# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 4 Dynamic Response Analysis
When the payload is transferred from satellite B to satellite A at a speed of 20 m/s, the attitudes of the satellite system for various payload masses are shown in Figure 7. In this scenario, the initial conditions were set to maintain a circular orbit, with the payload initially positioned on satellite B. Similar to Figure 6, Figure 7 shows that tether deformation increased as the payload mass increased. For relatively large payload masses, specifically, \(m_{v}=400\) and 500 kg, significant tether deformations were observed as the payload approached satellite A (Figure 7e,f). Unlike in Figure 6, where the payload moves clockwise relative to the line connecting satellites A and B, Figure 7 shows the payload moving counterclockwise in the direction of orbital motion. This movement causes the tether to deform accordingly. Figure 5.0 Initial position of the tethered satellites with the moving payload. \begin{table}
\begin{tabular}{l l l} \hline  & **Symbols** & **Values** \\ \hline Masses of satellites \(A\) and \(B\) & \(m_{A}\), \(m_{B}\) & 1000 kg \\ Mass of payload & \(m_{v}\) & 100 kg \\ Length of tether & \(L\) & 20 km \\ Mass density of tether & \(\rho\) & 1440 kg/m\({}^{3}\) \\ Cross-sectional area of tether & \(A\) & 20 mm\({}^{2}\) \\ Young’s modulus of tether & \(E\) & 131 GPa \\ Payload moving speed & \(V\) & 20 m/s \\ \hline \end{tabular}
\end{table} 
Table 1: Reference values for the physical properties and dimensions used in the dynamic analysis. The observed phenomenon in Figures 6 and 7, where the payload deviates from the straight line connecting satellites A and B and causes tether deformation, is attributed to the Coriolis effect. The Coriolis force is an inertial force that acts on objects moving relative to a rotating reference frame. In these figures, the reference frame, centered on the Earth, rotates counterclockwise along the line joining satellites A and B. When the payload moved outward from the origin of the reference frame, as shown in Figure 6, a clockwise Coriolis force acted on it. Conversely, when the payload moved towards the origin, as shown in Figure 7, a counterclockwise Coriolis force acted on it. Since the Coriolis force is proportional to both the payload mass and speed, the tether deformation increased with these factors. Next, the impact of payload speed on the attitude of the tethered satellites was analyzed. All material properties and dimensions used in the analysis, except for the payload speed, were the same as those listed in Table 1. The attitude of the tethered satellite system was investigated as the payload was transferred between satellites A and B at six different speeds, in the range of 10-60 m/s, in increments of 10 m/s. The attitudes during payload transfer from satellite A to satellite B at these speeds are illustrated in Figure 8,  
Figure 6: Attitudes of tethered satellites for different payload masses as the payload moves from satellite A to satellite B: (**a**) 10 kg, (**b**) 100 kg, (**c**) 200 kg, (**d**) 300 kg, (**e**) 400 kg, and (**f**) 500 kg. The green circle, red square, and blue triangle represent satellite A, satellite B, and the payload, respectively. Figure 7: Attitudes of tethered satellites for different payload masses as the payload moves from satellite B to satellite A: (**a**) 10 kg, (**b**) 100 kg, (**c**) 200 kg, (**d**) 300 kg, (**e**) 400 kg, and (**f**) 500 kg. The green circle, red square, and blue triangle represent satellite A, satellite B, and the payload, respectively. while Figure 9 shows the attitudes during the transfer from satellite B to satellite A. Both figures demonstrate that as the payload speed increased, the transfer time between the two satellites decreased, resulting in a reduction in the orbital distance of the satellite system. At a payload speed of 10 m/s, the tether connecting the satellites appears nearly straight (Figures 8a and 9a). However, at a speed of 20 m/s, the tether appears as a broken line with the payload as its vertex (Figures 8b and 9b).