# Research on the Dynamics of the Space Tubular Expandable Structure Driving Deployment Unit
## 2 Methods
### The Dynamic Fitting of the Parallel Configuration
These changes in displacements in the X and Y directions are caused by the reduction in tube diameter. Figure 12 illustrates the midpoint deflection of the thin-walled flexible beam structure in its free-end state. Figure 11: The \(Y\)-axis displacement component at the free end node. Figure 12: The mid-span deflection of the free end state. Figure 10: The \(X\)-axis displacement component at the free end node. Figure 12 shows that as the driving expansion unit transitions among the different stages of the tube, the displacement of the endpoints causes changes in the midpoint deflection at various locations. This observation indicates that during this motion process, the thin-walled flexible beam structure in its free-end state undergoes both tensile and bending deformations. Additionally, Figure 13 illustrates the changes in the displacement field of the thin-walled flexible beam structure in its free-end state. It can be found from Figure 13 that when the time interval is 0.2, the change of \(L_{3}\) changes from the initial state T = 0 to the time of T = 2.0 after realizing the function of driving and retracting, the thin-walled flexible beam element produces tension and deformation, resulting in the change of displacement in X and Y directions. Based on the free end, through the analysis of the motion state of the driving expansion unit in the space sleeve, combined with the free end and the constraint end, a complete working state is formed. Figure 14 illustrates the positional variations of the driving expansion unit at different time intervals in its complete state. At the four time intervals (0.25 s, 0.5 s, 0.75 s, and 1.0 s), the displacement in the X direction of the thin-walled flexible beam structure shows more noticeable changes compared to the displacement of the flexible beam structure from its initial position, while maintaining the same Y height. Similarly, when the X position remains constant, the displacement in the Y direction also undergoes significant changes, which align more closely with the structural characteristics of the thin-walled flexible beam structure. Figure 14: _Cont._  
Figure 13: The displacement field state at the free end. Figure 15 illustrates the trajectory variations of the axial expansion within the interior of the tube when the driving expansion unit is in an integrated state, considering both constrained and unconstrained conditions. The displacement analysis of the overall parallel structure utilizes a combination of constrained and unconstrained states (free end) as boundary conditions to better approximate the actual operating state and validate the motion trajectory. Figure 14: Pose variations of the driving deployment unit. (**a**) At the time of 0.25 s; (**b**) At the time of 0.50 s; (**c**) At the time of 0.75 s; (**d**) At the time of 1.00 s.  
Figure 15: Motion trajectory of the driving deployment unit. (**a**) Centroid displacement in the X-direction; (**b**) Centroid displacement in the Y-direction; (**c**) Centroid displacement in the Z-direction; (**d**) Trajectory rotation angle. Figure 15a,b illustrates the nodal displacements of the flexible beam structure and the thin-walled flexible beam structure using the absolute nodal coordinate formulation (ANCF) method. Within the time interval of 0 to 0.1 s, the displacement variations of their centroids in the x and y directions are not significant. This lack of significant displacement is attributed to the initial motion occurring on the inner surface of the first-level tube. As the diameter of the tube changes, the displacement gradually increases. However, the presence of inter-level discontinuous surfaces at the connections between the various levels of tubes results in observed peaks and valleys during this period. Figure 15c shows that the thin-walled flexible beam structure exhibits oscillations in the Z direction compared to the flexible beam structure. This is due to its enhanced flexibility and the effect of its own weight.