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Journal of Mechanical Science and Technology

, Volume 23, Issue 8, pp 2299–2307 | Cite as

Schauder fixed point theorem based existence of periodic solution for the response of Duffing’s oscillator

  • Ahmad Feyz Dizaji
  • Hossein Ali SepianiEmail author
  • Farzad Ebrahimi
  • Akbar Allahverdizadeh
  • Hasan Ali Sepiani
Article

Abstract

An initial-boundary value problem that is Duffing’s oscillator with time varying coefficients will be studied. Using Banach’s fixed-point theorem, the existence of periodic solution of the equation will be predicted. The method applied in this paper is the Schauder second fixed point theorem, which includes the response of structures under vibratory force systems. As an example, the dynamics of nonlinear simply supported rectangular thin plate under influence of a relatively moving mass is studied. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development vibration amplitude, which is Duffing’s oscillator. Finally, a parametric study is developed, after that some numerical examples are solved, and the validity of the present analysis is clearly shown.

Keywords

Banach’s theorem Large deformation of thin plates Moving loads Non-linear vibration Schauder fixed point theorem 

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References

  1. [1]
    J. R. Graef and L. Kong, A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems, Nonlinear Analysis, 66(1) (2007) 2389–2412.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    HABIB MÂAGLI, FATEN TOUMI, & MALEK ZRIBI, EXISTENCE OF POSITIVE SOLUTIONS FOR SOME POLYHARMONIC NONLINEAR BOUNDARY-VALUE PROBLEMS, Electronic Journal of Differential Equations, 2003(58) (2003) 58 1–19.Google Scholar
  3. [3]
    P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations, 190 (2003) 643–662.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    K. Latracha, M. A. Taoudib and A. Zeghal, Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations, J. Differential Equations, 221 (2006) 256–271.CrossRefMathSciNetGoogle Scholar
  5. [5]
    J. Benedikt, Uniqueness theorem for p-biharmonic equations, Electronic Journal of Differential Equations, 2002(53) (2002) 1–17.MathSciNetGoogle Scholar
  6. [6]
    B. J. McCartin, An alternative analysis of Duffing’s equation, SIAM Rev., 34 (1992) 482–491.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    K. B. Blair, C. M. Krousgrill and T. N. Farris, Harmonic balance and continuation techniques in the dynamic analysis of Duffing’s equation, J. Sound Vibration, 202 (1997) 717–731.CrossRefMathSciNetGoogle Scholar
  8. [8]
    P. J. Holmes, New Approaches to Nonlinear Problems in Dynamics, SIAM, Philadelphia, PA, 1981.Google Scholar
  9. [9]
    P. Habets and G. Metzen, Existence of periodic solutions of Duffing’s equation, J. Diff. Equs., 78 (1989) 1–32.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    H. Z. Wang and Y. Li, Existence and uniqueness of periodic solutions for Duffing equation, J. Diff. Equs., 108 (1994) 152–169.zbMATHCrossRefGoogle Scholar
  11. [11]
    C. W. Wang, Multiplicity of periodic solutions for Duffing equation under nonuniform non-resonance conditions, Proc. Amer. Math. Soc., 126 (1998) 1725–1732.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    T. R. Ding, R. Iannacci and F. Zanolin, On periodic solutions of sublinear Duffing equations, J. Math. Anal. Appl., 158 (1991) 316–332.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    T. Matsuo and A. Kishima, Numerical analysis of bifurcations in Duffing’s equation with hysteretic functions, Electron. Comm. Japan Part III Fund. Electron. Sci., 75 (1992) 61–71.CrossRefMathSciNetGoogle Scholar
  14. [14]
    R. Kosechi, The unit condition and global existence for a class of nonlinear Klein-Gordon equation, J. Diff. Equs., 100 (1992) 257–268.CrossRefGoogle Scholar
  15. [15]
    V. Geoggiev, Decay estimates for the Klein-Gordon equation, Comm. Partial Diff. Equs., 17 (1992) 1111–1139.CrossRefGoogle Scholar
  16. [16]
    T. Ozawa, K. Tsutaya and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two dimensional space, Math. Z., 222 (1996) 341–362.zbMATHMathSciNetGoogle Scholar
  17. [17]
    C. AVRAMESCU, A fixed point theorem for multivalued mappings, Electronic Journal of Qualitative Theory of Differential Equations.Google Scholar
  18. [18]
    P. Gao and B. L. Guo, The time-periodic solution for a 2D dissipative Klein-Gordon equation, J. Math. Anal. Appl., 296 (2004) 286–294.CrossRefMathSciNetGoogle Scholar
  19. [19]
    C.-Z. Bai and J.-X Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Applied Mathematics and Computation, 150 (2004) 611–621.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    J. CHU and P. J. TORRES, APPLICATIONS OF SCHAUDER’s FIXED POINT THEOREM TO SINGULAR DIFFERENTIAL EQUATIONS, Submitted exclusively to the London Mathematical Society, 2000 Mathematics Subject Classification, 34B15, 34D20.Google Scholar
  21. [21]
    P. J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007) 277 284.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Y. Guo, W. Shan and Weigao Ge, Positive solutions for second order m-point boundary value Problems, Journal of Computational and Applied Mathematics, 151 (2003) 415–424.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    D. H. Griffle, Applied Functional Analysis, Wiley, New York, 1985.Google Scholar
  24. [24]
    A. C. Ugural, Stresses in Plates and Shells, Wiley, New York, 1990.Google Scholar
  25. [25]
    M. Shadnam, M. Mofid and J.E. Akin, On the dynamic response of a rectangular plate with moving mass, Journal of Thin Walled Structures, 2001; 39(9) 797–806.CrossRefGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers and Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ahmad Feyz Dizaji
    • 1
  • Hossein Ali Sepiani
    • 1
    Email author
  • Farzad Ebrahimi
    • 1
  • Akbar Allahverdizadeh
    • 1
  • Hasan Ali Sepiani
    • 2
  1. 1.School of Mechanical EngineeringUniversity of TehranTehranIran
  2. 2.Department of Marine TechnologyAmirkabir University of TechnologyAmirkabirIran

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