Skip to main content
Log in

An efficient method for nonlinear aeroelasticy of slender wings

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


This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler–Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulation yields nonlinear integro-differentials aeroelastic equations. Using the Galerkin’s method and a mode summation technique, the governing equations will be solved by introducing a numerical method without the need to adding any aerodynamic state space variables and the corresponding equations related to these variables of the problem. The obtained equations are solved to predict the aeroelastic response of the problem. The obtained results for a test case are compared with those of some other works and show a good agreement between results.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others


x,y,z :

Undeformed coordinates system

v,w,θ :

Forward, vertical, and torsional motions

U,ρ :

Air speed and density of free stream

l,c,b :

Length, chord, and half-chord of wing

m,S α ,J α :

Mass per length, first and second moment of inertia per mass of wing

L,M e.a :

Lift and pitching moment distribution about elastic axis

a,x θ :

Distance coefficient of mid chord and center of gravity to elastic axis


Deformed coordinates system


Bending and torsional stiffness


Wagner function

ξ i ,η i ,β i :

Generalized coordinates


  1. Bisplinghoff, R.L., Ashley, H., Halfman, R.L.: Aeroelasticity. Addison-Wesley, Cambridge (1995)

    Google Scholar 

  2. Fung, Y.C.: An Introduction to the Theory of Aeroelasticity. Dover, New York (1969)

    Google Scholar 

  3. Woolston, D.S., Runyan, H.L., Andrews, R.E.: An investigation of certain types of structural nonlinearities on wing and control surface flutter. J. Aeronaut. Sci. 24, 57–63 (1957)

    Google Scholar 

  4. Shen, S.F.: An approximate analysis of nonlinear flutter problems. J. Aeronaut. Sci. 26, 25–32 (1959)

    MATH  Google Scholar 

  5. Kryloff, N., Bogoliuboff, N.: Introduction to Nonlinear Mechanics. Princeton University Press, Princeton (1947). Translation by Solomon Lifschitz

    Google Scholar 

  6. Hodges, D.H., Dowell, E.H.: Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. NASA TN D-7818 (1974)

  7. Lee, B.H.K., Leblanc, P.: Flutter analysis of a two-dimensional airfoil with cubic nonlinear restoring force. National Research Council of Canada, NAE-AN-36, NRC No. 25438 (1986)

  8. Lee, B.H.K., Leblanc, P.: Flutter analysis of a two-dimensional airfoil containing structural nonlinearities. National Research Council of Canada, LR-618, NRC No. 27833 (1987)

  9. Price, S.J., Lee, B.H.K., Alighanbari, H.: Post-instability behavior of a two-dimensional airfoil with a structural nonlinearity. J. Aircr. 31, 1395–1401 (1994)

    Article  Google Scholar 

  10. 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 

  11. Lee, B.H.K., Gong, L., Wong, Y.S.: Analysis and computation of nonlinear dynamic response of a two-degree-of-freedom system and its application in aeroelasticity. J. Fluids Struct. 11, 225–246 (1997)

    Article  Google Scholar 

  12. Strganac, T.W., Mook, D.T.: A numerical model of unsteady subsonic aeroelastic behavior. AIAA J. 28(5), 903–909 (1990)

    Article  Google Scholar 

  13. Preidikman, S., Mook, D.T.: Time-domain simulations of linear and nonlinear aeroelastic behavior. J. Vib. Control 6(8), 1135–1175 (2000)

    Article  Google Scholar 

  14. Hall, B.D., Preidikman, S., Mook, D.T., Nayfeh, A.H.: Novel strategy for suppressing the flutter oscillations of aircraft wings. AIAA J. 39(10), 1843–1850 (2001)

    Article  Google Scholar 

  15. Liu, L., Wong, Y.S., Lee, B.H.K.: Application of the center manifold theory in non-linear aeroelasticity. J. Sound Vib. 234(4), 641–659 (2000). doi:10.1006/jsvi.1999.2895. Available online at

    Article  MathSciNet  Google Scholar 

  16. Kim, K.: Nonlinear aeroelastic analysis of aircraft wing-with-store configurations. PhD Dissertation. Texas A & M University (2004)

  17. Tang, D., Dowell, E.H.: Experimental and theoretical study on aeroelastic response of high-aspect-ratio wings. AIAA J. 39(8), 1430–1441 (2001)

    Article  Google Scholar 

  18. Tang, D., Dowell, E.H.: Experimental and theoretical study of gust response for high-aspect-ratio wings. AIAA Journal 40(3), 419–429 (2002)

    Article  Google Scholar 

  19. Tang, D.M., Dowell, E.H.: Effects of geometric structural nonlinearity on flutter and limit cycle oscillations of high-aspect-ratio wings. J. Fluids Struct. 19, 291–306 (2004)

    Article  Google Scholar 

  20. Patil, M.J., Hodges, D.H.: Limit-cycle oscillations in high-aspect-ratio wings. J. Fluids Struct. 15, 107–132 (2001)

    Article  Google Scholar 

  21. Sadr Lahidjani, M.H., Haddadpour, H., Shams, Sh.: Nonlinear behavior of a high flexibility wing with long span considering large deflection. In: 45th AIAA/ASME/ASCE/AHS/ASC Structures, Palm Springs, CA, 19–22 April 2004. ID:1943

    Google Scholar 

  22. Abbas, L.K., Chen, Q., Marzocca, P., Milanese, A.: Non-linear aeroelastic investigations of store(s)-induced limit cycle oscillations. Proc. Inst. Mech. Eng., G J. Aerosp. Eng. 222(1), 63–80 (2008)

    Google Scholar 

  23. Arafat, H.N.: Nonlinear response of cantilever beams. Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA (1999)

  24. Malatkar, P.: Nonlinear vibrations of cantilever beams and plates. Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA (2003)

  25. Crespo da Silva, M.R.M., Glynn, C.C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams. I: equations of motion. J. Struct. Mech. 6, 437–448 (1978)

    Article  Google Scholar 

  26. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)

    MATH  Google Scholar 

  27. Mase, G.E.: Theory and Problem of Continuum Mechanics. Schaum’s Outline Series. McGraw-Hill, New York (1970)

    Google Scholar 

  28. Meirovitch, L.: Analytical Methods in Dynamics. Macmillan, New York (1967)

    Google Scholar 

  29. Jones, R.T.: The unsteady lift of a wing of finite aspect ratio. NACA report 681 (1940)

  30. Shams, Sh., Haddadpour, H., Sadr Lahidjani, M.H., Kheiri, M.: A direct method in computational aeroelasticity based on Wagner function. In: 25th International Congress of the Aeronautical Sciences, Hamburg, 10–13 January 2006. ID:542

    Google Scholar 

  31. Shams, Sh., Sadr Lahidjani, M.H., Haddadpour, H., Malekian, M.: Investigating of the nonlinear aeroelasticity behavior of an airfoil using a direct approach. In: The 7th Conference of Iranian Aerospace Society, Sharif University of Technology, Tehran, Iran, IAS-2008-ST1335, 19–21 February 2008

    Google Scholar 

  32. Haddadpour, H., Kouchakzadeh, M.A., Shadmehri, F.: Aeroelastic instability of aircraft composite wings in an incompressible flow. Compos. Struct. 83, 93–99 (2008). doi:10.1016/j.compstruct.2007.04.012

    Article  Google Scholar 

  33. Hodges, D.H., Pierce, G.A.: Introduction to Structural Dynamics and Aeroelasticity. Cambridge University Press, Cambridge (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to H. Haddadpour.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shams, S., Sadr, M.H. & Haddadpour, H. An efficient method for nonlinear aeroelasticy of slender wings. Nonlinear Dyn 67, 659–681 (2012).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: