# Research on the Dynamics of the Space Tubular Expandable Structure Driving Deployment Unit
## 2 Methods
### The Spatial Nodal Dynamic Description of the Thin-Walled Flexible Beam Structure
#### 2.2.1 Analysis of the Displacement Field in the Flexible Beam Structure  
The global position vector \(\mathbf{r}\) of any point on the space absolute coordinate flexible beam element can be described by the global shape function [34; 35] and the absolute node coordinates of the element as follows:  
\[\mathbf{r}=\mathbf{S}(x,y,z)\mathbf{e} \tag{1}\]  
where: \(\mathbf{S}\) is the element shape function matrix; \(x,y,z\) are the position coordinates of nodes in the global coordinate system; \(\mathbf{e}\) is the absolute node coordinate vector, which is represented by the node displacement vector and slope vector. Figure 6 shows the change of the global position vector \(\mathbf{r}\). Difference polynomial of displacement field of this element:  
\[\mathbf{r}= \begin{bmatrix}r_{1}\\ r_{2}\\ r_{3}\end{bmatrix} = \begin{bmatrix}a_{0}+a_{1}x+a_{2}y+a_{3}z+a_{4}xy+a_{5}xz+a_{6}x^ {2}+a_{7}x^{3}\\ b_{0}+b_{1}x+b_{2}y+b_{3}z+b_{4}xy+b_{5}xz+b_{6}x^{2}+b_{7}x^{3}\\ c_{0}+c_{1}x+c_{2}y+c_{3}z+c_{4}xy+c_{5}xz+c_{6}x^{2}+c_{7}x^{3}\end{bmatrix}  \tag{2}\]  
The displacement vector and gradient vector in node coordinates can be obtained through rigid body motion from the undeformed reference configuration. The three-dimensional beam element with absolute node coordinates is an isoparametric element [36,37]. Each element consists of 2 nodes, and each node has 12 node coordinates. When the beam element is undeformed, its length is denoted by \"1\". The absolute node coordinate vector of the element can be expressed as follows:  
\[\mathbf{e}= \begin{bmatrix}\mathbf{r}_{i}\mathbf{r}_{i,x}\mathbf{r}_{i,y}\mathbf{ r}_{i,z}\mathbf{r}_{i}\mathbf{r}_{i,x}\mathbf{r}_{i,y}\mathbf{r}_{i,z}\end{bmatrix} ^{\text{T}} \tag{3}\]  
where: \(\mathbf{r}_{i}\) and \(\mathbf{r}_{i}\) are the absolute position vectors of node k on unit j; \(\mathbf{r}_{x}\), \(\mathbf{r}_{y}\), and \(\mathbf{r}_{z}\) are the partial derivative vectors of element nodes \(i\) and \(j\) in the \(x\), \(y\), and \(z\) directions, respectively, which are used to describe the deformation of the beam cross-section in this direction. Figure 5: Space nodal dynamic description flowchart. Figure 6: The ANCF (absolute nodal coordinate formulation) coordinate vector. \(\mathbf{r}_{x}=\partial\mathbf{r}/\partial x,\mathbf{r}_{y}=\partial\mathbf{r}/ \partial y,\mathbf{r}_{z}=\partial\mathbf{r}/\partial z\). Thus, the overall node coordinate vector of the element can be written as:  
\[ \begin{array}{l}e=\left[r_{i1},r_{i2},r_{i3},\frac{\partial r_{i1}}{\partial x },\frac{\partial r_{i2}}{\partial x},\frac{\partial r_{i3}}{\partial x},\frac{ \partial r_{i1}}{\partial y},\frac{\partial r_{i2}}{\partial y},\frac{\partial r _{i3}}{\partial y},\frac{\partial r_{i1}}{\partial z},\frac{\partial r_{i2}}{ \partial z},\frac{\partial r_{i3}}{\partial z}\right.\\ \left.r_{j1},r_{j2},r_{j3},\frac{\partial r_{i1}}{\partial x},\frac{\partial r _{j2}}{\partial x},\frac{\partial r_{i3}}{\partial x},\frac{\partial r_{i1}}{ \partial x},\frac{\partial r_{i2}}{\partial y},\frac{\partial r_{i3}}{\partial y },\frac{\partial r_{i2}}{\partial z},\frac{\partial r_{i2}}{\partial z},\frac{ \partial r_{i2}}{\partial z}\right]\end{array}  \tag{4}\]  
It can be observed that each beam element has 24 node coordinates, which is also a disadvantage of the absolute node coordinates method. The large number of node coordinates results in reduced calculation efficiency. The element shape function matrix can be defined using the element interpolation polynomial and node coordinates:  
\[\mathbf{S}=\left[S_{1}I\;S_{2}I\;S_{3}I\;S_{4}I\;S_{5}I\;S_{6}I\;S_{7}I\;S_{8}I\right] \tag{5}\]  
where matrix I is a 3 \(\times\) 3 order identity matrix:  
\[ \begin{array}{l}S_{1}=1-3\xi^{2}+2\xi^{3},S_{2}=l(\xi-2\xi^{2}+\xi^{3})\\ S_{3}=l(\eta-\zeta\eta),S_{4}=l(\zeta-\zeta\zeta)\\ S_{5}=3\xi^{2}-2\xi^{3},S_{6}=l(-\xi^{2}+\xi^{3})\\ S_{7}=l\zeta\eta,S_{8}=l\zeta\zeta)\end{array}