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Research Papers

Vibration Control for Parametrically Excited Coupled Nonlinear Oscillators

[+] Author and Article Information
Attilio Maccari

 Via Alfredo Casella 3, 00013 Mentana, Rome, Italy

J. Comput. Nonlinear Dynam 3(3), 031010 (May 05, 2008) (8 pages) doi:10.1115/1.2908317 History: Received July 14, 2007; Revised October 23, 2007; Published May 05, 2008

A complex nonlinear system under state feedback control with a time delay corresponding to two coupled nonlinear oscillators with a parametric excitation is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. In the system without control, phase-locked solutions with period equal to the parametric excitation period are possible only if the oscillator amplitudes are equal, but they depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. It is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude and phase can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Stability chart (a) in the plane (Θ,ρ) for the nonlinear system (4) with the parameter values a=0.1, b=0.05, F=0.1, σ=−0.1, and Ω=1. The phase varies from 0 to 2π and the response from 0 to 1.5. White (black) regions stand for stable (unstable) solutions. Values of the vibration control terms A and B (b) and (c) in the plane (Θ,ρ).

Grahic Jump Location
Figure 2

Stability chart in the plane (Θ,ρ) for the nonlinear system (4) with the parameter values a=0.1, b=0.05, F=0.2, σ=−0.1, and Ω=1. The phase varies from 0 to 2π and the response from 0 to 1.5. White (black) regions stand for stable (unstable) solutions. Values of the vibration control terms A and B (b) and (c) in the plane (Θ,ρ).

Grahic Jump Location
Figure 3

Stability chart in the plane (Θ,ρ) for the nonlinear system (4) with the parameter values a=0.1, b=0.05, F=0.1, σ=0, and Ω=1. The phase varies from 0 to 2π and the response from 0 to 1.5. White (black) regions stand for stable (unstable) solutions. Values of the vibration control terms A and B (b) and (c) in the plane (Θ,ρ).

Grahic Jump Location
Figure 4

Stability chart in the plane (Θ,ρ) for the nonlinear system (4) with the parameter values a=0.1, b=0.05, F=0.1, σ=−0.15, and Ω=1. The phase varies from 0 to 2π and the response from 0 to 1.5. White (black) regions stand for stable (unstable) solutions. Values of the vibration control terms A and B (b) and (c) in the plane (Θ,ρ).

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