# Study on Dominant Frequency Attenuation of Blasting Vibration for Ultra-Small-Spacing Tunnel
## 5 Data Analysis
### Spectrum Analysis of Each Delay Sequence Vibration  
Analyzing the X-direction curve in Figure 8b, the entire vibration velocity waveform is divided into six curves based on the detonator delay time, as shown in Figure 10. The Arabic numerals above each waveform represent the delay sequence, and the dotted line corresponds to the setting time of the detonator initiation. The initiation time of each sequence does not fully correspond to the setting time, owing to the accuracy of the MS detonator and the variation of the seismic wave during propagation. The delay sequences of this vibration waveform are 1, 5, 7, 9, 11, and 13. Noticeably, the waveform and spectrum analysis in Figure 10 show that the vibration waveforms for all sequences have different characteristics (PPV, frequency, and duration). The characteristics of each delay sequence are listed in Table 5.  
As shown in Figure 10, the ZCF is depicted near the peak of each waveform and the DF is depicted at the peak amplitude in the power spectrum. Overall, the ZCF and the DF are increased as the delay sequence grows; however, it is not a completely linear trend. To better describe it, the MFs are calculated in Table 5, and three dominant frequencies (ZCF, DF, and MF) of each delay sequence are depicted in Figure 11. It is seen that the global frequency of the entire vibration is close to the DF of the cutting blastholes in MS1.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Blast Parameters**} & **MS1** & **MS5** & **MS7** & **MS9** & **MS11** & **MS13** \\  & **Delay** & **Delay** & **Delay** & **Delay** & **Delay** & **Delay** & **Delay** \\ \hline ZCF (Hz) & 57.0 & 66.4 & 67.8 & 82.0 & 76.8 & 130.5 \\ DF (Hz) & 29.9 & 60.3 & 45.9 & 81.6 & 78.1 & 91.8 \\ MF (Hz) & 93.7 & 111.8 & 107.9 & 111.9 & 99.1 & 119.0 \\ Number of holes & 16 & 8 & 22 & 16 & 22 & 42 \\ Explosive weight & 52.8 & 21.6 & 47.4 & 28.8 & 33.6 & 29.4 \\ amount (kg) & 24.2 & 83.9 & 55.3 & 91.7 & 77.0 & 146.5 \\ Effective vibration (ms) & 22.5 & 35.2 & 41.7 & 48.2 & 55.98 & 65.5 \\ EBVS (cm) & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Characteristics of all sequence waveforms in X–direction shown in Figure 8b.  
Figure 9: Power spectrum of vibration waveform in (**a**) Figure 8a, (**b**) Figure 8b.  
The dominant frequency of vibration is greatly influenced by the structure of the charge [28]. Understandably, in this study, the change in the dominant frequency of each sequence is independent of the charge structure because the charge structure is  
Figure 11: Relationship between frequency and delay blast sequencies.  
Figure 10: All sequence waveforms in X–direction shown in Figure 6b and their power spectra. (a) MS1, (b) MS5, (c) MS7, (d) MS9, (e) MS11, (f) MS13.  
the same in all sequences. There are two other theories for the propagation law of the dominant frequency. One is the blasting cavity theory [31]: for the same free surface or the same blast shapes, a large equivalent blasting vibration source (EBVS) implies a low vibration frequency. The other is the free surface theory [30]: multiple free surfaces cause superposition of blasting vibration waves and increase of frequency; for example, as the cutting blastholes are detonated under a confined boundary condition in the MS delay blasting, the dominant frequency of it is smaller than those of the subsequent blastholes. The abovementioned not-completely-linear trend of the dominant frequency with the increase in the delay sequence (as shown in Figure 11) can be explained by the free surface theory. In contrast, the curve fluctuates for some sequences with the same blast shapes conforms to the blasting cavity theory. Based on the blasting parameters (as shown in Figure 4), the EBVS at MS5 and MS7 have the same shapes but different sizes, and there is a negative correlation between the dominant frequency and the radius of the EBVS. This rule, which conforms to the blasting cavity theory, is also observed for MS9 and MS11.