Skip to main content
SpringerLink
Log in

The bifurcation analysis on the circular functionally graded plate with combination resonances

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The bifurcation and chaos of a clamped circular functionally graded plate is investigated. Considered the geometrically nonlinear relations and the temperature-dependent properties of the materials, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic excitation and thermal load are derived. The Duffing nonlinear forced vibration equation is deduced by using Galerkin method and a multiscale method is used to obtain the bifurcation equation. According to singularity theory, the universal unfolding problem of the bifurcation equation is studied and the bifurcation diagrams are plotted under some conditions for unfolding parameters. Numerical simulation of the dynamic bifurcations of the FGM plate is carried out. The influence of the period doubling bifurcation and chaotic motion with the change of an external excitation are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Yamanouchi, M., Hirai, T., Shiota, I.: Overall view of the P/M fabrication of functionally gradient materials. In: Proceedings of the First International Symposium on Functionally Gradient Materials, Sendai, Japan, pp. 59–64 (1990)

    Google Scholar 

  2. Noda, N.: Thermal stresses in functionally graded materials. J. Therm. Stresses 22, 477–512 (1999)

    Article  MathSciNet  Google Scholar 

  3. Reddy, J.N., Praveen, G.N.: Nonlinear transient thermo elastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35, 4457–4476 (1998)

    Article  MATH  Google Scholar 

  4. Liew, K.M., Yang, J., Kitipornchai, S.: Thermal post-buckling of laminated plates comprising FGM with temperature-dependent properties. J. Appl. Mech. 71, 839–850 (2004)

    Article  MATH  Google Scholar 

  5. Liew, K.M., Yang, J., Kitipornchai, S.: Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Int. J. Solids Struct. 40, 3869–3892 (2003)

    Article  MATH  Google Scholar 

  6. Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47, 663–684 (2000)

    Article  MATH  Google Scholar 

  7. Liew, K.M., He, X.Q., Kitipornchai, S.: Finite element method for the feedback control of FGM shells in the frequency domain via piezoelectric sensors and actuators. Comput. Methods Appl. Mech. Eng. 193, 257–273 (2004)

    Article  MATH  Google Scholar 

  8. Hosseini Kordkheili, S.A., Naghdabadi, R.: Geometrically non-linear thermo-elastic analysis of functionally graded shells using finite element method. Int. J. Numer. Methods Eng. 72(8), 964–986 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zhao, X., Lee, Y.Y., Liew, K.M.: Mechanical and thermal buckling analysis of functionally graded plates. Compos. Struct. 90, 161–171 (2009)

    Article  Google Scholar 

  10. Reddy, J.N., Praveen, G.N.: Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35(33), 4457–4476 (1998)

    Article  MATH  Google Scholar 

  11. Ng, T.Y., Lam, K.Y., Liew, K.M.: Effect of FGM materials on the parametric resonance of plate structures. Comput. Methods Appl. Mech. Eng. 19(8-10), 953–962 (2000)

    Article  Google Scholar 

  12. Yang, J., Shen, H.S.: Free Vibration and parametric resonance of shear deformable functionally graded cylindrical panels. J. Sound Vib. 261(5), 871–893 (2003)

    Article  Google Scholar 

  13. Yang, J., Shen, H.S.: Dynamic response of initially stressed functionally graded rectangular thin plates. Compos. Struct. 54(4), 497–508 (2001)

    Article  Google Scholar 

  14. Vel, S.S., Batra, R.C.: Three-dimensional exact solutions for the vibration of functionally graded rectangular plate. J. Sound Vib. 272, 703–730 (2004)

    Article  Google Scholar 

  15. Liew, K.M., He, X.Q., Ng, T.Y., Kitipornchai, S.: Finite element piezothermoelasticity analysis and the active control of FGM plates with integrated piezoelectric sensors and actuators. Comput. Mech. 31, 350–358 (2003)

    MATH  Google Scholar 

  16. Allahverdizadeh, A., Naei, M.H., Bahrami, M.N.: Nonlinear free and forced vibration analysis of thin circular functionally graded plates. J. Sound Vib. 310(4–5), 966–984 (2008)

    Article  Google Scholar 

  17. Zhao, X., Liew, K.M.: Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method. Comput. Methods Appl. Mech. Eng. 198, 2796–2811 (2009)

    Article  MATH  Google Scholar 

  18. Zhao, X., Lee, Y.Y., Liew, K.M.: Thermoelastic and vibration analysis of functionally graded cylindrical shells. Int. J. Mech. Sci. 51, 694–707 (2009)

    Article  Google Scholar 

  19. Hao, Y.X., Chen, L.H., Zhang, W.: Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. J. Sound Vib. 312(4–5), 862–892 (2008)

    Article  Google Scholar 

  20. Yang, J., Hao, Y.X., Zhang, W., Kitipornchai, S.: Nonlinear dynamic response of a functionally graded plate with a through-width surface crack. Nonlinear Dyn. 59, 207–219 (2010)

    Article  MATH  Google Scholar 

  21. Sridhar, S., Mook, D.T., Nayfeh, A.H.: Non-linear resonances in the forced responses of plates, part I: symmetric responses of circular plates. J. Sound Vib. 41, 359–373 (1975)

    Article  MATH  Google Scholar 

  22. Hadian, J., Nayfeh, A.H.: Modal interaction in circular plates. J. Sound Vib. 142, 279–292 (1990)

    Article  MathSciNet  Google Scholar 

  23. Feng, Z.C., Sethna, P.R.: Global bifurcations in the motion of parametrically excited thin plate. Nonlinear Dyn. 4, 389–408 (1993)

    Article  Google Scholar 

  24. Chang, S.I., Bajaj, A.K., Krousgrill, C.M.: Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dyn. 4, 433–60 (1993)

    Article  Google Scholar 

  25. Liu, C.F., Chen, G.T.: Geometrically nonlinear axisymmetric vibrations of polar orthotropic circular plates. Int. J. Mech. Sci. 3, 325–333 (1996)

    Article  Google Scholar 

  26. Zhang, W., Liu, Z.M., Yu, P.: Global dynamics of a parametrically and externally excited thin plate. Nonlinear Dyn. 24(3), 245–268 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Haterbouch, M., Benamar, R.: The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: Iterative and explicit analytical solution for non-linear transverse vibrations. J. Sound Vib. 265, 123–154 (2003)

    Article  Google Scholar 

  28. Haterbouch, M., Benamar, R.: The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part II: Iterative and explicit analytical solution for non-linear coupled transverse and in-plane vibrations. J. Sound Vib. 277, 1–30 (2004)

    Article  Google Scholar 

  29. Haterbouch, M., Benamar, R.: Geometrically nonlinear free vibrations of simply supported isotropic thin circular plates. J. Sound Vib. 280, 903–924 (2005)

    Article  Google Scholar 

  30. Akour, S.N., Nayfeh, J.F.: Nonlinear dynamics of polar-orthotropic circular plates. Int. J. Struct. Stab. Dyn. 6, 253–268 (2006)

    Article  Google Scholar 

  31. Zhang, W., Zhang, J.H., Yao, M.H.: The extended Melinkov method for non-autonomous nonlinear dynamical systems and application to multi-pulse chaotic dynamics of a buckled thin plate. Nonlinear Anal., Real World Appl. 11, 1442–1457 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  32. Touzé, C., Thomas, O., Amabili, M.: Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates. Int. J. Non-Linear Mech. 46, 234–246 (2011)

    Article  Google Scholar 

  33. Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, New York (1989)

    Book  MATH  Google Scholar 

  34. Touloukian, Y.S.: Thermophysical Properties of High Temperature Solid Materials. McMillan, New York (1967)

    Google Scholar 

  35. Shen, H.S.: Thermal postbulking behavior of shear deformable FGM plates with temperature-dependent properties. Int. J. Mech. Sci. 49(4), 466–47 (2007)

    Article  Google Scholar 

  36. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-Interscience, New York (1979)

    MATH  Google Scholar 

  37. Xu, K.Y., Cheng, C.J.: The Subharmonic bifurcation of a viscoelastic circular cylindrical shell. Nonlinear Dyn. 17, 159–171 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  38. Chen, F.Q., Liang, J.S., Chen, Y.S., et al.: Bifurcation analysis of an arch structure with parametric and forced excitation. Mech. Res. Commun. 34, 213–221 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. I. Springer, New York (1985)

    MATH  Google Scholar 

  40. Reddy, J.N., Chin, C.D.: Thermomechanical analysis of functionally graded cylinders and plates. J. Therm. Stresses 21(6), 593–629 (1998)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao, 066004, P.R. China

    Yuda Hu & Zhiqiang Zhang

Authors
  1. Yuda Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiqiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuda Hu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, Y., Zhang, Z. The bifurcation analysis on the circular functionally graded plate with combination resonances. Nonlinear Dyn 67, 1779–1790 (2012). https://doi.org/10.1007/s11071-011-0105-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0105-4

Keywords

Search

Navigation