## 2 Establishment of Prediction Model
### Metro Train-FST Coupled Model and Train Load Calculations  
A 3D metro-train-FST coupled analytical model was established in the frequency domain. As illustrated in Figure 2, the coupled model can be divided into training and tracking systems. In the train system, each carriage was simplified as a car body, two bogie frames and four wheelsets connected by primary and secondary suspension systems. Twenty-seven degrees of freedom (DOFs) were selected for each carriage in this model, comprising five DOFs for each car body and bogie frame, and three DOFs for each wheelset. The train consisted of six carriages. Figure 1: Flowchart for using the proposed prediction model. Based on the theory of multi-rigid-body dynamics and D'Alembert's principle, the equation of motion of each carriage can be expressed in the frequency domain as follows:  
\[\left(-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{K}\right)\hat{ \mathbf{U}}(\omega)=\hat{\mathbf{F}}(\omega) \tag{1}\]  
where \(\hat{\mathbf{U}}\) is the vehicle displacement vector of the vehicle in the frequency domain, which includes the displacement of the vehicle body, the two bogie frames and the four wheelsets, and the positive directions of the displacement of each component have been shown in Figure 2. \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass, damping and stiffness matrices of the carriage, respectively. \(\hat{\mathbf{F}}\) is the wheel-rail contact force vector. \"\(\cdots\)\" indicates that the physical quantities are in the frequency domain. By re-arranging Equation (1), the displacement response of the carriage can be obtained as follows:  
\[\hat{\mathbf{U}}(\omega) =\left(-\omega^{2}\mathbf{M}+\mathrm{i}\omega\mathbf{C}+\mathbf{ K}\right)^{-1}\hat{\mathbf{F}}(\omega) \tag{2}\] \[=\mathbf{A}_{c}(\omega)\hat{\mathbf{F}}(\omega)\]  
where \(\mathbf{A}_{c}(\omega)\) is the flexibility matrix of the vehicle. Based on the inter-relationship between the wheels and wheelsets of a train, by introducing transfer matrices, the flexibility matrix \(\mathbf{A}_{\mathrm{wheel}}^{\mathrm{vehicle}}\) of the wheels is derived; the detailed derivation process is shown in [51]. Furthermore, the relationship between the wheel displacement of the entire train and the corresponding wheel-rail interaction force at frequency \(\omega_{F}\) can be obtained as follows:  
\[\overline{\mathbf{U}}_{\mathrm{wheel}}^{\mathrm{vehicle}}(\omega_{F})=- \mathbf{A}_{\mathrm{wheel}}^{\mathrm{vehicle}}(\omega_{F})\overline{\mathbf{F}}_{ \mathrm{wr}}^{\mathrm{vehicle}}(\omega_{F}) \tag{3}\]  
where \(\overline{\mathbf{U}}_{\mathrm{wheel}}^{\mathrm{vehicle}}\) is the wheel-displacement vector for the entire train, \(\overline{\mathbf{F}}_{\mathrm{wr}}^{\mathrm{vehicle}}\) is the corresponding wheel-rail interaction force vector and \"-\" indicates that the directions of the wheel displacement and wheel-rail interaction force are opposite. For the FST system, a 3D FST model was established. The two rails were modelled as infinite Euler-Bernoulli beams, whereas the finite slabs were modelled as thin plates ([PERSON]'s theory) with free boundary conditions. The two rails were discretely fixed by fasteners on the slab, and the slabs were supported by discrete springs on the concrete base. Fasteners and springs were modelled as spring-damper elements. According to the periodic-infinite theory [52], the FST (Figure 3) can be regarded as a periodic-infinite structure in the longitudinal direction, and a slab with rails and spring-damper elements within its range can be regarded as the periodic element of the track. Figure 2: Schematic of metro-train–FST coupled model. Based on the [PERSON] beam theory and the [PERSON] thin plate theory, the equations of motion in the vertical direction for this basic element under the action of a moving unit harmonic load in the frequency domain are given as follows, which mainly includes the basic equations of the Euler beam [53], thin plate theory, and the terms related to fasteners and steel spring supports [54].