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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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Asymptotics and stabilization for dynamic models of nonlinear beams; pp. 150–155

(Full article in PDF format) doi: 10.3176/proc.2010.2.14


Authors

Fágner D. Araruna, Pablo Braz e Silva, Enrique Zuazua

Abstract

We prove that the von Kármán model for vibrating beams can be obtained as a singular limit of a modified Mindlin–Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth-order dispersive operator is added. We also show that the energy of solutions for this modified Mindlin–Timoshenko system decays exponentially, uniformly with respect to the parameter k, when suitable damping terms are added. As k → ∞, one deduces the uniform exponential decay of the energy of the von Kármán model.

Keywords

vibrating beams, Mindlin–Timoshenko system, von von Kármán system, singular limit, uniform stabilization.

References

  1. Araruna , F. D. and Zuazua , E. Controllability of the Kirchhoff system for beams as a limit of the Mindlin–Timoshenko system. SIAM J. Cont. Optim. , 2008 , 47 , 1909–1938.
doi:10.1137/060659934

  2. Araruna , F. D. , Braz e Silva , P. , and Zuazua , E. Asymptotic limits and stabilization for the 1D nonlinear Mindlin–Timoshenko system. J. Syst. Sci. Complexity (to appear).

  3. Doyle , J. F. Wave Propagation in Structures. Springer-Verlag , New York , 1997.

  4. Engelbrecht , J. , Berezovski , A. , Pastrone , F. , and Braun , M. Waves in microstructured materials and dispersion. Phil. Mag. , 2005 , 85 , 4127–4141.
doi:10.1080/14786430500362769

  5. Favini , A. , Horn , M. A. , Lasiecka , I. , and Tartaru , D. Global existence , uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation. Diff. Integ. Eqns , 1996 , 9 , 267–294.

  6. Lagnese , J. E. Boundary Stabilization of Thin Plates. SIAM , 1989.

  7. Lagnese , J. E. and Leugering , G. Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J. Diff. Eqns , 1991 , 91 , 355–388.
doi:10.1016/0022-0396(91)90145-Y

  8. Lagnese , J. E. and Lions , J.-L. Modelling Analysis and Control of Thin Plates. RMA 6 , Masson , Paris , 1988.

  9. Pazoto , A. , Perla Menzala , G. , and Zuazua , E. Stabilization of Berger–Timoshenko’s equation as limit of the uniform stabilization of the von Kármán system of beams and plates. Math. Model. Numer. Anal. , 2002 , 36 , 657–691.
doi:10.1051/m2an:2002029

10. Perla Menzala , G. and Zuazua , E. Explicit exponential decay rates for solutions of von Kármán’s system of thermoelastic plates. C. R. Acad. Sci. Paris , 1997 , 324 , 49–54.

11. Perla Menzala , G. and Zuazua , E. The beam equation as a limit of 1-D nonlinear von Kármán model. Appl. Math. Lett. , 1999 , 12 , 47–52.
doi:10.1016/S0893-9659(98)00125-6

12. Perla Menzala , G. and Zuazua , E. Timoshenko’s beam equation as limit of a nonlinear one-dimensional von Kármán system. Proc. Roy. Soc. Edinburgh , 2000 , 130A , 855–875.
doi:10.1017/S0308210500000470

13. Perla Menzala , G. and Zuazua , E. Timoshenko’s plate equation as a singular limit of the dynamical von Kármán system. J. Math. Pures Appl. , 2000 , 79 , 73–94.
doi:10.1016/S0021-7824(00)00149-5
 
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Current Issue: Vol. 68, Issue 4, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December