# Tunable Clamped–Guided Arch Resonators Using Electrostatically Induced Axial Loads

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Design and Principle of the Tunable Microresonators

_{0}). The two flexure beams—which are used to guide the microbeams—had length of 460 μm and width of 10 μm. The width of the four folded flexure beams suspended to the mass of Type 2 was 2.5 μm. Figure 3 shows a picture of one of the fabricated resonators of Type 2. Note that the handle and box layer has been totally removed in some selected areas of the structure of Type 2; hence in Figure 3, those areas show black color due to poor reflection of light. Note also that the focus for Type 2 structure resonance frequency tuning was by means of axial tensile stress, hence the compressive electrode was intentionally removed.

## 3. Finite Element Model

## 4. Results and Discussion

_{C}and V

_{T}, applied between the large mass and the corresponding compressive/tensile electrodes. In this measurement method, we applied a sudden impulse of an electrostatic load (V

_{DC}) on the beam and the actuation electrode (fixed electrode), allowing the beam to vibrate freely (ring down) until the motion dies out. To amplify the generated V

_{DC}voltage by the Micro System Analyzer (MSA), an amplifier was used, and then the voltage was applied between the beam and the fixed electrode.

_{0}) (which decreases the resonance frequencies), and the increase in the axial stress due to the axial tensile load (which increases the resonance frequency). However, the results in Figure 5 and Figure 6 show that—for the current designs—the axial stress effect dominates the reduction in stiffness due to the decrease in curvature, and thus results in increasing the resonant frequency. This dominant effect can be attributed to the low initial curvature of the arches compared to their length.

_{DC}] − f

_{0}[V

_{DC}= 0]) × 100/f

_{0}). For the third mode (Figure 8), the results show relatively less resonant frequency shift (3%). For structure D, Figure 7a shows a remarkable increase in the first mode resonant frequency from 15 to 39 kHz (160%). For the third mode (Figure 8), it increases from 79 to 102 kHz (29%) for a maximum tuning voltage of 110 V. With the arch beams length of 600 μm (Figure 7b), the first mode resonant frequency can be tuned from 42 to 46.65 kHz (11%) for structure C by increasing the electrostatic voltage up to 65 V. For structure G, the tuning range is about (23%) from 38.7 to 47.5 kHz for 80 V. It can be inferred that for type 2 (D and G) design, the resonant frequency can be tuned more than Type 1 (A, B, and C) design, both for the first and third mode.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the resonators. (

**a**) Type 1: a fixed-guided microbeam is suspended by two flexure beams with a large mass and electrostatic actuation electrodes for compressive or tensile axial force; (

**b**) Type 2: the mass is suspended by four flexure beams.

**Figure 4.**Finite element model (FEM) static simulation results of the x–displacement of the Type 1 actuator with L = 1000 μm, h = 1.8 μm, b

_{0}= 1.8 μm, b = 50 μm, and g = 5 μm under (

**a**) a compressive load of V

_{C}= 50 V and (

**b**) a tensile load of V

_{T}= 40 V.

**Figure 5.**FEM static simulation results of the x–displacement of the Type 2 actuator with L = 1000 μm, h = 1.8 μm, b

_{0}= 1.8 μm, b = 10 μm, and g = 2 μm under a tensile load of V

_{T}= 40 V.

**Figure 6.**Experimental setup: (

**a**) schematic and (

**b**) photograph of setup. MSA: Micro System Analyzer.

**Figure 7.**Measured change in resonant frequency with tensile axial load for various designs in Table 1 of Type 1 and Type 2 for the first resonant mode. (

**a**) L = 1000 μm; (

**b**) L = 600 μm.

**Figure 8.**Measured change in resonance frequency with tensile axial load for various designs in Table 1 for Type 1 and Type 2 for the third mode of the 1000 μm length beams.

**Figure 9.**Measurements and FE simulations of the resonant frequency tuning with the tensile axial load for structure B of Type 1 at (

**a**) first mode and (

**b**) third mode.

**Figure 10.**Measurements and FE simulations of the resonant frequency tuning with the tensile axial load for structure D of Type 2 at (

**a**) first mode and (

**b**) third mode.

**Figure 11.**Measurements of the change in resonant frequency of the first mode with bi-directional axial loads for Type 1. (

**a**) Structure A; (

**b**) Structure C.

Type 1 | L (μm) | h (μm) | b_{0} (μm) | b (μm) | g (μm) | f_{01} (kHz) |

A | 1000 | 1.85 | 1.8 | 50 | 5 | 14.74 |

B | 1000 | 1.85 | 2.6 | 50 | 5 | 14.87 |

C | 600 | 1.85 | 1.8 | 50 | 2 | 42.7@30V |

Type 2 | L (μm) | h (μm) | b_{0} (μm | b (μm) | g (μm) | f_{01} (kHz) |

D | 1000 | 1.85 | 2.6 | 20 | 2 | 15 |

E | 1000 | 1.85 | 2.6 | 10 | 2 | 14.2 |

F | 1000 | 1.85 | 2.6 | 10 | 5 | 14.5 |

G | 600 | 1.85 | 2.6 | 10 | 2 | 38.7 |

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**MDPI and ACS Style**

Alcheikh, N.; Ramini, A.; Hafiz, M.A.A.; Younis, M.I.
Tunable Clamped–Guided Arch Resonators Using Electrostatically Induced Axial Loads. *Micromachines* **2017**, *8*, 14.
https://doi.org/10.3390/mi8010014

**AMA Style**

Alcheikh N, Ramini A, Hafiz MAA, Younis MI.
Tunable Clamped–Guided Arch Resonators Using Electrostatically Induced Axial Loads. *Micromachines*. 2017; 8(1):14.
https://doi.org/10.3390/mi8010014

**Chicago/Turabian Style**

Alcheikh, Nouha, Abdallah Ramini, Md Abdullah Al Hafiz, and Mohammad I. Younis.
2017. "Tunable Clamped–Guided Arch Resonators Using Electrostatically Induced Axial Loads" *Micromachines* 8, no. 1: 14.
https://doi.org/10.3390/mi8010014