Aeroelastic Stability Estimation of Control Surfaces with Freeplay Nonlinearity

Aykut Tamer, Pierangelo Masarati


This work discusses the quantitative stability evaluation of aeroelastic problems with
freeplay in the control surfaces. Stability estimation of linear time invariant and linear time
periodic systems rely on eigenanalysis of state transition matrices and implies simplifications on
the problems governed by nonlinear non-autonomous equations. Lyapunov Characteristic Exponents directly provide quantitative information on the stability of nonlinear non-autonomous
dynamical systems. Stability estimation using Lyapunov Characteristic Exponents does not
require a special reference solution and is consistent with the eigensolution of linear time invariant and Floquet-Lyapunov analysis of linear time periodic systems. Thus, they represent
a natural generalization of conventional stability analysis. The Discrete QR method is used to
practically estimate the Lyapunov Characteristic Exponents. The method is applied to a
three-dimensional aeroelastic problems with freeplay in the control surface.


Freeplay; Lyapunov Characteristic Exponents; Nonlinear Aeroelasticity

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Raymond L. Bisplinghoff, Holt Ashley, and Robert L. Halfman. Aeroelasticity. Dover, New York, 1996.

Xiang Jinwu, Yan Yongju, and Li Daochun. Recent advance in nonlinear aeroelastic analysis and control of the aircraft. Chinese Journal of Aeronautics, 27(1):12{22, 2014. doi:10.1016/j.cja.2013.12.009.

Steven H. Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, Reading, Massachusetts, 1994.

Earl Dowell, John Edwards, and Thomas Strganac. Nonlinear aeroelasticity. Journal of Aircraft, 40(5):857-874, 2003.

B.H.K. Lee, S.J. Price, and Y.S. Wong. Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Progress in Aerospace Sciences, 27:205-334, 1999. doi:10.1016/j.paerosci.2009.08.002.

Jan R. Wright and Jonathan E. Cooper. Introduction to Aircraft Aeroelasticity and Loads. John Wiley & Sons, 2007.

J.E. Shigley and C.R. Mischke. Standard Handbook of Machine Design. McGraw-Hill, 2 edition, 1996.

Dan .B. Marghitu, editor. Mechanical Engineer's Handbook. Academic Press, 2001.

D. Eller. Friction, freeplay and flutter of manually controlled aircraft. In International Forum on Aeroelasticity and Structural Dynamics, Stockholm, Sweden, June 2007.

L. Ya. Adrianova. Introduction to Linear Systems of Differential Equations, volume 146 of Translations of Mathematical Monographs. American Mathematical Society, Providence, Rhode Island, 1995.

Giancarlo Benettin, Luigi Galgani, Antonio Giorgilli, and Jean-Marie Strelcyn. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. part 1: Theory. Meccanica, 15(1):9-20, March 1980. doi:10.1007/BF02128236.

Luca Dieci and Erik S. Van Vleck. Lyapunov and other spectra: a survey. In Donald Estep and Simon Taverner, editors, Collected lectures on the preservation of stability under discretization, chapter 11, pages 197{218. SIAM, Philadelphia, PA (USA), 2002.

Alan Wolf, Jack B. Swift, Harry L. Swinney, and John A. Vastano. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3):285-317, July 1985. doi:10.1016/0167-2789(85)90011-9.

S. J. Price, H. Ali Ghanbari, and B. H. K. Lee. The aeroelastic response of a two dimensional airfoil with bilinear and cubic structural nonlinearities. Journal of Fluids and Structures, 9:175-193, September 1995.

H. Dai, X. Yue, D. Xie, and S. N. Atluri. Chaos and chaotic transients in an aeroelastic system. Journal of Sound and Vibration, 333:7267-7285, December 2014. doi:10.1016/j.jsv.2014.08.034.

Piergiovanni Marzocca Liviu Librescu, Gianfranco Chiocchia. Implications of cubic physical/aerodynamic non-linearities on the character of the flutter instability boundary. International Journal of Non-Linear Mechanics, 38(2):173-199, March 2003. doi:10.1016/S0020-7462(01)00054-3.

A. Tamer and P. Masarati. Stability of nonlinear, time-dependent rotorcraft systems using lyapunov characteristic exponents. Journal of the American Helicopter Society, 61(2):14-23, 2016.

Pierangelo Masarati and Aykut Tamer. Sensitivity of trajectory stability estimated by lyapunov characteristic exponents. Aerospace Science and Technology, 47:501-510, 2015.

Nguyen Dinh Cong and Hoang Nam. Lyapunov's inequality for linear dierential algebraic equation. Acta Mathematica Vietnamica, 28(1):73-88, 2003.

Nguyen Dinh Cong and Hoang Nam. Lyapunov regularity of linear dierential algebraic equations of index 1. Acta Mathematica Vietnamica, 29(1):1-21, 2004.

Pierangelo Masarati. Estimation of Lyapunov exponents from multibody dynamics in differential-algebraic form. Proc. IMechE Part K: J. Multi-body Dynamics, 227(4):23-33, 2013. doi:10.1177/1464419312455754.

Karlheinz Geist, Ulrich Parlitz, and Werner Lauterborn. Comparison of dierent methods for computing Lyapunov exponents. Progressof Theoretical Physics, 83(5):875-893, May 1990. doi:10.1143/PTP.83.875.

Pierangelo Masarati and Aykut Tamer. The real schur decomposition estimates lyapunov characteristic exponents with multiplicity greater than one. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 230(4):568-578, 2016.

E.H. Dowell L.N. Virgin M.D. Conner, D.M. Tang. Nonlinear behavior of a typical airfoil section with control surface freeplay: a numerical and experimental study. Journal of Fluids and Structures, 11:89-109, January 1997.

John W. Edwards, Holt Ashley, and John V. Breakwell. Unsteady aerodynamic modeling for arbitrary motions. AIAA Journal, 17(4):365-374, April 1979. doi:10.2514/3.7348.

T. Theodorsen and I.E. Garrick. Nonstationary flow about a wing-aileron-tab combination including aerodynamic balance. Report 736, NACA, 1942.

J. Awrejcewicz, G. Kudra. Stability analysis and Lyapunov exponents of a multi-body mechanical system with rigid unilateral constraints. Nonlinear Analysis, 63:e909-e918, 2005. doi:10.1016/

R. Vasconcellos, A. Abdelkefi, F.D. Marques, and M.R. Hajj. Representation and analysis of control surface freeplay nonlinearity. Journal of Fluids and Structures, 31:79-91, 2012. doi:10.1016/j.jfluidstructs.2012.02.003.

Rui Vasconcellos, Abdessattar Abdelkefi, Muhammad R. Hajj, and Flavio Marques. Discontinuity induced bifurcation in aeroelastic systems with freeplay nonlinearity. In 22nd International Congress of Mechanical Engineering (COBEM'13), pages 2832-2843, Ribeirao Preto, SP, Brasil, November 3-7 2013.

Alfredo Medio and Marji Lines. Nonlinear Dynamics | A Primer. Cambridge University Press, 2001.



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