## 2 Finite Element Analysis of the Electromagnetic Track Launcher
### Finite Element Calculation Results  
#### 2.2.1 Armature Velocity  
The excitation current obtained by fitting the experimental current is used in the finite element calculation in this paper, which is basically consistent with the experimental current, and more accurate calculated results can be obtained. The expression of excitation current is as follows:  
\[I=\left\{ \begin{array}{l}I_{0}\sin(2.5\pi t)\ 0\leq t\leq 0.2\ \text{ms}\\ I_{0}\ e^{-0.61(t-0.2)}\ \ \ 0.2<t\leq 0.6\ \text{ms}\end{array} \right. \tag{9}\]  
where, \(I_{0}\) is the excitation current peak at 318 kA. The curves of excitation current and experimental one are shown in Figure 3:  
In the actual launching process, the armature is subjected to three forces, including electromagnetic thrust, friction force and air resistance. According to the balance of forces, the resultant force of the armature should be:  
\[F_{a}=F_{e}-f_{1}v_{\text{af}}^{2}-f_{2}m_{0}g \tag{10}\]  
where, \(F_{a}\) is the resultant force of the armature, \(F_{e}\) the electromagnetic thrust of the armature, \(f_{1}\) the wind resistance coefficient \(7.9\ \times\ 10^{-5}\ \text{kg/m}\), calculated by multiplying the coefficient of air resistance, air density and the windward area of the armature, \(v_{\text{af}}\) the armature velocity considering the influence of air resistance and friction force, \(f_{2}\) the sliding friction coefficient which is generally 0.5, \(m_{0}\) is the armature mass of 10 g, consistent with experimental armature mass. \(G\) is the acceleration due to gravity 9.8 m/s\({}^{2}\). Figure 3: Curves of excitation current and experimental one. \(F_{\text{a}}\) can also be expressed by deriving \(v_{\text{af}}\). \[F_{\text{a}}=m_{0}\frac{\text{d}v_{\text{af}}}{\text{d}t} \tag{11}\]  
In summary, considering the influence of air resistance and the friction force, the driving equation of the armature is written as follows:  
\[m_{0}\frac{\text{d}v_{\text{af}}}{\text{d}t}=F_{e}-fv_{\text{af}}^{2}-\mu_{0}m_ {0}g \tag{12}\]  
By solving Equation (12), \(\mathbf{v}_{\text{af}}\) can be obtained as follows:  
\[v_{\text{af}}=\sqrt{\frac{F_{e}-\mu_{0}m_{0}g}{f}}\cdot\left(\frac{e^{2t\frac{ \sqrt{(F_{e}-\mu_{0}m_{0}g)}t}{\mu_{0}}}-1}{e^{2t\frac{\sqrt{(F_{e}-\mu_{0}m_{0 }g)}t}{\mu_{0}}}+1}\right) \tag{13}\]  
When the air resistance and friction force are far less than \(F_{\text{e}}\), which means that \(\frac{fv^{2}}{F_{e}}\to 0\), \(\frac{\mu_{0}m_{0}g}{F_{e}}\to 0\), Equation (13) is changed as follows:  
\[v_{\text{af}}=\frac{F_{e}}{m_{0}}t \tag{14}\]  
According to the armature electromagnetic thrust calculated equation. \[F_{\text{e}}=\frac{1}{2}LI^{2} \tag{15}\]  
where, \(L\) is the inductance gradient, typically between 0.8 uH/m and 1.2 uH/m. It can be seen that when \(I\) reaches its peak value (318 kA), \(F_{\text{e}}\) can reach up to 5 \(\times\) 10\({}^{4}\) N.  
Because the air resistance and friction are very small compared with \(F_{\text{e}}\), the influence can be ignored. The LS-dyna finite element software is used for simulation to obtain the armature velocity, as shown in Figure 4:  
In order to provide data for the improved electromagnetic-thermal coupling model, multiple sets of finite element simulations are set up with the fixed track conductivity \(\sigma_{\text{g}}\) and the armature conductivity \(\sigma_{\text{d}}\), fixed armature conductivities \(\sigma_{\text{d}}\) and track conductivities \(\sigma_{\text{g}}\), respectively. The armature velocity calculated results are obtained and shown in Figure 5.  
\[F_{a}=F_{e}-f_{1}v_{\text{af}}^{2}-f_{2}m_{0}g \tag{10}\]0 Finite element calculated value of armature velocity. It can be seen that, when the track conductivity \(\sigma_{\rm g}\) is fixed, the armature velocity increases with the armature conductivity \(\sigma_{\rm d}\). This is due to the fact that the greater armature conductivity \(\sigma_{\rm d}\) causes a more uniform the magnetic field distribution, and a larger armature electromagnetic thrust and velocity.