Quick and Effective Modal and Flutter Analyses for Low Aspect Ratio Wings

Document Type : Original Article


1 Department of Mechanical Engineering, Malek-Ashtar University of Technology, Shahinshahr, Iran

2 Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran


In the present work, an analytical study is proposed to investigate the flutter behavior of low-aspect-ratio wings in subsonic flow. An equivalent plate model is used for structural modelling of a semi-monocoque main wing, consisting of ribs, skins, and spars. Legendre polynomials are used in the Rayleigh-Ritz method as trial functions, and the first-order shear deformation theory is utilized to formulate the structural deformation. Boundary conditions are enforced by applying proper artificial springs. A doublet point method is used to calculate the unsteady aerodynamic loads. Chordwise pressure coefficient distribution at the tip and root of a rectangular wing oscillating in pitching motion is calculated. Flutter analysis is performed using the k method. Instead of using the computationally expensive finite element method, the proposed approach is intended to achieve purposes of quick modelling and effective analysis in free vibration and flutter analyses of low-aspect-ratio wings for preliminary design applications. The effects of aspect ratio on the flutter behavior of wings in subsonic flow are investigated. The obtained results are validated with the results available in the literature.


[1]     Lin, K. J., Lu, P. J., and Tarn, J. Q., Flutter Analysis of Cantilever Composite Plates in Subsonic Flow, AIAA Journal, Vol. 27, No. 8, 1989, pp. 1102-1109, DOI: https://doi.org/10.2514/3.10228.
[2]     Giles, G., Equivalent Plate Modelling for Conceptual Design of Aircraft Wing Structures, Aircraft Engineering, Technology, and Operations Congress, AIAA, Los Angeles, 1995, DOI: https://doi.org/10.2514/6.1995-3945.
[3]     Haidar, M., Kamel, M., El Shabka, A., and Negm, H., Aeroelastic Investigation of Composite Plate Wing in Subsonic Flow, International Conference on Aerospace Sciences and Aviation Technology, The Military Technical College, Cairo, Vol. 15, 2013, pp. 1-14, DOI: 10.21608/asat.2013.22087.
[4]     Giles, G. L., Equivalent Plate Analysis of Aircraft Wing Box Sructures with General Planform Geometry, Journal of Aircraft, Vol. 23, No. 11, 1986, pp. 859-864, DOI: https://doi.org/10.2514/3.45393.
[5]     Giles, G. L., Further Generalization of an Equivalent Plate Representation for Aircraft Structural Analysis, Journal of Aircraft, Vol. 26, No. 1, 1989, pp. 67-74, DOI: https://doi.org/10.2514/3.45724.
[6]     Tizzi, S., Numerical Procedure for the Dynamic Analysis of Three-Dimensional Aeronautical Structures, Journal of Aircraft, Vol. 34, No. 1, 1997, pp. 120-130, DOI: https://doi.org/10.2514/2.2145.
[7]     Tizzi, S., Improvement of a Numerical Procedure for the Dynamic Analysis of Aircraft Structures, Journal of Aircraft, Vol. 37, No. 1, 2000, pp. 144-154, DOI: https://doi.org/10.2514/2.2574.
[8]     Livne, E., Equivalent Plate Structural Modelling for Wing Shape Optimization Including Transverse Shear, AIAA Journal, Vol, 32, No. 6, 1994, pp. 1278-1288, DOI: https://doi.org/10.2514/3.12130.
[9]     Livne, E., Navarro, I., Nonlinear Equivalent Plate Modelling of Wing-Box Structures, Journal of Aircraft, Vol. 36, No. 5, 1999, pp. 851-865, DOI: https://doi.org/10.2514/2.2519.
[10]  Demasi, L., Livne, E., Structural Ritz-Based Simple-Polynomial Nonlinear Equivalent Plate Approach: an Assessment, Journal of Aircraft, Vol. 43, No. 6, 2006, pp. 1685-1697, DOI: https://doi.org/10.2514/1.17466.
[11]  Kapania, R. K., Lovejoy, A. E., Free Vibration of Thick Generally Laminated Cantilever Quadrilateral Plates, AIAA Journal, Vol. 34, No. 7, 1996, pp. 1474-1486, DOI: https://doi.org/10.2514/3.13256.
[12]  Kapania, R. K., Liu, Y., Static and Vibration Analyses of General Wing Structures Using Equivalent-Plate Models, AIAA Journal, Vol. 38, No. 7, 2000, pp. 1269-1277, DOI: https://doi.org/10.2514/2.1098.
[13]  Krishnamurthy, T., Frequencies and Flutter Speed Estimation for Damaged Aircraft Wing Using Scaled Equivalent Plate Analysis, 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 18th AIAA/ASME/AHS Adaptive Structures Conference 12th, AIAA, Orlando, 2010, DOI: https://doi.org/10.2514/6.2010-2769.
[14]  Krishnamurthy, T., Tsai, F., Static and Aynamic Structural Response of an Aircraft Wing with Damage Using Equivalent Plate Analysis, 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 16th AIAA/ASME/AHS Adaptive Structures Conference, 10th AIAA Non-Deterministic Approaches Conference, 9th AIAA Gossamer Spacecraft Forum, 4th AIAA Multidisciplinary Design Optimization Specialists Conference, AIAA, Schaumburg, 2008, pp. 1- 18, DOI: 10.2514/6.2008-1967.
[15]  Na, Y. H., Shin, S., Equivalent-Plate Analysis for a Composite Wing with a Control Surface, Journal of Aircraft, Vol. 50, No. 3, 2013, pp. 853-862, DOI: https://doi.org/10.2514/1.C032020.
[16]  Jian, T., Changchuan, X., and Chao, Y., Flutter Analysis of Aircraft Wing Using Equivalent-Plate Models with Orthogonal Polynomials, Transactions of Nanjing University of Aeronautics and Astronautics, Vol. 32, No. 5, 2015, pp. 508-516, DOI: 10.16356/j.1005-1120.2015.05.508.
[17]  Fung, Y. C., An Introduction to the Theory of Aeroelasticity, Courier Dover Publications, New York, USA, 2008, pp. 186-187, ISBN: 0486469360. Kussner, H., General Airfoil Theory, NACA TM-979, 1941.
[18]  Watkins, C. E., Woolston, D. S., and Cunningham, H. J., A Systematic Kernel Function Procedure for Determining Aerodynamic Forces on Oscillating or Steady Finite Wings at Subsonic Speeds, NASA TR-R-48, 1959.
[19]  Redman, M., Rowe, W., Prediction of Unsteady Aerodynamic Loadings Caused by Leading Edge and Trailing Edge Control Surface Motions in Subsonic Compressible Flow, NASA CR-2543, 1975.
[20]  Albano, E., Rodden, W. P., A Doublet-Lattice Method for Calculating Lift Distributions on Oscillating Surfaces in Subsonic Flows, AIAA Journal, Vol. 7, No. 2, 1969, pp. 279-285, DOI: https://doi.org/10.2514/3.5086.
[21]  Blair, M., A Compilation of the Mathematics Leading to the Doublet Lattice Method, WL-TR-92-3028, 1992, DOI: 10.21236/ada256304.
[22]  Rodden, W. P., Taylor, P. F., and McIntosh Jr, S. C., Further Refinement of the Subsonic Doublet-Lattice Method, Journal of Aircraft, Vol. 35, No. 5, 1998, pp. 720-727, DOI: https://arc.aiaa.org/doi/10.2514/2.2382.
[23]  Ueda, T., Dowell, E., A New Solution Method for Lifting Surfaces in Subsonic Flow, AIAA Journal, Vol. 20, No. 3, 1982, pp. 348-355, DOI: https://arc.aiaa.org/doi/10.2514/3.7916.
[24]  Hollowell, S. J., Dugundji, J., Aeroelastic Flutter and Divergence of Stiffness Coupled, Graphite/Epoxy Cantilevered Plates, Journal of Aircraft, Vol. 21, No. 1, 1984, pp. 69-76, DOI: https://arc.aiaa.org/doi/10.2514/3.48224.
[25]  Vepa, R., Aeroelastic Analysis of Wing Structures Using Equivalent Plate Models, AIAA Journal, Vol. 46, No. 5, 2008, pp. 1216-1225, DOI: https://arc.aiaa.org/doi/10.2514/1.34928.
[26]  Saeed, S., Salman, S., Flutter Analysis of Hybrid Mtal-Composite Low Aspect Ratio Trapezoidal Wings in Supersonic Flow, Chinese Journal of Aeronautics, Vol. 30, No. 1, 2017, pp. 196-203, DOI: https://doi.org/10.1016/j.cja.2016.12.016.
[27]  Babin, T., Sangeetha, N., Flutter Analysis of Supersonic Low Aspect Ratio Composite Wings Using FSI Methodology, Advances in Manufacturing Processes, Springer, Singapore, 2019, pp. 361-372, DOI: https://doi.org/10.1007/978-981-13-1724-8_35.
[28]  Crawley, E. F., Curtiss, H. C., Peters, D. A., Scanlan, R. H., and Sisto, F., A Modern Course in Aeroelasticity, 4th ed, Springer, New York, USA, 1995, pp. 246, ISBN: 9401104999.
[29]  Abbas, M. K., Negm, H. M., and Elshafei, M. A., Flutter and Divergence Characteristics of Composite Plate Wing, International Conference on Aerospace Sciences and Aviation Technology, The Military Technical College, Cairo, Vol. 15,2013, pp. 1-21, DOI: 10.21608/ASAT.2013.22181.
[30]  Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., Aeroelasticity, 2nd ed, Courier Corporation, New York, USA, 2013, pp. 545-551, ISBN: 0486132439.
[31]  Na, Y. H., Kim, J. H., and Shin, S. J., Vibration Analyses of an Equivalent Plate Wing with an External Store, The Aeronautical Journal, Vol. 118, No. 1207, 2014, pp. 1090-1098, DOI: https://doi.org/10.1017/S0001924000009763.
[32]  Irani, S., Dehghani, R., Golparvar, H., and Hosseinian, A., Wing Flutter Analysis of Model with Experimental and Theory Method, Aerospace Mechanics Journal, Vol. 8, No. 3, 2012, pp. 69-76.