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Nonlinear Dynamics

, Volume 67, Issue 4, pp 2637–2649 | Cite as

Supercritical and subcritical Hopf bifurcation and limit cycle oscillations of an airfoil with cubic nonlinearity in supersonic\hypersonic flow

  • Hulun Guo
  • Yushu Chen
Original Paper

Abstract

In this paper, the Hopf bifurcations and limit cycle oscillations (LCOs) of an airfoil with cubic nonlinearity in supersonic\hypersonic flow are investigated. The harmonic balance method and multivariable Floquet theory are applied to analyze the LCOs of the airfoil. Four distinct cases of the LCOs response are detected in this system: (I) supercritical Hopf bifurcation, (II) a single subcritical Hopf bifurcation, (III) two subcritical Hopf bifurcations, and (IV) no Hopf bifurcation. Furthermore, the parameter variations domains separating the supercritical and subcritical Hopf bifurcations are presented using singularity theory.

Keywords

Supercritical and subcritical Hopf bifurcation LCO Harmonic balance method Floquet theory Singularity theory 

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References

  1. 1.
    Librescu, L., Chiocchia, G., Marzocca, P.: Implications of cubic physical/aerodynamic nonlinearities on the character of the flutter instability boundary. Int. J. Non-Linear Mech. 38(2), 173–199 (2003) MATHCrossRefGoogle Scholar
  2. 2.
    Librescu, L., Na, S., Marzocca, P., Chung, C.: Active aeroelastic control of 2-D wing-flap systems in an incompressible flow field. AIAA Paper 2003-1414 (2003) Google Scholar
  3. 3.
    Abbas, L.K., Qian, C., Marzocca, P., Zafer, G., Mostafa, A.: Active aerothermoelastic control of hypersonic double-wedge lifting surface. Chin. J. Aeronaut. 21, 8–18 (2008) CrossRefGoogle Scholar
  4. 4.
    Hopf, E.: Bifurcation of a periodic solution from a stationary solution of a system of differential equations. Berl. Math. Phys. Klasse, Sachsischen. Akad. Wiss. Leip. 94, 3–32 (1942) Google Scholar
  5. 5.
    Lee, B.H.K., Price, S.J., Wong, Y.S.: Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Prog. Aerosp. Sci. 35, 205–334 (1999) CrossRefGoogle Scholar
  6. 6.
    Holmes, P.J.: Bifurcations to divergence and flutter in flow-induced oscillations: a finite-dimensional analysis. J. Sound Vib. 53(4), 471–503 (1977) MATHCrossRefGoogle Scholar
  7. 7.
    Mastroddi, F., Morino, L.: Limit-cycle taming by nonlinear control with application to flutter. Aeronaut. J. 100(999), 389–396 (1996) Google Scholar
  8. 8.
    Dowell, E., John, E., Thomas, S.: Nonlinear aeroelasticity. J. Aircr. 40(5), 857–874 (2003) CrossRefGoogle Scholar
  9. 9.
    Woolston, D.S., Runyan, H.L., Byrdsong, T.A.: Some effects of system nonlinearities in the problem of aircraft flutter. NACA TN 3539 (1995) Google Scholar
  10. 10.
    Woolston, D.S., Runyan, H.L., Andrews, R.E.: An investigation of effects of certain types of structural nonlinearities on wing and control surface flutter. J. Aeronaut. Sci. 24, 57–63 (1957) Google Scholar
  11. 11.
    Liu, J.K., Zhao, L.C.: Bifurcation analysis of airfoils in incompressible flow. J. Sound Vib. 154(1), 117–124 (1992) MATHCrossRefGoogle Scholar
  12. 12.
    Lee, B.H.K., Jiang, L.Y., Wong, Y.S.: Flutter of an airfoil with a cubic restoring force. J. Fluids Struct. 13, 75–101 (1999) CrossRefGoogle Scholar
  13. 13.
    Lee, B.H.K., Liu, L., Chung, K.W.: Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces. J. Sound Vib. 281, 699–717 (2005) CrossRefGoogle Scholar
  14. 14.
    Liu, L., Dowell, E.H.: The secondary bifurcation of an aeroelastic airfoil motion: effect of high harmonics. Nonlinear Dyn. 37(1), 31–49 (2004) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Chen, Y.S., Leung, A.Y.T.: Bifurcation and Chaos in Engineering. Springer, London (1998) MATHCrossRefGoogle Scholar
  16. 16.
    Liu, L., Wong, Y.S., Lee, B.H.K.: Application of the centre manifold theory in non-linear aeroelasticity. J. Sound Vib. 234(4), 641–659 (2000) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Coller, B.D., Chamara, P.A.: Structural non-linearities and the nature of the classic flutter instability. J. Sound Vib. 277, 711–739 (2004) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ding, Q., Wang, D.L.: The flutter of an airfoil with cubic structural and aerodynamic non-linearities. Aerosp. Sci. Technol. 10, 427–434 (2006) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Yu, P., Chen, Z., Librescu, L., Marzocca, P.: Implications of time-delayed feedback control on limit cycle oscillation of a two-dimensional supersonic lifting surface. J. Sound Vib. 304, 974–986 (2007) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shahrzad, P., Mahzoon, M.: Limit cycle flutter of airfoils in steady and unsteady flows. J. Sound Vib. 256(2), 213–225 (2002) CrossRefGoogle Scholar
  21. 21.
    Ashley, H., Zartarian, G.: Piston theory—a new aerodynamic tool for the aeroelastician. J. Aeronaut. Sci. 23(12), 1109–1118 (1956) MathSciNetGoogle Scholar
  22. 22.
    Abbas, L.K., Chen, Q., O’Donnell, K., Valentine, D., Marzocca, P.: Numerical studies of a non-linear aeroelastic system with plunging and pitching freeplays in supersonic/hypersonic regimes. Aerosp. Sci. Technol. 11, 405–418 (2007) CrossRefGoogle Scholar
  23. 23.
    Thuruthimattam, B.J., Friedmann, P.P., McNamara, J.J., Powell, K.G.: Modeling approaches to hypersonic aeroelasticity. ASME International Mechanical Engineering Congress and Exposition, 2002–32943 (2001) Google Scholar
  24. 24.
    Friedmann, P.P., McNamara, J.J., Thuruthimattam, B.J., Nydick, I.: Aeroelastic analysis of hypersonic vehicles. J. Fluids Struct. 19, 681–712 (2004) CrossRefGoogle Scholar
  25. 25.
    Friedmann, P., Hammond, C.E., Woo, T.H.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numer. Methods Eng. 11, 1117–1136 (1977) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Ge, Z.M., Chen, H.H.: Bifurcations and chaotic motions in a rate gyro with a sinusoidal velocity about the spin axis. J. Sound Vib. 200(2), 121–137 (1997) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 1. Springer, New York (1985) MATHGoogle Scholar
  28. 28.
    Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 2. Springer, New York (1988) MATHCrossRefGoogle Scholar
  29. 29.
    Chen, Y.S., Langford, W.F.: The subharmonic bifurcation solution of nonlinear Mathieu’s equation and Euler dynamically buckling problem. Acta Mech. Sin. 20, 522–532 (1988) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of AstronauticsHarbin Institute of TechnologyHarbinChina

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