# Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture
## 4 Dynamic Response Analysis of the EDT System  
### Comparison of the ANCF Model with a Dumbbell Model  
Before analyzing the dynamic behavior of the EDT system, the dynamic responses of the proposed ANCF model are compared with those of the simplified dumbbell model. In the dumbbell model, the tether is idealized as massless and straight, with no deformation, resulting in a uniform Lorentz force acting along the entire tether length. Conversely, the ANCF model accounts for large tether deformations, causing variations in the Lorentz force direction on each tether element. These differences significantly impact the forces and behaviors observed in the EDT system. Therefore, a comparative analysis of the dumbbell and ANCF models is conducted to highlight the unique insights provided by the ANCF model. The parameters used for both models are as follows: the main satellite and sub-satellite masses are \(m_{m}=10,000\) kg and \(m_{n}=50\) kg, respectively; the tether length and diameter are \(L=10\) km and \(d=0.5\) mm. The main satellite operates at an altitude of 450 km. The ANCF model has a density of \(\rho=1450\) kg/m\({}^{3}\) and [PERSON]'s modulus of \(E=131\) GPa. The tether system undergoes circular orbital motion, with the initial velocity calculated using the following equation:  
\[v_{0}(s)=|\mathbf{r}(s)|\sqrt{GM/r_{c}^{3}} \tag{33}\]  
where \(r_{c}\) is the distance between the center of mass of the EDT system and the center of the Earth. During circular orbital motion, the dynamic behavior and electrodynamic characteristics of the ANCF model are compared with those of the dumbbell model. Figure 4 illustrates the orbital motion of both models over one orbital period in an equatorial orbit, with the main satellite and sub-satellite represented as green and red circles, respectively. To clearly illustrate the change in attitude, the images are created by enlarging the tether length relative to the orbital radius of the satellites. The ANCF model displays tether deformation due to the electrodynamic forces acting on the EDT system, whereas the dumbbell model represents the tether as a straight line. This distinction arises because the ANCF model exhibits a more subtle behavior than the dumbbell model by taking into account the flexibility and deformation of the tether. Figure 5 presents the average current generated by both models during circular orbital motion in an equatorial orbit. Despite minor discrepancies, the average current over one orbital period is similar for both models. The variations in current during orbital motion are attributed to changes in the magnetic field, which influence current generation as the system progresses along its orbit. The total computational time required to perform the simulations for Figures 4 and 5 is 631.7 min for the ANCF model and 315.8 min for the dumbbell model. As expected, the ANCF model requires significantly more computation time than the dumbbell model due to its higher complexity. The libration angles generated by the ANCF model and the dumbbell model are also compared. As depicted in Figure 6, the libration angle is defined as the smallest angle between the local vertical axis, which connects the main satellite to the center of Earth, and an imaginary line linking the main satellite and sub-satellite. Figure 7 compares the libration angles from both models, showing that the ANCF model exhibits larger amplitude variations. This is due to the ANCF model accounting for tether deformation effects induced by electrodynamic forces, unlike the dumbbell model, which treats the tether as a rigid body and neglects deformation effects. These comparisons demonstrate that the ANCF model provides a more accurate analysis of the dynamic responses of the EDT system, highlighting the importance of incorporating tether flexibility in the model. Figure 4: Comparison of the orbital motion of the EDT system: (**a**) ANCF model and (**b**) dumbbell model. Figure 5: Comparison of the average current generated by the ANCF and dumbbell models during circular orbital motion.