## 3 Simulation Analysis
### Frequency Response Analysis by the State Space Method  
Although finite element software is widely applied to the frequency response analysis, for more complex and large models, the finite element method involves a huge number of calculations and takes time to provide a solution due to the large model dimensions. In addition, the software function is limited, and it is inconvenient to observe the response contribution of each mode, so the finite element method cannot always meet the needs of engineering practice [17]. To efficiently and accurately solve practical engineering problems, the state space method can be used to calculate structural frequency response problems. According to the modal analysis results, the frequency range of frequency response analysis is confirmed to be [220 Hz, 1640 Hz], the damping is in the form of modal damping g = 0.02, and the state space model of the primary mirror assembly is established. Take the frequency transfer characteristics between Node 53,798-Node 60,490 as an example. Node 53,798 is the excitation input point, while node 60,490 is the response output point. Figure 6: Frequency response curve of the primary mirror. The state space method often uses the scientific computing software MATLAB for analysis and problem solving. The analysis steps are as follows:  
1. Data preparation. Perform modal analysis on the mirror component structure, extract the natural frequency of the appropriate order modal data from the obtained f06 result file; extract the eigenvalues of the system, as shown in Figure 7; extract the force input degrees of freedom, response output degrees of freedom, and form a modal matrix (take the displacement in the Z direction as an example, as shown in Equation (26)); and determine the frequency range based on the results of the modal analysis. 2. Establish a state space model. Read the data file, use Equations (20)-(23) to establish the state space model of the structure, and draw the total frequency transfer characteristic curve and the frequency transfer characteristic curve of each mode. 3. Reduce modes. Use Equation (24) to solve the DC gain values of each non-rigid body mode and sort them from largest to smallest. The mode corresponding to the larger DC gain value is retained, while the mode with the smaller DC gain value is discarded, and the state space model is regenerated by the retained mode to obtain a simplified system. 4. Obtain the result. Draw a Bode diagram of the simplified system to obtain the frequency transfer characteristic curve of nodes 53,798-60,490. According to mode reduction theory in Section 2.3, the DC gain calculation of each mode is shown in Equation (25) and the order of the modes is based on the DC gain: 6, 1, 7, 3, 2, 9, 5, 10, 4, 8. As can be seen from the analysis results in Figure 8, the DC gain does not increase with the increase in frequency; the DC gain of modes 9, 5, 10, 4, and 8 is small; and the contribution of modes 5, 10, 4, and 8 to the overall frequency response is small. The frequency response analysis curves in Figures 9-11 can also verify this point. It can be seen in Figure 11 that modes 6, 1, 7, 3, and 2, in that order, make a greater contribution, while the mode with a smaller DC gain makes a small contribution to the frequency response function. Figure 12 shows that the frequency response curve of only the 5 th-order mode with the largest DC gain is highly consistent with the frequency response curve of all modes. Currently, increasing the number of selected modes does not increase the calculation accuracy and reduces the calculation effectiveness:  
\[dc\_gain=|x_{n}(node\_force,\cdot)\cdot x_{n}(node\_dis,\cdot)|./w^{2} \tag{25}\]  
where \(x_{n}\) is the system modal matrix (as shown in Equation (26), which is composed of the Z direction displacement of the excitation input point and the response output point) and \(w=2\pi\cdot frequency\) can be obtained in the f06 file after modal analysis, as shown in Figure 7.  
Figure 7: F06 data file. Figure 8: DC value of each mode contribution versus mode number.