0
Research Papers

Approaches for Reduced-Order Modeling of Electrically Actuated von-Karman Microplates

[+] Author and Article Information
Shahid Saghir

Physical Science and Engineering Division,
King Abdullah University of Science
and Technology (KAUST),
4700 KAUST,
Thuwal 23955-6900, Saudi Arabia
e-mail: shahid.saghir@kaust.edu.sa

M.I. Younis

Physical Science and Engineering Division,
King Abdullah University of Science and
Technology (KAUST),
4700 KAUST,
Thuwal 23955-6900, Saudi Arabia
e-mail: mohammad.younis@kaust.edu.sa

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 21, 2016; final manuscript received July 14, 2016; published online September 1, 2016. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(1), 011011 (Sep 01, 2016) (12 pages) Paper No: CND-16-1092; doi: 10.1115/1.4034271 History: Received February 21, 2016; Revised July 14, 2016

This article presents and compares different approaches to develop reduced-order models for the nonlinear von-Karman rectangular microplates actuated by nonlinear electrostatic forces. The reduced-order models aim to investigate the static and dynamic behavior of the plate under small and large actuation forces. A fully clamped microplate is considered. Different types of basis functions are used in conjunction with the Galerkin method to discretize the governing equations. First, we investigate the convergence with the number of modes retained in the model. Then for validation purpose, a comparison of the static results is made with the results calculated by a nonlinear finite element model. The linear eigenvalue problem for the plate under the electrostatic force is solved for a wide range of voltages up to pull-in. Results among the various reduced-order modes are compared and are also validated by comparing to results of the finite-element model. Further, the reduced-order models are employed to capture the forced dynamic response of the microplate under small and large vibration amplitudes. Comparison of the different approaches is made for this case.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Vogl, G. W. , and Nayfeh, A. H. , 2003, “ A Reduced-Order Model for Electrically Actuated Clamped Circular Plates,” ASME Paper No. DETC2003/VIB-48530.
Younis, M. I. , Abdel-Rahman, E. M. , and Nayfeh, A. , 2003, “ A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS,” J. Microelectromech. Syst., 12(5), pp. 672–680. [CrossRef]
Zhao, X. , Abdel-Rahman, E. M. , and Nayfeh, A. H. , 2004, “ A Reduced-Order Model for Electrically Actuated Microplates,” J. Micromech. Microeng., 14(7), pp. 900–906. [CrossRef]
Nayfeh, A. H. , Younis, M. I. , and Abdel-Rahman, E. M. , 2005, “ Reduced-Order Models for MEMS Applications,” Nonlinear Dyn., 41(1–3), pp. 211–236. [CrossRef]
Batra, R. C. , Porfiri, M. , and Spinello, D. , 2008, “ Reduced-Order Models for Microelectromechanical Rectangular and Circular Plates Incorporating the Casimir Force,” Int. J. Solids Struct., 45(11–12), pp. 3558–3583. [CrossRef]
May, S. F. , and Smith, R. C. , 2009, “ Reduced-Order Model Design for Nonlinear Smart System Models,” Proc. SPIE, 7286, p. 72860B.
Younis, M. I. , 2011, MEMS Linear and Nonlinear Statics and Dynamics, Vol. 20, Springer, New York, pp. 1–11.
Machauf, A. , Nemirovsky, Y. , and Dinnar, U. , 2005, “ A Membrane Micropump Electrostatically Actuated Across the Working Fluid,” J. Micromech. Microeng., 15(12), pp. 2309–2316. [CrossRef]
Chao, P. C. , Chiu, C.-W. , and Tsai, C. , 2006, “ A Novel Method to Predict the Pull-In Voltage in a Closed Form for Micro-Plates Actuated by a Distributed Electrostatic Force,” J. Micromech. Microeng., 16(5), pp. 986–998. [CrossRef]
Nayfeh, A. H. , and Younis, M. I. , 2004, “ A New Approach to the Modeling and Simulation of Flexible Microstructures Under the Effect of Squeeze-Film Damping,” J. Micromech. Microeng., 14(2), pp. 170–181. [CrossRef]
Bertarelli, E. , Ardito, R. , Ardito, R. , Corigliano, A. , and Contro, R. , 2011, “ A Plate Model for the Evaluation of Pull-In Instability Occurrence in Electrostatic Micropump Diaphragms,” Int. J. Appl. Mech., 3(01), pp. 1–19. [CrossRef]
Ahmad, B. , and Pratap, R. , 2010, “ Elasto-Electrostatic Analysis of Circular Microplates Used in Capacitive Micromachined Ultrasonic Transducers,” IEEE Sens. J., 10(11), pp. 1767–1773. [CrossRef]
Porfiri, M. , 2008, “ Vibrations of Parallel Arrays of Electrostatically Actuated Microplates,” J. Sound Vib., 315(4), pp. 1071–1085. [CrossRef]
Srinivas, D. , 2012, “ Electromechanical Dynamics of Simply-Supported Micro-Plates,” Int. J. Comput. Eng. Res., 2(5), pp. 1388–1395.
Ng, T. Y. , Jiang, T. Y. , Li, H. , Lam, K. Y. , and Reddy, J. N. , 2004, “ A Coupled Field Study on the Non-Linear Dynamic Characteristics of an Electrostatic Micropump,” J. Sound Vib., 273(4–5), pp. 989–1006. [CrossRef]
Younis, M. I. , and Nayfeh, A. H. , 2007, “ Simulation of Squeeze-Film Damping of Microplates Actuated by Large Electrostatic Load,” ASME J. Comput. Nonlinear Dyn., 2(3), pp. 232–241. [CrossRef]
Faris, W. F. , 2003, “ Nonlinear Dynamics of Annular and Circular Plates Under Thermal and Electrical Loadings,” Ph.D dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA. https://vtechworks.lib.vt.edu/handle/10919/11100
Asghari, M. , 2012, “ Geometrically Nonlinear Micro-Plate Formulation Based on the Modified Couple Stress Theory,” Int. J. Eng. Sci., 51, pp. 292–309. [CrossRef]
Gholipour, A. , Farokhi, H. , and Ghayesh, M. H. , 2014, “ In-Plane and Out-of-Plane Nonlinear Size-Dependent Dynamics of Microplates,” Nonlinear Dyn., 79(3), pp. 1771–1785. [CrossRef]
Zand, M. M. , and Ahmadian, M. , 2007, “ Characterization of Coupled-Domain Multi-Layer Microplates in Pull-In Phenomenon, Vibrations and Dynamics,” Int. J. Mech. Sci., 49(11), pp. 1226–1237. [CrossRef]
Pursula, A. , Råback, P. , Lähteenmäki, S. , and Lahdenperä, J. , 2006, “ Coupled FEM Simulations of Accelerometers Including Nonlinear Gas Damping With Comparison to Measurements,” J. Micromech. Microeng., 16(11), pp. 2345–2354. [CrossRef]
Telukunta, S. , and Mukherjee, S. , 2006, “ Fully Lagrangian Modeling of MEMS With Thin Plates,” IEEE/ASME J. Microelectromech. Syst., 15(4), pp. 795–810. [CrossRef]
Jia, X. L. , Yang, J. , and Kitipornchai, S. , 2011, “ Pull-In Instability of Geometrically Nonlinear Micro-Switches Under Electrostatic and Casimir Forces,” Acta Mech., 218(1–2), pp. 161–174. [CrossRef]
Wang, B. , Zhou, S. , Zhao, J. , and Chen, X. , 2011, “ Pull-In Instability Analysis of Electrostatically Actuated Microplate With Rectangular Shape,” Int. J. Precis. Eng. Manuf., 12(6), pp. 1085–1094. [CrossRef]
Mohammadi, V. , Ansari, R. , Shojaei, M. F. , Gholami, R. , and Sahmani, S. , 2013, “ Size-Dependent Dynamic Pull-In Instability of Hydrostatically and Electrostatically Actuated Circular Microplates,” Nonlinear Dyn., 73(3), pp. 1515–1526. [CrossRef]
Mukherjee, S. , Bao, Z. , Roman, M. , and Aubry, N. , 2005, “ Nonlinear Mechanics of MEMS Plates With a Total Lagrangian Approach,” Comput. Struct., 83(10), pp. 758–768. [CrossRef]
Faris, W. F. , Abdel-Rahman, E. M. , and Nayfeh, A. H. , 2002, “ Mechanical Behavior of an Electrostatically Actuated Micropump,” AIAA Paper No. 2002-1303.
Zand, M. M. , and Ahmadian, M. , 2009, “ Vibrational Analysis of Electrostatically Actuated Microstructures Considering Nonlinear Effects,” Commun. Nonlinear Sci. Numer. Simul., 14(4), pp. 1664–1678. [CrossRef]
Fu, Y. , and Zhang, J. , 2009, “ Active Control of the Nonlinear Static and Dynamic Responses for Piezoelectric Viscoelastic Microplates,” Smart Mater. Struct., 18(9), p. 095037. [CrossRef]
Karimzade, A. , Moeenfard, H. , and Ahmadian, M. T. , “ Nonlinear Analysis of Pull-In Voltage for a Fully Clamped Microplate With Movable Base,” ASME Paper No. IMECE2012-89285.
Farokhi, H. , and Ghayesh, M. H. , “ Nonlinear Dynamical Behaviour of Geometrically Imperfect Microplates Based on Modified Couple Stress Theory,” Int. J. Mech. Sci., 90, pp. 133–144. [CrossRef]
Ghayesh, M. H. , and Farokhi, H. , 2015, “ Nonlinear Dynamics of Microplates,” Int. J. Eng. Sci., 86, pp. 60–73. [CrossRef]
Rahaeifard, M. , Ahmadian, M. , and Firoozbakhsh, K. , 2015, “ Vibration Analysis of Electrostatically Actuated Nonlinear Microbridges Based on the Modified Couple Stress Theory,” Appl. Math. Model., 39(1), pp. 6694–6704. [CrossRef]
Nayfeh, A. H. , and Mook, D. T. , 2008, Nonlinear Oscillations, Wiley, Hoboken, NJ.
Nayfeh, A. H. , and Pai, P. F. , 2008, Linear and Nonlinear Structural Mechanics, Wiley, Hoboken, NJ.
COMSOL, 2012, “ COMSOL Multiphysics,” COMSOL, Inc., Burlington, MA. https://www.comsol.com/products
Lobitz, D. , Nayfeh, A. , and Mook, D. , 1977, “ Non-Linear Analysis of Vibrations of Irregular Plates,” J. Sound Vib., 50(2), pp. 203–217. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

A schematic diagram of an electrically actuated fully clamped microplate

Grahic Jump Location
Fig. 2

Convergence of the static results with the number of transversal modeshapes retained in the reduced-order models. Variation of the maximum nondimensional deflection Wmax at the center of the microplate with the electrostatic voltage parameter α2Vdc2 when α=1 and α1=1 : (a) model I, (b) model II, (c) model III, (d) model IV, and (e) model V.

Grahic Jump Location
Fig. 3

A comparison of the maximum deflection Wmax(a2,b2) calculated by the reduced-order models with the results obtained from FE model implemented in COMSOL for various values of Vdc, until the pull-in instability: (a) model I, (b) model II, (c) model III, (d) model IV, and (e) model V

Grahic Jump Location
Fig. 4

The nondimensional fundamental natural frequency (λ=ωa2(ρ/D)) of a square microplate for different levels of Vdc until pull-in (stars). Comparison with the results computed by the FE model implemented in COMSOL (diamonds): (a) model II, (b) model III, (c) model IV, and (d) model V.

Grahic Jump Location
Fig. 5

Frequency response curves near non-dimensional fundamental natural frequency, maximum nondimensional deflection Wmax(0.5,0.5) of the microplate against actuating frequency Ω. The responses are captured at Vdc=1V, and (a) Vac=0.01V and (b) Vac=1V, when α=1, α1=1, α2=1, and a quality factor Q=1000.

Grahic Jump Location
Fig. 6

Mesh convergence study: convergence of the first natural frequency with the increasing number of elements

Grahic Jump Location
Fig. 7

In-plane displacement shape functions for model IV: (a) ψu(x,y) and (b) ψv(x,y)

Grahic Jump Location
Fig. 8

The first six symmetric-symmetric transversal modeshapes of a square microplate

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In