# Parameter Analysis and Experiment Validation of Deployment Characteristics of a Rectangular Tether-Net

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model of the Rectangular Tether-Net

#### 2.1. Overview of the Rectangular Tether-Net

#### 2.2. Model Description

#### 2.3. Establishment of the Tether-Net Dynamic Model

_{ij}of the element can be calculated as follows:

_{ij}is the deformed length of truss element H

_{ij}, parameter ${l}_{ij}^{0}$ is the initial length of the truss element and k

_{ij}is the equivalent stiffness of the truss element, which depends on the material of the net tether (Aramid III, in this study). The equivalent stiffness (k

_{ij}) is calculated as

_{ij}is the cross-sectional area of the tether.

#### 2.4. Contact between Net Tethers

_{A}and Ω

_{B}are the current configurations of two tethers. B

_{A}and B

_{B}are the boundary surfaces. ${x}_{p}^{t}$ is the coordinate of a specified point P on boundary surface B

_{A}at time t. dt is used to represent the distance between point P and point Q (Figure 6) on boundary surface B

_{B}, which can be expressed as

_{A}, the formula needs to be modified as follows [24]:

**n**of ${d}_{n}^{t}$ represents the distance along the normal direction, ${\mathit{n}}_{Q}^{t}$ is the unit normal vector at point Q on boundary surface B

_{B}at time t, ${d}_{n}^{t}>0$ indicates the separation of P point and surface B

_{B}, and $=0$ indicates that point P is in contact with surface B

_{B}during the collision of tethers. Given that Equation (4) should be valid for random points on the contact surface, the requirement of the non-penetration condition is given as

#### 2.5. Solution of the Tether-Net Dynamic Model

**M**is the mass matrix of the node, ${\mathit{P}}_{(\mathit{t})}$ is the external force applied to the node and ${\mathit{I}}_{(\mathit{t})}$ is the internal force of the element. The acceleration is explicitly integrated over time with the central difference method to solve the velocity and displacement through the following equations:

_{e}is the cross-sectional area of the element.

_{e}is the minimum element size, E is the elastic modulus and ρ is the material density. In the simulation model, the smallest element was on the side tether, and the value of $\Delta {t}_{\mathrm{stable}}$ was calculated to be 0.017 s. According to Equation (18), time step $\Delta t$ was set to 0.01 s.

#### 2.6. Elastic Moduli of Net Tethers

_{ij}) of the two different types of net tethers were calculated to be 0.196 mm

^{2}and 3.14 mm

^{2}, respectively; ${l}_{ij}^{0}$ indicates the initial length of the tether (100 mm) in the tensile experiments.

#### 2.7. Interior Ballistics of the Tether-Net System

## 3. Experimental Validation

#### 3.1. Prototype Design

#### 3.2. Comparison of Simulations and Experiments

_{i}. The calculation result was 3.55%, which is acceptable.

## 4. Parameter Analysis of the Rectangular Tether-Net

#### 4.1. Deployment Performance of the Rectangular Tether-Net

^{2}. Therefore, the shape-preserving distance was defined as the range of launch distance where the deployment area was over 102.4 m

^{2}. Once the dynamic model was established, the simulation of the rectangular tether-net system in the space environment was carried out. The configurations of the tether-net during deployment with the initial parameters are shown in Figure 14, the curves reflecting the deployment performance of the tether-net are shown in Figure 15 and the deployment parameters of the tether-net are shown in Table 3.

^{2}at the launch distance of 42.8 m, and the maximum expansion rate of the tether-net reached 95.6%. The deployment distance was 42.8 m around 1.8 s after launching, and the range of the shape-preserving distance was 38.5~48.9 m. Additionally, the deployment area gradually increased from 0.0 m

^{2}to 122.3 m

^{2}at first and then decreased to 1.5 m

^{2}due to the rebound of the tether-net. Therefore, with the current imposed parameters, the optimal distance between the launcher and the target should be selected as 42.8 m. Additionally, towing blocks 1, 5, 6 and 10 moved at the highest speed, followed by towing blocks 3 and 8, while towing blocks 2, 4, 7 and 9 were the slowest ones. This was due to the different weight of the tether-net driven by each towing block, with the tether-net driven by towing blocks 1, 5, 6 and 10 being the lightest.

#### 4.2. Influence of the Mass of Towing Blocks

^{2}to 128.7 m

^{2}. The maximum deployment lengths of the long and short sides increased from 16 m and 7.2 m to 16.6 m and 7.8 m, respectively. In addition, with the increase in towing block mass, the deployment distance and the shape-preserving distance were maintained around 42.4 m and in the range of 38.7~48.9 m, respectively. Based on the above results, it was concluded that as the mass of the towing blocks increased, better deployment performance of the tether-net was achieved. However, the propellant mass increased if the same launch speed was maintained, and this increased the chamber pressure and recoil force.

#### 4.3. Influence of the Launch Speed of Towing Blocks

^{2}to 135 m

^{2}. The maximum deployment lengths of the long and short sides increased from 13.8 m and 5.8 m to 17 m and 8 m, respectively. In addition, as shown in Figure 19 d, the range of the shape-preserving distance was reduced significantly with the decrease in launch speed. As the launch speed decreased from 40 m/s to 30 m/s, the range was reduced from 37.8~52.8 m to 38.5~48.9 m. When the launch speed was 20 m/s, the tether-net showed the lowest deployment area, with the maximum value not even reaching the area limit of 102.4 m

^{2}throughout the whole deployment process. According to the above analysis, the deployment performance of the tether-net was better at the highest launch speed; however, the recoil impulse increased from 3.3 Ns to 6.5 Ns with the increase in launch speed from 20 m/s to 40 m/s based on Equation (22), which further caused an increase in structural weight.

## 5. Conclusions

^{2}, and the maximum expansion rate was 95.6%. The deployment distance was 42.8 m, and the shape-preserving distance ranged from 38.5 m to 48.9 m.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Capture process of rectangular tether-net, (

**a**) t = 1 s, (

**b**) t = 2 s, (

**c**) t = 3 s, (

**d**) t = 4 s.

**Figure 12.**Comparison of tether-net configuration between simulations and experiments. (

**a**) On-site photos of on-ground experiments. (

**b**) Main view of simulation results. (

**c**) Top view of simulation results.

**Figure 14.**Configuration of tether-net during deployment with initial parameters. (

**a**) Main view. (

**b**) Top view.

**Figure 15.**Tether-net deployment performance. (

**a**) Launch distance. (

**b**) Deployment length of long side. (

**c**) Deployment length of short side. (

**d**) Deployment area.

**Figure 16.**Displacement of different towing blocks. (

**a**) Displacement–launch distance. (

**b**) Displacement–time.

**Figure 17.**Deployment performance with different towing block mass values. (

**a**) Launch distance. (

**b**) Deployment length of long side. (

**c**) Deployment length of short side. (

**d**) Deployment area.

**Figure 19.**Deployment performance with different launch speed values. (

**a**) Launch distance. (

**b**) Deployment length of long side. (

**c**) Deployment length of short side. (

**d**) Deployment area.

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

Net size, m^{2} | 16 × 8 |

Net mesh, m^{2} | 0.8 × 0.8 |

Diameter of main tethers, mm | 0.5 |

Diameter of side tethers, mm | 2 |

Diameter of towing tethers, mm | 2 |

Towing block mass, g | 170 |

Launch speed, m/s | 30 |

Launch angle, ° | 16 |

Tether Type | Elastic Modulus |
---|---|

Main tether | 748.7 MPa |

Towing tether | 59.5 MPa |

Side tether | 46.2 MPa |

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

Maximum deployment length of the long side, m | 16.3 |

Maximum deployment length of the short side, m | 7.5 |

Maximum deployment area, m^{2} | 122.3 |

Deployment distance, m | 42.8 |

Shape-preserving distance, m~m | 38.5~48.9 |

Parameter | 150 g | 170 g | 190 g |
---|---|---|---|

Maximum deployment length of the long side, m | 16 | 16.3 | 16.6 |

Maximum deployment length of the short side, m | 7.2 | 7.5 | 7.8 |

Maximum deployment area, m^{2} | 114.3 | 122.3 | 128.7 |

Deployment distance, m | 42.8 | 42.8 | 45 |

Shape-preserving distance, m~m | 39.3~48.2 | 38.5~48.9 | 38.7~48.9 |

Parameter | 20 m/s | 30 m/s | 40 m/s |
---|---|---|---|

Maximum deployment length of the long side, m | 13.8 | 16.3 | 17 |

Maximum deployment length of the short side, m | 5.8 | 7.5 | 8 |

Maximum deployment area, m^{2} | 80 | 122.3 | 135 |

Deployment distance, m | 42.8 | 42.8 | 45 |

Shape-preserving distance, m~m | 38.5~48.9 | 37.8~52.8 |

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## Share and Cite

**MDPI and ACS Style**

Yue, S.; Li, M.; Zhao, Z.; Du, Z.; Wu, C.; Zhang, Q.
Parameter Analysis and Experiment Validation of Deployment Characteristics of a Rectangular Tether-Net. *Aerospace* **2023**, *10*, 115.
https://doi.org/10.3390/aerospace10020115

**AMA Style**

Yue S, Li M, Zhao Z, Du Z, Wu C, Zhang Q.
Parameter Analysis and Experiment Validation of Deployment Characteristics of a Rectangular Tether-Net. *Aerospace*. 2023; 10(2):115.
https://doi.org/10.3390/aerospace10020115

**Chicago/Turabian Style**

Yue, Shuai, Mengsheng Li, Zhen Zhao, Zhonghua Du, Chunbo Wu, and Qingzhan Zhang.
2023. "Parameter Analysis and Experiment Validation of Deployment Characteristics of a Rectangular Tether-Net" *Aerospace* 10, no. 2: 115.
https://doi.org/10.3390/aerospace10020115