# Characteristics of Vibration Velocity Signal Using Liquid Carbon Dioxide Rock-Breaking Technology
## 2 Field Test
### Vibration Velocity Monitoring Results
#### 3.4.2 Frequency Domain Analysis  
An analysis of the damage caused by blasting should consider not only the peak value of the vibration velocity but also the vibration signal's frequency characteristics.  
Fourier transform is often used in time-frequency domain signal conversion. Discrete time signal \(x(n)\) can be converted to spectrum signal \(X(jw)\) by a Fourier transform, and the conversion formula is:  
\[X(jw)=\sum\limits_{n=-\infty}^{\infty}x(n)e^{-jwn} \tag{7}\]  
where \(j\) is an imaginary unit, and \(w\) is the digital frequency.  
Because blasting vibration signals have a limited length, the discrete Fourier transform is used in the actual processing. For vibration signal \(x(n)\) with length \(n\), the discrete Fourier transform formula is:  
\[X(k)=\sum\limits_{n=0}^{N-1}x(n)e^{-j(2\pi/N)kn} \tag{8}\]  
where \(X(k)\) (\(k\) = 0, 1, 2,..., \(N-1\)) is the discrete Fourier coefficient, representing the amplitude of the corresponding frequency.  
The main vibration frequency and frequency distribution characteristics of blasting vibrations can be analyzed by a discrete Fourier transform, and the energy distribution characteristics of the vibration signal in different frequency ranges can be further analyzed. The Fourier power spectrum can be calculated by Equation (9), and the cumulative power of the vibration signal can be calculated according to Equation (10).  
\[P_{k}=X_{k}\overline{X_{k}} \tag{9}\]  
\[C_{m}=\frac{\sum\limits_{k=0}^{m}X_{k}}{\sum\limits_{k=0}^{N-1}X_{k}} \tag{10}\]  
where \(\overline{X_{k}}\) is the conjugate complex numbers of Fourier coefficient \(X_{k}\), \(m\) = (1, 2,..., \(N-1\)).  
First, we compared the differences in the frequency domains of the vibration velocity signals generated by explosive blasting and L-CDRB. By using the scaled charge of the explosive method and the equivalent scaled charge of the L-CDRB method, monitoring points with approach equivalent scaled charges were selected for a Fourier spectrum analysis, as shown in Figure 10.  
Figure 10: Frequency domain comparison of vibration velocities using different test methods: (**a**) scaled charge approach 0.03; (**b**) scaled charge approach 0.08.  
Figure 10 shows that the dominant frequency of the explosive blasting vibration was higher, the frequency composition was richer, and the frequency range was wider. These results were related to the two tests' different detonation modes: explosive blasting for segmental detonation results in many interactions, such as superposition and reflection in the propagation process of the shock wave, meaning it has a more abundant frequency component. In L-CDRB, all the detonations occur at one time, so the frequency composition is relatively simple.  
Figure 11 shows the energy accumulation of the vibration velocity signal in different frequency bands. When the equivalent scaled charge approaches 0.03 and 0.08, 60% of the energy of the L-CDRB vibration velocity signal was concentrated in the range of 10-30 Hz, and the explosive blasting was mainly concentrated in the range of 30-60 Hz. Thus, the energy of the vibration velocity signal generated by the L-CDRB was mainly concentrated in the lower frequency bands.  
To further analyze the frequency characteristics of the vibration signal generated by L-CDRB, monitoring points were selected in the three tests at the same distance from the explosion zone's center. Figure 12 shows that larger filling amounts of liquid CO\({}_{2}\) led to higher main frequencies of the vibration velocities, and the vibration energy gradually moved from a low to a high-frequency band.  
Figure 11: Energy distribution comparison of vibration velocities using different test methods: **(a)** scaled charge approach 0.03; **(b)** scaled charge approach 0.08.  
Figure 12: Comparison of vibration velocities under different filling amounts: **(a)** spectrum; **(b)** energy distribution.