# Design and Structure Optimization of Arresting Gear Based on Magnetorheological Damper

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Structure and Numerical Modeling of MR Dampers

#### 2.1. Parametric Modeling of MR Dampers

#### 2.1.1. Parametric Modeling of Single-coil MR Damper

_{p}is the piston diameter, d

_{p}is the piston rod diameter, σ

_{p}is the width of flow path, L is the piston length, v

_{p}is the velocity of piston, η

_{p}is the apparent viscosity of MR fluid, and τ

_{py}is the shear yield strength of MR fluid.

_{py}in Formulas (1) and (2)) does not act on the whole length of the flow path. It is not accurate to calculate the Coulomb damping force generated by the liquid shear with L. So, L is split into L

_{p}and l

_{p}. The damping force and its composition are determined by the following formulas:

_{Zp}is the damping force, F

_{Vp}is the viscous damping force, F

_{Cp}is the Coulomb damping force, l

_{p}is the effective length of magnetic pole and L

_{p}is the effective length of flow path.

_{p}is the total flux in magnetic path; R

_{m}

_{0}R

_{m}

_{1}R

_{m}

_{2}are the piston reluctance, the air gap reluctance and the outer cylinder reluctance, respectively; l

_{pp}l

_{pc}are the average length of the magnetic circuit of the piston and the outer cylinder, respectively; μ

_{1}μ

_{2}μ

_{3}are the permeability of the piston, the MR fluid and the outer cylinder, respectively; and S

_{p}S

_{σ}S

_{c}are the magnetic circuit cross-sectional area of the piston, the flow path and the outer cylinder, respectively.

#### 2.1.2. Parametric Modeling of Double-Coil MR Damper

_{y}

_{1}is the shear yield strength of MR fluids in flow path l

_{1}and flow path l

_{2}under the impact of magnetic field, and τ

_{y}

_{3}is the shear yield strength of MR fluid in flow path l

_{3}under the impact of magnetic field.

_{1}= Φ

_{2}= 0.5Φ

_{3}. Then, the following formula can be obtained:

_{1}~R

_{4}are the piston reluctance; R

_{σ}R

_{σ2}are the air gap reluctance; R

_{5}is the outer cylinder reluctance; l

_{pp}l

_{pc}are the average length of the magnetic circuit of the piston and the outer cylinder, respectively; μ

_{1}μ

_{2}μ

_{3}are the permeability of the piston, the MR fluid and the outer cylinder, respectively; and S

_{p}S

_{σ}S

_{c}are the magnetic circuit cross-sectional area of the piston, the flow path and the outer cylinder, respectively.

## 3. Optimization Process

## 4. Introduction of Optimization Model

#### 4.1. Determine the Structure Parameters to Be Optimized

_{p}

_{min}(D

_{min}); piston rod diameter, d

_{p}(d); width of flow path, σ

_{p}(σ); effective length of magnetic pole, l

_{p}(l); effective length of piston, L

_{p}(L); and coil turns, N

_{p}(N). The number of coil turns is limited by its structural parameters. The depth of the coil embedded in the piston is r

_{p}

_{min}(r

_{min}), and the length is l

_{p}

_{min}(l

_{min}). The structural parameters to be optimized were obtained, as shown in Figure 6.

#### 4.2. Finite Element Simulation Model of Magnetic Field of Damper

#### 4.3. Boundary and Genetic Algorithm Optimization Model

_{p}= L

_{p}− l

_{p}

_{min}, and the effective length of the magnetic pole of the double-coil piston is l

_{2}= L

_{2}− 2l

_{2min}.

#### 4.4. UAV Arresting Dynamics Simulation Model

## 5. Optimization Results and Analysis

^{3}, and the double-coil piston volume is 0.333 m

^{3}. The double-coil piston volume is 28% larger than the single-coil piston volume. When the piston velocity is 1.8 m/s, the damping force variation range of the single-coil piston is 1,769,801 N, and that of the double-coil piston is 1,950,654 N, which is a 10.2% increase over the single-coil one.

^{2}. The adaptability of the MR damper and arresting system is verified by changing the mass of the UAV. The peaks of the UAV acceleration before and after structural optimization are shown in Table 3. The value of the parameters before optimization is the value that maximizes the damping force.

^{2}to 32.5 m/s

^{2}in the early stage of the arresting process, decreases to 15 m/s

^{2}within 0.8 s and then maintains a slow rise until the UAV is stopped. For a 6000 kg UAV, the peak arresting acceleration decreases to 24.8 m/s

^{2}, and then it drops to 19 m/s

^{2}and remains at this value until the UAV is stopped. As can be seen from Figure 10b and Figure 11b it takes 1.9 s for the arresting gear to stop the UAV of 4000 kg and 2.0 s for the UAV of 6000 kg. As shown in Figure 10c and Figure 11c, the arresting gear stops the 4000 kg UAV at 35.1 m, and the 6000 kg UAV needs 41.9 m to stop. Therefore, different coil arrangements have no significant impact.

^{2}to 15.0 m/s

^{2}. The double-coil damper has no obvious inflection point, and there is a small but not obvious decline after 2.1 s. The single-coil arresting gear takes 2.2 s to stop the UAV within 45.5 m. The double-coil arresting gear’s stopping time is 2.1 s, 4.5% lower than that of the single-coil one, and the arresting distance is 44.5 m, which is 2.2% shorter. When the UAV mass is 10,000 kg, the single-coil inflection point appears at 1.4 s, and the double-coil inflection point appears at 1.7 s, i.e., 21% delayed. It takes 2.4 s for the single-coil arresting gear to stop the UAV at 45.5 m. I confirm. The double-coil arresting gear takes 46.5 m to stop the UAV, so the distance is shortened by 2.2% compared with that of the single-coil arresting gear. The arresting time is 2.2 s, making it 9.1% shortened.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Parameters that need to be optimized: (

**a**) parameters to be optimized for single-coil damper and (

**b**) parameters to be optimized for double-coil damper.

**Figure 10.**Arresting performance comparison of 4000 kg UAV: (

**a**) UAV acceleration comparison, (

**b**) UAV velocity comparison and (

**c**) UAV displacement comparison.

**Figure 11.**Arresting performance comparison of 6000 kg UAV: (

**a**) UAV acceleration comparison, (

**b**) UAV velocity comparison and (

**c**) UAV displacement comparison.

**Figure 12.**Arresting performance comparison of 8000 kg UAV: (

**a**) UAV acceleration comparison, (

**b**) UAV velocity comparison and (

**c**) UAV displacement comparison.

**Figure 13.**Arresting performance comparison of 10,000 kg UAV: (

**a**) UAV acceleration comparison, (

**b**) UAV velocity comparison and (

**c**) UAV displacement comparison.

**Figure 15.**Acceleration simulation results: (

**a**) 4000 kg UAV acceleration comparison, (

**b**) 6000 kg UAV velocity comparison, (

**c**) 8000 kg UAV displacement comparison and (

**d**) 10,000 kg UAV displacement comparison.

Parameter | Symbol | Lower Limit | Upper Limit | Unit |
---|---|---|---|---|

Piston diameter | D_{p}(D) | 0.7 | 1.0 | m |

Depth of coil embedded in the piston | r_{p}_{min}(r_{min}) | 0.10 | 0.20 | m |

Width of flow path | σ_{p}(σ) | 0.006 | 0.015 | m |

Coil length of the single-coil damper | l_{p}_{min} | 0.20 | 0.40 | m |

Coil length of the double-coil damper | l_{min} | 0.10 | 0.20 | m |

Piston−length−to−diameter ratio | L_{p}/D_{p} | 0.6 | 1.0 |

Parameter | Single Coil | Double Coil | ||
---|---|---|---|---|

Value | Unit | Value | Unit | |

Piston radius | 0.389 | m | 0.376 | m |

Coil depth | 0.244 | m | 0.198 | m |

Width of flow path | 0.0149 | m | 0.0149 | m |

Coil length | 0.200 | m | 0.100 | m |

Ratio of piston length to diameter | 0.877 | 0.999 | ||

Strength of magnetic field | 1.757 | T | Middle 2.405 | T |

Sides 1.3 | ||||

Damping force | 3,869,462 | N | 4,050,563 | N |

Viscous damping force | 2,099,661 | N | 2,099,909 | N |

UAV Mass | Before Optimization | After Optimization | ||
---|---|---|---|---|

4000 kg (single-coil damper) | −40.5 | m/s^{2} | −32.5 | m/s^{2} |

6000 kg (single-coil damper) | −31.1 | m/s^{2} | −24.8 | m/s^{2} |

8000 kg (single-coil damper) | −28.8 | m/s^{2} | −21.0 | m/s^{2} |

10,000 kg (single-coil damper) | −26.8 | m/s^{2} | −20.0 | m/s^{2} |

4000 kg (double-coil damper) | −41.8 | m/s^{2} | −32.5 | m/s^{2} |

6000 kg (double-coil damper) | −32.6 | m/s^{2} | −24.8 | m/s^{2} |

8000 kg (double-coil damper) | −29.2 | m/s^{2} | −20.5 | m/s^{2} |

10,000 kg (double-coil damper) | −26.8 | m/s^{2} | −19.9 | m/s^{2} |

UAV Mass | Displacement | Arresting Time | Acceleration | |||
---|---|---|---|---|---|---|

MR | Hydraulic | MR | Hydraulic | MR | Hydraulic | |

4000 kg | 35.1 m | 31.8 m | 1.9 s | 1.7 s | −32.5 m/s^{2} | −37.4 m/s^{2} |

6000 kg | 41.9 m | 39.5 m | 2.0 s | 2.1 s | −24.8 m/s^{2} | −29.0 m/s^{2} |

8000 kg | 44.5 m | 45.4 m | 2.1 s | 2.4 s | −20.5 m/s^{2} | −24.1 m/s^{2} |

10,000 kg | 46.5 m | 50.1 m | 2.2 s | 2.6 s | −19.9 m/s^{2} | −20.8 m/s^{2} |

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**MDPI and ACS Style**

Hao, J.; Wang, Y.; Peng, Y.; Ma, H.; Wei, X.
Design and Structure Optimization of Arresting Gear Based on Magnetorheological Damper. *Aerospace* **2023**, *10*, 1019.
https://doi.org/10.3390/aerospace10121019

**AMA Style**

Hao J, Wang Y, Peng Y, Ma H, Wei X.
Design and Structure Optimization of Arresting Gear Based on Magnetorheological Damper. *Aerospace*. 2023; 10(12):1019.
https://doi.org/10.3390/aerospace10121019

**Chicago/Turabian Style**

Hao, Jiayu, Yifeng Wang, Yiming Peng, Hui Ma, and Xiaohui Wei.
2023. "Design and Structure Optimization of Arresting Gear Based on Magnetorheological Damper" *Aerospace* 10, no. 12: 1019.
https://doi.org/10.3390/aerospace10121019