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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,
Prince Sattam Bin Abdulaziz University,
P.O. Box 54,
e-mail: hany_samih@yahoo.com

A. T. EL-Sayed

Department of Basic Sciences,
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

Abstract

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 $Ω1≅ω1 and Ω2≅ω2$. 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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References

Feng, Z. C. , and Sethna, P. R. , 1993, “ Global Bifurcations in the Motion of Parametrically Excited Thin Plates,” Nonlinear Dyn., 4(4), pp. 389–408.
Chang, S. I. , Bajaj, A. K. , and Krousgrill, C. M. , 1993, “ Non-Linear Vibrations and Chaos in Harmonically Excited Rectangular Plates With One-to-One Internal Resonance,” Nonlinear Dyn., 4(5), pp. 433–460.
Ostiguy, G. L. , Samson, L. P. , and Nguyen, H. , 1993, “ On the Occurrence of Simultaneous Resonances in Parametrically-Excited Rectangular Plates,” ASME J. Vib. Acoust., 115(3), pp. 344–352.
Lai, S. K. , Lim, C. W. , Xiang, Y. , and Zhang, W. , 2009, “ On Asymptotic Analysis for Large Amplitude Nonlinear Free Vibration of Simply Supported Laminated Plates,” ASME J. Vib. Acoust., 131(5), p. 051010.
Zhang, W. , 2001, “ Global and Chaotic Dynamics for a Parametrically Excited Thin Plate,” J. Sound Vib., 239(5), pp. 1013–1036.
Zhang, W. , and Liu, Z. , 2001, “ Global Dynamics of a Parametrically and Externally Excited Thin Plate,” Nonlinear Dyn., 24(3), pp. 245–268.
Kim, C. H. , Lee, C. W. , and Perkins, N. C. , 2005, “ Nonlinear Vibration of Sheet Metal Plates Under Interacting Parametric and External Excitation During Manufacturing,” ASME J. Vib. Acoust., 127(1), pp. 36–43.
Shiau, L.-C. , and Kuo, S.-Y. , 2006, “ Free Vibration of Thermally Buckled Composite Sandwich Plates,” ASME J. Vib. Acoust., 128(1), pp. 1–7.
Hegazy, U. H. , 2010, “ Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonance,” ASME J. Vib. Acoust., 132(5), p. 051004.
Zhang, W. , and Li, S. B. , 2010, “ Resonant Chaotic Motions of a Buckled Rectangular Thin Plate With Parametrically and Externally Excitations,” Nonlinear Dyn., 62(3), pp. 673–686.
Sorokin, S. V. , and Ershov, O. A. , 2003, “ Forced and Free Vibrations of Rectangular Sandwich Plates With Parametric Stiffness Modulation,” J. Sound Vib., 259(1), pp. 119–143.
Lei, Y. , Xu, W. , Shen, J. , and Fang, T. , 2006, “ Global Synchronization of Two Parametrically Excited Systems Using Active Control,” Chaos, Solitons Fractals, 28(2), pp. 428–436.
Anlas, G. , and Elbeyli, O. , 2002, “ Nonlinear Vibrations of a Simply Supported Rectangular Metallic Plate Subjected to Transverse Harmonic Excitation in the Presence of a One-to-One Internal Resonance,” Nonlinear Dyn., 30(1), pp. 1–28.
Sayed, M. , and Mousa, A. A. , 2012, “ Second-Order Approximation of Angle-Ply Composite Laminated Thin Plate Under Combined Excitations,” Commun. Nonlinear Sci. Numer. Simul., 17(12), pp. 5201–5216.
Guo, X. Y. , and Zhang, W. , 2011, “ Nonlinear Dynamics of Composite Laminated Thin Plate With 1:2:3 Inner Resonance,” 2nd International Conference on Mechanic Automation and Control Engineering (MACE'11), Hohhot, China, July 15–17, pp. 7479–7482.
Udwadia, F. E. , and Wanichanon, T. , 2014, “ Control of Uncertain Nonlinear Multibody Mechanical Systems,” ASME J. Appl. Mech., 81(4), p. 041020.
Udwadia, F. E. , and Wanichanon, T. , 2014, “ A New Approach to the Tracking Control of Uncertain Nonlinear Multi-Body Mechanical Systems,” Nonlinear Approaches in Engineering Applications 2, Springer, New York, pp. 101–136.
Udwadia, F. E. , Wanichanon, T. , and Cho, H. , 2014, “ Methodology for Satellite Formation-Keeping in the Presence of System Uncertainties,” J. Guid. Control Dyn., 37(5), pp. 1611–1624.
Udwadia, F. E. , and Wanichanon, T. , 2013, “ On General Nonlinear Constrained Mechanical Systems,” Numer. Algebra, Control Optim., 3(3), pp. 425–443.
Udwadia, F. E. , and Wanichanon, T. , 2012, “ Explicit Equations of Motion of Constrained Systems With Applications to Multi-Body Dynamics,” Nonlinear Approaches in Engineering Applications, Springer, New York, pp. 315–348.
Udwadia, F. E. , and Kalaba, R. E. , 2000, “ Nonideal Constraints and Lagrangian Dynamics,” J. Aerosp. Eng., 13(1), pp. 17–22.
Udwadia, F. E. , 2008, “ Optimal Tracking Control of Nonlinear Dynamical Systems,” Proc. R. Soc. London, Ser. A, 464(2097), pp. 2341–2363.
Udwadia, F. E. , 1996, “ Equations of Motion for Mechanical Systems: A Unified Approach,” Int. J. Nonlinear Mech., 31(6), pp. 951–958.
Udwadia, F. E. , and Schutte, A. D. , 2010, “ Equations of Motion for General Constrained Systems in Lagrangian Mechanics,” Acta Mech., 213, pp. 111–129.
Nayfeh, A. H. , 1973, Perturbation Methods, Wiley, New York.
Nayfeh, A. H. , and Mook, D. T. , 1979, Nonlinear Oscillations, Wiley-Interscience, New York.
Chia, C. Y. , 1980, Non-Linear Analysis of Plate, McGraw-Hill, New York.

Figures

Fig. 1

The model of a rectangular thin plate and the coordinate system

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

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

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.

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.

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.

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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