## 3 Model Test
### Discussion  
Figure 7 displays the experimentally determined connection among the buried depth of perilous rock, the vibration frequency, and the mass of perilous rock.  
1. Correlation between the buried depth of the perilous rock and the natural vibration frequency.  
Figure 7 shows that the natural vibration frequency of the perilous rock grows non-linearly as the buried depth increases.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Test Factor** & **Average** & **Unit** \\ \hline natural density (\(\gamma\)) & 1.92 & g/cm\({}^{3}\) \\ internal friction angle (\(\varphi\)) & 20.0 & \({}^{\circ}\) \\ coefficient of subgrade & 6.5 \(\times\) 104 & kN/m\({}^{3}\) \\ reaction (k) & 7.0 & kN/m\({}^{2}\) \\ cohesion (C) & 7.0 & kN/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Foundation soil parameters.  
Figure 6: Test scheme.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **Buried Depth/cm** & **Measured Frequency/Hz** & **Buried Depth Calculation/cm** & **Error/\%** \\ \hline A & 8 & 13.6 & 7.49 & 6.37\% \\ A & 16 & 23.2 & 16.72 & 4.51\% \\ A & 24 & 31.8 & 24.61 & 2.52\% \\ A & 32 & 39.2 & 31.18 & 2.56\% \\ B & 6 & 16.5 & 6.34  
By transforming Equation (17), the relationship between the buried depth of the perilous rock and the natural vibration frequency could be obtained as:  
\[\mathrm{h}=\frac{4\pi^{2}\mathrm{M}\mathrm{f}_{\mathrm{d}}^{2}(\mathrm{a}/ \mathrm{H_{s}}^{2}+1)}{\mathrm{kb}(1-\xi^{2})(\mathrm{a}/\mathrm{H_{s}})^{2}} \tag{18}\]  
Equation (18) demonstrates that as the buried depth of the perilous rock increases, the frequency of its natural vibrations grows steadily in a quadratic fashion. Additionally, the theoretical model has an average inaccuracy of 3.82% when comparing the experimentally observed buried depth of the perilous rock model to the predicted buried depth, showing that there is a strong theoretical relationship between the frequency of the natural vibrations and their buried depth, as proposed in this work. Unlike the manual method to measure the buried depth of the perilous rock, this paper, by measuring its natural vibration frequency, provides the theoretical basis and new technical support for its early identification, stability evaluation, and monitoring and early warning.  
Relationship between the natural vibration frequency and the perilous rock mass.  
Figure 7 shows that, for a given natural vibration frequency, a lighter model has a shallower buried depth than a heavier one. When the models have the same natural vibration frequency, the buried depths of the models fall in the following order: \(\mathrm{A}>\mathrm{B}>\mathrm{C}\). If two rocks of equal mass are buried at the same depth, the vibration frequency of a perilous rock with a larger mass is lower than that with a smaller mass. The order of the natural vibration frequency of the models under the same buried depth is \(\mathrm{C}>\mathrm{B}>\mathrm{A}\).  
Equation (17) leads to the following conclusion on the link between the perilous rock's bulk and its inherent vibration frequency:  
\[\mathrm{M}=\frac{\mathrm{kb}\mathrm{h}(1-\xi^{2})(\mathrm{a}/\mathrm{H_{s}}) ^{2}}{4\pi^{2}\mathrm{f}_{\mathrm{d}}^{2}(\mathrm{a}/\mathrm{H_{s}}^{2}+1)} \tag{19}\]  
The natural vibration frequency reduces quadratically with the enhancing mass of the perilous rock, as shown by Formula (19). Data gathered from experiments involving the magnitude of perilous rocks and the frequency of natural vibrations confirm the accuracy of this result.  
Figure 7: Relationship between natural vibration frequency and buried depth.