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Research Papers

# Fully Lagrangian Modeling of Dynamics of MEMS With Thin Beams—Part I: Undamped Vibrations

[+] Author and Article Information
Ranajay Ghosh

Department of Theoretical and Applied Mechanics, Cornell University, Thurston Hall, Ithaca, NY 14853

Subrata Mukherjee

Department of Theoretical and Applied Mechanics and Sibley School of Mechanical and Aerospace Engineering, Cornell University, Kimball Hall, Ithaca, NY 14853

Also called the membrane assumption.

J. Appl. Mech 76(5), 051007 (Jun 18, 2009) (10 pages) doi:10.1115/1.3086785 History: Received July 18, 2008; Revised January 05, 2009; Published June 18, 2009

## Abstract

Micro-electro-mechanical systems (MEMSs) often use beam or plate shaped conductors that can be very thin—with $h/L≈O(10–2–10–3)$ (in terms of the thickness $h$ and length $L$ of the beam or side of a square plate). Such MEMS devices find applications in microsensors, micro-actuators, microjets, microspeakers, and other systems where the conducting beams or plates oscillate at very high frequencies. Conventional boundary element method analysis of the electric field in a region exterior to such thin conductors can become difficult to carry out accurately and efficiently—especially since MEMS analysis requires computation of charge densities (and then surface traction) separately on the top and bottom surfaces of such beams. A new boundary integral equation has been proposed to handle the computation of charge densities for such high aspect ratio geometries. In the current work, this has been coupled with the finite element method to obtain the response behavior of devices made of such high aspect ratio structural members. This coupling of electrical and mechanical problems is carried out using a Newton scheme based on a Lagrangian description of the electrical and mechanical domains. The numerical results are presented in this paper for the dynamic behavior of the coupled MEMS without damping. The effect of gap between a beam and the ground, on mechanical response of a beam subjected to increasing electric potential, is studied carefully. Damping is considered in the companion paper (Ghosh and Mukherjee, 2009, “Fully Lagrangian Modeling of Dynamics of MEMS With Thin Beams—Part II: Damped Vibrations  ,” ASME J. Appl. Mech.76, p. 051008).

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## Figures

Figure 1

Parallel plate resonator: geometry and detail of the parallel plate fingers from Ref. 1

Figure 2

Deformable clamped beam over a fixed ground plate

Figure 3

Notation used in boundary integral equations

Figure 4

Two parallel conducting beams

Figure 5

Evaluations of angles

Figure 6

Response behavior of MEMS beam. Pull in behavior.

Figure 7

Response behavior of MEMS beam. Competing nonlinearities.

Figure 8

Vibration under a dc bias

Figure 9

Beat phenomenon for near natural frequency excitation

Figure 10

Amplitude-frequency response of a MEM beam

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