# Natural Frequencies of a Tethered Satellite System
## 3 Dynamic Responses of the ANCF Model
The libration angle \(\theta\) can be obtained by  
\[\theta=\tan^{-1}\frac{\overline{y}_{N+1}}{\overline{x}_{N+1}}. \tag{39}\]  
The displacements \(\overline{\overline{x}}_{e}\) and \(\overline{\overline{y}}_{e}\), from which both the orbital and libration motions are removed, can be obtained by applying the following coordinate transformation to the displacements of Equation (38):  
\[ \begin{cases}\overline{\overline{x}}_{e}\\ \overline{y}_{e}\end{cases} = \begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix}\begin{cases}\overline{\overline{x}}_{e}\\ \overline{\overline{y}}_{e}\end{cases} \;\;\text{for}\;e=1,\;2,\;\cdots,\;N+1. \tag{40}\]  
Denoting the longitudinal and transverse deformations of the tether at node \(e\) by \(u_{e}\) and \(w_{e}\), respectively, these deformations at time \(t\) are given by  
\[u_{e}(t)=\overline{\overline{x}}_{e}(t)-\overline{\overline{x}}_{e}(0),\;\;\; w_{e}(t)=\overline{\overline{y}}_{e}(t)-\overline{\overline{y}}_{e}(0). \tag{41}\]  
We numerically investigated the dynamic responses of the libration angle as well as the transverse and longitudinal deformations of the tether. The tether was discretized into 100 uniform elements (i.e., \(N=100\)) to construct the ANCF model. The dimensions and physical properties of the tethered satellite system used in the numerical analysis are identical to those of the dumbbell model (\(m_{1}=m_{2}=1000\) kg and \(L=10\) km). However, the density, Young's modulus, and cross-sectional area of the tether are given by \(\rho=1440\) kg/m\({}^{3}\), \(E=131\) GPa, \(A=3.1416\) mm\({}^{2}\), and \(I=0.7854\) mm\({}^{4}\), respectively. The initial conditions for the numerical integration used to determine the dynamic responses are as follows: \(r_{1}(0)=6600\) km, \(r_{2}(0)=6610\) km, \(\theta(0)=3^{\circ}\), \(\dot{\varphi}(0)=0\), \(\dot{r}_{1}(0)=\dot{r}_{2}(0)=0\), \(\dot{\theta}(0)=0\), and \(\dot{\psi}(0)=1.176\times 10^{-3}\) rad/s. The reference values for the physical properties and dimensions of the satellite system are summarized in Table 1. The initial transverse deformation of the tether was assumed to be zero, while the initial longitudinal deformation was considered stretched, resulting from the combined effects of the gravitational gradient and the centrifugal force difference between the two satellites. The initial longitudinal deformation was calculated based on the tension provided by Equation (42), discussed in Section 4. The tether system was initialized with a libration angle of \(3^{\circ}\) and an orbital velocity corresponding to a circular orbit. Numerical calculations were performed using the Newmark method with a time step size of 0.002 s. To address the nonlinearity of Equation (37), the Newton-Raphson method was employed at each time step. The dynamic response for the libration angle of the satellite system was calculated using Equation (39) and is presented in Figure 9. Figure 9a illustrates the dynamic response, while Figure 9b shows the corresponding frequency spectrum derived from the dynamic response in Figure 9a. The red number in Figure 9b identifies the frequency at the peak of the spectrum. As expected, the libration angle exhibits harmonic motion, as shown in Figure 9a, with a peak frequency of 0.0020 rad/s, as depicted in Figure 9b. This value closely matches the second natural frequency of the dumbbell model, \(\omega_{2}=2.0371\times 10^{-3}\) rad/s. The dynamic responses of transverse and longitudinal deformations at node 31 of the ANCF model are presented in Figures 10 and 11, respectively. The tether, with a total length of 10 km, was discretized into 100 elements, making node 31 correspond to a point 3 km away from satellite \(m_{1}\) in the direction of satellite \(m_{2}\). As shown in Figure 10b, the dominant peak frequencies in the dynamic response for the transverse deformation of the tether are 0.0020 and 0.0215 rad/s.