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Chaotic patterns in aeroelastic signals. (English) Zbl 1190.94018

Summary: This work presents the analysis of nonlinear aeroelastic time series from wing vibrations due to airflow separation during wind tunnel experiments. Surrogate data method is used to justify the application of nonlinear time series analysis to the aeroelastic system, after rejecting the chance for nonstationarity. The singular value decomposition (SVD) approach is used to reconstruct the state space, reducing noise from the aeroelastic time series. Direct analysis of reconstructed trajectories in the state space and the determination of Poincaré sections have been employed to investigate complex dynamics and chaotic patterns. With the reconstructed state spaces, qualitative analyses may be done, and the attractors evolutions with parametric variation are presented. Overall results reveal complex system dynamics associated with highly separated flow effects together with nonlinear coupling between aeroelastic modes. Bifurcations to the nonlinear aeroelastic system are observed for two investigations, that is, considering oscillations-induced aeroelastic evolutions with varying freestream speed, and aeroelastic evolutions at constant freestream speed and varying oscillations. Finally, Lyapunov exponent calculation is proceeded in order to infer on chaotic behavior. Poincaré mappings also suggest bifurcations and chaos, reinforced by the attainment of maximum positive Lyapunov exponents.

MSC:

94A12 Signal theory (characterization, reconstruction, filtering, etc.)
37M10 Time series analysis of dynamical systems
37N35 Dynamical systems in control

Software:

TSTOOL
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Full Text: DOI EuDML

References:

[1] P. P. Friedmann, “The renaissance of aeroelasticity and its future,” in Proceedings of the International Forum on Aeroelasticity and Structural Dynamics (CEAS ’97), pp. 19-49, Rome, Italy, June 1997.
[2] I. E. Garrick, “Aeroelasticity-frontiers and beyond,” AIAA Journal of Aircraft, vol. 13, no. 9, pp. 641-657, 1976.
[3] L. E. Ericsson and J. P. Reding, “Fluid dynamics of unsteady separated flow. Part II. Lifting surfaces,” Progress in Aerospace Sciences, vol. 24, no. 4, pp. 249-356, 1987.
[4] B. H. K. Lee, S. J. Price, and Y. S. Wong, “Nonlinear aeroelastic analysis of airfoils: bifurcation and chaos,” Progress in Aerospace Sciences, vol. 35, no. 3, pp. 205-334, 1999. · doi:10.1016/S0376-0421(98)00015-3
[5] E. H. Dowell and D. Tang, “Nonlinear aeroelasticity and unsteady aerodynamics,” AIAA Journal of Aircraft, vol. 40, no. 9, pp. 1697-1707, 2002.
[6] H. Alighanbari and B. H. K. Lee, “Analysis of nonlinear aeroelastic signals,” AIAA Journal of Aircraft, vol. 40, no. 3, pp. 552-558, 2003.
[7] E. F. Sheta, V. J. Harrand, D. E. Thompson, and T. W. Strganac, “Computational and experimental investigation of limit cycle oscillations of nonlinear aeroelastic systems,” AIAA Journal of Aircraft, vol. 39, no. 1, pp. 133-141, 2002.
[8] J. W. Edwards, “Computational aeroelasticity,” in Structural Dynamics and Aeroelasticity, A. K. Noor and S. L. Venner, Eds., vol. 5 of Flight Vehicle Materials, Structures and Dynamics-Assessment and Future Directions, pp. 393-436, ASME, New York, NY, USA, 1993.
[9] J. G. Leishman and T. S. Beddoes, “A semi-empirical model for dynamic stall,” Jornal of the American Helicopter Society, vol. 34, no. 3, pp. 3-17, 1989.
[10] F. D. Marques, Multi-layer functional approximation of non-linear unsteady aerodynamic response, Ph.D. thesis, University of Glasgow, Glasgow, UK, 1997.
[11] J. S. Bendat and A. G. Piersol, Random Data: Analysis & Measurement Procedures, John Wiley & Sons, New York, NY, USA, 2nd edition, 1986. · Zbl 0662.62002
[12] T. Schreiber and A. Schmitz, “Surrogate time series,” Physica D, vol. 142, no. 3-4, pp. 346-382, 2000. · Zbl 1098.62551 · doi:10.1016/S0167-2789(00)00043-9
[13] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, UK, 2nd edition, 2004. · Zbl 1050.62093
[14] D. S. Broomhead and G. P. King, “Extracting qualitative dynamics from experimental data,” Physica D, vol. 20, no. 2-3, pp. 217-236, 1986. · Zbl 0603.58040 · doi:10.1016/0167-2789(86)90031-X
[15] A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics, John Wiley & Sons, New York, NY, USA, 1995. · Zbl 0848.34001
[16] R. M. G. Vasconcellos, Reconstru de espa \ccos de estados aeroelásticos por decomposi em valores singulares, M.S. thesis, Universidade de São Paulo-EESC-USP, São Paulo, Brazil, 2007.
[17] R. C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, The Clarendon Press, Oxford University Press, New York, NY, USA, 2nd edition, 2000. · Zbl 1015.37001
[18] F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898, pp. 366-381, Springer, Berlin, Germany, 1981. · Zbl 0513.58032 · doi:10.1007/BFb0091924
[19] N. J. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Physical Review Letters, vol. 45, no. 9, pp. 712-716, 1980. · doi:10.1103/PhysRevLett.45.712
[20] F. D. Marques, E. M. Belo, V. A. Oliveira, J. R. Rosolen, and A. R. Simoni, “On the investigation of state space reconstruction of nonlinear aeroelastic response time series,” Shock and Vibration, vol. 13, no. 4-5, pp. 393-407, 2006.
[21] D. Kugiumtzis and N. Christophersen, “State space reconstruction: method of delays vs singular spectrum approach,” Research Report 236, Department of Informatics, University of Oslo, Oslo, Norway, 1997.
[22] M. Casdagli, S. Eubank, D. Farmer, and J. Gibson, “State space reconstruction in the presence of noise,” Physica D, vol. 51, no. 1-3, pp. 52-98, 1991. · Zbl 0736.62075 · doi:10.1016/0167-2789(91)90222-U
[23] M. A. Athanasiu and G. P. Pavlos, “SVD analysis of the magnetospheric AE index time series and comparison with low-dimensional chaotic dynamics,” Nonlinear Processes in Geophysics, vol. 8, no. 1-2, pp. 95-125, 2001.
[24] U. ParlitzJ. A. K. Suykens and J. Vandewalle, “Nonlinear time-series analysis,” in Nonlinear Modeling-Advanced Black-Box Techniques, pp. 209-239, Kluwer Academic Publishers, Boston, Mass, USA, 1998.
[25] J. Theiler, B. Galdrikian, A. Longtin, S. Eubank, and J. D. Farmer, “Using surrogate data to detect nonlinearity in time series,” in Nonlinear Modeling and Forecasting, vol. 12 of SFI Studies in the Sciences of Complexity, pp. 163-188, Addison-Wesley, Reading, Mass, USA, 1992. · Zbl 1194.37144
[26] C. Merkwirth, U. Parlitz, and W. Lauterborn, “TSTOOL-a software package for nonlinear time series analysis,” in Proceedings of the International Workshop on Advanced Black-Box Techniques for Nonlinear Modeling, J. A. Suykens and J. Vandewalle, Eds., Katholieke Universiteit Leuven, Leuven, Belgium, July 1998.
[27] S. Sato, M. Sano, and Y. Sawada, “Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high-dimensional chaotic systems,” Progress of Theoretical Physics, vol. 77, no. 1, pp. 1-5, 1987. · doi:10.1143/PTP.77.1
[28] P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D, vol. 9, no. 1-2, pp. 189-208, 1983. · Zbl 0593.58024 · doi:10.1016/0167-2789(83)90298-1
[29] T. Yal\ccinkaya and Y.-C. Lai, “Phase characterization of chaos,” Physical Review Letters, vol. 79, no. 20, pp. 3885-3888, 1997.
[30] F. D. Marques, E. M. Belo, V. A. Oliveira, J. R. Rosolen, and A. R. Simoni, “Non-linear phenomena analysis of stall-induced aeroelastic oscillations,” in Proceedings of the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, vol. 6, pp. 4507-4513, Palm Springs, Calif, USA, April 2004.
[31] A. R. Simoni, Análise de séries temporais esperimentais não lineares, Ph.D. thesis, Escola de Engenharia de São Carlos, Universidade de São Paulo, São Paulo, Brazil, 2007.
[32] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, vol. 16, no. 3, pp. 285-317, 1985. · Zbl 0585.58037 · doi:10.1016/0167-2789(85)90011-9
[33] F. D. Marques, R. M. G. Vasconcellos, and A. R. Simoni, “Analysis of an experimental aeroelastic system through nonlinear time series,” in Proceedings of the International Symposium on Dynamic Problems of Mechanics (DINAME ’09), Angra dos Reis, Brazil, March 2009.
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