# Natural Frequencies of a Tethered Satellite System
## 3 Dynamic Responses of the ANCF Model
\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{30}\]  
where  
\[ \begin{array}{llll}N_{1}=(1/4)(\xi-1)^{2}(\xi+2),&N_{2}=(h_{e}/8)(\xi-1)^{2 }(\xi+1),\\ N_{3}=(1/4)(\xi+1)^{2}(2-\xi),&N_{2}=(h_{e}/8)(\xi+1)^{2}(\xi-1),\end{array}  \tag{31}\]  
in which  
\[\xi=2(s-s_{e})/h_{e}-1,\ h_{e}=s_{e+1}-s_{e}. \tag{32}\]  
After discretizing the tether into \(N\) elements, the displacement vector of Equation (28) was substituted into the kinetic energy of Equation (25) and the potential energy of Equation (26). Using these energy expressions, the equations of motion were derived from [PERSON]'s equation. The derived discretized equations of motion can be written as  
\[\mathbf{m}_{e}\ddot{\mathbf{d}}_{e}+\mathbf{k}_{e}\mathbf{d}_{e}+\mathbf{q}_{ e}=0,\,\text{for}\ e=1,\,2,\ \cdots,N, \tag{33}\]  
where \(\mathbf{m}_{e}\) is the element mass matrix, \(\mathbf{k}_{e}\) is the element stiffness matrix, and \(\mathbf{q}_{e}\) is the element nonlinear force vector. These matrices and vectors can be represented by  
\[\mathbf{m}_{e}=\delta_{e1}m_{1}\Big{[}\mathbf{N}^{\mathsf{T}}\mathbf{N}\Big{]} _{\xi=-1}+\delta_{eN}m_{2}\Big{[}\mathbf{N}^{\mathsf{T}}\mathbf{N}\Big{]}_{ \xi=1}+\frac{1}{2}\rho Ah_{e}\int_{-1}^{1}\mathbf{N}^{\mathsf{T}}\mathbf{N} \mathbf{d}\xi, \tag{34}\]  
\[\mathbf{k}_{e}=\int_{-1}^{1}\bigg{(}\frac{2 EI}{h_{e}}\mathbf{N}^{\mathsf{T}} \mathbf{N}^{\mathsf{T}}-\frac{EA}{h_{e}}\mathbf{N}^{\mathsf{T}}\mathbf{N}^{ \mathsf{T}}\mathbf{N}\bigg{)}\,d\xi, \tag{35}\]  
\[ \begin{array}{llll}\mathbf{q}_{e}&=GM\bigg{\{}\delta_{e1}m_{1}\bigg{[} \bigg{(}\mathbf{d}_{e}^{\mathsf{T}}\mathbf{N}^{\mathsf{T}}\mathbf{N}\mathbf{d }_{e}\bigg{)}^{-3/2}\mathbf{N}^{\mathsf{T}}\mathbf{N}\bigg{]}_{\xi=-1}+\delta _{eN}m_{2}\bigg{[}\bigg{(}\mathbf{d}_{e}^{\mathsf{T}}\mathbf{N}^{\mathsf{T}} \mathbf{N}\mathbf{d}_{e}\bigg{)}^{-3/2}\mathbf{N}^{\mathsf{T}}\mathbf{N} \bigg{]}_{\xi=1}\bigg{\}}\mathbf{d}_{e}\\ &+\frac{1}{2}\rho h_{e}GM\int_{-1}^{1}\bigg{(}\mathbf{d}_{e}^{\mathsf{T}} \mathbf{N}^{\mathsf{T}}\mathbf{N}\mathbf{d}_{e}\bigg{)}^{-3/2}\mathbf{N}^{ \mathsf{T}}\mathbf{N}\mathbf{d}_{e}\mathbf{d}\xi+\frac{4E}{h_{e}^{2}}\int_{-1 }^{1}\mathbf{d}_{e}^{\mathsf{T}}\mathbf{N}^{\mathsf{T}}\mathbf{N}\mathbf{d}_{e }\mathbf{N}^{\mathsf{T}}\mathbf{N}\mathbf{d}_{e}\mathbf{d}\xi.\end{array}  \tag{36}\]  
By assembling the element mass matrix, element stiffness matrix, and element nonlinear force vector, the following global matrix-vector equation was obtained:  
\[\mathbf{M}\ddot{\mathbf{d}}+\mathbf{K}\mathbf{d}+\mathbf{Q}=0, \tag{37}\]  
where \(\mathbf{d}\) is the global displacement vector, \(\mathbf{M}\) is the global mass matrix, \(\mathbf{K}\) is the global stiffness matrix, and \(\mathbf{Q}\) is the global nonlinear force vector. The dynamic responses for  
Figure 8: Discretized ANCF model of the tethered satellite system. the displacements in the \(x\)- and \(y\)-directions, as well as the change in these displacements with respect to arc length, can be computed by applying a time integration method to Equation (37). Since Equation (37) represents a nonlinear equation of motion, the dynamic responses were obtained using the Newmark method with the Newton-Raphson method. The dynamic responses for the libration angle and tether deformation can be obtained from the displacements derived from Equation (37). To eliminate the orbital motion from the computed displacements \(x_{e}\) and \(y_{e}\), the following coordinate transformation was applied:  
\[ \begin{cases}\overline{x}_{e}\\ \overline{y}_{e}\end{cases} = \begin{bmatrix}\cos\psi&\sin\psi\\ -\sin\psi&\cos\psi\end{bmatrix}\begin{cases}x_{e}-x_{1}\\ y_{e}-y_{1}\end{cases} \;\;\text{for}\;e=1,\;2,\;\cdots,\;N+1. \tag{38}\]  
The above coordinate transformation removes the orbital motion of the satellite system from all nodes, leaving only the libration motion centered around satellite \(m_{1}\).