Abstract
Aiming at the nondestructive recovery of ultra-high-speed armature of electromagnetic railgun, a mathematical model of fluid medium for recovery is established based on fluid damping theory. At the same time, the fluid-solid coupling finite element model is constructed, and then the numerical calculation is carried out. Furthermore, the recovery characteristics of different armature speed under different density medium are analyzed, and then the relationship between the armature initial velocity and the maximum density of medium is given. Additionally, the overload limit of aluminum armature is obtained. Finally, the overall scheme of variable density segments is proposed. The experimental system is built and verification experiments are carried out. The research results show that by using piecewise recovery method, low-density medium is selected for the front part and high-density medium is selected for the rear part, the problem of too long distance or too much overload when a single medium for recovery can be solved. Meanwhile, the nondestructive recovery of armature can be realized, and the appearance of armature can be maintained.
1. Introduction
As a kind of ultra-high-speed kinetic energy weapon, electromagnetic railgun is a hot research topic all over the world [1, 2]. In recent years, researchers from various countries have carried out researches on the launching mechanism, parameters matching, and power supply integration of electromagnetic railgun and achieved some achievements. However, the lifetime of electromagnetic rails still needs further study. There exist sliding electric contact between the rails and armature during the launching process, which will cause extreme mechanical impact [3–5]. By studying the surface morphology of armature and rails after launch, the characteristics of sliding electric contact can be analyzed, thus providing support for improving the service life of rails [6, 7]. Due to the high initial velocity of the armature launched by the electromagnetic railgun, it will be broken or deformed under the action of the end effect, which brings difficulties to the nondestructive recovery of the ultra-high-speed armature [8]. Thus, the analysis of the armature after launch is impossible.
In the aspect of projectile recovery, some relevant theoretical and experimental researches have been carried out at home and abroad, mainly focusing on explosive shaped projectile (EFP), high-speed flying body recovery test, projectile recovery test based on air damping, etc., which provides experience for the nondestructive recovery technology of ultra-high-speed projectile of electromagnetic railgun [9–14]. However, in these experiments, the recovery speed of the projectile is small, the distance of the recovery device is long, and the nondestructive recovery of the armature surface cannot be realized. Therefore, the study of nondestructive recovery of ultra-high-speed armature of electromagnetic railgun is very important.
2. Principle of Nondestructive Recovery of Ultra-high-speed Projectile
The essence of nondestructive recovery of projectile is to decelerate the projectile through buffer materials of different densities under the condition of safety overload, and finally make its velocity decay to zero. As to the ultra-high-speed projectile launched by electromagnetic railgun, its muzzle velocity is about 2500 m/s. Therefore, it is difficult to use traditional materials to meet the design requirements. Low-density fluid can be used as the main medium of ultra-high-speed projectile nondestructive recovery of electromagnetic railgun, and its principle is shown in Figure 1.
Because of the ultra-high-speed and short time of projectile passing through the recovery medium, the trajectory of projectile in recovery medium is approximately a straight line, and the stress on the horizontal direction of projectile is analyzed. According to the damping theory of high-speed projectile in fluid, the resistance of projectile in fluid is proportional to the square of projectile velocity, which can be expressed as follows:where, is the initial muzzle velocity of projectile, t is time, and b is the attenuation coefficient, representing the velocity attenuation characteristics of the projectile in the medium. In the fluid medium, the attenuation coefficient b can be expressed as follows:where ρfluid represents the density of the fluid, A is the effective cross-sectional area of the projectile, CD is the damping coefficient, and m is the mass of the projectile. The reciprocal of the attenuation coefficient b is usually called as the characteristic attenuation length, which is represented by a. Its physical meaning is the displacement of the projectile in the medium when the velocity attenuation is equal to the 1/e of initial velocity. Therefore, a can be expressed as follows:where ρprojectile is the equivalent density and l is the effective length of projectile.
According to the analysis above, the mathematical model of ultra-high-speed projectile recovery theory is as follows:
After integrating both sides of (4), it can be expressed as follows:where x is the displacement of the projectile in the medium, is the instantaneous velocity of the projectile, is the initial velocity of the projectile, and the velocity of the projectile after passing through the recovery medium can be obtained from (5).
It can be seen from equations (1)–(3) that the overload of high-speed projectile at a certain moment in the fluid is proportional to the square of the velocity of the projectile at that moment and inversely proportional to the characteristic attenuation length. For the nondestructive recovery of projectiles, the most important is to make the projectile velocity effectively decrease within a limited distance and to ensure that the overload of the projectile during the recovery process does not exceed the maximum overload it can withstand.
Considering the engineering practice, the characteristic attenuation length a is not more than 10 m. According to (1), when the initial velocity is 2500 m/s, the maximum overload is about 62500 g. At the same time, it can be seen from the above analysis that the shorter the characteristic attenuation length, the greater the overload of the projectile. Thus, as to the high-speed projectile, the density of fluid should be low enough to decelerate the projectile.
3. Simulation Analysis of Nondestructive Recovery of Ultra-high-speed Armature for Electromagnetic Railgun
In order to further study the characteristics of ultra-high-speed armature nondestructive recovery, the recovery of U-shaped aluminum armature is simulated with dynamic finite element software, which is very important to the ultra-high-speed armature recovery experiment of electromagnetic railgun.
3.1. Establishment of Simulation Model
It can be seen from the analysis above that the fluid medium is used in the nondestructive recovery of ultra-high-speed U-shaped aluminum armature of electromagnetic railgun. The simulation model is a fluid-solid problem. Furthermore, it also covers the problem of large displacement deformation. Thus, simulation model is composed of three parts: armature, air domain, and recovery medium domain. The armature uses Lagrangian solid elements and the constant stress solid element algorithm in SECTION_SOLID is used. At the same time, the air domain and the recovery medium domain use solid Euler elements and ALE multisubstance element algorithm. The width and height of the armature is 20 mm, and the width and height of the air domain and the recovery medium domain is 100 mm. An independent boundary without reflection is applied around the air domain and the recovery media domain. The height and width of the fluid domain are divided into 15 parts in meshing, and the middle and both ends are sparse. In order to ensure the uniformity and compactness of the mesh at the head of the armature, the mesh of the curved part is divided into 20 parts. The simulation model is shown in Figure 2.
The material of the armature is an aluminum alloy and is defined by keyword MAT_PLASTIC_KINEMATIC. The parameters are shown in Table 1.
The material model of air medium is described by the equation of state, namely,where P is the pressure in unit initial volume, C0∼C6 are constants, E is the internal energy of air, μ is the specific volume, if μ < 0, C2μ2, and C6μ2 are set to 0, where ρ is the air density, ρ0 is the initial density of air, and is the relative volume. The state equation can be used to simulate the gas conforming to the equation of state of γ law. The coefficients are set as C0 = C1 = C2 = C3 = C6, C4 = C5 = γ−1, where γ is unit calorific rate.
As to air, C0 = C1 = C2 = C3 = C6, C4 = C5 = 0.4, its density is 1.29 kg/m3, the initial relative volume is 1.0.
Water medium is represented by MAT_NULL material model and EOS_GRUNEISEN equation of state, namely,where C is the shock wave velocity , that is, the intercept of curve (velocity unit), which is sometimes called sound propagation velocity in the medium as it is numerically the same as sound propagation velocity in the medium. S1, S2, and S3 are the coefficients of the slope of curve, γ0 is Gruneisen constant, α is a constant, which is the first order volume correction of γ0. Meanwhile, the μ can be expressed as follows:where is the relative volume.
The parameters of the recovery medium are shown in Table 2.
3.2. Verification of the Simulation Model
In order to verify the accuracy of the simulation model, the test results and simulation results of the flat-head projectile provided in the literature are used for comparison and verification [15]. In the experiment, the length of the flat-head projectile is 26 mm, and the projectile diameter is 12 mm. The velocity and displacement of the projectile change with time are obtained in the experiment. In order to verify the accuracy of the simulation model above, the simulation model is established the same as the parameters of the projectile in the above experiment, as shown in Figure 3. The initial velocity of the projectile is 75.4 m/s, 118.8 m/s, and 142.7 m/s, respectively, and then the simulation of the model is carried out. With the simulation results, the curves of the armature velocity and displacement with time are obtained, as shown in Figure 4. Meanwhile, the experimental results are shown in Figure 5.
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It can be seen from Figures 4 and 5 that the speed of the projectile decreases obviously when it passes through the water medium, and the greater the initial speed, the faster the attenuation. The simulation results are in good agreement with the experimental results of the velocity and displacement variation characteristics, and it can be considered that the simulation model can be used to analyze the nondestructive recovery of the ultra-speed armature.
3.3. Recovery Characteristics Analysis of High-Speed U-Shaped Armature
During the nondestructive recovery of the armature, the structure should not be deformed. High-density recovery medium can decelerate the armature quickly, but will increase the probability of armature deformation. Low-density recovery medium will decrease the deformation effectively, but the recovery time and recovery distance will be more significant. Therefore, it is crucial to determine the maximum medium density, which can just make the armature deformation not occur.
Whether the deformation occurs depends on whether the maximum overload exceeds the yield stress of the material. Therefore, the maximum overload subjected on the armature during the recovery process can be used to determine whether deformation occurs, so as to determine the maximum density of recovery medium with different armature speed. In the following research, the initial speeds of the armature were set as 2500 m/s, 1500 m/s, 1000 m/s, and 400 m/s, respectively. The maximum overload on the armature is obtained by changing the density of the recovery medium, and then compared with the armature material properties.
When the armature initial velocity is 2500 m/s and the density of medium is 20 kg/m3, 30 kg/m3, and 35 kg/m3, the simulation results when the armature is subjected to the maximum overload are shown in Figure 6.
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By analyzing the simulation results, the maximum stress of the armature is 245 MPa in Figure 6(a), the value is 269 MPa in Figure 6(b), and 294 MPa in Figure 6(c). According to the armature material properties, the yield stress of the armature is 280 MPa. The maximum stress of the armature is smaller than the yield stress when the medium density is 20 kg/m3, so the deformation will not occur. However, the armature stress is 35 MPa less than the yield stress, the medium density can be further increased. The maximum stress of the armature is slightly larger than the yield stress when the medium density is 35 kg/m3, so the deformation occurs. When the medium density is 30 kg/m3, the maximum stress of the armature is just smaller than the yield stress, and the deformation just does not occur. Therefore, it can be preliminarily determined that the maximum medium density of nondestructive recovery of the U-shaped aluminum armature with the initial velocity of 2500 m/s is about 30 kg/m3.
When the armature initial velocity is 1500 m/s and the density of medium is 50 kg/m3, 80 kg/m3, and 100 kg/m3, the simulation results when the armature is subjected to the maximum overload are shown in Figure 7.
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By analyzing the simulation results, the maximum stress of the armature is 251 MPa in Figure 7(a), the value is 272 MPa in Figure 7(b), and 297 MPa in Figure 7(c). Compared with the yield stress of the armature material, the maximum stress is far smaller than the yield stress when the medium density is 50 kg/m3. The maximum stress exceeds the yield stress and the deformation occurs when the medium density is 100 kg/m3. When the medium density is 80 kg/m3, the maximum stress is slightly smaller than the yield stress and the armature just does not deform, so it can be preliminarily determined that the maximum medium density of nondestructive recovery of the U-shaped aluminum armature with the initial velocity of 1500 m/s is about 80 kg/m3.
When the armature initial velocity is 1000 m/s and the density of medium is 100 kg/m3, 150 kg/m3, and 200 kg/m3, the simulation results when the armature is subjected to the maximum overload are shown in Figure 8.
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By analyzing the simulation results, the maximum stress of the armature is 241 MPa in Figure 8(a), the value is 265 MPa in Figure 8(b), and 284 MPa in Figure 8(c). As similar to the above analysis, when the medium density is 150 kg/m3, the maximum stress is slightly smaller than the yield stress and the armature just does not deform, so it can be preliminarily determined that the maximum medium density of nondestructive recovery of the U-shaped aluminum armature with the initial velocity of 1000 m/s is about 150 kg/m3.
When the armature initial velocity is 400 m/s and the density of medium is 900 kg/m3, 1000 kg/m3, and 1100 kg/m3, the simulation results when the armature is subjected to the maximum overload are shown in Figure 9.
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By analyzing the simulation results, the maximum stress of the armature is 261 MPa in Figure 9(a), the value is 278 MPa in Figure 9(b), and 304 MPa in Figure 9(c) When the medium density is 1000 kg/m3, the maximum stress is slightly smaller than the yield stress and the armature just does not deform, so it can be preliminarily determined that the maximum medium density of nondestructive recovery of U-shaped aluminum armature with the initial velocity of 400 m/s is about 1000 kg/m3.
According to the research and analysis above, it can be drawn that when the armature speed is 2500 m/s, 1500 m/s, 1000 m/s, and 400 m/s, the maximum medium density that can be selected for the nondestructive recovery of armature is about 30 kg/m3, 80 kg/m3, 150 kg/m3, and 1000 kg/m3, respectively. The relationship between armature initial velocity and maximum medium density is shown in Figure 10.
As can be seen from the analysis results, the density of the medium decreases with the increase of the armature speed, and the decrease rate becomes smaller and smaller. Furthermore, according to the simulation results, the relationship between the maximum density ρmax of the recovery medium and armature velocity can be obtained by data fitting as follows:
It can be seen from the analysis that the nondestructive recovery of ultra-high velocity armature should meet certain requirements, that is, the maximum density of the recovery medium and the speed of the armature should satisfy the Equation (9).
Meanwhile, according to the simulation results, the acceleration curves of armatures with different speed in maximum nondestructive recovery medium density can be obtained, as shown in Figure 11.
It can be seen from Figure 11 that the maximum overload of the moment when the armature with different speed enters the corresponding maximum density medium is basically the same, about 8 × 105 g. Therefore, without considering other components of the ultra-high-speed projectile of the electromagnetic railgun, the overload limit that the U-shaped aluminum armature of the electromagnetic railgun can withstand in the recovery process is about 8 × 105 g. As long as the overload of the armature in the recovery process does not exceed this limit, the nondestructive recovery can be theoretically achieved.
4. Design of Nondestructive Recovery Scheme of Ultra-high-speed Armature for Electromagnetic Railgun
For the U-shaped armature with a muzzle velocity of 2500 m/s, and under the condition that its maximum overload is not more than 8 × 105 g, a four-stage recovery scheme is designed according to the nondestructive recovery principle. The parameters of the recovery scheme are shown in Table 3.
The recovery scheme consists of four sections: dust mist section, waterfall section, still water section, and submerged fiber section. In the dust mist section, the water mist is stirred by the air pump through the air duct and dispersed into the air to make the density of the recovered medium in this section reach 71 kg/m3. In the waterfall section, the waterfall surface is perpendicular to the movement direction of the armature, and the armature passes through each waterfall in turn. The thickness of the waterfall and the distance between the two adjacent waterfall surfaces are smaller than the length of the armature. As a special variable density medium, the thickness of the waterfall and the distance between adjacent waterfalls are controlled so that the equivalent density is about 300 kg/m3. In the still water section, the recovery medium is 1000 kg/m3. In the submerged fiber section, the water is filled with fibers, and the fiber distribution is gradually dense from front to back, with the densest distribution at the end.
The change of armature velocity with displacement in the recovery process is shown in Figure 12. As can be seen from the figure, when the maximum overload in the process of armature recovery is 8 × 105 g, the nondestructive recovery of the armature can be achieved within a distance of no more than 2 m.
The design of the nondestructive recovery scheme is based on the condition that the maximum overload of the armature in the recovery process is 8 × 105 g, and this overload is obtained from the critical deformation condition of the armature in the last section. Furthermore, it is the overload limit that the U-shaped aluminum armature can withstand. However, in the engineering application, the length of each recovery section should be extended to some extent.
5. Verification Experiment of the Nondestructive Recovery of Armature
In order to verify the validity of theoretical analysis and numerical simulation, a U-shaped armature recovery test system is constructed, and U-shaped armature recovery test is carried out. The U-shaped armature is launched by the electromagnetic railgun, and then it is recovered by the recovery test chamber. As to the limit of the pulsed power source of the railgun, the maximum muzzle velocity of the armature does not exceed 500 m/s. Therefore, according to the simulation results, water medium can be used for nondestructive recovery. To ensure the sealing of the water medium, the entrance of the armature launching into the test chamber is sealed with polyvinyl chloride plastic film, as shown in Figure 13 [16]. The inlet seal will bulge outward under water pressure.
A high-speed camera is used to capture the motion state of U-shaped armature in water and B probe is used to measure the muzzle velocity of the armature. According to the test, when the charging voltage of the pulsed power source of railgun is 7000 V, the velocity of the armature is about 452.3 m/s, and the motion state of the armature in the recovery medium is shown in Figure 14. As it is shown in Figure 14, affected by the launching and machining accuracy, when the high-speed armature moves to pierce the sealing film, it is difficult to accurately focus the central part of the spherical bulge of the film, so that the ballistic trajectory is deflected.
In order to verify the effectiveness of the test results, according to the simulation model established above, the initial velocity of the armature is set as 452.5 m/s, and other settings were consistent with the simulation model above. The displacement and velocity obtained by simulation are compared and analyzed with the data measured in the test, as shown in Figure 15. The test results are obtained from high-speed photography, and the position of the camera is the initial position in calculation, so the starting point of the speed is 160 m/s. It can be seen from Figure 15 that the change trend of the experimental results and the simulation results are basically consistent, and it verifies the validity of the theoretical analysis and numerical simulation.
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The armature before launch and after recovery is shown in Figure 16. It can be seen from Figure 16 that after recovery, the overall structure of the armature is preserved without obvious deformation. In addition, the surface ablation and melting morphology of the recovered armature are very well preserved, and no other damage is caused by the end effect. That is of great significance for the study of ultra-high-speed sliding electrical contact performance between the armature and the rails.
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6. Conclusion
Through theoretical analysis, numerical simulation, and experimental test, the nondestructive recovery technology of ultra-high-speed armature of electromagnetic railgun is studied. The results show that the mathematical model and finite element simulation model can be used for the design and analysis of nondestructive recovery scheme of ultra-high-speed projectile. The relationship between the initial velocity of the armature and the maximum density of recovery medium is given, and the overload limit of U-shaped aluminum armature for nondestructive recovery is obtained. In addition, the higher the armature speed, the lower the density of the recovery medium, the smaller the overload value of the armature, but the deceleration effect is not obvious, and the recovery distance is too long.
On the basis of the method of subsection variable density medium recovery, low-density medium is used in the front part and high-density medium is used for the rear part, the problem of too long distance or too large overload in single medium recovery can be solved to a certain extent. Furthermore, the nondestructive recovery of ultra-high-speed armature can be realized and good appearance of the armature can be maintained.
Data Availability
Both system mode parameters and simulation results used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by PLA Army Engineering University and Beijing University of Technology.