# Nonlinear Dynamic Behavior of a Bi-Axial Torsional MEMS Mirror with Sidewall Electrodes

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## Abstract

**:**

## 1. Introduction

## 2. Static Simulation

#### 2.1. Operation Principles

#### 2.2. System Model

#### 2.2.1. Electrostatic Forces and Torques

#### Sidewall Electrodes

#### Mechanical Spring Force

#### 2.3. Static Simulation Results

## 3. Dynamic Simulation

#### 3.1. Equations of Motion

#### 3.2. Dynamic Simulation Results

#### 3.3. Analytical Explanation of Secondary Resonances

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix: Integral Boundaries

## References

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**Figure 1.**(

**a**) Parametric dimensions of the structure; (

**b**) 3D model of the mirror elements and quadrants (Adapted from [19]).

**Figure 2.**Serpentine spring dimensions (μm): ${l}_{o}=10$, ${l}_{p}=110$, ${l}_{f}=120$, ${l}_{i}=120$, ${w}_{f}=160$, $w=3\phantom{\rule{3.33333pt}{0ex}}$μm (width), and ${t}_{b}=12\phantom{\rule{3.33333pt}{0ex}}$μm (thickness).

**Figure 3.**The mirror rotated about the X-axis by α and Sidewalls 12 and 22 projected on the $YZ$ plane.

**Figure 4.**The projection of the mirror plate and the gimbal and sidewall electrodes on the $YZ$ plane.

**Figure 5.**(

**a**) Static simulation of rotation angle β versus differential voltage V

_{diff}when V

_{bias}= 55 V; (

**b**) static simulation of rotation angle α versus differential voltage V

_{diff}versus differential voltage V

_{diff}when V

_{bias}= 55 V; (results presented by yellow and red colors are adapted from [19]) (

**c**) vertical displacement of the mirror when V

_{bias}= 55 V obtained using the present model.

**Figure 6.**(

**a**) Experimental and numerical comparison for the frequency response of angle alpha when ${V}_{DC}=\phantom{\rule{3.33333pt}{0ex}}55\phantom{\rule{4pt}{0ex}}\mathrm{V}$ and ${V}_{ac}=25\phantom{\rule{4pt}{0ex}}\mathrm{V}$; experimental and numerical comparison for frequency response of the angle beta when ${V}_{DC}=55\phantom{\rule{4pt}{0ex}}\mathrm{V}$ and ${V}_{ac}=15\phantom{\rule{4pt}{0ex}}\mathrm{V}$; (results presnted by solid blue and dash red lines are adpated from [19]) (

**b**) the frequency response of the mirror for vertical displacement is numerically simulated when ${V}_{DC}=55\phantom{\rule{4pt}{0ex}}\mathrm{V}$ and ${V}_{ac}=25\phantom{\rule{4pt}{0ex}}\mathrm{V}$.

**Figure 7.**(

**a**) The secondary resonance softening behavior of the beta angle when ${V}_{DC}=115\phantom{\rule{4pt}{0ex}}\mathrm{V}$ and ${V}_{ac}$ changes from 0 to 200 V; (

**b**) the secondary resonance of beta angle when ${V}_{ac}=100\phantom{\rule{4pt}{0ex}}\mathrm{V}$ and ${V}_{DC}$ changes from 0 to 200 V.

**Figure 8.**The effect of the decreased damping ratio on the nonlinear dynamic behavior of the beta angle when ${V}_{DC}=45\phantom{\rule{4pt}{0ex}}\mathrm{V}$ and ${V}_{ac}=65\phantom{\rule{4pt}{0ex}}\mathrm{V}$ for ${\zeta}_{2}=0.001$.

**Figure 9.**Phase portraits of the beta angle for ${V}_{ac}=100\phantom{\rule{4pt}{0ex}}\mathrm{V}$, ${V}_{DC}=115\phantom{\rule{4pt}{0ex}}\mathrm{V}$ and ${\zeta}_{2}=0.0208$. (

**a**) Primary resonance. These phase portraits are plotted for initial values of $x\left(0\right)=0$, $\dot{x}=0$. (

**b**) Superharmonic resonance of order two. (

**c**) Superharmonic resonance of order three.

© 2016 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ozdogan, M.; Towfighian, S.
Nonlinear Dynamic Behavior of a Bi-Axial Torsional MEMS Mirror with Sidewall Electrodes. *Micromachines* **2016**, *7*, 42.
https://doi.org/10.3390/mi7030042

**AMA Style**

Ozdogan M, Towfighian S.
Nonlinear Dynamic Behavior of a Bi-Axial Torsional MEMS Mirror with Sidewall Electrodes. *Micromachines*. 2016; 7(3):42.
https://doi.org/10.3390/mi7030042

**Chicago/Turabian Style**

Ozdogan, Mehmet, and Shahrzad Towfighian.
2016. "Nonlinear Dynamic Behavior of a Bi-Axial Torsional MEMS Mirror with Sidewall Electrodes" *Micromachines* 7, no. 3: 42.
https://doi.org/10.3390/mi7030042