## 3 Graphical Analysis Method of the Mode Shape  
At the low frequency, the number of guided wave modes in the rail is relatively small. With the increase of the frequency, the number increases gradually. Expressing the vector matrix data of the mode shapes in a graphical way can display the vibration characteristics of each mode more intuitively. Furthermore, processing is more efficient with a common image processing method. Given a frequency, \(f\), the eigenvector, \(\mathbf{U}\), can be solved as Equation (5). It contains the displacement information of each node of all modes as Equation (6):  
\[\mathbf{U}_{n}^{m}(x,y,z)=\left[ \begin{array}{cccc}u_{1x}^{1}&u_{1x}^{2}&\cdots& u_{1x}^{m}\\ u_{1y}^{1}&u_{1y}^{2}&\cdots&u_{1y}^{m}\\ u_{1z}^{1}&u_{1z}^{2}&\cdots&u_{1z}^{m}\\ \vdots&\vdots&\cdots&\vdots\\ u_{nx}^{1}&u_{nx}^{2}&\cdots&u_{nx}^{m}\\ u_{ny}^{1}&u_{ny}^{2}&\cdots&u_{ny}^{m}\\ u_{nz}^{1}&u_{nz}^{2}&\cdots&u_{nz}^{m}\\ \end{array} \right] \tag{6}\]  
The eigenvector, \(\mathbf{U}_{n}^{m}(x,y,z)\), includes the mode shape information of m modes. There are \(n\) nodes, and each node has displacements in the \(x\), \(y\), and \(z\) directions in the discrete rail cross-section as shown in Figure 2. For a specific mode, the eigenvector solves the displacement values of \(n\) discrete nodes of all rail cross-sections. Taking the frequency of 35 kHz as an example, 20 guided wave modes can propagate in the rail, as shown in Table 1.  
The shapes of mode 2, mode 3, mode 7, and mode 10 with phase velocities of 1983.80 m/s, 2286.06 m/s, 2737.57 m/s, and 3297.15 m/s, respectively, are plotted in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline
**Mode** & **Phase Velocity (m/s)** & **Group Velocity (m/s)** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 1: Phase velocity and group velocity for each mode at the frequency of 35 kHz. According to the vibration mode information of the guided wave modes in the rail, the main vibration position and energy concentration point can be analyzed to determine the excitation position and direction of the mode. The orthogonal relationship between the modes can also be analyzed. However, these processing methods require related data processing on the mode matrix data without a fixed algorithm theory, and researchers need to design related algorithms according to the data characteristics. At present, image processing technology has been widely used in various industries. There are many image processing methods, such as the histogram, binarization, gradient method, and so on. If the vibration mode vector data is converted into an image, the guided wave modes in the rail can be analyzed by the image processing method directly. The conversion idea is shown in Figure 8, which converts the displacement value of any point in the triangle element into RGB pixel values. Figure 8: The conversion relationship between node displacements and RGB pixel values. Figure 7: Guided wave mode diagrams at the frequency of 35 kHz. (**a**) The phase velocity of mode 2 is 1984.05 m/s; (**b**) the phase velocity of mode 3 is 2286.06 m/s; (**c**) the phase velocity of mode 7 is 2737.57 m/s; (**d**) the phase velocity of mode 10 is 3297.15 m/s. The conversion process includes the following steps:  
1. According to the rail size, the original image is generated, and the initial RGB pixel value is 0. 2. Set the DPI value to generate the coordinate values of all the pixel points. 3. Find the dot in the triangle elements of all coordinate values. 4. Solve the displacement value of the current pixel point in the triangle elements by the difference equation. 5. The displacement values of all the pixel points are converted into [0, 255] RGB pixel values by normalization. 6. Plot the vibration mode image of the rail, RGB color, or gray image. The width of the CHN60 rail base is 150 mm, and the height is 176 mm. The DPI value of the generated vibration image is set to 96. There are about 4 pixel points of 1 mm, and the image resolution is set to 601 \(\times\) 705.