## 3 Modal Experiment and Analysis of U-Section Rubber Outer Windshield Structure
### Modal Analysis  
After data of modal test collection, LMS PolyMAX module was used for mode extraction. The peaks with relatively concentrated S value (the mode with stable frequency, damping value and vector) were selected as the mode frequency, and the corresponding mode shapes of each order were extracted. In the modal test of the lateral part of the outer windshield structure, the results of the test under three different excitation points also have a good agreement. Based on the data measured at these excitation points, the first nine natural frequencies and damping ratios of the lateral part of the outer windshield can be obtained, as shown in Table 1, and the corresponding mode shapes of the first six modes are shown in Figure 12. The common point of the first four modes is large amplitude in the middle and small amplitude on both sides (connecting with the end wall of the train). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode** & **Natural Frequency (Hz)** & **Damping Ratio (\%)** \\ \hline
1 & 12.02 & 6.25 \\
2 & 14.25 & 6.92 \\
3 & 18.31 & 6.37 \\
4 & 23.66 & 8.63 \\
5 & 26.72 & 6.21 \\
6 & 28.41 & 5.36 \\
7 & 31.78 & 5.72 \\
8 & 33.46 & 5.70 \\
9 & 39.57 & 4.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modal frequency and damping ratio of outer windshield. Figure 11: U-section rubber outer windshield structure and lateral part of the measurement points (red number) and excitation points (yellow five-pointed star) schematic. **(a)** The layout diagram of the modal experiment of the outer windshield. **(b)** The bottom section of the outer windshield. **(c)** The bottom of outer windshield. **(d)** The measurement points (red number) and excitation points (yellow five-pointed star). 4 Establishment of U-Section Rubber Outer Windshield Flow-Induced Vibration Response Analysis Method  
In this section, based on the modal superposition method, the time course curves of the aerodynamic loads on the outer windshield surface in Section 2 and the modal parameters of the outer windshield structure in Section 3 are used as inputs to build a simulation model of the flow-induced vibration of the U-section rubber outer windshield. The dynamic response of the model is solved using the Runge-Kutta method. In addition to their mathematical formulation, the advantages of using each of these methods are also presented. Flow-Induced Vibration Model Based on Mode Superposition Method of U-Section Rubber Outer Windshield Structure  
The differential equation of motion for the outer windshield system can be expressed as  
\[[\mathbf{M}]\big{\{}\ddot{\mathbf{x}}\big{\}}+[\mathbf{C}]\big{\{}\dot{\mathbf{x}}\big{\}}+[\bm {K}]\big{\{}\mathbf{x}\big{\}}=\{\mathbf{f}\}, \tag{3}\]  
Figure 12: The first four mode shapes of the lateral part of the outer windshield structure. (**a**) \(\omega_{1}=12.02\) Hz. (**b**) \(\omega_{2}=14.25\) Hz. (**c**) \(\omega_{3}=18.31\) Hz. (**d**) \(\omega_{4}=23.66\) Hz. (**e**) \(\omega_{5}=26.72\) Hz. (**f**) \(\omega_{6}=28.41\) Hz. Nodes 1–20 correspond to the modal test points in Figure 11d and different colored lines represent the outer outline of the windshield structure. the general eigenvalue problem if the mass matrix is the unit matrix, and therefore the modal vectors can be normalized. The generalized mass matrix corresponding to the ith order modal vector is  
\[[\mathbf{M}_{i}]=[\mathbf{u}_{i}]^{T}[\mathbf{M}][\mathbf{u}_{i}]. \tag{14}\]  
\([\mathbf{u}_{i}]\) is \(i\)th order modal vector.