# Natural Frequencies of a Tethered Satellite System
## 1 Introduction  
The study of tethered satellite systems represents a critical area of research in space engineering, offering significant potential for advancing satellite deployment, on-orbit servicing, and space exploration missions. Tethered satellite systems comprise satellites interconnected by a flexible tether, forming a dynamically coupled system where the interactions between the rigid-body motions of the satellites and the vibrational behavior of the tether play a pivotal role in system stability. The dynamics of such systems are inherently complex due to gravitational gradient forces, centrifugal effects, and the flexibility of the tether. Ensuring the stability of tethered satellite systems is essential, as uncontrolled vibrations or resonance phenomena can lead to mission failure or even system disintegration. Consequently, understanding the natural frequencies of these systems is a prerequisite for designing stable and reliable tethered satellite operations. Satellite systems can be broadly categorized into conventional satellites and tethered satellite systems. Each type poses unique challenges in structural dynamics, stability, and control. Research on vibration mitigation in conventional satellites has focused on minimizing disturbances caused by internal mechanisms or external environments. For instance, [PERSON] et al. [1] developed a dual micro-vibration isolation system for optical satellites, achieving significant vibration attenuation and improving imaging stability. [PERSON] al. [2] addressed vibration issues in space robots with flexible appendages, proposing a closed-loop control system to manage rigid-flexible coupling dynamics, validated by experimental results. These studies highlight the importance of robust control strategies and isolation mechanisms in enhancing satellite performance. To ensure structural integrity and avoid resonance, researchers have extensively studied natural frequency optimization in conventional satellites. [PERSON] et al. [3] employed hybrid optimization algorithms to incorporate natural frequency constraints in satellite layout design, improving both stability and performance. [PERSON] et al. [4] proposed a lightweight strategy for adjusting natural frequencies by modifying structural stiffness, validated through finite element and experimental analyses. [PERSON] et al. [5] introduced a simplified method for estimating the natural frequencies of micro-satellites, enabling effective early-stage design without complex computational requirements. [PERSON] and [PERSON] [6] investigated the coupling of pitch motion, axial vibration, and orbital motion, demonstrating the critical influence of natural frequency ratios on dynamic behaviors in large space structures. Tethered satellite systems, however, present unique challenges due to their inherently flexible structures and complex coupled dynamics. [PERSON] et al. [7] explored methods for minimizing deviations and vibrations during maneuver planning, proposing optimization-based approaches to suppress high-order vibrations. [PERSON] et al. [8] introduced a friction self-excitation mechanism for controlling longitudinal vibrations in tethered systems, validated through ground experiments. [PERSON] and [PERSON] [9] applied optimization techniques to enhance active damping, focusing on energy dissipation during longitudinal vibration suppression. [PERSON] et al. [10] analyzed nonlinear vibrations of tethered systems under station-keeping conditions, providing insights into the physical causes of high-frequency peaks. Modal analysis and frequency characterization have been central to understanding tethered satellite dynamics. [PERSON] et al. [11] presented an orbit-attitude-vibration coupled dynamic model for tethered solar power satellites, identifying dependencies on parameters like orbital radius and bus mass. [PERSON] and [PERSON] [12] compared different tethered system models for space debris removal, deriving analytical solutions for natural frequencies under various configurations. [PERSON] et al.