# Natural Frequencies of a Tethered Satellite System
## 4 Natural Frequencies of a Tensioned Beam Model
With longer tether lengths, this stiffness decreases, leading to a reduction in longitudinal natural frequencies, as shown in Figure 13c. Furthermore, for tether lengths of 10, 30, 50, 70, and 90 km, the main peak frequencies in the frequency spectrum were obtained by applying the Fourier transform on the dynamic responses of the transverse and longitudinal displacements computed using the ANCF model. These peak frequencies, marked as blue squares in Figure 13b,c, show good agreement with the solid red lines. This consistency demonstrates that the natural frequencies determined using the proposed method can accurately predict the main frequency components of the dynamic responses of satellite system without necessitating the complex computations associated with the ANCF model. Figure 13: First natural frequencies of the tethered satellite system for the variation of the tether length: (**a**) average tension of the tether, (**b**) natural frequency of the transverse motion, and (**c**) natural frequency of the longitudinal motion. Next, the effect of orbital radius on the transverse and longitudinal natural frequencies of the tether was analyzed. All physical properties and dimensions, except for the orbital radius of satellite \(m_{1}\), were assigned the same values as those listed in Table 1. The natural frequencies were investigated by varying the orbital radius from 6500 km to 50,000 km. The tension, transverse natural frequencies, and longitudinal natural frequencies are plotted in Figure 14a,b,c, respectively. As in the previous analysis, Figure 14b,c demonstrates that the natural frequencies obtained from the eigenvalue problem are in close agreement with the peak frequencies of the frequency spectrum computed using the ANCF model. The results in Figure 14 indicate that as the orbital radius increases, both the tension and transverse natural frequency decrease, while the longitudinal natural frequency remains constant. Figure 14a shows that the average tension decreases with increasing orbital radius, which in turn reduces the transverse natural frequency, as illustrated in Figure 14b. On the other hand, the longitudinal vibrations of the tether are independent of tension and are primarily influenced by the masses of the two satellites. Since the satellite masses remain constant, the longitudinal natural frequency is unaffected by changes in orbital radius and maintains a constant value, as shown in Figure 14c. The effect of simultaneously increasing the masses of satellites \(m_{1}\) and \(m_{2}\) from 100 to 10,000 kg on the transverse and longitudinal natural frequencies of the tether was investigated. All material properties and dimensions, except for the masses of \(m_{1}\) and \(m_{2}\), were maintained as the values listed in Table 1. It is observed in Figure 15 that the natural frequencies obtained from the eigenvalue problem closely match the peak frequencies of the frequency spectrum derived from the dynamic response computed with the ANCF model. Figure 14: First natural frequencies of the tethered satellite system for the variation of the orbital radius of \(m_{1}\): (**a**) average tension of the tether, (**b**) natural frequency of the transverse motion, and (**c**) natural frequency of the longitudinal motion. As the masses of \(m_{1}\) and \(m_{2}\) increase, the gravitational gradient and the difference in centrifugal forces grow, resulting in an increase in the average tension of the tether (Figure 15a). This increase in tension leads to a rise in the transverse natural frequency (Figure 15b). However, since the longitudinal vibrations are independent of tension and are influenced by the masses at both ends of the tether, the longitudinal natural frequency decreases as the masses increase, as shown in Figure 15c. Finally, with all material properties and dimensions fixed as given in Table 1, the effect of increasing only one satellite mass, either \(m_{1}\) or \(m_{2}\), from 100 to 10,000 kg on the natural frequencies of the tether was examined.