## 4 Discussion  
### Energy Calculation Model for Hard Roof Considering Mining Speed and Analysis of Influencing Factors  
In calculating the initial collapse step distance and cyclic collapse step distance of the roof, two commonly used models are the beam fracture model and the plate fracture model [26]. Generally, both the initial and cyclic collapse step distances of the roof are relatively small compared to the length of the working face. Therefore, a rock beam of unit width (1 m) can be taken from the middle of the working face, and the plane strain theory can be applied to analyze the stress on the roof. (1) During the Initial Fracture Period  
Based on the relevant theories of material mechanics, a beam model for the roof during the initial fracture period is established (as shown in Figure 9). The normal stress (\(\sigma\)) at any point A within the beam is:  
\[\sigma=My/J_{z} \tag{2}\]  
where \(M\) is the bending moment at the cross-section where the point is located, \(y\) is the distance of the point from the neutral axis of the cross-section, and \(J_{z}\) is the moment of inertia about the neutral axis. Figure 8: Proportion of acoustic emission events in each stage under different loading rates. Since the width of the beam is considered to be of unit width, the moment of inertia of the beam section occurs at the point where \(J_{x}=h^{3}/12\) (where \(h\) is the thickness of a single layer of the rock stratum). Therefore, the normal stress \(\sigma\) at any point A can be expressed as \(\sigma=12 My/h^{3}\). Assuming that solid coal exists on both sides before the initial fracture of the roof and that the rock beam is embedded between the overlying rock stratum and the coal seam, it is therefore treated as a fixed support beam for calculation and analysis. The maximum bending moment forms at the ends of the beam, \(M_{\text{max}}=-qd^{2}/12\). Accordingly, the maximum tensile stress at that location is:  
\[\sigma_{\text{max}}=\frac{12\times\frac{1}{12}qd^{2}\times\frac{h}{2}}{h^{3}}= \frac{qd^{2}}{2h^{2}} \tag{3}\]  
We have taken the mining speed and mining duration in the model to be \(V_{\text{m}}\) and \(t_{\text{m}}\), respectively, while in actual production scenarios the face mining speed, mining duration and face length are \(V_{\text{p}}\), \(t_{\text{p}}\), and \(L\), respectively. Assuming that the shearer maintains a uniform speed while cutting back and forth before the initial fracture, and the working face advances at a constant speed, then the span of the rock beam before the initial fracture in the model can be represented as \(d=V_{\text{m}}t_{\text{m}}\). In actual production scenarios, \(d=V_{\text{p}}t_{\text{p}}\), leading to \(V_{\text{m}}t_{\text{m}}=V_{\text{p}}t_{\text{p}}\). Therefore, the mining speed in the model is:  
\[V_{\text{m}}=\frac{t_{\text{p}}}{t_{\text{m}}}V_{\text{p}} \tag{4}\]  
Let the coefficient \(\beta=t_{\text{p}}/t_{\text{m}}\). Since the width of the rock beam in the model is a unit width of 1 m, and the working face length is \(L\), the time spent cutting coal within the unit width of the rock beam accounts for \(1/L\) of the entire working face range. Therefore, \(\beta=L\), and based on actual conditions, \(\beta\) is taken to be 194.