## 3 Numerical Simulation of Underground Tunnel Vibration Waves
### Blasting Vibration Analysis
#### 3.3.2 Blasting Vibration Velocity Analysis  
In response to the numerical simulation results, we conducted a study on the impact of blasting near tunnels, with a focus on analyzing the attenuation pattern of the vibration velocity with varying distances from the blast point. The distance from the explosion center to measurement point A is 12.60 m, The measuring point is located at the intersection of the two tunnels and is in contact with the air, leading to rapid attenuation of the vibration velocity when propagated to this point. When analyzing the pattern of vibration velocity changes with the distance from the blast point, we disregarded the data at inflection points. We investigated the propagation pattern of blasting vibration waves in tunnels, determining the maximum single charge and maximum safe distance for blasting operations. Regression analysis was performed on the simulated values, and the Sadovsky formula was used:  
\[V=K\left(\frac{\sqrt[3]{Q}}{R}\right)^{\alpha} \tag{2}\]  
where \(V\) represents the vibration velocity of the medium particle (cm/s); \(Q\) denotes the maximum charge weight of a single explosive segment (kg); \(R\) is the distance from the explosion center (m); and \(K\) and \(\alpha\) are coefficients and attenuation exponents related to the terrain and geological conditions between the blast area and the measurement point.  
Under conditions of in situ stress of 1 MPa and 5 MPa, the maximum values of vibration velocity in the X, Y, and Z directions and the aggregate velocity at measurement points A to G were extracted from the simulated results. The vibration velocities at measurement points B to G were then fitted using the Sadovsky formula. The fitting results are shown in Table 7.  
In the above Table 7, \(r\) is the coefficient of determination for the fitting. According to the analysis of the data in Table 7, it is evident that the vibration velocity generated by blasting is influenced by the in situ stress. Under unchanged conditions, when the in situ stress is 5 MPa, the maximum vibration velocity values at each measuring point are significantly higher than when the in situ stress is 1 MPa. When the distance from the explosion center is between 10 and 20 m, the vibration velocity in all directions rapidly decreases after reaching its peak. However, when this distance exceeds 25 m, the rate of decrease in vibration velocity slows down.  
For measuring points B-G, the propagation law of the vibration velocity in all directions conforms to the referenced [PERSON]'s theoretical formula, with a fitting correlation coefficient ranging from 0.92 to 0.97. The high correlation coefficient confirms the reliability of the vibration velocity propagation law in the simulation results using the FEM-SPH algo  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{
\begin{tabular}{c} **R** \\ **(m)** \\ \end{tabular} } & \multicolumn{2}{c}{**X**} & \multicolumn{2}{c}{**Y**} & \multicolumn{2}{c}{**Z**} & \multicolumn{2}{c}{**Aggregate Velocity**} \\  & \multicolumn{2}{c}{**(cm/s)**} & \multicolumn{2}{c}{**(cm/s)**} & \multicolumn{2}{c}{**(cm/s)**} & \multicolumn{2}{c}{**Magnitude (cm/s)**} \\  & **1 MPa** & **5 MPa** & **1 MPa** & **5 MPa** & **1 MPa** & **5 MPa** & **1 MPa** & **5 MPa** \\ \hline
12.6 & 3.95 & 4.20 & 1.87 & 2.04 & 1.17 & 1.62 & 4.23 & 4.32 \\
13.34 & 8.21 & 8.51 & 5.88 & 5.59 & 5.68 & 6.51 & 9.56 & 9.97 \\rithm. Based on the blasting parameters of underground blasting projects, such as the blast point and maximum single charge, we can rapidly and effectively calculate the vibration velocity generated by blasting and predict the propagation of vibration waves, thereby providing strong support for the safety engineering design of underground blasting.