An internal ballistic model of electromagnetic railgun based on PFN coupled with multi-physical field and experimental validation

Benfeng Gu, Haiyuan Li, Baoming Li

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

Keywords:Internal ballistic modeling Electromagnetic rail gun Multi-physics field coupling Experimental validation PFN

ABSTRACT To accelerate the practicality of electromagnetic railguns, it is necessary to use a combination of threedimensional numerical simulation and experiments to study the mechanism of bore damage.In this paper,a three-dimensional numerical model of the augmented railgun with four parallel unconventional rails is introduced to simulate the internal ballistic process and realize the multi-physics field coupling calculation of the rail gun,and a test experiment of a medium-caliber electromagnetic launcher powered by pulse formation network(PFN)is carried out.Various test methods such as spectrometer,fiber grating and high-speed camera are used to test several parameters such as muzzle initial velocity, transient magnetic field strength and stress-strain of rail.Combining the simulation results and experimental data,the damage condition of the contact surface is analyzed.© 2023 China Ordnance Society.Publishing services by Elsevier B.V.on behalf of KeAi Communications Co.Ltd.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

The internal ballistics process of electromagnetic railgun is a transient dynamic launching process under the action of coupled multi-physical fields, which instantly generates huge electromagnetic force, Joule heat, etc.along with high-speed friction and impact between the contact area, which will lead to wear and erosion [1,2].Due to the extreme working environment of the internal ballistic process, the changes in the characteristics of the inner multi-physical fields such as electromagnetic, thermal, and force are difficult to be measured by experiments,so the simulation calculation for the launch process of the rail gun has become an important tool to study the mechanism of the internal ballistic process [3-5].Among the current simulation calculation models for electromagnetic rail guns, the one with high accuracy and acceptance of the calculation results is the EMAP3D software developed by Kuo-Ta Hsieh et al.of the University of Texas in 1995[6],which is based on the set of Maxwell's equations expressed by a magnetic vector potential, electrical scalar potential for multiphysics field calculation [7], and in subsequent developments based on this different algorithms were introduced to achieve efficient computation of sliding electrical contacts [8-10].

In this paper, the accurate internal ballistic simulation and the introduction of velocity frequency together with the general finite element software to establish the internal ballistic model of the rail gun allows the calculation of the electromagnetic-thermal-force coupling analysis, the multi-physical field characteristics of the sliding electric contact process, the damage analysis of the launch combined with the experimental results, and the optimization of the structure.

2.Simulation and experimental setup

2.1.Simulation model

The differential equation of armature motion can be described as

where mais the armature mass, Lp′is the propulsive inductance gradient of the rail,the propulsive inductance gradient is generally similar to the high frequency inductance gradient of the rail,which can be obtained by Maxwell high frequency eddy current field simulation, I is the loaded excitation current, Fairis the air resistance, and f is the friction between armature and rails.The air resistance is

ρ0is the air density,A is the cross section area of inner bore,τ is the bore section perimeter, and the Cfis air resistance coefficient obtained by simulating the aerodynamic characteristics of the projectile and armature integration.The calculation of the friction force depends on the pressure of the armature on the rail,including the loading preload and the electromagnetic pressure, both of which should be no less than I/100 during the launching process according to Marshall's "1 g per amp" rule to maintain sufficient contact pressure between the armature rail so as to avoid the contact transition between armature and rail[11].According to the refraction law of the electric field strength vector line and the current density vector line [12], so armature friction and pressure can be obtained by

where laand t are the armature tail length and separation of inner rails, α is the angle between the vector line of physical quantities inside the armature and the normal to the contact surface,c is the interference of armature [13].The friction coefficient between the armature and the rail can be expressed as

where Eaand Erare the elastic modulus of the armature and the rail, respectively, c is the armature excess considering the average wear variation due to Joule and frictional heat [14], hr2is the thickness of the inner rail, and S is the caliber of the rail gun.

2.2.Velocity frequency and multi-physics field calculations

During the sliding electrical contact, the new part of rails are continuously accelerated into the loop [15], and the skin depth of the contact area can be equal to the step current diffusion depth[16].

The magnetic quasi-static field equations obtained from Maxwell equations are the controlling equations for electromagnetic launch, and the magnetic diffusion equations with the velocity term is

where h is the finite element mesh length in the armature motion direction which is limited to the velocity,thus ensuring the validity of the velocity frequency results in areas other than the contact surface of armature and rail.

2.3.Launch parameters

The model of experiment and simulation is a 20 mm caliber augmented railgun with round bore, and the materials of rail and armature are 7075 aluminum alloy and OFHC copper.In this paper,the parameters of the launcher are shown in Fig.1, and the launching calculation parameters are shown in Table 1.

The excitation current and muzzle voltage used in experiments and simulations are shown in Fig.2.

Fig.1.Structure of rail gun.

Table 1 Launch calculation parameters.

Fig.2.Rail current and muzzle voltage curve.

The reverse fluctuation of the voltage in Fig.2 is due to the secondary rails,around which the magnetic field is generated at the moment of discharge,and the inner rail in the direction of armature and the muzzle forms an open ring and is cross-linked to the magnetic field generated by the secondary rails, producing a negative induced voltage.

2.4.Experimental testing

The launch and test system (Fig.3) mainly includes launch system,high power pulse power supply,launch control system,and test system.

One day, therefore, he went to the king, and said that the eleven brothers who had come to the palace a little while ago, and served as stablemen, could do a great deal more than they pretended

A zone intercept device was used to measure the initial velocity,and B-dot probes were used to measure the velocity and magnetic induction.The atomic emission spectroscopy temperature measurement method was used to the measure the transient temperature in the bore, and this non-interventional measurement method has a simple optical path,a large temperature range,and a high measurement accuracy, which is widely used in the measurement of plasma [19].Transition is the phenomenon that the contact of rails and armature changes from good electrical contact suddenly to arc contact.The measurement of the bore transient temperature can provide further insight into the bore environment in both the transition and non-transition cases,and thus provide an experimental reference for the study of the transition.The experiments are based on the fiber-optic grating sensing principle to measure the dynamic stress-strain of the rail to avoid the effect of electromagnetic interference [20].

3.Results and discussion

Fig.3.Launch and test system of railgun.

Fig.4.Friction coefficient curve.

Fig.5.Simulated and experimental velocity displacement curves.

The friction coefficient is shown in Fig.4,the time stages of it is determined by the ideal speed and the simulated current.The electromagnetic thrust on the armature is less than the maximum static friction at the moment from 0 ms to 0.12 ms when the armature is still,and the dynamic friction coefficient is 0 during this time; start-up stage is from 0.12 ms to 0.27 ms, due to the maximum static friction at the starting stage is greater than the sliding friction and the rapid increase of current,the accumulation of Joule heat makes the interference of armature melting which produces lubrication effect on the rail;from 0.27 ms to 1.75 ms,the previous current rise is relatively small,but the armature is still in the high current and high speed stage,the erosion and wear on the contact surface form the formulation of aluminum melt film leads to a small decrease in the coefficient of friction, the current continues to drop and the friction coefficient shows a significant decrease after 0.4 ms; from 1.75 ms to 1.95 ms, the current drop leads to the instability of the aluminum melt film,the thickness of the armature tail decreases, the mechanical interference disappears, the contact state between armature and rails deteriorates,and the friction coefficient rises,and according to the analysis of the time of the voltage jump at the muzzle, the transition occurs after 1.6 ms in practice [21], which is close to 1.75 ms.

The calculation of the armature velocity is mainly obtained by the Internal ballistic simulation of the railgun,and the comparison of the calculated and experimental results of the speed and stroke after considering the erosion is shown in Fig.5.

The simulations show that the muzzle initial velocity is 1204 m/s and the experimental result is 1171 m/s with an error of 2.8%.The error is caused by the friction coefficient calculated by ideal velocity and simulated current and the larger delayed peak current of the simulated current.With the current decreases, the armature velocity levels off, then the velocity decreases due to the rise in the coefficient of friction.Due to the smaller caliber and excitation current in this simulation,the above factors have less influence on the final numerical calculation, thus the velocity frequency is obtained by the calculated velocity for eddy current field calculation.The magnetic induction intensity of the pivot rail during the launch of the rail gun is shown in Fig.6.

The initial moment is at low speed,so the velocity skin effect is negligible [22].The magnetic field is mainly concentrated at the inner edge of the armature and rail, and the concentration of magnetic energy at the armature throat is more obvious.As the current increases to the peak stage, the magnetic field diffusion depth cannot keep up with the increasing armature speed,and the magnetic energy near the back end of the armature can only be concentrated on the rail surface.The magnetic field distribution in the contact part of the armature rail is affected by the velocity skin effect, and the magnetic energy in the armature throat is concentrated towards the edge.The magnetic induction strength of the armature tail wall gradually increases, and the magnetic field is spread in a band-like trend on the rail at the back end of the armature.The closer to the armature tail,the smaller the magnetic field diffusion depth.In addition,the distribution characteristics of the current and the magnetic field are consistent.

Fig.7.B-probe position.

Fig.8.Magnetic flux density measured by B-probe: (a) Magnetic flux density in x direction; (b) Magnetic flux density in z direction.

Fig.9.Frictional heat flow curve.

The mounting position of the B-dot probe on the tube is shown in Fig.7.

By integrating the B-probe signal, the variation process of the magnetic field at the location of the probe can be found.The flux density in the x-direction at the installation location of the B-probe is shown in Fig.8(a) x-direction flux density is obtained by integrating the armature current B-probe signal and dividing it by the coil constant(the product of the number of turns N and the crosssectional area A).

Fig.8(b)shows the flux density in the z-direction at the location of the probe obtained from the signal of the rail current B-dot probe, which is closer to the launch axis, and the currents in the upper and lower orbits produce a magnetic field in the same direction at this location,so the peak flux density measured is higher than that of the armature current B probe.As the current decreases,the flux density measured by each probe becomes progressively smaller.

Fig.10.Temperature.

Fig.11.Light intensity signal of four spectral lines of copper atoms: (a) Full Spectrogram; (b) Partial enlarged detail.

The frictional heat flux is shown in Fig.9.

The temperature field results calculated with frictional heat flux are shown in Fig.10.

It can be seen that from 1.7 ms (Fig.11(a)), the obvious light intensity appears, indicating that a stronger arc is generated only after the armature is out of the muzzle.The partial enlarged detail of the spectrum before 1.68 ms is given in Fig.11(b).

In the beginning,the position change of armature is small,while the current increases rapidly to its maximum value and Joule heat is the main source of heat.The high heat zone appears later because during the launching process, the armature flips over causing one side close to the rail,and frictional heat is the main source of heat in that period,and The experimental result is qualitatively consistent with the simulated (Fig.10).

It is generally believed that rail surface defects and dynamic response on the rail are important causes of rail gouging [23,24].The gouging is related to the microscopic scratches or particles on the rails,and teardrop-shaped pits are generated on the rails when the armature reaches a specific speed related to the hardness of the rail due to the extremely high pressure under dynamic loading,frictional heat, and debris cut into the sliding contact gap et al.Frictional and Joule heat can also soften or even melt the rail surface and eventually cause damage to the rail under the simultaneous action of high speed impact and stress concentration, Figs.12(a)and 12(b) show that rail stresses and frictional heat rise after the transition occurs, Fig.13 shows that dynamic stresses on the contact surface are concentrated at the edge of the armature while moving towards the armature tail wall with the maximum value with motion.

The armature reconstructed by CCD photographs at three different angles is shown in Fig.14,and the armature waist is more severely worn.

The fiber optic probes are arranged on the rail,and the positions of the measurement points are shown in Fig.15.

The dynamic strain measurements of the tube in the experiment are shown in Fig.16.

The strain trend at the first two measurement points is similar to the results of the vortex field coupled with the static structural field,and the strain increases around 1.5 ms(see Fig.17).According to Fig.14, the tail of it is thin on one side with asymmetric wear,indicating that it oscillated during the launching and the lateral force was not uniform, resulting in the armature deflecting to one side of the rail and squeezing the rail surface causing a sudden increase in the strain value on one side of the rail.Around 1.5 ms,the thermal stress may have played a major role.The distribution is similar to that of magnetic induction, which can be identified in Fig.6,especially on the rail surface,where even small current peaks can generate large thermal stresses.According to the velocity simulation curve,the velocity after 1 ms is nearly smooth,but still in the rising phase, the current density on the rail surface is more concentrated, thus it will cut obvious drag marks on the armature and rail edges,and according to the simulation results of Fig.16,the structural stress on the armature at the later stage is concentrated at the edge of the tail wall.

Fig.12.Dynamic rail stress and frictional heat curves.

Fig.13.Dynamic stress on armature contact surface.

Fig.14.Photograph of armature before launching and reconstructed morphology.

The rails of the 20 mm augmented railgun were destroyed later in the serial test,so the pictures of the damage can not demonstrate the different levels of damage at different stages.We use diagrams(Fig.18) from other experiments we did before to verify this phenomenon.

As a result, although the rail strain value is larger in the later stage,the damage to the rail surface is less extensive.

4.Conclusions

Through the detailed velocity analysis of the inner ballistic process combined with the existing finite element simulation software,the simple inner ballistic modeling of small and medium caliber electromagnetic orbital launchers can be realized.The experimental validation shows that the model has some practical significance.The multi-physics field simulation calculation of the inner ballistic process was in high agreement with the experimental results before the transition occurred,and the results were different from the actual ones at a later stage due to the complex mechanism of the rail gun launching process, but some of the phenomena could be explained to each other based on the relationship between the multi-physics fields.Then the simulation of the model can be used to verify the accuracy of the results of the self-developed program of sliding electrical contact.

Fig.16.Dynamic strain measurement results of the tube.

Fig.17.Stress distribution at the moment of maximum stress in armature structure.

Fig.15.Positions of dynamic strain measurement points.

Fig.18.Rail wear of 40 mm caliber square-bore railgun.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.