## 2 SAFE Method for Solving Vibration Mode Data of a Rail  
### SAFE Method  
The SAFE method is used to achieve the mode shape information of guided wave modes in waveguide media by establishing the wave equation and solving the eigenvectors. The method only needs finite element discretization on the cross-section of the waveguide medium, while the displacements along the wave propagation direction are expressed by harmonic motion. In this paper, we consider a CHN60 rail, which is a 60 kg/m China rail. In Figure 1, the cross-section of the rail is defined as the \(y\)-\(z\) plane, and the wave propagation direction is the \(x\) direction.  
Assuming that the displacement field of the guided wave propagating in the \(x\) direction is a harmonic motion, the displacement, \(u\), of any point in the waveguide medium is expressed as Equation (1) by a spatial function [24]:  
\[\mathbf{u}(x,y,z,t)=\left[\begin{array}{c}u_{x}(x,y,z,t)\\ u_{y}(x,y,z,t)\\ u_{z}(x,y,z,t)\end{array}\right]=\left[\begin{array}{c}U_{x}(y,z)\\ U_{y}(y,z)\\ U_{z}(y,z)\end{array}\right]e^{i(\xi x-\omega t)} \tag{1}\]  
where \(\xi\) is the wavenumber and \(\omega\) is the frequency.  
Using the Partial Differential Equation (PDE) tool of MATLAB (version R2009a, the MathWorks, Inc., Natick, MA, USA), the cross section of the CHN60 rail is discretized by the triangular element. There are 177 nodes and 255 elements in the cross-section of the CHN60 rail as shown in Figure 2. Each element has three nodes and each node has three degrees of freedom, which correspond to displacements in the \(x\), \(y\), and \(z\) directions, respectively.  
After the finite element discretization on the cross-section of the waveguide medium, the displacement of any point in the element is expressed as a form of the shape function [24] as Equation (2):  
\[u^{(e)}(x,y,z,t)=\left[\begin{array}{c}\sum\limits_{k=1}^{n}N_{k}(y,z)U_{kk} \\ \sum\limits_{k=1}^{k}N_{k}(y,z)U_{kk}\\ \sum\limits_{k=1}^{n}N_{k}(y,z)U_{kk}\end{array}\right]^{(e)} \tag{2}\]  
where \(U_{kk}\), \(U_{kk}\), and \(U_{kk}\) are the displacements of the nodes, \(\mathbf{N}(y,z)\) is the shape function matrix, \(\mathbf{q}^{(e)}\) is the node displacement vector, and \(n=3\) is the number of nodes in a triangular element. In Equation (2), the shape function is defined as:  
Figure 1: SAFE model of wave propagation.  
Figure 2: Discretization of a rail with triangle elements.  
\[\mathbf{N}(y,z)=\left[\begin{array}{ccccccccc}N_{1}&&&N_{2}&&&\cdots&&&N_{n}\\ &N_{1}&&&N_{2}&&&\cdots&&&N_{n}\\ &&N_{1}&&&N_{2}&&&\cdots&&&N_{n}\\ &&&N_{1}&&&N_{2}&&&\cdots&&&N_{n}\\ \end{array}\right] \tag{3}\]  
\[\mathbf{q}^{(e)}=\left[\begin{array}{ccccccccc}U_{x1}&U_{y1}&U_{z1}&U_{x2}&U_{y2 }&U_{z2}&\cdots&U_{xn}&U_{ym}&U_{zn}\\ \end{array}\right]^{T} \tag{4}\]  
Based on the Hamiltonian principle, the general homogenization wave equation of ultrasonic guided waves in the CHN60 rail is derived [24] as Equation (5):  
\[\left[\mathbf{K}_{1}+i\xi\mathbf{K}_{2}+\xi^{2}\mathbf{K}_{3}-\omega^{2}\mathbf{M}\right]_{M} \mathbf{U}=0 \tag{5}\]  
where \(\mathbf{K}_{1}\), \(\mathbf{K}_{2}\), and \(\mathbf{K}_{3}\) are the stiffness matrix; and \(\mathbf{M}\) and \(\mathbf{U}\) are the mass matrix and nodal displacement vector, respectively  
According to Equation (5), for a given wavenumber value, \(\xi\), the value of the frequency, \(\omega\), is calculated by calculating the eigenvalues. Thus, the relationship between the wavenumber and the frequency is determined. Subsequently, the phase velocity dispersion curves of the ultrasonic guided waves in teh CHN60 rail are depicted in Figure 3.