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Research Papers

# Fully Lagrangian Modeling of Dynamics of MEMS With Thin Beams—Part II: Damped 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

J. Appl. Mech 76(5), 051008 (Jun 18, 2009) (9 pages) doi:10.1115/1.3086786 History: Received July 18, 2008; Revised January 05, 2009; Published June 18, 2009

## Abstract

Micro-electro-mechanical systems (MEMS) often use beam or plate shaped conductors that are very thin with $h/L≈O(10−2–10−3)$ (in terms of the thickness $h$ and length $L$ of a beam or side of a square plate). A companion paper (Ghosh and Mukherjee, 2009, “Fully Lagrangian Modeling of Dynamics of MEMS With Thin Beams—Part I: Undamped Vibrations,” ASME J. Appl. Mech., 76, p. 051007) addresses the coupled electromechanical problem of MEMS devices composed of thin beams. A new boundary element method (BEM) is coupled with the finite element method (FEM) by Ghosh and Mukherjee, and undamped vibrations are addressed there. The effect of damping due to the surrounding fluid modeled as Stokes flow is included in the present paper. Here, the elastic field modeled by the FEM is coupled with the applied electric field and the fluid field, both modeled by the BEM. As for the electric field, the BEM is adapted to efficiently handle narrow gaps between thin beams for the Stokes flow problem. The coupling of the various fields is carried out using a Newton scheme based on a Lagrangian description of the various domains. Numerical results are presented for damped vibrations of MEMS beams.

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## Figures

Figure 1

Two parallel beams in a surrounding Stokes fluid

Figure 2

Deformable clamped beam over a fixed ground plate

Figure 3

Horizontal traction on plates for plane Couette flow

Figure 4

Vertical traction on plates moving vertically toward each other

Figure 5

Horizontal traction on plates moving vertically toward each other

Figure 6

Compressibility influence on convergence

Figure 7

Damped response for a dc bias of 0.5 V

Figure 8

Damped vibrations for various Poisson parameters (dc bias=0.5 V)

Figure 9

Damped vibration for ac 0.5 cos(0.5ΩNatt)V

Figure 10

Damped vibration for ac+dc bias of 0.5+0.05 cos(0.5ΩNatt)V

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