# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 4 Dynamic Response Analysis
The initial attitude of the tethered satellites with a moving payload is such that satellites \(A\) and \(B\), along with the Earth's center, are aligned in a straight line, as shown in Figure 5. In this figure, \(r_{A}\) and \(r_{B}\) represent the orbital radii from the Earth's center to satellites \(A\) and \(B\), respectively, with \(r_{A}(0)=6600\) km and \(r_{B}(0)=6620\) km. The initial tensile deformation of the tether was considered by taking into account the gravitational forces acting on the satellites and tether, as well as the centrifugal force resulting from the orbital motion around the Earth. Assume the payload starts at satellite A and moves to satellite B at a constant speed of \(V=20\) m/s. The angular speed \(\dot{\psi}\) for the satellite system to maintain a circular orbit when the payload is located at satellite \(A\) can be determined under the condition that the total gravitational and centrifugal forces of the satellite system are equal. The angular speed is given by  
\[\dot{\psi}=\sqrt{F_{\rm G}/M_{r}}, \tag{36}\]  
where  
\[F_{\rm G}=GM\bigg{[}\frac{\rho A(r_{B}-r_{A})}{r_{A}r_{B}}+\frac{m_{A}+m_{z}} {r_{A}^{2}}+\frac{m_{B}}{r_{B}^{2}}\bigg{]}, \tag{37}\]  
\[M_{r}=\frac{1}{2}\rho A\Big{(}r_{B}^{2}-r_{A}^{2}\Big{)}+(m_{A}+m_{v})r_{A}+m_ {B}r_{B}. \tag{38}\]  
According to the equation shown above, the satellite system shown in Figure 5 orbits in a circular path with an initial orbital angular speed of \(\dot{\psi}(0)=1.1745\times 10^{-3}\) rad/s. Consequently, the initial linear velocities of satellites \(A\) and \(B\), as well as the payload, were calculated by multiplying their orbital radii by the initial angular speed. Similarly, the initial velocities of the mass points on the tether were calculated to establish the initial conditions. The reference values for the material properties and dimensions used in the  
Figure 4: Comparison of orbital radii: results from the current study using the ANCF model versus results from the previous study [17] using the two-piece dumbbell model. dynamic response analysis are summarized in Table 1. Unless otherwise specified, these values are used in the calculations presented in this section. First, the influence of the payload mass on the attitude of the tethered satellite system was analyzed. The dynamic responses of the satellite system were computed for various payload masses when the payload moved at \(V=20\) m/s, as shown in Figure 6. In Figure 6, similar to Figure 3, satellite A is represented by a green circle, satellite B by a red square, the moving payload by a blue triangle, and the tether by a black line. These symbols are used to describe the attitudes of the tethered satellite system with a moving payload in subsequent discussions. The changes in the attitude of the tethered satellite system during the transfer of the payload from satellite A to satellite B for various payload masses (\(m_{v}\)) of 10, 100, 200, 300, 400, and 500 kg are shown in Figure 6a,b,c,d,e,f, respectively. The attitudes of the satellite system at 10 different moments, spaced at equal time intervals during the transfer, are illustrated in Figure 6. The satellite system orbits the Earth in a counterclockwise direction, as indicated by the arrow in Figure 6a. When the payload mass was relatively small (\(m_{v}=10\) kg), the tethers appeared almost straight, as shown in Figure 6a. However, as the payload mass increased, the tether deformation became more pronounced, as shown in Figure 6b-f. The payload moved clockwise relative to the line connecting satellites A and B, in contrast to the counterclockwise orbital motion. For relatively large payload masses, as shown in Figure 6e and f, the tether deformations become significant. As the payload approaches satellite B, the deformation becomes so severe that the tether experiences slack and rebound due to the payload's tumbling, resulting in an unstable attitude.