# Simulation Study on Ground Vibration Reduction Measures of the Elevated Subway Line
## 3 Numerical Results and Discussion
### Effect of Fastener Stiffness on Ground Vibration  
The vibration velocity decreases from 0.08 mm/s to 0.07 mm/s when the train is running at 65 km/h. The time domain results and frequency domain results of the ground vibration velocity with different fasteners are shown in Figure 12. From Figure 12, it can be seen that as the vertical stiffness of the fasteners decreases, the trend of the time domain results of the ground vibration velocity is consistent and the vibration peak decreases. The frequency domain results show that the ground vibration is mainly concentrated at 8-50 Hz, the peak vibration value decreases, and the peak vibration frequency shifts from 23 Hz to 31 Hz.  
The parameters of different vibration-reducing fasteners and the results of ground vibration are shown in Table 5. The vibration velocity under two kinds of vibration-reducing fasteners still cannot satisfy the vibration limit requirements.  
The RMS value of ground vibration acceleration can be obtained according to ISO 2631 [32] as shown in the following equation:  
\[a_{w,\varepsilon}=[\frac{\sum{a_{w}}^{2}.T_{i}}{\sum{T_{i}}}]\frac{1}{2} \tag{4}\]  
where \(a_{w,\varepsilon}\) is the equivalent vibration quantity (root-mean-square acceleration in m/s\({}^{2}\)), and \(a_{w,i}\) is the amount of vibration at time \(T_{i}\) (root-mean-square acceleration expressed in m/s\({}^{2}\)).  
The time domain results and frequency domain results of ground vibration acceleration for different schemes are given in Figure 13.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multirow{2}{*}{**Speed (km/h)**} & \multicolumn{5}{c}{**Fastener parameters**} \\ \cline{2-7}  & & **Vertical** & **Lateral** & **Vertical Damping** & **Lateral Damping** & **Vibration** \\  & & **Stiffness (N/m)** & **Stiffness (N/m)** & **(Ns/m)** & **(Ns/m)** & **Velocity (mm/s)** \\ \hline Case 1 & 65 & \(12.07\times 10^{6}\) & \(13.5\times 10^{6}\) & \(1361.12\) & \(947.27\) & 0.08 \\ Case 2 & 65 & \(10\times 10^{6}\) & \(32\times 10^{6}\) & \(1861.12\) & \(1474.27\) & 0.07 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Calculation results of ground vibration under different fasteners.  
Figure 12: Ground vibration results under different vibration reduction schemes: (**a**) time domain results; (**b**) frequency domain results.  
As can be seen from Figure 13, the ground vibration acceleration tends to be consistent in the time domain, and the peak vibration acceleration decreases as the vertical stiffness of the fastener decreases. The RMS value of ground vibration acceleration is 15.63 mm/s\({}^{2}\) when using case 1 and 13.61 mm/s\({}^{2}\) when using case 2. The RMS value of ground vibration acceleration decreases by 2 mm/s\({}^{2}\). In the frequency domain results, the peak acceleration decreases, but the dominant frequency of the vibration acceleration is shifted. Combined with the vibration velocity, it can be found that the values of vibration velocity and acceleration RMS have decreased, and the peak frequency of vibration velocity and acceleration has been shifted, which is not obvious for the ground vibration damping.