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Research Papers

Active Control of a Rectangular Thin Plate Via Negative Acceleration Feedback

[+] Author and Article Information
H. S. Bauomy

Department of Mathematics,
Faculty of Science,
Zagazig University,
Zagazig 44519, Egypt;
Department of Mathematics,
College of Arts and Science in Wadi Addawasir,
Prince Sattam Bin Abdulaziz University,
P.O. Box 54,
Wadi Addawasir 11991, Saudi Arabia
e-mail: hany_samih@yahoo.com

A. T. EL-Sayed

Department of Basic Sciences,
Modern Academy for Engineering and
Technology,
Mokatem 11585, Egypt
e-mail: ashraftaha211@yahoo.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 6, 2015; final manuscript received March 23, 2016; published online May 13, 2016. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 11(4), 041025 (May 13, 2016) (12 pages) Paper No: CND-15-1282; doi: 10.1115/1.4033307 History: Received September 06, 2015; Revised March 23, 2016

In this paper, the dynamic oscillation of a rectangular thin plate under parametric and external excitations is investigated and controlled. The motion of a rectangular thin plate is modeled by coupled second-order nonlinear ordinary differential equations. The formulas of the thin plate are derived from the von Kármán equation and Galerkin's method. A control law based on negative acceleration feedback is proposed for the system. The multiple time scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to the second-order approximations. One of the worst resonance case of the system is the simultaneous primary resonances, where Ω1ω1andΩ2ω2. From the frequency response equations, the stability of the system is investigated according to the Routh–Hurwitz criterion. The effects of the different parameters are studied numerically. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. The simulation results are achieved using matlab 7.0 software. A comparison is made with the available published work.

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Figures

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Fig. 1

The model of a rectangular thin plate and the coordinate system

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Fig. 2

Nonresonant time response solution at selected values: α1=β1=5.125, α2=β2=7.375, ω1=8,ω2=9,Ω1=4,Ω2=4.5,f1=0.4,f2=0.25,μ=0.4, F1=4, F2=1.5, G1=4, and G2=3.0

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Fig. 3

Simultaneous primary resonance case Ω1≅ω1 and  Ω2≅ω2 : (a) system without controller, (b) system with negative linear velocity feedback controller, (c) system with negative quadratic velocity feedback controller, (d) system with negative cubic velocity feedback controller, and (e) system with negative acceleration feedback controller

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Fig. 4

Theoretical frequency response curves: α1=5.125, α2=7.375, ω1=8,μ=0.4, F1=4, G1=0.009, and a2=0.5. (a) Frequency response curve of a1, (b) effect of the excitation force F1, (c) effect of the natural frequency ω1, (d) effect of the damping coefficient μ, (e) effect of the nonlinear parameter α1, (f) effect of the nonlinear parameter α2, and (g) effect of the gain G1.

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Fig. 5

Theoretical frequency response curves: β1=5.125, β2=7.375, ω2=9,μ=0.4, F2=3, G2=0.04, and a1=0.05. (a) Frequency response curve of a2, (b) effect of the excitation force F2, (c) effect of the natural frequency ω2, (d) effect of the damping coefficient μ, (e) effect of the nonlinear parameter β1, (f) effect of the nonlinear parameter β2, and (g) effect of the gain G2.

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Fig. 6

Excitation–response curve of simultaneous primary resonance for α1=5.125, α2=7.375, ω1=8,μ=0.4, σ1=0.5, G1=0.009, and a2=0.5. (a) Excitation–response curves (a1 against F1), (b) effect of the natural frequency ω1, (c) effect of the damping coefficient μ, (d) effect of the detuning parameter σ1, (e) effect of the nonlinear parameter α1, (f) effect of the nonlinear parameter α2, and (g) effect of the gain G1.

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Fig. 7

Excitation–response curve of simultaneous primary resonance for β1=5.125, β2=7.375, ω2=9,μ=0.4, σ2=2, G2=0.04, and a1=0.1. (a) Excitation–response curves (a2 against F2), (b) effect of the natural frequency ω2, (c) effect of the damping coefficient μ, (d) effect of the detuning parameter σ2, (e) effect of the nonlinear parameter β1, (f) effect of the nonlinear parameter β2, and (g) effect of the gain G2.

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