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Research Papers: Modeling

# Dynamics of a Duffing Oscillator With Two Time Delays in Feedback Control Under Narrow-Band Random Excitation

[+] Author and Article Information
Yanfei Jin

Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, P.R.C.yanfeijin@nuaa.edu.cn

Haiyan Hu

Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, P.R.C.

J. Comput. Nonlinear Dynam 3(2), 021205 (Feb 04, 2008) (7 pages) doi:10.1115/1.2833890 History: Received May 28, 2007; Revised August 17, 2007; Published February 04, 2008

## Abstract

The paper presents analytical and numerical results of the primary resonance of a Duffing oscillator with two distinct time delays in the linear feedback control under narrow-band random excitation. Using the method of multiple scales, the first-order and the second-order steady-state moments of the primary resonance are derived. For the case of two distinct time delays, the appropriate choices of the combinations of the feedback gains and the difference between two time delays are discussed from the viewpoint of vibration control and stability. The analytical results are in well agreement with the numerical results.

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Copyright © 2008 by American Society of Mechanical Engineers
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## Figures Figure 1

Stable region of the steady-state solution in (ζe,σe) plane Figure 2

Stable region of the first-order and the second-order steady-state moments in (ζe,σe) plane Figure 3

The variation of ζe with ψ for ζ>∣u±v∣: ζ=0.05, ∣u∣=0.01, ∣v∣=0.03 (thick curves: τ=0, dotted curves: τ=π∕4, dash curves: τ=π∕2) Figure 4

The variation of ζe with ψ for ζ>∣u±v∣: ζ=0.05, ∣u∣=0.03, ∣v∣=0.01 (thick curves: τ=0, dotted curves: τ=π∕4, dash curves: τ=π∕2) Figure 5

αm with ζ=0.05, u=0.01, v=−0.03 for τ1>τ2: (a) as a function of ψ, thick curves: τ=0, dotted curves: τ=π∕4, dash curves: τ=π∕2; (b) as a function of τ, thick curves: ψ=0, dotted curves: ψ=π∕4, dash curves: ψ=π∕2 Figure 6

αm with ζ=0.05, u=0.03, v=−0.01 for τ2>τ1: (a) as a function of ψ, thick curves: τ=0, dotted curves: τ=π∕4, dash curves: τ=π∕2; (b) as a function of τ, thick curves: ψ=0, dotted curves: ψ=π∕4, dash curves: ψ=π∕2 Figure 7

Amplitude-frequency response curves with ζ=0.05, f=0.5, μ=0.05: (a) three sets of time delays, u=−0.01, v=−0.03; (b) four sets of feedback gains, τ1=π, τ2=π∕2 Figure 8

Frequency response of Eq. 1 (f=0.5, u=0.01, v=−0.01, τ1=π∕2, τ2=π, γ1=0.1, x(t)=4, ẋ(t)=−4, t∊[−τ,0])

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