Pull-in Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force

[+] Author and Article Information
Slava Krylov

Department of Solid Mechanics Materials and Systems, Faculty of Engineering, Tel Aviv University, Ramat Aviv, 69978, Israele-mail: vadis@eng.tau.ac.il

Ronen Maimon

RAFAEL, P.O. Box 2250 (M1), 31021 Haifa, Israel

J. Vib. Acoust 126(3), 332-342 (Jul 30, 2004) (11 pages) doi:10.1115/1.1760559 History: Received February 01, 2003; Revised September 01, 2003; Online July 30, 2004
Copyright © 2004 by ASME
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Grahic Jump Location
Microbeam of the type tested
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Potential energy for different voltages and zero initial conditions (1) V1=0.85VPI (2) V1=VDPI (3) V1=VPI
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Phase portrait for step function excitation, zero initial conditions and zero DC voltage: (1) V1=0.85VPI<VDPI (3) V1=VPI. The homoclinic orbit, (2), corresponds to the dynamic pull-in voltage V1=VDPI.
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Phase portrait for step function excitation and non-zero DC voltage: (1)-V*=0.85VPI<VDPI (2) V*=0,V1=VDPI (3) V*=0.988VPI. The dashed curve corresponds to the DC voltage equal to the static pull-in voltage V*=VPI.
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Time history for zero initial velocity u̇0=0 and step function excitation near the pull-in point q*=0.95qPI;V⁁*=0.9987. The dashed curve corresponds to the approximate expression, Eq. (35). (1) V⁁1=0.7546(1−V⁁*) and corresponds to the critical (separatrix) value given by Eqs. (31) (2) V⁁1=0.7357(1−V⁁*) and leads to the periodic motion (3) V⁁1=0.7471(1−V⁁*) and corresponds to the critical value given by Eq. (35). The nondimensional period of the system is τ⁁=5.11.
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Response of the undamped beam to step-function excitation V1=VDPI for single mode model (solid curve) and three mode model (dashed curve): velocity versus displacement of the end of the beam is shown.
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Response of the beam to a low frequency triangular wave with 500 μs rise and 1000 μs period (a) time history; the signal voltage is normalized to the signal maximal voltage (b) displacement-voltage dependence
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Response of the beam to a square wave of 500 μs period. Results correspond to voltages of 20, 22.5, 25, 27 and 28 V (a) time history for one period (b) response to suddenly applied voltages (c) free damped vibrations after the voltage release
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Linear equivalent of squeeze damping factor as function of displacement, Eq. (39). The dashed line corresponds to the case when the shift in the eigenfrequency is neglected.
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Response to the square wave excitation of different voltages (a) V=26 V (b) V=28 V. Dotted curve corresponds to experimental data. Model results are obtained for ζ=11%.
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Influence of the damping on the dynamic pull-in (a) normalized dynamic pull-in voltage (b) deflection of the end point of the beam at dynamic pull-in point normalized to the value for the undamped system. Dashed curve corresponds to the case of pure linear damping.
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Time history for the accurate single-mode model (curve 1) and damping dominated mass-less model (curve 2). Curve 3 corresponds to the approximate expression, Eq. (40)
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Comparison of experimental data with model results (dashed curve) near the instability point (a) time history (b) phase plot. Experimental data correspond to the measured dynamic pull-in voltage V=29.44 V. Model data are obtained for ζL=11%,V=29.44 V (curve 1) and V=30.3 V (curve 2)
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Phase portrait for step function excitation of the damped system, Eq. (25). (1) V1<VDPI (2) V=VDPI of the damped system (3) V1=VPI. The dashed curve is the separatrix orbit corresponding to the undamped system.




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