# Research on the Dynamics of the Space Tubular Expandable Structure Driving Deployment Unit
## 2 Methods
### The Dynamic Fitting of the Parallel Configuration  
To effectively enhance stability, we arrange the thin-walled flexible beam structures in parallel within the spatial telescope tube, forming a parallel configuration and constituting the variable-radius internal driving device, as illustrated in Figure 8.  
To achieve the sequential expansion and driving functions of each level of the spatial telescope tube, we employ a dual-layer symmetrical structure with an angle of 120\({}^{\circ}\). This is accomplished using six identical thin-walled flexible beam triangular unit configurations, as shown in Figure 9, which represents the overall model of the thin-walled flexible beam triangular units. Additionally, from Figure 8, it can be observed that the upper and lower layers of the structure consist of thin-walled flexible beam triangular units on the same side, forming a new triangular configuration that constitutes one-third of the overall variable-radius internal driving device structure. As the variable-radius internal driving device is axially symmetrical, we take a partial structure as an example to derive the kinematics of the parallel configuration of the driving expansion units. Figure 8: Schematic diagram of the parallel symmetrical structure. Taking the dynamic equation in Section 2.2 as an example, the new triangular configuration is further deduced. The section of the longest thin-walled flexible beam element with the same triangular configuration of the upper and lower thin-walled flexible beams is set as A\({}_{4}\) and A\({}_{5}\). The length is \(l_{4}\) and \(l_{5}\), and the material density is \(\rho\), which are respectively substituted into Formula (22) to form:  
\[\left\{ \begin{array}{l}Q_{y4}=\int zdA_{A},\ Q_{y5}=\int zdA_{5}\\ Q_{24}=\int\limits_{AA}^{AA}ydA_{A},\ Q_{z5}=\int\limits_{A5}^{A5}ydA_{5}\\ I_{yy4}=\int\limits_{AA}^{AA}z^{2}dA_{A},\ I_{yy5}=\int\limits_{A5}^{A5}z^{2}dA _{5}\\ I_{z24}=\int\limits_{A}^{y}y^{2}dA_{A},\ I_{z5}=\int\limits_{A5}^{y}ydA_{5}\\ I_{yz4}=\int\limits_{AA}^{\infty}yzdA_{A},\ I_{yz5}=\int\limits_{A5}^{A5}yzdA _{5}\end{array} \right. \tag{29}\]  
After substituting Equation (29) into Equation (15) and obtaining the unit mass matrix, we employ the direct integration method for dynamic analysis. First, we obtain the nodal accelerations, and then, utilizing Equations (27) and (28), we determine the nodal velocities and displacements of the new triangular configuration. Figures 10 and 11 illustrate the displacement variations of the driving expansion unit in the parallel structure. We observe that the displacement variations are more uncertain at the free end compared to the constrained end. This uncertainty arises from the thin-walled flexible beam structure's behavior, which exhibits \"free end\" conditions when not subjected to radial loads but acts as \"constrained end\" conditions on the discontinuous surface when under radial loads. These conditions are not independent but occur as part of a complete process, where the \"free end\" temporarily appears due to the unique nature of the internally driven unit with a variable diameter moving within the spatial sleeve involving a transitional discontinuous surface (step-like). Consequently, there is a possibility of momentary failure due to axial friction and radial preloading. Although the occurrence of such a situation is low in probability, considering the overall stability and reliability of the structure [44], its uncertainty is greater than that of the \"constrained end\" conditions. Therefore, our analysis focuses on the displacement components under the \"free end\" condition. Figure 9: Overall configuration of the triangular element. Figure 10 shows that the displacement in the X-direction of the local coordinate system increases negatively with axial expansion, while the radial distance increases positively. Additionally, Figure 11 demonstrates that the displacement in the Y-direction gradually increases over time, exhibiting positive growth in the axial distance.