# Dynamic Behavior and Libration Control of an Electrodynamic Tether System for Space Debris Capture
## 2 Dynamics of a Flexible EDT System
Using the one-dimensional Green-Lagrange strain, the strain in Equation (4) is given by  
\[\varepsilon=\frac{1}{2}\left(\textbf{r}\cdot\textbf{r}^{\prime}-1\right) \tag{5}\]  
where the prime (\(\prime\)) denotes differentiation with respect to \(s\). The work done by non-conservative forces such as atmospheric drag, solar radiation pressure, electrodynamic force, damping force, and so on should be considered; however, for simplicity, in this study only the work done by the electrodynamic and damping forces is considered. The virtual work done by these forces can be represented as  
\[\delta\textbf{W}=\int_{0}^{L}\textbf{F}_{l}\cdot\delta\textbf{r}\text{d}s- \int_{0}^{L}EA\alpha\dot{\varepsilon}\delta\varepsilon\text{d}s \tag{6}\]  
where \(\delta\) is the variation notation, \(\textbf{F}_{l}\) is the electrodynamic force, and \(\delta\textbf{r}\) the virtual displacement of the tether. The electromagnetic force in the EDT system is generated by the interaction between the current flowing through the tether and the magnetic field of Earth as the tether moves through the magnetic field. In this study, the 13 th IGRF Earth magnetic field model is used to compute electromagnetic force. The magnetic field vector of Earth, **B**, derived from the IGRF model, may be expressed using Legendre polynomials as follows:  
\[\textbf{B}=\left\{B_{r},\ B_{\phi^{\prime}},\ B_{\theta}\right\}^{\text{T}} \tag{7}\]  
where \(B_{r}\), \(B_{\phi^{\prime}}\), and \(B_{\lambda}\) are the components of the magnetic field strength in spherical coordinates:  
\[B_{r}=\sum_{n=1}^{N}\sum_{m=0}^{n}\bigg{(}\frac{R}{r}\bigg{)}^{n+2}(n+1)P_{n} ^{m}\cos\phi^{\prime}(G_{n}^{m}\cos m\lambda+H_{n}^{m}\sin m\lambda) \tag{8}\]  
\[K=\frac{1}{2}m_{m}\dot{\mathbf{r}}_{m}\cdot\dot{\mathbf{r}}_{m}+\frac{1}{2}m_{ n}\dot{\mathbf{r}}_{n}\cdot\dot{\mathbf{r}}_{n}+\frac{1}{2}\int_{0}^{L}\rho A\dot{ \mathbf{r}}\cdot\dot{\mathbf{r}}\mathrm{d}s \tag{2}\]0  
\[K=\frac{1}{2}m_{m}\dot{\mathbf{r}}_{m}\cdot\dot{\mathbf{r}}_{m}+\frac{1}{2}m_{ n}\dot{\mathbf{r}}_{n}\cdot\dot{\mathbf{r}}_{n}+\frac{1}{2}\int_{0}^{L}\rho A\dot{ \mathbf{r}}\cdot\dot{\mathbf{r}}\mathrm{d}s \tag{2}\]1in which \(r\) is the geocentric radius, \(R\) is the radius of the Earth, \(\phi^{\prime}\) is the co-latitude, \(\lambda\) is the longitude, \(P_{n}^{m}\) is the Schmidt semi-normalized associated Legendre functions of degree \(n\) and order \(m\), and \(G_{n}^{m}\) and \(H_{n}^{m}\) are Gaussian coefficients. As the EDT system traverses the magnetic field of Earth, the EMF induced in the tether is determined by  
\[K=\frac{1}{2}m_{m}\dot{\mathbf{r}}_{m}\cdot\dot{\mathbf{r}}_{m}+\frac{1}{2}m_{ n}\dot{\mathbf{r}}_{n}\cdot\dot{\mathbf{r}}_{n}+\frac{1}{2}\int_{0}^{L}\rho A\dot{ \mathbf{r}}\cdot\dot{\mathbf{r}}\mathrm{d}s \tag{2}\]2  
where \(\mathbf{v}\) denotes the relative velocity vector between the tether and the magnetic field and \(\mathbf{e}_{t}\) is the unit vector directed from the sub-satellite to the main satellite. This induced EMF causes the EDT system to develop partial positive and negative charges relative to the surrounding ionospheric plasma. The positively charged section of the tether attracts free electrons, while the negatively charged section attracts ions. Ultimately, electrons flowing along the tether are emitted through an electron emitter positioned at the tether end, resulting in an induced current within the EDT system. The current \(I\) and voltage \(\Phi\) generated in the EDT are characterized by the OML theory as follows:  
\[\frac{\mathrm{d}I}{\mathrm{d}s}=\left\{ \begin{matrix}q_{e}N_{e}d\sqrt{2q_{e} \Phi/m_{e}}&\text{for}\ \ \Phi>0\\ -\mu q_{e}N_{e}d\sqrt{-2q_{e}\Phi/m_{e}}&\text{for}\ \ \Phi<0\end{matrix} \right.