# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 2 Modeling of a Tethered Satellite System
Similarly, the potential energy of the tethered satellite system can be written as  
\[U=\frac{1}{2}{\int_{0}^{L}}\,EA\epsilon^{2}\mathrm{d}s-GM\bigg{(}\frac{m_{A}}{| \mathbf{r}_{A}|}+\frac{m_{B}}{|\mathbf{r}_{B}|}+\frac{m_{v}}{|\mathbf{r}_{v}|}+ \int_{0}^{L}\,\frac{\rho}{|\mathbf{r}|}\mathrm{d}s\bigg{)}, \tag{5}\]  
where \(G\) is the universal gravitational constant, \(M\) is the Earth's mass, \(E\) is the Young's modulus of the tether, and \(\epsilon\) is the strain of the tether. The values of \(G\) and \(M\) are given by \(G=6.6743\times 10^{-11}\) m\({}^{3}\)/kg s\({}^{2}\) and \(M=5.9722\times 10^{24}\) kg, respectively. Using the one-dimensional Lagrange strain, the strain \(\epsilon\) can be represented as  
\[\epsilon=\frac{1}{2}\bigg{[}\bigg{(}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s} \bigg{)}^{\mathrm{T}}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}-1\bigg{]}. \tag{6}\]  
In this study, the damping effect of the tether is neglected for simplicity. The tethered satellite system with a moving payload was discretized using the ANCF. The tether was discretized into \(N\) uniform two-node elements, as shown in Figure 2a, where the numbers represent the global node numbers. The \(e\)-th element consists of nodes \(e\) and \(e+1\). Satellites A and B are positioned at nodes \(1\) and \(N+1\), respectively, whereas the payload moves along the \(i\)-th element. Figure 2a shows that the moving payload is located between nodes \(i\) and \(i+1\). In Figure 2b, \(s_{e}\) is the arc length from satellite \(A\) to node \(e\), \(x_{e}\) and \(y_{e}\) are the global nodal coordinates of node \(e\), and \(\theta_{e}^{x}\) and \(\theta_{e}^{y}\) are the derivatives of \(x\) and \(y\) with respect to \(s\) at \(s=s_{e}\), respectively. The position vector of a point inside element \(e\), denoted by \(\mathbf{r}_{e}\), can be represented as  
\[\mathbf{r}_{e}(s,t)=\mathbf{N}(s)\mathbf{d}_{e}(t)\ \ \ \text{for}\ s_{e}\leq s \leq s_{e+1}, \tag{7}\]  
where \(\mathbf{N}\) is a \(2\times 8\) shape function matrix, and \(\mathbf{d}_{e}\) is an \(8\times 1\) element nodal coordinate vector given by  
\[\mathbf{d}_{e}=\Big{\{}x_{e},\ y_{e},\ \theta_{e}^{x},\ \theta_{e}^{y},\ x_{e+1}, \ y_{e+1},\ \theta_{e+1}^{x},\ \theta_{e+1}^{y}\Big{\}}^{\mathrm{T}}. \tag{8}\]  
Using the coordinate transformation given by  
\[\mathbf{r}(s,t)=\genfrac{\{}{\}}{0.0 pt}{}{x(s,t)}{y(s,t)}. \tag{1}\]0  
where \(\xi\) is the parametric coordinate and \(h\) is the constant element length defined by \(h=s_{e+1}-s_{e}\), the position vector of a point inside element can be rewritten as  
\[\mathbf{r}(s,t)=\genfrac{\{}{\}}{0.0 pt}{}{x(s,t)}{y(s,t)}. \tag{1}\]1  
\[\mathbf{r}(s,t)=\genfrac{\{}{\}}{0.0 pt}{}{x(s,t)}{y(s,t)}. \tag{1}\]2 Discretization of tethered satellites with a moving payload: (**a**) ANCF model and (**b**) two-node element. The numbers represent the nodal points of the discretized model. where  
\[\mathbf{N}(\xi)= \begin{bmatrix}N_{1}&0&N_{2}&0&N_{3}&0&N_{4}&0\\ 0&N_{1}&0&N_{2}&0&N_{3}&0&N_{4}\end{bmatrix} , \tag{11}\]  
with  
\[N_{1}=(1/4)(\xi-1)^{2}(\xi+2),\quad N_{2}=(h_{e}/8)(\xi-1)^{2}(\xi+1),\]  
\[N_{3}=(1/4)(\xi+1)^{2}(2-\xi),\quad N_{2}=(h_{e}/8)(\xi+1)^{2}(\xi-1). \tag{12}\]  
Next, we consider the position and velocity vectors of the moving payload as it travels along the tether at a constant speed \(V\). In this scenario, both the nodal coordinate vector and shape function matrix become functions of time. When the moving payload is located between nodes \(i\) and \(i+1\), its position vector can be represented as  
\[\mathbf{r}_{v}(\xi(t),t)=\mathbf{N}(\xi(t))\mathbf{d}_{i}(t). \tag{13}\]  
From Equation (9), the time derivative of the parametric coordinate is given by \(\dot{\xi}=2V/h\). Differentiating the position vector with respect to time yields the velocity vector of the moving payload:  
\[\dot{\mathbf{r}}_{v}=(2V/h)\mathbf{N}^{\prime}\mathbf{d}_{i}+\mathbf{N}\dot{ \mathbf{d}}_{i}, \tag{14}\]  
where the prime (\({}^{\prime}\)) represents differentiation with respect to the parametric coordinate \(\xi\).