# Research on a Rail Defect Location Method Based on a Single Mode Extraction Algorithm
## 2 An Accurate Modal Identification Method  
### Basic Characteristics of Ultrasonic Guided Waves in Rails  
As a result of the complex cross-sectional structure of rails, the number of guided wave modes propagating in a rail is large. To detect the internal defects of rails based on ultrasonic guided wave technology, the most important thing is to grasp the fundamental characteristics of ultrasonic guided waves in rails, such as the frequency, wave number, phase velocity, group velocity, mode shape, and other information, so as to analyze the propagation characteristics of the guided waves. The dispersion curves of ultrasonic guided waves in rails can be obtained with the semi-analytical finite element method. Taking the rail laid on Beijing-Shanghai high-speed railway in China as the research object [19], the finite element method was used to discretize the cross-section. It was assumed that the guided waves propagate along the longitudinal direction of the rail in the form of harmonic vibrations. The wave equation was established based on the finite element method. The eigenvalues and eigenvectors were obtained by solving the eigenvalue equation. The eigenvalues contain the information of frequencies and wave numbers, and the eigenvectors contain the information of mode shapes. Thus, the dispersion curves and mode shapes of guided waves in the CHN60 rail were obtained. First, the coordinate system of the CHN60 rail was established, as shown in Figure 1.  
The wave number of guided waves is \(\xi\) and the frequency is \(\omega\). The displacement, stress, and strain of each node in the rail can be expressed as Equation (1). \[u= \begin{bmatrix}u_{x}&u_{y}&u_{z}\end{bmatrix} ^{T}\]  
\[\sigma= \begin{bmatrix}\sigma_{x}&\sigma_{y}&\sigma_{z}&\sigma_{yz}&\sigma_{xz }&\sigma_{xy}\end{bmatrix} ^{T} \tag{1}\]  
Figure 1: CHN60 rail coordinate system. \[\varepsilon=\left[\varepsilon_{x}\quad\varepsilon_{y}\quad\varepsilon_{z}\quad \gamma_{yz}\quad\gamma_{xz}\quad\gamma_{xy}\right]^{T}\]  
\(\varepsilon_{x}\), \(\varepsilon_{y}\), and \(\varepsilon_{z}\) are normal strains, and \(\gamma_{yz}\), \(\gamma_{xz}\), and \(\gamma_{xy}\) are shear strains. In the elastic range of materials, stress and strain conform to Hooke's Law. It can be expressed as \(\sigma=C\varepsilon\), where \(C\) is the elastic constant matrix of a rail. The relationship between the strain and displacement at any node in the rail is expressed in a matrix form as follows:  
\[\varepsilon=\left[L_{x}\frac{\partial u}{\partial x}+L_{y}\frac{\partial u}{ \partial y}+L_{z}\frac{\partial u}{\partial z}\right] \tag{2}\]  
Therefore, in the form  
\[L_{x}= \begin{bmatrix}1&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&1\\ 0&1&0\end{bmatrix} L_{y}= \begin{bmatrix}0&0&0\\ 0&1&0\\ 0&0&0\\ 0&0&1\\ 0&0&0\\ 1&0&0\end{bmatrix} L_{z}= \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&1\\ 0&1&0\\ 1&0&0\\ 0&0&0\end{bmatrix}  \tag{3}\]  
The SAFE method was used to get the dispersion curves of the ultrasonic guided waves which propagate in the form of harmonics in the longitudinal direction of the rail. Therefore, only the finite element discretization of the rail cross-section was needed. The displacement of any discrete node in the rail can be calculated as shown in Equation (4), where \(x\) is the longitudinal coordinate of the rail. \[u(x,y,z,t)= \begin{bmatrix}u_{x}(x,y,z,t)\\ u_{y}(x,y,z,t)\\ u_{z}(x,y,z,t)\end{bmatrix} = \begin{bmatrix}U_{x}(y,z)\\ U_{y}(y,z)\\ U_{z}(y,z)\end{bmatrix} e^{-i(\xi x-\omega t)} \tag{4}\]  
The triangular element was selected to discretize the cross-section of the CHN60 rail, and 177 nodes and 255 elements were obtained, as shown in Figure 2. Seven nodes are circled in the figure to illustrate the signal extraction nodes in the subsequent modal identification method. The rail cross-section was discretized by triangular elements. The displacement of any particle in an element can be obtained by the displacement of the nodes and the shape function. The strain and  
Figure 2: Discretization of the cross-section of the CHN60 rail.