# Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture
## 2 Dynamics of a Flexible EDT System  
As depicted in Figure 1, the EDT system, consisting of a main satellite, a sub-satellite, a bare tether, and an electron emitter, generates an electromotive force (EMF) as it orbits Earth, interacting with the magnetic field of Earth. This interaction induces a potential difference along the tether, enabling the collection of electrons from the surrounding plasma environment. These electrons are then transported through the tether and emitted via an electron emitter of an electric field emission type installed on the sub-satellite. The current flowing through the tether interacts with the magnetic field of Earth to generate a Lorentz force. In a passive tether system, this Lorentz force opposes the direction of the tether motion, contributing to the overall functionality of the system. As shown in Figure 2, the main satellite has a mass \(m_{m}\), the sub-satellite has a mass \(m_{n}\), and the tether has a length \(L\). Both satellites are treated as point masses, with the mass moment of inertia neglected. Due to the tether length being substantially greater than its diameter, it is modeled as a string. The position along the tether is denoted by the variable \(s\), where \(s=0\) corresponds to the location of the main satellite and \(s=L\) corresponds to the location of the sub-satellite. A global coordinate system is established to describe the position of the EDT system in orbit, with the center of Earth as its origin \(O\). The \(X\) axis points towards the vernal equinox, the \(Z\) axis towards the North Pole, and the \(Y\) axis is determined by the right-hand rule. The position vector \(\mathbf{r}\), representing a point along the tether, is a function of arc length \(s\) and time \(t\), expressed as follows:  
\[\mathbf{r}(s,t)=x(s,t)\mathbf{i}+y(s,t)\mathbf{j}+z(s,t)\mathbf{k} \tag{1}\]  
where \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are unit vectors along the \(X\)-, \(Y\)-, and \(Z\)-axis directions, respectively. The equation of motion for the EDT system is derived using the [PERSON]'s equation. First, the kinetic energy of the EDT system is the sum of the kinetic energies of the main satellite, the sub-satellite, and the tether. The total kinetic energy of the EDT system is given by  
\[K=\frac{1}{2}m_{m}\dot{\mathbf{r}}_{m}\cdot\dot{\mathbf{r}}_{m}+\frac{1}{2}m_{ n}\dot{\mathbf{r}}_{n}\cdot\dot{\mathbf{r}}_{n}+\frac{1}{2}\int_{0}^{L}\rho A\dot{ \mathbf{r}}\cdot\dot{\mathbf{r}}\mathrm{d}s \tag{2}\]  
Figure 1: Working principle of an EDT system. Figure 2: Dynamic model of EDT system. where the superposed dot represents differentiation with respect to time, \(\rho\) and \(A\) are the density and cross-sectional area of the tether, respectively, and \(\textbf{r}_{m}\) and \(\textbf{r}_{n}\) are the position vectors of the main satellite and sub-satellite. The first two terms in Equation (2) represent the kinetic energies of the satellites and the last term represents the kinetic energy of the tether. The potential energy of the EDT system is derived as the sum of gravitational potential energy and strain energy. Using the Kelvin-Voigt model [30], the stress \(\sigma\) within the tether can be expressed as a function of the strain \(\varepsilon\) and strain rate \(\dot{\varepsilon}\) as follows:  
\[\sigma=E\big{(}\varepsilon+\alpha\dot{\varepsilon}\big{)} \tag{3}\]  
where \(E\) and \(\alpha\) are [PERSON]'s modulus and the internal damping coefficient of the tether, respectively. The total potential energy of the EDT system may be written as  
\[U=\frac{1}{2}\int_{0}^{L}EA\varepsilon^{2}\text{d}s-GM\bigg{(}\frac{m_{m}}{| \textbf{r}_{m}|}+\frac{m_{n}}{|\textbf{r}_{n}|}+\int_{0}^{L}\frac{\rho A}{| \textbf{r}|}\text{d}s\bigg{)} \tag{4}\]  
where \(G\) is the gravitational constant and \(M\) is the mass of the Earth. The values of \(G\) and \(M\) are given by \(G=6.673\times 10^{-11}\) N \(\cdot\) m\({}^{2}\)/kg\({}^{-2}\) and \(M=5.979\times 10^{24}\) kg, respectively.