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On Aspects of Asymptotics for Plate Equations

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Abstract

In this paper some initial-boundary value problems for plate equations will be studied. These initial-boundary value problems can be regarded as simple models describing free oscillations of plates on elastic foundations or of plates to which elastic springs are attached on the boundary. It is assumed that the foundations and springs have a different behavior for compression and for extension. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For specific parameter values it turns out that complicated internal resonances occur.

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References

  1. Moore, K. S., ‘Large torsional oscillations in a suspension bridge: Multiple periodic solutions to a nonlinear wave equation’, SIAM Journal on Mathematical Analysis 33(6), 2002, 1411–1429.

    Article  Google Scholar 

  2. Lazer, A. C. and McKenna, P. J., ‘Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis’, SIAM Review 32, 1990, 537–578.

    Article  Google Scholar 

  3. Lazer, A. C. and McKenna, P. J., ‘Large scale oscillatory behavior in loaded asymmetric systems’, Annales Inst. Henri Poincaré Anal. Non Lineaire 4, 1987, 243–274.

    Google Scholar 

  4. McKenna, P. J. and Walter, W., ‘Nonlinear oscillations in a suspension bridges’, Archive for Rational Mechanics and Analysis 98, 1987, 167–177.

    Article  Google Scholar 

  5. McKenna, P. J. and Walter, W., ‘Traveling waves in a suspension bridge’, SIAM Journal on Applied Mathematics 50, 1990, 702–715.

    Article  Google Scholar 

  6. Choi, Y. S. and McKenna, P. J., ‘The study of a nonlinear suspension bridge equation by a variational reduction method’, Applicable Analysis 50, 1993, 73–92.

    Google Scholar 

  7. Chen, Y. and McKenna, P. J., ‘Traveling waves in a nonlinear suspended beam: Theoretical results and numerical observations’, Journal of Differential Equations 136, 1997, 325–355.

    Article  Google Scholar 

  8. Humphreys, L. D. and McKenna, P. J., ‘Multiple periodic solutions for a nonlinear suspension bridge equation’, IMA Journal of Applied Mathematics 63, 1999, 37–49.

    Article  Google Scholar 

  9. Ahmed, N. U. and Biswas, S. K., ‘Mathematical modeling and control of large space structures with multiple appendages’, Journal of Mathematics and Computational Modeling (10), 1988, 891–900.

    Google Scholar 

  10. Ahmed, N. U. and Harbi, H., 1998, ‘Mathematical analysis of dynamical models of suspension bridges’, SIAM Journal on Applied Mathematics 58(3), 1988, 853–874.

    Article  Google Scholar 

  11. van Horssen, W. T., ‘An asymptotic theory for a class of initial-boundary value problems for weakly nonlinear wave equations with an application to a model of the galloping oscillations of overhead transmission lines’, SIAM Journal on Applied Mathematics 48, 1988, 1227–1243.

    Article  Google Scholar 

  12. van Horssen, W. T., ‘Asymptotics for a class of semilinear hyperbolic equations with an application to a problem with quadratic nonlinearity’, Nonlinear Analysis 19, 1992, 501–530.

    Article  Google Scholar 

  13. van Horssen, W. T. and van der Burgh, A. H. P., ‘On initial-boundary value problems for weakly semilinear telegraph equations. Asymptotic theory and applications’, SIAM Journal on Applied Mathematics 48, 1988, 719–736.

    Article  Google Scholar 

  14. Boertjens, G. J. and van Horssen, W. T., ‘On mode interactions for a weakly nonlinear beam equation’, Nonlinear Dynamics 17, 1998, 23–40.

    Article  Google Scholar 

  15. Boertjens, G. J. and van Horssen, W. T., ‘An asymptotic theory for a weakly nonlinear beam equation with a quadratic perturbation’, SIAM Journal on Applied Mathematics 60(2), 2000, 602–632.

    Article  Google Scholar 

  16. van Horssen, W. T. and Zarubinskaya, M. A., ‘On an elastic dissipation model for a cantilevered beam’, Quarterly of Applied Mathematics 61(3), 2002, 565–573.

    Google Scholar 

  17. Zarubinskaya, M. A. and van Horssen, W. T., ‘On the free vibrations of a rectangular plate with two opposite sides simply supported and the other sides attached to linear springs’, Journal of Sound and Vibration 278(4–5), 2004, 1081–1093.

    Article  Google Scholar 

  18. van Horssen, W. T., ‘On the applicability of the method of separation of variables for partial difference equations’, Journal of Difference Equations and Applications 8(1), 2002, 53–60.

    Article  Google Scholar 

  19. Zarubinskaya, M. A. and van Horssen, W. T., ‘On the vibration of a simply supported square plate on a weakly nonlinear elastic foundation’, Reports of the Department of Applied Mathematical Analysis, Delft, http://ta.twi.tudelft.nl, 2004.

  20. Leissa, A. W., ‘The free vibrations of rectangular plate’, Journal of Sound and Vibration 31(3), 1973, 257–293.

    Google Scholar 

  21. Gorman, D. J., Free Vibration Analysis of Rectangular Plates, Elsevier, New York, 1982.

    Google Scholar 

  22. Gorman, D. J., Vibration Analysis of Plates by the Superposition Method, World Scientific, Singapore, 1999.

    Google Scholar 

  23. Park, J., Siegmund, T., and Mongeau, L., ‘Analysis of the flow-induced vibrations of viscoelastically supported rectangular plates’, Journal of Sound and Vibration 261, 2003, 225–245.

    Article  Google Scholar 

  24. Russel, D. L., ‘On the positive root of the fourth derivative operator’, Quaterly of Applied Mathematics XVI(4), 1991, 751–773.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematical Analysis, Faculty of Information Technology and Systems, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    M. A. Zarubinskaya & W. T. Van Horssen

Authors
  1. M. A. Zarubinskaya
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  2. W. T. Van Horssen
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Correspondence to M. A. Zarubinskaya.

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Zarubinskaya, M.A., Horssen, W.T.V. On Aspects of Asymptotics for Plate Equations. Nonlinear Dyn 41, 403–413 (2005). https://doi.org/10.1007/s11071-005-1396-0

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  • DOI: https://doi.org/10.1007/s11071-005-1396-0

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