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On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency

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Abstract

In the present work, we prove that there exists a relation between a physical inconsistence known as second spectrum of frequency or non-physical spectrum and the exponential decay of a dissipative Timoshenko system where the damping mechanism acts on angle rotation. The so-called second spectrum is addressed into stabilization scenario and, in particular, we show that the second spectrum of the classical Timoshenko model can be truncated by taking a damping mechanism. Also, we show that dissipative Timoshenko type systems which are free of the second spectrum [based on important physical and historical observations made by Elishakoff (Advances mathematical modeling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010), Elishakoff et al. (ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11 2015) and Elishakoff et al. (Int J Solids Struct 109:143–151, 2017)] are exponential stable for any values of the coefficients of system. In this direction, we provide physical explanations why weakly dissipative Timoshenko systems decay exponentially according to equality between velocity of wave propagation as proved in pioneering works by Soufyane (C R Acad Sci 328(8):731–734, 1999) and also by Muñoz Rivera and Racke (Discrete Contin Dyn Syst B 9:1625–1639, 2003). Therefore, the second spectrum of the classical Timoshenko beam model plays an important role in explaining some results on exponential decay and our investigations suggest to pay attention to the eventual consequences of this spectrum on stabilization setting for dissipative Timoshenko type systems.

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References

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

    Article  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 Brese–Timoshenko equations. Acta Mech. 66(1–4), 39–59 (1987)

    Article  MATH  Google Scholar 

  3. Almeida Júnior, D.S., Santos, M.L., Muñoz Rivera, J.E.: Stability to 1-D thermoelastic Timoshenko beam acting on shear force. Zeitschrift für Angewandte Mathematik und Physik 65, 1233–1249 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Almeida Júnior, D.S., Muñoz Rivera, J.E.: Stability criterion to explicit finite difference applied to the Bresse system. Afrika Matematika 26(5–6), 761–778 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Anderson, R.A.: Flexural vibration in uniform beams according to the Timoshenko theory. J. Appl. Mech. 75, 504–510 (1953)

    MATH  Google Scholar 

  8. Andreaus, U., Dell’Isola, F., Porfiri, M.: Piezoelectric passive distributed controllers for beam flexural vibrations. J. Vib. Control 10(5), 625–659 (2004)

    MATH  Google Scholar 

  9. Barr, A.D.S.: Parametric vibration in beams. In: Proceedings of the 14th Canadian Congress of Applied Mechanics, Queen’s University, Kingston, ON. vol. 1. pp. 3–9 (1993)

  10. Bhaskar, A.: Elastic waves in Timoshenko beams: the ’lost’ and ’found’ of an eigenmode. Proc. R. Soc. A Math. Phys. Eng. Sci. 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. Boussouira, F.A., Rivera, M.R., Almeida júnior, D.S.: Stability to weak dissipative Bresse system. J. Math. Anal. Appl. 374, 481–498 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  14. Campelo, A.D.S., Almeida Júnior, D.S., Santos, M.L.: Stability to the dissipative Reissner–Mindlin–Timoshenko acting on displacement equation. Eur. J. Appl. Math. 27(2), 157–193 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Campelo, A.D.S., Almeida Júnior, D.S., Santos, M.L.: Stability of dissipative Reissner–Mindlin–Timoshenko plates: a sharp result. Eur. J. Appl. Math. 27 (2017). https://doi.org/10.1017/S0956792517000092

  16. Casas, P.S., Quintanilla, R.: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32(6), 652–658 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cazzani, A., Stochino, F., Turco, E.: On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Zeitschrift für Angewandte Mathematik und Physik 67(2), 24 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cattaneo, C.: Sulla conduzione de calore. Atti del Seminario Matematico e Físico della Universita di Moderna 3, 3–21 (1948)

    Google Scholar 

  19. Cattaneo, C.: Sur une forme d’équation de la chaleur eliminant le paradoxe d’une propagation instantantanée. Comptes Rendus de l’Académie des Sciences 247, 431–433 (1958)

    MATH  Google Scholar 

  20. Cazzani, A., Stochino, F., Turco, E.: On the whole spectrum of Timoshenko beams. Part II: further applications. Zeitschrift für Angewandte Mathematik und Physik 67(2), 25 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. Zeitschrift für Angewandte Mathematik und Mechanik 96(10), 1220–1244 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chan, K.T., Wang, X.Q., So, R.M.C., Reid, S.R.: Superposed standing waves in a Timoshenko beam. Proc. R. Soc. A Math. Phys. Eng. Sci. 458, 83–108 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chervyakov, A.M., Nesterenko, V.V.: Is it possible to assign physical meaning to field theory with higher derivatives? Phys. Rev. D 48(12), 5811–5817 (1993)

    Article  Google Scholar 

  24. Downs, B.: Transverse vibration of a uniform simply supported Timoshenko beam without transverse deflection. Trans. ASME J. Appl. Mech. 43, 671–674 (1976)

    Article  MATH  Google Scholar 

  25. Davies, R.M.: The frequency of transverse vibration of a loaded fixed-free bar III. The effect of rotatory inertia of the bar. Philos. Mag. 23(155), 563–573 (1937)

  26. 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 

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

    Google Scholar 

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

  29. Elishakoff, I., Lubliner, E.: Random vibration of a structure via classical and non-classical theories. In: Proceedings of the IUTAM Symposium on Probabilistic Mechanics of Structures. Springer, New York (1985)

  30. Elishakoff, I.: An equation both more consistent and simpler than the Bresse–Timoshenko equation. In: Gilat, R., Banks-Sills, L. (eds.) Advanced in Mathematical Modeling and Experimental Methods for Materials and Structures, The jacob aboudi volume, Springer, Berlin. pp. 249–254 (2010)

  31. 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 

  32. 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 

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

    MathSciNet  MATH  Google Scholar 

  34. Fernandez Sare, H.D., Racke, R.: On the stability of damped Timoshenko systems: Cattaneo versus Fourier law. Arch. Ration. Mech. Anal. 194(1), 221–251 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  36. Hagedorn, P., DasGupta, A.: Vibration and Waves in Continuous Mechanical Systems. Wiley, England (2007)

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  38. Huang, T.C.: The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. J. Appl. Mech. 28, 579–584 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  39. Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 62, 375–391 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lagnese, J.E., Lions, J.L.: Modeling, Analysis and Control of Thin Plates. Collection RMA, Masson, Paris (1988)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  42. Leseduarte, M.C., Magaña, A., Quintanilla, R.: On the time decay of solutions in porous-thermo-elasticity of type III. Discrete Contin. Dyn. Syst. Ser. B 13(2), 375–391 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. 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 

  44. Liu, Z., Rao, B.: Energy decay rate of the thermoelastic Bresse system. Zeitschrift für angewandte Mathematik und Physik 60(1), 54–69 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Magaña, A., Quintanilla, R.: On the exponential decay of solutions in one-dimensional generalized porous-thermo-elasticity. Asymptot. Anal. 49(3,4), 173–187 (2006)

    MathSciNet  MATH  Google Scholar 

  46. Manevich, A.I.: Dynamics of Timoshenko beam on linear and nonlinear foundation phase relations, significance of the second spectrum, stability. J. Sound Vib. 344, 209–220 (2015)

    Article  Google Scholar 

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

    Google Scholar 

  48. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. 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 

  53. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Öncü, T.S., Moodie, T.B.: On the constitutive relations for second sound in elastic solids. Arch. Ration. Mech. Anal. 121(1), 87–99 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  56. Parshad, R.D., Quansah, E., Black, K., Beauregard, M.: Biological control via ecological damping: an approach that attenuates non-target effects. Math. Biosci. 273, 23–44 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  60. 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 

  61. 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 

  62. 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)

  63. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  69. Straughan, B.: Heat Waves, Applied Mathematical Sciences, vol. 177. Springer, Berlin (2011)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  71. Trefethen, L.N.: Group velocity in finite difference schemes. SIAM Rev. 24(2), 113–136 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  72. Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 6(41/245), 744–746 (1921)

    Article  Google Scholar 

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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

  2. Interdisciplinary Innovation Laboratory (LabX), Faculty of Sciences, Federal University of Pará, Raimundo Santana Street, s/n, Salinópolis, Pará, 68721-000, Brazil

    A. J. A. Ramos

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  1. D. S. Almeida Júnior
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Dedicated to Prof. Marcelo Moreira Cavalcanti on the occasion of his 60th Birthday.

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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). https://doi.org/10.1007/s00033-017-0881-x

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