# 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
#### 2.2.2 Analysis of the Displacement Field in the Thin-Walled Flexible Beam Structure  
Building upon the flexible beam structure, we conduct a displacement field analysis for the flexible beam with thin-walled characteristics. The primary objective is to achieve both driving transmission and pre-tensioning transmission while maintaining lightweight characteristics. In contrast to the displacement field analysis of the flexible beam structure in the previous Section 2.2.1, this section employs the middle surface characteristics to more accurately describe the displacement field variation of the flexible beam structure with thin-walled characteristics. To better illustrate the concept, Figure 7 illustrates a schematic diagram of the thin-walled flexible beam structure.  
As shown in Figure 4, since the wall thickness \(h\) of the thin-walled flexible beam element is much smaller than the section width \(w\), according to [PERSON]'s thin plate theory, the displacement field of the thin-walled flexible beam element can be determined from its midplane. In the global coordinate system, the displacement field of the position vector \(r\) of any point on the middle plane of the thin-walled flexible beam element at time \(t\) is given by the following Equation (7):  
\[\mathbf{r}(x,y,t)=\sum_{k=0}^{n}\mathbf{r}_{k}(x,t)L_{k,n}(y) \tag{7}\]  
Figure 7: Schematic diagram of thin-walled flexible beam element.  
where: \(\mathbf{r}_{k}\) is the \(k\)-th axis of the middle plane. The \(k\)-th axis in the thin-walled beam element can be expressed by the absolute node coordinate method under three-dimensional coordinates as follows:  
\[\mathbf{r}_{k}(x,t)=\mathbf{S}(x)\mathbf{e}_{k}(t) \tag{8}\]  
where: \(\mathbf{r}_{k}(x,t)\) is the function of the parameter \(x\) and time \(t\) in the local coordinates, where \(\mathbf{S}(x)\) is still the shape function matrix, and \(\mathbf{e}_{k}(t)\) is the value of the node coordinate matrix of the \(k\)-th axis at time \(t\). Where \(\mathbf{e}_{k}(t)\) is:  
\[\mathbf{e}_{k}=\left[\begin{array}{c|c}\mathbf{r}_{k}^{\mathsf{T}}\Big{|}_ {x=0}\frac{\partial\mathbf{r}_{k}^{\mathsf{T}}}{\partial x}\Big{|}_{x=0} \mathbf{r}_{k}^{\mathsf{T}}\Big{|}_{x=l}\frac{\partial\mathbf{r}_{k}^{ \mathsf{T}}}{\partial x}\Big{|}_{x=l}\end{array}\right]^{\mathsf{T}} \tag{9}\]  
In the formula, \(T\) is the transposition and \(l\) is the length of the thin-walled flexible beam element. In Formula (7), the basis function \(L_{k,n}\) defines the corresponding centerline on the middle plane of the thin-walled flexible beam element, which is a function of the local coordinate parameter \(y\). The interpolation form of basis function \(L_{k,n}\) is:  
\[L_{k,n}=\prod_{\begin{subarray}{c}i=0\\ i\
eq k\end{subarray}}^{n}\frac{(y-y_{i})}{(y_{k}-y_{i})} \tag{10}\]  
where: \(y_{i}\) is the node coordinate of the \(i\)-th axis. Through the above displacement field of thin-walled flexible beam element, for different \(n\)-order Lagrangian interpolation, there is corresponding thin-walled flexible beam element model \(L_{n}\), and the coordinate \(q\) of the element node is:  
\[\mathbf{q}=\left[\mathbf{e}_{0}^{\mathsf{T}}\ \mathbf{e}_{1}^{\mathsf{T}}\ \cdots\ \mathbf{e}_{n}^{\mathsf{T}}\right] \tag{11}\]  
The unit shape function is:  
\[\mathbf{N}(x,y)=\left[L_{0,n}(y)S(x)L_{1,n}(y)S(x)\ \cdots\ L_{n,n}(y)S(x)\right] \tag{12}\]  
Through Equations (11) and (12), the position vector of any point on the middle plane of the thin-walled flexible beam element is:  
\[\mathbf{r}(x,y,t)=\mathbf{N}(x,y)\mathbf{q}(t) \tag{13}\]