## 2 Theoretical Analysis Model and Calculation Method
### Vibration Transmission Characteristics of Urban Track Transit Bridge Lines  
#### Vibration Energy Transfer Characteristics of Integral Ballast Bed Damping Track  
Figure 4 displays the power flow transmission characteristics of a track bridge line with an integral ballast bed. Based on the power flow spectrum curve, the following conclusions can be drawn.  
Firstly, the initial peak of the ballast bed and the roof of the box girder bridge are evident at approximately 250 Hz. The natural frequency of the first mode of vertical track bending is 200 Hz. Consequently, a coupling effect exists between the track, integral ballast bed, and the top plate of the box girder bridge at 200 Hz.  
Secondly, the second peak in power flow is observed in the track, integral ballast bed, box girder bridge roof, and bottom plate at around 600 Hz. Among these, the integral ballast bed exhibits the highest cumulative power flow, reaching approximately 300 dB, while the lowest cumulative power flow is found in the bottom plate, at approximately 175 dB.  
Thirdly, within the frequency domain below 2000 Hz, notable power flow peaks are additionally present near 800 Hz, 1200 Hz, and 1400 Hz.  
Fourthly, within the entire line system, the power flow distribution diminishes in the sequence of the integral ballast bed, bridge roof, track, and bridge bottom plate. Among these, the power flow characteristics of the integral ballast bed exhibit the most significance, indicating a substantial accumulation of vibration energy within the ballast bed.  
During the transfer of vibration energy within the track structure, certain structural elements may either consume or store a portion of the energy. As a result, potential issues  
Figure 4: Power flow of integral ballast bed.  
can arise, such as the attenuation of vibration energy or the amplification of vibrations in specific components due to energy storage. The fastener, functioning as a component that links the track and the integral ballast bed, exerts a noticeable impact on the power flow of both the track and the ballast. The disparity in power flow between the track and the ballast can serve as an indicator of the influence of the fastener on power flow transmission. In the lower frequency range, the amplitude of attenuation of vibration energy in the fastener and ballast increases as the frequency rises. The highest attenuation occurs near 100 Hz and 300 Hz. In the middle- and high-frequency bands, the amplitude of attenuation of vibration energy decreases as the frequency increases. This indicates that vibration energy within the low-frequency range is more readily absorbed or stored within the fastener system. Through a comparison of the power flow transmission characteristics between the bridge roof and the floor, it becomes evident that the power flow behaviors of the bridge roof and the floor significantly diverge within the low-frequency range. This observation suggests that, in the low-frequency band, the vibration energy primarily concentrates in the bridge roof. Moreover, it underscores the fact that, within the low-frequency range, the bridge roof constitutes the principal source of low-frequency sound in the bridge structure. This characteristic aligns with the structural focus of the current study.  
Based on the above analysis, it is evident that the power flow characteristics of the entire track bed are more significant than those of the track. In the vibration transmission process, the fastener system serves as a crucial connection component closely related to the track bed and the track. Hence, the relevant parameters of the fastener system are expected to have a significant impact on the power flow transmission characteristics. Consequently, the effects of fastener stiffness and damping parameters on power flow transmission characteristics will be further analyzed and systematically studied in the following paragraphs.