Quantitative Aeroelastic Stability Prediction of Wings Exhibiting Nonlinear Restoring Forces

Year 2023, Volume: 6 Issue: 2, 76 - 86, 07.08.2023

Aykut Tamer

https://doi.org/10.33187/jmsm.1214586

Cited By: 1

Abstract

In engineering practice, eigen-solution is used to assess the stability of linear dynamical systems. However, the linearity assumption in dynamical systems sometimes implies simplifications, particularly when strong nonlinearities exist. In this case, eigen-analysis requires linerisation of the problem and hence fails to provide a direct stability estimation. For this reason, a more reliable tool should be implemented to predict nonlinear phenomena such as chaos or limit cycle oscillations. One method to overcome this difficulty is the Lyapunov Characteristic Exponents (LCEs), which provide quantitative indications of the stability characteristics of dynamical systems governed by nonlinear time-dependent differential equations. Stability prediction using Lyapunov Characteristic Exponents is compatible with the eigen-solution when the problem is linear. Moreover, LCE estimations do not need a steady or equilibrium solution and they can be calculated as the system response evolves in time. Hence, they provide a generalization of traditional stability analysis using eigenvalues. These properties of Lyapunov Exponents are very useful in aeroelastic problems possessing nonlinear characteristics, which may significantly alter the aeroelastic characteristics, and result in chaotic and limit cycle behaviour. A very common nonlinearity in flexible systems is the nonlinear restoring force such as cubic stiffness, which would substantially benefit from using LCEs in stability assessment. This work presents the quantitative evaluation of aeroelastic stability indicators in the presence of nonlinear restoring force. The method is demonstrated on a two-dimensional aeroelastic problem by comparing the system behaviour and estimated Lyapunov Exponents.

Keywords

Aeroelasticity, Lyapunov Exponents, Nonlinear Dynamical Systems

References

• [1] R. L. Bisplinghoff, H. Ashley, R. L. Halfman, Aeroelasticity, Dover, New York, 1996.
• [2] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Perseus Books, Reading, Massachusetts, 1994.
• [3] E. Dowell, J. Edwards, T. Strganac, Nonlinear aeroelasticity, J. Aircraft, 40(5)(2003), 857-874.
• [4] B.H.K. Lee, S.J. Price, Y.S. Wong, Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos, Prog. Aero. Sci., 35(3)(1999), 205-334.
• [5] X. Jinwu, Y. Yongju, L. Daochun, Recent advance in nonlinear aeroelastic analysis and control of the aircraft, Chin J. Aeronaut., 27(1)(2014),12-22.
• [6] L. Adrianova, Introduction to Linear Systems of Differential Equations, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, 1995.
• [7] G. Benettin, L. Galgani, A. Giorgilli, J. 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)(1980), 9-20.
• [8] L. Dieci, Jacobian free computation of Lyapunov exponents. J. Dynam. Differential Equations, 14(3)(2002), 697-717.
• [9] A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Determining Lyapunov exponents from a time series, Phys. D., 16(3)(1985), 285–317.
• [10] S. J. Price, H. A. Ghanbari, B. H. K. Lee. The aeroelastic response of a two-dimensional airfoil with bilinear and cubic structural nonlinearities, J. Fluids Struct., 9(1995), 175–193.
• [11] H. Dai, X. Yue, D. Xie, S. N. Atluri, Chaos and chaotic transients in an aeroelastic system, J. Sound Vib., 333(2014), 7267–7285.
• [12] L. Librescu, G. Chiocchia, P. Marzocca, Implications of cubic physical/aerodynamic non-linearities on the character of the flutter instability boundary, Int. J. Non Linear Mech., 38(2)(2003), 173–199.
• [13] A. Tamer, P. Masarati, Stability of nonlinear, time-dependent rotorcraft systems using Lyapunov characteristic exponents, J. Am. Helicop. Soc., 61(2)(2016), 14-23.
• [14] P. Masarati, A. Tamer, Sensitivity of trajectory stability estimated by Lyapunov characteristic exponents, Aerosp. Sci. Technol., 47(2015), 501-510.
• [15] N. D. Cong, H. Nam, Lyapunov’s inequality for linear differential algebraic equation, Acta Math. Vietnam, 28(1)(2003), 73–88.
• [16] A. Tamer, P. Masarati, Generalized quantitative stability analysis of time-dependent comprehensive rotorcraft systems, Aerospace, 9(1)(2022), 10.
• [17] P. Masarati, Estimation of Lyapunov exponents from multibody dynamics in differential-algebraic form, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 227(1)(2013), 23-33.
• [18] G. Dimitriadis, Introduction to Nonlinear Aeroelasticity, Aerospace, Wiley, Chichester, West Sussex, U.K., 2017.
• [19] A. Tamer, Aeroelastic response of aircraft wings to external store separation using flexible multibody dynamics, Machines, 9(3)(2021), 61.
• [20] B. H. K. Lee, L.Y. Jiang, Y. S. Wong, Flutter of an airfoil with a cubic restoring force, J. Fluids Struct., 13(1999), 75–101.
• [21] D. Li, S. Guo, J. Xiang, Aeroelastic dynamic response and control of an airfoil section with control surface nonlinearities, J. Sound Vib., 329(2010), 4756–4771.
• [22] A. Abdelkefi, R. Vasconcellos, A. H. Nayfeh, M. R. Hajj, An analytical and experimental investigation into limit-cycle oscillations of an aeroelastic system, Nonlinear Dyn., 71(2013), 159-173.
• [23] B.H.K. Lee, L. Liu, K.W. Chung. Airfoil motion in subsonic flow with strong cubic nonlinear restoring forces, J. Sound Vib., 281(2005), 699–717.
• [24] K. Geist, U. Parlitz, W. Lauterborn, Comparison of different methods for computing Lyapunov exponents, PTEP. Prog. of Theor. Phys, 83(5)(1990), 875–893.
• [25] P. Masarati, A. 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)(2016), 568–578.
• [26] D. Hodges, G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge University Press, Cambridge, England, 2002.
• [27] J. R. Wright, J. E. Cooper, Introduction to Aircraft Aeroelasticity and Loads, John Wiley & Sons, 2007.
• [28] W.A. Silva, R.E. Bartels, Development of reduced-order models for aeroelastic analysis and flutter prediction using the cfl3dv6.0 code, J. Fluids Struct., 19(6)(2004), 729-745.
• [29] J. W. Edwards, H. Ashley, J. V. Breakwell, Unsteady aerodynamic modeling for arbitrary motions, AIAA Journal, 17(4)(1979), 365-374.
• [30] A. Medio, M. Lines, Nonlinear Dynamics — A Primer, Cambridge University Press, 2001.