Skip to main content
SpringerLink
Log in

Flutter, limit cycle oscillation, bifurcation and stability regions of an airfoil with discontinuous freeplay nonlinearity

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper is devoted to study a two-dimensional airfoil oscillating in pitch and plunge degrees of freedom. A nonlinear analysis is performed to investigate the effects of a discontinuous freeplay nonlinearity in pitch on the response of the airfoil system. In fact we show that in the presence of freeplay, the air velocity has a direct effect on the pitch vibrations of the airfoil system. Namely, it can generate the flutter leading to the limit cycle oscillation for the airfoil. With the aid of a fixed point of the Poincar\(\acute{\text {e}}\) map of the system and numerical findings, we determine the flutter and the limit cycle oscillation of that. The frequency, period of the limit cycle oscillation of pitch motion and the flutter speed are calculated. Tangent points are also computed, and it is shown that these points cannot be two-fold singularities for the system. Furthermore, by using the theoretical techniques of discontinuous systems, we will obtain parametric regions for the existence of grazing bifurcation (global bifurcation). The existence of grazing bifurcation helps us to display that for some values of the air velocity, different transitions or sudden jumps can occur in the system’s response. Numerical results demonstrate that these transitions are accompanied by the appearance and disappearance of a tangential contact between the trajectory and the switching boundaries. Also they can cause a change in the response of the pitch motion from simply periodic to double periodic (periodic-2). Moreover, stability regions for the airfoil system with freeplay will be found. The property of these stability regions is that inside them there exist no flutter and limit cycle oscillation. Some numerical examples are given which are in good agreement with our theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Afsharnezhad, Z., Karimi Amaleh, M.: Continuation of the periodic orbits for the differential equation with discontinuous right hand side. J. Dyn. Differ. Equ. 23, 71–92 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alighanbari, H., Price, S.J.: The post-Hopf-bifurcation response of an airfoil in incompressible two-dimensional flow. Nonlinear Dyn. 10(4), 381–400 (1996)

    Article  Google Scholar 

  3. Asjes, D. C.: Nonlinear analysis of a two- and three-degree-of-freedom aeroelastic system with rotational stiffness free-play, Doctoral thesis, Iowa state University (2015)

  4. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A.B., Olivar Tos, G., Piiroinen, P.T.: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50(4), 629–701 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. di Bernardo, M., Hogani, S.J.: Discontinuity-induced bifurcations of piecewise smooth dynamical systems. Philos. Trans. R. Soc. A 368, 4915–4935 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, M., Liu, W.-F., Liu, J.-K.: Bifurcation and chaos of airfoil with multiple strong nonlinearities. App. Math. Mech. (English Edition) 34(5), 627–636 (2013)

    Article  Google Scholar 

  7. Candon, M., Ogawa, H.i, Carrese, R., Marzocca, P.: Identification of Nonlinear Aeroelastic Behavior of a Wing with Pitching and Plunging Freeplay via Higher-Order Spectra Analysis. In: AIAA SciTech, pp. 1953 (2016)

  8. Colombo, A., Jeffrey, M.R.: The two-fold singularity of nonsmooth flows: leading order dynamics in n-dimensions. Phys. D 263, 1–10 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dercole, F., Rossa, F.D., Colombo, A., Kuznetsov, Y.A.: Two degenerate boundary equilibrium bifurcations in planar Filippov systems. SIAM J. Appl. Dyn. Syst. 10(4), 1525–1553 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. FangQi, C., LiangQiang, Z., YuShu, C.: Bifurcation and chaos of an airfoil with cubic nonlinearity in incompressible flow. Sci. China Technol. Sci. 54(8), 1954–1965 (2011)

    Article  MATH  Google Scholar 

  11. Filippov, A.F.: Differential Equations with Discontinuous Right Hand Sides. Kluwer Academic Publishers, Dordrecht (1988)

    Book  MATH  Google Scholar 

  12. Fung, Y.C.: An Introduction to The Theory of Aeroelasticity. Wiley, New York (1955)

    Google Scholar 

  13. Guckenheiner, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, Berlin (1993)

    Google Scholar 

  14. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of the Hopf Bifurcation. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  15. Hosham, Hany A.: Cone-like Invariant Manifolds for Nonsmooth Systems, PhD thesis, University of Cologne, Germany (2011)

  16. Irani, S., Sazesh, S., Molazadeh, V.R.: Flutter analysis of a nonlinear airfoil using stochastic approach. Nonlinear Dyn. 84, 1735–1746 (2016)

    Article  MathSciNet  Google Scholar 

  17. Kevadiya, M., Vaidya, H.: 2D Analysis Of Naca 4412 Airfoil. Int. J. Innov. Res. Sci. Eng. Technol. 2, 1686–1691 (2013)

  18. Kukučka, P.: Jumps of the fundamental solution matrix in discontinuous systems and applications. Nonlinear Anal. 66, 2529–2546 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Küpper, T., Moritz, S.: Generalized Hopf bifurcation for non-smooth planar systems. Philos. Trans. R. Soc. Lond. A 359, 2483–2496 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kuznetsov, Y.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. Bifurcat. Chaos 13, 2157–2188 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leine, R.I., van Campenb, D.H.: Bifurcation phenomena in non-smooth dynamical systems. Eur. J. Mech. A/Solids 25, 595–616 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Leine, R.I., van Campenb, D.H., van de Vrande, B.L.: Bifurcations in Nonlinear Discontinuous Systems. Nonlinear Dyn. 23, 105–164 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, J.-K., Zhao, L.-C.: Bifurcation analysis of airfoils in incompressible flow. J. Sound Vib. 154(1), 117–124 (1992)

    Article  MATH  Google Scholar 

  24. Liu, L., Dowell, E.H.: Harmonic balance approach for an airfoil with a freeplay control surface. AIAA J. 43(4), 802–815 (2005)

    Article  Google Scholar 

  25. Monfared, Z., Dadi, Z.: Analyzing panel flutter in supersonic flow by Hopf bifurcation. Iran. J. Numer. Anal. Optim. 4(2), 1–14 (2014)

    MATH  Google Scholar 

  26. Perco, L.: Differential Equations and Dynamical Systems. Springer, Berlin (1991)

    Book  Google Scholar 

  27. Süli, E.: Numerical solution of ordinary differential equations. University of Oxford, Lecture note (2013)

  28. Price, S.J., Alighanbari, H., Lee, B.H.K.: The aeroelastic response of a two-dimensional airfoil with bilinear and cubic structural nonlinearities. J. Fluids Struct. 9, 175–193 (1995)

    Article  Google Scholar 

  29. Price, S.J., Lee, B.H.K., Alighanbari, H., An analysis of the post-instability behaviour of a two-dimensional airfoil with a structural nonlinearity. 34th Structures, Structural Dynamics and Materials Conference (1993)

  30. Vasconcellos, R., Abdelkefi, A., Hajj, M.R., Marques, F.D., Discontinuity induced bifurcation in aeroelastic systems with freeplay nonlinearity. 22nd International Congress of Mechanical Engineering (COBEM 2013), Brazil (2013)

  31. Vasconcellos, R., Abdelkefi, A., Hajj, M.R., Marques, F.D.: Grazing bifurcation in aeroelastic systems with freeplay nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 19, 1611–1625 (2014)

    Article  MathSciNet  Google Scholar 

  32. Wang, D., Chen, Y., Hao, Z., Cao, Q.: Bifurcation analysis for vibrations of a turbine blade excited by air flows. Sci. China Technol. Sci. 59(8), 1217–1231 (2016)

    Article  Google Scholar 

  33. Yang, Z.C., Zhao, L.C.: Analysis of limit cycle flutter of an airfoil in incompressible flow. J. Sound Vib. 123(1), 1–13 (1988)

    Article  Google Scholar 

  34. Zhang, S.-J., Wen, G.-L., Peng, F., Liu, Z.-Q.: Analysis of limit cycle oscillations of a typical airfoil section with freeplay. Acta. Mech. Sin. 29(4), 583–592 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhao, D.-M., Zhang, Q.-C.: Bifurcation and chaos analysis for aeroelastic airfoil with freeplay structural nonlinearity in pitch. Chin. Phys. B 19(3), 030518 (2010)

    Article  Google Scholar 

  36. Zhao, H., Cao, D., Zhu, X.: Aerodynamic flutter and limit cycle analysis for a 2-D wing with pitching freeplay in the supersonic flow. In: 3rd International Symposium on System Control in Aeronautics and Astronautics (ISSCAA) (2010)

  37. Zhao, L.C., Yang, Z.C.: Chaotic motions of an airfoil with non-linear stiffness in incompressible flow. J. Sound Vib. 138(2), 245–254 (1990)

    Article  MATH  Google Scholar 

  38. Zou, Y., Küpper, T., Beyn, W.J.: Generalized Hopf bifurcation for planar Filippov systems continuous at the origin. J. Nonlinear Sci. 16, 159–177 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

    Z. Monfared & Z. Afsharnezhad

  2. Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    J. A. Esfahani

Authors
  1. Z. Monfared
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. Afsharnezhad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. A. Esfahani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. Afsharnezhad.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Monfared, Z., Afsharnezhad, Z. & Esfahani, J.A. Flutter, limit cycle oscillation, bifurcation and stability regions of an airfoil with discontinuous freeplay nonlinearity. Nonlinear Dyn 90, 1965–1986 (2017). https://doi.org/10.1007/s11071-017-3773-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3773-x

Keywords

Search

Navigation