# Train-Induced Vibration Analysis and Isolation Trench Measures in Metro Depot Structures
## 3 Propagation Law of Train-Induced Vibration in Metro Depot
### Vibration Response Analysis of the Floor in the Over-Track Building  
Figure 11 shows the peak accelerations of all floor slabs from the 2 nd to the 9 th floors of the over-track building. It can be observed from Figure 11 that the overall peak accelerations of slabs 3 and 13 are relatively large, followed by slabs 1, 2, 4, 12, 14, and 15. The peak accelerations of slabs 5 to 11 are comparatively smaller. The maximum accelerations of slabs 1, 2, 3, 13, 14, and 15 are 0.0182 m/s\({}^{2}\), 0.0144 m/s\({}^{2}\), 0.0312 m/s\({}^{2}\), 0.0395 m/s\({}^{2}\), 0.0175 m/s\({}^{2}\), and 0.0180 m/s\({}^{2}\), respectively. Figure 8: Location of verification points. Figure 10: Vibration level of 1/3 octave band comparison: (**a**) verification point 1; (**b**) verification point 2. Figure 9: Acceleration time history comparison: (**a**) verification point 1; (**b**) verification point 2. To further investigate the differences in vibration responses between different floor slabs, modal analysis was conducted on floor slabs 1 and 3 as examples. Table 6 presents the first five vertical natural frequencies obtained from the modal analysis of floor slabs 1 and 3. The vibration in different frequency bands exhibits distinct variation trends during its transmission from the structural column to the center of the floor slab. In this study, the acceleration amplification factor is used to reflect these differences. The specific formula is shown as follows:  
\[A_{a}=\frac{A_{m}}{A_{c}} \tag{3}\]  
where \(A_{a}\) represents the acceleration amplification factor; \(A_{m}\) denotes the \(1/3\) octave band acceleration root mean square (RMS) value at the midpoint of the floor slab; and \(A_{c}\) refers to the \(1/3\) octave band acceleration RMS value at the corner point of the floor slab. Figure 12 illustrates the locations of different vibration measurement points on the floor slab. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Model Order**} & \multicolumn{2}{c}{**Natural Frequency (Hz)**} \\ \cline{2-3}  & \multicolumn{2}{c}{**Floor Slab 1**} & \multicolumn{2}{c}{**Floor Slab 3**} \\ \hline
1 & 45.7 & 29.4 \\
2 & 49.4 & 44.3 \\
3 & 49.7 & 44.4 \\
4 & 54.7 & 51.5 \\
5 & 56.1 & 54.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Floor slab modes. Figure 11: Peak acceleration of floor slabs. Figure 12: Location of vibration points on the floor slab. Figure 13 shows the acceleration amplification factors in the 1/3 octave band ranging from 2 Hz to 80 Hz for floor slabs 1 and 3 of the over-track building. It can be observed that the vibration of floor slab 1 is amplified in the frequency range of 31.5 Hz to 80 Hz. The amplification is most significant at the frequencies of 50 Hz and 63 Hz. For floor slab 3, vibration amplification occurs across the frequency range of 5 Hz to 80 Hz. The amplification is particularly pronounced at the frequencies of 20 Hz, 25 Hz, 31.5 Hz, and 80 Hz. Considering the vibration frequency range of 2 Hz to 80 Hz, the overall amplification trend of floor slab 3 is stronger than that of floor slab 1. The area of floor slab 1 is 21.76 m\({}^{2}\), while the area of floor slab 3 is 36.72 m\({}^{2}\). As shown in Figure 11, the peak acceleration of floor slab 3 is generally greater than that of floor slab 1. This indicates that the acceleration response of the floor slab is related to its area. As demonstrated by the modal analysis of the floor slabs in Table 6, under the condition of equal slab thickness, larger floor spans result in lower vertical stiffness. Consequently, the lower-order vertical natural frequency of the slab decreases. This increases the overlap between the natural frequency range of the slab and the dominant frequency range of vibration induced by train operations, making resonances more likely to occur. In the vibration response analysis of the floor slab of the over-track building, the analysis can be conducted not only from the perspective of acceleration response but also by considering Z-vibration levels and vibration levels in the 1/3 octave band.