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Research Papers

Frequency Analysis of Linearly Coupled Modes of MEMS Arrays

[+] Author and Article Information
Prashant N. Kambali, Gyanadutta Swain

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Sangareddy, Medak 502285, India

Ashok Kumar Pandey

Department of Mechanical and
Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Sangareddy, Medak 502285, India
e-mail: ashok@iith.ac.in

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 14, 2015; final manuscript received January 5, 2016; published online February 5, 2016. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 138(2), 021017 (Feb 05, 2016) (9 pages) Paper No: VIB-15-1384; doi: 10.1115/1.4032517 History: Received September 14, 2015; Revised January 05, 2016

Microelectromechanical system (MEMS) based arrays have been employed to increase the bandwidth and sensitivity of many sensors and actuators. In this paper, we present an approximate model to demonstrate the tuning of in-plane and out-of-plane frequencies of MEMS arrays consisting of fixed–fixed beams. Subsequently, we apply the Galerkin's method with single approximate mode to obtain the reduced-order static and dynamic equations. Corresponding to a given direct current (DC) voltage, we first solve the static equations and then obtain corresponding frequencies from the dynamic equation for single beam and arrays of multibeams. We compare the model with available experimental results. Later, we show the influence of different frequency tuning parameters such as the initial tensions, fringing field coefficients and the variable inter beam gaps between the microbeam and electrodes to control the coupling region and different modal frequencies of the beam. Finally, we obtain a compact model which can be used in optimizing the bandwidth and sensitivity of microbeams array.

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Figures

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Fig. 1

(a) Side view of N fixed–fixed beams of width B, thickness H are separated from the side electrodes, E1 and E2, and the ground electrode Eg by distance d; (b) top view of N beams and each having a length of L. Here, ith beam is separated from its neighboring beams by gaps of gi−1 and gi, respectively.

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Fig. 2

(a) Displacement of the beam in two different directions are represented by y and z and (b) the corresponding forces are represented by Qz and Qy

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Fig. 3

Comparison of experimental and analytical results for (a) single beam with gap ratios r0 = 1 and r1 = 1.55 and (b) three beam arrays with gap ratios r0 = 1, r1 = 1.35, r2 = 2.45, and r3 = 2.25. Here, rn=gn/g0, n=0…N.

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Fig. 4

Variation of in-plane and out-of-plane frequencies with DC bias beyond crossing region of (a) single beam and (b) an array of three beams

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Fig. 5

Tuning of frequencies of two modes and coupling regions in an array of three beams with g0 = 2 μm, g1 = 2.7 μm, g2 = 4.9 μm, g3 = 4.5 μm, k1 = 0.135, 0.45, 1, k2 = 1, 1, 1 k3 = 2.5, 4, 2.5 (a) for N1 = N0, N2 = 1.04N0, N3 = 1.05N0, and d = 15 μm and (b) for N1 = N2 = N3 = N0, and d = 15 μm. In both the cases, L = 500 μm, B = 4 μm, H = 200 nm, E = 3.183 × 1010 N/m2, ρ = 3234.2 kg/m3, and N0 = 43.81 μN/m.

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Fig. 6

Tuning of frequencies of two modes and coupling regions in an array of three beams with g0 = 2 μm, g1 = 2.7 μm, g2 = 4.9 μm, g3 = 4.5 μm, k1 = 0.135, 0.45, 1, k2 = 1, 1, 1 k3 = 2.5, 4, 2.5 (a) for N1 = N0, N2 = 1.04N0, N3 = 1.05N0, and d = 5000 μm and (b) for N1 = N2 = N3 = N0, and d = 5000 μm. In both the cases, L = 500 μm, B = 4 μm, H = 200 nm, E = 3.183 × 1010 N/m2, ρ = 3234.2 kg/m3, and N0 = 43.81 μN/m.

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Fig. 7

Tuning of frequencies of two modes and coupling regions in an array of three beams with g0 = 2.2 μm, g1 = 3 μm, g2 = 5 μm, g3 = 5.5 μm, k1 = 0.1, 0.6, 1, k2 = 1, 1, 1 k3 = 4, 4, 3, d = 20 μm (a) for N1 = N0, N2 = 1.06N0, N3 = 1.07N0 and (b) For N1 = N2 = N3 = N0. In both the cases, L = 500 μm, B = 4 μm, H = 200 nm, E = 3.183 × 1010 N/m2, ρ = 3234.2 kg/m3, and N0 = 43.81 μN/m.

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Fig. 8

Tuning of frequencies of two modes and coupling regions with gn varying from 4 to 5 μm, k1 = 1, k2 = 2.5 k3 = 3, d = 500 μm (a) for an array of ten beam, with initial tension Nn varying N0 to 1.019N0; (b) for an array of 20 beam, with initial tension Nn varying N0 to 1.039N0; and (c) for an array of 39 beam, with initial tension Nn varying 1.08N0 to 1.11N0. In all the cases, L = 500 μm, B = 4 μm, H = 200 nm, E = 3.183 × 1010 N/m2, ρ = 3234.2 kg/m3, and N0 = 43.81 μN/m.

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