# Natural Frequencies of a Tethered Satellite System
## 2 Natural Frequencies of the Dumbbell Model
\begin{array}{ll}b&=G^{2}M^{2}\big{(}3\overline{r}^{2}+3L\overline{r}+L^{2} \big{)}\big{[}\big{(}m_{1}^{3}+m_{2}^{3}\big{)}\overline{r}^{3}(\overline{r}+ L)^{3}+m_{1}m_{2}^{2}(\overline{r}+L)\big{(}3\overline{r}^{5}+6L\overline{r}^{4} -3L^{2}\overline{r}^{3}\\ &-8L^{3}\overline{r}^{2}-7L^{4}\overline{r}-2L^{5}\big{)}+m_{1}^{2}m_{2} \overline{r}\big{(}3\overline{r}^{5}+9L\overline{r}^{4}+3L^{2}\overline{r}^{3 }-7L^{3}\overline{r}^{2}-9L^{4}\overline{r}-3L^{5}\big{)}\big{]}\big{/}\big{(} m_{1}\\ &+m_{2}\big{)}\tilde{r}^{6}(\overline{r}+L)^{6}(m_{1}\overline{r}+m_{2} \overline{r}+m_{2}L)^{2}].\end{array}  \tag{18}\]  
The solutions of Equation (16) are given by  
\[x=0,\ x=\Big{(}-a\pm\sqrt{a^{2}-4b}\Big{)}/2, \tag{19}\]  
The conditions for \(x=\Big{(}-a\pm\sqrt{a^{2}-4b}\Big{)}/2\) to be negative are as follows:  
\[a>0,\ b>0,\ a^{2}-4b>0 \tag{20}\]  
Since the physical properties and dimensions used in this study satisfy the above conditions, the solutions of Equation (16) consist of zero and two distinct negative real numbers. Given the substitution \(x=\lambda^{2}\), the corresponding solutions for \(\lambda\) are zeros and two purely imaginary complex conjugate pairs. Consequently, the roots of Equation (15) can be expressed as \(0\), \(0\), \(\pm\mathrm{i}\omega_{1}\), and \(\pm\mathrm{i}\omega_{2}\). Therefore, the tethered satellite system has three natural frequencies: zero, \(\omega_{1}\), and \(\omega_{2}\). When the two satellites have the same mass, i.e., when \(m_{1}=m_{2}\), the coefficients \(a\) and \(b\) are reduced to  
\[a=\frac{GM\big{(}8\overline{r}^{2}+8L\overline{r}+3L^{2}\big{)}}{\overline{r} ^{2}(\overline{r}+L)^{2}(2\overline{r}+L)}, \tag{21}\]  
\[b=\frac{G^{2}M^{2}\big{(}2\overline{r}^{2}+2L\overline{r}+L^{2}\big{)}\big{(}3 \overline{r}^{2}+3L\overline{r}+L^{2}\big{)}\Big{(}2\overline{r}^{4}+4L \overline{r}^{3}-2L^{2}\overline{r}^{2}-4L^{3}\overline{r}-L^{4}\Big{)}}{ \overline{r}^{6}(\overline{r}+L)^{6}(2\overline{r}+L)^{2}}. \tag{22}\]  
The natural frequencies of the tethered satellite system due to variations in orbital radius and system parameters were analyzed when the satellite system orbits circularly the Earth. The reference physical properties used in the analysis are \(G=6.6743\times 10^{-11}\) m\({}^{3}\)/kg s\({}^{2}\), \(M=5.9722\times 10^{24}\) kg, \(m_{1}=m_{2}=1000\) kg, \(\overline{r}=6600\) km, and \(L=10\) km. Unless otherwise stated, these reference values were used in the analysis. First, the effect of tether length on the natural frequencies of the satellite system was examined. Remaining all parameters constant except for the tether length, the natural frequencies were computed as the tether length increased from 1 to 100 km, as depicted in Figure 2. In this figure, the red, black, and blue lines represent \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{c}\), respectively. It is observed that increasing the tether length \(L\) results in a decrease in both natural frequencies \(\omega_{1}\) and \(\omega_{2}\), as well as in the orbital angular velocity \(\omega_{c}\). When the masses of the two satellites are equal and constant, and the orbital radius of satellite \(m_{1}\) is held fixed, an increase in tether length effectively increases the orbital radius of the satellite system. This occurs because the orbital radius of the mass center of the system increases as the tether length increases. Next, we analyzed the effect of orbital radius on the natural frequencies of the satellite system. Other physical properties, except for the orbital radius, are the same as the above reference values. As mentioned before, the orbital angular velocity is determined by Equation (6) to maintain the circular orbit. When the masses of the two satellites are equal, the orbital angular velocity is given by  
\[\omega_{c}=\sqrt{\frac{GM\big{(}2\overline{r}^{2}+2L\overline{r}+L^{2}\big{)} }{\overline{r}^{2}(\overline{r}+L)^{2}(\overline{r}+L)}}. \tag{23}\]  
The natural frequencies were analyzed for the variation of the orbital radius \(\overline{r}\) from 6500 to 50,000 km.