# Natural Frequencies of a Tethered Satellite System
## 5 Conclusions  
In this study, we investigated the natural frequencies of tethered satellite systems, focusing on both the rigid-body motion of the satellites and the deformation of the tether. To evaluate the natural frequencies of rigid-body motion, a dumbbell model of the tethered satellite system was developed, and the nonlinear equations of motion were derived using [PERSON]'s equations. These nonlinear equations were linearized near the equilibrium configuration through the application of the perturbation method, resulting in an eigenvalue problem. From the eigenvalue problem, the natural frequencies associated with orbital and libration motions were determined, and the effects of key system parameters--including tether length, orbital radius, and satellite masses--on these frequencies were systematically analyzed. To assess the dynamic response arising from tether deformation, an ANCF model of the tethered satellite system was constructed. The Newmark time integration method was employed to compute dynamic responses, and the dominant frequency components were identified through frequency spectrum analysis. Additionally, considering the influence of gravitational gradient and centrifugal forces, the tether was modeled as a tensioned beam with masses at its ends. An eigenvalue problem was formulated for this configuration, enabling the analytical calculation of natural frequencies without relying on the computational complexity of the ANCF-based approach. The results demonstrated that the analytically derived natural frequencies closely matched the dominant frequency components obtained from the dynamic responses computed using the ANCF model. Furthermore, the effects of system parameters on the natural frequencies for the transverse and longitudinal vibrations of the tether were thoroughly examined. The key findings derived from this study are summarized as follows:  
1. The first natural frequency obtained from the eigenvalue analysis of the dumbbell model corresponds closely to the orbital angular velocity of the satellite system, while the second natural frequency represents the libration frequency. 2. The first natural frequency, associated with the orbital angular velocity, decreases as the tether length, orbital radius, or the mass of the satellite farther from Earth increases, but it increases with the mass of the satellite closer to Earth. 3. The second natural frequency, representing the libration frequency, decreases with increasing tether length, orbital radius, or the mass of the satellite closer to Earth, but increases with the mass of the satellite farther from Earth. 4. In the dynamic response for the transverse motion of the tether, the dominant frequency components are the transverse natural frequency and the libration frequency. 5. In the dynamic response for the longitudinal motion of the tether, the dominant frequency components are the longitudinal natural frequency and the libration frequency. 6. The transverse natural frequency of the tether decreases with increasing tether length or orbital radius but increases with the satellite masses. 7. The longitudinal natural frequency of the tether is independent of the orbital radius but decreases with increasing tether length or satellite masses. Conceptualization, J.C.; methodology, Y.Y. and W.J. ; software, Y.Y. and W.J. ; validation, Y.Y. and W.J. ; formal analysis, J.C. and W.J. ; investigation, Y.Y. and W.J. ; resources, Y.Y. and W.J. ; data curation, Y.Y. ; writing--original draft preparation, Y.Y. ; writing--review and editing, W.J. and J.C.; visualization, Y.Y. ; supervision, J.C.; project administration, J.C.; funding acquisition, J.C. All authors have read and agreed to the published version of the manuscript. **Funding:** This work was supported by a grant from the National Research Foundation of Korea (NRF), funded by the Korean Government (MEST) (NRF-2021R1A2C2007979). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.