## 2 Numerical Simulation of Armature Electromagnetic Force
### Dynamic Characteristics and Numerical Solution Stability of Armature Electromagnetic Force  
The armature electromagnetic force and movement velocity obtained by numerical simulation are shown in Figure 3. Before the armature velocity reaches 500 m/s, the stable calculation of the armature electromagnetic force and velocity can be realized; when the armature velocity increases to 500 m/s, the calculated values of the armature electromagnetic force and velocity begin to become unstable, that is, the \"pseudo-oscillation\" problem of the numerical solution occurs. The numerical simulation results of the electromagnetic field on the rail section at 1.2 ms are shown in Figure 4. The rail section at 10 mm from the tail of the armature is selected as the observation angle. Under the joint action of the skin effect, proximity effect, and velocity skin effect, the magnetic flux density tends to be concentrated on the inner surface of the rail with a maximum value of 8.61 T, and the current density tends to be concentrated at two sharp corners of the inner surface of the rail, and the maximum current density is 8.84 \(\times\) 10\({}^{8}\) A/m\({}^{2}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Values** \\ \hline Armature mass/g & \(m_{a}\) & 125.00 \\ Friction coefficient & \(\mu\) & 0.11 \\ Mechanical preloading pressure/N & \(F_{N,p}\) & 5600 \\ Specific heat ratio of the air & \(\gamma\) & 1.4 \\ Initial air density/(kg/m\({}^{3}\)) & \(\rho_{0}\) & 1.29 \\ Armature cross-sectional area/m\({}^{2}\) & \(S\) & 0.00084 \\ Perimeter of armature section/m & \(L\) & 0.116 \\ Viscous friction coefficient & \(C_{f}\) & 0.003 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of armature kinetic equation. Figure 2: Waveform diagram of trapezoidal excitation current. The numerical simulation results of the electromagnetic field on the armature horizontal cross-section at 1.2 ms are shown in Figure 5. The horizontal cross-section in the middle of the armature is selected as the observation angle. The magnetic flux density tends to be concentrated at the throat position inside the armature with a maximum value of 16.5 T;  
Figure 4: The numerical simulation results of the electromagnetic field on rail section at 1.2 ms. (**a**) Observation angle. (**b**) Magnetic flux density on rail section. (**c**) Current density on rail section. Figure 3: Calculated values of armature electromagnetic force and armature movement velocity. the current density tends to be concentrated in the inner throat and outer fillet position of the armature, and the maximum current density occurs at the armature outer fillet position of 4.35 \(\times\) 10\({}^{9}\) A/m\({}^{2}\). The electromagnetic field governing Equation (1) of the 3D electromagnetic railgun is the convection-diffusion equation, and the velocity is reflected in the convection term of the equation. \(P_{e}\) is the dimensionless number that reflects the stability of the numerical solution of this equation. Additionally, the \(P_{e}\) is calculated as  
Figure 5: The numerical simulation results of the electromagnetic field on armature horizontal cross-section at 1.2 ms. (**a**) Observation angle. (**b**) Magnetic flux density on armature horizontal cross-section. (**c**) Current density distribution on armature horizontal cross-section. \[P_{e}=\frac{convection\ rate}{diffusion\ rate}=h_{c}\times\frac{\sigma v}{\big{/}\mu}=h_{c} \sigma v\mu \tag{4}\]  
In the formula, \(h_{c}\) is the cell grid size. From Equation (3), we can know that \(P_{e}\) is proportional to the cell grid size, velocity, conductivity, and relative permeability. When \(v=500\) m/s, for \(h_{c}=7\times 10^{-5}\) in the armature region, \(P_{e}=0.97\) is calculated based on the material properties of the armature in Table 1. With the increase in velocity, the effect of the convection term is enhanced, and the \(P_{e}\) number increases gradually.