# Natural Frequencies of a Tethered Satellite System
## 2 Natural Frequencies of the Dumbbell Model  
The dynamic characteristics of a tethered satellite system were analyzed using a dumbbell model. This system consists of two satellites with masses \(m_{1}\) and \(m_{2}\), connected by a non-deformable tether of length \(L\), as illustrated in Figure 1. In this study, the satellites were assumed as point masses without mass moments of inertia, and the mass of the tether was neglected. Since the dumbbell model in Figure 1 can be treated as a rigid body, when it moves on the \(xy\)-plane including the center of the Earth \(O\), the satellite system has three degrees of freedom. The generalized coordinates of the system may be selected as the orbital radius \(r\) of satellite \(m_{1}\), the true anomaly \(\psi\), and the libration angle \(\theta\). The equations of motion for the dumbbell model of the tethered satellites can be derived using [PERSON]'s equations. After expressing the position and velocity vectors of satellites \(m_{1}\) and \(m_{2}\) in terms of the generalized coordinates \(r\), \(\psi\), and \(\theta\), the kinetic energy and gravitational potential energy of the system were calculated. Substituting the kinetic and potential energies into [PERSON]'s equation yields the following nonlinear equations of motion:  
\[ \begin{array}{c}(m_{1}+m_{2})\left(\ddot{r}-r\dot{\psi}^{2}\right)-m_{2}L \left(\ddot{\psi}+\dot{\theta}\right)^{2}\cos\theta-m_{2}L\left(\ddot{\psi}+ \ddot{\theta}\right)\sin\theta+\frac{GMm_{1}}{r^{2}}\\ \qquad\qquad\qquad\qquad\qquad\qquad+\frac{GMm_{2}\left(r+1\cos\theta\right) }{\left(r^{2}+2 rL\cos\theta+L^{2}\right)^{3/2}}=0,\end{array}  \tag{1}\]  
\[ \begin{array}{c}\left(m_{1}+m_{2}\right)\left(r^{2}\ddot{\psi}+2r\dot{r} \dot{\psi}\right)+m_{2}L^{2}\left(\ddot{\psi}+\ddot{\theta}\right)+m_{2}L \left(2r\ddot{\psi}+2\dot{r}\dot{\psi}+\ddot{r}\ddot{\theta}\right)\cos\theta \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad-m_{2}L\left(\ddot{r}+2r\dot{\psi} \dot{\theta}+r\dot{\theta}^{2}\right)\sin\theta=0,\end{array}  \tag{2}\]  
\[ \begin{array}{c}m_{2}L\left(\ddot{\psi}+\ddot{\theta}\right)+m_{2}\left(r \dot{\psi}+2r\dot{\psi}\right)\cos\theta-m_{2}\left(\ddot{r}-r\dot{\psi}^{2} \right)\sin\theta\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{GMm_{2}r\sin\theta}{ \left(r^{2}+2 rL\cos\theta+L^{2}\right)^{3/2}}=0,\end{array}  \tag{3}\]  
where the superposed dot represents differentiation with respect to time, \(G\) is the universal gravitational constant, and \(M\) is the mass of the Earth. For the detailed derivation process, please refer to [PERSON] et al. [15]. Using the perturbation method, the natural frequencies of the dumbbell model with small libration motion were determined when the satellite system had circular orbit around  
Figure 1: Dumbbell model of tethered satellites. the Earth. The orbital radius, orbital angular velocity, and libration angle when the satellite system is in equilibrium state of the circular orbit are denoted by \(\overline{r}\), \(\omega_{c}\), and \(\overline{\theta}\), respectively. Representing the perturbed values of \(r\), \(\dot{\psi}\), and \(\theta\) from the equilibrium state are represented by \(\Delta r\), \(\Delta\dot{\psi}\), and \(\Delta\theta\), the orbital radius, the orbital velocity, and the libration angle can be expressed by the following equations:  
\[r=\overline{r}+\Delta r,\ \dot{\psi}=\omega_{c}+\Delta\dot{\psi},\ \theta= \overline{\theta}+\Delta\theta. \tag{4}\]  
By substituting Equation (4) into Equations (1)-(3) and expanding the resulting nonlinear equations in Taylor's series around the equilibrium state under the assumption that \(\Delta r\), \(\Delta\dot{\psi}\), and \(\Delta\theta\) are small, the conditions for equilibrium and the linearized equations of motion were obtained.