# Natural Frequencies of a Tethered Satellite System
## 3 Dynamic Responses of the ANCF Model  
In contrast to the previous section, this section models the tether as a slender beam to account for both tether deformation and mass. Figure 7 illustrates the tethered satellite system, where two satellites, \(m_{1}\) and \(m_{2}\), are connected by a flexible tether of length \(L\) and move within the \(xy\)-plane. The position vectors of satellites \(m_{1}\) and \(m_{2}\) from the center of Earth are denoted as \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\), respectively. The libration angle \(\theta\) of the satellite system with a flexible tether is defined as the angle between the line connecting the two satellites and the line from the center of Earth to satellite \(m_{1}\). The position vector \(\mathbf{r}\) of a point on the  
Figure 6: Dynamic responses of the libration angle \(\theta\) for the dumbbell model for different initial libration angles: (**a**) \(\theta(0)=1^{\"}\), (**b**) \(\theta(0)=10^{\"}\), and (**c**) \(\theta(0)=30^{\"}\). tether is a function of the arc length coordinate \(s\) from satellite \(m_{1}\) to that point and time \(t\), and is expressed as  
\[\mathbf{r}=\{x(s,t),\ y(s,t)\}^{\mathrm{T}}. \tag{24}\]  
To derive the equations of motion for the tethered satellite system using [PERSON]'s equation, it is necessary to compute the kinetic and potential energies of the system. First, the kinetic energy of the satellite system is given by  
\[T=\frac{1}{2}m_{1}\dot{\mathbf{r}_{1}}\cdot\dot{\mathbf{r}_{1}}+\frac{1}{2}m_{2 }\dot{\mathbf{r}_{2}}\cdot\dot{\mathbf{r}_{2}}+\frac{1}{2}\rho A\int_{0}^{L} \dot{\mathbf{r}}\cdot\dot{\mathbf{r}}\mathrm{d}s, \tag{25}\]  
where \(\rho\) and \(A\) are the density and cross-sectional area of the tether, respectively. The first two terms of Equation (25) represent the kinetic energy of satellites \(m_{1}\) and \(m_{2}\), while the last term represents the kinetic energy of the tether. On the other hand, the potential energy of the system can be expressed as  
\[U=-GM\bigg{(}\frac{m_{1}}{|\mathbf{r}_{1}|}+\frac{m_{2}}{|\mathbf{r}_{2}|}+ \int_{0}^{L}\frac{\rho A}{|\mathbf{r}|}\mathrm{d}s\bigg{)}+\frac{1}{2}EA\int_ {0}^{L}\varepsilon^{2}\mathrm{d}s+\frac{1}{2}EI\int_{0}^{L}\kappa^{2}\mathrm{ d}s, \tag{26}\]  
where \(E\) and \(I\) are Young's modulus and the area moment of inertia of the tether, respectively; \(\varepsilon\) and \(\kappa\) are the Lagrange strain and curvature of the tether given by  
\[\varepsilon=\frac{1}{2}\bigg{[}\bigg{(}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d }s}\bigg{)}^{\mathrm{T}}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}-1\bigg{]}, \kappa=\bigg{|}\frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}s^{2}}\bigg{|}. \tag{27}\]  
The first term of Equation (26) represents the gravitational potential energy of the satellites and tether, while the second and third terms represent the longitudinal and bending strain energies, respectively. To account for both the orbital motion of the satellite system around Earth and the deformation of the tether, this study employed the ANCF. For detailed derivation of ANCF, see [PERSON] et al. (2018) and [PERSON] (2019). The model was discretized into \(N\) beam elements with the node numbering scheme illustrated in Figure 8. In this figure, the nodes corresponding to satellites \(m_{1}\) and \(m_{2}\) are \(1\) and \(N+1\), respectively. The value of the arc length coordinate at node \(e\) is denoted as \(s_{e}\), and the values of the coordinates \(x\) and \(y\) at this node are denoted as \(x_{e}\) and \(y_{e}\). The displacement vector \(\mathbf{r}\) within element \(e\) can be expressed as  
\[\mathbf{r}(s,t)=\mathbf{N}(s)\mathbf{d}_{e}(t)\ \text{for}\ s_{e}\leq s\leq s_{e+1}, \tag{28}\]  
Figure 7: Satellites connected by a flexible tether. where the vector \(\mathbf{d}_{e}\) and matrix \(\mathbf{N}\) are defined by  
\[\mathbf{d}_{e}=\big{\{}x_{e},\ y_{e},\ (\partial x/\partial s)_{e^{\prime}}( \partial y/\partial s)_{e^{\prime}}x_{e+1},\ y_{e+1},\ (\partial x/\partial s)_{e+1},\ (\partial y/\partial s)_{e+1}\big{\}}^{ \mathsf{T}}, \tag{29}\]  
\[\mathbf{N}=