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Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate

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Abstract

The multi-pulse orbits and chaotic dynamics of a simply supported laminated composite piezoelectric rectangular plate under combined parametric excitation and transverse excitation are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motion for the laminated composite piezoelectric rectangular plate are derived from von Karman-type equation and third-order shear deformation plate theory of Reddy. The two-degree-of-freedom dimensionless equations of motion are obtained by using the Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plate. The four-dimensional averaged equation in the case of primary parametric resonance and 1:3 internal resonances is obtained by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plate. The analysis of the global dynamics indicates that there exist multi-pulse jumping orbits in the perturbed phase space of the averaged equation. Based on the averaged equation obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plate are also found by numerical simulation. The results obtained above mean the existence of the chaos in the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plate.

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References

  1. Reddy J N. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. Boca Raton, Florida: CRC Press, 2004

    MATH  Google Scholar 

  2. Reissner E. On transverse bending of plates, including the effect of transverse shear deformation. Int J Solids Struct, 1975, 11: 569–573

    Article  MATH  Google Scholar 

  3. Reddy J N. A simple higher-order theory for laminated composite plates. J Appl Mech, 1984, 5: 745–752

    Article  Google Scholar 

  4. Kapuria S, Achary G G S. A coupled consistent third-order theory for hybrid piezoelectric plates. Compos Struct, 2005, 70: 120–133

    Article  Google Scholar 

  5. Kapuria S, Achary G G S. An efficient higher order zigzag theory for laminated plates subjected to thermal loading. Int J Solids Struct, 2004, 41: 4661–4684

    Article  MATH  Google Scholar 

  6. Zhu L F, Chattopadhyay A, Goldberg R K. Nonlinear transient response of strain rate dependent composite laminated plates using multiscale simulation. Int J Solids Struct, 2006, 43: 2602–2630

    Article  MATH  Google Scholar 

  7. Moita J M S, Soares C M M, Soares C A M. Active control of forced vibrations in adaptive structures using a higher order model. Compos Struct, 2005, 71: 349–355

    Article  Google Scholar 

  8. Dimitris V, Dimitris A S. Coupled buckling and postbuckling analysis of active laminated piezoelectric composite plates. Int J Solids Struct, 2004, 41: 1519–1538

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  10. Lee S J, Reddy J N. Non-linear response of laminated composite plates under thermomechanical loading. Int J Non-Linear Mech, 2005, 40: 971–985

    Article  MATH  Google Scholar 

  11. Krommer M, Irschik H. A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect. J Acta Mech, 2000, 141: 51–69

    Article  MATH  Google Scholar 

  12. Heidary F, Eslami M R. Dynamic analysis of distributed piezothermoelastic composite plate using first order shear deformation theory. J Therm Stresses, 2004, 27: 587–605

    Article  Google Scholar 

  13. Ribeiro P. Thermally induced transitions to chaos in plate vibrations. J Sound Vibr, 2007, 299: 314–330

    Article  Google Scholar 

  14. Zhang W. Global and chaotic dynamics for a parametrically excited thin plate. J Sound Vibr, 2001, 239: 1013–1036

    Article  Google Scholar 

  15. Ye M, Sun Y H, Zhang W, et al. Nonlinear oscillations and chaotic dynamics of an antisymmetric cross-ply laminated composite rectangular thin plate under parametric excitation. J Sound Vibr, 2005, 287: 723–758

    Article  Google Scholar 

  16. 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-Tech Sci, 2009, 52: 731–742

    Article  MATH  MathSciNet  Google Scholar 

  17. Guo X Y, Zhang W, Yao M H. Nonlinear dynamics of angle-ply composite laminated thin plate with third-order shear deformation. Sci China Tech Sci, 2010, 53: 612–622

    Article  MATH  Google Scholar 

  18. 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, 2010, 211: 23–47

    Article  MATH  Google Scholar 

  19. Haller G, Wiggins S. Orbits homoclinic to resonance: The Hamiltonian. Physica D, 1993, 66: 298–346

    Article  MATH  MathSciNet  Google Scholar 

  20. Haller G, Wiggins S. Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrodinger equation. Physica D, 1995, 85: 311–347

    Article  MATH  MathSciNet  Google Scholar 

  21. Haller G, Wiggins S. N-pulse homoclinic orbits in perturbations of resonant Hamiltonian systems. Arch Rational Mech Anal, 1995, 130: 25–101

    Article  MATH  MathSciNet  Google Scholar 

  22. Haller G, Wiggins S. Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems. Physica D, 1996, 90: 319–365

    Article  MATH  MathSciNet  Google Scholar 

  23. Haller G. Chaos Near Resonance. New York: Springer-Verlag, 1999

    Book  MATH  Google Scholar 

  24. Malhotra N, Sri Namachchivaya N, McDonald R J. Multipulse orbits in the motion of flexible spinning discs. J Nonlinear Sci, 2002, 12: 1–26

    Article  MATH  MathSciNet  Google Scholar 

  25. Yao M H, Zhang W. Multi-pulse Shilnikov orbits and chaotic dynamics in nonlinear nonplanar motion of a cantilever beam. Int J Bifurcation Chaos, 2005, 15: 3923–3952

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang W, Yao M H. Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt. Chaos Solitons Fractals, 2006, 28: 42–66

    Article  MATH  MathSciNet  Google Scholar 

  27. McDonald R J, Sri Namachchivaya N. Pipes conveying pulsating fluid near a 0:1 resonance: Global bifurcations. J Fluids Struct, 2005, 21: 665–687

    Google Scholar 

  28. Yao M H, Zhang W. Shilnikov-type multipulse orbits and chaotic dynamics of a parametrically and externally excited rectangular thin plate. Int J Bifurcation Chaos, 2007, 17: 851–875

    Article  MATH  MathSciNet  Google Scholar 

  29. Zhang W, Yao M H. Theories of multi-pulse global bifurcations for high-dimensional systems and applications to cantilever beam. Int J Modern Phys B, 2008, 22: 4089–4141

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhang W, Gao M J, Yao M H, Yao Z G. Higher-dimensional chaotic dynamics of a composite laminated piezoelectric rectangular plate. Sci China Ser G-Phys Mech Astron, 2009, 52: 1989–2000

    Article  Google Scholar 

  31. Nayfeh A H, Mook D T. Nonlinear Oscillations. New York: Weily, 1979

    MATH  Google Scholar 

  32. Yu P, Zhang W, Bi Q S. Vibration analysis on a thin plate with the aid of computation of normal forms. Int J Non-Linear Mech, 2001, 36: 597–627

    Article  MATH  Google Scholar 

  33. Li J, Wang D, Zhang W. General forms of the simplest normal form of Bogdanov-Takens singularities. Dyn Continuous Discrete Impulsive Syst, 2001, 8: 519–530

    MATH  MathSciNet  Google Scholar 

  34. Chen G T, Della Dora J. An algorithm for computing a new normal form for dynamical systems. J Symbolic Comput, 2000, 29: 393–418

    Article  MATH  MathSciNet  Google Scholar 

  35. Elphick C, Tirapegui E, Brachet M E, Coullet P, Iooss G. A simple global characterization for normal forms of singular vector fields. Physica D, 1987, 29: 95–117

    Article  MATH  MathSciNet  Google Scholar 

  36. Chow S N, Li C, Wang D. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge: Cambridge University Press, 1994

    Book  MATH  Google Scholar 

  37. 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 Vibr, 2004, 278: 949–974

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China

    MingHui Yao, Wei Zhang & ZhiGang Yao

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  1. MingHui Yao
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  2. Wei Zhang
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  3. ZhiGang Yao
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Correspondence to Wei Zhang.

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Yao, M., Zhang, W. & Yao, Z. Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate. Sci. China Technol. Sci. 54, 2064–2079 (2011). https://doi.org/10.1007/s11431-011-4472-3

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  • DOI: https://doi.org/10.1007/s11431-011-4472-3

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