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TECHNICAL PAPERS

A Reduced-Order Dynamic Model of Nonlinear Oscillating Devices

[+] Author and Article Information
R. Xu, K. Komvopoulos

Department of Mechanical Engineering, University of California, Berkeley, CA 94720

J. Dyn. Sys., Meas., Control 129(4), 514-521 (Jan 22, 2007) (8 pages) doi:10.1115/1.2745858 History: Received October 09, 2005; Revised January 22, 2007

A reduced-order dynamic model is presented for nonlinear devices subjected to in-plane oscillatory motion. Comparisons between numerical and finite element results demonstrate that the nonlinear behavior of a planar resonator can be predicted accurately by the derived dynamic model with significantly less computation. Simulation results illustrate the effects of nonlinear stiffness, damping ratio, electrostatic driving force, and device dimensions on the nonlinear dynamic behavior. The analysis yields two possible stable responses, depending on the initial rotation angle and rotation rate. The present dynamic model can be easily modified to analyze the nonlinear response of various planar resonators.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Scanning electron micrograph of a two-beam device with 100×2×2μm flexure beams. Comb drives denoted by A are used to electrostatically actuate the device, while comb drives denoted by B are used to capacitively sense the rotation of the device during oscillation.

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Figure 2

Resonant modes of the undamped vibration of a two-beam device with 30×2×2μm flexure beams obtained from the FEM analysis: (a) three-dimensional FEM model, (b) sixth (out-of-plane) resonant mode (19.96kHz), (c) seventh (in-plane) resonant mode (41.0kHz), and (d) eighth (out-of-plane) resonant mode (48.66kHz). The undeformed shape is also shown to facilitate the observation of each consecutive resonant mode.

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Figure 3

(a) FEM model of a two-beam device with 30×2×2μm flexure beams showing the distributed electrostatic force acting in the plane of oscillation and (b) detailed mesh of the anchor-beam region enclosed by dashed lines in (a)

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Figure 11

Effect of flexure-beam width on the dynamic response of a two-beam device with 100×4(±0.2)×2μm flexure beams for ξ=0.2%, Vdr=10V, and Vbias=25V: (a) amplitude and (b) phase shift

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Figure 10

Dynamic response of a two-beam device with 30×2×2μm flexure beams for ξ=0.2%, Vdr=10V, and Vbias=30V: (a) amplitude and (b) phase shift

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Figure 9

(a) Resonant amplitude of a two-beam device with 100×4×2μm flexure beams versus bias voltage for ξ=0.2% and Vdr=10V and (b) effect of bias voltage on the amplitude response of a two-beam device with 100×4×2μm flexure beams for ξ=1% and Vdr=10V

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Figure 8

Effect of bias voltage on the dynamic response of a two-beam device with 100×4×2μm flexure beams for ξ=0.2% and Vdr=10V: (a) amplitude and (b) phase shift

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Figure 7

Phase plot of a two-beam device with 100×4×2μm flexure beams for ξ=0.2%, ωdr=16.5kHz, Vdr=10V, and Vbias=25V: (a) three-dimensional phase plot and (b) projection of phase plot. (The rotation angle and the rotation rate have been normalized by the maximum rotation angle and the maximum rotation rate, respectively.)

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Figure 6

Effect of damping ratio on the dynamic response of a two-beam device with 100×4×2μm flexure beams for Vdr=10V and Vbias=25V: (a) amplitude and (b) phase shift

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Figure 5

Time responses of a two-beam device with 30×2×2μm flexure beams for ξ=1% and ωdr=41kHz obtained with the FEM analysis and the reduced-order dynamic model for (a) small and (b) large amplitude of rotation angle

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Figure 4

Normalized effective stiffness of two-beam devices with (a)30×2×2μm and (b)100×4×2μm flexure beams as a function of rotation angle

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