The railgun used in this study consists of convex rails made of copper C18150 and an armature made of 7075 aluminum alloy material, the shells material made of steel and the insulator is made of a composite material of glass fibers and epoxy resin. The geometry structure and mesh model of the railgun are shown in Fig. 1. The armature and rails material parameters are shown in Table 1.

The railgun used in this study consists of convex rails made of copper C18150 and an armature made of 7075 aluminum alloy. The shell material is steel, and the insulator is made of a composite material of glass fibers and epoxy resin. The geometry structure and mesh model of the railgun are shown in Fig. 1. The armature and rail material parameters are shown in Table 1.

The simulation is meshed using eight-node hexahedral cells with a total of 111,660 elements and 128,086 nodes and the maximum size of mesh is 0.5 mm. The armature and rails surface meshes are refined, as shown in Fig. 1.

In the model, the armature length is 40 mm (of which the armature arm length is 23.5 mm), the maximum interference is 0.18 mm, the initial position of the armature is 50 mm from the end of the rails, the rails spacing is 13.72 mm at the narrowest point, and the armature motion distance is 100 mm in the simulation. Specific structural parameters are also labeled in Fig. 1.

The internal ballistic process of the railgun is affected by the coupled electric-magnetic-thermal-structural fields. In order to study the changes in the physical properties of the railgun, we analyze the coupling relationship between the multi-fields, as shown in Fig. 2.

For the electromagnetic field, since the frequency range of the pulsed power supply is in kHz (the frequency range of the railgun power supply can be obtained by joint time-frequency analysis), we neglect the displacement current and solve Maxwell’s equations under the assumption of a magnetically quasi-static field. Using the A-φ-A format of Maxwell’s equations and considering the Lorentz effect on current density, that is J=σ(E+v×B), we obtain the electromagnetic field governing equations: (1)∇2A−μσ(∂A∂t+∇ϕ)+μσv×(∇×A)=−μJs, (2)∇⋅(−σ∂A∂t−σ∇ϕ+σv×(∇×A))=0,where A is the magnetic vector potential, φ is the electric scalar potential, μ and σ are the magnetic permeability and electric conductivity of the material, respectively, J_s is the source current density, and v is the velocity of conductor motion.

The numerical calculation is less stable due to the presence of v. Therefore, Lagrange coordinates are used to describe the moving conductor, in which the velocity term v will be transformed from explicit to implicit. In turn, several key physical quantities that relate to the performance of the railgun can be derived: (3)J=−1μ0∇2A, (4)Bx=∂A∂y,By=−∂A∂x.

For the general operating environment of an electromagnetic railgun, it can be assumed that air does not conduct electricity. Therefore, the electromagnetic governing equations in the air region degenerate into Laplace’s equations: (5)∇2A=0.

Considering the force characteristics of the railguns and the fact that the materials of the railguns are all non-ferromagnetic, we use the Maxwell tensor method to solve the Lorentz force equation: (6)Fem=J×B=−σ(∂A∂t+∇ϕ)×(∇×A).

For the thermal field of the railgun, we use Fourier’s law to describe the heat transfer phenomena in the railgun due to the short operating time of the railgun:

(7)ρc(∂T∂t+v∇T)−∇⋅(κ∇T)=Q,where κ is the thermal conductivity, ρ is the density of the materials, c is the specific heat capacity, T is the temperature, and Q is the density of the heat source. Similarly, in the Lagrange coordinate system, the term v in Eq. (7) will be changed to implicit. The heat sources for railguns include Joule heat Qj=J2/σ and frictional heat Qf=fv, where f is friction. In the simulation, the static and dynamic friction coefficients are taken as 0.3 and 0.15, respectively.

The Joule heat consists of two parts, the first is heat generated by the metal resistance, the other is the Joule heat generated by the contact resistance. The heat generated by the contact resistance acts partially on the rails, and this ratio is expressed by the coefficient D_r: (8)Dr=κrρrcrκrρrcr+κaρaca,

where the subscripts a and r correspond to the armature and rails, respectively. The ratio of contact resistance heat obtained by the armature is D_a = 1 − D_r.

The contact resistance was calculated according to the Contact Layer model (CLM model) [15]: (9)ρclc=ρac(HsoftP)m,

where ρ_c is the contact resistivity, l_c is the thickness of the nominal contact layer, ρ_a is the arithmetic mean of the resistivities of the two contact materials, H_soft is the hardness of the armature, and P is the contact pressure. m and c are empirical contact constants measured experimentally, the values of m and c are related to material properties. For railguns, m = 0.63 and c = 9.45 × 10⁻⁴.

For simulation, the current shown in Fig. 3 is loaded into the breech end of the rails as a source term (in the form of average current density). Fixed constraint is applied to the outer surface of the rails and the armature is in contact with the rails. The ambient temperature is set to 273 K.

Fig. 3 shows the current and muzzle voltage waveforms measured after the launch test used in this paper. It can be seen that the muzzle voltage rises sharply at 0.514 ms, indicating that an increase in resistance between the armature and rails occurs at this time, suggesting that an electrical breakdown (also known as a transition) occurs at this moment. It is important to note that the muzzle voltage is different from the electrode voltage: a sudden increase in muzzle voltage indicates a change in the contact state, but does not imply an increase in the electrode voltage. In addition, at 0.968 ms, a greater rate of decrease in pulse current can be seen, indicating that the armature is out of the chamber at this point. The pulsed current waveforms in Fig. 3 are also the currents used for the simulation. According to joint time-frequency analysis, at the rise edge of the current, the frequency of the current decreases rapidly from 3900 to 3200 Hz, and at the drop edge of the current, the frequency of the current is maintained at about 3200 Hz.

Reliable contact between the armature and rails is achieved by an interference fit. Due to the stiffness and shape of the armature, the trail end of the armature arm cannot be fitted to the rails, which is known as contact separation (called initial separation). Fig. 4 illustrates the distribution of contact pressure on the armature surface and armature deformation after assembly. It is clear that the contact pressure is concentrated in the throat of the armature and the deformation of the armature arm occurs mainly at the tail end of the armature arm.

Fig. 4b,c shows the deformation on the straight line L₁ and L₂ in Fig. 4a and it can be seen that the contact separation occurs in the latter part of the armature arm (after 21 mm), and the maximum separation reaches 0.05 mm, the deformation of L₂ is larger than that of L₁, and the phenomenon of contact separation occurs earlier in L₂ than in L₁.

Once the armature is loaded (preloaded), it is powered through the pulse current shown in Fig. 3, and the armature advances under electromagnetic force. Fig. 5 shows the simulation curves of the armature kinematic characteristics and the armature. The time and velocity of the armature when it is out of the chamber, as measured by the light screen, are also labeled in Fig. 5 with blue color blocks.

The total simulation time is 1 ms and the current peaks at 167.23 kA at 0.23 ms. From Fig. 5a, it can be seen that the armature acceleration increases rapidly at the moment of 0.92 ms, at which time the armature is out of the chamber. The armature velocity (out of the chamber) obtained by simulation is 201.72 m/s and the armature velocity (out of the chamber) obtained by test is 194.5 m/s.

Fig. 6 illustrates the magnetic flux density and current flow lines near the armature/rail at 0.8 ms to explain how the negative electromagnetic leads to electromagnetic contact separation at the drop edge of the current. Notice that the magnetic field strength is greatest behind the armature (in the circuit I). As the current decreases, the magnetic flux in the circuit I decreases and, according to Lezn Law, induced current will be generated in circuit I in the opposite direction of the pulsed current. The current flow on the armature is plotted in Fig. 6b, where the flow lines of the induced and pulsed currents can be seen. It can be observed that the induced current interacting with the magnetic field will produce electromagnetic force F_neg in the opposite direction of the contact force F_pos. F_neg will change the force state of the armature, inducing the armature arm to separate from the rails. That is, based on the initial separation, the electromagnetic force F_pos first causes the armature arm to be fully affixed to the rails (full face contact). When the current reaches the drop edge, F_neg will separate the armature from the rails (electromagnetic separation) as the electromagnetic force decreases and the negative electromagnetic force appears.

Further, we observe the distribution of x-direction electromagnetic forces (the coordinate system is shown in Fig. 1) on the armature as shown in Fig. 7. We showed only half of the armature to make observation more easily. It can be seen that at the rising stage of the current, there is almost no opposite electromagnetic force on the armature arm; at the drop stage of the current, the opposite electromagnetic force is mainly concentrated on the two sides of the tail of the armature arm. In addition, the armature head under the influence of eddy current also has a lower density of opposite electromagnetic force.

In addition, we counted the change of contact force with time, as shown in Fig. 8, where F_c represents the contact force, F_t is the electromagnetic force in the z-direction whose effect is to thrust the armature forward, F_np and F_nn are the electromagnetic force in x-positive direction and in x-negative direction on half of the armature (the effect of F_np is to fit the armature arm to the rails and the effect of F_nn is to separate the armature arm from the rails). Notice that the contact force cannot be considered equal to F_np since we use elastic materials.

It can be seen that the electromagnetic force follows essentially the same trend as the current. Although F_nn is much smaller than F_np, this force still causes contact separation due to the more concentrated distribution of F_nn. Corresponding to the change of armature acceleration, F_t decreases rapidly around 0.92 ms. While F_nn, F_np, and F_c decrease around 0.82 ms, which is due to the fact that at this time the armature starts to move out of the chamber, and the contact area between the armature and the rails decreases rapidly.

Fig. 9 shows the dynamic process of contact separation during armature movement in the chamber and Fig. 10 shows the statistics of parameters about contact separation, where SepP_max and SepP_min are the maximum and minimum values of the distance from the armature tail for the position at which the contact separation occurs, PreP_max is the distance from the armature tail for the position at which the maximum value of the contact pressure occurs, Dis_max and Dis_aver indicate the maximum and the average values of the separation, respectively.

The area labeled green (from 0.08 to 0.52 ms) in Fig. 10 is the area where no contact separation occurs. It is noted that under the action of the interference fit, the armature appears to be in contact separation with the rails when the armature loading is completed; after energization, the armature is quickly and completely affixed to the rails under the action of the electromagnetic force, and the contact separation disappears (full face contact); After 0.52 ms, contact separation reappeared and remained until the armature left the rails. The following phenomenon deserves to be noted: (1) Despite the long full-face contact time (0.08–0.52 ms), the distance of armature movement at this stage is very short due to the slow velocity of the armature(0.07–36 mm); (2) Contact separation always occurs at the tail of the armature and decreases gradually from the tail to the head (along z-positive direction) and from the edge to the center (from y→∞ to y = 0). In addition, the length of the separation region (in the z-direction) remains essentially constant; (3) After energization, the magnitude of separation decreases dramatically; in the electromagnetic separation stage, the magnitude of separation is maintained at about half of the initial magnitude of separation. In fact, the area of the separation region is much smaller than the area of the initial separation region, despite the electromagnetic separation under F_nn.

The armature-rail contact is an electrical contact with high temperatures and high loads, and the electrical and thermal characteristics are important contact parameters for railguns. Fig. 11 shows the temperature distribution on the armature at different times, and Fig. 12 shows the temperature distribution on the rails. Fig. 12a–c is the results on line (a), line (b) and line (c), respectively. The legend in Fig. 12b indicates the correspondence between curve color and time, and it holds true for Fig. 12a,c.

Obviously, the maximum temperature on the armature is much greater than that on the rails but doesn’t exceed the melting point of 7075 aluminum alloy. In a general railgun, the armature almost always melts, but we use shorter rails and smaller currents to mitigate the melting (since the melting affects the size of the contact gap and is difficult to calculate accurately). On the armature, the temperature maximum occurs initially at the throat (as shown in Fig. 11b ②) and then shifts to the tail (as shown in Fig. 11d ③) of the armature. On the contact surface, a high-temperature region appeared in the middle section (as shown in Fig. 11a ①) of the contact surface at 0.04 ms; after 0.25 ms, the high-temperature region appeared on both sides of the middle section (as shown in Fig. 11f ④) of the contact surface. The tail of the armature remains high temperature throughout, but after 0.64 ms, the maximum temperature in this region decreases. The change of temperature distribution on the contact surface is affected by the contact separation: before 0.08 ms, the armature tail is in the contact separation state, and region ① has the highest contact pressure, so the frictional heat is the most; after 0.52 ms, the armature tail separates from the rails again, so the maximum temperature decreases.

Fig. 12 shows that the maximum temperature on the rails does not reach the melting point, indicating that the rails were not melted during launching. The location of the highest temperature on the rails always lags behind the armature, and the faster the armature the longer the lag. Notice that the temperature at the location on the rails in contact with the armature is always lower than the armature’s temperature. This is because the current density on the armature is always greater than that on the rails. Considering the short operating time of the railgun, the armature and rails cannot reach thermal equilibrium, so we do not force the armature and rails temperatures to be equal.

Fig. 13 shows the distribution of potential differences on the contact surface. The contact separation occurs only at the tail of the armature, and therefore the potential difference occurs only at the same place, and we have plotted only the tail of the armature 17.5 mm. It can be noticed that the distribution of the potential difference is very inhomogeneous and is always greater in the edge regions than in the center region. In the initial separation stage (before 0.08 ms), the maximum value of the potential difference increases with time. In the electromagnetic separation stage (after 0.52 ms), the maximum value of the potential difference increases slowly with time; the minimum value of the potential difference remains stable and close to 0 both in the initial separation and electromagnetic separation stages. Figs. 11 and 13 together illustrate that the maximum values of the potential difference and contact separation are more stable during the electromagnetic separation stage. Considering the small potential difference at the initial separation stage, we can determine that if electrical breakdown can be emitted, it will only occur at the electromagnetic separation stage and shortly after the start of electromagnetic separation. Moreover, once the electrical breakdown condition is satisfied, the electrical breakdown will continue.