# Natural Frequencies of a Tethered Satellite System
## 2 Natural Frequencies of the Dumbbell Model
Figure 3 presents the natural frequencies for this variation, with Figure 3a,b corresponding to the natural frequencies \(\omega_{1}\) and \(\omega_{2}\), respectively. The two lines for \(\omega_{1}\) and \(\omega_{c}\) nearly coincide, appearing as a single line, because \(\omega_{1}\) is closely associated with orbital motion. The eigenvector corresponding to the natural frequency \(\omega_{1}\) is \(\mathbf{X}_{1}=\{\Delta r,\ \Delta\Phi,\Delta\theta\ \}^{\mathrm{T}}=\{1,\ 0,\ 0\ \}^{\mathrm{T}}\). This implies that \(\omega_{1}\) represents the natural frequency associated with variation in the orbital radius. This variation arises from elliptical orbital motion deviating from a circular orbit, with their period determined by the orbital angular velocity. Consequently, \(\omega_{1}\) is strongly linked to the orbital motion. As depicted in the figure, both \(\omega_{1}\) and \(\omega_{c}\) decrease as the orbital radius increases, reflecting the increase in the orbital period of the satellite with increasing radius. The second natural frequency \(\omega_{2}\), shown in Figure 3b also decreases with increasing orbital radius and remains greater than \(\omega_{1}\). At \(\overline{r}=6600\) km, \(\omega_{1}\) is \(1.1761\times 10^{-3}\) rad/s and \(\omega_{2}\) is \(2.0371\times 10^{-3}\) rad/s, indicating  
Figure 2: Natural frequencies of the dumbbell model for the variation of the tether length, \(L\): (**a**) first natural frequency and (**b**) second natural frequency. that \(\omega_{2}\) is 1.732 times greater than \(\omega_{1}\). Note that \(\omega_{2}\) is related to the libration motion of the tethered satellite system. Figure 4 illustrates the variation of the natural frequencies and orbital angular velocity as a function of \(m_{1}\), with all other physical parameters fixed at the reference values. Specifically, the mass of satellite \(m_{2}\) is set to 1000 kg, while the mass \(m_{1}\) varies from 100 to 10,000 kg. As shown in Figure 4, as \(m_{1}\) increases, both the first natural frequency \(\omega_{1}\) and the orbital angular velocity \(\omega_{c}\) increase, whereas the second natural frequency \(\omega_{2}\) decreases. As \(m_{1}\) increases while keeping \(m_{2}\) fixed, the mass center of the tethered satellite system moves toward \(m_{1}\), reducing the orbital radius of the mass center, which causes \(\omega_{1}\) and \(\omega_{c}\) to increase. On the other hand, as the mass \(m_{1}\) close to Earth becomes larger relative to the mass \(m_{2}\) far from Earth, the system becomes more stable, resulting in a longer libration period and thus a decrease in \(\omega_{2}\). In contrast, when other parameters are fixed as the reference values and the mass \(m_{2}\) of the satellite located farther from Earth is increased, the variations in the first natural frequency \(\omega_{1}\), the second natural frequency \(\omega_{2}\), and the orbital angular velocity \(\omega_{c}\) are shown in Figure 5. As shown in this figure, as \(m_{2}\) increases, \(\omega_{1}\) and \(\omega_{c}\) decrease but \(\omega_{2}\) increases. This phenomenon occurs because an increase in \(m_{2}\) while maintaining \(m_{1}\) fixed results in an increase in the orbital radius for the mass center of the satellite system. The increase in the orbital radius leads to a decrease in both the first natural frequency and orbital angular velocity, as shown in Figure 5a. However, when the mass \(m_{2}\) of the satellite farther from Earth is larger than the mass \(m_{1}\) of the satellite closer to Earth, the satellite system becomes relatively unstable, decreasing the period of libration and causing an increase in the second natural frequency, as depicted in Figure 5b. Figure 3: Natural frequencies of the dumbbell model for the variation of the orbital radius \(\bar{r}\): (**a**) first natural frequency and (**b**) second natural frequency. Finally, when the masses of the two satellites were equal, we examined the effects of satellite masses on the natural frequencies of the tethered satellite system.