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Research Papers

# Bifurcation and Chaotic Analysis of Aeroelastic Systems

[+] Author and Article Information
Cheng-Chi Wang

National Chin-Yi University of Technology,
No.57, Sec. 2, Zhongshan Rd.,
Taiping Dist., Taichung 41170, Taiwan

Chieh-Li Chen

Department of Aeronautic and Astronautics,
National Cheng Kung University,
Tainan 70101, Taiwan

Her-Terng Yau

Department of Electrical Engineering,
National Chin-Yi University of Technology,
No.57, Sec. 2, Zhongshan Rd.,
Taiping Dist., Taichung 41170, Taiwan

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 17, 2012; final manuscript received June 9, 2013; published online September 12, 2013. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 9(2), 021004 (Sep 12, 2013) (13 pages) Paper No: CND-12-1207; doi: 10.1115/1.4025124 History: Received November 17, 2012; Revised June 09, 2013

## Abstract

The dynamic behavior of aeroelastic systems is governed by a complex interaction among inertial, elastic, and aerodynamic forces. To prevent system instability, the interaction among these forces must be properly understood. Accordingly, the present study utilizes the differential transformation method (DTM) to examine the nonlinear dynamic response of a typical aeroelastic system (an aircraft wing) under realistic operating parameters. The system behavior and onset of chaos are interpreted by means of bifurcation diagrams, Poincaré maps, power spectra, and maximum Lyapunov exponent plots. The results reveal the existence of a complex dynamic behavior comprising periodic, quasi-periodic and chaotic responses. It is shown that chaotic motion occurs at specific intervals for different trailing edge and leading edge angles with changing initial conditions. The results presented in this study provide a useful guideline for the design of aircraft wings and confirm the validity of the DTM method as a design and analysis tool for aeroelastic systems in general.

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## Figures

Fig. 1

Aeroelastic system model with two degrees of freedom [2]

Fig. 2

Phase portraits at β = −4.293, −4.292, −4.289, and −4.044 (Figs. 2.1a-2.4a); and power spectra of pitch angle α (Figs. 2.1b-2.4b)

Fig. 3

Bifurcation diagrams versus β at α (0) = 5.0, α·(0)=0, h (0) = 0.02, h·(0)=0, γ = 46: (a) α(nT), (b) α· (nT), (c) h(nT), and (d) h· (nT)

Fig. 4

Poincaré maps of system at value of angle of trailing edge: (a) β = −4.293, (b) −4.292, (c) −4.289, and (d) −4.044(−4.5 ≤ β ≤ −4.0)

Fig. 5

Maximum Lyapunov exponents of system at different values of β with four initial values of pitch angle

Fig. 6

Bifurcation diagrams versus β at α (0) = 10.0, α·(0)=0, h (0) = 0.02, h·(0)=0, γ = 45: (a) α(nT), (b) α· (nT), (c) h(nT), and (d) h· (nT)

Fig. 7

Maximum Lyapunov exponents of system at (a) β = −4.203, (b) −4.202, (c) −4.199, and (d) −3.96

Fig. 8

Maximum Lyapunov exponents of system at different values of β with four initial values of plunge displacement

Fig. 9

Phase portraits at γ = −1.373, −1.372, −1.366, and 1.373 (Figs. 9.1a-9.4a); and power spectra of pitch angle α (Figs. 9.1b-9.4b)

Fig. 10

Bifurcation diagrams versus γ atα (0) = 9.0, α·(0)=0, h (0) = 0.02, h·(0)=0, β = 0: (a) α(nT), (b) α· (nT), (c) h(nT), and (d) h· (nT)

Fig. 11

Maximum Lyapunov exponents of system at (a) γ = −1.372, and (b) − 1.366

Fig. 12

Maximum Lyapunov exponents of system at different values of γ with four initial values of pitch angle

Fig. 13

Bifurcation diagrams versus γ at α (0) = 10.0, α·(0)=0, h (0) = 0.01, h·(0)=0, β = 0: (a) α(nT), (b) α· (nT), (c) h(nT), and (d) h· (nT)

Fig. 14

Maximum Lyapunov exponents of system at (a) γ = −1.373, (b) −1.367, (c) −2.0, and (d) 1.373

Fig. 15

Maximum Lyapunov exponents of system at different values of γ with four initial values of plunge displacement

Fig. 16

Maximum Lyapunov exponents of system over the interval of −50.0 ≤ γ ≤ 50.0, and −18.0 ≤ β ≤ 18.0 at α (0) =1.0, α·(0) = 0, h (0) = 0.01, h·(0)= 0

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