Skip to main content
SpringerLink
Log in

Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The global bifurcations and multi-pulse chaotic dynamics of a simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations are investigated in this paper for the first time. The formulas of the laminated composite piezoelectric rectangular plate are derived by using the von Karman-type equation, the Reddy’s third-order shear deformation plate theory and the Galerkin’s approach. The extended Melnikov method is improved to enable us to analyze directly the non-autonomous nonlinear dynamical system, which is applied to the non-autonomous governing equations of motion for the laminated composite piezoelectric rectangular thin plate. The results obtained here indicate that multi-pulse chaotic motions can occur in the laminated composite piezoelectric rectangular thin plate. Numerical simulation is also employed to find the multi-pulse chaotic motions of the laminated composite piezoelectric rectangular thin plate.

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. Tzou H.S., Ye R.: Pyroelectric and thermal strain effects of piezoelectric (PVDF and PZT) devices. Mech. Syst. Signal Process. 10, 459–469 (1996)

    Article  Google Scholar 

  2. Reddy J.N.: On laminated composite plates with integrated sensors and actuators. Eng. Struct. 21, 568–593 (1999)

    Article  Google Scholar 

  3. Ye R., Tzou H.S.: Control of adaptive shells with thermal and mechanical excitations. J. Sound Vib. 231, 1321–1338 (2000)

    Article  Google Scholar 

  4. Gopinathan S.V., Varadan V.V., Varadan V.K.: A review and critique of theories for piezoelectric laminates. Smart Mater. Struct. 9, 24–48 (2000)

    Article  Google Scholar 

  5. Oh I.K., Han J.H., Lee I.: Postbuckling and vibration characteristics of piezolaminated composite plate subject to thermo-piezoelectric loads. J. Sound Vib. 233, 19–40 (2000)

    Article  Google Scholar 

  6. Shen H.S.: Thermal postbuckling of shear-deformable laminated plates with piezoelectric actuators. Compos. Sci. Technol. 61, 1931–1943 (2001)

    Article  Google Scholar 

  7. Moita J.M.S., Soares C.M.M., Soares C.A.M.: Geometrically non-linear analysis of composite structures with integrated piezoelectric sensors and actuators. Compos. Struct. 57, 253–261 (2002)

    Article  Google Scholar 

  8. Varelis D., Saravanos D.A.: Coupled buckling and postbuckling analysis of active laminated piezoelectric composite plates. Int. J. Solids Struct. 41, 1519–1538 (2004)

    Article  MATH  Google Scholar 

  9. Varelis D., Saravanos D.A.: Mechanics and finite element for nonlinear response of active laminated piezoelectric plates. AIAA J. 42, 1227–1235 (2004)

    Article  Google Scholar 

  10. Liew K.M., He X.Q., Tan M.J., Lim H.K.: Dynamic analysis of laminated composite plates with piezoelectric sensor/ actuator patches using the FSDT mesh-free method. Int. J. Mech. Sci. 46, 411–431 (2004)

    Article  MATH  Google Scholar 

  11. Shen H.S.: Nonlinear bending analysis of unsymmetric cross-ply laminated plates with piezoelectric actuators in thermal environments. Compos. Struct. 63, 167–177 (2004)

    Article  Google Scholar 

  12. Ishihara M., Noda N.: Nonlinear dynamic behavior of a piezothermoelastic laminated plate with anisotropic material properties. Acta Mech. 166, 103–118 (2003)

    Article  MATH  Google Scholar 

  13. Huang X.L., Shen H.S.: Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators. Int. J. Mech. Sci. 47, 187–208 (2005)

    Article  MathSciNet  Google Scholar 

  14. Cheng J.Q., Wang B., Du S.Y.: A theoretical analysis of piezoelectric/composite anisotropic laminate with larger-amplitude deflection effect, Part I: fundamental equations. Int. J. Solids Struct. 42, 6166–6180 (2005)

    Article  MATH  Google Scholar 

  15. Rabinovitch O.: Geometrically nonlinear behavior of piezoelectric laminated plates. Smart Mater. Struct. 14, 785–798 (2005)

    Article  Google Scholar 

  16. Ye M., Sun Y.H., Zhang W., Zhan X.P., Ding Q.: Nonlinear oscillations and chaotic dynamics of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation. J. Sound Vib. 287, 723–758 (2005)

    Article  Google Scholar 

  17. Zhang W., Song C.Z., Ye M.: Further studies on nonlinear oscillations and chaos of a symmetric cross-ply laminated thin plate under parametric excitation. Int. J. Bifurcation Chaos 16, 325–347 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Huang J.H., Liu Y.C.: Electroelastic response of a laminated composite plate with piezoelectric sensors and actuators. ASCE J. Eng. Mech. 132, 889–897 (2006)

    Article  Google Scholar 

  19. Abe A., Kobayashi Y., Yamada G.: Nonlinear dynamic behaviors of clamped laminated shallow shells with one-to-one internal resonance. J. Sound Vib. 304, 957–968 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

  21. Zhu F.H., Fu Y.M.: Nonlinear dynamic response and fatigue damage evolution for piezoelectric laminated plates with matrix cracks. Theor. Appl. Fract. Mech. 49, 291–304 (2008)

    Article  Google Scholar 

  22. Zhang W., Yao Z.G., Yao M.H.: Periodic and chaotic dynamics of composite laminated piezoelectric rectangular plate with one-to-two internal resonance. Sci. China Ser. E: Technol. Sci. 52, 731–742 (2008)

    Article  Google Scholar 

  23. Wiggins S.: Global Bifurcations and chaos Analytical Methods. Springer, New York (1988)

    MATH  Google Scholar 

  24. Kovacic G., Wiggins S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation. Physica D 57, 185–225 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kaper T.J., Kovacic G.: Multi-bump orbits homoclinic to resonance bands. Trans. Am. Math. Soc. 348, 3835–3887 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Camassa R., Kovacic G., Tin S.K.: A Melnikov method for homoclinic orbits with many pulse. Arch. Ration. Mech. Anal. 143, 105–193 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Zhang W., Yao M.H., Zhang J.H.: Using extended Melnikov method to study multi-pulse global bifurcations and chaos of a cantilever beam. J. Sound Vib. 319, 541–569 (2009)

    Article  Google Scholar 

  28. Zhang J.H., Zhang W., Yao M.H., Guo X.Y.: Multi-pulse Shilnikov chaotic dynamics for a non-autonomous buckled thin plate under parametric excitation. Int. J. Nonlinear Sci. Numer. Simul. 9, 381–394 (2008)

    Google Scholar 

  29. Wiggins S.: Introduction to applied nonlinear dynamical systems and chaos, 2nd edn. Springer, New York (2000)

    Google Scholar 

  30. Yao, Z.G., Zhang, W., Chen, L.H., Gao, M.J.: Bifurcations and chaos of composite laminated piezoelectric rectangular plate with one-to-three internal resonance. Eur. J. Mech. A-Solid (2008, submitted)

  31. Zhang W., Wang F.X., Zu J.W.: Computation of normal forms for high dimensional nonlinear systems and application to nonplanar motions of a cantilever beam. J. Sound Vib. 278, 949–974 (2004)

    Article  MathSciNet  Google Scholar 

  32. Zhang W., Chen Y., Cao D.X.: Computation of normal forms for eight-dimensional nonlinear dynamical system and application to a viscoelastic moving belt. Int. J. Nonlinear Sci. Numer. Simul. 7, 35–58 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mechanical Engineering, Beijing University of Technology, 1000124, Beijing, People’s Republic of China

    Wei Zhang, J. H. Zhang, M. H. Yao & Z. G. Yao

Authors
  1. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. H. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. H. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Z. G. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, W., Zhang, J.H., Yao, M.H. et al. Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Acta Mech 211, 23–47 (2010). https://doi.org/10.1007/s00707-009-0210-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-009-0210-3

Keywords

Search

Navigation