Advertisement

On the quintic nonlinear Schrödinger equation created by the vibrations of a square plate on a weakly nonlinear elastic foundation and the stability of the uniform solution

  • Ben T. NoharaEmail author
  • Akio Arimoto
Article
  • 45 Downloads

Abstract

Plates are common structural elements of most engineering structures, including aerospace, automotive, and civil engineering structures. The study of plates from theoretical perspective as well as experimental viewpoint is fundamental to understanding of the behavior of such structures. The dynamic characteristics of plates, such as natural vibrations, transient responses for the external forces and so on, are especially of importance in actual environments. In this paper, we conside the envelope surface created by the vibrations of a square plate on a weakly nonliner elastic foundation and analyze the stability of the uniform solution of the governing equation for the envelope surface. We derive the two-dimensional equation that governs the spatial and temporal evolution of the envelope surface on cubic nonlinear elastic foundation. The fact that the governing equation becomes the quintic nonlinear Schrödinger equation is shown. Also we obtain the stability condition of the uniform solution of the quintic nonlinear Schrödinger equation.

Key words

elastic foundation envelope nearly monochromatic waves perturbation Schrödinger equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S.P. Timoshenko, Theory of Plates and Shells. McGraw-Hill, New York, 1940.zbMATHGoogle Scholar
  2. [2]
    S.P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells McGraw-Hill, Singapore, 1970.Google Scholar
  3. [3]
    A.C. Ugural, Stresses in Plates and Shells. McGraw-Hill, New York, 1981.Google Scholar
  4. [4]
    A.W. Leissa, Vibration of Plates. NASA-Sp-160, 1969.Google Scholar
  5. [5]
    H.N. Chu and G. Herrmann, Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. Journal of Applied Mechnics,23 (1956), 532–540.zbMATHMathSciNetGoogle Scholar
  6. [6]
    M.M. Hrabok and T.M. Hrudey, A review and catalog of plate bending finite elements. Computers and Structures,19 (1984), 479–495.CrossRefGoogle Scholar
  7. [7]
    R.C. Averill and J.N. Reddy, Behavior of plate elements based on the first-order shear deformation theory. Engineering Computations,7 (1990), 57–74.CrossRefGoogle Scholar
  8. [8]
    J.N. Reddy, An Introduction to the Finite Element Method, second edition. McGraw-Hill, New York, 1993.Google Scholar
  9. [9]
    R. Haberman, Elementary Applied Partial Differential Equations. Prentice Hall, Englewood Cliff, NJ, 1983.zbMATHGoogle Scholar
  10. [10]
    Y. Goda, Numerical experiments on wave statistics with spectral simulation. Report Port Harbour Research Institute,9 (1970), 3–57.Google Scholar
  11. [11]
    S.K. Chakrabarti, R.H. Snider and P.H. Feldhausen, Mean length of runs of ocean waves. Journal of Geophysical Research,79 (1974), 5665–5667.CrossRefGoogle Scholar
  12. [12]
    M.S. Longuet-Higgins, Statistical properties of wave groups in a random sea-state. Philosophical Transactions of the Royal Society of London, Series A,312 (1984), 219–250.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Physics Review Letters,17 (1966), 996–998.CrossRefGoogle Scholar
  14. [14]
    G.P. Agrawal, Fiber-Optic Communication System, second edition. Wiley, New York, 1997.Google Scholar
  15. [15]
    B.T. Nohara, Governing equations of envelope surface created by directional, nearly monochromatic waves. Journal of Society of Industrial and Applied Mathematics,13 (2003), 75–86, (in Japanese).MathSciNetGoogle Scholar
  16. [16]
    B.T. Nohara, Derivation and consideration of governing equations of the envelope surface created by directional, nearly monochromatic waves. International Journal of Nonlinear Dynamics and Chaos in Engineering Systems,31 (2003), 375–392.zbMATHMathSciNetGoogle Scholar
  17. [17]
    B.T. Nohara, Governing equations of envelope surface created by nearly bichromatic waves, Propagating on an Elastic Plate and Their Stability. Japan Journal of Industrial and Applied Mathematics,22 (2005), 87–109.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    B.T. Nohara and A. Arimoto, The stability of the governing equation of envelope surface created by nearly bichromatic waves propagating on an elastic plate. Nonlinear Analysis,63 (2005), 2197–2208.CrossRefGoogle Scholar
  19. [19]
    B.T. Nohara, A. Arimoto and T. Saigo, Governing equations of envelopes created by nearly bichromatic waves and relation to the nonlinear Schrödinger equation. Chaos, Soliton and Fractals, 2006, to appear.Google Scholar
  20. [20]
    B.T. Nohara and T. Saigo, Numerical simulations of the envelope created by nearly bichromatic waves. Proceeding on COMSOL Multiphysics Conference, Boston, USA, October 23–25, 2005, 383–386.Google Scholar
  21. [21]
    M.A. Zarubinskaya and W.T. van Horssen, On the vibration on a simply supported square plate on a weakly nonlinear elastic fooundation. International Journal of Nonlinear Dynamics and Chaos in Engineering Systems,40 (2005), 35–60.zbMATHGoogle Scholar
  22. [22]
    A.H. Nayfeh, Perturbation Methods. Wiley, New York, 2002.Google Scholar
  23. [23]
    A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations. Wiley, New York, 1979.zbMATHGoogle Scholar
  24. [24]
    T.B. Benjamin and J.E. Feir, The disintegration of wavetrains on deep water, Part 1, Theory. Journal of Fluid Mechnics,27 (1967), 417–430.zbMATHCrossRefGoogle Scholar

Copyright information

© JJIAM Publishing Committee 2007

Authors and Affiliations

  1. 1.Department of Electronic and Computer EngineeringTokyoJapan
  2. 2.Department of MathematicsMusashi Institute of TechnologyTokyoJapan

Personalised recommendations