Application of the Generalized Differential Quadrature Method to the Study of Pull-In Phenomena of MEMS Switches

Sadeghian, Hamed; Rezazadeh, Ghader; Osterberg, P.M.

doi:10.1109/jmems.2007.909237

Cited by 132 publications

(70 citation statements)

References 25 publications

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“…The relative errors of results obtained from ALM are comparable to those in [17] and [16], for all cases of finite element approximations. In fact, it appears …”

Section: Numerical Testssupporting

confidence: 53%

Augmented Lagrangian Methods for Numerical Solutions to Higher Order Differential Equations

Li¹

2017

JAMP

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A large number of problems in engineering can be formulated as the optimization of certain functionals. In this paper, we present an algorithm that uses the augmented Lagrangian methods for finding numerical solutions to engineering problems. These engineering problems are described by differential equations with boundary values and are formulated as optimization of some functionals. The algorithm achieves its simplicity and versatility by choosing linear equality relations recursively for the augmented Lagrangian associated with an optimization problem. We demonstrate the formulation of an optimization functional for a 4th order nonlinear differential equation with boundary values. We also derive the associated augmented Lagrangian for this 4th order differential equation. Numerical test results are included that match up with well-established experimental outcomes. These numerical results indicate that the new algorithm is fully capable of producing accurate and stable solutions to differential equations.

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“…The relative errors of results obtained from ALM are comparable to those in [17] and [16], for all cases of finite element approximations. In fact, it appears …”

Section: Numerical Testssupporting

confidence: 53%

Augmented Lagrangian Methods for Numerical Solutions to Higher Order Differential Equations

Li¹

2017

JAMP

View full text Add to dashboard Cite

show abstract

“…For this, the generalized differential quadrature ͑GDQ͒ algorithm was employed to solve the nonlinear differential equation extracted from the variational calculus of energy. 17 By solving Eq. ͑1͒, the instability point or the pull-in voltage can be determined.…”

Section: ẽ Tmentioning

confidence: 99%

“…The pull-in behavior depends on the interaction of the electrostatic load ͑generated by the applied voltage͒, the stiffness of the cantilever and the geometry. 17 Through careful fabrication and precise measurement, the high stiffness sensitivity of the pull-in voltage enables a confident Ẽ measurement.…”

mentioning

confidence: 99%

Characterizing size-dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull-in instability

Sadeghian

Yang

Goosen

et al. 2009

Applied Physics Letters

142

103

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This letter presents the application of electrostatic pull-in instability to study the size-dependent effective Young's Modulus Ẽ ͑ϳ170-70 GPa͒ of ͓110͔ silicon nanocantilevers ͑thickness ϳ1019-40 nm͒. The presented approach shows substantial advantages over the previous methods used for characterization of nanoelectromechanical systems behaviors. The Ẽ is retrieved from the pull-in voltage of the structure via the electromechanical coupled equation, with a typical error of Յ12%, much less than previous work in the field. Measurement results show a strong size-dependence of Ẽ . The approach is simple and reproducible for various dimensions and can be extended to the characterization of nanobeams and nanowires.

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“…The equations are simplifications of the general case given in [11,12] and assume there is no residual stress present in the beams. In addition, the equations are only valid in the small deflection regime (linear elastic mechanics) which is valid for d/w ≤ 1 [15]. The beam equations adapted from [13] and [14] are given in Equation 3 (cantilever) and Equation 4 (fixed-fixed).…”

Section: Verificationmentioning

confidence: 99%

Reducing Pull-In Voltage by Adjusting Gap Shape in Electrostatically Actuated Cantilever and Fixed-Fixed Beams

et al. 2010

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A gap with variable geometry is presented for both cantilever beam and fixed-fixed beam actuators as a method to reduce the pull-in voltage while maintaining a required displacement. The method is applicable to beams oriented either in a plane parallel to or perpendicular to a substrate, but is most suitable for vertically oriented (lateral) beams fabricated with a high aspect ratio process where variable gap geometry can be implemented directly in the layout. Finite element simulations are used to determine the pull-in voltages of these modified structures. The simulator is verified against theoretical pull-in voltage equations as well as previously published finite element simulations. By simply varying the gap in a linear fashion the pull-in voltage can be reduced by 37.2% in the cantilever beam case and 29.6% in the fixed-fixed beam case over a structure with a constant gap. This can be reduced a further 4.8% by using a polynomial gap shape (n = 4/3) for the cantilever beam and 1.2% for the fixed-fixed beam by flattening the bottom of the linearly varying gap.

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Application of the Generalized Differential Quadrature Method to the Study of Pull-In Phenomena of MEMS Switches

Cited by 132 publications

References 25 publications

Augmented Lagrangian Methods for Numerical Solutions to Higher Order Differential Equations

Augmented Lagrangian Methods for Numerical Solutions to Higher Order Differential Equations

Characterizing size-dependent effective elastic modulus of silicon nanocantilevers using electrostatic pull-in instability

Reducing Pull-In Voltage by Adjusting Gap Shape in Electrostatically Actuated Cantilever and Fixed-Fixed Beams

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