## 5 Numerical Examples
### Deployable Spatial Truss Structure Unit  
The dynamics of a deployable truss structure unit, which can be employed to construct large space systems [34], was investigated in this section. As shown in Figure 7, the structure unit is an assembly composed of three square frames, eight cross-bars, and eight diagonal rods joined to one another. The cross-bars and the diagonal rods on both sides are jointed with square frames with only the rotations around the \(\xi_{2}\)-axis released, while the above and below diagonal rods are clamped at corresponding points on square frames. During the deployment process, the diagonal rods on both sides can slide between two ends of the carrier rods through a sliding joint. The geometrical and material parameters can be found in Table 4.  
In the first test case, the first (left) square frame was fixed initially, and the structure unit was kept with \(\beta_{i}=5^{\circ}\) (\(i=1,\,2,\,3,\,4\)). Then, the structure unit was driven with the gravity 0.005 g along X-axis. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Value** \\ \hline Side length of square frame & 2 m \\ Length of cross-bar & 4 m \\ Length of diagonal rod & \(\sqrt{20}\) m \\ Inside diameter of all rods & 0.055 m \\ Outside diameter of all rods & 0.060 m \\ Young’s modulus & 2.1e11 N/m\({}^{2}\) \\ Poisson coefficient & 0.3 \\ Mass density & 7850 kg/m\({}^{3}\) \\ Gravity & 9.81 m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters for the deployable truss structure unit. Figure 6: The trajectories of the mass displayed in different planes: (a) X–Z plane and (b) Y–Z plane. Figure 7: The deployed configuration of the structure unit. The calculation was conducted with the time step \(h=0.1\) s and a single 2-order NURBS finite element per beam. When \(\beta_{i}=90^{\circ}\) (\(i=1,2,3,4\)), as shown in Figure 7, the sliding joints were locked in the top end of carrier beams, and the calculation was terminated through a constraint of \(\zeta\). For comparison, the reproductive simulation based on MSC.ADAMS and ANSYS were conducted. As suggested by [35; 36], the modal neutral files (.mnf files) was firstly obtained from ANSYS and was then imported into ADAMS/FLEX module, which allows us to model flexible parts based on its modal frequency. The beam members were modeled with the specified parameters in Table 4 using Solid45 element in ANSYS. The X-position of point D during the motion calculated from proposed formulation were of good agreement with MSC.ADAMS, as displayed in Figure 8. It was demonstrated that the present formulation was appropriate for the deployment dynamic analysis of the spatial truss structure. In the second case, the initial configuration and material parameters were the same with the first case, while the gravity was ignored. The structure was set to deploy in \(t_{f}=20\) s through control function applied in \(\beta_{i}\) (\(i=1,2,3,4\)) synchronously with the time step \(h=0.005\) s, and the simulation was terminated at 30 s. Equations (54)-(57) are the linear control function, polynomial control function, cycloid control function, and cosine control function with \(\beta_{0}=5^{\circ}\), \(\beta_{f}=90^{\circ}\), respectively. \[\beta_{i}(t)=(\beta_{f}-\beta_{0})\frac{t}{t_{f}}\ (i=1,2,3,4) \tag{54}\]  
\[\beta_{i}(t)=(\beta_{f}-\beta_{0})[6(\frac{t}{t_{f}})^{5}-15(\frac{t}{t_{f}}) ^{4}+10(\frac{t}{t_{f}})^{3}]+\beta_{0}\ (i=1,2,3,4) \tag{55}\]  
\[\beta_{i}(t)=(\beta_{f}-\beta_{0})[\frac{t}{t_{f}}-\frac{1}{2\pi}\sin(\frac{ 2\pi t}{t_{f}})]+\beta_{0}\ (i=1,2,3,4) \tag{56}\]  
\[\beta_{i}(t)=(\beta_{f}-\beta_{0})[\frac{1}{2}-\frac{9}{16}\cos(\frac{\pi t}{ t_{f}})+\frac{1}{16}\cos(\frac{3\pi t}{t_{f}})]+\beta_{0}\ (i=1,2,3,4) \tag{57}\]  
To test the convergence of element, 4 and 10 finite elements using 2-order NURBS basic function were employed at first. As displayed in Figure 9, the motions of D in X direction obtained from two discrete ways agreed well, which demonstrated that the results had converged.