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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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Figures

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Fig. 1

Motion of the wing under aerodynamic load

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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. 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. 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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