# Natural Frequencies of a Tethered Satellite System
## 2 Natural Frequencies of the Dumbbell Model
The equilibrium conditions for the satellite system to have circular orbit are given by  
\[\overline{\theta}=0, \tag{5}\]  
\[\omega_{c}=\sqrt{\frac{GM\left[m_{1}/\overline{r}^{2}+m_{2}/\left(\overline{r }+L\right)^{2}\right]}{\left[m_{1}\overline{r}+m_{2}(\overline{r}+L)\right]}}. \tag{6}\]  
The equations of motion linearized in the neighborhood of the equilibrium state can be represented by  
\[ \begin{array}{c}(m_{1}+m_{2})\Delta\bar{r}-2[m_{1}\overline{r}+m_{2}( \overline{r}+L)]\omega_{c}\Delta\dot{\psi}-2m_{2}L\omega_{c}\Delta\dot{\theta }\\ -\left\{2 GM\left[\frac{m_{1}}{\overline{r}^{2}}+\frac{m_{2}}{\left(\overline{r }+L\right)^{3}}\right]+(m_{1}+m_{2})\omega_{c}^{2}\right\}\Delta r=0,\end{array}  \tag{7}\]  
\[\left[m_{1}\overline{r}^{2}+m_{2}(\overline{r}+L)^{2}\right]\Delta\dot{\psi}+ m_{2}L(\overline{r}+L)\Delta\dot{\theta}+2[m_{1}\overline{r}+m_{2}(\overline{r }+L)]\omega_{c}\Delta\dot{r}=0, \tag{8}\]  
\[m_{2}L(\overline{r}+L)\Delta\ddot{\psi}+m_{2}L^{2}\Delta\ddot{\theta}+2m_{2}L \omega_{c}\Delta\dot{r}+m_{2}L\overline{r}\left[\omega_{c}^{2}-\frac{GM}{ \left(\overline{r}+L\right)^{3}}\right]\Delta\theta=0. \tag{9}\]  
To derive the eigenvalue problem, Equations (7)-(9) can be expressed as the following matrix-vector equation:  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{G}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=0, \tag{10}\]  
where \(\mathbf{x}\) is the displacement vector, \(\mathbf{M}\) is the mass matrix, \(\mathbf{G}\) is the gyroscopic matrix, and \(\mathbf{K}\) is the stiffness matrix. These vectors and matrices are given by  
\[\mathbf{x}=\{\Delta r,\ \ \Delta\psi,\ \ \Delta\theta\}^{\mathrm{T}}, \tag{11}\]  
\[\mathbf{M}=\left[ \begin{matrix}m_{1}+m_{2}&0&0\\ 0&m_{1}\overline{r}^{2}+m_{2}(\overline{r}+L)^{2}&m_{2}L(\overline{r}+L)\\ 0&m_{2}L(\overline{r}+L)&m_{2}L^{2}\end{matrix} \right], \tag{12}\]  
\[\mathbf{G}=\left[ \begin{matrix}0&-2[m_{1}\overline{r}+m_{2}(\overline{r}+L)] \omega_{c}&-2m_{2}L\omega_{c}\\ 2[m_{1}\overline{r}+m_{2}(\overline{r}+L)]\omega_{c}&0&0\\ 2m_{2}L\omega_{c}&0&0\end{matrix} \right], \tag{13}\]  
\[\mathbf{K}=\left[ \begin{matrix}-2 GM\left[\frac{m_{1}}{\overline{r}^{2}}+\frac {m_{2}}{\left(\overline{r}+L\right)^{3}}\right]-(m_{1}+m_{2})\omega_{c}^{2}&0 &0\\ 0&0&0\\ 0&0&m_{2}L\overline{r}\left[\omega_{c}^{2}-\frac{GM}{\left(\overline{r}+L \right)^{3}}\right]\end{matrix} \right]. \tag{14}\]  
As seen in Equations (12)-(14), the mass matrix \(\mathbf{M}\) and stiffness matrix \(\mathbf{K}\) are symmetric while the gyroscopic matrix \(\mathbf{G}\) is skew-symmetric. The natural frequencies can be obtained from the eigenvalue problem of the linearized equations near the equilibrium state. Assuming the solution of Equation (10) as \(\mathbf{x}=\mathbf{X}\mathbf{e}^{\lambda t}\), where \(\mathbf{X}\) is the eigenvector and \(\lambda\) is the eigenvalue, the eigenvalue can be computed from  
\[\text{det}\Big{(}\lambda^{2}\mathbf{M}+\lambda\mathbf{G}+\mathbf{K}\Big{)}=0. \tag{15}\]  
Using the symbolic manipulation software _Mathematica_[16], Equation (15) can be expressed as a polynomial in \(\lambda\). By substituting \(\lambda^{2}\) with \(x\), the equation can be rewritten as  
\[x\Big{(}x^{2}+ax+b\Big{)}=0, \tag{16}\]  
where  
\[ \begin{array}{ll}a&=GM\big{[}m_{1}^{2}\overline{r}(2\overline{r}+L)\big{(}2 \overline{r}^{2}+2L\overline{r}+L^{2}\big{)}+m_{2}^{2}(\overline{r}+L)\big{(}2 \overline{r}^{2}+2L\overline{r}+L^{2}\big{)}\\ &+m_{1}m_{2}\big{(}8\overline{r}^{4}+16L\overline{r}^{3}+8L\overline{r}^{2} -L^{4}\big{)}\big{]}\big{/}\big{[}(m_{1}+m_{2})\overline{r}^{3}(\overline{r}+ L)^{3}(m_{1}\overline{r}+m_{2}\overline{r}+m_{2}L)\big{]},\end{array}  \tag{17}\]  
\[