# An Ultrasonic Guided Wave Mode Selection and Excitation Method in Rail Defect Detection
## 2 Mode Selection Based on Semi-Analytical Finite Element (SAFE) Method  
In order to study the mode selection method of guided waves in rail defect detection, it is necessary to obtain the mode shape data. Many studies have proved that SAFE method is suitable to solve the propagation characteristics of guided waves in waveguide, such as plates [16], cylinders [17], rails [18], etc. SAFE method is also used in this paper to solve the mode shape data of CHN60 rail, which is a 60 kg/m China rail. The CHN60 rail coordinate system is shown in Figure 1.  
The displacement \(u\), stress \(\sigma\) and strain \(\epsilon\) of each particle in the rail can be expressed as Equation (1):  
\[\mathbf{u}=\left[ \begin{array}{cccccc}u_{x}&u_{y}&u_{z}\end{array}  \right]^{T}\] \[\mathbf{\sigma}=\left[ \begin{array}{cccccc}\sigma_{x}&\sigma_{y}& \sigma_{z}&\sigma_{yz}&\sigma_{xz}&\sigma_{xy}\end{array} \right]^{T} \tag{1}\] \[\mathbf{\varepsilon}=\left[ \begin{array}{cccccc}\varepsilon_{x}& \varepsilon_{y}&\varepsilon_{z}&\gamma_{yz}&\gamma_{xz}&\gamma_{xy}\end{array}  \right]^{T}\]  
Figure 1: Coordinate system of a rail. Based on SAFE theory, it is assumed that the displacement field of the guided wave propagation direction is harmonic [19]. The cross section of the rail is discretized into 255 triangular elements and 177 nodes, as shown in Figure 2.  
After discretization, the strain energy \(\phi\) of each triangular element can be written as Equation (2):  
\[\phi=\int\limits_{V_{e}}\delta(\epsilon^{(e)T})C\epsilon^{(e)}dV_{e}=\delta q^{ (e)^{T}}(\mathbf{K}_{1}-i\xi\mathbf{K}_{2}+\xi^{2}\mathbf{K}_{3})q^{(e)} \tag{2}\]  
where \(q^{(e)}\) is the nodal unknown displacement of each triangular element; \(\xi\) is the wavenumber; \(\mathbf{K}_{1}\), \(\mathbf{K}_{2}\), and \(\mathbf{K}_{3}\) are stiffness matrices; \(C\) is the elastic constant matrix of the rail. By substituting strain energy and potential energy at any node in the rail cross section into [PERSON]'s formula, the general homogeneous wave equation of guided waves can be derived as Equation (3) [19]:  
\[\left[\mathbf{K}_{1}+i\xi\mathbf{K}_{2}+\xi^{2}\mathbf{K}_{3}-\omega^{2}\mathbf{M}\right]_{M} \mathbf{U}=0 \tag{3}\]  
where \(\mathbf{M}\) is the mass matrix; \(\mathbf{U}\) is the nodal displacements; \(\omega\) is frequency. By solving the eigenproblem of Equation (3), the corresponding wavenumber \(\xi\) can be obtained for a given frequency \(\omega\). The triangular element nodal displacements \(\mathbf{U}\) are calculated from the eigenvalue problem. And the strain fields are reconstructed from Equation (2). At the frequency of 35 kHz, the nodal displacement \(q^{(e)}\), stiffness matrix \(\mathbf{K}_{1}\), \(\mathbf{K}_{2}\), and \(\mathbf{K}_{3}\), and wavenumber \(\xi\) are substituted into the Equation (2), and the strain energy of each mode in each triangular element can be acquired. The strain energy distribution diagram of each mode is normalized as shown in Figure 3.  
At the frequency of 35 kHz, there are 20 kinds of guided wave modes propagating in the rail. Our expected mode is that the strain energy is concentrated in a single part of the rail. From Figure 3, we can see that mode 1, mode 2, mode 3, mode 7, mode 10, and mode 20 have such characteristics while strain energy of the other modes is distributed at three rail parts. In Figure 4a,b, the strain energy of mode 7 and mode 20 is concentrated on the rail head. In Figure 4c, it can be seen that the strain energy of mode 3 is focused on the rail web. In Figure 4d, Figure 4e,f, the strain energy of mode 1, mode 2, and mode 10 is concentrated on the rail base. Therefore, mode 7 and mode 20 can be selected to detect rail head defects, similarly, mode 3 for rail web and mode 1, mode 2 and mode 10 for rail base. Figure 2: Discretization of triangle elements. Figure 4: _Cont._  
Figure 3: Strain energy distribution diagram of modes.