## 2 Establishment of Prediction Model
### Metro Train-FST Coupled Model and Train Load Calculations
And especially in the frequency band below 100 Hz, there is basically no \(\omega_{F}\) established in the frequency domain and using wheel-rail Hertz linear contact could satisfy the requirement of the research of the train-induced environment vibration, the Hertz linear spring is used to connect the wheels and the rails in the train-FST coupled model in the frequency domain. When the excitation frequency of the track irregularity is \(\omega_{F}\), the wheel-rail interaction forces of the wheels of the \(k\)th wheelset can be expressed as follows:  
\[\left(-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{K}\right)\hat{ \mathbf{U}}(\omega)=\hat{\mathbf{F}}(\omega) \tag{1}\]6  
\[\left(-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{K}\right)\hat{ \mathbf{U}}(\omega)=\hat{\mathbf{F}}(\omega) \tag{1}\]7  
where \(k_{\text{z}k}^{\text{L}}=\frac{3}{2c_{\text{r}}}(P_{0})^{\frac{1}{3}}\), \(G\) is the contact constant and \(P_{0}\) is the static wheel-rail interaction force of the \(k\)th wheel. For the entire train, the wheel-rail interaction forces can be calculated using Equation (12):  
\[\left(-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{K}\right)\hat{ \mathbf{U}}(\omega)=\hat{\mathbf{F}}(\omega) \tag{1}\]8  
where \(\mathbf{k}_{\mathrm{C}}=\mathrm{diag}\left(\mathbf{k}_{\mathrm{Z}}^{\mathrm{L}},\mathbf{k}_{\mathrm{Z}}^{\mathrm{R}}\right)\) and \(\mathbf{k}_{\mathrm{Z}}^{\mathrm{L}}=\mathbf{k}_{\mathrm{Z}}^{\mathrm{R}}= \mathrm{diag}\left(k_{\mathrm{z}\mathrm{z}\mathrm{z}\mathrm{z}}^{\mathrm{L}},k_ {\mathrm{z}\mathrm{z}}^{\mathrm{L}},\cdots,k_{\mathrm{z}\mathrm{z}\mathrm{z} \mathrm{z}}^{\mathrm{L}}\right)\). By substituting Equations (3), (7) and (10) into Equation (13), the dynamic wheel-rail interaction force at frequency \(\omega_{F}\) can be obtained, as expressed by Equation (14). \[ \begin{array}{rl}\overline{\mathbf{F}}_{\mathrm{wr}}^{\mathrm{vehicle}}(\omega_{F})&=\mathbf{k}_{\mathrm{C}}\left[- \mathbf{A}_{\mathrm{wheel}}^{\mathrm{vehicle}}(\omega_{F})\overline{\mathbf{F} }_{\mathrm{wr}}^{\mathrm{vehicle}}(\omega_{F})-\mathbf{A}_{\mathrm{rail}}( \omega_{F})\overline{\mathbf{F}}_{\mathrm{wr}}^{\mathrm{vehicle}}(\omega_{F})- \overline{\mathbf{U}}_{\mathrm{rough}}(\omega_{F})\right]\\ &=-\left[\mathbf{A}_{\mathrm{wheel}}^{\mathrm{vehicle}}(\omega_{F})+ \mathbf{A}_{\mathrm{rail}}(\omega_{F})+\mathbf{k}_{\mathrm{C}}^{-1}(\omega_{ F})\right]^{-1}\overline{\mathbf{U}}_{\mathrm{rough}}(\omega_{F})\end{array}  \tag{14}\]  
After determining the wheel-rail interaction forces in the frequency domain, the vibration response of the FST can be calculated using the solution method when the moving harmonic load acts on the track, as mentioned earlier. Because the track structure was excluded from the tunnel-soil FE model, the steel-spring support reaction force must be calculated as the excited force for the FE model. The supporting force of the \(i\)th spring under the \(j\)th slab in the frequency domain is obtained as follows:  
\[\hat{F}_{\mathrm{s}i,j}^{z}\left(x_{\mathrm{s}}^{k},y_{\mathrm{s}i,j},\omega, \omega_{F}\right)=k_{\mathrm{sz}}^{*}a_{\mathrm{s}i,j}^{z}\left(x_{\mathrm{s}} ^{k},y_{\mathrm{s}i,j},\omega,\omega_{F}\right) \tag{15}\]  
where \(\hat{u}_{\mathrm{s}i,j}^{z}\left(x_{\mathrm{s}}^{k},y_{\mathrm{s}i,j},\omega, \omega_{F}\right)\) indicates that the displacement response of the track with response frequency \(\omega\) in the frequency domain at point \(\left(x_{\mathrm{s}}^{k},y_{\mathrm{s}i,j}\right)\) when the wheel-rail interaction forces with excitation frequency \(\omega_{F}\) act on the rails. Based on the periodic-infinite structure theory, the steel-spring-supporting force at any point can be calculated from its mapping point in the periodic element of the FST.