## 5 Analysis Results
### Effect of Drag Springs  
This paper mainly studies about the effect of drag springs to improve the structural dynamics for this PDSR. According to the numerical simulations, the first ten natural vibration frequencies of Model 1 and 2 are listed in Table 5 and their mode shapes are shown in the Figure 7.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequency Order** & **Test** & **Simulation** \\ \hline
1 & 0.8163 & 0.7847 \\
2 & 0.8629 & 0.8301 \\
3 & 0.8723 & 0.8533 \\
4 & 0.9007 & 0.8819 \\
5 & 0.9206 & 0.9268 \\
6 & 0.9600 & 0.9858 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Natural frequencies (Hz) of test and simulation for Model 1.  
Figure 6: Test model by full-model vibration exciter method.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequency Order** & **Model 1** & **Model 2** \\  & **with Drag Springs** & **without Drag Springs** \\ \hline
1 & 0.7847 & 0.3331 \\
2 & 0.8301 & 0.3360 \\
3 & 0.8533 & 0.3419 \\
4 & 0.8819 & 0.3631 \\
5 & 0.9268 & 0.7635 \\
6 & 0.9858 & 1.2923 \\
7 & 1.0077 & 1.2948 \\
8 & 1.0359 & 3.8519 \\
9 & 1.0896 & 3.8637 \\
10 & 1.1364 & 3.8809 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Natural frequencies (Hz) of Mode 1 and 2.  
Figure 7: _Cont._Figure 7: _Cont._  
Figure 7: Cont.  
In order to clearly illustrate the vibration mode, the first order mode shapes of Model 1 and 2 in 3D visualization are shown in Figure 8 as an example. It is shown that the petal vibration mode has the relationship with the deployment mechanism as Figure 1a,b. Because two orthogonal revolute pins of the deployment mechanism are set by aforementioned CRM. The DOF of pin rotation is released and locked by the rotation limiter and the stiffness of torsion spring. Hence, it is demonstrated that the petal in Figure 8b easily occurs the upwarp with maximum displacement 3.86 mm as the first order mode shape, while the five petals connected with drag springs in Figure 8a have the smaller vibration with maximum displacement 2.20 mm. Other orders have the similar petal vibration mode.  
For the mode shapes of Model 1 with the drag springs, several adjacent petals have the vibrations together in all ten modes in Figure 7a and the difference of these modes is the different petal combinations. The main reason is the function of drag spring by providing the tension stiffness between two adjacent petals. On the contrary, in Figure 7b, the mode shapes of Model 2 without drag springs are all the single-petal vibration. The first five orders of Model 2 are one single-petal vibration in different locations, and the last five orders are two single-petal vibration with different combinations.  
Figure 7: First ten mode shapes of the reflector.  
From Table 5, the natural frequencies of the first five orders for Model 2 are much smaller than Model 1, while the rests of five frequencies for Model 2 are larger than Model 1. Combined with the difference of mode shapes for Model 1 and 2 in Figure 7, therefore, Model 2 has much more dispersed frequency domain and more flexible dynamic property in lower frequency domain. This performance is very averse to the reflector structure in deployed working state. Especially suffering from external disturbance, the model without drag springs may cause responses in a very broad frequency domain so as to hardly maintain the reflector accuracy.  
By contrast, Model 1 with drag springs has a totally different dynamic property. The drag springs connect every adjacent two petals together and provide the circumferential tensions to make petals collaborative work. In view of structural dynamics, the drag springs provide the reflector with extra tension stiffness to improve the natural frequencies. The connection of drag springs brings about the multi-petal vibration mode so as to narrow down its frequency domain. It can help to effectively conduct vibration isolation measures to improve the reflector accuracy.