# Structural Behavior of a Multi-Layer Based Microbeam Actuator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{DC}) in Volt. For the double microbeam configuration, we consider dielectric layers for each electric field.

- For the single-microbeam based actuator:$$\frac{{d}^{4}w}{d{x}^{4}}={\alpha}_{1}\mathrm{\Gamma}\frac{{d}^{2}w}{d{x}^{2}}+\frac{{\alpha}_{2}{V}_{DC}{}^{2}}{{\left(1-w\right)}^{2}}$$$$w(0)=0,\text{}w(1)=0,\text{}\frac{dw}{dx}(0)=0,\text{}\frac{dw}{dx}(1)=0,$$
- For the double-microbeams based actuator:$$\{\begin{array}{l}\text{Lower Microbeam}\to \frac{{d}^{4}{w}_{1}}{d{x}^{4}}={\alpha}_{3}{\mathrm{\Gamma}}_{1}\frac{{d}^{2}{w}_{1}}{d{x}^{2}}+\frac{{\alpha}_{4}{V}_{DC}{}^{2}}{{\left(1-{w}_{1}\right)}^{2}}-\frac{{\alpha}_{4}{V}_{DC}{}^{2}}{{\left({d}_{2}/{d}_{1}+{w}_{1}-{w}_{2}\right)}^{2}}\\ \text{Upper Microbeam}\to \frac{{d}^{4}{w}_{2}}{d{x}^{4}}={\alpha}_{3}{\mathrm{\Gamma}}_{2}\frac{{d}^{2}{w}_{2}}{d{x}^{2}}+\frac{{\alpha}_{4}{V}_{DC}{}^{2}}{{\left({d}_{2}/{d}_{1}+{w}_{1}-{w}_{2}\right)}^{2}}\end{array}$$$$\{\begin{array}{l}{w}_{1}(0)=0,\text{\hspace{1em}}{w}_{1}(1)=0,\text{\hspace{1em}}\frac{d{w}_{1}}{dx}(0)=0,\text{\hspace{1em}}\frac{d{w}_{1}}{dx}(1)=0,\\ {w}_{2}(0)=0,\text{\hspace{1em}}{w}_{2}(1)=0,\text{\hspace{1em}}\frac{d{w}_{2}}{dx}(0)=0,\text{\hspace{1em}}\frac{d{w}_{2}}{dx}(1)=0,\end{array}$$

## 3. Reduced-Order Model (ROM)

_{i}, f

_{i}, and g

_{i}are time-independent constants and ${\varphi}_{i}\left(x\right)$ are trial functions assumed to be the linear mode shapes of a clamped-clamped beam. To solve for the time-independent unknowns constants k

_{i}, f

_{i}, and g

_{i}, it is essential to substitute Equation (10) into Equations (6)–(9), then multiply the outcome by ${\varphi}_{j}\left(x\right)$, and finally integrate the outcome from x = 0 to x = 1 while using the orthogonality of the mode shapes functions of a clamped-clamped beam. Following the previous procedure, we get the following reduced-order modeling equations bot both assumed actuators as follows:

- For the single-microbeam based actuator:$$\begin{array}{l}{\displaystyle {\int}_{x=0}^{1}{\varphi}_{j}(x){\displaystyle \sum _{i=1}^{N}{k}_{i}{\varphi}_{i}^{iv}(x)}dx}={\displaystyle {\int}_{x=0}^{1}{\varphi}_{j}(x)\frac{{\alpha}_{2}{V}_{DC}^{2}}{{\left(1-{\displaystyle \sum _{i=1}^{N}{k}_{i}{\varphi}_{i}(x)}\right)}^{2}}}dx\\ +{\displaystyle {\int}_{x=0}^{1}{\varphi}_{j}(x)\left({\alpha}_{1}{{\displaystyle {\int}_{0}^{1}\left({\displaystyle \sum _{i=1}^{N}{k}_{i}{\varphi}_{i}^{\prime}(x)}\right)}}^{2}dx\right){\displaystyle \sum _{i=1}^{N}{k}_{i}{\varphi}_{i}^{\u2033}(x)}}dx\end{array}$$
- For the double-microbeams based actuator:$$\begin{array}{l}\text{Lower Microbeam}\to {\displaystyle {\int}_{x=0}^{1}{\varphi}_{j}(x){\displaystyle \sum _{i=1}^{N}{f}_{i}{\varphi}_{i}^{iv}(x)}dx}={\displaystyle {\int}_{x=0}^{1}{\varphi}_{j}\left(\left({\alpha}_{3}{{\displaystyle {\int}_{0}^{1}\left({\displaystyle \sum _{i=1}^{N}{f}_{i}{\varphi}_{i}^{\prime}(x)}\right)}}^{2}dx\right){\displaystyle \sum _{i=1}^{N}{f}_{i}{\varphi}_{i}^{\u2033}(x)}\right)}dx+\\ {\alpha}_{4}{V}_{DC}^{2}\left({\displaystyle {\int}_{x=0}^{1}\frac{{\varphi}_{j}(x)}{{\left(1-{\displaystyle \sum _{i=1}^{N}{f}_{i}{\varphi}_{i}(x)}\right)}^{2}}dx}-{\displaystyle {\int}_{x=0}^{1}\frac{{\varphi}_{j}(x)}{{\left({d}_{2}/{d}_{1}+{\displaystyle \sum _{i=1}^{N}{f}_{i}{\varphi}_{i}(x)}-{\displaystyle \sum _{i=1}^{N}{g}_{i}{\varphi}_{i}(x)}\right)}^{2}}dx}\right)\end{array}$$$$\begin{array}{l}\text{Upper Microbeam}\to {\displaystyle {\int}_{x=0}^{1}{\varphi}_{j}(x){\displaystyle \sum _{i=1}^{N}{g}_{i}{\varphi}_{i}^{iv}(x)}dx}={\alpha}_{3}{\displaystyle {\int}_{x=0}^{1}{\varphi}_{j}\left(\left({{\displaystyle {\int}_{0}^{1}\left({\displaystyle \sum _{i=1}^{N}{g}_{i}{\varphi}_{i}^{\prime}(x)}\right)}}^{2}dx\right){\displaystyle \sum _{i=1}^{N}{g}_{i}{\varphi}_{i}^{\u2033}(x)}\right)}dx+\\ +{\alpha}_{4}{V}_{DC}^{2}{\displaystyle {\int}_{x=0}^{1}\left(\frac{{\varphi}_{j}(x)}{{\left({d}_{2}/{d}_{1}+{\displaystyle \sum _{i=1}^{N}{f}_{i}{\varphi}_{i}(x)}-{\displaystyle \sum _{i=1}^{N}{g}_{i}{\varphi}_{i}(x)}\right)}^{2}}\right)}dx\end{array}$$

## 4. Static Analysis

_{i}in Equation (11), whereas for double microbeam we solve for f

_{i}, and g

_{i}in the coupled Equation (12a,b), and this can be done by several methods like the harmonic balance method coupled with the asymptotic numerical method [33,34], which enables the capture of stable and unstable branches or using the so-called Newton’s method. We adopted the latter approach by using the command FindRoot in Mathematica software. Then increase the number of assumed modes in the ROM by one. The previous steps are to be repeated until the maximum deflection variation with V

_{DC}for both microbeams is converging. The maximum deflection of the upper microbeam is presented versus the applied DC voltage in Figure 2 for the case study of Table 1. It can be noted from the graph that when the number of modes is increased the solution is varying slightly, until convergence is reached at almost three modes in the ROM.

- Case 1: d
_{1}= d_{2}(where both initial gaps are set equal to 1.25 μm) - Case 2: d
_{1}> d_{2}(where d_{1}= 1.5 μm and d_{2}= 1.0 μm) - Case 3: d
_{1}< d_{2}(where d_{1}= 1.0 μm and d_{2}= 1.5 μm)

#### 4.1. Case 1 (d_{1} = d_{2})

#### 4.2. Case 2 (d_{1} > d_{2})

#### 4.3. Case 3 (d_{1} < d_{2})

_{1}< d

_{2}there is no such tendency. This strange behavior may be because the lower microbeam in this case is very close to the fixed electrode, and so the distance between the two microbeams will be higher. This makes the force between the two microbeams lower, and hence the upper microbeam will be far from the pull-in instability allowing it to still vibrate safely.

## 5. Natural Frequencies and Mode Shapes

_{i}, f

_{i}and g

_{i}are the time-independents unknown constants that were calculated in the static analysis part and β(t), μ

_{i}(t) and ν

_{i}(t) are time-dependent unknown functions. Details about the derivation of the single-microbeam based actuator can found in [25]. We will focus next on the derivation of the equations governing the eigenvalue problem of the double-microbeams based actuator. Subsequently, and since we are in the process of developing the linear eigenvalue problem, the nonlinear electrostatic force terms in the derived ROM equations are to be linearized using the so-called Taylor series expansion. Therefore, neglecting all of the higher order terms to obtain:

_{1}= d

_{2}. The natural frequencies were obtained by using three symmetric modes and were displayed versus the applied voltage as shown in Figure 8. The obtained results indicate that all of the higher-order natural frequencies are insensitive with the applied voltage, with the exception of the fundamental one (lower frequency). The fundamental natural frequency starts at its maximum position (when no electrical load is applied) and then decreases gradually with an increase in the applied voltage until it gets close to the pull-in voltage, at which point it drops sharply to zero. In addition, it can be noted that each odd natural frequency, when paired with the consecutive one are the same, except for the fundamental frequency, especially at higher applied DC voltages.

_{1}> d

_{2}). Moreover, the other two cases reach the pull-in voltage at about the same value (23 Volt) and accordingly their fundamental natural frequency drops to zero or close to it.

_{1}= d

_{2}at V

_{DC}= 2 Volt are presented in Figure 10 and Figure 11, respectively. The results show that for the first mode shape Ф

_{1}the coupled modes (Ф

_{11}and Ф

_{12}) are opposite to each other, sharing an out-of-phase motion. Conversely, for the second mode shape (Ф

_{2}) the two coupled modes (Ф

_{21}and Ф

_{22}) have almost the same magnitude as the first in an absolute value and both of them share an in-phase-motion.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of a parallel-plates based DC electrostatic actuator assuming: (

**a**) a single-microbeam; (

**b**) a double-microbeams configuration.

**Figure 2.**Convergence of the upper microbeam maximum static deflection assuming the case study of Table 1.

**Figure 3.**Variation of the maximum static deflection with the DC voltage for the lower and upper microbeams and for the case when d

_{1}= d

_{2}.

**Figure 4.**Variation of the maximum static deflection with the DC voltage for the lower and upper microbeams and for the case when d

_{1}> d

_{2}.

**Figure 5.**Variation of the maximum static deflection with the DC voltage for the lower and upper microbeams and for the case when d

_{1}< d

_{2}.

**Figure 6.**Comparison between the variations of the upper microbeam maximum static deflections with the DC voltage for all three cases with the single-microbeam case.

**Figure 7.**Comparison between the static profiles of (

**a**) the single-microbeam configuration at V

_{DC}= 23 Volt with the double microbeam configuration just before the pull-in voltage for the case of: (

**b**) d

_{1}= d

_{2}; (

**c**) d

_{1}> d

_{2}; (

**d**) d

_{1}< d

_{2}(Note: the length (L) in the above sketches is not to the scale).

**Figure 8.**Variation of the first six natural frequencies with the applied DC voltage for the case of the double-microbeams based actuator and for d

_{1}= d

_{2}.

**Figure 9.**Variation of the fundamental natural frequencies with the applied DC voltage for the case of the double-microbeams based actuator and for three different cases of the microbeams initial air gaps.

Parameter | Value | Parameter | Value |
---|---|---|---|

Beam Length (L) | 150 µm | Effective young‘s modulus (E) | 124 GPa |

Beam thickness (h) | 1.0 µm | Density (ρ) | 2332 kg/m^{3} |

Beam width (b) | 4.0 µm | Air gap depth (d_{1} and d_{2}) | 1.25 µm |

**Table 2.**Comparison of pull-in voltage of all cases shown in Figure 7.

Assumed Structure | Pull-in Voltage | |
---|---|---|

Double-microbeams | Case d_{1} = d_{2} | 23 Volt |

Case d_{1} > d_{2} | 15 Volt | |

Case d_{1} < d_{2} | 23 Volt | |

Single-microbeam | 236 Volt |

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**MDPI and ACS Style**

Alofi, A.; Ouakad, H.M.; Tausiff, M.
Structural Behavior of a Multi-Layer Based Microbeam Actuator. *Actuators* **2016**, *5*, 22.
https://doi.org/10.3390/act5030022

**AMA Style**

Alofi A, Ouakad HM, Tausiff M.
Structural Behavior of a Multi-Layer Based Microbeam Actuator. *Actuators*. 2016; 5(3):22.
https://doi.org/10.3390/act5030022

**Chicago/Turabian Style**

Alofi, Abdulrahman, Hassen M. Ouakad, and Mohammad Tausiff.
2016. "Structural Behavior of a Multi-Layer Based Microbeam Actuator" *Actuators* 5, no. 3: 22.
https://doi.org/10.3390/act5030022