Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-27T22:48:21.382Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic nonlinear aeroelastic analysis of membrane aerofoils with uncertain properties and aerodynamic loadings

Published online by Cambridge University Press:  26 August 2025

O. S. Hussein Open the ORCID record for O. S. Hussein [Opens in a new window]
O. S. Hussein*
Affiliation:
Aerospace Engineering Department, Cairo University, Giza, Egypt
*
Email: aero.omarshukry@cu.edu.eg
Get access Rights & Permissions [Opens in a new window]

Abstract

Membrane aerofoils are used for the design of small unmanned air vehicles which have gained interest in the past few years. This paper deals with the nonlinear uncertain aeroelastic analysis of an elastically supported membrane aerofoil. The uncertainties in the aerofoil aerodynamic coefficients are estimated due to five uncertain input parameters, which are the initial tension coefficient, the membrane elastic modulus, the stiffness coefficients of the two supporting springs at the trailing edge and the leading edge, and the fifth parameter is the free stream angle-of-attack. Both static uncertain aeroelasticity and dynamic aeroelasticity for a sinusoidal gust loading are considered. A detailed novel parametric analysis is performed to assess the effect of each parameter. The analysis is carried out using a nonlinear aeroelastic finite element method, which is based on the Theodorsen’s unsteady aerodynamics theory. The polynomial chaos expansion method is used for the uncertainty quantification process and for the sensitivity analysis. Also, the Karhunen-Loéve expansion is used to model the random field of the elastic modulus. The interesting results of the analysis show that the effect of each uncertain input depends on the values of the other parameters and that the initial tension is the key parameter. The type of the probability density functions (or histograms) of the aerodynamic coefficients can vary from a Gaussian distribution to an exponential-like distribution.

Keywords

Membrane airfoilaeroelasticityuncertainty quantificationstatic loadinggust loading

Information

Type
Research Article
Information
The Aeronautical Journal , First View , pp. 1 - 27
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Royal Aeronautical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Beran, P., Stanford, B. and Schrock, C. Uncertainty Quantification in Aeroelasticity, Annu. Rev. Fluid Mech, 2016, 49, pp 361–386. https://doi.org/10.1146/annurev-fluid-122414-034441 CrossRefGoogle Scholar
Pettit, C.L. Uncertainty quantification in aeroelasticity: Recent results and research challenges, J Aircr, 2004, 41, pp 1217–1229. https://doi.org/10.2514/1.3961 CrossRefGoogle Scholar
Bijl, H., Lucor, D., Mishra, S. and Schwab, C., eds., Uncertainty Quantification in Computational Fluid Dynamics, Springer International Publishing, Cham, 2013. https://doi.org/10.1007/978-3-319-00885-1 CrossRefGoogle Scholar
Tiomkin, S. and Raveh, D.E. A review of membrane-wing aeroelasticity, Prog Aerosp Sci, 2021, 126, p 100738. https://doi.org/10.1016/j.paerosci.2021.100738 CrossRefGoogle Scholar
Lian, Y., Shyy, W., Viieru, D. and Zhang, B. Membrane wing aerodynamics for micro air vehicles, Prog Aerosp Sci, 2003, 39, pp 425–465. https://doi.org/10.1016/S0376-0421(03)00076-9 CrossRefGoogle Scholar
Jackson, P.S. A simple model for elastic two-dimensional sails, AIAA J, 1983, 21, pp 153–155. https://doi.org/10.2514/3.60106 CrossRefGoogle Scholar
Newman, B.G. and Low, H.T. Two-dimensional impervious sails: experimental results compared with theory, J Fluid Mech, 1984, 144, pp 445–462. https://doi.org/10.1017/S0022112084001683 CrossRefGoogle Scholar
Thwaites, B. The aerodynamic theory of sails. I. Two-dimensional sails, Proc Royal Soc London A Math Phys Sci, 1961, 261, pp 402–422.Google Scholar
Newman, B.G. The aerodynamics of flexible membranes, Proc Indian Acad Sci (Eng Sci), 1982, 5, pp 107–130. https://doi.org/10.1007/BF02845068 CrossRefGoogle Scholar
Newman, B.G. and Païdoussis, M.P. The stability of two-dimensional membranes in streaming flow, J Fluids Struct, 1991, 5, pp 443–454. https://doi.org/10.1016/0889-9746(91)90437-T CrossRefGoogle Scholar
Newman, B.G. Aerodynamic theory for membranes and sails, Prog Aerosp Sci, 1987, 24, pp 1–27. https://doi.org/10.1016/0376-0421(87)90005-4 CrossRefGoogle Scholar
Nielsen, J.N. Theory of flexible aerodynamic surfaces, J Appl Mech, 1963, 30, pp 435–442. https://doi.org/10.1115/1.3636575 CrossRefGoogle Scholar
Greenhalgh, S., Curtiss, H.C. and Smith, B. Aerodynamic properties of a two-dimensional inextensible flexible airfoil, AIAA J, 1984, 22, pp 865–870. https://doi.org/10.2514/3.8701 CrossRefGoogle Scholar
Scott, R., Bartels, R. and Kandil, O. An aeroelastic analysis of a thin flexible membrane, In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, 2007. https://doi.org/10.2514/6.2007-2316 CrossRefGoogle Scholar
Sygulski, R. Stability of membrane in low subsonic flow, Int J Non Linear Mech, 2007, 42, p 196. https://doi.org/10.1016/j.ijnonlinmec.2006.11.012 CrossRefGoogle Scholar
Hu, H., Tamai, M. and Murphy, J.T. Flexible-membrane airfoils at low Reynolds Numbers, J Aircr, 2008, 45, pp 1767–1778. https://doi.org/10.2514/1.36438 CrossRefGoogle Scholar
Tiomkin, S. and Raveh, D.E. On the stability of two-dimensional membrane wings, J Fluids Struct, 2017, 71, pp 143–163. https://doi.org/10.1016/j.jfluidstructs.2017.03.003 CrossRefGoogle Scholar
Tiomkin, S. and Raveh, D. On membrane-wing stability in laminar flow, J Fluids Struct 2019, 91, p 102694. https://doi.org/10.1016/j.jfluidstructs.2019.102694 CrossRefGoogle Scholar
Alioli, M., Masarati, P., Morandini, M., Albertani, R. and Carpenter, T. Modeling effects of membrane tension on dynamic stall for thin membrane wings, Aerosp Sci Technol, 2017, 69, pp 419–431. https://doi.org/10.1016/j.ast.2017.07.008 CrossRefGoogle Scholar
Alon Tzezana, G. and Breuer, K. Thrust, drag and wake structure in flapping compliant membrane wings, J Fluid Mech, 2019, 862, pp 871–888. https://doi.org/10.1017/jfm.2018.966 CrossRefGoogle Scholar
Mavroyiakoumou, C. and Alben, S. Large-amplitude membrane flutter in inviscid flow, J Fluid Mech, 2020, 891, p A23. https://doi.org/10.1017/jfm.2020.153 CrossRefGoogle Scholar
Mavroyiakoumou, C. and Alben, S. Eigenmode analysis of membrane stability in inviscid flow, Phys Rev Fluids, 2021, 6, p 043901. https://doi.org/10.1103/PhysRevFluids.6.043901 CrossRefGoogle Scholar
Mavroyiakoumou, C. and Alben, S. Dynamics of tethered membranes in inviscid flow, J Fluids Struct, 2021, 107, p 103384. https://doi.org/10.1016/j.jfluidstructs.2021.103384 CrossRefGoogle Scholar
Mavroyiakoumou, C. and Alben, S. Membrane flutter in three-dimensional inviscid flow, J Fluid Mech, 2022, 953, p A32. https://doi.org/10.1017/jfm.2022.957 CrossRefGoogle Scholar
Tiomkin, S. and Jaworski, J.W. Unsteady aerodynamic theory for membrane wings, J Fluid Mech, 2022, 948, p A33. https://doi.org/10.1017/jfm.2022.682 CrossRefGoogle Scholar
Hussein, O.S. Aeroelastic analysis of membrane airfoils and flexible-chord airfoils with trailing-edge flaps, Acta Mech, 2023, 234, pp 4487–4508. https://doi.org/10.1007/s00707-023-03618-y CrossRefGoogle Scholar
Hussein, O.S. Nonlinear FE aeroelastic analysis of membrane airfoils with fixed and elastic supports, Thin Walled Struct, 2023, 184, p 110544. https://doi.org/10.1016/j.tws.2023.110544 CrossRefGoogle Scholar
Hussein, O.S. Aeroelasticity of membrane airfoils and flexible-chord airfoils with permeable trailing sections, Appl Math Model, 2025, 141, p 115947. https://doi.org/10.1016/j.apm.2025.115947 CrossRefGoogle Scholar
Gehrke, A., Richeux, J., Uksul, E. and Mulleners, K. Aeroelastic characterisation of a bio-inspired flapping membrane wing, Bioinspir Biomim, 2022, 17, p 065004. https://doi.org/10.1088/1748-3190/ac8632 CrossRefGoogle ScholarPubMed
Tapaswini, S. and Chakraverty, S. Non-probabilistic solution of uncertain vibration equation of large membranes using adomian decomposition method, Sci World J, 2014, 2014, pp 1–11. https://doi.org/10.1155/2014/308205 CrossRefGoogle ScholarPubMed
Tapaswini, S., Chakraverty, S. and Behera, D. Uncertain vibration equation of large membranes, Eur Phys J Plus, 2014, 129, p 251. https://doi.org/10.1140/epjp/i2014-14251-7 CrossRefGoogle Scholar
Li, D., Zheng, Z., Tian, Y., Sun, J., He, X. and Lu, Y. Stochastic nonlinear vibration and reliability of orthotropic membrane structure under impact load, Thin Walled Struct, 2017, 119, pp 247–255. https://doi.org/10.1016/j.tws.2017.06.008 CrossRefGoogle Scholar
Li, D., Zheng, Z.-L., Yang, R. and Zhang, P. Analytical solutions for stochastic vibration of orthotropic membrane under random impact load, Materials, 2018, 11, p 1231. https://doi.org/10.3390/ma11071231 CrossRefGoogle ScholarPubMed
Liu, X., Zhao, X., Adhikari, S. and Liu, X. Stochastic dynamic stiffness for damped taut membranes, Comput Struct, 2021, 248, p 106483. https://doi.org/10.1016/j.compstruc.2021.106483 CrossRefGoogle Scholar
Li, D., Lai, Z., Liu, C., Guo, J., Yang, X. and Guan, M. Random vibration of pretensioned rectangular membrane structures under heavy rainfall excitation, Thin Walled Struct, 2021, 164, p 107856. https://doi.org/10.1016/j.tws.2021.107856 CrossRefGoogle Scholar
Liu, C., Pan, R., Deng, X., Xie, H., Liu, J. and Wang, X. Random vibration and structural reliability of composite hyperbolic–parabolic membrane structures under wind load, Thin Walled Struct, 2022, 180, p 109878. https://doi.org/10.1016/j.tws.2022.109878 CrossRefGoogle Scholar
Iwasa, T., Fujibayashi, S., Katsumata, N. and Higuchi, K. Upper-bound vibration response spectrum computation for deployable membrane structures with uncertainty effects, Thin Walled Struct, 2024, 198, p 111706. https://doi.org/10.1016/j.tws.2024.111706 CrossRefGoogle Scholar
Kasimala, N.R. and Chakraverty, S. Forward and inverse problems of time-fractional vibration equation of large membranes in uncertain environment, J Vib Eng Technol, 2024, 12, pp 497–514. https://doi.org/10.1007/s42417-024-01429-6 CrossRefGoogle Scholar
Kasimala, N.R. and Chakraverty, S. Vibration equation of large membranes in uncertain environment, J Vib Eng Technol, 2024, 12, pp 207–220. https://doi.org/10.1007/s42417-024-01411-2 CrossRefGoogle Scholar
Wang, X. and Mao, X. Non-overshooting sliding mode for UAV control, Aeronaut J, 2024, 128, pp 2483–2529. https://doi.org/10.1017/aer.2024.50 CrossRefGoogle Scholar
Rezaei, M., Fazelzadeh, S.A., Mazidi, A., Friswell, M.I. and Khodaparast, H.H. Fuzzy uncertainty analysis and reliability assessment of aeroelastic aircraft wings, Aeronaut J, 2020, 124, pp 786–811. https://doi.org/10.1017/aer.2020.2 CrossRefGoogle Scholar
Borello, F., Cestino, E. and Frulla, G. Structural uncertainty effect on classical wing flutter characteristics, J Aerosp Eng, 2010, 23, pp 327–338. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000049 CrossRefGoogle Scholar
Badcock, K.J., Timme, S., Marques, S., Khodaparast, H., Prandina, M., Mottershead, J.E., Swift, A., Da Ronch, A. and Woodgate, M.A. Transonic aeroelastic simulation for instability searches and uncertainty analysis, Prog Aerosp Sci, 2011, 47, pp 392–423. https://doi.org/10.1016/j.paerosci.2011.05.002 CrossRefGoogle Scholar
Tartaruga, I., Cooper, J.E., Georgiou, G. and Khodaparast, H. Flutter uncertainty quantification for the S4T model, In 55th AIAA Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics, Grapevine, Texas, 2017. https://doi.org/10.2514/6.2017-1653 CrossRefGoogle Scholar
Hussein, O.S. Deterministic and peridynamic operator-based uncertain nonlinear aeroelastic analyses of in-plane functionally graded panels with elastic boundary supports and yawed supersonic flow, J Braz Soc Mech Sci Eng, 2024, 46, p 720. https://doi.org/10.1007/s40430-024-05289-9 CrossRefGoogle Scholar
Hussien, O. and Mulani, S.B. Two-dimensional optimization of functionally graded material plates subjected to buckling constraints, In 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, Grapevine, Texas, 2017. https://doi.org/10.2514/6.2017-1546 CrossRefGoogle Scholar
Ji, T.Y., Chu, W.L. and Guo, Z.T., Uncertainty quantification and sensitivity analysis of blade geometric deviation on compressor performance, Aeronaut J, 2024, 128, pp 2945–2963. https://doi.org/10.1017/aer.2024.65 CrossRefGoogle Scholar
Khodaparast, H.H., Mottershead, J.E. and Badcock, K.J. Propagation of structural uncertainty to linear aeroelastic stability, Comput Struct, 2010, 88, pp 223–236. https://doi.org/10.1016/j.compstruc.2009.10.005 CrossRefGoogle Scholar
Hussein, O.S. Probabilistic and fuzzy nonlinear discontinuous aeroelastic analysis of in-plane FG panels in supersonic flow with mechanical and thermal in-plane loadings, Arab J Sci Eng, 2024, 49, pp 2327–2344. https://doi.org/10.1007/s13369-023-08209-6 CrossRefGoogle Scholar
Hussein, O.S. Optimization and uncertain nonlinear vibration of pre/post-buckled in-plane functionally graded metal nanocomposite plates, J Vib Eng Technol, 2023, 12, pp 2091–2110. https://doi.org/10.1007/s42417-023-00969-7 CrossRefGoogle Scholar
Hao, Y., Ma, C. and Hu, Y. Nonlinear stochastic flutter analysis of a three-degree-of-freedom wing in a two-dimensional flow field under stochastic perturbations, Aerosp Sci Technol, 2023, 138, p 108323. https://doi.org/10.1016/j.ast.2023.108323 CrossRefGoogle Scholar
Hao, Y., Guo, X.Y. and Fu, Y.B. Stochastic flutter analysis of a three-degree-of-freedom airfoil under vertical turbulence disturbance, Int J Non Linear Mech, 2024, 158, p 104584. https://doi.org/10.1016/j.ijnonlinmec.2023.104584 CrossRefGoogle Scholar
Ghommem, M., Hajj, M.R. and Nayfeh, A.H. Uncertainty analysis near bifurcation of an aeroelastic system, J Sound Vib, 2010, 329, pp 3335–3347. https://doi.org/10.1016/j.jsv.2010.02.028 CrossRefGoogle Scholar
Yun, H. and Han, J. Robust flutter analysis of a nonlinear aeroelastic system with parametric uncertainties, Aerosp Sci Technol, 2009, 13, pp 139–149. https://doi.org/10.1016/j.ast.2008.08.001 CrossRefGoogle Scholar
Sarkar, S., Witteveen, J.A.S., Loeven, A. and Bijl, H. Effect of uncertainty on the bifurcation behavior of pitching airfoil stall flutter, J Fluids Struct, 2009, 25, pp 304–320. https://doi.org/10.1016/j.jfluidstructs.2008.06.006 CrossRefGoogle Scholar
Bisplinghoff, R.L., Ashley, H. and Halfman, R.L. Aeroelasticity, Dover Publications Inc., New York, 1996.Google Scholar
Crestaux, T., Le Matre, O. and Martinez, J.-M. Polynomial chaos expansion for sensitivity analysis, Reliab Eng Syst Saf, 2009, 94, pp 1161–1172. https://doi.org/10.1016/j.ress.2008.10.008 CrossRefGoogle Scholar
Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M. and Tarantola, S. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index, Comput Phys Commun, 2010, 181, pp 259–270. https://doi.org/10.1016/j.cpc.2009.09.018 CrossRefGoogle Scholar
Alexanderian, A. A brief note on the Karhunen-Loe’ve expansion, 2015. https://doi.org/10.48550/arXiv.1509.07526 CrossRefGoogle Scholar
Uncertainty Quantification Test Problems (n.d.). https://www.sfu.ca/~ssurjano/uq.html Google Scholar