# Application of Absolute Nodal Coordinate Formulation in Calculation of Space Elevator System
## 2 Tether Modelling
\(\omega^{2}\) can be taken from Equation (15) and integrate Equation (13), Equation (14) along the length of the element to obtain:  
\[\vec{Q}_{g}^{\varepsilon}=-\int_{0}^{L}\frac{\mu\rho A}{\vec{r}(x)^{2}} \mathrm{d}x \tag{16}\]  
\[\vec{Q}_{\varepsilon}^{\varepsilon}=\int_{0}^{L}\frac{\mu\rho A}{R_{g}^{3}} \mathbf{C}\vec{r}(x)\mathrm{d}x \tag{17}\]  
Then, for an element with constant density \(\rho\) and a cross-sectional area \(A\), the generalized structural force \(\vec{Q}_{s}^{\varepsilon}\) can be shown as:  
\[\vec{Q}_{s}^{\varepsilon}=\vec{Q}_{g}^{\varepsilon}+\vec{Q}_{\varepsilon}^{ \varepsilon}=\mu\rho A\int_{0}^{L}(\frac{\mathbf{C}\vec{r}(x)}{R_{g}^{3}}- \frac{1}{\vec{r}(x)^{2}})\mathrm{d}x \tag{18}\]  
The external forces acting on the rope mainly include concentrated load and distributed load. The generalized force of the concentrated load \(\vec{P}\) acting on \(x_{p}\) can be presented as:  
\[\vec{Q}_{\varepsilon 1}^{\varepsilon}=\mathbf{S}(x_{p})^{\intercal}\vec{P} \tag{19}\]The generalized force of distributed load \(\vec{F}\) can be presented as:  
\[\vec{Q}_{e2}^{e}=\int_{0}^{L}\mathbf{S}(x)^{\intercal}\vec{F}\mathrm{d}x \tag{20}\]  
The system mass matrix \(\mathbf{M}\) can be assembled from the mass matrix of each element using the same method as the finite element method. The system kinetic energy is shown as:  
\[T=\vec{q}^{\intercal}\mathbf{M}\vec{q} \tag{21}\]  
where \(\vec{q}\) is the generalized coordinate vector composed of the coordinates \(\vec{q}_{e}\) of each node. The assemblies of the generalized force \(Q_{e}\), \(Q_{s}\), \(Q_{e1}\) and \(Q_{e2}\) are slightly simpler, because they are one-dimensional vectors. As a result of the use of generalized coordinates to describe the dynamical state of rope elements, the dynamic model of steel cable capture mechanism, which is a constrained multi-body system, can be expressed by a unified system equation (differential-algebraic equation) on the basis of the Lagrange equation:  
\[\frac{\mathrm{d}}{\mathrm{d}t}(\frac{\partial T}{\partial\vec{q}})^{\intercal }+\frac{\partial U_{e}}{\partial\vec{q}}=\vec{Q}_{s}+\vec{Q}_{e1}+\vec{Q}_{e2} \tag{22}\]  
In the static calculation, the quantity related to the time derivative in the dynamic equation can be directly eliminated, and the dynamic equation degenerates into:  
\[\vec{Q}_{\varepsilon}^{x}=m_{x}^{\varepsilon}\omega^{2}\mathbf{C}\vec{r}(x) \tag{14}\]0  
The setting of the boundary conditions is also the same as the finite element method. The initial value \(\vec{q}_{e0}\) of each node is shown as:  
\[\vec{q}_{e0}= \begin{bmatrix}r_{x}&0&0&1&0&0\end{bmatrix} ^{\intercal} \tag{24}\]  
where \(r_{x}\) is the absolute coordinate of \(X\)-axis of the node. Since the rope is fixed at the anchor point, the 6-degrees-of-freedom of anchor point node is set to 0, and (the number of nodes \(-\) 1) \(\times\) 6 equations can be obtained. Let  
\[F(\vec{q})=\vec{Q}_{e}(\vec{q})-\vec{Q}_{s}(\vec{q})+\vec{Q}_{e1}(\vec{q})+ \vec{Q}_{e2}(\vec{q})=0 \tag{25}\]  
The Jacobian matrix of \(F(\vec{q})\) should be obtained by numerical methods for the complicated expression of the generalized elastic force; then, the generalized coordinates \(\vec{q}\) can be solved by the Newton Raphson iterative method, and the absolute coordinates of each node can be obtained by Equation (2).