An approximate analytical solution to the pull-in voltage of a micro bridge with an elastic boundary

We establish the deflection functions of electrostatically actuated micro beams by an approximate finite element method (AFEM), in which the beam and the electrostatic load are discretized. The beam is replaced with a series of beam elements by traditional FEM. Using the total differential of the distributed electrostatic force, we represent such a force with the nodal forces on the beam elements. We calculate the deformation curve of the beam by gradually loading voltage in small increments, and pull-in behavior is identified when the convergence of the deflection iteration cannot be achieved after voltage increment. The proposed method considers the effect of deformation on stiffness by establishing a new equivalent stiffness matrix for each voltage step based on of the results of previous steps. Through this approach, we prevent the approximate errors of the stiffness matrix from accumulating. The AFEM results on micro beams with different geometries indicate good agreement with those obtained by other studies and those derived using commercial FE software.

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“…Thus,

\overset{\approx}{E}

is equal to E /(1‐ λ 2 ), where E and λ are the Young's modulus and Poisson's ratio of the beam material, respectively. When b < 5 h ,

\overset{\approx}{E}

is simply equal to E .…”

Section: Methodsmentioning

confidence: 99%

Modeling the pull‐in behavior of electrostatically actuated micro beams by an approximate finite element method

Zhou

Shen

Guo

et al. 2013

Int J Numerical Modelling

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“…This difference of the deflection due to the anticlastic curvature may affect the results of the extracted residual stress, and it will be discussed in Section V-A. The anchor supports in surface-micromachined MEMS beams cannot be assumed ideally rigid, and additionally, these supports are not always vertical as assumed in most beam models [27], [36]- [38]. Instead, they can be inclined as illustrated in Fig.…”

Section: A Comprehensive Reduced-order Models and Characterization Omentioning

confidence: 99%

“…17(a). Using a boundary torsion stiffness calculated for vertical supports in [38], the mean value of extracted residual stresses for 87 beams increases to 9.6 MPa ( Fig. 17(b)).…”

Section: Uncertainties Due To Non-ideal Effects Modelingmentioning

confidence: 99%

Residual Stress Extraction of Surface-Micromachined Fixed-Fixed Nickel Beams Using a Wafer-Scale Technique

Zeng

Kovacs

Garg

et al. 2015

J. Microelectromech. Syst.

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This paper reports on the extraction of residual stress in surface-micromachined nickel thin films of electrostatically actuated fixed-fixed beams using a wafer-scale technique. The distribution of residual stress for 87 beams on a 4-in quarter wafer piece is presented. The residual stress (σ 0 ) is determined from the best fit of the displacement-voltage curves predicted by a computationally efficient model to the experimental data. The nondestructive and automated measurements are taken at room temperature and directly at the beam itself without any additional test structures. The model employed incorporates the nonideal effects of inclined supports, nonflat initial beam profiles, and fringing fields. The extracted residual stress values vary between −12.8 and 13.6 MPa (negative values are for compressive stresses and positive ones for tensile stresses). The residual stresses for these 87 beams follow a nearly normal distribution with a mean value of −1.7 MPa and a standard deviation of 5.9 MPa, which represents the variability of the residual stresses across the wafer. Detailed uncertainty analysis has been conducted, and it reveals that inaccurate modeling of the nonideal effects will result in significant errors in the extracted residual stress. Although demonstrated on nickel thin films, this technique can be applied to other metallic thin films.[ 2014-0344]Index Terms-Microelectromechanical systems (MEMS), residual stress, fixed-fixed beams, fringing fields, inclined supports, non-flat beam profiles, reduced-order numerical model, displacement-voltage measurements, wafer-scale techniques.

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“…For example, one can utilize the electrostatic deformations of a microbeam or a diaphragm to achieve sensing or actuation purposes. [1][2][3] Moreover, the pull-in voltage of micro test structures can be used to extract the material properties such as Young's modulus and residual stress. 4,5 The design of micromachined electrostatic devices is complex due to the electromechanical coupling effect.…”

Section: Introductionmentioning

confidence: 99%

“…A thorough literature survey in the related topics can be referred to our previous work. 3 Among the aforementioned analytical models, only few published studies 8,9 consider the initial curling of the beam induced by stress gradient, while the rest are meant for straight beams. Hu and Wei 10 further improved the previous model 8 by considering the fringing field capacitance and showed superior accuracy.…”

Section: Introductionmentioning

confidence: 99%

Electromechanical behavior of the curled cantilever beam

Chuang

Lee

et al. 2009

J. Micro/Nanolith. MEMS MOEMS

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We propose an approximate analytical solution to the pull-in voltage of a microcurled cantilever beam. The analytical model considers the realistic situations, which include stress gradient, nonideal boundary conditions, and fringing field capacitance. The proposed analytical model can be used at wafer level for extracting the Young's modulus of the thin film of which the cantilever beam is made. The approximate analytical solution is obtained based on the Euler's beam model and the minimum energy method. The accuracy of the proposed model is verified to be more accurate than the other published models. The model presented in this work can be used for wafer-level evaluation of the material properties through simple electrical testing and is also expected to find use in the design of microelectromechanical devices.

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An approximate analytical solution to the pull-in voltage of a micro bridge with an elastic boundary

Cited by 18 publications

References 25 publications

Modeling the pull‐in behavior of electrostatically actuated micro beams by an approximate finite element method

Modeling the pull‐in behavior of electrostatically actuated micro beams by an approximate finite element method

Residual Stress Extraction of Surface-Micromachined Fixed-Fixed Nickel Beams Using a Wafer-Scale Technique

Electromechanical behavior of the curled cantilever beam

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