## 4 Simulation Experiment and Mathematical Model Validation
### Critical Value Simulation of Boundary Conditions for Driving Under Pressure Dynamic Model
Analyzing Figure 13, it can be seen that under the condition that the specific pressure of the base of support 1 against the floor is \(P_{12d}\) = 0.2 MPa when the pushing force of the moving cylinder exceeds 908 kN, the friction force between support 1 and the surrounding rock reaches its maximum value, and support 1 begins to move. The friction force between support 2 and the surrounding rock continues to increase with the pushing force of the moving cylinder, and support 2 remains stationary. At this time, support 1 can normally drive under pressure. When the pushing force of the moving cylinder exceeds 1031 kN, the friction force between support 2 and the surrounding rock reaches its maximum value, and support 2 begins to move. At this time, support 1 cannot drive under pressure. Compared with the theoretical values, the maximum error of the maximum friction force between support 1 and support 2 and the surrounding rock obtained by simulation is 1.2%, less than 5%. When support 2 is in the state of driving under pressure, it is known from the value range of the specific pressure of the base of support 2 against the floor, 0.038 MPa \(<P_{22d}<\) 0.17 MPa, that support 2 is in the state of driving under pressure when the specific pressure \(P_{22d}\) of the base of support 2 against the floor is taken in the range of 0.038 MPa to 0.17 MPa; support 2 cannot drive under pressure when the specific pressure \(P_{22d}\) of the base of support 2 against the floor is taken in the range of 0.17 MPa to 0.2 MPa. Therefore, two cases, \(P_{22d}\) = 0.1 MPa and \(P_{22d}\) = 0.19 MPa, are used for the simulation analysis. When support 1 is stably supporting, its base is under the specific pressure P\({}_{128}\) = 0.2 MPa. Assuming \(F_{1}\) = \(F_{2}\) and \(F_{3}\) = \(F_{4}\), it can be calculated from Equations (2) and (5) that each supporting cylinder of support 1 has a working resistance of approximately 376 kN, and the maximum friction force generated by contact with the surrounding rock is approximately 908 kN. When \(P_{22d}\) = 0.1 MPa, the specific pressure of the top beam of support 2 against the roof, P\({}_{21d}\), is approximately 0.082 MPa, satisfying the support conditions. Each supporting cylinder of support 2 has a working resistance of approximately 198 kN, and the maximum friction force generated by contact with the surrounding rock is 492 kN. When \(P_{22d}\) = 0.19 MPa, the specific pressure of the top beam of support 2 against the roof, P\({}_{21d}\), is approximately 0.17 MPa, satisfying the support conditions. Currently, each supporting cylinder of support 1 has a working resistance of approximately 401 kN, and the maximum friction force generated by contact with the surrounding rock is 979 kN. A gradually increasing pushing force is applied synchronously to the four moving cylinders for simulation, and the variation in the displacement of support 1 and support 2, as well as the friction between them and the surrounding rock with the resultant pushing force, are shown in Figures 14 and 15. Analyzing Figure 14, it can be seen that under the condition that the specific pressure of the base of support 2 against the floor is \(P_{12d}\) = 0.1 MPa, when the pulling force exceeds 496 kN, the friction force between support 2 and the surrounding rock reaches its maximum value, and support 2 begins to move. The friction force between support 1 and the surrounding rock continues to increase with the pulling force of the moving cylinder, and support 1 remains stationary. At this time, support 2 can normally drive under pressure. When the pulling force exceeds 908 kN, the friction force between support 1 and the surrounding rock reaches its maximum value, and support 1 begins to move. At this time, support 2 cannot drive under pressure.