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Research Papers

# Modeling MEMS Resonators Past Pull-In

[+] Author and Article Information
Chandrika P. Vyasarayani1

Systems Design Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canadacpvyasar@engmail.uwaterloo.ca

Eihab M. Abdel-Rahman, John McPhee, Stephen Birkett

Systems Design Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(3), 031008 (Feb 01, 2011) (7 pages) doi:10.1115/1.4002835 History: Received August 30, 2009; Revised October 09, 2010; Published February 01, 2011; Online February 01, 2011

## Abstract

In this paper, we develop a mathematical model of an electrostatic MEMS (Micro-Electro-Mechanical systems) beam undergoing impact with a stationary electrode subsequent to pull-in. We model the contact between the beam and the substrate using a nonlinear foundation of springs and dampers. The system partial differential equation is converted into coupled nonlinear ordinary differential equations using the Galerkin method. A numerical solution is obtained by treating all nonlinear terms as external forces. We use the model to predict the contact length, natural frequencies, and mode shapes of the beam past pull-in voltage as well as the dynamic response of a shunt switch in a closing and opening sequence.

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Copyright © 2011 by American Society of Mechanical Engineers
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## Figures

Figure 3

Comparison of the natural frequencies obtained analytically and experimentally (19). Ω1(Vdc) is the fundamental natural frequency of the beam as a function of dc voltage and Ω1(0) is the frequency at Vdc=0. The results are for β=1 consistent with the experimental system (19).

Figure 4

Fifth to tenth symmetric mode shapes (a) obtained from the analytical expression given by Eq. 9 and (b) obtained by mirroring the left-half into the right-half of the modes. Note that the y-axis scale is arbitrary.

Figure 5

Contact length using 5-mode to 11-mode approximations

Figure 6

Static deflection of the beam past pull-in using 5-mode, 10-mode, and 11-mode approximations at Vdc=60 V

Figure 7

Midpoint beam deflection as a function of dc voltage (Vdc)

Figure 8

The contact length of the beam with the substrate as a function of dc voltage

Figure 9

Variation in natural frequency with dc voltage

Figure 10

Variation in the first mode shape with dc voltage

Figure 11

Variation in the second mode shape with dc voltage

Figure 13

Snapshots of the beam deformation

Figure 2

Foundation model of the substrate

Figure 1

Fixed-fixed MEMS beam

Figure 12

(a) Response of the beam to the voltage waveform; (b) magnified view of the beam motion during contact

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