Prediction and Control of Chaos in Nonlinear Rectangular Micro-Plate on the Elastic Foundation

Document Type : Research Article

Authors

Department of Mechanical Engineering, Quchan University of Technology, Quchan, Iran

Abstract

In this study, nonlinear dynamics of non-classical Kirchhoff microplate is analyzed and chaotic behavior is predicted and controlled by designing the robust adaptive fuzzy controller. Virtual displacement principle is employed to derive the governing equation of micro-plate resting on a nonlinear elastic foundation. In the governing equation, von-Karman geometric nonlinearity and couple stress theory are considered. Then eigenvalue governing equation is solved for fully simply supported boundary conditions and results are validated. In the next step, considering harmonic excitation of the first mode, the micro-plate forced vibration equation is derived using the Galerkin method. Regardless of modal interaction, the chaos threshold is then investigated. Homoclinic orbits of the unperturbed system are plotted and stable and unstable manifold transversely cut that is criteria to predict chaos according to Melnikov’s method are studied. Using the maximum Lyapunov exponents numerical method, size-dependent chaos is also locally identified. Phase portrait, Poincare mapping and time response are plotted for different values of size ratios and the significant effect of size on the chaotic behavior of micro-plats is presented. Subsequently, designing the robust adaptive fuzzy controllers, chaotic vibrations are completely eliminated from the system and the robust adaptive fuzzy controller is introduced as an effective method for controlling chaos in these systems.

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References

[1] S. Chaterjee, G. Pohit, A large deflection model for the pull-in analysis of electrostatically actuated microcantilever beams, Journal of sound and vibration, .689-969 (9002) (4)223.
[2] A. Tocchio, A. Caspani, G. Langfelder, Mechanical and electronic amplitude-limiting techniques in a MEMS resonant accelerometer, IEEE Sensors Journal, 12)6( )2012( 1719-1725.
[3] L.J. Currano, M. Yu, B. Balachandran, Latching in a MEMS shock sensor: Modeling and experiments, Sensors and Actuators A: Physical, 159)1( )2010( 41-50.
[4] M.H. Mahdavi, A. Farshidianfar, M. Tahani, S. Mahdavi, H. Dalir, A more comprehensive modeling of atomic force microscope cantilever, Ultramicroscopy, 109)1( )2008( 54-60.
[5] M. Kahrobaiyan, M. Asghari, M. Rahaeifard, M. Ahmadian, Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress  theory, International Journal of Engineering Science, 48(12) (2010) 1985-1994.
[6]  R. Mestrom, R. Fey, J. Van Beek, K. Phan, H. Nijmeijer, Modelling the dynamics of a MEMS resonator: simulations and experiments, Sensors and Actuators A: Physical, 142(1) (2008) 306-315.
[7]     M. Roukes, Nanoelectromechanical systems face the future, Physics World, 14(2) (2001) 25.
[8]   V. Logeeswaran, F.E. Tay, M. Chan, F.S. Chau, Y.C. Liang, 2f method for the measurement of resonant frequency and Q-factor of micromechanical transducers, in:  Design, Test, Integration, and Packaging of MEMS/MOEMS, International Society for Optics and Photonics, 4755 (2002), 584-594.
[9]  J.A. Harley, E.M. Chow, T.W. Kenny, Design of resonant beam transducers: An axial force probe for atomic force microscopy, in:  Micro-ElectroMechanical Systems: ASME Intl. ME Congress and Exposition, 1998, 274-252.
[10]  Y.C. Wang, S.G. Adams, J.S. Thorp, N.C. MacDonald, P. Hartwell, F. Bertsch, Chaos in MEMS, parameter estimation and its potential application, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 45)10( )1998( 1013-1020.
[11]   J. Awrejcewicz, V. Krysko, I. Papkova, A. Krysko, Routes to chaos in continuous mechanical systems. Part 1: Mathematical models and solution methods, Chaos, Solitons & Fractals, 45(6) (2012) 687-708.
[12]   A. Krysko, J. Awrejcewicz, I. Papkova, V. Krysko, Routes to chaos in continuous mechanical systems: Part 2. Modelling transitions from regular to chaotic dynamics, Chaos, Solitons & Fractals, 45(6) (2012) 709-720.
[13] J. Awrejcewicz, A. Krysko, I. Papkova, V. Krysko, Routes to chaos in continuous mechanical systems. Part 3: The Lyapunov exponents, hyper, hyperhyper and spatial–temporal chaos, Chaos, Solitons & Fractals, 45(6) (2012) 721-736.
[14]    X. Yang, P. Sethna, Local and global bifurcations in parametrically excited vibrations of nearly square plates, International journal of Non-linear Mechanics, 26(2) (1991) 199-220.
[15]    X. Yang, P.R. Sethna, Non-linear phenomena in forced vibrations of a nearly square plate: antisymmetric case, Journal of Sound and Vibration, 155(3)(1992) 413-441.
[16]    J. Awrejcewicz, E.Y. Krylova, I. Papkova, V.A. Krysko, Regular and chaotic dynamics of flexible plates, Shock and Vibration, (2014 ) 1-8.
[17]    C. Touzé, O. Thomas, M. Amabili, Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates, International Journal of nonlinear Mechanics, 46(1) (2011) 234-246.
[18]    M. Amabili, A. Sarkar, M. Païdoussis, Chaotic vibrations of circular cylindrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method, Journal of Sound and Vibration, 290(3-5) (2006) 736-762.
[19]    Y. Hao, L. Chen, W. Zhang, J. Lei, Nonlinear oscillations, bifurcations and chaos of functionally   graded materials plate, Journal of Sound and Vibration,312(4-5)(2008) 862-892.
[20]    Q. Ma, D.R. Clarke, Size dependent hardness of silver single crystals, Journal of Materials Research, 10(4)(1995)853-863.
[21]    J. Stölken, A. Evans, A microbend test method for measuring the plasticity length scale, Acta Materialia, 46(14) (1998) 5109-5115.
[22]    A.C. Chong, D.C. Lam, Strain gradient plasticity effect in indentation hardness of polymers, Journal of Materials Research, 14(10) )(999) 4103-4110.
[23]    F. Yang, A. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10) (2002) 2731-2743.
[24]    S. Park, X. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 16(11) (2006) 2355.
[25]       A. Andakhshideh, S. Maleki, H. Karamad, Sizedependent nonlinear vibration of non-uniform microbeam with various boundary conditions, Modares Mechanical Engineering, 18(9) (2019) 189- 198.(in Persian)
[26]       S. Salehi, O. Rahmani, S.A. Hoseini, Free and forced vibration analysis of Kelvin-Voigt viscoelastic rectangular nanoplate based on the modified couple stress theory, Amirkabir Journal of Mechanical Engineering, (2018). (in Persian)
[27]       M.H. Ghayesh, H. Farokhi, Nonlinear dynamics of microplates, International Journal of Engineering Science, 86 (2015) 60-73.
[28]       H. Farokhi, M.H. Ghayesh, Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory, International Journal of Mechanical Sciences, 90 (2015) 133-144.
[29]       H. Ramezannejad Azarboni, H. Keshavarzpour, M. Rahimzadeh, Nonlocal analysis of chaotic vibration, primary and super-harmonic resonance of single walled carbon nanotube in thermal environment, Amirkabir Journal of Mechanical Engineering, (2018). (in Persian)
[30]       E.M. Miandoab, A. Yousefi-Koma, H.N. Pishkenari, F. Tajaddodianfar, Study of nonlinear dynamics and chaos in MEMS/NEMS resonators, Communications in Nonlinear Science and Numerical Simulation, 22(1)(2015) 611-622.
[31]       D.G. Bassinello, A.M. Tusset, R.T. Rocha, J.M. Balthazar, Dynamical Analysis and Control of a Chaotic Microelectromechanical Resonator Model, Shock and Vibration, (2018).
[32]       S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer Science & Business Media, (2003).
[33]       A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16(3) (1985) 285-317.
[34]    H. Tourajizadeh, M. Kariman, M. Zamanian, B. Firouzi, Optimal Control of Electrostatically Actuated Micro-Plate Attached to the End of Microcantilever, Amirkabir Journal of Mechanical Engineering, 49(4) (2018) 805-818. (in Persian)
[35]    A.M. Tusset, F.C. Janzen, R.T. Rocha, J.M. Balthazar, On an Optimal Control Applied in MEMS Oscillator with Chaotic Behavior including Fractional Order, Complexity, 2018 (2018) 1-12.
[36]    H. Vaghefpour, H. Arvin, Y. Tadi Bani, Tip Tracking Control of Piezoelectric Nano-Actuator with Flexoelectric Size-Dependent Theory, Amirkabir Journal of Mechanical Engineering,  (2018). (in Persian)
[37]    A. Poursamad, A.H. Davaie-Markazi, Robust adaptive fuzzy control of unknown chaotic systems, Applied Soft Computing, 9(3) (2009) 970-976.
[38]    J.N. Reddy, Theory and analysis of elastic plates and shells, CRC press, 2006.
[39]    S. Wiggins, Global bifurcations and chaos: analytical methods, Springer Science & Business Media, (2013).
[40]    A.W. Leissa, The free vibration of rectangular plates, Journal of sound and vibration, 31(3) (1973) 257-293.
[41]    A.R. Zeni, J.A. Gallas, Lyapunov exponents for a Duffing oscillator, Physica D: Nonlinear Phenomena,89(1-2)(1995)71-82.
Amirkabir Journal of Mechanical Engineering
Volume 52, Issue 5
August 2020
Pages 1285-1300

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