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Refinements to the study of electrostatic deflections: theory and experiment

Published online by Cambridge University Press:  14 December 2012

N. D. BRUBAKER ,
J. I. SIDDIQUE ,
E. SABO ,
R. DEATON  and
J. A. PELESKO
N. D. BRUBAKER
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA email: brubaker@math.udel.edu, esabo@udel.edu, rdeaton@udel.edu, pelesko@math.udel.edu
J. I. SIDDIQUE
Affiliation:
Department of Mathematics, Pennsylvania State University, York Campus, 1031 Edgecomb Avenue, York, PA 17403, USA email: ji15@psu.edu
E. SABO
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA email: brubaker@math.udel.edu, esabo@udel.edu, rdeaton@udel.edu, pelesko@math.udel.edu
R. DEATON
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA email: brubaker@math.udel.edu, esabo@udel.edu, rdeaton@udel.edu, pelesko@math.udel.edu
J. A. PELESKO
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA email: brubaker@math.udel.edu, esabo@udel.edu, rdeaton@udel.edu, pelesko@math.udel.edu
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Abstract

To study electrostatic actuation, researchers commonly use a setup proposed by G. I. Taylor in [Proc. R. Soc. Lond. Ser. A, 306 (1968), pp. 423–434]. It consists of soap film held at a distance h above a rigid plate so that when a voltage difference is applied between the two components, the top film deflects towards the bottom plate. The most striking feature of this system is when the voltage difference exceeds a critical value V*, the electrostatic forces dominate the surface forces and the soap film gets ‘pulled-into’ or collapses onto the bottom plate. This so-called ‘pull-in’ instability is a ubiquitous feature of electrostatic actuation and as a result, has been the subject of many studies. Recently, Siddique et al. [J. Electrostatics, 69 (2011), pp. 1–6] measured the value of V* as a function of the separation distance and found that the standard prediction breaks down as h increases. Here, we continue the work done in [N. D. Brubaker and J. A. Pelesko, European J. Appl. Math., 22 (2011), pp. 455–470] by investigating the cause of this discrepancy. Specifically, we model the effect of gravity on the generalized version of Taylor's model and study whether it provides the proper correction to the predicted value of V*. In doing so, we derive two nonlinear eigenvalue value problems and investigate their solutions sets.

Keywords

Electrostatic actuationPull-in voltageNonlinear elliptic PDEPrescribed mean curvatureNonlinear eigenvalue
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 3 , June 2013 , pp. 343 - 370
Copyright
Copyright © Cambridge University Press 2012

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