A method for predicting the superimposed distribution of flying space of multi-buried concave fragmentation bomb fragments

Abstract

The application provides a multi-projectile concave demolition bomb fragment scattering space superposition distribution prediction method, which describes the spatial distribution of fragments through the number and probability of fragments, and obtains the spatial distribution of a multi-projectile fragment field after superposition through single-projectile fragment field analysis. A large number of experiments show that the curve shape in the fragment space distribution test of the demolition bomb is similar to a normal distribution curve type, but it is also found that there are more fragment scattering distributions in the tail of the concave demolition bomb, which are mostly concentrated in 170-180 degrees, because there are more fragments at the connection between the cylindrical part and the concave part. The previous research on the fragment scattering formula ignores the fragments in the tail of the bomb, so the single-projectile fragment space distribution is modified, and the modified multi-projectile concave demolition bomb fragment scattering space superposition distribution prediction formula is obtained by combining the test data, and the prediction result has a small error compared with the test result.

Description

Technical Field

[0001] This invention belongs to the field of ammunition damage, and in particular, it is a method for predicting the spatial superposition distribution of fragments from multiple base-concave high-explosive fragmentation projectiles. Background Technology

[0002] When calculating the spatial distribution of fragments formed by multiple projectile fields of pre-fragmented warheads, the spatial distribution of fragments formed by a single projectile field is first analyzed. This spatial distribution can be described by the number and probability of fragments. When a pre-fragmented warhead explodes, each fragment deviates from the normal direction upon ejection. Taylor proposed a basic idea for predicting the characteristics of static fragment distribution, which Shapiro then applied in detail. Shapiro assumed that the warhead consists of a series of rings arranged continuously, with the centers of all rings located on the axis of symmetry of the projectile. The detonation wave originates from the detonation point and propagates outward in the form of a spherical wavefront. Previous experimental results show that the curve shape in previous warhead fragment spatial distribution experiments resembles a normal distribution curve.

[0003]

[0004] ρ N (θs) is the probability density of the number of fragment pairs in space, θ S Let θ0 be the spatial angle of the fragment at radius R, and θ0 be the spatial angle θ. S The mathematical expectation of σ is usually close to π / 2; σ is the space angle θ. S The mean squared error.

[0005] Experimental statistics revealed a significant amount of fragmentation at the tail end of the projectile, concentrated primarily at 170-180°. This is because the junction between the cylindrical section and the concave base contains more fragments, and previous fragmentation formulas did not consider fragments generated at the base of the projectile, leading to substantial errors in predicting the spatial distribution of fragments in concave-base grenades. Summary of the Invention

[0006] The purpose of this invention is to provide a method for predicting the spatial superposition distribution of fragments from multiple concave explosive fragmentation munitions, so as to achieve prediction of the spatial distribution of explosive fragmentation field.

[0007] The technical solution to achieve the purpose of this invention is as follows:

[0008] A method for predicting the spatial superposition distribution of fragments from a concave-shaped high-explosive anti-tank projectile, including the spatial distribution of multiple projectile fragments:

[0009]

[0010] Where N(θ) i ~θ i+1 ) represents the number of fragments, N1 and N2 represent the total number of fragments in the two projectiles respectively, ρ represents the distance between the projectiles, and θ represents the distance between the projectiles.i and θ i+1 Represents the lower and upper limits of the spatial angle of a projectile dispersion region; denoted by θ. j and θ j+1 θ represents the lower and upper limits of the spatial angle of the dispersion region of the other projectile, respectively; S Let θ0 be the spatial angle of the fragment at radius R, and θ0 be the spatial angle θ. S The mathematical expectation; σ is the spatial angle θ S The mean square error, where a and b are parameters introduced to fit the fragments at the tail end of the projectile; θ k and θ k+1 This represents the lower and upper limits of the spatial angle of a portion of the dispersion area of ​​multiple projectiles.

[0011] A method for predicting the spatial superposition distribution of fragments from a concave-shaped high-explosive anti-tank projectile, including the spatial distribution of single-projectile fragments:

[0012]

[0013] Where N(θ) i ~θ i+1 ) represents the number of fragments, N0 represents the total number of fragments formed by the entire projectile, and θ represents the number of fragments formed. i and θ i+1 θ represents the lower and upper limits of the spatial angle of the scattering region. S Let θ0 be the spatial angle of the fragment at radius R, and θ0 be the spatial angle θ. S The mathematical expectation; σ is the spatial angle θ S The mean square error, where a and b are parameters introduced to fit the fragments at the tail end of the projectile; θ k and θ k+1 This represents the lower and upper limits of the spatial angle of a portion of the dispersion area of ​​a single projectile.

[0014] The significant advantages of this invention compared to existing technologies are:

[0015] (1) Previous fragmentation formulas ignored the fragments generated at the tail of the projectile, while the fragmentation spatial distribution prediction formula of the present invention takes into account the fragments generated at the bottom concave of the tail of the projectile, thus reducing the error.

[0016] (2) The fragment dispersion spatial distribution prediction formula of the present invention can predict the spatial distribution of multiple fragment fields superimposed, and has wider applicability. Attached Figure Description

[0017] Figure 1 This is a flowchart for predicting the spatial superposition and distribution of fragments from multiple concave-bottomed explosive fragmentation munitions.

[0018] Figure 2 This is a schematic diagram of a multi-explosive static detonation test.

[0019] Figure 3The images on the witness board show the distribution of fragments. (a to d) are the distribution charts of fragments scattered at 60-120 degrees for a single projectile, 60-120 degrees for two projectiles, 60-120 degrees for three projectiles, and 170-180 degrees (tail end of the projectile), respectively.

[0020] Figure 4 This is a diagram showing the spatial dispersion distribution results of multiple projectile fragments. Detailed Implementation

[0021] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0022] When calculating the spatial distribution of fragments formed by multiple projectile fields of pre-formed fragments, first analyze the spatial distribution of fragments formed by a single projectile field, and then consider the superposition of fragment fields from multiple projectiles. Specific steps are as follows: Figure 1 As shown.

[0023] I. Prediction of Spatial Distribution of Single-Shot Fragments

[0024] 1. Calculate the probability density of the number of fragments in space:

[0025] Let the spatial angle θ S to θ S +dθ S The number of fragments within this sphere is dN, where dθ S Let N be a spatial angular infinitesimal element, and let N0 be the number of fragments formed by the entire projectile. Then, the probability density of the number of fragments in space is ρ. N (θ s )for:

[0026]

[0027] 2. Calculate the number density of the spherical solid angle per unit area of ​​the fragment:

[0028] Take the unit solid angle of the sphere as dΩ = sinθ s ·dθ s ·dΨ, where dΨ is the unit angle in the circumferential direction of the projectile, then the number density of the spherical solid angle of the fragment is g. N (θ s )for:

[0029]

[0030] Then we have:

[0031]

[0032] 3. Calculate the number of fragments in a given dispersion area:

[0033] Previous experimental results show that the curve shape in previous fragment spatial distribution experiments is similar to a normal distribution curve.

[0034]

[0035] θ S Let θ0 be the spatial angle of the fragment at radius R, and θ0 be the spatial angle θ. S The mathematical expectation of σ is usually close to π / 2; σ is the space angle θ. S The root mean square error of a certain scattering region θ i ~θ i+1 Number of fragments N(θ) i ~θ i+1 It can be determined using the following formula:

[0036]

[0037] Where θ i and θ i+1 This represents the lower and upper limits of the spatial angle of the scattering area.

[0038] II. Prediction of Spatial Distribution of Multiple Projectile Fragments

[0039] 1. Relative position coordinate transformation of multiple projectiles:

[0040] First, establish a Cartesian coordinate system, with the Y-axis representing the trajectory of the projectiles and the Z-axis representing the direction perpendicular to the ground. Let the origin be the center of mass of one projectile; then the relative coordinates of the other projectile are (Δx, Δy, Δz). Then, transform the Cartesian coordinate system into a cylindrical coordinate system, and we have...

[0041] Δx=ρcosθ

[0042] Δy=ρsinθ

[0043] Δz=z

[0044] Where ρ represents the distance between projectiles projected onto the ground, θ represents the azimuth angle between multiple projectiles, and z represents the distance between projectiles projected onto the Z-axis.

[0045] 2. Multi-projectile fragmentation calculation

[0046] Using θ i and θ i+1 Represents the lower and upper limits of the spatial angle of a projectile dispersion region; denoted by θ. j and θ j+1 N1 and N2 represent the lower and upper limits of the spatial angle of the dispersion area of ​​the other projectile, respectively; N1 and N2 represent the total number of fragments of the two projectiles, respectively.

[0047] (1) Multiple bullet fragment scenes do not stack.

[0048] When the relative polar coordinate distance between the centroids of multiple projectiles along the projectile axis is greater than the coverage angle of the fragmentation angle of a single projectile, the fragmentation fields of the multiple projectiles do not superimpose, and thus there is a spatial density distribution of multiple projectile fragments:

[0049]

[0050] or

[0051]

[0052] Where N(θ) i ~θ i+1 () represents the number of fragments in the other bullet.

[0053] (2) Multiple bullet fragmentation scenes superimposed

[0054] When the relative polar coordinate distance between the centroids of multiple projectiles along the projectile axis is less than the coverage angle of the fragmentation angle of a single projectile, the multiple projectile fragment fields are superimposed within a certain fragmentation angle range. Assuming the polar coordinate system of the first projectile is the reference system, the spatial density distribution of the multiple projectile fragments is as follows:

[0055]

[0056]

[0057] III. Correction of the Prediction Method for Fragmentation of Multiple Bullets with Concave Bottom

[0058] 1. Verification of the spatial distribution characteristics of multiple projectiles scattered

[0059] To study the spatial distribution characteristics of multiple projectiles, the following arrangements were made: Figure 2The static explosion test shown determines the spatial distribution of fragments by measuring the impact perforation points on the witness plate. The sample is placed on a wooden plank support with its center of mass 1.75m above the ground. The test was set up for single, double, and triple simultaneous detonation. Considering the accuracy of the 155mm base-concave fragmentation projectile's impact point, the spacing between multiple projectiles was set to 3m. The target plate was made of 3mm thick low-carbon steel, and the target material was 3mm Q235 steel. Each grid was 50cm x 50cm, and each steel plate was divided into 28 grids: 4 horizontal and 7 vertical. The target numbers decreased sequentially from left to right; each target corresponded to a 5.7-degree central angle with the detonation point as the center, and each grid corresponded to a 1.4-degree central angle. The projectile was aimed at steel plate #1, with the detonation point pointing towards the middle of targets 20-21, at a height of 1.75m, and the closest distance to the steel plate was 20m. Considering the accuracy of the dynamic CEP impact point, the spacing between multiple projectiles was set to 3m. In addition, a high-speed camera was placed 50m away from the blast point. The high-speed camera system was a Fastcam nltima APX model manufactured by Photron, with a shooting rate set to 24,000 frames / s during the test to capture the instantaneous velocity of fragments penetrating the target during the explosion-driven process. The requirement was to extract the perforation details of the target plate after multiple consecutive strikes. The test design is shown in the table below.

[0060]

[0061] The evolution of the penetration of a multiple-missile warhead into the target back under explosion-driven conditions was recorded by high-speed photography. The experimental results show that a significant flash was produced after the explosion. The flash initially intensified and then gradually weakened over time. The initial flash time was set to 0 ms. Furthermore, each high-speed photograph in the image was scaled proportionally, using the size of the Q235 target plate in the background as a reference, allowing for the measurement of the fragmentation direction.

[0062] Following the multi-shot impact test, the perforation of the Q235 steel target fragments was processed. Using the abrupt changes in the grayscale values ​​of pixels at the edges of the crater areas on the witness plate, image segmentation, target region identification, and region shape extraction were performed. Image-Pro Plus (IPP) and ImageJ image analysis software were used to differentiate the obtained crater contours on the witness plate, and natural fragment perforation and pre-formed fragment perforation were classified and statistically analyzed. The target plate was processed using Image-Pro to obtain a grayscale bitmap of fragment perforation. Pre-drawn area lines on the witness plate were used to divide and count the angles of the pre-formed fragment perforations. The impact situations under continuous dynamic multi-shot impacts during the test were statistically analyzed.

[0063] The number of fragments formed on the witness plate under the six experimental groups was statistically analyzed, and the experimental results are as follows: Figure 3As shown, the spatial distribution of fragment quantity basically conforms to the normal distribution curve of static fragment dispersion. Under single-projectile impact, the fragments formed by the warhead in the 19-23 range are relatively concentrated, with most fragments falling within this range. Under multiple-projectile detonation, the overlapping area in the 16-27 range is relatively dense, representing the spatial dispersion superposition area under continuous impact from multiple-projectile fragmentation fields. Figure 3 The spatial distribution characteristics of fragments formed by multiple projectiles were analyzed. Fragments formed by the warhead's dispersion angle on the Q235 target at 20m were mainly concentrated in the range of 82–103°, representing the fragmentation distribution characteristics of a single projectile. Fragments formed by multiple projectiles were mainly concentrated in the ranges of 71–113° and 59–124°, with the spatial overlap interval located between 59–124°, consistent with experimental data from a rectangular target. Figure 3 The consistency indicates that under the concentrated dispersion area of ​​warhead fragments, the spatial distribution of the multi-projectile fragment field is the sum of the distributions of multiple single-projectile fragment fields under their spatial positions, and the superposition relationship follows a normal distribution, which satisfies the superposition requirement and provides a way of thinking for the study of the continuous strike law of warhead multi-projectiles.

[0064] 2. Spatial distribution correction of multi-elastic fragments with concave base

[0065] Experimental statistics revealed that numerous fragments were scattered at the tip of the projectile, mostly concentrated at 170-180°. Figure 3 As shown in (d), this is because there are more fragments at the connection between the cylindrical part and the concave bottom, which is different from the fragment scattering formula studied previously. In addition, there is a case of multiple fragments superimposed at the connection. Therefore, based on the experimental results, the prediction formula for the spatial superposition distribution of multiple fragment fields is modified. It can be seen that the fragment distribution at the tail end of the projectile grows exponentially. By introducing parameters a and b to fit the fragments at the tail end of the projectile, the modified spatial distribution of single fragments is obtained as follows.

[0066]

[0067] Where θ k and θ k+1 This represents the lower and upper limits of the spatial angle of a portion of the dispersion area of ​​multiple projectiles.

[0068] When the relative polar coordinate distance between the centroids of multiple projectiles along the projectile axis is less than the coverage angle of the fragmentation angle of a single projectile, the multiple projectile fragment fields are superimposed within a certain fragmentation angle range. Assuming the polar coordinate system of the first projectile is the reference system, the spatial density distribution of the multiple projectile fragments is as follows:

[0069]

[0070]

[0071] Where θ k and θ k+1This represents the lower and upper limits of the spatial angle of a portion of the dispersion area of ​​multiple projectiles.

[0072] Substituting the experimental results into the above formula for fitting, we obtain parameters a = 0.73, b = 1.38, R0. 2 =0.9952, 9 iterations, residual sum of squares 0.72, DF value 6, mean square 0.12, fitting error 4.8%, meeting the model error requirements. Therefore, there is a prediction function for the spatial density distribution of multiple fragments:

[0073]

[0074]

[0075] The results of the prediction model calculation were compared with the statistical test results of the spatial dispersion of single and double projectile rectangular targets. The comparison results are as follows: Figure 4 As shown, excluding individual statistical errors, the prediction model calculation results are in good agreement with the experimental statistics. The calculated value of the spatial distribution of multiple fragments has an error of 10.8% compared with the experimental statistical value, which meets the prediction requirements.

Claims

1. A method for predicting the spatial superposition distribution of fragments from a concave-bottom explosive fragmentation projectile, characterized in that, Spatial distribution of multiple projectile fragments: in N1 and N2 represent the total number of fragments in the two projectiles, respectively, ρ represents the distance between the projectiles, and θ represents the number of fragments. i and θ i+1 Represents the lower and upper limits of the spatial angle of a projectile dispersion region; denoted by θ. j and θ j+1 These represent the lower and upper limits of the spatial angle of the dispersion area of ​​the other projectile, respectively; θ S Let θ0 be the spatial angle of the fragment at radius R, and θ0 be the spatial angle θ. S The mathematical expectation; σ is the spatial angle θ S The mean square error, where a and b are parameters introduced to fit the fragments at the tail end of the projectile; θ k and θ k+1 This represents the lower and upper limits of the spatial angle of a portion of the dispersion area of ​​multiple projectiles.

2. A method for predicting the spatial superposition distribution of fragments from a concave-bottom explosive fragmentation projectile, characterized in that, Spatial distribution of single-projectile fragments: in θ represents the number of fragments, N0 represents the total number of fragments formed by the entire projectile, and θ represents the number of fragments formed by the projectile. i and θ i+1 θ represents the lower and upper limits of the spatial angle of the scattering region. S Let θ0 be the spatial angle of the fragment at radius R, and θ0 be the spatial angle θ. S The mathematical expectation; σ is the spatial angle θ S The mean square error, where a and b are parameters introduced to fit the fragments at the tail end of the projectile; θ k and θ k+1 This represents the lower and upper limits of the spatial angle of a portion of the dispersion area of ​​a single projectile.

3. The method for predicting the spatial superposition distribution of fragments from a concave-shaped explosive fragmentation projectile according to claim 1 or 2, characterized in that, The spatial distribution of fragments was determined by measuring the impact perforation location of fragments on the witness plate through static explosion tests. The perforation of the target plate under continuous multiple bomb strikes was extracted, and the test results were substituted into the formula for fitting to obtain the values ​​of the introduced parameters.

4. The method for predicting the spatial superposition distribution of fragments from a concave-shaped explosive fragmentation projectile according to claim 1 or 2, characterized in that, Introduce parameters a=0.73, b=1.38.

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