# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 4 Dynamic Response Analysis  
Before examining the dynamic responses of the satellite system, the responses using the proposed ANCF in this paper were verified through comparison with those obtained using the two-piece dumbbell model from a previous study [17]. When the payload speed is very low, the deflection of the tether remains relatively small. In this case, the response of the ANCF model in this study is nearly identical to that of the two-piece dumbbell model, which does not consider tether deformation. For this purpose, the payload speed was selected as \(V=1\) m/s. The material properties and dimensions used for verification were as follows: the masses of satellites A and B were \(m_{A}=m_{B}=1000\) kg, and the mass of the moving payload was \(m_{v}=100\) kg. The length, mass density, cross-sectional area, and Young's modulus of the tether were \(L=10\) km, \(\rho=1440\) kg/m\({}^{3}\), \(A=2\) mm\({}^{2}\), and \(E=131\) GPa, respectively. The initial conditions for verification were \(|\mathbf{r}_{A}(0)|=6600\) km, \(|\mathbf{r}_{B}(0)|=6610\) km, \(\psi(0)=0\), and \(\dot{\psi}=1.177\times 10^{-3}\) rad/s. All material properties and dimensions were identical to those in Ref. [17], except for the density, cross-sectional area, and Young's modulus of the tether, which were not specified in Ref. [17]. For validation, we compared the attitudes of the tethered satellite system calculated using the ANCF and two-piece dumbbell models. Figure 3a shows the attitudes derived from the ANCF model, whereas Figure 3b presents those derived from the two-piece dumbbell model [17]. In Figure 3, satellite A is depicted as a green circle, satellite B as a red square, the moving payload as a blue triangle, and the tether as a black line. The orbital direction of the satellite system was counterclockwise. For clarity when observing attitude changes, the orbital radius of satellite A is shown at its actual distance, and the length of the tether is magnified by a factor of 1000. The attitudes obtained using the ANCF model exhibited minimal deviation from those obtained using the two-piece dumbbell model in the previous study [17]. In Figure 4, the orbital radius of satellite A, obtained from the ANCF model, is compared with that obtained from the two-piece dumbbell model. In this figure, the orbital  
Figure 3: Comparison of the attitudes of tethered satellites with a moving payload: (**a**) ANCF model and (**b**) two-piece dumbbell model [17]. The green circle, red square, and blue triangle represent satellite A, satellite B, and the payload, respectively. radius from the current study, obtained using the ANCF, model is plotted as a red solid line, whereas the radius from the previous study [17], obtained using the two-piece dumbbell model, is plotted as a blue dotted line. As illustrated in Figure 4, the orbital radii derived from both models were nearly identical. This similarity is due to the payload moving at a low velocity and the small cross-sectional area of the tether, resulting in negligible mass effects from the tether. These comparisons validate the use of the proposed method for obtaining dynamic responses using the ANCF model. We now proceed to analyze the dynamic responses of a tethered satellite system with a moving payload as it orbits the Earth in a circular orbit. For this analysis, the material properties and dimensions were the same as those used in the previous validation, except for the tether length, cross-sectional area, and payload speed. Unlike in the previous validation, the tether length and cross-sectional area used in this analysis were \(L=20\) km and \(A=20\) mm\({}^{2}\), respectively, and the payload speed was \(V=20\) m/s.