Research Papers

Analysis of a Two-Dimensional Aeroelastic System Using the Differential Transform Method

[+] Author and Article Information
Xuechuan Wang

National Key Laboratory
of Aerospace Flight Dynamics,
Northwestern Polytechnical University,
Xi'an 710072, China;
Department of Mechanical Engineering,
Texas Tech University,
Apt. 1423,
701 N Indiana Avenue,
Lubbock, TX
e-mail: xc.wang@ttu.edu

Xiaokui Yue

National Key Laboratory of Aerospace
Flight Dynamics,
Northwestern Polytechnical University,
Youyi West Road, #251 Mail Box,
Xi'an 710072, China
e-mail: xkyue@nwpu.edu.cn

Honghua Dai

National Key Laboratory
of Aerospace Flight Dynamics,
Northwestern Polytechnical University,
Youyi West Road, #251 Mail Box,
Xi'an 710072, China
e-mail: hhdai@nwpu.edu.cn

Jianping Yuan

National Key Laboratory
of Aerospace Flight Dynamics,
Northwestern Polytechnical University,
Youyi West Road, #251 Mail Box,
Xi'an 710072, China
e-mail: jyuan@nwpu.edu.cn

1Corresponding authors.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 21, 2015; final manuscript received June 21, 2016; published online August 9, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 11(6), 061012 (Aug 09, 2016) (10 pages) Paper No: CND-15-1103; doi: 10.1115/1.4034123 History: Received April 21, 2015; Revised June 21, 2016

This paper investigates the nonlinear responses of a typical two-dimensional airfoil with control surface freeplay and cubic pitch stiffness in an incompressible flow. The differential transform (DT) method is applied to the aeroelastic system. Due to the nature of this method, it is capable of providing analytical solutions in forms of Taylor series expansions in each subdomain between two adjacent sampling points. The results demonstrate that the DT method can successfully detect nonlinear aeroelastic responses such as limit cycle oscillations (LCOs), chaos, bifurcation, and flutter phenomenon. The accuracy and efficiency of this method are verified by comparing it with the RK (Henon) method. In addition to ordinary differential equations (ODEs), the DT method is also a powerful tool for directly solving integrodifferential equations. In this paper, the original aeroelastic system of integrodifferential equations is handled directly by the DT method. With no approximation or simplification imposed on the integral terms of aerodynamic function, the resulted solutions are closer to representing the real dynamical behavior.

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Fig. 1

Schematic diagram for airfoil section with control surface

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Fig. 2

The response curve of the system for δ=0: (a) U=10 m/s and (b) U=25 m/s. Solid lines: RK-Henon method and cross: DT method.

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Fig. 3

The oscillation of the system: δ=2deg and U=10 m/s. RK-Henon method and DT method.

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Fig. 4

State-space model results, bifurcation diagrams for (a) pitch angle, (b) flap deflection, and (c) plunge displacement

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Fig. 5

Results obtained by the RK (Henon) method and the DTmethod. (a) Poincare maps and (b) amplitude spectra of flapdeflection for U=9  m/s. RK-Henon method and DT method.

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Fig. 6

Phase planes of flap motion for U=8 m/s. (a) Results of original system by DT method and (b) results of state-space model by RK (Henon) method.

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Fig. 7

Original aeroelastic model results, bifurcation diagrams for flap deflection

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Fig. 8

Frequency responses of Theodorsen's function C(s) (gray), Jone's approximation PJ(s) (dashed red), and three-term Pade approximation P3(s) (red), respectively

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Fig. 9

Results obtained with three-term Pade approximation. (a) Bifurcation diagram and phase plane for (b) U=10 m/s.



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