Nonlinear Dynamics

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

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

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.


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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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