## 2 Theoretical Analysis Model and Calculation Method
### Effect of Fastener Parameters on System Power Flow Characteristics
#### 3.2.1 Effect of Fastener Stiffness on Power Flow Transmission Characteristics  
Figure 5 illustrates the spectrum curve depicting the variation of power flow for the track, integral ballast bed, bridge roof, and bridge bottom plate in relation to changes in fastener stiffness.  
With the fastener damping set at 40 kN/s/m and unchanged, the power flow of each subsystem is individually analyzed for fastener stiffness values of 20 kN/mm, 40 kN/mm, and 60 kN/mm. Indeed, the figure demonstrates that the power flow characteristics of the track are significantly impacted by the stiffness of the fastener within the frequency range of 0-1200 Hz. In essence, there is a high sensitivity between track power flow and fastener stiffness, particularly within the middle- and low-frequency domain. The track power flow reaches its maximum at 1160 Hz, with power flow peaks for the three stiffness values registering at 297 dB, 277 dB, and 256 dB. Regarding the influence of fastener stiffness on ballast power flow, it is primarily concentrated within the frequency range of 100-700 Hz, peaking at 600 Hz. However, it is worth noting that the power flow of the entire ballast remains consistent within the high-frequency band. This indicates that the power flow of the whole ballast at high frequency is not affected by the stiffness of the fastener.  
The power flow characteristics of the box girder bridge roof within the 0-400 Hz range display significant variation, with changes in fastener stiffness. Within the 0-200 Hz range, a higher fastener stiffness facilitates more effective transfer of vibration energy to the bridge roof. At 230 Hz, the power flow of the bridge roof reaches its maximum value. In the 200-400 Hz frequency band, the impact of fastener stiffness on the power flow of the bridge roof becomes more pronounced as the fastener stiffness increases. For fastener stiffness values of 20 kN/mm and 40 kN/mm, the power flow of the bridge roof is similar, with an increase of less than 10%. Beyond 400 Hz, fastener stiffness has minimal influence on the power flow of the bridge roof, and the spectrum curve remains relatively consistent.  
The influence of fastener stiffness on the power flow of both the bridge bottom plate and bridge roof plate exhibits similar patterns. However, a notable distinction arises in the fact that fastener stiffness has a more pronounced effect on the power flow of the bridge bottom plate within the low-frequency band. The frequency range where fastener stiffness affects power flow is primarily between 0 and 270 Hz. Within this range, the power flow of the bridge bottom plate increases with higher fastener stiffness. When the fastener stiffness increases from 20 kN/mm to 40 kN/mm, the power flow increases by approximately 10%. Similarly, when the fastener stiffness increases from 40 kN/mm to 60 kN/mm, the power flow experiences an approximately 50% increase. These outcomes underscore the significant impact of fastener stiffness on the power flow of the bridge bottom plate in the low-frequency range. Within the frequency range of 270-520 Hz, fastener stiffness exerts minimal influence on the bridge bottom plate's power flow. For  
Figure 5: Power flow of track, integral ballast bed, bridge roof, and bridge bottom plate varies with the stiffness of coupler: (**a**) Power flow of track; (**b**) Power flow of integral ballast bed; (**c**) Power flow of bridge roof; (**d**) Power flow of bridge bottom plate.  
frequencies exceeding 520 Hz, the impact of fastener stiffness on the bridge becomes minor, and power flow characteristics tend to stabilize. With fastener stiffness values of 20 kN/mm, 40 kN/mm, and 60 kN/mm, the power flow of the bridge stabilizes at 102 dB, 105 dB, and 148 dB, respectively.