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
Grahic Jump Location
Potential energy for different voltages and zero initial conditions (1) V1=0.85VPI (2) V1=VDPI (3) V1=VPI
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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%.
Grahic Jump Location
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.
Grahic Jump Location
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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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.
Grahic Jump Location
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)
Grahic Jump Location
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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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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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.
Grahic Jump Location
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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