## 3 Characteristics and Mechanisms of Bridge Vibration and Structure-Borne Noise
### Characteristics of Bridge Vibration and Radiated Noise  
Taking the condition of the freight EMU passing through the bridge at 350 km/h as an example, the characteristics of wheel-rail forces, bridge vibration, and bridge radiation noise under conditions of empty and loaded containers are analyzed. Figure 3a shows the comparison of vertical wheel-rail forces in the time domain. It can be seen that the wheel-rail forces under a loaded containers condition are significantly larger than that of an empty containers condition, and the wheel-rail forces of the two conditions fluctuate around 68 kN and 88 kN, respectively, but the corresponding fluctuation ranges are close. Figure 3b shows the comparison of acceleration in the frequency domain at the center position of the deck. It can be seen that the overall trends of accelerations under two conditions are consistent in the range below 200 Hz with main frequencies of 71.2 Hz, in which the accelerations of loaded containers condition at frequency ranges below 15 Hz and above 165 Hz, and at frequencies such as 60.8 Hz, 92.0 Hz, and 137.2 Hz, they are slightly larger than that of empty containers condition, with a maximum difference of about 0.06 m/s\({}^{2}\). Figure 2: Model validation results: (**a**) arrangement of test points; (**b**) comparison of bridge acceleration in the time domain; (**c**) comparison of bridge acceleration RMS in the frequency domain; (**d**) comparison of 1/3 octave band SPLs. Figure 4 shows the comparison of bridge acoustic powers under two load conditions. Similar to the bridge vibration results, the acoustic powers of the two conditions are close with the same main frequencies of 71 Hz, and both the maximum acoustic powers are around 125.9 dB. The main frequencies of simulated acoustic power are consistent with that in Ref. [29]. In the frequency range below 40 Hz and above 120 Hz, the bridge acoustic power of the loaded containers condition is slightly larger than that of the empty containers condition, and the maximum increase is about 2.4 dB. Moreover, the standard noise test point N1 (1.2 m above the rail and 7.5 m away from the center of the near-track rail line [30]) and field points N2-N6 in Figure 1 are selected as typical field points, and the characteristics of field point SPLs under two load conditions are compared, as shown in Figure 5. It can be seen that the SPLs of the two load conditions are consistent at the same field point, and the main frequencies of SPLs at points N1-N6 are at 80 Hz with the maximum SPL around 83-88 dB. The prominent differences between the SPLs under the two conditions lie in the frequency range of 20 Hz and below, in which the SPLs under the loaded containers condition are larger than that of the empty containers condition, and the maximum difference is about 2 dB. In addition, the SPLs under loaded container conditions are also slightly larger than those under empty container conditions at the frequency range above 125 Hz. To further study the panel contribution of the bridge to the above field points, taking the loaded containers condition as an example, Figure 6 shows contributions of the left flange plate (P1), deck (P2), right flange plate (P3), left web plate (P4), right web plate (P5), and bottom plate (P6) to the SPLs of each field point. As shown in Figure 6a, the main frequency of SPLs at the field point N1 is around 65-90 Hz, in which the deck, right flange plate, right web plate, and bottom plate contribute most to the SPLs. The prominent SPLs at the field point N2 are concentrated in the frequency ranges of 2-10 Hz and 65-90 Hz,  
Figure 4: Comparison of bridge acoustic powers under different load conditions. Figure 3: Dynamic responses of the vehicle–track–bridge system under different load conditions: (**a**) vertical wheel–rail forces; (**b**) bridge deck accelerations. and the main contribution plates are the deck and bottom plate, as shown in Figure 6b.