## 2 Establishment of Prediction Model
### Metro Train-FST Coupled Model and Train Load Calculations
In Equations (4)-(6), \(\hat{u}_{\rm r}^{\rm L}\big{(}x_{\rm r}^{\rm L},y,\omega\big{)}\) and \(\hat{u}_{\rm r}^{\rm R}\big{(}x_{\rm r}^{\rm R},y,\omega\big{)}\) are the vertical displacements of the left and right rails, abbreviated as \(\hat{u}_{\rm r}^{\rm L}\) and \(\hat{u}_{\rm r}^{\rm R}\), respectively. \(\hat{u}_{\rm s}(x,y,\omega)\) is the vertical displacement of the slab, abbreviated as \(\hat{u}_{\rm s}\):  
\[E_{\rm r}I_{\rm r}\frac{\partial^{4}\hat{u}_{\rm r}^{\rm L}}{ \partial y^{4}}-m_{\rm r}\omega^{2}\hat{u}_{\rm r}^{\rm L}+k_{\rm r}^{*}\sum_{ a=1}^{N_{\rm r}}\Big{(}\hat{u}_{\rm r}^{\rm L}-\hat{u}_{\rm s}\Big{)}\delta(y-y_{ \rm n})\delta\Big{(}x-x_{\rm r}^{\rm L}\Big{)}=\frac{1}{v}e^{i\frac{\omega_{ \rm F}-\omega}{v}(y-y_{0})}\delta\Big{(}x-x_{\rm r}^{\rm L}\Big{)} \tag{4}\] \[E_{\rm r}I_{\rm r}\frac{\partial^{4}\hat{u}_{\rm r}^{\rm R}}{ \partial y^{4}}-m_{\rm r}\omega^{2}\hat{u}_{\rm r}^{\rm R}+k_{\rm r}^{*}\sum_{ a=1}^{N_{\rm r}}\Big{(}\hat{u}_{\rm r}^{\rm R}-\hat{u}_{\rm s}\Big{)}\delta(y-y_{ \rm n})\delta\Big{(}x-x_{\rm r}^{\rm R}\Big{)}=\frac{1}{v}e^{i\frac{\omega_{ \rm F}-\omega}{v}(y-y_{0})}\delta\Big{(}x-x_{\rm r}^{\rm R}\Big{)}\] (5) \[\frac{E_{\rm s}\hat{u}_{\rm s}^{\rm L}}{12\big{(}1-v_{\rm s}^{2} \big{)}}\Big{(}\frac{\partial^{4}\hat{u}_{\rm s}}{\partial x^{4}}+2\frac{ \partial^{4}\hat{u}_{\rm s}}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4} \hat{u}_{\rm s}}{\partial y^{4}}\Big{)}-\rho_{\rm s}h_{\rm s}\omega^{2}\hat{u} _{\rm s}-k_{\rm r}^{*}\sum_{a=1}^{N_{\rm r}}\big{(}\hat{u}_{\rm r}^{\rm L}- \hat{u}_{\rm s}\big{)}\delta(y-y_{\rm n})\delta\big{(}x-x_{\rm r}^{\rm L}\big{)}-\] (6) \[\left(-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{K}\right)\hat{ \mathbf{U}}(\omega)=\hat{\mathbf{F}}(\omega) \tag{1}\]0  
where \(E_{\rm r}\) and \(I_{\rm r}\) are the elastic modulus and moment of inertia of the rail, respectively. \(m_{\rm r}\) is the mass per unit length of the rail. \(k_{\rm r}^{*}=k_{\rm r}+i\omega c_{\rm r}\), where \(k_{\rm r}\) and \(c_{\rm r}\) are the supporting stiffness and damping of the fastener, respectively. \(N_{\rm r}\) is the number of fasteners under the rail in a basic element. \(y_{\rm n}\) is the y-coordinate of the \(a\)th fastening. \(E_{\rm s}\), \(\
u_{\rm s}\), \(\rho_{\rm s}\) and \(h_{\rm s}\) are the elastic modulus, Poisson's ratio, mass density and slab thickness, respectively. \(k_{\rm s}^{*}=k_{\rm s}+i\omega c_{\rm s}\), where \(k_{\rm s}\) and \(c_{\rm s}\) are the stiffness and damping of springs under the slab, respectively. \(N_{\rm s}\) is the number of supports under the slab along the y-coordinate in the basic element, and \(\big{(}x_{\rm s}^{\rm L},y_{\rm sb}\big{)}\) and \(\big{(}x_{\rm s}^{\rm R},y_{\rm sb}\big{)}\) are the positions of the steel-spring supports under the slab. The mathematical mode superposition method, integral-transform method and properties of the Dirac Function were used to obtain the dynamic response of the track under moving harmonic loads. The detailed solution process can be found in [55]. By introducing and organising the rail dynamic flexibility matrix, the rail displacement vector at each wheel-rail interaction point under moving wheel-rail interaction forces at frequency \(\omega_{F}\) can be expressed as follows. \[\left(-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{K}\right)\hat{ \mathbf{U}}(\omega)=\hat{\mathbf{F}}(\omega) \tag{1}\]1  
Because the typical frequency of metro train-induced environmental vibrations is lower than 100 Hz, the Hertzian contact theory was applied, assuming that the wheel-rail contact point is at the central axis of the rail head and that the wheel-rail dynamic interaction is linear [56]. In addition, when a train moves on the track, the wheel-rail interaction forces are excited by the track irregularities. In the train-FST coupled model established here, the track irregularities can be simulated via the trigonometric series approach based on the power spectrum density (PSD). The PSD of the track irregularity was the same for each wheel.