# Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture
## 4 Dynamic Response Analysis of the EDT System
### Influence of Tether Length and Diameter on EDT System Performance  
We examine the effects of the tether length and diameter on the response and electrodynamic characteristics of the EDT system. These parameters significantly impact the mechanical and electrical properties of the system, including the electron collection area, stiffness, and mass. The reference values of the EDT system properties used in the analysis are summarized in Table 1. The analysis considered tether lengths of 5, 10, and 20 km, and tether diameters of 0.5, 1, and 1.5 mm.  
\begin{table}
\begin{tabular}{l l l} \hline \hline  & **Symbols** & **Values** \\ \hline Mass of main satellite & \(m_{m}\) & 10,000 kg \\ \hline Mass of sub-satellite & \(m_{n}\) & 50 kg \\ \hline Length of tether & \(L\) & 10 km \\ \hline Diameter of tether & \(d\) & 0.5 mm \\ \hline Mass density of tether & \(\rho\) & 1450 kg/m\({}^{3}\) \\ \hline Young’s modulus of tether & \(E\) & 131 GPa \\ \hline \hline \end{tabular}
\end{table}
Table 1: Reference values of the EDT system properties.  
Figure 6: Definition of the libration angle of the EDT system.  
Figure 7: Comparison of the libration angle between the ANCF model and dumbbell model.  
Figure 8 illustrates the orbital motion of the EDT system for different tether lengths. In Figure 8a, the 5 km tether shows minimal deformation, with the main satellite and sub-satellite maintaining a stable circular orbit. As depicted in Figure 8b, increasing the tether length to 10 km results in noticeable deformation and the onset of libration angles within the system. Figure 8c shows that at a tether length of 20 km, the deformation is more pronounced, with longer tethers exhibiting enhanced interactions with the magnetic field of Earth, generating higher currents and Lorentz forces.  
Figure 9 presents the average current and Lorentz force for the three tether lengths. In Figure 9a, the 5 km tether exhibits limited interaction with the magnetic field of Earth, generating low current. The 10 km tether produces a higher current, with an increasing trend as the tether length extends. The 20 km tether generates the highest current due to the expanded electron collection area. Similarly, Figure 9b shows the Lorentz force increasing with tether length. The difference in Lorentz force between the 10 and 20 km tethers is more pronounced than the current difference, as the Lorentz force is proportional to both the current and tether length.  
The effects of tether diameter on dynamic behavior and electrodynamic characteristics are shown in Figure 10, which depicts the orbital motion of the EDT system for tether diameters of 0.5, 1, and 1.5 mm. The tether diameter influences the mechanical stiffness and mass, affecting tether deformation and dynamic response. As the diameter increases, more pronounced tether behavior is observed (Figure 10), leading to greater instability in the orbit of the sub-satellite and increased tether deformation  
Figure 11 illustrates the average current and Lorentz force for varying tether diameters. Figure 11a shows that the average current increases with larger diameters. For diameters of 0.5 mm and 1 mm, the current remains relatively stable. However, at a diameter of 1.5 mm, significant tether deformation leads to intervals of reduced current. In Figure 11b, a similar trend can be observed for the Lorentz force. While it grows with the current, tether deformation at larger diameters causes intervals of sharply reduced Lorentz force. At 1.5 mm, pronounced tether vibrations toward the end of the orbit cause more significant variations in the Lorentz force. In conclusion, both tether length and diameter substantially impact the dynamic response and electrodynamic characteristics of the EDT system. Longer and thicker tethers increase the current generation and Lorentz force but also lead to greater tether deformation and dynamic instability.