanics. Dover Publication, New York (1964)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  37. Liu, Z., Rao, B.: Energy decay rate of the thermoelastic Bresse system. Z. Angew. Math. Phys. 60(1), 54–69 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Malacarne, A., Muñoz Rivera, J.E.: Lack of exponential stability to Timoshenko system with viscoelastic Kelvin–Voigt type. Z. Angew. Math. Phys. 67, 67 (2016)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  40. Messaoudi, S.A., Said-Houari, B.: Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl. 348(1), 298–307 (2008)

    MathSciNet  MATH  Google Scholar 

  41. Muñoz Rivera, J.E., Racke, R.: Mildly dissipative nonlinear Timoshenko systems: global existence and exponential stability. J. Math. Anal. Appl. 276(1), 248–278 (2002)

    MathSciNet  MATH  Google Scholar 

  42. Muñoz Rivera, J.E., Racke, R.: Global stability for damped Timoshenko systems. Discrete Contin. Dyn. Syst. Ser. B. 9(6), 1625–1639 (2003)

    MathSciNet  MATH  Google Scholar 

  43. Muñoz Rivera, J.E., Racke, R.: Timoshenko systems with indefinite damping. J. Math. Anal. Appl. 341(2), 1068–1083 (2008)

    MathSciNet  MATH  Google Scholar 

  44. Nesterenko, V.V.: A theory for transverse vibrations of the Timoshenko beam. J. Math. Anal. Appl. 57(4), 669–677 (1993)

    MathSciNet  MATH  Google Scholar 

  45. Nicholson, J.W., Simmonds, J.G.: Timoshenko beam theory is not always more accurate than elementary beam theory. J. Appl. Mech. 44(2), 337–338 (1977)

    MATH  Google Scholar 

  46. Prathap, G.: The two frequency spectra of Timoshenko beams: a re-assessment. J. Sound Vib. 90(3), 443–445 (1983)

    MathSciNet  Google Scholar 

  47. Prüss, J.: On the spectrum of \(C_0\)-semigroups. Trans. Am. Math. Soc. 284(2), 847–857 (1984)

    MATH  Google Scholar 

  48. Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16(4), 487–491 (2003)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  50. Rayleigh, J.W.S.: The Theory of Sound, pp. 1842–1919. Macmillan Publications, London (1877)

    Google Scholar 

  51. 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(5), 535–541 (2005)

    MathSciNet  MATH  Google Scholar 

  52. Roux, A., van der Merwe, A.J., van Rensburg, N.F.J.: Elastic waves in a Timoshenko beam with boundary damping. Wave Motion 57, 194–206 (2015)

    MathSciNet  MATH  Google Scholar 

  53. Russell, D.L.: Controllability and stabilization theory for linear partial differential equations. Recent progress and open problems. SIAM Rev. 20, 639–739 (1978)

    MathSciNet  MATH  Google Scholar 

  54. 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)

    MathSciNet  MATH  Google Scholar 

  55. Shi, D.-H., Feng, D.-X.: Exponential decay of Timoshenko beam with locally distributed feedback. In: Proceeding of the 99’IFAC World Congress. F, Beijing (1999)

  56. Smith, R.W.M.: Graphical representation of Timoshenko beam modes for clamped-clamped boundary conditions at high frequency: beyond transverse deflection. Wave Motion 45(6), 785–794 (2008)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  58. 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 

  59. Stephen, N.G.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 80(4), 578–582 (1982)

    Google Scholar 

  60. Stephen, N.G.: The second frequency spectrum of Timoshenko beams theory: further assessment. J. Sound Vib. 292(1–2), 372–389 (2006)

    MATH  Google Scholar 

  61. Stephen, N.G., Puchegger, S.: On the valid frequency range of Timoshenko beam theory. J. Sound Vib. 297(3–5), 1082–1087 (2006)

    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(2), 186–222 (1953)

    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, 744–746 (1921)

    Google Scholar 

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Acknowledgements

The authors are grateful to the referees for their constructive remarks, which have enhanced the presentation of this paper.

Funding

D. S. Almeida Júnior thanks the CNPq for financial support through the Project “Stabilization for Timoshenko systems from second spectrum point of view”—PNPD/Capes/InctMat/LNCC 88887.351763/2019-00. A. J. A. Ramos thanks the CNPq for financial support through the Project: “Asymptotic stabilization and numerical treatment for carbon nanotubes”—CNPq Grant 310729/2019-0. M. L. Santos thanks the CNPq for financial support through the Projects CNPq Grant 303026/2018-9 and CNPq Grant PDS114563/2018-7. Conselho Nacional de Desenvolvimento Científico e Tecnológico. Grant No. 310423/2016-3.

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Authors and Affiliations

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

    D. S. Almeida Júnior & M. L. Santos

  2. Faculty of Mathematics, Federal University of Pará, Raimundo Santana Street, s/n, Salinópolis, Pará, 68721-000, Brazil

    A. J. A. Ramos & M. L. Cardoso

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

    A. Soufyane

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  1. D. S. Almeida Júnior
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Almeida Júnior, D.S., Ramos, A.J.A., Soufyane, A. et al. Issues related to the second spectrum, Ostrogradsky’s energy and the stabilization of Timoshenko–Ehrenfest-type systems. Acta Mech 231, 3565–3581 (2020). https://doi.org/10.1007/s00707-020-02730-7

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