# The Effect of the Anisotropy of Single Crystal Silicon on the Frequency Split of Vibrating Ring Gyroscopes

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## Abstract

**:**

## 1. Introduction

## 2. Anisotropic Elastic Properties of SCS

**R**to describe the transformation from the default coordinate system. Matrices

**R**s of coordinate systems of (100) and (111) wafers are presented as

## 3. Working Principle and Modal Simulation of VRG

#### 3.1. Working Principle

#### 3.2. Modal Simulation

**Anisotropic**option and the

**Voigt**option are chosen in the lists of the

**Solid model**section and the

**Material data ordering**section, respectively, and the matrix in Equation (5) is typed in the table of

**Elasticity matrix**in the

**Settings**window for

**Linear Elastic Material**in the interface of COMSOL. When it comes to the (111) wafer, the procedure is the same, but the matrix is changed to Equation (6).

#### 3.2.1. Width of Circular Ring

#### 3.2.2. Radius of Circular Ring

## 4. Fabrication and Test

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Coordinate systems and overhead views of the (100) and (111) silicon wafers: (

**a**) Coordinate systems of two wafers, (

**b**) Overhead views of two wafers.

**Figure 3.**Resonant frequencies and frequency splits of rings with different widths: (

**a**) (100) ring, (

**b**) (111) ring.

**Figure 4.**Resonant frequencies and frequency splits of rings with different radii: (

**a**) (100) ring, (

**b**) (111) ring.

**Figure 6.**The fabrication process of VRG: (

**a**) Photo-etching, (

**b**) Bonding area etching, (

**c**) Metal deposition, (

**d**) Anodic bonding, (

**e**) Thinning and polishing, (

**f**) Dry etching and structure release.

**Figure 8.**Second harmonic frequency data of four (100) VRG samples: (

**a**) #1 VRG, (

**b**) #2 VRG, (

**c**) #3 VRG, (

**d**) #4 VRG.

**Figure 9.**Second harmonic frequency data of four (111) VRG samples: (

**a**) #1′ VRG, (

**b**) #2′ VRG, (

**c**) #3′ VRG, (

**d**) #4′ VRG.

Orthotropic Elasticity Values | (100) Wafer | (111) Wafer |
---|---|---|

${E}_{x},{E}_{y},{E}_{z}$ (GPa) | 169, 169, 130 | 169, 169, 188 |

${\upsilon}_{yz},{\upsilon}_{zx},{\upsilon}_{xy}$ | 0.36, 0.28, 0.064 | 0.16, 0.18, 0.26 |

${G}_{yz},{G}_{zx},{G}_{xy}$ (GPa) | 79.4, 79.4, 50.9 | 57.8, 57.8, 66.9 |

Structure Parameter | Value ($\mathsf{\mu}$m) |
---|---|

Radius of Ring | 3000 |

Width of Ring | 80 |

Radius of Anchor | 750 |

Width of Support Beam | 20 |

Thickness of Ring | 120 |

(100) VRG | (111) VRG | |||||||
---|---|---|---|---|---|---|---|---|

$\#\mathbf{1}$ | $\#\mathbf{2}$ | $\#\mathbf{3}$ | $\#\mathbf{4}$ | $\#{\mathbf{1}}^{\prime}$ | $\#{\mathbf{2}}^{\prime}$ | $\#{\mathbf{3}}^{\prime}$ | $\#{\mathbf{4}}^{\prime}$ | |

${f}_{1}$ (Hz) | 9871.1 | 9865.7 | 9685.4 | 9913.2 | 10,018.4 | 10,096.2 | 9917.7 | 10,183.4 |

${f}_{2}$ (Hz) | 9451.3 | 9453.6 | 9290.5 | 9461.7 | 10,016.3 | 10,091.5 | 9923.4 | 10,172.3 |

$|\Delta f|$ (Hz) | 419.8 | 412.1 | 394.9 | 451.5 | 2.1 | 4.7 | 5.7 | 11.1 |

$|\overline{\Delta f}|$ (Hz) | 419.6 | 5.9 | ||||||

$\sigma \left(\right|\Delta f\left|\right)$ (Hz) | 23.7 | 3.8 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Qin, Z.; Gao, Y.; Jia, J.; Ding, X.; Huang, L.; Li, H.
The Effect of the Anisotropy of Single Crystal Silicon on the Frequency Split of Vibrating Ring Gyroscopes. *Micromachines* **2019**, *10*, 126.
https://doi.org/10.3390/mi10020126

**AMA Style**

Qin Z, Gao Y, Jia J, Ding X, Huang L, Li H.
The Effect of the Anisotropy of Single Crystal Silicon on the Frequency Split of Vibrating Ring Gyroscopes. *Micromachines*. 2019; 10(2):126.
https://doi.org/10.3390/mi10020126

**Chicago/Turabian Style**

Qin, Zhengcheng, Yang Gao, Jia Jia, Xukai Ding, Libin Huang, and Hongsheng Li.
2019. "The Effect of the Anisotropy of Single Crystal Silicon on the Frequency Split of Vibrating Ring Gyroscopes" *Micromachines* 10, no. 2: 126.
https://doi.org/10.3390/mi10020126