**3. Experimental Results**

#### *3.1. Amplitude Variation of Interface Vibration under Impact Load*

3.1.1. Amplitude Variation of Interface Vibration under Single Point Excitation

When a single point was excited, the dynamic variation laws of the vibration amplitude of each interface obtained from the four sensors were similar; the amplitude time-history curves of 2#, 4# and 6# action points excited separately are shown in Figures 3–5. It can be seen from Figure 3 that when the 2# action point was excited alone, the amplitude curve (2#-1 in Figure 3) of No. 1 reached the first extreme value s1 = −0.005 mm when t1 = 15 ms and reached a peak s6 = 0.014 mm when t6 = 51 ms. When 4# action point was excited alone (Figure 4), the amplitude curve of No. 1 (2#-1 in Figure 4) reached the first extreme value s1 = −0.004 mm when t1 = 12 ms and the peak s6 = 0.015 mm when t6 = 49 ms. Similarly, when 6# action point was excited alone (Figure 5), the amplitude curve of No. 1 (6#-1 in Figure 5) reached the first extreme value s1 = −0.001 mm when t1 = 7 ms and the peak value s6 = 0.005 mm when t6 = 18 ms. The extreme points of amplitude curves of single point excitation (2#, 4# and 6#) and their corresponding time can be seen in Table 2. The extreme values of the amplitude curves of these three action points showed two stages as follows: first increasing and then decreasing; the vibration curves of 2# and 4# action points reached the peak after 2.25 cycles, and that of 6# action point the peak after 1.25 cycles.


**Figure 3.** Amplitude time–history curves under 2# action point excited alone. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

**Figure 4.** Amplitude time–history curves under 4# action point excited alone. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

**Figure 5.** Amplitude time-history curves under 6# action point excited alone. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

#### 3.1.2. Amplitude Variation of Interface Vibration under Multi-Point Excitation

When 2# and 4# action points were excited step-by-step, the curve of No. 1 (24#-1 in Figure 6) reached the first extreme value s1 = −0.01 mm when t1 = 11 ms, the peak s2 = 0.029 mm when t2 = 14 ms, and the second peak s6 = 0.021 mm when t6 = 46 ms. Compared with No. 3 and No. 4, No. 1 and No. 2 were close to the action points, and there were complex micro vibrations during step-by-step excitation (24#-1 and 24#-2 in Figure 6), but the whole dynamic change of all measuring points were similar. When 6# and 7# action points were excited synchronously, the curve of No. 1 (67#-1 in Figure 7) reached the first extreme value s1 = −0.001 mm when t1 = 12 ms, the peak s6 = 0.016 mm when t6 = 48 ms, and the whole dynamic change of all measuring points were also similar. The extreme values of the amplitude curves of these two groups of action points showed two stages as follows: first increasing and then decreasing, and the vibration curves reached the peak after 2.25 cycles. The extreme points of amplitude curves of multi-point excitation (2# and 4#, 6# and 7#) and their corresponding time could be seen in Table 3.

**Figure 6.** Amplitude time-history curves under 2# and 4# action points excited step by step. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

**Figure 7.** Amplitude time-history curves under 6# and 7# action points excited synchronously. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.


**Table 3.** The extreme points of amplitude curves of two groups of action points (2# and 4#, 6# and 7#).

### *3.2. Amplitude Attenuation Law of Interface Vibration under Impact Load*

#### 3.2.1. Dynamic Attenuation Law of Amplitude under Single Point Excitation

The analysis results of the amplitude extreme value under single point excitation showed that the dynamic changes were similar. The extreme values distribution of the test curves when 6# action point was excited alone can be seen in Table 4, and the curve– fitting results are shown in Figure 8. It could be concluded that the dynamic attenuation of amplitude under single point excitation conformed to the law of exponential variation y = y0 + Aexp (x/k), and the fitting degree was as high as 0.99.


**Figure 8.** Amplitude attenuation law under 2# action point excitation. (**a**) Overlapping display of the four sensor signals; (**b**) Curve-fitting results of curve decay extreme value.

3.2.2. Dynamic Attenuation Law of Amplitude under Multi-Point Excitation

The analysis results of the amplitude extreme value under multi–point excitation showed that the dynamic changes were similar. The extreme values distribution of the test curves when 6# and 7# action points were excited synchronously can be seen in Table 5, and the curve–fitting results are shown in Figure 9. It could be concluded that the dynamic attenuation of amplitude under single point excitation also conformed to the law of exponential variation y = y0 + Aexp (x/k), and the fitting degree was as high as 0.99.

**Table 5.** Amplitude extreme points of No. 1–4 under 6# and 7# action points excited synchronously.


**Figure 9.** Amplitude attenuation law under 6# and 7# action points excited synchronously. (**a**) Overlapping display of the four sensor signals; (**b**) Curve-fitting results of curve decay extreme value.

#### **4. Amplitude-Frequency Distribution of Interface Vibration under Impact Load**

#### *4.1. Amplitude-Frequency Distribution of Interface Vibration under Single Point Excitation*

The amplitude–frequency distributions when the 2#, 4# and 6# action points were excited separately are shown in Figures 10–12. When the 2# action point was excited, the amplitude of No. 1 (2#-1 in Figure 10) reached the peak 7.2 × <sup>10</sup>−<sup>3</sup> mm at P1 = 50 Hz. When the 4# action point was excited, the amplitude of No. 1 (4#-1 in Figure 11) reached the peak 7.5 × <sup>10</sup>−<sup>3</sup> mm at P1 = 50 Hz. When the 6# action point was excited, the amplitude of No. 1 (6#-1 in Figure 12) reached the peak 1.6 × <sup>10</sup>−<sup>3</sup> mm at P1 = 53.7 Hz. The amplitude variations of No. 2~4 were all similar to that of No. 1.

**Figure 10.** Amplitude–frequency distribution of interface vibration under 2# action point excitation. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

#### *4.2. Amplitude-Frequency Distribution under Multi-Point Excitation Step-by-Step*

The amplitude–frequency distributions when 2# and 4# action points were excited step-by-step are shown in Figure 13. The amplitude of No. 1 (24#-1 in Figure 13) reached the peak 5.4 × <sup>10</sup>−<sup>3</sup> mm at P1 = 52.1 Hz, compared with single point excitation, the vibration complexity of multi-point excitation step-by-step was relatively high, and the amplitude variations of No. 2~4 were all similar to that of No. 1.

**Figure 11.** Amplitude–frequency distribution of interface vibration under 4# action point excitation. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

**Figure 12.** Amplitude–frequency distribution of interface vibration under 6# action point excitation. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

**Figure 13.** Amplitude–frequency distribution under 2# and 4# action points excited step-by-step. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

#### *4.3. Amplitude-Frequency Distribution under Multi-Point Synchronous Excitation*

The amplitude–frequency distributions when 2# and 4# action points were excited synchronously are shown in Figure 14. The amplitude of No. 1 (67#-1 in Figure 14) reached the peak 5.1 × <sup>10</sup>−<sup>3</sup> mm at P1 = 48.9 Hz, the second peak 1.3 × <sup>10</sup>−<sup>3</sup> mm at P2 = 92.4 Hz, and the amplitude variations of No. 2~4 were all similar to that of No. 1.

**Figure 14.** Amplitude–frequency distribution under 6# and 7# action points excited synchronously. (**a**) Overlapping display of the four sensor signals; (**b**) Separate display of the four sensor signals.

### **5. Effective Vibration Modes and Predominant Frequency of Interface Vibration under Impact Load**

Based on Hilbert Huang transform (HHT), the interface vibration waveforms were decomposed by EEMD [20–24]; combined with energy formula: ! <sup>∞</sup> <sup>−</sup><sup>∞</sup> x2(t)dt, the energy distributions and marginal spectrum of the decomposed waveforms were obtained [25].

Under 2# action point excitation, the decomposition result of the vibration waveform is shown in Figure 15, which was decomposed into five vibration modes (IMF1~IMF5), and the residual res <10−<sup>3</sup> (Figure 15a). The energy proportions of modes IMF1, IMF2 and IMF3 were relatively high (Figure 15b), which were the effective vibration modes. Among them, the energy proportion of IMF2 was the highest, accounting for about 96% of the total energy, which was the main vibration mode. By analyzing the marginal spectrum of the original waveform and effective vibration modes (Figure 15c,d), it could be concluded that the vibration frequency of the original waveform was mainly concentrated in P1~P3 = 39.9~89.8 Hz, and the predominant frequency corresponding to the peak was P2 = 52.9 Hz. The predominant frequencies of effective vibration modes (IMF1, IMF2 and IMF3) were P4 = 236.8 Hz, P5 = 50.2 Hz and P6 = 35.9 Hz respectively.

Under 4# action point excitation, the decomposition result of vibration waveform are shown in Figure 16, which was decomposed into six vibration modes (IMF1~IMF6), and the residual res <10−<sup>3</sup> (Figure 16a). The energy proportions of modes IMF1, IMF2 and IMF3 were relatively high (Figure 16b), which were the effective vibration mode. Among them, the energy proportion of IMF2 was the highest, accounting for about 95% of the total energy, which was the main vibration mode. By analyzing the marginal spectrum of the original waveform and effective vibration modes (Figure 16c,d), it could be concluded that the vibration frequency of the original waveform was mainly concentrated in P1~P3 = 38.9~108.8 Hz, and the predominant frequency corresponding to the peak was P2 = 50.4 Hz. The predominant frequencies of effective vibration modes (IMF1, IMF2 and IMF3) were P4 = 152.4 Hz, P5 = 49.2 Hz and P6 = 47 Hz respectively.

**Figure 15.** The waveform decomposition under 2# action point excitation. (**a**) The EEMD decomposition of the interface vibration waveform; (**b**) Energy distribution of the decomposed waveform; (**c**) Effective vibration modes; (**d**) Marginal spectral amplitude.

**Figure 16.** The waveform decomposition under 4# action point excitation. (**a**) The EEMD decomposition of the interface vibration waveform; (**b**) Energy distribution of the decomposed waveform; (**c**) Effective vibration modes; (**d**) Marginal spectral amplitude.

Under 6# action point excitation, the decomposition result of vibration waveform was shown in Figure 17, which was decomposed into six vibration modes (IMF1~IMF6), and the residual res <10−<sup>3</sup> (Figure 17a). The energy proportions of modes IMF1, IMF2 and IMF3 were relatively high (Figure 17b), which were the effective vibration modes. Among them, the energy proportion of IMF2 was the highest, accounting for about 92% of the total energy, which was the main vibration mode. By analyzing the marginal spectrum of the original waveform and effective vibration modes (Figure 17c,d), it could be concluded that the vibration frequency of the original waveform was mainly concentrated in P1~P3 = 37.8~92.7 Hz, and the predominant frequency corresponding to the peak was P2 = 52.3 Hz. The predominant frequencies of effective vibration modes (IMF1, IMF2 and IMF3) were P4 = 91.3 Hz, P5 = 47.7 Hz and P6 = 47.7 Hz respectively.

**Figure 17.** The waveform decomposition under 6# action point excitation. (**a**) The EEMD decomposition of the interface vibration waveform; (**b**) Energy distribution of the decomposed waveform; (**c**) Effective vibration modes; (**d**) Marginal spectral amplitude.

Under 2# and 4# action points excitation step-by-step, the decomposition result of the vibration waveform is shown in Figure 18, which was decomposed into seven vibration modes (IMF1~IMF7), and the residual res <10−<sup>4</sup> (Figure 18a). The energy proportions of modes IMF1, IMF2 and IMF3 were relatively high (Figure 18b), which were the effective vibration mode. Among them, the energy proportion of IMF2 was the highest, accounting for about 86% of the total energy, which was the main vibration mode. By analyzing the marginal spectrum of the original waveform and effective vibration modes (Figure 18c,d), it could be concluded that the vibration frequency of the original waveform was mainly concentrated in P1~P3 = 37.5~99.9 Hz, and the predominant frequency corresponding to the peak was P2 = 49.4 Hz. The predominant frequencies of effective vibration modes (IMF1, IMF2 and IMF3) were P4 = 82.7 Hz, P5 = 48.6 Hz and P6 = 46.5 Hz, respectively.

**Figure 18.** The waveform decomposition under 2# and 4# action points excited step-by-step. (**a**) The EEMD decomposition of the interface vibration waveform; (**b**) Energy distribution of the decomposed waveform; (**c**) Effective vibration modes; (**d**) Marginal spectral amplitude.

Under 6# and 7# action points synchronous excitation, the decomposition result of vibration waveform was shown in Figure 19, which was decomposed into six vibration modes (IMF1~IMF6), and the residual res <10−<sup>3</sup> (Figure 19a). The energy proportions of modes IMF1, IMF2 and IMF3 were relatively high (Figure 19b), which were the effective vibration modes. Among them, the energy proportion of IMF2 was the highest, accounting for about 85% of the total energy, which was the main vibration mode. By analyzing the marginal spectrum of the original waveform and the effective vibration modes (Figure 19c,d), it could be concluded that the vibration frequency of the original waveform was mainly concentrated in P1~P3 = 24.8~90.2 Hz, and the predominant frequency corresponding to the peak was P2 = 45.7 Hz. The predominant frequencies of effective vibration modes (IMF1, IMF2 and IMF3) were P4 = 201.3 Hz, P5 = 45.6 Hz and P6 = 47.4 Hz, respectively.

It could be seen that IMF1, IMF2 and IMF3 were effective vibration modes under single point excitation and multi-point excitation (synchronous/step-by-step). Among them, the energy of IMF2 accounted for the highest proportion (85–94%), which was the main vibration mode, and its predominant frequencies were mostly concentrated in 45.6~50.2 Hz. It could be concluded that IMF2 played a decisive role in the vibration process, so the predominant frequencies of coal–rock and rock–rock interfaces vibration under impact load were also concentrated in this range, and the vibration in this frequency range had an important effect on the dynamic response, damage and failure of coal and rock mass. Of course, in the actual conditions, the range of the actual predominant frequencies could be obtained by converting according to the size and mechanical properties of coal and rock mass.

**Figure 19.** The waveform decomposition under 6# and 7# action points excited synchronously. (**a**) The EEMD decomposition of the interface vibration waveform; (**b**) Energy distribution of the decomposed waveform; (**c**) Effective vibration modes; (**d**) Marginal spectral amplitude.

#### **6. Conclusions**

Under single and multi–point excitation (synchronous/step-by-step), the dynamic changes of amplitude curves of each interface vibration were obtained from the four sensors (No. 1~4) were similar, and the extreme value showed two stages as follows: first increase and then attenuation, most of which required 2.25 cycles to reach the peak. The dynamic attenuation of amplitude conformed to the law of exponential variation y = y0 + Aexp (x/k).

Based on FFT transform, the time–history curves of interface vibration under single point and multi-point excitation (synchronous/step-by-step) were analyzed, and the predominant frequencies distribution of amplitude were obtained. Among them, the two predominant frequencies were P1 = 48.9~53.7 Hz and P2 = 92.4 Hz.

The time–history curves of interface vibration under single point and multi-point excitation (synchronous/step-by-step) were decomposed by EEMD to obtain a total of component, 5. modes (IMF), of which IMF1, IMF2 and IMF3 contained high energy and were effective vibration modes.

The energy of IMF2 accounted for the highest proportion (85–94%), which was the main vibration mode, and its predominant frequencies were concentrated in 45.6~50.2 Hz, which overlapped with that of the original waveform to a great extent. The vibration of IMF2 played a decisive role in the vibration process and had an important effect on the dynamic response, damage, and failure of coal and rock mass. In actual conditions, the range of the actual predominant frequencies can be obtained by converting according to the size and mechanical properties of coal and rock mass.

In this paper, the characteristics of vibration frequency and vibration amplitude of coal rock vibration response signal were mainly studied. The vibration frequency study of coal rock vibration signals can help the monitoring and early warning of coal and gas protrusion. The study of vibration amplitude can help to determine the danger area of coal rock vibration. The vibration frequency characteristics and amplitude characteristics of the dynamic response signals of coal rocks have rarely been studied by previous authors; there are even fewer studies on the dynamic response characteristics of multi-layer coal rocks. In this paper, we studied the dynamic response characteristics of the five–seam coal rock body, which is closer to the complex reality. However, the study only performs vibration response analysis for similar experiments in this paper. Further studies are needed to verify whether the dynamic response of coal rock in different scenarios also conforms to the vibration law summarized in this study. Further research is needed to investigate the dynamic response of coal rocks under different sizes and mechanical properties of the actual site.

**Author Contributions:** Conceptualization, F.L.; methodology, F.L.; software, G.W. and B.R.; validation, F.L. and G.W.; formal analysis, G.W. and J.T.; investigation, F.L.; resources, F.L.; data curation, F.L. and G.W.; writing—original draft preparation, F.L.; writing—review and editing, F.L., G.W. and G.X.; visualization, F.L., G.W. and Z.C.; supervision, F.L.; project administration, F.L.; funding acquisition, F.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the financial support from National Natural Science Foundation of China grant number [52064046, 51804311]; China Scholarship Council (CSC); and the Fundamental Research Funds for the Central Universities grant number [2020YJSAQ13].

**Data Availability Statement:** The data presented in this paper is freely available from the corresponding author upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**


#### **References**


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