Nonlinear Dynamics

, Volume 67, Issue 4, pp 2483–2503 | Cite as

Nonlinear dynamics analysis of a two-dimensional thin panel with an external forcing in incompressible subsonic flow

  • Peng Li
  • Yiren Yang
  • Wei Xu
Original Paper


Based on the potential theory of incompressible flow and the energy method, a two-dimensional simply supported thin panel subjected to external forcing and uniform incompressible subsonic flow is theoretically modeled. The nonlinear cubic stiffness and viscous damper in the middle of the panel is considered. Transformation of the governing partial differential equation to a set of ordinary differential equations is performed through the Galerkin method. The stability of the fixed points of the panel system is analyzed. The regions of different motion types of the panel system are investigated in different parameter spaces. The rich dynamic behaviors are presented as bifurcation diagrams, phase-plane portraits, Poincaré maps and maximum Lyapunov exponents based on carefully numerical simulations.


Potential theory Subsonic flow Parameter spaces Bifurcation Chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Schetz, J.A.: Aerodynamics of high-speed trains. Annu. Rev. Fluid Mech. 33, 371–414 (2001) CrossRefGoogle Scholar
  2. 2.
    Jeromr, C.R.: A train for the 21st century. Rail Int., 2–8 (1994) Google Scholar
  3. 3.
    Raghunathan, R.S., Kim, H.D., Setoguchi, T.: Aerodynamics of high-speed railway trains. Prog. Aerosp. Sci. 38, 469–514 (2002) CrossRefGoogle Scholar
  4. 4.
    Dowell, E.H.: Flutter of a buckled plate as an example of chaotic motion of a deterministic autonomous system. J. Sound Vib. 85, 333–344 (1982) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kobayashi, S.: Flutter of a simply supported rectangular panels in a supersonic flow-two dimensional panel flutter, I simply supported, II clamped panels. Trans. Jpn. Soc. Aeron. Space Sci. 5, 79–119 (1962) Google Scholar
  6. 6.
    Koo, K.N., Hwang, W.S.: Effects of hysteretic and aerodynamic damping on supersonic panel flutter of composite plates. J. Sound Vib. 273, 569–583 (2004) CrossRefGoogle Scholar
  7. 7.
    Yuen, S.W., Lau, S.L.: Effects of inplane load on nonlinear panel flutter by incremental harmonic balance method. AIAA J. 29, 1472–1479 (1991) CrossRefGoogle Scholar
  8. 8.
    Chen, D.L., Yang, Y.R., Fan, C.G.: Nonlinear flutter of a two-dimension thin plate subjected to aerodynamic heating by differential quadrature method. Acta Mech. Sin. 24, 45–50 (2008) CrossRefGoogle Scholar
  9. 9.
    Dugundji, J., Dowell, E.H., Perkin, B.: Subsonic flutter of panels on continuous elastic foundations. AIAA J. 1, 1146–1154 (1963) CrossRefGoogle Scholar
  10. 10.
    Kornecki, A.: Static and dynamic instability of panels and cylindrical shells in subsonic potential flow. J. Sound Vib. 32, 251–263 (1974) MATHCrossRefGoogle Scholar
  11. 11.
    Kornecki, A., Dowell, E.H., O’Brien, J.: On the aeroelastic instability of two-dimensional panels in unform incompressible flow. J. Sound Vib. 47, 163–178 (1974) CrossRefGoogle Scholar
  12. 12.
    Guo, C.Q., Paidoussis, M.P.: Stability of rectangular plates with free side-edges in two-dimensional inviscid channel flow. J. Appl. Mech. 67, 171–176 (2000) MATHCrossRefGoogle Scholar
  13. 13.
    Awrejcewicz, J., Krysko, V.A., Narkaitis, G.G.: Bifurcations of a thin plate-strip excited transversally and axially. Nonlinear Dyn. 32, 187–209 (2003) MATHCrossRefGoogle Scholar
  14. 14.
    Akour, S.N., Nayfeh, J.F.: Nonlinear dynamics of polar-orthotropic circular plates. Int. J. Struct. Stab. Dyn. 6, 253–268 (2006) CrossRefGoogle Scholar
  15. 15.
    Wang, L., Ni, Q., Huang, Y.Y.: Bifurcations and chaos in forced cantilever system with impacts. J. Sound Vib. 296, 1068–1078 (2006) CrossRefGoogle Scholar
  16. 16.
    Jing, Z.J., Wang, R.Q.: Complex dynamic in Duffing system with two external forcings. Chaos Solitons Fractals 25, 399–411 (2005) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Huang, J.C., Jing, Z.J.: Bifurcations and chaos in three-well Duffing system with one external forcing. Chaos Solitons Fractals 40, 1449–1466 (2009) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kenfack, A.: Bifurcation structure of two coupled periodically driven double-well Duffing oscillators. Chaos Solitons Fractals 15, 205–218 (2003) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Ye, M., Chen, Y.S.: Period doubling bifurcation in elastic bulked beam under parametric excitation. Acta Mech. Solida Sin. 19, 361–164 (1998) Google Scholar
  20. 20.
    Zhang, W., Huo, Q.Z.: Bifurcation of nonlinear oscillation system under combined parametric and forcing excitation. Acta Mech. Sin. 23, 464–474 (1991) MathSciNetGoogle Scholar
  21. 21.
    Zhang, W., Liu, Z.M., Yu, P.: Global dynamic of a parametrically and externally excited thin plate. Nonlinear Dyn. 24, 245–268 (2001) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Awrejcewicz, J.: Bifurcations and Chaos in Simple Dynamical Systems. World Scientific, Singapore (1989), 126 pp. Google Scholar
  23. 23.
    Awrejcewicz, J.: Bifurcations and Chaos in Coupled Oscillators. World Scientific, Singapore (1991), 245 pp. Google Scholar
  24. 24.
    Awrejcewicz, J., Krysko, V.A.: Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells. World Scientific, Berlin (2003), 428 pp. MATHCrossRefGoogle Scholar
  25. 25.
    Bisplinghoff, R.L., Ashley, H., Halfman, R.L.: Aeroelasticity. Addison-Wesley, Cambridge (1955) MATHGoogle Scholar
  26. 26.
    Dowell, E.H.: Aeroelasticity of Plates and Shells. Nordhoff, Leyden (1975) MATHGoogle Scholar
  27. 27.
    Zhang, J.Y., Yang, Y.R., Zeng, J.: An algorithm criterion for Hopf bifurcation and its application in vehicle dynamic. Acta Mech. Sin. 32, 596–605 (2000) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.National Traction Power LaboratorySouthwest Jiaotong UniversityChengduP.R. China
  2. 2.School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduP.R. China

Personalised recommendations