# Design and Thermal Stability Analysis of Swing Micro-Mirror Structure for Gravitational Wave Observatory in Space

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Design Process of Swing Micro-Mirror Structure

- Pointing jitter is better than 10 nrad/√Hz;
- The first-order natural frequency is greater than 80 Hz.

#### 2.2. Structure Design of the Swing Micro-Mirror Structure

#### 2.2.1. Mechanical Structure Design of the Mechanism

#### 2.2.2. Material Selection of the Mechanism

_{6}Al

_{4}V is chosen as the material of flexible hinges. For other parts, Titanium Alloy with high yield strength and low density is selected as their material. The properties of different materials are shown in Table 2.

#### 2.2.3. Lightweight Design of the Mechanism

#### 2.3. The Index of Stability Analysis of the Swing Micro-Mirror Structure

#### 2.3.1. Research on Fitting Algorithm of Mirror Shape

#### 2.3.2. Research on Integrated Algorithm of Mirror Angle Deviation

#### 2.3.3. Research on the Solving Method of ASD of Mirror Angle Deviation

## 3. Finite Element Analysis of the Swing Micro-Mirror Structure

#### 3.1. Modal Analysis

#### 3.1.1. Theory of Modal Analysis

#### 3.1.2. Results of Modal Analysis

#### 3.2. Thermal Stability Analysis

#### 3.2.1. Theory of Thermal Analysis

#### 3.2.2. Establishment of the Model of Thermal Stability Analysis

#### 3.2.3. Finite Element Analysis Results of Thermal Stability

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**3D model of the assembly of the swing micro-mirror structure: (

**a**) right view; (

**b**) main view; (

**c**) left view; (

**d**) upper view.

**Figure 2.**3D model of the parts of the swing micro-mirror structure: (

**a**) framework; (

**b**) mirror cell; (

**c**) mirror fixed plate; (

**d**) mirror; (

**e**) flexible hinge A; (

**f**) flexible hinge B; (

**g**) flexible arm; (

**h**) actuator connecting plate; (

**i**) actuator keep plate; (

**j**) actuator; (

**k**) supporting foot A; (

**l**) supporting foot B.

**Figure 5.**Amplitude spectral density (ASD) corresponding to four temperature variation functions: (

**a**) 0.03 °C; (

**b**) 0.05 °C; (

**c**) 0.1 °C; (

**d**) 0.5 °C.

**Figure 8.**Amplitude spectral density function in three heating directions: (

**a**) Heating on the inner surface perpendicular to the X-axis; (

**b**) Heating on the inner surface perpendicular to the Y-axis; (

**c**) Heating on the inner surface perpendicular to the Z-axis.

Materials | Parts |
---|---|

Invar | framework, mirror cell, supporting foot A, supporting foot B |

Titanium Alloy | mirror fixed plate, flexible arm, actuator connecting plate, actuator keep plate |

Ti_{6}Al_{4}V | flexible hinge A, flexible hinge B |

Zerodur | mirror |

Material | Density (kg/m^{−3}) | Coefficient of Thermal Expansion (c ^{−1}) | Young’s Modulus (MPa) | Poisson’s Ratio | Tensile Yield Strength (MPa) | Compressive Yield Strength (MPa) |
---|---|---|---|---|---|---|

Invar | 8050 | 5.00 × 10^{−7} | 141,000 | 0.29 | 274 | 274 |

Titanium Alloy | 4620 | 9.40 × 10^{−6} | 96,000 | 0.36 | 930 | 930 |

Ti_{6}Al_{4}V | 4540 | 7.89 × 10^{−6} | 110,000 | 0.34 | 860 | 970 |

Zerodur | 2530 | 1.00 × 10^{−7} | 90,300 | 0.24 | 358.90 | 500.83 |

Model | Weight (kg) | Size mm $\mathbf{Length}\text{}\times \text{}\mathbf{Width}\text{}\times \text{}\mathbf{Height}$ |
---|---|---|

Initial | $6.33$ | $170\text{}\times \text{}143\text{}\times \text{}102$ |

Final | $2.05$ | $160\text{}\times \text{}128\text{}\times \text{}94$ |

Mode No. | Frequency (Hz) |
---|---|

1 | 247.55 |

2 | 452.90 |

3 | 890.04 |

4 | 1066.00 |

5 | 1267.20 |

6 | 1381.90 |

No. | Initial Temperature (°C) | Cycle (s) | Temperature Variation (°C) | Functions |
---|---|---|---|---|

1 | 22 | 120 | 0.03 | $T=22+0.03\text{}\times \text{}\mathrm{sin}\left(3t\right)$ |

2 | 22 | 120 | 0.05 | $T=22+0.05\text{}\times \text{}\mathrm{sin}\left(3t\right)$ |

3 | 22 | 120 | 0.1 | $T=22+0.1\text{}\times \text{}\mathrm{sin}\left(3t\right)$ |

4 | 22 | 120 | 0.5 | $T=22+0.5\text{}\times \text{}\mathrm{sin}\left(3t\right)$ |

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**MDPI and ACS Style**

Zheng, K.; Xu, M.
Design and Thermal Stability Analysis of Swing Micro-Mirror Structure for Gravitational Wave Observatory in Space. *Machines* **2021**, *9*, 104.
https://doi.org/10.3390/machines9050104

**AMA Style**

Zheng K, Xu M.
Design and Thermal Stability Analysis of Swing Micro-Mirror Structure for Gravitational Wave Observatory in Space. *Machines*. 2021; 9(5):104.
https://doi.org/10.3390/machines9050104

**Chicago/Turabian Style**

Zheng, Kunyao, and Mingming Xu.
2021. "Design and Thermal Stability Analysis of Swing Micro-Mirror Structure for Gravitational Wave Observatory in Space" *Machines* 9, no. 5: 104.
https://doi.org/10.3390/machines9050104