## 4 Case Analysis
### Case 2  
In case 1, the armature velocity is low. In order to verify the applicability of the extrapolation prediction method under high-speed armature movement, the 40 mm \(\times\) 50 mm medium-caliber railgun developed by the Agency for Defense Development (ADD) is taken as the research object [27]. Using the launch data of test number #26, where the armature mass is 300 g and it flies out after accelerating in the bore for 4.0 ms. The waveform diagram of the excitation current is shown in Figure 15.  
Figure 14: Comparison of armature movement velocity.  
Figure 13: Integrated armature electromagnetic force under standard armature conductivity in case 1.  
Establishing the finite element simulation model of the electromagnetic railgun. The numerical simulation results of the armature electromagnetic force under different armature conductivities are obtained when 2.50 \(\times\) 10\({}^{7}\) S/m is used as the standard armature conductivity in this case model, as shown in Figure 16. Under the standard conductivity, the numerical solution of the armature electromagnetic force appears as \"pseudo-oscillation\" after 1.41 ms, and the lower the armature conductivity, the later the \"pseudo-oscillation\" occurs, and the stable numerical simulation of the whole launch process can be realized under 40% standard armature conductivity.  
Through the above numerical simulation, 9500 groups of sample data under different armature conductivities of the stable stage are obtained. These sample data are normalized and divided into a training set and a test set. The training set contains the 0-1.13 ms sample data under the standard armature conductivity and all the sample data under low conductivity; the test set is the 1.13-1.41 ms sample data under the standard armature conductivity. Using the training set to train the DBN-DNN model, and then using the test  
Figure 16: Simulation calculation value of armature electromagnetic force under different armature conductivities in case 2.  
Figure 15: Waveform diagram of excitation current in case 2.  
set to test the DBN-DNN model after training. The average MAPE of five training tests is 0.56%.  
The DBN-DNN model, which is closest to the average value of MAPE in five training tests, is used to predict the armature electromagnetic force of the pseudo-oscillation stage (1.41-4.0 ms) under the standard conductivity, and the extrapolated prediction value of the armature electromagnetic force is obtained. Then, superimposed with the calculated value of the stable stage under the standard conductivity (0-1.41 ms) in Figure 16, the integrated armature electromagnetic force of the whole launch process (0-4.0 ms) is obtained, as shown in Figure 17.  
According to the armature kinetic equation and the integrated armature electromagnetic force under the standard armature conductivity obtained in Figure 17, the armature exit velocity is calculated as 2029.2 m/s. It is 1.03% smaller than the experimental measurement value 2050.3 m/s, which can meet the needs of practical engineering calculation and tests the performance of the armature electromagnetic force extrapolation prediction of the DBN-DNN model.