## 4 Space-Time Dynamic Analysis of Forced Vibration of Thin Plate
### Analysis Results and Comparison  
For the thin plate in Figure 10, let its length \(a\) and width \(b\) be 0.8 m, thickness \(h\) = 0.01 m, elastic modulus \(E\) = 1.5 \(\times\) 10\({}^{11}\) N/m\({}^{2}\), Poisson's ratio \(v\) = 0.2, density \(\rho\) = 8000 kg/m\({}^{3}\), the value of initial velocity and initial displacement be 0, and the time-dependent uniform load \(q\)(\(x\),\(y\),\(t\)) = 10,000\(\sin\)\((30t)\) N/m\({}^{2}\). Firstly, all displacements in a time step are calculated by using orthogonal grids of 5 \(\times\) 5 \(\times\) 9 and 7 \(\times\) 7 \(\times\) 9, respectively, where the CGL orthogonal grids are 5 \(\times\) 5 and 7 \(\times\) 7 on the \(x\)-\(y\) plane and 9 uniformly distributed spatial nodes in the t direction. Let the time step length be 0.16 s, the displacement obtained by DCM at each time step is compared with the analytical solution and the relevant reference solution. The comparison results are shown in Table 4. Figure 12 shows the global displacement of the 7 \(\times\) 7 \(\times\) 9 orthogonal grid corresponding to the eight moments in Table 4 as a reference. Figure 11: Three-dimensional orthogonal mesh. Figure 12: _Cont._According to the data in Table 4, the error between the finite element solution and the analytical solution in the literature is roughly maintained at 1-3%. The grid of 21 \(\times\) 21 is used in space, and the stepwise method is used in time domain. The minimum step of segmentation is 0.01 s in FEM. In this paper, the error between the results obtained by DCM with the orthogonal grid of 5 \(\times\) 5 \(\times\) 9 and the analytical solution is less than one percent and is uniformly distributed over time. When the orthogonal grid of 7 \(\times\) 7 \(\times\) 9 is used, the error between the displacement solution and the analytical solution is kept within one thousandth. Next, the total time T of the above load action is set to be 3.2 s, and the orthogonal grid of 5 \(\times\) 5 \(\times\) 9 is used for calculation. The length of each time step is set to 0.16 s, so a total of 20 time steps are required. As a reference, the finite element software Abaqus is used to calculate. Three-dimensional shell elements are selected, the number of elements is 20 \(\times\) 20, and the  
It can be seen from Figures 13 and 14 that the calculation results of DC method with a 5 \(\times\) 5 \(\times\) 9 orthogonal grid based on 0.16 s time step are consistent with the calculation results of DCM with a 21 \(\times\) 21 grid based on 0.01 s and 0.05 s time steps, indicating the correctness of the method used in this paper. In addition, it can be found from Figure 15 that when the time step of 0.1 s is adopted by FEM, the displacement no longer keeps sinusoidal variation, but is in a state of amplitude increasing first and then decreasing, which indicates that the calculation results of FEM with this time step are already in an unstable state, and the correct results cannot be obtained. Table 5 shows the computation time of FEM and DCM in this case. When the time step of FEM is 0.1 s, the computation time is 17 s, but the correct result is not obtained. When the time step decreases, the computation time increases. The calculation time of DCM under two orthogonal grids is less than 1 s, and extremely accurate results can be obtained. Based on the above comparison with FEM, the following conclusions can be obtained: when dealing with the dynamic response problem on the plate with long duration, DCM can obtain more accurate results with less computational time. Figure 14: DC solution and finite element solution of time-history displacement of the center point (\(T\) = 3.2 s, FEM \(\Delta T\) = 0.05 s). Figure 15: DC solution and finite element solution of time-history displacement of the center point (\(T\) = 3.2 s, FEM \(\Delta T\) = 0.1 s).