Abstract
A strictly nonlinear state feedback control law is designed for an aeroelastic system to eliminate subcritical limit cycle oscillations. Numerical continuation techniques and harmonic balance methods are employed to generate analytical estimates of limit cycle oscillation commencement velocity and its sensitivity with respect to the introduced control parameters. The obtained estimates are used in a multiobjective optimization framework to generate optimal control parameters which maximize the limit cycle oscillation commencement velocity while minimizing the control cost. Numerical simulations are used to show that the assumed nonlinear state feedback law with the optimal control parameters successfully eliminates any existing subcritical limit cycle oscillations by converting it to supercritical limit cycle oscillations, thereby guaranteeing safe operation of the system in its flight envelope.
Similar content being viewed by others
References
Lee, B.H.K., Price, S.J., Wong, Y.S.: Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Prog. Aerosp. Sci. 35(3), 205–334 (1999)
Dowell, E., Edwards, J., Strganac, T.: Nonlinear aeroelasticity. J. Aircr. 40(5), 857–874 (2003)
Woolston, D.S., Runyan, H.L., Byrdsong, T.A.: Some effects of system nonlinearities in the problem of aircraft flutter, Issue 3539 of Technical note. National Advisory Committee for Aeronautics, United States (1955). http://digital.library.unt.edu/ark:/67531/metadc57724/
Woolston, D.S., Runyan, H.L., Andrews, R.E.: An investigation of effects of certain types of structural nonhnearities on wing and control surface flutter. J. Aeronaut. Sci. 24(1), 57–63 (1957)
Librescu, L., Marzocca, P.: Advances in the linear/nonlinear control of aeroelastic structural systems. Acta Mech. 178(3–4), 147–186 (2005)
Nayfeh, A.H.: On direct methods for constructing nonlinear normal modes of continuous systems. J. Vib. Control 1(4), 389–430 (1995)
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)
Ghommem, M., Nayfeh, A.H., Hajj, M.R.: Control of limit cycle oscillations of a two-dimensional aeroelastic system. Math. Probl. Eng. (2010). doi:10.1155/2010/782457
Liu, L., Dowell, Earl H.: Harmonic balance approach for an airfoil with a freeplay control surface. AIAA J. 43(4), 802–815 (2005)
Dimitriadis, G.: Continuation of higher-order harmonic balance solutions for nonlinear aeroelastic systems. J. Aircr. 45(2), 523–537 (2008)
Vakakis, A.F.: Analysis and Identification of Linear and Nonlinear Normal Modes in Vibrating Systems. Ph.D. Thesis, California Institute of Technology (1991)
Jiang, D., Pierre, C., Shaw, S.W.: Nonlinear normal modes for vibratory systems under harmonic excitation. J. Sound Vib. 288(4), 791–812 (2005)
Emory, C.W., Patil, M.J.: Predicting limit cycle oscillation in an aeroelastic system using nonlinear normal modes. J. Aircr. 50(1), 73–81 (2013)
Shukla, H., Patil, M.: Control of limit cycle oscillation amplitudes in nonlinear aerelastic systems using nonlinear normal modes. In: AIAA Atmospheric Flight Mechanics Conference (2016)
Shahrzad, P., Mahzoon, M.: Limit cycle flutter of airfoils in steady and unsteady flows. J. Sound Vib. 256(2), 213–225 (2002)
Liu, J.-K., Zhao, L.-C.: Bifurcation analysis of airfoils in incompressible flow. J. Sound Vib. 154(1), 117–124 (1992)
Jiang, L.Y., Lee, B.H.K., Wong, Y.S.: Flutter of an airfoil with a cubic restoring force. J. Fluids Struct. 13(1), 75–101 (1999)
Lee, B.H.K., Liu, L., Chung, K.W.: Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces. J. Sound Vib. 281(3), 699–717 (2005)
Guo, H., Chen, Y.: Supercritical and subcritical hopf bifurcation and limit cycle oscillations of an airfoil with cubic nonlinearity in supersonic\(\backslash \) hypersonic flow. Nonlinear Dyn. 67(4), 2637–2649 (2012)
Theodorsen, T., Mutchler, WH.: General theory of aerodynamic instability and the mechanism of flutter, NACA Report 496. National Advisory Committee for Aeronautics, United States (1935)
Marzocca, P., Librescu, L., Silva, W.A.: Flutter, postflutter, and control of a supersonic wing section. J. Guid. Control Dyn. 25(5), 962–970 (2002)
Lee, I., Kim, S.-H.: Aeroelastic analysis of a flexible control surface with structural nonlinearity. J. Aircr. 32(4), 868–874 (1995)
Holmes, P.J.: Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis. J. Sound Vib. 53(4), 471–503 (1977)
Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems and bifurcations of vector fields. J. Appl. Mech. 51(4), 947 (1984). doi:10.1115/1.3167759
Liu, L., Dowell, E.H.: The secondary bifurcation of an aeroelastic airfoil motion: effect of high harmonics. Nonlinear Dyn. 37(1), 31–49 (2004)
Hall, K.C., Ekici, K., Thomas, J.P., Dowell, E.H.: Harmonic balance methods applied to computational fluid dynamics problems. Int. J. Comput. Fluid Dyn. 27(2), 52–67 (2013)
Allgower, E.L., Georg, K.: Numerical Continuation Methods: An Introduction, vol. 13. Springer, Berlin (2012)
Marsden, J.E, Hoffman, M.J: Elementary Classical Analysis. W.H. Freeman & Company, New York (1993)
Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. Appl. Bifurc. Theory 1(38), 359–384 (1977)
Keller, H.B.: Global homotopies and newton methods. Recent Adv. Numer. Anal. 41, 73–94 (1978)
Keller, H.B.: Lectures on numerical methods in bifurcation problems. Appl. Math. 217, 50 (1987)
Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: Matcont: a matlab package for numerical bifurcation analysis of odes. ACM Trans. Math. Softw. (TOMS) 29(2), 141–164 (2003)
Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms, vol. 16. Wiley, London (2001)
Haftka, R.T., Gürdal., Z.: Elements of Structural Optimization, vol. 11. Springer, Berlin (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shukla, H., Patil, M.J. Nonlinear state feedback control design to eliminate subcritical limit cycle oscillations in aeroelastic systems. Nonlinear Dyn 88, 1599–1614 (2017). https://doi.org/10.1007/s11071-017-3332-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-017-3332-5