# Application of Absolute Nodal Coordinate Formulation in Calculation of Space Elevator System
## 2 Tether Modelling  
The typical space elevator system model comprises a tether, climbers and a zenith anchor. It can be considered that the anchor point of the tether is at a fixed location while the system oscillation has little impact on the floating platform. The centrifugal force would pull the tether perpendicular to the Earth's axis while the tether would also droop toward the equator owing to gravity, which points to the center of the Earth. The space elevator system model is depicted in Figure 1. The center of the Earth \(O\) is assumed to be inertially fixed and is used as the origin of the inertial frame. \(O_{0}\) is the anchor point of the tether on the earth's equator. The \(X\) axis points from point \(O\) to point \(O_{0}\) and is perpendicular to the ground. The \(Z\) axis points to the North Pole along the Earth's rotation axis. The \(Y\) axis is perpendicular to the \(X\) axis and \(Z\) axis, \(\omega\) is the angular velocity of the system is equal to the rotation rate of the Earth. \(L_{T}\) is the nominal (unstressed) length of the tether. \(m_{c}\) is the mass of the counterweight at the end of the tether. \(\theta\) is the angle between the projection of the rope of the space elevator system in the \(XY\) plane and the \(X\) axis. \(\phi\) is the angle between the projection of the rope of the space elevator system in the \(XY\) plane and itself. The rope of the space elevator system is a kind of slender flexible body, in which the scale of one dimension (length) is much larger than the other two dimensions (section), so it can be modeled based on beam element. ANCF beam elements can generally be divided into two categories. The first one is the beam element without considering the transverse shear effect. This kind of element describes the displacement and deformation field of the beam through the node position coordinates and their longitudinal derivatives. Due to the lack of transverse derivatives, it is usually called the gradient default beam element. The other one is the beam element considering the transverse shear effect. This kind of element introduces the transverse derivative of the node; thus, the section can be deformed. The section size of the rope of the space elevator system is much smaller than the length, so the shear effect can be ignored. In this paper, a flexible rope model is established based on the rope beam element with gradient default beam element. The model of ANCF rope element is shown in Figure 2. \(O-XYZ\) is the absolute coordinate system of space. \(L\) is the length of the rope element. \(x\) is the coordinate of the element in the length direction. \(\vec{q_{1}}\) and \(\vec{q_{2}}\) are the generalized coordinates of the two nodes of the rope element. \(\vec{r}(x)\) is the absolute coordinate of the point on the element, whose element coordinate is \(x\). Figure 1: Space elevator system model. Figure 2: Model of ANCF rope element. The node coordinates of the gradient default rope beam element are composed of the node position and its derivative to the axial element coordinates. For a three-dimensional two node element with length \(L\), the node coordinates can be expressed as:  
\[\vec{q}_{e}= \begin{bmatrix}\vec{q}_{1}&\vec{q}_{2}\end{bmatrix} ^{\intercal}= \begin{bmatrix}\vec{r}^{\intercal}(x=0)&\vec{r}_{x}^{\intercal}(x=0)&\vec{r}^{ \intercal}(x=L)&\vec{r}_{x}^{\intercal}(x=L)\end{bmatrix} ^{\intercal} \tag{1}\]  
The displacement field function of the element can be obtained by cubic Hermite interpolation of the node coat dinates of the element:  
\[\vec{r}(x,t)=\mathbf{S}(x)\vec{q}_{e}(t) \tag{2}\]  
where \(\mathbf{S}(x)\) is the interpolation basis function, that is, the element shape function, expressed as follows:  
\[\mathbf{S}(x)= \begin{bmatrix}S_{1}(x)\mathbf{I_{3}}&S_{2}(x)\mathbf{I_{3}}&S_{ 3}(x)\mathbf{I_{3}}&S_{4}(x)\mathbf{I_{3}}\end{bmatrix}  \tag{3}\]  
where \(\mathbf{I_{3}}\) is third order identity matrix, and \(S_{i}(x)\) can be presented as:  
\[