## 3 Numerical Simulations  
### Dynamic Coupling Model of Roof-Top Coal-Hydraulic Support  
The findings of the damage and movement status of the surrounding rock of the actual working face are difficult to obtain by field testing. Consequently, the numerical simulation technology realizes the combination of roof-top coal-hydraulic support and obtains the damage and movement status of the roof and the bearing characteristics of hydraulic support in different roof instability fields. The dynamic coupling model of roof-top coal-hydraulic support is established, as shown in Figure 7, which comprises the upper immediate roof, lower immediate roof, top coal, and hydraulic support. According to Table 1, the immediate roof is composed of sandy mudstone with a thickness of 12 m and fine sandstone with a thickness of 8 m. The thickness of top coal and the height of mechanical mining are both 5 m.  
Figure 6: The changing trend of the working resistance of hydraulic support under the deceleration state of the rotary instability at the coal wall. To shorten the solution time and guarantee computational precision, the hydraulic support is simplified, ignoring minor supporting components such as the side guard plate and guard plate. Meanwhile, the coal caving process is not taken into account by the numerical simulation, and the tail beam has been taken out of the model because it is not the main bearing component [47]. The spring damping system is used to replace the column [48]. The model was meshed using HyperMesh. Because the accuracy of mesh division affects the simulation results, surrounding rock and the hydraulic support are separated for mesh processing. The mesh size of the surrounding rock is controlled at 200 mm and divided into the hexahedral mesh. The hydraulic support is divided into the tetrahedral mesh, and the mesh size is controlled at 30 mm. The simulation parameters of the model are set in LS-PREPOST. To simulate the influence of roof instability on the working face, the model only applies full constraints at the bottom of the model in the vertical direction. From the above analysis, the fracture line of the upper immediate roof is located in the extension line of the coal wall and the roof cutting. The grid on the fracture line is removed, and then the degree of freedom of the rotation center is constrained to simulate the different instability forms of the upper immediate roof. To simplify the roof strata and not ignore the role of the main roof, the weight of the main roof is applied to the upper immediate roof as an external load. The rigid area is defined at the shaft hole, and the hinge joints of each component are constrained to form a rotating pair. The surface-to-surface contact is set between the contact types between the rock strata and the top coal and the canopy. Table 2 shows the mechanical properties of the overlying strata based on the working face's overlying strata structure. The Holmquist-Johnson-Cook model, often known as the HJC model, is frequently used to examine the mechanical behavior of rock [49; 50; 51], and adds the keyword MAT_ADD_EROSION to simulate the failure of the material [52]. The failure criterion of rock is based on reference [53]. Based on the HJC model theory, the HJC model parameters of the overlying strata are shown in Table 3. The material properties of hydraulic support are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Overlying Strata** & \begin{tabular}{c} **Density** \\ **(kg/m3)** \\ \end{tabular} & \begin{tabular}{c} **Bulk Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Shear Modulus** \\ **(GPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Tensile Strength** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Compressive Strength** \\ **(MPa)** \\ \end{tabular} \\ \hline Fine sandstone & 2530 & 18.9 & 14.3 & 4.8 & 122.3 \\ Sandy mudstone & 2460 & 3.59 & 2.92 & 3.9 & 31 \\ Coal & 1440 & 1.82 & 0.89 & 1.2 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of the overlying strata.