# Static and Dynamic Analysis of Electrostatically Actuated MEMS Shallow Arches for Various Air-Gap Configurations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{DC}is assumed to be applied between the micro-beam and the stationary electrode resulting into an electrostatic attractive force, which pushes the flexible micro-beam towards the stationary one. Figure 1 portrays four stationary electrodes actuation arrangements offering four different ways of designs and informative comparisons. The first design with flexible and stationary electrodes are separated with non-uniform air-gap (Case 1: lower actuation, Figure 1a, and Case 2: upper actuation, Figure 2b), and the second design has both flexible curved micro-beam and its actuating electrodes separated with a uniform air-gap (Case 1: lower actuation, Figure 2a, and Case 2: upper actuation, Figure 2b). It is worth noting here that for Case 2 in both arrangements, as the DC load increases the movable electrode stroke increases until reaching the pull-in instability. In contrary, in the Case 1, the COSINE-shaped micro-beam might undergo both the snap-through and pull-in instabilities together or directly undergoes the pull-in instability.

^{3}/12 are, respectively, its cross-sectional area and second moment of inertia. Several equations can be assumed to designate the initial curved profile of the investigated micro-beam [29]. Here, we assume the below COSINE-shaped expression:

_{i}. This results in a system of nonlinear algebraic equations in terms of those coefficients, which is then solved numerically using the Newton-Raphson method as it is simple to be implemented and relatively less computationally expensive. Each equilibrium point can be stable or unstable and this query of stability is very important for any dynamical systems [2,36]. The entire static curve is computed by conducting a sweep through its constitutive equilibrium points [36] but before proceeding further, it should be noted that the stability analysis in the below would be a local one because the original nonlinear system will be linearized to compute the eigenvalue problem [2]. Formerly, the eigenvalue problem is solved to get the proper microbeam natural frequencies and this through numerically computing the eigenvalues of the Jacobian matrix of the generated ROM. For this, a Jacobian matrix-based eigenvalue problem is first constructed from the ROM equations to compute the in-plane flexure natural frequencies of the micro-beam at a given DC voltage after substitution of the equilibrium solutions into the Jacobian matrix [2].

## 3. Results and Discussion

## 4. Static Analysis

## 5. Eigenvalue Problem Analysis

_{0}, are displayed in Figure 7 and Figure 8 for the upper and lower actuation arrangements, respectively. Figure 7 shows that for all cases of b

_{0}, the natural frequencies decrease and the fundamental one drops to zero when snap-through and/or pull-in instability occurs, Figure 7a–c. For higher b

_{0}(3 and 4 µm cases), a decreasing trend is noted for the three frequencies and with only a unique drop in the fundamental when the micro-beam reaches the snap-through instability. For the case of b

_{0}= 2 µm, the micro-beam undergoes both snap-through and pull-in instability with two drops to zero in its lowest fundamental frequency, and with a decreasing trend before snap-through and increasing trend after the snap-through.

_{0}cases. The fundamental frequency, Figure 8a shows an increase for a wider range DC voltage and then drops to zero only near the pull-in state. In addition, as b

_{0}is increased, the increase of the fundamental frequency is showing a little decline, therefore less frequency tunability in the three lowest natural frequencies, Figure 8a–c.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Rebeiz, G.M. RF MEMS: Theory, Design, and Technology; John Wiley & Sons: Hoboken, NJ, USA, 2004. [Google Scholar]
- Younis, M.I. MEMS Linear and Nonlinear Statics and Dynamics; Springer Science and Business Media LLC: Berlin, Germany, 2011. [Google Scholar]
- Yang, S.; Xu, Q. A review on actuation and sensing techniques for MEMS-based microgrippers. J. Micro-Bio Robot.
**2017**, 13, 1–14. [Google Scholar] [CrossRef] - Pimpin, A.; Charoenbunyarit, I.; Srituravanich, W. Material and performance characterization of Z-shaped nickel electrothermal micro-actuators. Sens. Actuators A Phys.
**2017**, 253, 49–58. [Google Scholar] [CrossRef] - Jia, X.L.; Yang, J.; Kitipornchai, S.; Lim, C.W. Pull-in instability and free vibration of electrically actuated poly-SiGe graded micro-beams with a curved ground electrode. Appl. Math. Model.
**2012**, 36, 1875–1884. [Google Scholar] [CrossRef] - Derakhshani, M.; Berfield, T.A. Snap-Through and Mechanical Strain Analysis of a MEMS Bistable Vibration Energy Harvester. Shock. Vib.
**2019**, 2019, 1–10. [Google Scholar] [CrossRef] [Green Version] - Guan, C.; Zhu, Y. An electrothermal microactuator with Z-shaped beams. J. Micromech. Microeng.
**2010**, 20, 085014. [Google Scholar] [CrossRef] - Tajalli, S.A.; Tajalli, S.M. Wavelet based damage identification and dynamic pull-in instability analysis of electrostatically actuated coupled domain microsystems using generalized differential quadrature method. Mech. Syst. Signal Process.
**2019**, 133, 106256. [Google Scholar] [CrossRef] - Lin, W.-H.; Zhao, Y.-P. Pull-in Instability of Micro-switch Actuators: Model Review. Int. J. Nonlinear Sci. Numer. Simul.
**2008**, 9, 175–184. [Google Scholar] [CrossRef] [Green Version] - Song, Y.-H.; Yoon, J.-B. Micro and Nanoelectromechanical Contact Switches for Logic, Memory, and Power Applications. In Nano Devices and Circuit Techniques for Low-Energy Applications and Energy Harvesting; KAIST Research Series; Springer Science and Business Media LLC: Berlin, Germany, 2015; pp. 65–117. [Google Scholar]
- Hung, E.; Senturia, S. Extending the travel range of analog-tuned electrostatic actuators. J. Microelectromech. Syst.
**1999**, 8, 497–505. [Google Scholar] [CrossRef] - Moghimi Zand, M.; Rashidian, B.; Ahmadian, M.T. Contact time study of electrostatically actuated microsystems. Sci. Iran. Trans. B Mech. Eng.
**2010**, 17, 348–357. [Google Scholar] - Ouakad, H.M.; Al-Qahtani, H.M.; Hawwa, M.A. Influence of squeeze-film damping on the dynamic behavior of a curved micro-beam. Adv. Mech. Eng.
**2016**, 8, 1687814016650120. [Google Scholar] [CrossRef] [Green Version] - Najar, F.; Ghommem, M.; Abdel-Rahman, E. Arch microbeam bifurcation gas sensors. Nonlinear Dyn.
**2021**, 104, 923–940. [Google Scholar] [CrossRef] - Alneamy, A.M.; Heppler, G.R.; Abdel-Rahman, E.M.; Khater, M.E. On Design and Analysis of Electrostatic Arch Micro-Tweezers. J. Vib. Acoust.
**2020**, 143, 1–23. [Google Scholar] [CrossRef] - Goll, C.; Bacher, W.; Büstgens, B.; Maas, D.; Menz, W.; Schomburg, W.K. Microvalves with bistable buckled polymer diaphragms. J. Micromech. Microeng.
**1996**, 6, 77–79. [Google Scholar] [CrossRef] - Kim, C.; Marsland, R.; Blick, R.H. The Nanomechanical Bit. Small
**2020**, 16, e2001580. [Google Scholar] [CrossRef] [PubMed] - Cao, T.; Hu, T.; Zhao, Y. Research Status and Development Trend of MEMS Switches: A Review. Micromachines
**2020**, 11, 694. [Google Scholar] [CrossRef] - Hu, N.; Burgueño, R. Buckling-induced smart applications: Recent advances and trends. Smart Mater. Struct.
**2015**, 24, 063001. [Google Scholar] [CrossRef] - Zhang, Y.; Wang, Y.; Li, Z.; Huang, Y.; Li, D. Snap-Through and Pull-In Instabilities of an Arch-Shaped Beam Under an Electrostatic Loading. J. Microelectromech. Syst.
**2007**, 16, 684–693. [Google Scholar] [CrossRef] [Green Version] - Das, K.; Batra, R.C. Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems. J. Micromech. Microeng.
**2009**, 19, 035008. [Google Scholar] [CrossRef] - Zand, M.M. The Dynamic Pull-In Instability and Snap-Through Behavior of Initially Curved Microbeams. Mech. Adv. Mater. Struct.
**2012**, 19, 485–491. [Google Scholar] [CrossRef] - Hu, Y.; Yang, J.; Kitipornchai, S. Snap-through and pull-in analysis of an electro-dynamically actuated curved micro-beam using a nonlinear beam model. J. Sound Vib.
**2013**, 332, 3821–3832. [Google Scholar] [CrossRef] - Chen, X.; Meguid, S. On the parameters which govern the symmetric snap-through buckling behavior of an initially curved microbeam. Int. J. Solids Struct.
**2015**, 66, 77–87. [Google Scholar] [CrossRef] - Li, L.; Chew, Z.J. Microactuators: Design and technology. In Smart Sensors and MEMS; Woodhead Publishing: Sawston, UK, 2018; pp. 315–354. [Google Scholar]
- Ouakad, H.M.; Younis, M.I. The dynamic behavior of MEMS arch resonators actuated electrically. Int. J. Nonlinear Mech.
**2010**, 45, 704–713. [Google Scholar] [CrossRef] - Hajjaj, A.Z.; Alcheikh, N.; Ramini, A.; Al Hafiz, M.A.; Younis, M.I. Highly Tunable Electrothermally and Electrostatically Actuated Resonators. J. Microelectromech. Syst.
**2016**, 25, 440–449. [Google Scholar] [CrossRef] [Green Version] - Ben Hassena, M.A.; Samaali, H.; Ouakad, H.M.; Najar, F. 2D electrostatic energy harvesting device using a single shallow arched microbeam. Int. J. Nonlinear Mech.
**2021**, 132, 103700. [Google Scholar] [CrossRef] - Krylov, S.; Ilic, B.R.; Schreiber, D.; Seretensky, S.; Craighead, H. The pull-inbehavior of electrostatically actuated bistable microstructures. J. Micromech. Microeng.
**2008**, 18, 055026–055046. [Google Scholar] [CrossRef] - Shen, X.; Chen, X. Mechanical Performance of a Cascaded V-Shaped Electrothermal Actuator. Int. J. Adv. Robot. Syst.
**2013**, 10, 379. [Google Scholar] [CrossRef] - Enikov, E.T.; Kedar, S.S.; Lazarov, K. Analytical model for analysis and design of V-shaped thermal microactuators. J. Microelectromech. Syst.
**2005**, 14, 788–798. [Google Scholar] [CrossRef] - Alcheikh, N.; Ramini, A.; Al Hafiz, M.A.; Younis, M.I. Tunable Clamped–Guided Arch Resonators Using Electrostatically Induced Axial Loads. Micromachines
**2017**, 8, 14. [Google Scholar] [CrossRef] [Green Version] - Medina, L.; Gilat, R.; Krylov, S. Enhanced Efficiency of Electrostatically Actuated Bistable Microswitches Using Bow-Like Operation. IEEE/ASME Trans. Mechatron.
**2020**, 25, 2409–2415. [Google Scholar] [CrossRef] - Younis, M.; Abdel-Rahman, E.; Nayfeh, A.; Younis, M.; Abdel-Rahman, E.; Nayfeh, A. A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst.
**2003**, 12, 672–680. [Google Scholar] [CrossRef] - Ouakad, H.M. Electrostatic fringing-fields effects on the structural behavior of MEMS shallow arches. Microsyst. Technol.
**2018**, 24, 1391–1399. [Google Scholar] [CrossRef] - Simitses, G.J.; Hodges, D.H. Fundamentals of Structural Stability; Elsevier BV: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Medina, L.; Gilat, R.; Ilic, B.R.; Krylov, S. Experimental dynamic trapping of electrostatically actuated bistable micro-beams. Appl. Phys. Lett.
**2016**, 108, 073503. [Google Scholar] [CrossRef] [PubMed] - Das, K.; Batra, R.C. Symmetry breaking, snap-through and pull-in instabilities under dynamic loading of microelectromechanical shallow arches. Smart Mater. Struct.
**2009**, 18, 115008. [Google Scholar] [CrossRef] [Green Version] - Medina, L.; Gilat, R.; Krylov, S. Symmetry breaking in an initially curved micro beam loaded by a distributed electrostatic force. Int. J. Solids Struct.
**2012**, 49, 1864–1876. [Google Scholar] [CrossRef] [Green Version] - Slocum, J.; Kamrin, K.; Slocum, A. A Buckling Flexure-Based Force-Limiting Mechanism. J. Mech. Robot.
**2019**, 11, 041004-29. [Google Scholar] [CrossRef]

**Figure 1.**Schematics of electrostatically actuated in-plane COSINE-shaped resonators. (

**a**) Case 1 and (

**b**) Case 2: the micro-beams and the stationary electrodes are separated with non-uniform air-gap (g). (

**c**) Case 1 and (

**d**) Case 2: the micro-beams and the electrodes are separated with a uniform g.

**Figure 2.**The static displacement of micro-beams with a uniform and a non-uniform g under DC electrostatic actuation (V

_{DC}) for (

**a**) Case 1 and (

**b**) Case 2 (solid: stable, dashed: unstable).

**Figure 3.**The static displacement of the micro-beam of Table 1 with a uniform g under V

_{DC}for various values of initial rise (b

_{0}) for (

**a**) Case 1 and (

**b**) Case 2 (solid: stable, dashed: unstable).

**Figure 4.**The static displacement with a uniform g and for various values of length (L) under V

_{DC}for (

**a**) Case 1 and (

**b**) Case 2 (solid: stable, dashed: unstable).

**Figure 5.**The variation of the (

**a**) first, (

**b**) second, (

**c**) third, and (

**d**) fourth in-plane flexure natural frequencies for Case 1 as varying V

_{DC}and for uniform and non-uniform air-gaps (g).

**Figure 6.**The variation of the (

**a**) first, (

**b**) second, (

**c**) third, and (

**d**) fourth in-plane flexure natural frequencies for Case 2 as varying V

_{DC}and for uniform and non-uniform g.

**Figure 7.**The variation of the (

**a**) first, (

**b**) second, and (

**c**) third in-plane flexure natural frequencies for Case 1 as varying V

_{DC}with uniform g and for various values of b

_{0}.

**Figure 8.**The variation of the (

**a**) first, (

**b**) second, and (

**c**) third in-plane flexure natural frequencies for Case 2 as varying V

_{DC}with uniform g and for various values of b

_{0}.

**Figure 9.**The variation of the (

**a**) first, (

**b**) second, and (

**c**) third in-plane flexure natural frequencies for Case 1 as varying V

_{DC}with uniform g and for various values of L.

**Figure 10.**The variation of the (

**a**) first, (

**b**) second, and (

**c**) third in-plane flexure natural frequencies for Case 2 as varying V

_{DC}with uniform g and for various values of L.

**Table 1.**Cosine-shaped micro-beam (material and geometrical properties of doped single-Crystal Silicon) for types 1 and 2 of Figure 1.

Material Properties | Value | Geometrical Properties | Value |
---|---|---|---|

Mass density $\rho $ (kg/m^{3}) | 2332 | Thickness h (µm) | 2 |

Effective Young’s modulus $E$ (GPa) | 154 | Width b (µm) | 25 |

Poisson’s ratio | 0.28 | Length L (µm) | 800 |

Initial rise b_{0} (µm) | 3 | ||

Vertical air-gap g (µm) | 8 |

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Ouakad, H.M.; Alcheikh, N.; Younis, M.I.
Static and Dynamic Analysis of Electrostatically Actuated MEMS Shallow Arches for Various Air-Gap Configurations. *Micromachines* **2021**, *12*, 930.
https://doi.org/10.3390/mi12080930

**AMA Style**

Ouakad HM, Alcheikh N, Younis MI.
Static and Dynamic Analysis of Electrostatically Actuated MEMS Shallow Arches for Various Air-Gap Configurations. *Micromachines*. 2021; 12(8):930.
https://doi.org/10.3390/mi12080930

**Chicago/Turabian Style**

Ouakad, Hassen M., Nouha Alcheikh, and Mohammad I. Younis.
2021. "Static and Dynamic Analysis of Electrostatically Actuated MEMS Shallow Arches for Various Air-Gap Configurations" *Micromachines* 12, no. 8: 930.
https://doi.org/10.3390/mi12080930