# Ground-Borne Vibration Model in the near Field of Tunnel Blasting
## 2 Dimensional Analysis
### Method
 \tag{5}\]  
According to the Buckingham \(\pi\) theorem, the function can be obtained as:  
\[f(\pi_{1},\pi_{2},\pi_{3},\pi_{4},\pi_{5},\pi_{6},\pi_{7},\pi_{8})=0 \tag{6}\]  
The relationship equation is:  
\[\ln\dfrac{v}{RT^{-1}}=\partial_{0}+\partial_{1}\ln\dfrac{\rho_{r}}{QR^{-3}}+ \partial_{2}\ln\dfrac{C}{RT^{-1}}+\partial_{3}\ln\dfrac{b}{R}+\partial_{4}\ln \dfrac{l}{R}+\partial_{5}\ln n+\partial_{6}\dfrac{A}{R^{2}}+\partial_{7}\ln \dfrac{b}{a} \tag{7}\]  
\[\dfrac{V}{RT^{-1}}=K(\dfrac{\rho_{r}}{QR^{-3}})^{\gamma_{1}}(\dfrac{C}{RT^{-1 }})^{\gamma_{2}}(\dfrac{b}{R})^{\gamma_{3}}(\dfrac{l}{R})^{\gamma_{4}}(n)^{ \gamma_{5}}(\dfrac{A}{R^{2}})^{\gamma_{6}} \tag{8}\]  
\[V=K(\dfrac{\rho_{r}}{QR^{-3}})^{\gamma_{1}}(\dfrac{C}{RT^{-1}})^{\gamma_{2}}( \dfrac{b}{R})^{\gamma_{3}}(\dfrac{l}{R})^{\gamma_{4}}(n)^{\gamma_{5}}(\dfrac{ A}{R^{2}})^{\gamma_{6}}(\dfrac{b}{a})^{\gamma_{7}} \tag{9}\]  
According to the actual engineering experience, the complexity of practical problems, statistical influencing factors, the difficulty of monitoring data, the cost of obtaining data, the feasibility of the final scheme, and the collinearity caused by too many influencing factors are comprehensively considered to simplify the problem. Some secondary factors and effect repeatability factors are deleted, and only the following main factors that have a direct impact on the vibration in the near field of blasting are considered: maximum charge per delay \(Q\), initiation time per delay \(T\), initial density \(\rho_{r}\), distance from the detonation center \(R\), and the number of free surfaces \(n\). Then, the above solution changes into the following process:  
\[A= \begin{bmatrix}1&0&0&0&1&0\\ 0&1&0&1&-3&0\\ 0&0&1&-1&0&0\end{bmatrix} \begin{array}{c}\text{M}\\ \text{L}\\ \text{T}\end{bmatrix} \tag{10}\]  
Three independent zero-dimensional quantities are obtained:  
\[ \begin{cases}\pi_{1}=&R^{-1}TV\\ \pi_{2}=&Q^{-1}R^{3}\rho_{r}\\ \pi_{3}=&n\end{cases}  \tag{11}\]  
According to the Buckingham \(\pi\) theorem, the function can be obtained as:  
\[f(\pi_{1},\pi_{2},\pi_{3})=0 \tag{12}\]The new relationship equation is:  
\[\ln\frac{V}{RT^{-1}}=\partial_{0}+\partial_{1}\ln(\frac{\rho_{r}}{QR^{-3}})+ \partial_{2}\ln(n) \tag{13}\]  
\[\frac{V}{RT^{-1}}=K(\frac{\rho_{r}}{QR^{-3}})^{\partial_{1}}(n)^{\partial_{2}} \tag{14}\]  
\[V=KT^{-1}(\frac{\rho_{r}}{QR^{-3}})^{\gamma_{1}}(n)^{\gamma_{2}} \tag{15}\]  
A reasonable explanation of the blasting vibration prediction Formula (15): The blasting vibration velocity is inversely proportional to the blasting delay time. The longer the blasting delay time, the smaller the superposition energy at a certain point, and the smaller the rock vibration caused. This is consistent with the facts. Since the blasting vibration velocity is directly proportional to the maximum charge per detonating period, it can be concluded that \(\gamma_{1}\) is negative, so that the blasting vibration velocity is inversely proportional to the third power of the distance, which is also in line with reality. When other conditions remain unchanged and only the rock density is different, if the maximum charge per detonating period is the same, the force acting on the rock is the same, so the displacement caused by the force acting on the rock with low density will be relatively large. Under the condition of the same action time, since the velocity is the ratio of displacement to time, the vibration velocity caused by blasting will be relatively high in the rock with low density. Therefore, the blasting vibration velocity is inversely proportional to the density of the rock. Three factors not considered in the Sadovsky formula are added to Formula (15), which will further improve the accuracy of blasting vibration prediction.