Skip to main content

A successive robust flutter prediction technique for aeroelastic systems using µ method


In this work, a successive robust flutter prediction technique is developed by coupling nominal analysis, ground vibration test, wind tunnel test, uncertainty model updation and robust analysis based on the structured singular value method to predict the worst flutter boundary of a swept back wing in transonic flow regime. Here, uncertainties in both structural and unsteady aerodynamics parameters are considered in the generalized coordinates. These uncertainties are introduced in the nominal aeroelastic system in a linear fractional transformation framework. The magnitudes of structural uncertainties are estimated based on the difference in natural frequencies between ground vibration test and nominal analysis. The magnitudes of aerodynamic uncertainties are estimated using a model updation technique based on the structured singular value method considering the difference in damping values between wind tunnel test and nominal analysis. The capability of the present successive robust flutter prediction technique is investigated by estimating the robust flutter boundary of a swept back wing in transonic flow regime. From the results, it is observed that the uncertainty model updation provides a reasonable estimate of aerodynamic uncertainty magnitude. Further, the present flutter prediction approach gives a good estimate of transonic flutter boundary (transonic dip) by successively updating the aerodynamic uncertainty bounds using wind tunnel data for various set of test Mach numbers.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. Pettit CL (2004) Uncertainty quantification in aeroelasticity: recent results and research challenges. J Aircr 41(5):1217–1229

    Article  Google Scholar 

  2. Danowsky BP, Chrstos JR, Klyde DH, Farhat C, Brenner M (2010) Evaluation of aeroelastic uncertainty analysis methods. J Aircr 47(4):1266–1273

    Article  Google Scholar 

  3. Khodaparast HH, Mottershead JE, Badcock KJ (2010) Propagation of structural uncertainty to linear aeroelastic stability. Comput Struct 88(3–4):223–236

    Article  Google Scholar 

  4. Pitt DM, Haudrich DP, Thomas MJ, Griffin KE (2008) Probabilistic aeroelastic analysis and its implications on flutter margin requirements. Report No. AIAA-2008–2198

  5. Beran PS, Pettit CL, Millman DR (2006) Uncertainty quantification of limit-cycle oscillations. J Comput Phys 217(1):217–247

    Article  Google Scholar 

  6. Bruno L, Canuto C, Fransos D (2009) Stochastic aerodynamics and aeroelasticity of a flat plate via generalized polynomial chaos. J Fluids Struct 25(7):1158–1176

    Article  Google Scholar 

  7. Xiaowen J, Guoqing H, Yan-Gang Z (2020) Probabilistic flutter analysis of bridge considering aerodynamic and structural parameter uncertainties. J Wind Eng Ind Aerodyn 201:1041–1068

    Google Scholar 

  8. Mannini C, Bartoli G (2015) Aerodynamic uncertainty propagation in bridge flutter analysis. Struct Saf 52:29–39

    Article  Google Scholar 

  9. Cheng J, Xiao R (2005) Probabilistic free vibration and flutter analyses of suspension bridges. Eng Struct 27:1509–1518

    Article  Google Scholar 

  10. Wu S, Livne E (2017) Alternative aerodynamic uncertainty modeling approaches for flutter reliability analysis. AIAA J 55(8):2808–2823

    Article  Google Scholar 

  11. Kumar S, Onkar AK, Maligappa M (2019) Frequency domain approach for probabilistic flutter analysis using stochastic finite elements. Meccanica 54:2207–2225

    MathSciNet  Article  Google Scholar 

  12. Kumar S, Onkar AK, Manjuprasad M (2020) Stochastic modelling and reliability analysis of wing flutter. J Aerosp Eng 33(5):1–18

    Article  Google Scholar 

  13. Zheng Y, Qiu Z (2019) An efficient method for flutter stability analysis of aeroelastic systems considering uncertainties in aerodynamic and structural parameters. Mech Syst Signal Process 126:407–426

    Article  Google Scholar 

  14. Lokatt M (2017) Aeroelastic flutter analysis considering modelling uncertainties. J Fluids Struct 74:247–262

    Article  Google Scholar 

  15. Haiwei Y, Jinglong H (2009) Robust flutter analysis of a nonlinear aeroelastic system with parametric uncertainties. Aerosp Sci Technol 13:139–149

    Article  Google Scholar 

  16. Lind R, Brenner M (1999) Robust aeroservoelastic stability analysis. Springer, London

    Book  Google Scholar 

  17. Lind R (2002) Match-point solutions for robust flutter analysis. J Aircr 39(1):91–99

    MathSciNet  Article  Google Scholar 

  18. Borglund D (2004) The µ-k method for robust flutter solutions. J Aircr 41(5):1209–1216

    Article  Google Scholar 

  19. Borglund D (2008) Robust eigenvalue analysis using the structured singular value: the µ-p Flutter method. AIAA J 46(11):2806–2813

    Article  Google Scholar 

  20. Bueno DD, Goes LCS, Goncalves PJP (2015) Flutter analysis including structural uncertainties. Meccanica 50:2093–2101

    MathSciNet  Article  Google Scholar 

  21. Huang R, Hu H, Zhao Y (2014) Nonlinear reduced-order modeling for multiple-input/multiple-output aerodynamic systems. AIAA J 52(6):1219–1231

    Article  Google Scholar 

  22. Xiong C, Wang L, Liu G, Shi Q (2019) An iterative dimension-by-dimension method for structural interval response prediction with multidimensional uncertain variables. Aerosp Sci Technol 86:572–581

    Article  Google Scholar 

  23. Iannelli A, Marcos A, Bombardieri R, Cavallaro R (2020) Linear fractional transformation co-modeling of high-order aeroelastic systems for robust flutter analysis. Eur J Control 54:49–63

    MathSciNet  Article  Google Scholar 

  24. Chen X, Qiu Z, Wang X, Li Y, Wang R (2017) Uncertain reduced-order modeling for unsteady aerodynamics with interval parameters and its application on robust flutter boundary prediction. Aerosp Sci Technol 71:214–230

    Article  Google Scholar 

  25. Yates E, Carson Jr (1987) AGARD Standard Aeroelastic Configurations for Dynamic Response I - Wing 445.6. Tech. Rep. NASA-TM-100492

  26. Aeroelastic Analysis User’s Guide, MSC Nastran (2017)

  27. MATLAB Robust Control Toolbox User’s Guide, The MathWorks, Inc. (2019)

Download references


This work was supported by Council and Scientific and Industrial Research (CSIR), India under Focused Based Research (FBR) program. The author would like to thank The Head, STTD and The Director, CSIR-NAL for their encouragement and support during the course of the work.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Amit Kumar Onkar.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Onkar, A.K. A successive robust flutter prediction technique for aeroelastic systems using µ method. Meccanica 56, 2613–2629 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Uncertainty modelling
  • Model updation
  • Robust flutter
  • Laplace domain
  • Structured singular value
  • AGARD wing
  • ATW model
  • Wind tunnel test
  • Ground vibration test