## 2 Theoretical Analysis Model and Calculation Method
### Impact of Fastener Parameters on the Power Flow Transfer Rate of the Entire System  
Figure 8 presents the power flow transfer rate between the components of the integral ballast bed for fastener stiffness values of 20 kN/mm, 40 kN/mm, and 60 kN/mm. The influence of fastener stiffness on power flow transfer varies across different frequency bands. Figure 7: Average vibration energy level of integral roadbed and box girder bridge. As shown in Figure 8a, in the frequency range of 0-100 Hz and 650-1600 Hz, the downward vibration energy transmitted by the track increases with the rise in fastener stiffness. However, within the 100-650 Hz frequency band, the power flow transfer of track strengthens as the design value of fastener stiffness decreases. This indicates that the stiffness design within this frequency band should not be excessively small, as it could weaken the damping effect of the fastener system. Figure 8b shows that the impact of fastener stiffness on the power flow transfer rate from the ballast bed to the bridge roof primarily occurs in the frequency band above 700 Hz. Within the 100-700 Hz range, vibration energy transfers more readily downward with increasing fastener stiffness. However, in the frequency range of 700-1600 Hz, a smaller fastener stiffness results in a stronger power flow transfer efficiency. Figure 8c shows that the influence of fastener stiffness on the power flow transfer rate in the box girder bridge is comparatively weaker than in the track structure. Fastener stiffness mainly affects the high-frequency transmission of the box girder bridge. In summary, the stiffness of the fastener system significantly influences the vibration energy transfer within the integral ballast. Considering that the vibration noise of the bridge  
Figure 8: Impact of different fastener stiffness values on power flow transfer of the bridge system: (**a**) Power transfer rate from the track to overall ballast with different fastener stiffness values; (**b**) Power transfer rate form the integral ballast bed to the bridge roof with different fastener stiffness values; (**c**) Power flow transfer rate from the bridge roof to bridge bottom with different fastener stiffness values. primarily concentrates in the low-frequency band, the fastener stiffness design should not be excessively small when considering the vibration noise suppression of the bridge. Figure 9 illustrates the influence of the fastener system's damping on the vibration energy transfer within the bridge system for fastener damping of 30 kN\(\cdot\)s/m, 40 kN\(\cdot\)s/m, and 50 kN\(\cdot\)s/m. As seen in Figure 9a, the damping of fasteners has minimal impact on the vibration energy transfer from the track to the ballast within the 0-650 Hz frequency range, with a more pronounced effect observed in the frequency band above 650 Hz. This suggests that greater damping in the fastener system promotes the transmission of high-frequency vibration energy. From Figure 9b, it is evident that within the frequency band above 650 Hz, fastener damping exerts little influence on the energy transfer rate of the ballast bed to the bridge. However, during this period, vibration energy accumulates within the ballast bed, potentially affecting its service performance. Figure 9c illustrates that fastener damping significantly affects the transfer of vibration energy within the box girder bridge. Within the 0-812 Hz damping range, the fastener damping is inversely proportional to the overall power flow transfer rate. This suggests  
Figure 9: Impact of different fastener damping levels on power flow transfer of the bridge system: (**a**) Power transfer rate from the track to overall ballast with different fastener damping levels; (**b**) Power transfer rate form the integral ballast bed to the bridge roof with different fastener damping levels; (**c**) Power flow transfer rate from the bridge roof to bridge bottom with different fastener damping levels.