## 3 Characteristics and Mechanisms of Bridge Vibration and Structure-Borne Noise
### Mechanism Analysis of Bridge Radiated Noise  
Based on the above analysis, the mechanism of bridge radiated noise is further investigated in this section. Figure 7 shows the bridge modal shapes and vibration clouds at peak frequencies of the bridge acoustic power, namely 43 Hz, 71 Hz, and 106 Hz. It can be seen  
Figure 5: Comparisons of 1/3 octave band SPLs under different loading conditions at field points (**a**) N1; (**b**) N2; (**c**) N3; (**d**) N4; (**e**) N5; and (**f**) N6. Figure 6: The panels’ sound pressure contributions under the loaded containers condition at field points (**a**) N1; (**b**) N2; (**c**) N3; (**d**) N4; (**e**) N5; and (**f**) N6. that the bridge modal shape at 43.0 Hz presents local vibration of the deck and the bottom plate distributed along the centerline of plates, and there are six and three longitudinal bending half-waves on the deck and the bottom plate, respectively. By comparing the vibration cloud and modal shape around 43 Hz, it can be observed that the deck of the bridge resonates significantly induced by wheel loads, with a maximum acceleration of about 0.56 m/s\({}^{2}\). At the frequency of 70.1 Hz, the bridge modal shape mainly exhibits the local vibration of the deck near the center line, and six longitudinal and two lateral bending half-waves appear. The vibration cloud at 71.2 Hz presents intense local vibration near the center of the deck, and the maximum acceleration attains 2.06 m/s\({}^{2}\). Moreover, prominent local vibration at the bottom plate and the left and right flange plates can be observed, which is larger than that of the deck at 43 Hz. The bridge modal shape at 105.5 Hz is similar to that at 70.1 Hz, but the high-frequency vibration is more significant at 105.5 Hz due to the higher modal order, with eleven longitudinal and two lateral bending half-waves. The vibration cloud at 105.9 Hz appears several intense vibration zones overlapped with the local vibration area of the modal shape in the deck. The maximum acceleration is about 0.42 m/s\({}^{2}\), which is much smaller than that at 71.2 Hz. [PERSON] et al. [31] indicate that the local resonance of structures is one of the main causes of bridge radiation noise. The maximum local vibration of the deck at 71.2 Hz corresponds to the maximum bridge acoustic power at 71 Hz, and the maximum acceleration of the deck at 43.4 Hz is slightly larger than that at 105.9 Hz, but the local vibrations of the left and right web plates at 105.9 Hz is more intense, leading to the close acoustic powers at these two frequencies, which exactly coincides with the acoustic power results at peak frequencies in Figure 4.  
Figure 7: Characteristics of the bridge vibrations near peak frequencies of the acoustic power. Figure 8 shows the acoustic radiation efficiency of the bridge in the frequency range of 0-200 Hz. It can be seen that there exist obvious peaks of the acoustic radiation efficiency at 71 Hz and 106 Hz, while the corresponding value at 43 Hz is relatively small. By combining the bridge acoustic power in Figure 4, it can be analyzed that both the maximum acoustic power and acoustic radiation efficiency lead to maximum SPLs around 71 Hz. The bridge acoustic powers around 105 Hz are close to that of around 43 Hz, but the larger acoustic radiation efficiency causes greater SPLs around 105 Hz. The bridge is generally considered as multiple rectangular thin plates with different thicknesses in the research of bridge radiated noise, and the ideal thin plate theory is introduced to study the characteristics of acoustic radiation efficiency. Based on the acoustic radiation theory of bending waves in a finite plate, the critical frequency of the plate is [32; 33] as follows:  
\[f_{c}=c^{2}\sqrt{12\rho(1-\sigma^{2})/E}/2\pi h \tag{8}\]  
where \(\rho\) is the density of the plate; \(\sigma\) is [PERSON]'s ratio; \(E\) is [PERSON]'s modulus; \(h\) is the height of the plate.