## 4 Simulation Experiment and Mathematical Model Validation
### Critical Value Simulation of Boundary Conditions for Driving Under Pressure Dynamic Model  
Import the model constructed in Section 2.3 of the TSR into the Adams View 2018 software. The driving under pressure process of support 1 and support 2 is simulated and analyzed using the Adams software to obtain the pushing force of the moving cylinder and the critical values of the movement of support 1 and support 2 to verify the accuracy of the boundary conditions of the driving under pressure dynamic model. The relevant parameters of the simulation model are consistent with those described in Section 4.1. The contact stiffness between the surrounding rock and the TSR is \(1\times 108\) N/m, the damping coefficient is \(1\times 10^{5}\) Ns/m, and the penetration depth is 0.1 mm. When support 1 is in the state of driving under pressure, it is known from the value range of the specific pressure of the base of support 1 against the floor, 0.056 MPa \(<P_{12d}<\) 0.22 MPa, that the specific pressure \(P_{12d}\) of the base of support 1 against the floor falls within the range of the maximum allowable specific pressure of 0.2 MPa for the contact between the TSR and the surrounding rock, enabling driving under pressure action. Therefore, two cases, \(P_{12d}=\) 0.1 MPa and \(P_{12d}=\) 0.2 MPa, are used for the simulation analysis. When support 2 is stably supporting, its base is under the specific pressure \(P_{22s}=\) 0.2 MPa. Assuming \(F_{1}=F_{2}\) and \(F_{3}=F_{4}\), it can be calculated from Equations (2) and (5) that the working resistance of each supporting cylinder of support 2 is approximately 423 kN, and the maximum friction force generated by contact with the surrounding rock is approximately 1033 kN. When \(P_{12d}=\) 0.1 MPa, the specific pressure  
Figure 11: Finite element simulation results of support 2 under a load of 0.18 MPa applied to the top beam: (**a**) stress cloud of support 1; (**b**) stress cloud of the floor under support 1.  
of the top beam of support 1 against the roof, \(P_{11d}\), is approximately 0.045 MPa, satisfying the support conditions. Currently, each supporting cylinder of support 1 has a working resistance of approximately 175.5 kN, and the maximum friction force generated by contact with the surrounding rock is 427 kN. When \(P_{12d}\) = 0.2 MPa, the specific pressure of the top beam of support 1 against the roof, \(P_{11d}\), is approximately 0.1 MPa, satisfying the support conditions. Currently, 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 907 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 12 and 13. Analyzing Figure 12, it can be seen that under the condition that the specific pressure of the base of support 1 against the floor is \(P_{12d}\) = 0.1 MPa, when the pushing force of the moving cylinder exceeds 426 kN, the friction force between support 1 and the surrounding  
Figure 12: Critical values of motion for support 1 and support 2 at \(P_{12d}\) = 0.1 MPa. Figure 13: Critical values of motion for support 1 and support 2 at \(P_{12d}\) = 0.2 MPa. 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 1021 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.