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Research Papers

Modeling of Initially Curved Beam Structures for Design of Multistable MEMS

[+] Author and Article Information
Matthew D. Williams

 Interdisciplinary Microsystems Group, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250, mdwilli@ufl.edu

Fred van Keulen

Structural Optimization and Computational Mechanics Group,  Department of Precision and Microsystems Engineering, Faculty of Mech., Maritime, & Material Eng., Delft University of Technology, 2628 CD Delft, The Netherlands, a.vankeulen@tudelft.nl

Mark Sheplak

Interdisciplinary Microsystems Group,  Department of Mech. and Aero. Engineering, University of Florida, Gainesville, FL 32611-6250, sheplak@ufl.edu

J. Appl. Mech. 79(1), 011006 (Nov 14, 2011) (11 pages) doi:10.1115/1.4004711 History: Received July 21, 2010; Revised April 18, 2011; Posted July 28, 2011; Published November 14, 2011; Online November 14, 2011

This article describes a mathematical model and two solution methodologies for efficiently predicting the equilibrium paths of an arbitrarily shaped, precurved, clamped beam. Such structures are common among multistable microelectromechanical systems (MEMS). First, a novel polynomial-based solution approach enables simultaneous solution of all equilibrium configurations associated with an arbitrary mechanical loading pattern. Second, the normal flow algorithm is used to negotiate the particularly complex nonlinear equilibrium paths associated with electrostatic loading and is shown to perform exceptionally well. Overall, the techniques presented herein provide designers with general and efficient computational frameworks for studying the effects of loading, shape, and imperfections on beam behavior. Sample problems motivated from switch and actuator applications in the literature demonstrate the methodologies’ utility in predicting the nonlinear equilibrium paths for structures of practical importance.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

A clamped, precurved beam under arbitrary mechanical loading and electrostatic loading

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Figure 2

Example of the normal flow algorithm in which the corrector iterations converge to the equilibrium path R  = 0 normal to Davidenko flow lines

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Figure 3

A precurved beam in a microscale switch application

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Figure 4

Equilibrium paths (polynomial-based approach and normal flow - - -) in terms of buckling mode amplitudes for a perfect beam (b1  = 8.66) under central point load

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Figure 5

Equilibrium paths (polynomial-based approach and normal flow - - -) of an imperfect beam nominally in first buckling mode shape (b1  = 8.66): (a) with no imperfection, b2  = 0; (b) with small asymmetric imperfection, b2  = 0.05; (c) with large asymmetric imperfection, b2  = 0.5

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Figure 6

(a) Equilibrium paths of a perfect beam in first buckling mode shape (b1  = 8.66) under central point load compared with FEA results for imperfect variants. (b) Zoomed in, with associated beam shapes are found in the inset.

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Figure 7

Equilibrium paths of a perfect beam in first-buckling mode shape (b1  = 8.66) under constant distributed loading compared with FEA: (a) equilibrium path; (b) zoomed in

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Figure 8

Equilibrium paths for a curved beam actuated electrostatically (g0  = 6)

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