Research Papers

Flutter Control of a Two-Degrees-of-Freedom Airfoil Using a Nonlinear Tuned Vibration Absorber

[+] Author and Article Information
Arnaud Malher

IMSIA (Institute of Mechanical Sciences and
Industrial Applications),
Université Paris-Saclay,
828 Bd des Maréchaux,
Palaiseau Cedex 91732, France
e-mails: arnaud.malher@ensta-paristech.fr,

Cyril Touzé

IMSIA (Institute of Mechanical Sciences
and Industrial Applications),
Université Paris-Saclay,
828 Bd des Maréchaux,
Palaiseau Cedex 91732, France

Olivier Doaré

IMSIA (Institute of Mechanical Sciences and
Industrial Applications),
Université Paris-Saclay,
828 Bd des Maréchaux,
Palaiseau Cedex 91732, France

Giuseppe Habib, Gaëtan Kerschen

Space Structures and Systems Laboratory,
Department of Aerospace and
Mechanical Engineering,
University of Liège,
Liège 4000, Belgium

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 28, 2016; final manuscript received February 14, 2017; published online May 4, 2017. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 12(5), 051016 (May 04, 2017) (11 pages) Paper No: CND-16-1456; doi: 10.1115/1.4036420 History: Received September 28, 2016; Revised February 14, 2017

The influence of a nonlinear tuned vibration absorber (NLTVA) on the airfoil flutter is investigated. In particular, its effect on the instability threshold and the potential subcriticality of the bifurcation is analyzed. For that purpose, the airfoil is modeled using the classical pitch and plunge aeroelastic model together with a linear approach for the aerodynamic loads. Large amplitude motions of the airfoil are taken into account with nonlinear restoring forces for the pitch and plunge degrees-of-freedom. The two cases of a hardening and a softening spring behavior are investigated. The influence of each NLTVA parameter is studied, and an optimum tuning of these parameters is found. The study reveals the ability of the NLTVA to shift the instability, avoid its possible subcriticality, and reduce the limit cycle oscillations (LCOs) amplitude.

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Vipperman, J. S. , Clark, R. L. , Conner, M. , and Dowell, E. H. , 1998, “ Experimental Active Control of a Typical Section Using a Trailing-Edge Flap,” J. Aircr., 35(2), pp. 224–229. [CrossRef]
Karpel, M. , 1982, “ Design for Active Flutter Suppression and Gust Alleviation Using State-Space Aeroelastic Modeling,” J. Aircr., 19(3), pp. 221–227. [CrossRef]
Ko, J. , Kurdila, A. J. , and Strganac, T. W. , 1997, “ Nonlinear Control of a Prototypical Wing Section With Torsional Nonlinearity,” J. Guid., Control, Dyn., 20(6), pp. 1181–1189. [CrossRef]
Heeg, J. , 1993, “ Analytical and Experimental Investigation of Flutter Suppression by Piezoelectric Actuation,” NASA Langley Research Center, Hampton, VA, Technical Report No. NASA-TP-3241.
Dowell, E. H. , 2004, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, Boston, MA.
Malher, A. , Doaré, O. , and Touzé, C. , 2017, “ Influence of a Hysteretic Damper on the Flutter Instability,” J. Fluids Struct., 68, pp. 356–369. [CrossRef]
Kwon, S.-D. , and Park, K.-S. , 2004, “ Suppression of Bridge Flutter Using Tuned Mass Dampers Based on Robust Performance Design,” J. Wind Eng. Ind. Aerodyn., 92(11), pp. 919–934. [CrossRef]
Vaurigaud, B. , Manevitch, L. I. , and Lamarque, C.-H. , 2011, “ Passive Control of Aeroelastic Instability in a Long Span Bridge Model Prone to Coupled Flutter Using Targeted Energy Transfer,” J. Sound Vib., 330(11), pp. 2580–2595. [CrossRef]
Pourzeynali, S. , and Datta, T. , 2002, “ Control of Flutter of Suspension Bridge Deck Using TMD,” Wind Struct., 5(5), pp. 407–422. [CrossRef]
Den Hartog, J. P. , 1934, Mechanical Vibrations, McGraw-Hill Book Company, New York.
Gattulli, V. , Di Fabio, F. , and Luongo, A. , 2004, “ Nonlinear Tuned Mass Damper for Self-Excited Oscillations,” Wind Struct., 7(4), pp. 251–264. [CrossRef]
Lee, Y. S. , Vakakis, A. , Bergman, L. , McFarland, D. M. , and Kerschen, G. , 2007, “ Suppressing Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers, Part 1: Theory,” AIAA J., 45(3), pp. 693–711. [CrossRef]
Lee, Y. S. , Kerschen, G. , McFarland, D. M. , Hill, W. J. , Nichkawde, C. , Strganac, T. W. , Bergman, L. A. , and Vakakis, A. F. , 2007, “ Suppressing Aeroelastic Instability Using Broadband Passive Targeted Energy Transfers—Part 2: Experiments,” AIAA J., 45(10), pp. 2391–2400. [CrossRef]
Gendelman, O. V. , 2001, “ Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators,” Nonlinear Dyn., 25(1–3), pp. 237–253. [CrossRef]
Vakakis, A. F. , and Gendelman, O. , 2001, “ Energy Pumping in Nonlinear Mechanical Oscillators—Part II: Resonance Capture,” ASME J. Appl. Mech., 68(1), pp. 42–48. [CrossRef]
Vakakis, A. F. , Manevitch, L. I. , Gendelman, O. , and Bergman, L. , 2003, “ Dynamics of Linear Discrete Systems Connected to Local, Essentially Non-Linear Attachments,” J. Sound Vib., 264(3), pp. 559–577. [CrossRef]
Vakakis, A. F. , Gendelman, O. V. , Bergman, L. A. , McFarland, D. M. , Kerschen, G. , and Lee, Y. S. , 2008, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Vol. 156. Springer, New York.
Gendelman, O. V. , Vakakis, A. F. , Bergman, L. A. , and McFarland, D. M. , 2010, “ Asymptotic Analysis of Passive Nonlinear Suppression of Aeroelastic Instabilities of a Rigid Wing in Subsonic Flow,” SIAM J. Appl. Math., 70(5), pp. 1655–1677. [CrossRef]
Hubbard, S. A. , Fontenot, R. L. , McFarland, D. M. , Cizmas, P. G. , Bergman, L. A. , Strganac, T. W. , and Vakakis, A. F. , 2014, “ Transonic Aeroelastic Instability Suppression for a Swept Wing by Targeted Energy Transfer,” J. Aircr., 51(5), pp. 1467–1482. [CrossRef]
Viguié, R. , and Kerschen, G. , 2009, “ Nonlinear Vibration Absorber Coupled to a Nonlinear Primary System: A Tuning Methodology,” J. Sound Vib., 326(3), pp. 780–793. [CrossRef]
Habib, G. , Detroux, T. , Viguié, R. , and Kerschen, G. , 2015, “ Nonlinear Generalization of Den Hartog's Equal-Peak Method,” Mech. Syst. Signal Process., 52–53, pp. 17–28. [CrossRef]
Habib, G. , and Kerschen, G. , 2016, “ A Principle of Similarity for Nonlinear Vibration Absorbers,” Physica D: Nonlinear Phenom., 332(1), pp. 1–8. [CrossRef]
Habib, G. , and Kerschen, G. , 2015, “ Suppression of Limit Cycle Oscillations Using the Nonlinear Tuned Vibration Absorber,” Proc. R. Soc. London, 471(2176), p. 20140976. [CrossRef]
Grappasonni, C. , Habib, G. , Detroux, T. , and Kerschen, G. , 2016, “ Experimental Demonstration of a 3D-Printed Nonlinear Tuned Vibration Absorber,” Nonlinear Dynamics (Conference Proceedings of the Society for Experimental Mechanics Series), Vol. 1, Springer, Cham, Switzerland, pp. 173–183.
Benacchio, S. , Malher, A. , Boisson, J. , and Touzé, C. , 2016, “ Design of a Magnetic Vibration Absorber With Tunable Stiffnesses,” Nonlinear Dyn., 85(2), pp. 893–911. [CrossRef]
Lacarbonara, W. , and Cetraro, M. , 2011, “ Flutter Control of a Lifting Surface Via Visco-Hysteretic Vibration Absorbers,” Int. J. Aeronaut. Space Sci., 12(4), pp. 331–345.
Lee, B. H. K. , Jiang, L. , and Wong, Y. , 1998, “ Flutter of an Airfoil With a Cubic Nonlinear Restoring Force,” AIAA Paper No. AIAA-98-1725
Pettit, C. L. , and Beran, P. S. , 2003, “ Effects of Parametric Uncertainty on Airfoil Limit Cycle Oscillation,” J. Aircr., 40(5), pp. 1004–1006. [CrossRef]
Bisplinghoff, R. L. , and Ashley, H. , 1962, Principles of Aeroelasticity, Wiley, New York.
Fung, Y. C. , 2002, An Introduction to the Theory of Aeroelasticity, Dover Publications, Mineola, NY.
Lee, Y. S. , Vakakis, A. F. , Bergman, L. A. , McFarland, D. M. , and Kerschen, G. , 2005, “ Triggering Mechansms of Limit Cycle Oscillations Due to Aeroelastic Instability,” J. Fluids Struct., 21(5–7), pp. 485–529. [CrossRef]
Amandolèse, X. , Michelin, S. , and Choquel, M. , 2013, “ Low Speed Flutter and Limit Cycle Oscillations of a Two-Degree-of-Freedom Flat Plate in a Wind Tunnel,” J. Fluids Struct., 43, pp. 244–255. [CrossRef]
Dimitriadis, G. , and Li, J. , 2009, “ Bifurcation Behavior of Airfoil Undergoing Stall Flutter Oscillations in Low-Speed Wind Tunnel,” AIAA J., 47(11), pp. 2577–2596. [CrossRef]
Kuznetsov, Y. A. , 2013, Elements of Applied Bifurcation Theory, Vol. 112, Springer, New York.
Guckenheimer, J. , and Holmes, P. , 2013, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York.
Manneville, P. , 1995, “ Dissipative Structures and Weak Turbulence,” Chaos—The Interplay Between Stochastic and Deterministic Behaviour, Springer, New York, pp. 257–272.
Iooss, G. , and Adelmeyer, M. , 1998, Topics in Bifurcation Theory and Applications, World Scientific, Singapore.
Gai, G. , and Timme, S. , 2016, “ Nonlinear Reduced-Order Modelling for Limit-Cycle Oscillation Analysis,” Nonlinear Dyn., 84(2), pp. 991–1009. [CrossRef]
Touzé, C. , Thomas, O. , and Chaigne, A. , 2004, “ Hardening/Softening Behaviour in Non-Linear Oscillations of Structural Systems Using Non-Linear Normal Modes,” J. Sound Vib., 273(1), pp. 77–101. [CrossRef]
Doedel, E. J. , Paffenroth, R. C. , Champneys, A. R. , Fairgrieve, T. F. , Kuznetsov, Y. A. , Sandstede, B. , and Wang, X. , 2002, “ Auto 2000: Continuation and Bifurcation Software for Ordinary Differential Equations,” Concordia University, Montreal, QC, Canada, Technical Report.
Woolston, D. S. , Runyan, H. L. , and Andrews, R. E. , 1957, “ An Investigation of Effects of Certain Types of Structural Nonlinearities on Wing and Control Surface Flutter,” J. Aeronaut. Sci., 24(1), pp. 57–63. [CrossRef]


Grahic Jump Location
Fig. 1

Sketch of the two degrees-of-freedom airfoil (main structure) coupled with the NLTVA (absorber)

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Fig. 2

Reduced flutter velocity given by the Routh–Hurwitz criterion as a function of the NLTVA-reduced eigenfrequency γ, and for a damping ratio ζ = 0.15. The gray area corresponds to the stability of the system.

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Fig. 3

Cartography of the reduced flutter velocity given by the criterion e5 as a function of the NLTVA-reduced frequency γ and damping ratio ζ

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Fig. 4

Evolution of the optimal flutter velocity gain GŨf caused by the NLTVA versus xα, rα2, and Ω2

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Fig. 5

Estimation of GŨf using the fitted parameters ζoptfit and γoptfit versus xα, rα2 and Ω2

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Fig. 6

Bifurcation diagram of pitch and plunge mode for the system without absorber (a) and (b) and in the presence of an NLTVA without nonlinearities (c) and (d)

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Fig. 7

Bifurcation diagram of (a) pitch and (b) plunge mode for four increasing values of ξ. The solid lines correspond to the stable solutions and the dashed line to the unstable ones. LP stands for limit point and NS for Neimark–Sacker bifurcation point.

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Fig. 8

Influence of the NLTVA with ξ = 0 and ξ = 0.217 on the bifurcation diagram of the 2dofs airfoil

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Fig. 9

Time series simulations corresponding to Fig. 8 with and without absorber at Ũ=1.1 (a)–(c) and Ũ=1.4 (d)–(f). The NLTVA is set such as ξ = 0.217. The initial conditions are equal to zero except for the initial amplitude of the pitch mode, which is set equal to α(0) = 10 deg for (a)–(c) and to α(0) = 6.10−3 deg for (d)–(f).

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Fig. 10

Bifurcation diagram of (a) and (c) pitch and (b) and (d) plunge mode for increasing values of ξ. The solid lines correspond to the stable solutions and the dashed line to the unstable ones. The plots (c) and (d) detail the bifurcation diagram around the criticality for ξ ≥ −0.15. LP stands for limit point and NS for Neimark–Sacker bifurcation point.




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