# Study on Dominant Frequency Attenuation of Blasting Vibration for Ultra-Small-Spacing Tunnel
## 5 Data Analysis  
### Difference in Dominant Frequency Distributions on Primary Support and Secondary Lining  
The blasting vibration waves on the primary support and the secondary lining were monitored. The relationships between the SD and the frequencies (DF and ZCF) were illustrated in Figure 6. In the data of the primary support, the dominant frequencies in the Y- (longitudinal) and X-directions (vertical) show some regularity, whereas the regularity of those in the Z-direction (transverse) is poor. The dominant frequencies are mainly distributed within 200 Hz, and the distribution range of the ZCFs generally exceeds 10 Hz, whereas a part of the DFs is widely distributed below 10 Hz. In the range of the SD less than 5.0 kg/m\({}^{3}\), the dominant frequency of the vibration wave is greater than 100 Hz; and in the range of more than 10.0 kg/m\({}^{3}\), the high frequencies part attenuates to the range of less than 50 Hz. The prediction of the dominant frequency was extensively developed by the SD [31; 34]. The variations in the upper and lower bound values of the dominant frequency on the primary support with the SD can be expressed as power functions (as shown in Figure 6 and listed in Table 4). The change in the upper bound can be understood as the attenuation of the high-frequency part of the dominant frequency, in which the dominant frequencies in the X- and Y-directions decay rapidly, whereas that in the Z-direction (transverse) decays gradually. The characteristic that the dominant  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No. **} & \multirow{2}{*}{**Test No. **} & \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Measurements**} & \multirow{2}{*}{**SD** (kg/m\({}^{3}\))} & \multirow{2}{*}{**RD (m)**} & \multicolumn{2}{c}{**X–Directions**} & \multicolumn{2}{c}{**Y–Directions**} & \multicolumn{2}{c}{**Z–Directions**} \\  & & & & **Df** & **ZCF** & **Df** & **ZCF** & **Df** & **ZCF** & **Df** & **ZCF** \\  & & & & & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** & **(Hz)** \\ \hline \multirow{2}{*}{52} & \multirow{2}{*}{11} & Primary & Point A & 5.7 & 20.1 & 33.0 & 36.4 & 49.0 & 25.6 & 93.0 & 27.0 \\  & & support & & & & & & & & & \\  & & Primary & & & & & & & & & \\  & & support & & & & & & & & & \\  & & Primary & &  
frequency gradually attenuates with the increase in the distance is mainly caused by the damping of the surrounding rock and the high-frequency absorption by the nonlinear properties [4]. An alternative explanation for these phenomena may be that the vibration waves close to the blast are more affected by characteristics of blast design, while at larger distances the propagating medium dominates the vibration response [4]. The difference of the dominant frequency distributed on the primary support and lining is reflected in two aspects. One is concerning the frequency range of the vibration wave distribution on the lining is not as wide as that on the primary support; another is the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frequency Types** & **X–Direction** & **Y-Direction** & **Z–Direction** \\ \hline \multirow{3}{*}{ZCF} & \multirow{2}{*}{upper bound:} & \(ZCF=427.6\times SD^{-0.97}\), & \(ZCF=814.6\times SD^{-1.23}\), & \(ZCF=187.43\times SD^{-0.14}\), \\  & & \(R^{2}=0.91\) & \(R^{2}=0.96\) & \(R^{2}=0.91\) \\  & & \(ZCF=173.6\times SD^{-1.85}\), & \(ZCF=61.2\times SD^{-0.60}\), & \(ZCF=179.1\times SD^{-1.93}\), \\  & & \(R^{2}=0.87\) & \(R^{2}=0.76\) & \(R^{2}=0.92\) \\  & & \(DF=1082.2\times SD^{-1.41}\), & \(DF=2862.9\times SD^{-2.18}\), & \(DF=429.7\times SD^{-0.68}\), \\  & & \(R^{2}=0.94\) & \(R^{2}=0.77\) & \(R^{2}=0.67\) \\  & & \(DF=17.1\times SD^{-0.51}\), & \(DF=7.0\times SD^{-0.33}\), & \(DF=10.5\times SD^{-0.73}\), \\  & & \(R^{2}=0.13\) & \(R^{2}=0.16\) & \(R^{2}=0.27\) \\ \hline \hline \end{tabular} Note: Underlined correlative coefficient \(R^{2}\) is less than 0.60, i.e., formula only describes mathematical data, which cannot guide practical engineering.  
\end{table}