A comprehensive analysis on the discretization method of the equation of motion in piezoelectrically actuated microbeams

• Journal title : Smart Structures and Systems
• Volume 16, Issue 5,  2015, pp.891-918
• Publisher : Techno-Press
• DOI : 10.12989/sss.2015.16.5.891
Title & Authors
A comprehensive analysis on the discretization method of the equation of motion in piezoelectrically actuated microbeams
Zamanian, M.; Rezaei, H.; Hadilu, M.; Hosseini, S.A.A.;
Abstract
In many of microdevices a part of a microbeam is covered by a piezoelectric layer. Depend on the application a DC or AC voltage is applied between upper and lower side of the piezoelectric layer. A common method in many of previous works for evaluating the response of these structures is discretizing by Galerkin method. In these works often single mode shape of a uniform microbeam i.e. the microbeam without piezoelectric layer has been used as comparison function, and so the convergence of the solution has not been verified. In this paper the Galerkin method is used for discretization, and a comprehensive analysis on the convergence of solution of equation that is discretized using this comparison function is studied for both clamped-clamped and clamped-free microbeams. The static and dynamic solution resulted from Galerkin method is compared to the modal expansion solution. In addition the static solution is compared to an exact solution. It is denoted that the required numbers of uniform microbeam mode shapes for convergence of static solution due to DC voltage depends on the position and thickness of deposited piezoelectric layer. It is shown that when the clamped-clamped microbeam is coated symmetrically by piezoelectric layer, then the convergence for static solution may be obtained using only first mode. This result is valid for clamped-free case when it is covered by piezoelectric layer from left clamped side to the right. It is shown that when voltage is AC then the number of required uniform microbeam shape mode for convergence is much more than the number of required mode in modal expansion due to the dynamic effect of piezoelectric layer. This difference increases by increasing the piezoelectric thickness, the closeness of the excitation frequency to natural frequency and decreasing the damping coefficient. This condition is often indefeasible in microresonator system. It is concluded that discreitizing the equation of motion using one mode shape of uniform microbeam as comparison function in many of previous works causes considerable errors.
Keywords
microbeam;piezoelectric actuation;Galerkin method;convergence;
Language
English
Cited by
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Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium,;

Smart Structures and Systems, 2016. vol.18. 6, pp.1125-1143
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A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions, Physica E: Low-dimensional Systems and Nanostructures, 2017, 85, 285
2.
Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium, Smart Structures and Systems, 2016, 18, 6, 1125
References
1.
Abramovich, H. (1998), "Deflection control of laminated composite beams with piezoceramic layers-closed form solutions", Compos. Struct., 43(3), 217-231.

2.
Azizi, S., Rezazadeh, G., Ghazavi, M. and Khadem, S.E. (2011), "Stabilizing the pull-in instability of an electro-statically actuated micro-beam using piezoelectric actuation", Appl. Math. Model., 35(10), 4796-4815.

3.
Azizi, S., Ghazavi, M., Khadem, S.E., Rezazadeh, G. and Cetinkaya, C. (2013), "Application of piezoelectric actuation to regularize the chaotic response of an electrostatically actuated micro-beam", Nonlinear Dynam., 73(1-2), 853-867.

4.
Bashash, S., Salehi-Khojin, A. and Jalili, N. (2008), "Forced Vibration Analysis of Flexible Euler-Bernoulli Beams with Geometrical Discontinuities", Proceedings of the American Control Conference, Washington, June.

5.
Chen, C., Hu, H. and Dai, L. (2013), "Nonlinear behavior and characterization of a piezoelectric laminated microbeam system", Commun Nonlinear Sci. Numer Simulat., 18(5), 1304-1315.

6.
Dick, A.J., Balachandran, B., DeVoe, D.L. and Mote, J. (2006), "Parametric identification of piezoelectric microscale resonators", J. Micromech. Microeng., 16 (8), 1593-1601.

7.
Ghazavi, M., Rezazadeh, G. and Azizi, S. (2010), "Pure parametric excitation of a micro cantilever beam actuated by piezoelectric layers, Microsyst Technol", Appl. Math. Model., 4 (12), 4196-4207.

8.
Gopinathan, S., Varadan, V.V. and Varadan, V.K. (2000), "A review and critique of theories for piezoelectric laminates", Smart Mater. Struct., 9(1), 24-48.

9.
Hagedorn, P. and DasGupta, A. (2007), Vibrations And Waves In Continuous Mechanical Systems, John Wiley& Sons, Chichester, England.

10.
Korayem, M.H. and Ghaderi, R. (2013), "Vibration response of a piezoelectrically actuated microcantilever subjected to tip-sample interaction", Sci. Iran., 20(1), 195-206.

11.
Li, H., Preidikman, S., Balachandran, B. and Mote, J. (2006), "Nonlinear free and forced oscillations of piezoelectric microresonators", J. Micromech. Microeng., 16(2), 356-367.

12.
Li, H. and Balachandran, B. (2006), "Buckling and Free Oscillations of Composite Microresonators", IEEE J. MEMS., 15(1), 42-51.

13.
Mahmoodi, S.N. and Jalili, N. (2007), "Non-linear vibrations and frequency response analysis of piezoelectricallydriven microcantilevers", Int. J. Nonlinear Mech., 42(4), 577-587.

14.
Mahmoodi, S.N., Afshari, M. and Jalili, N. (2007), "Nonlinear vibrations of piezoelectric microcantilevers for biologically-induced surface stress sensing", Commun Nonlinear Sci. Numer Simul., 13(9), 1964-1977.

15.
Mahmoodi, S.N. and Jalili, N. (2008), "Coupled flexural-torsional nonlinear vibrations of piezoelectrically actuated microcantilevers with application to friction force microscopy", J. Vib. Acoust., 130 (6), 061003-1-10.

16.
Mahmoodi, S.N., Jalili, N. and Ahmadian, M. (2010), "Subharmonics analysis of nonlinear flexural vibrations of piezoelectrically actuated microcantilevers", Nonlinear Dynam., 59(3), 397-409.

17.
Preidikman, S. and Balachandran, B. (2006), "Semi-analytical tool based on geometric nonlinearities for microresonator design", J. Micromech. Microeng., 16(3), 512-525.

18.
Raeisi Fard, H., Nikkhah Bahrami, M. and Yousefi-Koma, A. (2014), "Mechanical characterization of electrostatically and piezoelectrically actuatedmicro-switches, including curveture and piezoelectric nonlinearities", J. Mech. Sci. Technol., 28(1), 263-272.

19.
Rezazadeh, G., Fathalilou, M. and Shabani, R. (2009), "Static and dynamic stabilities of a microbeam actuated by a piezoelectric voltage", Microsyst. Technol., 15(12), 1785-1791.

20.
Shooshtari, A., Hoseini, S.M., Mahmoodi, S.N. and Kalhori, H. (2012), "Analytical solution for nonlinear free vibrations of viscoelastic microcantilevers covered with a piezoelectric layer", Smart Mater. Struct., 21(7), 075015 (10pp).

21.
Simu, U. and Johansson, S. (2002), "Fabrication of monolithic piezoelectric drive units for a miniature robot", J. Micromech. Microeng., 12(5), 582-89.

22.
Wang, F., Tang, G.J. and Li, D.K. (2007), "Accurate modeling of a piezoelectric composite beam", Smart Mater. Struct., 16(5), 1595-602.

23.
Younis, M.I. (2011), EMS Linear And Nonlinear Statics And Dynamics, Springer, New York, USA.

24.
Zamanian, M., Khadem, S.E. and Mahmoodi, S.N. (2008), "The effect of a piezoelectric layer on the mechanical behavior of an electrostatic actuated microbeam", Smart Mater. Struct., 17(6), 065024 (15pp).