Yingtao Xu, Bo Tang, Baoming Li

The National Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China

Keywords:Railgun Momentum conservation Recoil Surface force

ABSTRACT In this paper, the momentum conservation equation in the longitudinal direction for simple railgun was deduced. Then, a three-dimensional model was established and the finite element method was utilized to simulate the problem. Based on the simulation, some results such as the surface force on the electromagnetic field, the electromagnetic force on the conductor in the longitudinal direction were obtained. Besides, the surface force density on different longitudinal section of railgun was analyzed and discussed in details.The results showed that the momentum of the railgun system was conserved when the surface force of the electromagnetic field was contained. Moreover, large amount of longitudinal force was transmitted to the breech by the electromagnetic field in the form of surface force. The exact position and distribution of recoil were related to the current input device.

1. Introduction

In the electromagnetic railgun system, the armature moves forward by a large electromagnetic force.According to momentum conservation,the launcher should be subject to a reverse force.The position and size of the reverse force are critical to the design of the electromagnetic railgun system.

In the previous studies of other researchers, it is always controversial to use Ampere or Lorentz force to calculate the longitudinal force on the rails. Graneau.P insisted that Ampere force should be utilized to calculate the longitudinal forces on the rails[1e14]. Calculations results indicated that the force acts near the armature. This phenomenon has also been found by experiments.Su Ching-Chuan et al. used the Lorentz force to calculate the longitudinal force on the rails[15,16].Sadedin et al.have done a lot of research on longitudinal force in electromagnetic field [17e24]. In addition,Marshall and Woods insisted that there is no longitudinal force on the rails. However, Cavalleri found that the two methods are equivalent[25,26].Besides,C.J.Carpenter calculated that there is a large amount of reverse momentum and stress in the air between the rails [27]. Allen believed that electromagnetic stress could be used to calculate the longitudinal force on the rails and the recoil located on the breech [18].

In previous research on the momentum of railgun,the research targets were mainly rails and armature,while the electromagnetic fields were ignored.

In this paper, based on Maxwell's equations, momentum conservation equations for rails,armature,and electromagnetic field in electromagnetic railgun systems were derived. A threedimensional model of the railgun was established, and the finite element method was used to simulate electromagnetic rail gun at the starting time.The surface force on the electromagnetic field and the electromagnetic force on the conductor of the electromagnetic railgun were obtained,and the relationship of the two forces were analyzed in details.

2. Equations and simulation

2.1. Momentum conservation equation in railgun

The space V is filled with an isotropic medium, the dielectric constant is ε,the magnetic permeability is m,the charge density is r,the current density is J,the surface is S,and dS is the surface unit of S.

The force density of the charge is:

In Maxwell's equations:

According to Eq. (2) and Eq. (3), Eq. (1) could be rewritten as:

In Maxwell's equations:

So,

Because,

Eq. (8) can be transformed into,

Assume that,

Eq. (11) can be simplified as,

f is the electromagnetic force density of the entity in space V,is the electromagnetic force Fmof the entity in space V,which is equal to the derivation of the mechanical momentum(Gm).

g is the momentum density of the electromagnetic field,gdV is the derivation of the electromagnetic field momentum(Ge)in V.Feis the electromagnetic force acting on the field of space V.

Eq.(17)is the momentum conservation equation in V,and Gmt Geis the total momentum of space V, which includes mechanical momentum and electromagnetic field momentum.FSis the surface force of V subjected to external electromagnetic fields.

2.2. Model and parameter

Fig.1 is the schematic diagram of a simple railgun.I is the source current.The material of the rails is copper,and the material of the armature is aluminum. In Fig. 2, a simple railgun is located in a space VABCD-IJKLfilled with air. The symmetry axis of the railgun locate along x coordinate,and the origin of coordinate locate in the center of surface ABCD.And the railgun breech is located on plane S0. The distance from plane Sxto plane S0is x.

The space between Sxand S0is defined as Vx?VABCD-EFGH.The outer surfaces of Vxare named as S0and Sxin the x direction,S+yand Styin the y direction, S-zand Stzin the z direction respectively.For the space Vxin Fig.2,According to in Eq.(15),edSTx?FSxis an integral of surface pressure over along x direction. It presented the surface force in the x direction acting on V by the external electromagnetic field. FSxcontains 6 components. The plane direction is defined as positive when pointing to the outside of Vx.

Fig.1. Schematic of simple railgun.

Fig. 2. Schematic diagram of railgun calculation model.

According to the right side of Eq.(18),the surface force in the x direction of Vxfrom different surfaces could be calculated.

By calculating the values of the three forces FSx, Fmxand Fexunder different Vx, the momentum distribution of the system at different longitudinal positions can be obtained. Further, by analyzing the momentum changes at the rails, the armature and the field, the longitudinal force distribution of the railgun system can be obtained.

2.3. The governing equation

In a railgun, the displacement current density is much smaller than the conduction current density and therefore negligible. The A-F method is used to express the Maxwell equations with Coulomb gauge. The control equations are presented below.

Where, F is the scalar potential, A is the vector magnetic potential, JSis the source current density, V is the vector differential operator, m is the magnetic permeability, s is the conductivity.Current density J, magnetic flux density B, and electromagnetic density f are:

A (x, t t△t) is the current magnetic vector at position x, and A(x0,t)is the last time magnetic vector at position x0.The conductors are surrounded by air domains and the magnetic vector potentials at the far field position are set to zero. An electric scalar potential boundary condition is imposed on one end face of the rails.

According to Ref.[28],the material of the armature is aluminum,and the material of the rail is cooper. The parameters of the simulation model is shown in Table 1. The finite element method was used in this paper to solve the equation above based on ANSYS electromagnetic solver. Initial conditions include initial source current and boundary conditions. Source current is applied to the left surface of the rails and the boundary condition was set to Dirichlet boundary. Mesh generation was performed in the model using tetrahedral elements.The rails is divided by 78464 elements along the length,The armature is divided by 13845 elements along the length,and has 228469 elements in total finite-element model.The electromagnetic field results calculated by ANSYS are substituted into Eq. (15) for further calculation to obtain the momentum distribution of the simple railgun.

3. Results and discussion

3.1. Forces balance on the railgun system

Fig.3 were curves of FSx,Fmx,Fexand I2at different times when x?90 mm. And curve 1 was the total surface forces (FSx) of Vxin the x direction. Curve 2 was the x-direction electromagnetic force(Fex) of the electromagnetic field in Vx. Curve 3 was the total x-direction electromagnetic force (Fmx) of the conductors in Vx.Curve 4 represented the squared loaded source current.

Table 1 Parameters of the simulation model.

Fig. 3. The curves of FSx, Fmx, Fex and I2 at different times (x?90 mm).

It can be observed that curve 1 and curve 3 are completely coincident, and both they are proportional to the squared source current.Besides,the values of curve 2 are almost equal to 0 at each moment.

Table 2 presented the specific values of FSx, Fmx, Fexat the moment of the dotted line A in Fig.3.The external surface force was 2938.6 N,and the electromagnetic force of field was 6.37e-16 N.The force on the conductor was 2938.7 N. Compared with the external surface force FSxand the electromagnetic force Fmxon the conductor, the electromagnetic force Fexof the electromagnetic field in Vxcan be ignored.The results showed that the momentum equation of railgun showed in Eq.(17)was conserved at any times.

Fig. 4 was the curves of FSx, Fmx, Fexwhen x changes at time t?5 ms. While x changes, the volume and surface of Vxchanges,and the three forces of Vxchange accordingly.Curve 1 represented the FSx,curve 2 was Fmx,and curve 3 was Fex.Fig.4 showed that the FSxcurve is the same as the Fmxcurve,and the value of Fexwas still close to zero. The results showed that the momentum equation of railgun showed in Eq. (17) was conserved at any position.

Both Figs.3 and 4 showed that compared with FSxand Fmx,Fexcould be ignored.Therefore,according to Eq.(17),the volume force Fmxcould be characterized by the surface force FSx. Furthermore,FSxis composed of 6 parts.And,Table 3 listed the contributions of each surface to FSx.

Table 3 showed that of the six surfaces of Vxonly FSxeS0T and FSxeSxT contribute to FSx,and the values of the other surfaces were almost zero. When x changed, the position of surface S0did not change, the surface force on S0will not change as well. Therefore,the force FSxeSxT on surface Sxis closely related to the volume forceFmx. This relationship was shown in Fig. 5.

Table 2 The values of FSx, Fmx, Fex at t?5 ms and x?90 mm.

Fig. 4. The curves of FSx, Fmx, Fex at different x position (t?5 ms).

Table 3 The values of the 6 components of FSx at t?5 ms and x?80 mm.

Fig. 5. Curves of surface force FSxeSxT and volume force Fmx.

In general, when x varies between 0 mm and 70 mm, the x-direction surface force remains the same.The conductor in this space is the rail. It showed that the rails from 0 mm to 70 mm have no longitudinal force.

When x in 70e95 mm, FSxeSxT is rapidly reduced, and at the same time, the electromagnetic force of the conductor Fmxis rapidly increased. Therefore the reduced surface force at Sxturns into an electromagnetic force within Vx.

From 95 mm to 100 mm, FSxeSxT rises slightly to 0, and this segment FSxeSxT is less than 0,indicating that the conductor here is subjected to a reverse electromagnetic force.

3.2. Detailed analysis of the surface force

When x changed, the media surface Sxlocated would change.Further analysis of the surface force distribution on Sxis needed.The surface Sxwas divided into the following four areas shown in Fig.6.Also,V can be divided into V1,V2,V3 and V4 according to the same division.

Table 4 showed the entities in different areas of Sxwhen x changes. According to the division, FSxwas rewritten as follows.

Fig. 6. Schematic diagram of cross-sectional area division on Sx.

Table 4 The entities at different areas of Sx.

Fig. 7 showed the curves of surface force in different areas of FSxeS1T and FSxeS2T varies with x when t?5 ms. Here, curve 1 and curve 2 represented the force act on the rail areas, curve 3 represented the force on the area between the rails, and curve 4 represented the force on the air domain outside the rails.

Fig. 8 represented the curves of electromagnetic force in different volume area.Curve 1 and curve 2 represented the force on the rails,and curve 3 was the force on the armature.

Fig. 7. The curves of surface force in different areas of Sx varies with x.

Fig. 8. The curves of electromagnetic force on different areas of Sx varies with x.

In Fig. 7, when x was between 0 mm and 40 mm, FSxeS1T and FSxeS2T were small compared with FSxeS3T and FSxeS4T, which indicated that the surface force FSxis mainly concentrated in the air domain. During 40 mm-80mm, FSxeS1T and FSxeS2T remained constant, FSxeS3T increased, and FSxeS4T decreased, while the sum remains constant. It is indicated that surface force were concentrated from the external air domain to the space between the rails.Further, it revealed the flow of electromagnetic field momentum.

When x is between 80 mm and 95 mm,FSxeS1T,FSxeS2T,FSxeS3T and FSxeS4T all fell sharply in Fig. 7, however, FmxeV3T is rising rapidly in Fig.8.According to Table 4,it is the armature body in this area. It is indicated that the surface force in the air was converted into the electromagnetic force on the armature.In addition,there is a small longitudinal force on the rails here according to Fig. 8.

When x is between 95 mm and 100 mm, the armature head located here.In addition,FmxeV3T reduced a little,which indicated that there was a reverse longitudinal force preventing the armature moving forward. This result was consistent with the simulation results of Zizhou Su about armature [15,16].

Fig.9 showed the distribution of surface force densityon Sxat different longitudinal positions.

The surface force represents the force of external electromagnetic field of space V on internal electromagnetic field of space V.The surface force on electromagnetic is different from the electromagnetic force on the conductor. It is the process quantity of the interaction between the external electromagnetic field of space V and the internal electromagnetic field of V, and it is an internal force between electromagnetic field. From these pictures, we can get how the momentum transferred in electromagnetic field in the longitudinal direction.

It could be seen that the surface force density gradually concentrated from the outside air to the area between the rails. In addition, the surface force disappeared instantly after passing the armature.

A large longitudinal surface force was generated by the source current at the end surface of the breech. The surface force transmitted to the armature through the electromagnetic field. Around the armature, the surface force was primarily converted into the longitudinal magnetic force on the armature. That is, the longitudinal force on the armature was balanced with the surface force of the electromagnetic field on the end section of the breech.The rails had almost no longitudinal force.

3.3. Recoil analysis of electromagnetic gun

Recoil force is the reverse force received by the electromagnetic gun system during the launching process.

The reverse longitudinal force is mainly concentrated on the field around the breech. However, in the actual structure of the railgun,the current input structure could not be idealized as shown in Fig.1. Different current input structures produce different electromagnetic field distribution at the breech, which also leads to inconsistent distribution of recoil. However, it is certain that the main recoil does not act on the rails.

4. Conclusion

The momentum conservation equations for simple railguns were derived in this paper. A 3-D model of railgun was built, and the model was simulated using the finite element method. The electromagnetic force and the surface force of the railgun were obtained. The relationship between surface force and electromagnetic force was analyzed in details.The conclusions of the analysis are summarized as follows.

A) For the simple railgun, the momentum of the system is conserved when the electromagnetic field is considered.

Fig. 9. The surface force density distribution on different Sx.

B) The longitudinal force on the armature is balanced with the surface force of the electromagnetic field at the breech.

C)The specific position and distribution of recoil were related to the current input device.