## 2 Mapping Relationship Between Rail Deformation and Bridge Deformation  
The track irregularity is classified by the length of the wave: the short wave, the medium wave, and the long wave. The long wave irregularity primarily stems from different foundational deformations, including the topographic relief, the roadbed settlement, and the bridge deformation. When scholars explore the correlation between the train vibration response and the track irregularity in the long-span bridge, it is crucial to initially establish the mapping relationship between the long-span bridge deformation and the track irregularity. There are more types of the track structure used in China's high-speed railway, mainly the ballasted track and the ballastless track. Ballastless tracks include the longitudinal ballastless track and the unit ballastless track. Finite element models of ballasted tracks, longitudinal ballastless tracks, and unit ballastless tracks were established by the ANSYS. The rail is modeled by the BEAM188 element. The fastener is simulated by a WJ-8-type fastener with the three-way spring element. The fastener is modelled with the COMBIN14 element in the vertical and transverse directions, and the COMBIN39 element in the longitudinal direction. The sleeper of the ballasted track is simulated by the BEAM188 element. The roadbed is simulated by the three-way spring element. For the ballastless track, the track slab and base slab are simulated using the SOLID 65 element. Their connection is modeled by the nonlinear spring element with the appropriate longitudinal, vertical, and lateral stiffness [17]. The finite element model is illustrated in Figure 1.  
The deformation of the bridge can be analyzed by decomposing it into various wavelength deformations through the Fourier transform. To examined the difference between the rail deformation and the bridge deformation, continuous cosine waves with different wavelengths ranging from 0.5 m to 120 m are applied to the bottom of the track. Figure 1: Cont. Figure 2 shows the difference between the rail deformation and the bridge deformation of the ballasted track (BT), the longitudinal-connected track (LT), and the unit slab-type ballastless track (USBT) influenced by continuous cosine waves with varying wavelengths. The figure reveals that, when the bridge deformation wavelength exceeds 6 m, 10 m, and 16 m for the BT, the LT, and the USBT, the difference in deformation between the bridge and the track is less than 1 mm. Conversely, for the wavelength of the bridge deformation below 6 m, 10 m, and 16 m, a substantial difference is observed in both track and bridge deformation. Figure 3 shows the rail deformation and the bridge deformation under the cosine wave excitation of the 2 m wavelength. For the BT, LT, and USBT, their differences of the deformation, respectively, are 8.97 mm, 11.73 mm, and 11.27 mm. For the BT, the LT, and the USBT, the 1:1 mapping relationship of the deformation between the rail and the bridge can be established when the bridge deformation, respectively, exceeds 6 m, 10 m, and 16 m. The rail deformation precisely mirrors the bridge deformation. However, when the bridge deformation wavelength is less than 6 m, 10 m, and 16 m, a significant difference in the deformation emerges between the track and the bridge. When the rail deformation cannot mirror the bridge deformation, it is necessary to establish a comprehensive rail-bridge model to accurately capture the rail deformation. Figure 1: Finite models of track structure: (**a**) the ballasted track; (**b**) the longitudinal ballastless track; and (**c**) the unit ballastless track. Figure 2: The relative deformation between rails and bridges. Figure 3: The vertical deformation between the rail and bridge: (**a**) BT; (**b**) LT; and (**c**) USBT.