Nonlinear Dynamics

, Volume 86, Issue 2, pp 1359–1379 | Cite as

Stability analysis of a composite laminated piezoelectric plate subjected to combined excitations

  • M. Sayed
  • A. A. Mousa
  • Ibrahim Hassan Mustafa
Original Paper


The aim of this paper is to discuss the stability of a symmetric cross-ply composite laminated piezoelectric plate subject to combined excitations. Multiple timescale perturbation method is implemented to solve the nonlinear governing equations including the second-order approximation. The case of 1:1:3 internal resonance and primary resonance case is investigated. The stability of the system is discussed using frequency, force response curves. A bifurcation analysis was performed using the amplitude of parametric excitation force as the bifurcation parameter. It is found that there are two Hopf bifurcation points: the first one is at \(f_{11}=5.592\), and the other one is at \(f_{11}=13.96\). It is observed that the system is dominated by the periodic attractor in the ranges of \((5.592< f_{11 }< 7.21)\) and \((13.31< f_{11 }< 13.96)\). The system is enriched with period doublings which are the main way leading to chaotic behavior. Some recommendations regarding the parameters limit of the dynamic system are reported.


Composite laminated plate Multiple timescales Resonance Stability 



The authors would like to express their gratitude to the editor and referees for their encouragement and constructive comments in revising the paper.


  1. 1.
    Oñate, E.: Composite laminated plates, lecture notes on numerical methods in engineering and sciences. Struct. Anal. Finite Elem. Method Linear Stat. 2, 382–437 (2013)Google Scholar
  2. 2.
    Marjanović, M., Vuksanović, D.: Layer wise solution of free vibrations and buckling of laminated composite and sandwich plates with embedded delaminations. Compos. Struct. 108, 9–20 (2014)CrossRefGoogle Scholar
  3. 3.
    Ghannadpour, S.A.M., Ovesy, H.R., Zia-Dehkordi, E.: An exact finite strip for the calculation of initial post-buckling stiffness of shear-deformable composite laminated plates. Compos. Struct. 108, 504–513 (2014)CrossRefGoogle Scholar
  4. 4.
    Mantari, J.L., Oktem, A.S., Soares, C.G.: Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory. Compos. Struct. 94(1), 37–49 (2011)CrossRefGoogle Scholar
  5. 5.
    Ganapathi, M., Mondal, B., Prakash, T., Kalyani, A.: Free vibration analysis of simply supported composite laminated panels. Compos. Struct. 90(1), 100–103 (2009)CrossRefGoogle Scholar
  6. 6.
    Khan, A.H., Patel, B.P.: Nonlinear forced vibration response of bimodular laminated composite plates. Compos. Struct. 108, 524–537 (2014)CrossRefGoogle Scholar
  7. 7.
    Chang, S.I., Bajaj, A.K., Krousgrill, C.M.: Nonlinear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dyn. 4, 433–460 (1993)CrossRefGoogle Scholar
  8. 8.
    Zhang, W.: Global and chaotic dynamics for a parametrically excited thin plate. J. Sound Vib. 239, 1013–1036 (2001)CrossRefGoogle Scholar
  9. 9.
    Noor, A.K., Peters, J.M.: Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique. Comput. Methods Appl. Mech. Eng. 29(3), 271–295 (1981)CrossRefMATHGoogle Scholar
  10. 10.
    Ye, M., Lu, J., Zhang, W., Ding, Q.: Local and global nonlinear dynamics of a parametrically excited rectangular symmetric cross-ply laminated composite plate. Chaos Solitons Fractals 26, 195–213 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Nath, Y., Prithviraju, M., Mufti, A.A.: Nonlinear statics and dynamics of antisymmetric composite laminated square plates supported on nonlinear elastic subgrade. Commun. Nonlinear Sci. Numer. Simul. 11(3), 340–354 (2006)CrossRefMATHGoogle Scholar
  12. 12.
    Guo, X.Y., Zhang, W., Yao, M.: Nonlinear dynamics of angle-ply composite laminated thin plate with third-order shear deformation. Sci. China Technol. Sci. 53, 612–622 (2010)CrossRefMATHGoogle Scholar
  13. 13.
    Tien, W.-M., Sri Namachchivaya, N.: Non-linear dynamics of a shallow arch under periodic excitation-I. 1:2 internal resonance. Int. J. Nonlinear Mech. 29, 349–366 (1994)CrossRefMATHGoogle Scholar
  14. 14.
    Sayed, M., Mousa, A.A.: Vibration, stability, and resonance of angle-ply composite laminated rectangular thin plate under multiexcitations. Math. Probl. Eng. 13, 26 (2013)MathSciNetMATHGoogle Scholar
  15. 15.
    Zhang, W., Yang, J., Hao, Y.: Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory. Nonlinear Dyn. 59, 619–660 (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Harras, B., Benamar, R., White, R.G.: Geometrically nonlinear free vibration of fully clamped symmetrically laminated rectangular composite plates. J. Sound Vib. 251(4), 579–619 (2002)CrossRefGoogle Scholar
  17. 17.
    Zhang, W., Li, S.B.: Resonant chaotic motions of a buckled rectangular thin plate with parametrically and externally excitations. Nonlinear Dyn. 62, 673–686 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Eissa, M., Sayed, M.: A comparison between passive and active control of non-linear simple pendulum Part-I. Math. Comput. Appl. 11, 137–149 (2006)MathSciNetMATHGoogle Scholar
  19. 19.
    Eissa, M., Sayed, M.: A comparison between passive and active control of non-linear simple pendulum part-II. Math. Comput. Appl. 11, 151–162 (2006)MathSciNetMATHGoogle Scholar
  20. 20.
    Eissa, M., Sayed, M.: Vibration reduction of a three DOF nonlinear spring pendulum. Commun. Nonlinear Sci. Numer. Simul. 13, 465–488 (2008)CrossRefMATHGoogle Scholar
  21. 21.
    Sayed, M.: Improving the mathematical solutions of nonlinear differential equations using different control methods, Ph.D. Thesis, Menoufia University, Egypt (2006)Google Scholar
  22. 22.
    Amer, Y.A., Bauomy, H.S., Sayed, M.: Vibration suppression in a twin-tail system to parametric and external excitations. Comput. Math. Appl. 58, 1947–1964 (2009)CrossRefMATHGoogle Scholar
  23. 23.
    Sayed, M., Hamed, Y.S.: Stability and response of a nonlinear coupled pitch-roll ship model under parametric and harmonic excitations. Nonlinear Dyn. 64, 207–220 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sayed, M., Kamel, M.: Stability study and control of helicopter blade flapping vibrations. Appl. Math. Model. 35, 2820–2837 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sayed, M., Kamel, M.: 1:2 and 1:3 internal resonance active absorber for nonlinear vibrating system. Appl. Math. Model. 36, 310–332 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zhang, W., Song, C., Ye, M.: Further studies on nonlinear oscillations and chaos of a symmetric cross-ply laminated thin plate under parametric excitation. Int. J. Bifurc. Chaos 16(02), 325–347 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rafiee, M., He, X.Q., Liew, K.M.: Non-linear dynamic stability of piezoelectric functionally graded carbon nanotube-reinforced composite plates with initial geometric imperfection. Int. J. Nonlinear Mech. 59, 37–51 (2014)CrossRefGoogle Scholar
  28. 28.
    Zhang, W., Zhang, J.H., Yao, M.H., Yao, Z.G.: Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Acta Mech. 211, 23–47 (2010)CrossRefMATHGoogle Scholar
  29. 29.
    Mousa, A.A., Sayed, M., Eldesoky, I.M., Zhang, W.: Nonlinear stability analysis of a composite laminated piezoelectric rectangular plate with multi-parametric and external excitations. Int. J. Dyn. Control 2, 494–508 (2014)CrossRefGoogle Scholar
  30. 30.
    Yao, Z.G., Zhang, W., Chen, L.H.: Periodic and chaotic oscillations of laminated composite piezoelectric rectangular plate with 1:2:3 internal resonances. In: Proceedings of the 5th International Conference on Nonlinear Mechanics, Shanghai, pp. 720–725 (2007)Google Scholar
  31. 31.
    Zhang, W., Hao, W.L.: Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate. Nonlinear Dyn. 73, 1005–1033 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)MATHGoogle Scholar
  33. 33.
    Nayfeh, A.H.: Nonlinear Interactions. Wiley-Inter-Science, New York (2000)Google Scholar
  34. 34.
    Nayfeh, A.H., Mook, D.T.: Perturbation Methods. Wiley, New York (1973)Google Scholar
  35. 35.
    Burden, R.L., Faires, J.D.: Numerical Analysis. Numerical Solutions of Nonlinear Systems of Equations. Thomson Brooks/Cole, Belmount (2005)MATHGoogle Scholar
  36. 36.
    Zhang, W., Gao, M., Yao, M., Yao, Z.: Higher-dimensional chaotic dynamics of a composite laminated piezoelectric rectangular plate. Sci. China Ser. G 52(12), 1989–2000 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • M. Sayed
    • 1
    • 3
  • A. A. Mousa
    • 2
    • 3
  • Ibrahim Hassan Mustafa
    • 4
  1. 1.Department of Engineering Mathematics, Faculty of Electronic EngineeringMenoufia UniversityMenoufEgypt
  2. 2.Department of Basic Engineering Sciences, Faculty of EngineeringMenoufia UniversityShibin El-KomEgypt
  3. 3.Department of Mathematics and Statistics, Faculty of ScienceTaif UniversityTaifKingdom of Saudi Arabia
  4. 4.Biomedical Engineering Department, Faculty of Engineering at HelwanHelwan UniversityCairoEgypt

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