Skip to main content
SpringerLink
Log in

Complex dynamics of a conceptual airfoil structure with consideration of extreme flight conditions

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

An aircraft in practice serves under extreme flight conditions that will have a substantial impact on its flight safety. Understanding dynamics of airfoil structure of an aircraft subjected to severe load conditions is thus extremely valuable and necessary. In this study, we will explore the complicated dynamical behaviors of a conceptual airfoil excited by an external harmonic force and an extreme random load. Importantly, such an extreme random load is portrayed by a non-Gaussian Lévy noise with a heavy-tailed feature. Bistable behaviors of the deterministic airfoil system are performed firstly from amplitude–frequency response and basin of attraction. Then, the effects of the extreme random load on the airfoil system are thoroughly investigated. Interestingly, within the bistable regime, the extreme random load can lead to stochastic transition and stochastic resonance. Due to its heavy-tailed nature, the Lévy noise would increase the possibility of a highly unexpected stochastic transition behavior between desirable low-amplitude and catastrophic high-amplitude oscillations compared with the Gaussian scenario. Such vibration patterns might damage or destroy the airfoil structure, which will put an aircraft in great danger. All the findings would be helpful in ensuring the flight safety and enhancing the strength and reliability of airfoil structure operating at extreme flight conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data availability

The authors declare that the data supporting the findings of this study are available within the article.

References

  1. Liu, Q., Xu, Y., Kurths, J., Liu, X.C.: Complex nonlinear dynamics and vibration suppression of conceptual airfoil models: a state-of-the-art overview. Chaos 32, 062101 (2022)

    MathSciNet  Google Scholar 

  2. Lee, B.H.K., Price, S.J., Wong, Y.S.: Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos. Prog. Aerosp. Sci. 35, 205–334 (1999)

    Google Scholar 

  3. Sales, T.P., Pereira, D.A., Marques, F.D., Rade, D.A.: Modeling and dynamic characterization of nonlinear non-smooth aeroviscoelastic systems. Mech. Syst. Signal Process. 116, 900–915 (2019)

    Google Scholar 

  4. Meehan, P.A.: Flutter prediction of its occurrence, amplitude and nonlinear behaviour. J. Sound Vib. 535, 117117 (2022)

    Google Scholar 

  5. Jonsson, E., Riso, C., Lupp, C.A., Cesnik, C.E.S., Martins, J.R.R.A., Epureanu, B.I.: Flutter and post-flutter constraints in aircraft design optimization. Prog. Aerosp. Sci. 109, 100537 (2019)

    Google Scholar 

  6. Liu, Q., Xu, Y., Kurths, J.: Active vibration suppression of a novel airfoil model with fractional order viscoelastic constitutive relationship. J. Sound Vib. 432, 50–64 (2018)

    Google Scholar 

  7. Jiffri, S., Fichera, S., Mottershead, J.E., Da Ronch, A.: Experimental nonlinear control for flutter suppression in a nonlinear aeroelastic system. J. Guid. Control Dyn. 40, 1925–1938 (2017)

    Google Scholar 

  8. Sinou, J.J.: Flutter instability and active aeroelastic control with time delay for a two-dimensional airfoil. Eur. J. Mech. A Solids 92, 104465 (2022)

    MathSciNet  MATH  Google Scholar 

  9. Kirschmeier, B.A., Gianikos, Z., Gopalarathnam, A., Bryant, M.: Amplitude annihilation in wake-influenced aeroelastic limit-cycle oscillations. AIAA J. 58, 4117–4127 (2020)

    Google Scholar 

  10. Liu, Q., Xu, Y., Li, Y.G., Kurths, J., Liu, X.C.: Fixed-interval smoothing of an aeroelastic airfoil model with cubic or free-play nonlinearity in incompressible flow. Acta Mechanica Sinica 37, 1168–1182 (2021)

    MathSciNet  Google Scholar 

  11. Tripathi, D., Vishal, S., Bose, C., Venkatramani, J.: Stall-induced fatigue damage in nonlinear aeroelastic systems under stochastic inflow: numerical and experimental analyses. Int. J. Non-Linear Mech. 142, 104003 (2022)

    Google Scholar 

  12. Ma, J.Z., Xu, Y., Li, Y.G., Tian, R.L., Ma, S.J., Kurths, J.: Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions. Appl. Math. Mech. (English Edition) 42, 65–84 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Ma, J.Z., Xu, Y., Xu, W., Li, Y.G., Kurths, J.: Slowing down critical transitions via Gaussian white noise and periodic force. Sci. China Technol. Sci. 62, 2144–2152 (2019)

    Google Scholar 

  14. Zhang, X.Y., Xu, Y., Liu, Q., Kurths, J.: Rate-dependent tipping-delay phenomenon in a thermoacoustic system with colored noise. Sci. China Technol. Sci. 63, 2315–2327 (2020)

    Google Scholar 

  15. Venstra, W.J., Westra, H.J., Van Der Zant, H.S.: Stochastic switching of cantilever motion. Nat. Commun. 4, 2624 (2013)

    Google Scholar 

  16. Xu, Y., Gu, R.C., Zhang, H.Q., Xu, W., Duan, J.Q.: Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Phys. Rev. E 83, 056215 (2011)

    Google Scholar 

  17. Ricci, F., Rica, R.A., Spasenović, M., Gieseler, J., Rondin, L., Novotny, L., Quidant, R.: Optically levitated nanoparticle as a model system for stochastic bistable dynamics. Nat. Commun. 8, 15141 (2017)

    Google Scholar 

  18. Zhang, T.T., Jin, Y.F., Zhang, Y.X.: Performance improvement of the stochastic-resonance-based tri-stable energy harvester under random rotational vibration. Theor. Appl. Mech. Lett. 12, 100365 (2022)

    Google Scholar 

  19. Mei, R.X., Xu, Y., Li, Y.G., Kurths, J.: Characterizing stochastic resonance in a triple cavity. Philos. Trans. R. Soc. A 379, 20200230 (2021)

    MathSciNet  Google Scholar 

  20. Poirel, D.C., Price, S.J.: Structurally nonlinear fluttering airfoil in turbulent flow. AIAA J. 39, 1960–1968 (2001)

    Google Scholar 

  21. Poirel, D., Mendes, F.: Experimental small-amplitude self-sustained pitch-heave oscillations at transitional reynolds numbers. AIAA J. 52, 1581–1590 (2014)

    Google Scholar 

  22. Xu, Y., Liu, Q., Guo, G.B., Xu, C., Liu, D.: Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance. Nonlinear Dyn. 89, 1579–1590 (2017)

    MathSciNet  Google Scholar 

  23. Liu, Q., Xu, Y., Xu, C., Kurths, J.: The sliding mode control for an airfoil system driven by harmonic and colored Gaussian noise excitations. Appl. Math. Model. 64, 249–264 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Liu, Q., Xu, Y., Kurths, J.: Bistability and stochastic jumps in an airfoil system with viscoelastic material property and random fluctuations. Commun. Nonlinear Sci. Numer. Simul. 84, 105184 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Chassaing, J.C., Lucor, D., Trégon, J.: Stochastic nonlinear aeroelastic analysis of a supersonic lifting surface using an adaptive spectral method. J. Sound Vib. 331, 394–411 (2012)

    Google Scholar 

  26. Raaj, A., Venkatramani, J., Mondal, S.: Synchronization of pitch and plunge motions during intermittency route to aeroelastic flutter. Chaos 29, 043129 (2019)

    MATH  Google Scholar 

  27. Raaj, A., Mondal, S., Jagdish, V.: Investigating amplitude death in a coupled nonlinear aeroelastic system. Int. J. Non-Linear Mech. 129, 103659 (2021)

    Google Scholar 

  28. Ma, J.Z., Liu, Q., Xu, Y., Kurths, J.: Early warning of noise-induced catastrophic high-amplitude oscillations in an airfoil model. Chaos 32, 033119 (2022)

    MathSciNet  Google Scholar 

  29. Venkatramani, J., Kumar, S.K., Sarkar, S., Gupta, S.: Physical mechanism of intermittency route to aeroelastic flutter. J. Fluids Struct. 75, 9–26 (2017)

    Google Scholar 

  30. Venkatramani, J., Sarkar, S., Gupta, S.: Intermittency in pitch-plunge aeroelastic systems explained through stochastic bifurcations. Nonlinear Dyn. 92, 1225–1241 (2018)

    Google Scholar 

  31. Abdallah, I., Natarajan, A., Sørensen, J.D.: Impact of uncertainty in airfoil characteristics on wind turbine extreme loads. Renew. Energy 75, 283–300 (2015)

    Google Scholar 

  32. Amaral, F.R.D., Himeno, F.H.T., Pagani Jr, C.D.C., de Medeiros, M.A.F.: Slat noise from an MD30P30N airfoil at extreme angles of attack. AIAA J. 56, 964–978 (2018)

    Google Scholar 

  33. Leknys, R.R., Arjomandi, M., Kelso, R.M., Birzer, C.: Dynamic- and post-stall characteristics of pitching airfoils at extreme conditions. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 232, 1171–1185 (2018)

    Google Scholar 

  34. Miao, W.P., Li, C., Wang, Y.B., Xiang, B., Liu, Q.S., Deng, Y.H.: Study of adaptive blades in extreme environment using fluid-structure interaction method. J. Fluids Struct. 91, 102734 (2019)

    Google Scholar 

  35. Rudy, S.H., Sapsis, T.P.: Prediction of intermittent fluctuations from surface pressure measurements on a turbulent airfoil. AIAA J. 60, 4174–4190 (2022)

    Google Scholar 

  36. Haghpanah, B., Shirazi, A., Salari-Sharif, L., Izard, A.G., Valdevit, L.: Elastic architected materials with extreme damping capacity. Extrem. Mech. Lett. 17, 56–61 (2017)

    Google Scholar 

  37. Misseroni, D., Barbieri, E., Pugno, N.M.: Extreme deformations of the cantilever Euler Elastica under transverse aerodynamic load. Extrem. Mech. Lett. 42, 101110 (2021)

    Google Scholar 

  38. Zhao, D., Li, Y.G., Xu, Y., Liu, Q., Kurths, J.: Extreme events in a class of nonlinear Duffing-type oscillators with a parametric periodic force. Eur. Phys. J. Plus 137, 314 (2022)

    Google Scholar 

  39. Zheng, X.J.: Extreme mechanics. Theor. Appl. Mech. Lett. 10, 1–7 (2020)

    Google Scholar 

  40. Xu, Y., Li, Y.G., Zhang, H., Li, X.F., Kurths, J.: The switch in a genetic toggle system with Lévy noise. Sci. Rep. 6, 31505 (2016)

    Google Scholar 

  41. Zhang, X.Y., Xu, Y., Liu, Q., Kurths, J., Grebogi, C.: Rate-dependent tipping and early warning in a thermoacoustic system under extreme operating environment. Chaos 31, 113115 (2021)

    MathSciNet  Google Scholar 

  42. Ditlevsen, P.D.: Observation of \(\alpha \)-stable noise induced millennial climate changes from an ice-core record. Geophys. Res. Lett. 26, 1441–1444 (1999)

    Google Scholar 

  43. Yang, A.J., Wang, H., Zhang, T.H., Yuan, S.L.: Stochastic switches of eutrophication and oligotrophication: Modeling extreme weather via non-Gaussian Lévy noise. Chaos 32, 043116 (2022)

    Google Scholar 

  44. Xiao, Y.Q., Li, Q.S., Li, Z.N., Chow, Y.W., Li, G.Q.: Probability distributions of extreme wind speed and its occurrence interval. Eng. Struct. 28, 1173–1181 (2006)

    Google Scholar 

  45. Shen, X.R., Zhang, H., Xu, Y., Meng, S.X.: Observation of alpha-stable noise in the laser gyroscope data. IEEE Sens. J. 16, 1998–2003 (2015)

    Google Scholar 

  46. Joelson, M., Néel, M.C.: On alpha stable distribution of wind driven water surface wave slope. Chaos 18, 033117 (2008)

    MATH  Google Scholar 

  47. Xu, Y., Zan, W.R., Jia, W.T., Kurths, J.: Path integral solutions of the governing equation of SDEs excited by Lévy white noise. J. Comput. Phys. 394, 41–55 (2019)

    MathSciNet  MATH  Google Scholar 

  48. Kim, D.H., Lee, I.: Transonic and low-supersonic aeroelastic analysis of a two-degree-of-freedom airfoil with a freeplay non-linearity. J. Sound Vib. 234, 859–880 (2000)

    Google Scholar 

  49. Liu, G., Wang, L., Liu, J.K., Chen, Y.M., Lu, Z.R.: Identification of an airfoil-store system with cubic nonlinearity via enhanced response sensitivity approach. AIAA J. 56, 4977–4987 (2018)

    Google Scholar 

  50. Monfared, Z., Afsharnezhad, Z., Esfahani, J.A.: Flutter, limit cycle oscillation, bifurcation and stability regions of an airfoil with discontinuous freeplay nonlinearity. Nonlinear Dyn. 90, 1965–1986 (2017)

    MATH  Google Scholar 

  51. Leung, A.Y.T., Guo, Z.J.: Residue harmonic balance for two-degree-of-freedom airfoils with cubic structural nonlinearity. AIAA J. 49, 2607–2615 (2011)

    Google Scholar 

  52. Blanc, F., Roux, F.X., Jouhaud, J.C.: Harmonic-balance-based code-coupling algorithm for aeroelastic systems subjected to forced excitation. AIAA J. 48, 2472–2481 (2010)

    Google Scholar 

  53. Feudel, U., Grebogi, C.: Why are chaotic attractors rare in multistable systems? Phys. Rev. Lett. 91, 134102 (2003)

    Google Scholar 

  54. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Rep. 637, 1–50 (2016)

  55. Brezetskyi, S., Dudkowski, D., Kapitaniak, T.: Rare and hidden attractors in Van der Pol-Duffing oscillators. Eur. Phys. J. Spec. Top. 224, 1459–1467 (2015)

    Google Scholar 

  56. Yue, X.L., Lv, G., Zhang, Y.: Rare and hidden attractors in a periodically forced Duffing system with absolute nonlinearity. Chaos Solitons Fractals 150, 111108 (2021)

    MathSciNet  MATH  Google Scholar 

  57. Menck, P.J., Heitzig, J., Marwan, N., Kurths, J.: How basin stability complements the linear-stability paradigm. Nat. Phys. 9, 89–92 (2013)

    Google Scholar 

  58. Brzeski, P., Lazarek, M., Kapitaniak, T., Kurths, J., Perlikowski, P.: Basin stability approach for quantifying responses of multistable systems with parameters mismatch. Meccanica 51, 2713–2726 (2016)

    MathSciNet  Google Scholar 

  59. Schultz, P., Menck, P.J., Heitzig, J., Kurths, J.: Potentials and limits to basin stability estimation. N. J. Phys. 19, 023005 (2017)

    MATH  Google Scholar 

  60. Li, Z.P., Zhang, P., Pan, T.Y., Li, Q.S., Zhang, J.: Study on effects of thickness on airfoil-stall at low reynolds numbers by cusp-catastrophic model based on GA(W)-1 airfoil. Chin. J. Aeronaut. 33, 1444–1453 (2020)

    Google Scholar 

  61. Li, Z.P., Zhang, P., Pan, T.Y., Li, Q.S., Zhang, J., Dowell, E.H.: Catastrophe-theory-based modeling of airfoil-stall boundary at low Reynolds numbers. AIAA J. 56, 36–45 (2018)

    Google Scholar 

  62. Schoenmakers, S., Feudel, U.: A resilience concept based on system functioning: a dynamical systems perspective. Chaos 31, 053126 (2021)

    MathSciNet  Google Scholar 

  63. Wang, Z.Q., Xu, Y., Yang, H.: Lévy noise induced stochastic resonance in an FHN model. Sci. China Technol. Sci. 59, 371–375 (2016)

    Google Scholar 

  64. Rice, S.O.: Mathematical analysis of random noise. Bell Syst. Techn. J. 23, 282–332 (1944)

    MathSciNet  MATH  Google Scholar 

  65. Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic resonance. Rev. Mod. Phys. 70, 223 (1998)

    Google Scholar 

  66. Xu, Y., Wu, J., Du, L., Yang, H.: Stochastic resonance in a genetic toggle model with harmonic excitation and Lévy noise. Chaos Solitons Fractals 92, 91–100 (2016)

    MathSciNet  MATH  Google Scholar 

  67. Galtier, T., Gupta, S., Rychlik, I.: Crossings of second-order response processes subjected to LMA loadings. J. Prob. Stat. 2010, 752452 (2009)

    MathSciNet  MATH  Google Scholar 

  68. Zhang, Y.X., Jin, Y.F.: Colored Lévy noise-induced stochastic dynamics in a tri-stable hybrid energy harvester. J. Comput. Nonlinear Dyn. 16, 041005 (2021)

    Google Scholar 

Download references

Funding

This work was partly supported by the NSF for Distinguished Young Scholars of China (Grant No. 52225211) and the NSF of China (Grant No. 12072264). The first author thanks the support of China Scholarship Council.

Author information

Authors and Affiliations

  1. Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo, 152-8552, Japan

    Qi Liu

  2. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China

    Qi Liu, Yong Xu & Yongge Li

  3. MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China

    Yong Xu

Authors
  1. Qi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongge Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

QL was involved in conceptualization, methodology, investigation, software, validation, formal analysis, writing—original draft, and writing–review and editing. YX was involved in conceptualization, supervision, project administration, funding acquisition, and writing—review and editing. YL was involved in supervision and writing—review and editing.

Corresponding author

Correspondence to Yong Xu.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Q., Xu, Y. & Li, Y. Complex dynamics of a conceptual airfoil structure with consideration of extreme flight conditions. Nonlinear Dyn 111, 14991–15010 (2023). https://doi.org/10.1007/s11071-023-08636-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08636-y

Keywords

Search

Navigation