# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 3 Discretized Equations of Motion
The potential energy, as given by Equation (5), can be rewritten as  
\[U=\frac{1}{2}\sum_{e=1}^{N}\int_{s_{e}}^{s_{e+1}}EA\epsilon_{e}^{2}\mathrm{d}s -GM\left(\frac{m_{A}}{|\mathbf{r}_{1}|}+\frac{m_{B}}{|\mathbf{r}_{N+1}|}+\frac {m_{v}}{|\mathbf{r}_{v}|}+\sum_{e=1}^{N}\int_{s_{e}}^{s_{e+1}}\frac{\rho}{| \mathbf{r}_{e}|}\mathrm{d}s\right). \tag{17}\]  
Since the magnitude of the position vector (representing the distance from the center of the Earth) is the square root of the product of the vector's transpose and itself, we express the magnitude of \(\mathbf{r}_{e}\) as  
\[|\mathbf{r}_{e}|=\left(\mathbf{d}_{e}^{\mathrm{T}}\mathbf{N}^{\mathrm{T}} \mathbf{N}\mathbf{d}_{e}\right)^{1/2}. \tag{18}\]Furthermore, the Lagrange strain, as defined by Equation (6) can be represented by  
\[\varepsilon_{e}=\frac{1}{2}\bigg{(}\frac{4}{h^{2}}\mathbf{d}_{e}^{\mathrm{T}} \mathbf{N}^{\prime\mathrm{T}}\mathbf{N}^{\prime}\mathbf{d}_{e}-1\bigg{)}. \tag{19}\]  
Using Equations (18) and (19), the potential energy from Equation (17), can be written as  
\[U=\frac{1}{16}EAh\sum_{e=1}^{N}\int_{-1}^{1}\bigg{(}\frac{4}{h^{2}}\mathbf{d}_ {e}^{\mathrm{T}}\mathbf{N}^{\prime\mathrm{T}}\mathbf{N}^{\prime}\mathbf{d}_{e }-1\bigg{)}^{2}\mathrm{d}\xi\]  
\[-\frac{1}{2}GM\rho h\sum_{e=1}^{N}\int_{-1}^{1}\bigg{(}\mathbf{d}_{e}^{ \mathrm{T}}\mathbf{N}^{\mathrm{T}}\mathbf{N}\mathbf{d}_{e}\bigg{)}^{-1/2} \mathrm{d}\xi-GMm_{A}\bigg{[}\mathbf{d}_{1}^{\mathrm{T}}\big{(}\mathbf{N}^{ \mathrm{T}}\mathbf{N}\big{)}_{\xi=-1}\mathbf{d}_{1}\bigg{]}^{-1/2}\]  
\[-GMm_{B}\bigg{[}\mathbf{d}_{N}^{\mathrm{T}}\big{(}\mathbf{N}^{\mathrm{T}} \mathbf{N}\big{)}_{\xi=1}\mathbf{d}_{N}\bigg{]}^{-1/2}-GMm_{v}\bigg{[}\mathbf{ d}_{i}^{\mathrm{T}}\big{(}\mathbf{N}^{\mathrm{T}}\mathbf{N}\big{)}_{\xi=-\xi(i)} \mathbf{d}_{i}\bigg{]}^{-1/2}. \tag{20}\]  
The discretized equations of motion can be derived by substituting the kinetic and potential energies from Equations (16) and (20), respectively, into the following Lagrange equation:  
\[K=\frac{1}{4}\sum_{e=1}^{N}\rho A\dot{\mathbf{d}}_{e}^{\mathrm{T}}\left(\int_ {-1}^{1}\mathbf{N}^{\mathrm{T}}\mathbf{N}\mathrm{d}_{e}^{x}\right)\dot{ \mathbf{d}}_{e}+\frac{1}{2}m_{A}\dot{\mathbf{d}}_{1}^{\mathrm{T}}\left( \mathbf{N}^{\mathrm{T}}\mathbf{N}\right)_{\xi=-1}\dot{\mathbf{d}}_{1}\]0  
where \(W\) represents the work of non-conservative forces. In this study, external forces such as solar radiation pressure and atmospheric drag are neglected, resulting in zero work resulting from non-conservative forces. For more detailed information on ANCF, refer to [PERSON] [28]. The derived equations of motion can be expressed as  
\[K=\frac{1}{4}\sum_{e=1}^{N}\rho A\dot{\mathbf{d}}_{e}^{\mathrm{T}}\left(\int_ {-1}^{1}\mathbf{N}^{\mathrm{T}}\mathbf{N}\mathrm{d}_{e}^{x}\right)\dot{ \mathbf{d}}_{e}+\frac{1}{2}m_{A}\dot{\mathbf{d}}_{1}^{\mathrm{T}}\left( \mathbf{N}^{\mathrm{T}}\mathbf{N}\right)_{\xi=-1}\dot{\mathbf{d}}_{1}\]1  
where \(\mathbf{m}_{e}\) is the element mass matrix, \(\mathbf{g}_{e}\) is the element gyroscopic matrix due to the moving payload, \(\mathbf{k}_{e}\)  
\[K=\frac{1}{4}\sum_{e=1}^{N}\rho A\dot{\mathbf{d}}_{e}^{\mathrm{T}}\left(\int_ {-1}^{1}\mathbf{N}^{\mathrm{T}}\mathbf{N}\mathrm{d}_{e}^{x}\right)\dot{ \mathbf{d}}_{e}+\frac{1}{2}m_{A}\dot{\mathbf{d}}_{1}^{\mathrm{T}}\left( \mathbf{N}^{\mathrm{T}}\mathbf{N}\right)_{\xi=-1}\dot{\mathbf{d}}_{1}\]2  
where \(\mathbf{m}_{e}\), \(\mathbf{g}_{e}\), and \(\mathbf{k}_{e}\) are \(8\times 8\) square matrices, and \(\mathbf{q}_{e}\) is an \(8\times 1\) column vector.