# Natural Frequencies of a Tethered Satellite System
## 4 Natural Frequencies of a Tensioned Beam Model
By integrating Equation (42) with respect to \(x\) over the tether length \(L\) and dividing the result by \(L\), the average stress can be expressed as  
\[\overline{\sigma}=\frac{GM\rho}{L}\ln\frac{r_{1}}{r_{1}+L}-\frac{\rho\omega_{ c}^{2}\Big{[}\big{(}r_{1}+L\big{)}^{3}-r_{1}^{3}\Big{]}}{6L}+C, \tag{49}\]  
As is well known, the equations of motion for the transverse and longitudinal vibrations of a tether modeled as a tensioned beam are given by [20]  
\[\rho A\frac{\partial^{2}w}{\partial t^{2}}+EI\frac{\partial^{4}w}{\partial x^ {4}}-\overline{\sigma}A\frac{\partial^{2}w}{\partial x^{2}}=0, \tag{50}\]  
\[\rho A\frac{\partial^{2}u}{\partial t^{2}}-EA\frac{\partial^{2}u}{\partial x^ {2}}=0. \tag{51}\]  
The boundary conditions at the ends of the tether, where the satellites \(m_{1}\) and \(m_{2}\) are attached, are given by  
\[EI\frac{\partial^{2}w}{\partial x^{2}}=0,\ \ \ EI\frac{\partial^{3}w}{\partial x^ {3}}-\overline{\sigma}A\frac{\partial^{2}w}{\partial x^{2}}=m_{1}\frac{ \partial^{2}w}{\partial t^{2}},\ \ \ EA\frac{\partial u}{\partial x}=m_{1}\frac{ \partial^{2}u}{\partial t^{2}}\ \ \ \mbox{at}\ x=0, \tag{52}\]  
\[EI\frac{\partial^{2}w}{\partial x^{2}}=0,\ -EI\frac{\partial^{3}w}{\partial x^ {3}}+\overline{\sigma}A\frac{\partial^{2}w}{\partial x^{2}}=m_{2}\frac{ \partial^{2}w}{\partial t^{2}},\ \ -EA\frac{\partial u}{\partial x}=m_{2}\frac{ \partial^{2}u}{\partial t^{2}}\ \ \ \mbox{at}\ x=L, \tag{53}\]  
The eigenvalue problem was derived using the linear equations of motion presented in Equations (50) and (51) and the boundary conditions in Equations (52) and (53). This study adopted the eigenvalue problem derivation procedure proposed by [PERSON] [21] and [PERSON] [20]. Using the reference values listed in Table 1, the first four lowest natural frequencies for the transverse and longitudinal vibrations of the tether connecting the two satellites were calculated and are presented in Table 2. The first natural frequencies of the transverse and longitudinal motions, 0.0216 and 0.2858 rad/s, respectively, closely match the peak frequencies of 0.0215 and 0.2858 rad/s observed in Figures 10b and 11b. This agreement confirms the validity and applicability of the eigenvalue problem derived in this study. It is both insightful and valuable to investigate the effects of system parameter variations on the transverse and longitudinal natural frequencies of the tether. First, we \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Mode**} & **Transverse Natural Frequency** & **Longitudinal Natural Frequency** \\  & **(Rad/s)** & **(Rad/s)** \\ \hline
1 st & 0.0216 & 0.2858 \\
2 nd & 0.0428 & 3.0237 rad/s \\
3 rd & 0.0641 & 6.0066 rad/s \\
4 th & 0.0855 & 8.9984 rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of the tether of the satellite system. examined the impact of tether length variation on these natural frequencies. The natural frequencies were computed using the physical properties and dimensions listed in Table 1, with tether length being the only variable parameter. When varying the tether length from 1 to 100 km, the average tension and the transverse and longitudinal natural frequencies are plotted as solid lines in Figure 13a,b,c, respectively. The natural frequencies were calculated through the eigenvalue problem (EVP) described above. As the tether length increases, the gravitational gradient and the difference in centrifugal forces acting on the two satellites also increase, resulting in a rise in the average tension, as shown in Figure 13a. The transverse natural frequency, however, is more significantly influenced by the tether length than by the tension. Consequently, the transverse natural frequency decreases with increasing tether length, as illustrated in Figure 13b. In contrast, as indicated by Equation (50), the longitudinal natural frequencies are independent of the tension but are affected by the effective axial stiffness of the tether.