Current unsteady aerodynamic modeling methods at high angle of attack usually use stable vibration test data at multiple frequencies to predict stable hysteresis loop. However, the rapid maneuvering process of aircraft, such as post-stall maneuver, cannot be a constant and stable vibration, but an unstable dynamic process. Therefore, the aerodynamics would not reach a stable hysteresis loop, but would always be in the initial unstable process of entering the hysteresis loop. The vibration theory analysis shows that the dynamic response process of unstable aerodynamics has the unstable and stable stages. The traditional modeling method focuses on the stable stage, while the actual maneuvering process of aircraft is in the unstable stage. Based on the Least Squares Support Vector Machine (LS-SVM), an excitation input suitable for nonlinear system identification is introduced to model unsteady aerodynamic forces of any motion in the amplitude and frequency ranges at high angle of attack. After completing the model training, the method is applied to predict the lift coefficient, drag coefficient, and pitching moment coefficient of a wing at high angle of attack with different reference states in pitching motion. The results show that not only the stable hysteresis loop is accurately predicted, but also the initial unstable process of entering the hysteresis loop is accurately predicted. In addition, the results also show that the reference state has significant influence on the characteristics of aerodynamics in the initial process. Further validation also shows that modeling based on the stable hysteresis loop data can only predict the stable hysteresis loop, and cannot predict the unstable process of entering the hysteresis loop.
[1] GHOREYSHI M, JIRASEK A, CUMMINGS R M. Computational investigation into the use of response functions for aerodynamic-load modeling[J]. AIAA Journal, 2012, 50(6):1314-1327.
[2] MCCRACKEN A J, KENNETT D J, BADCOCK K J, et al. Assessment of tabular models using CFD:AIAA-2013-4978[R]. Reston:AIAA, 2013.
[3] GHOREYSHI M, JIRASEK A, CUMMINGS R M. Reduced order unsteady aerodynamic modeling for stability and control analysis using computational fluid dynamics[J]. Progress in Aerospace Sciences, 2014, 71:167-217.
[4] LUCIA D J, BERAN P S, SILVA W A. Reduced-order modeling:New approaches for computational physics[J]. Progress in Aerospace Sciences, 2004, 40(1-2):51-117.
[5] 汪清, 钱炜祺, 丁娣. 飞机大迎角非定常气动力建模研究进展[J]. 航空学报, 2016, 37(8):2331-2347. WANG Q, QIAN W Q, DING D. A review of unsteady aerodynamic modeling of aircrafts at high angles of attack[J]. Acta Aeronautica et Astronautica Sinica, 2016, 37(8):2331-2347(in Chinese).
[6] SILVA W A. Application of nonlinear systems theory to transonic unsteady aerodynamic responses[J]. Journal of Aircraft, 1993, 30(5):660-668.
[7] 陈森林, 高正红, 饶丹. 基于多小波的Volterra级数非定常气动力建模方法[J]. 航空学报, 2018, 39(1):121379. CHEN S L, GAO Z H, RAO D. Unsteady aerodynamics modeling method using Volterra series based on multiwavelet[J]. Acta Aeronautica et Astronautica Sinica, 2018, 39(1):121379(in Chinese).
[8] GOMAN M, KHRABROV A. State-space representation of aerodynamic characteristics of an aircraft at high angles of attack[J]. Journal of Aircraft, 1994, 31(5):1109-1115.
[9] VINOGRADOV Y A, ZHUK A N, KOLINKO K A, et al. Mathematical simulation of dynamic effects of unsteady aerodynamics due to canard flow separation delay[J]. TsAGI Science Journal, 2011, 42(5):655-668.
[10] 龚正, 沈宏良. 非定常气动力非线性微分方程建模方法[J]. 航空学报, 2011, 32(1):83-90. GONG Z, SHEN H L. Unsteady aerodynamic modeling method using nonlinear differential equations[J]. Acta Aeronautica et Astronautica Sinica, 2011, 32(1):83-90(in Chinese).
[11] WANG Q, HE K F, QIAN W Q, et al. Unsteady aerodynamics modeling for flight dynamics application[J]. Acta Mechanica Sinica, 2012, 28(1):14-23.
[12] 汪清, 蔡金狮. 飞机大攻角非定常气动力建模与辨识[J]. 航空学报, 1996, 17(4):391-398. WANG Q, CAI J S. Unsteady aerodynamic modeling and identification of airplane at high angles of attack[J]. Acta Aeronautica et Astronautica Sinica, 1996, 17(4):391-398(in Chinese).
[13] GHOREYSHI M, JIRASEK A, CUMMINGS R M. Computational approximation of nonlinear unsteady aerodynamics using an aerodynamic model hierarchy[J]. Aerospace Science and Technology, 2013, 28(1):133-144.
[14] BRANDON J M, MORELLI E A. Nonlinear aerodynamic modeling from flight data using advanced piloted maneuvers and fuzzy logic:NASA-TM-2012-217778[R]. Washington, D.C.:NASA, 2012.
[15] WANG Q, QIAN W, HE K. Unsteady aerodynamic modeling at high angles of attack using support vector machines[J]. Chinese Journal of Aeronautics, 2015, 28(3):659-668.
[16] CHEN G, ZUO Y, SUN J, et al. Support-vector-machine-based reduced-order model for limit cycle oscillation prediction of nonlinear aeroelastic system[J]. Mathematical Problems in Engineering, 2012.
[17] VAPNIK V. The nature of statistical learning theory[M]. New York:Springer-Verlag, 1995:161-206.
[18] BURGES C J C. A tutorial on support vector machines for pattern recognition[J]. Data Mining and Knowledge Discovery, 1998, 2(2):121-167.
[19] SMOLA A J, SCHOLKOPF B. A tutorial on support vector regression[J]. Statistics and Computing, 2004, 14(3):199-222.
[20] SUYKENS J A K, VANDEWALLE J. Least squares support vector machine classifiers[J]. Neural Processing Letters, 1999, 9(3):293-300.
[21] MORELLI E, DERRY S, SMITH M. Aerodynamic parameter estimation for the X-43A (hyper-x) from flight data:AIAA-2005-5921[R]. Reston:AIAA, 2005.
[22] MORELLI E. Flight-test experiment design for characterizing stability and control of hypersonic vehicles[J]. Journal of Guidance, Control, and Dynamics, 2009, 32(3):949-959.
[23] CHENG C M, PENG Z K, ZHANG W M, et al. Wavelet basis expansion-based Volterra kernel function identification through multilevel excitations[J]. Nonlinear Dynamics, 2014, 76(2):985-999.
[24] TRONCHIN L. The emulation of nonlinear time-invariant audio systems with memory by means of Volterra series[J]. Journal of the Audio Engineering Society, 2012, 60(12):984-996.
[25] TISCHLER M B, REMPLE R K. 飞机和旋翼机系统辨识:工程方法和飞行试验案例[M]. 张怡哲, 左军毅, 译. 北京:航空工业出版社, 2012:75-76. TISCHLER M B, REMPLE R K. Aircraft and rotorcraft system identification:Engineering methods with flight-test examples[M]. ZHANG Y Z, ZUO J Y, translated. Beijing:Aviation Industry Press, 2012:75-76(in Chinese).
[26] PRAZENICA R J, KURDILA A J. Multiwavelet constructions and Volterra kernel identification[J]. Nonlinear Dynamics, 2006, 43(3):277-310.
[27] CHEN S, GAO Z. Unsteady aerodynamics modeling using Volterra series expansed by basis function[C]//2018 Asia Conference on Mechanical Engineering and Aerospace Engineering. Wuhan:EDP Sciences, 2018.
[28] SCHROEDER M. Synthesis of low-peak-factor signals and binary sequences with low autocorrelation[J]. IEEE Transactions on Information Theory, 1970, 16(1):85-89.
[29] IGNATYEV D I, KHRABROV A N. Neural network modeling of unsteady aerodynamic characteristics at high angles of attack[J]. Aerospace Science and Technology, 2015, 41:106-115.
[30] 史志伟, 王峥华, 李俊成. 径向基神经网络在非线性非定常气动力建模中的应用研究[J]. 空气动力学学报, 2012, 30(1):108-112. SHI Z W, WANG Z H, LI J C. The research of RBFNN in modeling of nonlinear unsteady aerodynamics[J]. Acta Aerodynamica Sinica, 2012, 30(1):108-112(in Chinese).
[31] CAWLEY G C, TALBOT N L C. Preventing over-fitting during model selection via bayesian regularisation of the hyper-parameters[J]. Journal of Machine Learning Research, 2007, 8:8841-861.
[32] NASA. CFL3D[CP]. https://github.com/nasa/cfl3d.
[33] TINOCO E N, BRODERSEN O, KEYE S, et al. Summary of data from the sixth AIAA CFD drag prediction workshop:CRM cases 2 to 5:AIAA-2017-1208[R]. Reston:AIAA, 2017.
[34] MANI M, RIDER B J, SCLAFANI A J, et al. Reynolds-averaged Navier-Stokes technology for transonic drag prediction:A Boeing perspective[J]. Journal of Aircraft, 2014, 51(4):1118-1134.
[35] PARK M A, LAFLIN K R, CHAFFIN M S, et al. CFL3D, FUN3D, and NSU3D contributions to the fifth drag prediction workshop[J]. Journal of Aircraft, 2014, 51(4):1268-1283.
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