Research Papers

Time Domain Approaches to the Stability Analysis of Flexible Dynamical Systems

[+] Author and Article Information
Jielong Wang

Beijing Aeronautical Science and
Technology Research Institute,
Commercial Aircraft Corporation of China, Ltd.,
Beijing 102211, China
e-mail: wangjielong@comac.cc

Xiaowen Shan, Bin Wu

Beijing Aeronautical Science and
Technology Research Institute,
Commercial Aircraft Corporation of China, Ltd.,
Beijing 102211, China

Olivier A. Bauchau

Department of Aerospace Engineering,
University of Maryland,
College Park, MD 20742

1Corresponding author.

Manuscript received March 23, 2015; final manuscript received September 11, 2015; published online November 13, 2015. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 11(4), 041003 (Nov 13, 2015) (9 pages) Paper No: CND-15-1075; doi: 10.1115/1.4031675 History: Received March 23, 2015; Revised September 11, 2015

This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dynamical systems and the predictions of the two approaches are found to be in close agreement.

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Hochstadt, H. , 1964, Differential Equations, Dover Publications, New York.
Nayfeh, A. , and Mook, D. , 1979, Nonlinear Oscillations, Wiley, New York.
Wang, X. , 1998, “ The Method of Generalized Floquet Theory Applied to Flap-Lag Dynamics With Inflow,” M.S. thesis, Washington University, Saint Louis, MO.
Bauchau, O. , and Wang, J. , 2007, “ Stability Evaluation and System Identification of Flexible Multibody Systems,” Multibody Syst. Dyn., 18(1), pp. 95–106. [CrossRef]
Bauchau, O. , and Wang, J. , 2010, “ Efficient and Robust Approaches for Rotorcraft Stability Analysis,” J. Am. Helicopter Soc., 55(1), p. 012005. [CrossRef]
Udwadia, F. , and von Bremen, H. , 2002, “ Comparison of Lyapunov Characteristic Exponents for Continuous Dynamical Systems,” Z. Angew. Math. Phys., 53(1), pp. 123–146. [CrossRef]
Geist, K. , Parlitz, U. , and Lauterborn, W. , 1990, “ Comparison of Different Methods for Computing Lyapunov Exponents,” Prog. Theor. Phys., 83(5), pp. 875–893. [CrossRef]
Bauchau, O. , and Wang, J. , 2008, “ Efficient and Robust Approaches to the Stability Analysis of Large Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3(1), p. 011001. [CrossRef]
Bauchau, O. , 2011, Flexible Multibody Dynamics, Springer, New York.
Wang, J. , and Li, Z. , 2015, “ Implementation of HHT Algorithm for Numerical Integration of Multibody Dynamics With Holonomic Constraints,” J. Nonlinear Dyn., 80(1–2), pp. 817–825. [CrossRef]
Golub, G. , and van Loan, C. , 1989, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD.
Arnoldi, W. , 1951, “ The Principle of Minimized Iterations in the Solution of the Matrix Eigenvalue Problem,” Q. Appl. Math., 9, pp. 17–29.
Wolf, A. , Swift, J. B. , Swinney, H. L. , and Vastano, J. A. , 1985, “ Determining Lyapunov Exponents From a Time Series,” Phys. D, 16(3), pp. 285–317. [CrossRef]
Goland, M. , 1945, “ The Flutter of a Uniform Cantilever Wing,” ASME J. Appl. Mech., 12(4), pp. A197–A208.
Wang, J. , 2015, “ Implementation of Geometrically Exact Beam Element for Nonlinear Dynamics Modeling,” Multibody Syst. Dyn., 1(1), pp. 1–16.
Yates, E. , 1958, “ Calculation of Flutter Characteristics for Finite-Span Swept or Unswept Wings at Subsonic and Supersonic Speeds by a Modified Strip Analysis,” NACA Research Memorandum, Technical Report No. RM L57L10.
Barmby, J. , Cunningham, H. , and Garrick, I. , 1951, “ Study of Effects of Sweep on the Flutter of Cantilever Wings,” NACA Report, Technical Report No. NR 1014.
Theodorsen, R. , 1949, “ General Theory of Aerodynamic Instability and the Mechanism of Flutter,” NACA Report, Technical Report No. 496.
Scherer, J. , 1968, “ Experimental and Theoretical Investigation of Large Amplitude Oscillation Foil Propulsion Systems,” U.S. Army Engineering Research and Development Laboratories, Technical Report No. TR-662-1-F.
Wang, J. , Rodriguez, J. , and Keribar, R. , 2010, “ Integration of Flexible Multibody Systems Using Radau IIA Algorithms,” ASME J. Comput. Nonlinear Dyn., 5(4), p. 041008. [CrossRef]
Bolotin, V. , 1963, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press Limited, Oxford, UK.


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Fig. 1

Motion of the wing under aerodynamic load

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Fig. 4

Curve fitting of the optimized signals (vn = 500 ft/s). Original signal (+) and reconstructed signal (∘).

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Fig. 5

Parametric excitation of a simply supported beam

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Fig. 2

Frequency (top figure) and damping (bottom figure) characteristics (vn = 500 ft/s). Modes 1 (□), 2 (∘), 3 (◇), and 4 (*).

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Fig. 3

Stability characteristics versus far-field flow velocity for the partial Floquet (◇) and LCEs (□) approaches. Predicted flutter boundary: partial Floquet (dashed vertical line) and LCEs (solid vertical line).

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Fig. 6

Time histories of q1 (top figure) and q˙1 (bottom figure) for ODE45 (solid line) and the exponential approximation (dashed line). Figures on the right show the normalized error between the two predictions.

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Fig. 7

Damping (top figure) and spectral radius (bottom figure) of the dominant eigenvalue of the system versus excitation frequency, for μ = 0.15. Floquet classical analysis: solid line and present approach by matlab: dashed line.

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Fig. 8

Time history of natural frequency (top figure) and structural damping (bottom figure) for ϵ = 40.56. Floquet classical analysis: solid line (□) and proposed approach: dashed line (◇).

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Fig. 9

Damping (top figure) and norm (bottom figure) of the maximum eigenvalue of the system versus excitation frequency, for ϵ = 40.56. Floquet classical analysis: solid line and present approach: dashed line.




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