Velocity Compensation in Shaped Charges

DEGREE PROJECT IN ENGINEERING SCIENCES, FIRST CYCLE, 15 CREDITS

STOCKHOLM, SWEDEN 2021

Velocity

Compensation in Shaped Charges

KTH Bachelor Thesis Report

Anmol Bhullar

Abstract

Shaped charges (SC) have been used as a means of explosives in military and civilian use for decades. Thus, there is a substantial amount of research behind this area. However, as this is a sensitive subject much of this research is not publicly available.

This thesis will look at how one can use asymmetries in SC’s to velocity compensate the jet formation. Velocity compensation is required when the SC is perpendicular to the projectile direction, hence, leading to an angled jet which decreases the penetration potential.

The asymmetries that were investigated are

• off­center detonation

• angled liner

• displaced wave shaper

• displaced wave shaper & angled liner.

The 3D explosive simulation was conducted in IMPETUS AFEA solver and to compare the performance of these asymmetries the position and velocity of the jet were measured. To create a baseline a simulation without any asymmetries was used.

The off­center detonation showed some velocity compensating characteristics at the tip of the jet. However, as the jet progressed it converged towards the reference.

Angled liner simulations were conducted with an angle of 0.5 degrees and 1 degree and these asymmetries behaved vastly differently. Angled Liner 0.5 degrees had a greater jet angle but a greater quantity of the jet particles were concentrated around one point increasing the penetration potential. A general characteristic that angled liner displaced was the fact that it had desirable velocity compensating traits all through the jet.

Displaced Wave Shaper, like off­center detonation, showed promising velocity compensating attributes at the tip of the jet, however, it too converged towards

the reference on the later part of the jet.

When combining the displaced wave shaper and angled liner asymmetries the desire was to also combine their velocity compensating traits, i.e achieving the displaced wave shaper’s tip compensation with the angled liner’s total compensation. Unfortunately, this was not achieved. The tip, again, showed promising velocity compensating attributes but the rest of the jet converged towards the reference.

Conclusively, angled liner shows the highest potential for compensating the velocity and allowed the most amount of jet particles to be concentrated around one point increasing the penetration potential.

Sammanfattning

Riktad sprängverkan eller RSV har använts som ett explosivt medel inom militärt och civilt bruk i decennier. Därför finns det också en del forskning inom området; men eftersom ämnet är känsligt så är mycket av den här forskningen hemligstämplad.

Det här kandidatexamensarbetet kommer titta på hur inbyggda asymmetrier i RSV:en kan hastighetskompensera jetstrålen. Hastighetskompensering är nödvändigt när själva RSVen är vinkelrät mot projektilens riktning, det här leder då en till en vinklad jetstråle som minskar penetrations potentialen.

Asymmetrierna som undersöks är

• snedtändning

• vinklad kona

• förflyttad vågformare

• förflyttad vågformare & vinklad kona.

3D­simuleringen utfördes i programmet IMPETUS AFEA solver och för att kunna göra en jämförelse mellan asymmetrierna mättes jetstrålens hastighet och position i X­,Y­,Z­riktning. För att skapa någon typ av referens utfördes också en simulering utan några inbyggda asymmetrier.

Snedtändningen visade på hastighetskompenserande egenskaper på toppen av jetstrålen men resten av jetstrålen betedde sig som referensen om inte sämre.

Simuleringarna med vinklad kona utfördes med 2 vinklar, 0.5 grader och 1 grad och dessa asymmetrier betedde sig märkvärt annorlunda. Vinklad kona 0.5 graders jetstråle hade en större vinkel men fler ut av partiklarna var koncentrerade kring en punkt vilket ökar penetrations potentialen. Genomgående för bägge vinklad kona simuleringarna var att de uppvisade en lovande hastighetskompenserande förmåga genom hela jetstrålen.

Förflyttad vågformare uppvisade lovande hastighetskompenserande förmågor vid toppen av jetstrålen men precis som snedtändning konvergerade resten av strålen

mot referensen.

Sedan så kombinerades förflyttad vågformare och vinklad kona just för att gifta deras egenskaper. Med det här så försökte man att förena vågformarens förmåga att hastighetskompensera vid toppen av jetstrålen med vinklad konas förmåga att hastighetskompensera resten av strålen. Tyvärr så uppstod inte det här scenariot.

Toppen av jetstrålen uppvisade bra hastighetskompenserande förmågor men resten av jetstrålen konvergerade mot referensen.

Sammanfattningsvis så kan man dra slutsatsen att av asymmetrierna som undersöktes så uppvisade vinklad kona störst potential. Vinklad kona hade en relativt liten vinkel samtidigt som en stor del av partiklarna var koncentrerade kring en punkt för att öka penetrations potentialen.

Acknowledgements

I would like to acknowledge our supervisor at SAAB Dynamics AB, Victor Björkgren who made this thesis possible and helped me throughout the whole process; without him this thesis would simply not been possible.

I would also like to acknowledge Ricardo Vinuesa who was our supervisor at KTH.

Ricardo supported us and gave us direction and without his help we would have been lost.

Author

Anmol Bhullar, bhullar@kth.se School of Engineering Sciences KTH Royal Institute of Technology

Examiner

Gunnar Tibert

School of Engineering Sciences KTH Royal Institute of Technology

Supervisor

Victor Björkgren

Specialist/Explosive Modelling SAAB Dynamics

Supervisor

Ricardo Vinuesa

School of Engineering Sciences KTH Royal Institute of Technology

Contents

1 Introduction 1

1.1 Background . . . 1

1.2 Problem Formulation . . . 3

1.3 Purpose . . . 4

2 Literature Review 5 2.1 Asymmetries in Earlier Reports . . . 5

3 Technical Background 7 3.1 Model Setup . . . 7

3.2 Theory . . . 8

4 Methodology 11 4.1 Workflow . . . 11

4.2 Analysis Parameters . . . 14

5 Results 15 5.1 Reference . . . 15

5.2 Angled Liner . . . 16

5.3 Off­center Detonation . . . 18

5.4 Wave Shaper Displaced 1 mm . . . 20

5.5 Wave Shaper & Angled Liner . . . 21

5.6 Compiled Results . . . 22

6 Conclusions 24 6.1 Discussion . . . 24

6.2 Compiled Conclusion . . . 25

6.3 Future Work . . . 26

References 27

1 Introduction

Shaped Charges (SC) have been in use for decades as a means of an explosion for military use but also for more civilian use as in mining. The concept of using a cylindrical shaped shell filled with a highly explosive material and a cavity is generally a well researched area among public and private institutions around the globe. However, as the subject is a highly sensitive one, this research is often hidden and only used internally.

1.1 Background

Figure 1.1: Shaped Charge

A SC consists of the elements above were it is traveling in the direction of the cone. A traditional SC takes this shape for several reasons, first and foremost these projectiles pierce through the air at high speeds and having an aerodynamic shape is vital to reach the target with the correct velocity. Further, a converging shape such as a cone allows for the force to be concentrated on a smaller area, thus increasing the pressure and penetration. The cavity in the explosive is often lined with a thin layer of copper which during impact deforms and creates a jet which can reach up to 10 km/s and reach peak pressures of 200 GPa.[5]

When the detonator is fired a pressure wave propagates as can be seen in figure 1.2, as this pressure reaches the liner in yellow the liner collapses towards the center axis of the liner cone under immense pressure. This collapse occurs from the rear to the front. Due to the fact that this pressure wave transcends the yield strength of the liner material this jet that gets formed by the deformed liner behaves like

Figure 1.2: Detonation wave without 1,2 & 6 in figure 1.1

an incompressible fluid. The course of this impact can be seen in figure 1.3.

Figure 1.3: Jet formation

1.2 Problem Formulation

Figure 1.4: Viper Shaped Charge

To limit the scope of this thesis we are only going to look at 3,4 & 5 in figure 1.1.

Thus the model in question is figure 1.4 were the dimensions are taken from the well established and generic viper shaped charge.

Figure 1.5: Perpendicular Shaped Charge

As can be seen in figure 1.1 the liner and explosive casing follow the symmetrical axis of the projectile, however, there are scenarios where the target is perpendicular to the symmetry axis of the projectile. In these scenarios the shaped charge is perpendicular to the projectile, see figure 1.5. Such a setup adds further

complications as in this scenario the projectile’s acceleration is perpendicular to the acceleration of the jet, increasing the risk of missing the target. Thus, the velocity of the jet has to be compensated to hit the target, hence the name of the thesis, velocity compensation.

1.3 Purpose

There are several paths one could take to compensate the velocity of the jet formation to insure that the jet hits the target. This thesis will look at how one could use asymmetries in the shaped charge to increase the likelihood of the jet in figure 1.3 penetrating the same target as the projectile is traveling in the perpendicular direction. Such asymmetries are interesting as they are easy to incorporate into the production of shaped charges.

2 Literature Review

As mentioned in the introduction the public material on this topic is limited hence this section is limited too. Further, the material related to the subject is either created experimentally or in 2D­models. Therefore, the experimental studies have a limited scope and the studies conducted in 2D­models are limited in nature.

2.1 Asymmetries in Earlier Reports

The studies researching asymmetries are conducted for different reasons; some are done to better understand how manufacturing defects affect the performance of the SC and other to understand how planned asymmetries affect performance.

No reports could be found on asymmetries in SC’s which needed to be velocity compensated, nonetheless, the reports found were still interesting to study to better understand what performance criteria and methodologies other researches used.

2.1.1 The influence of asymmetries in shaped charge performance

In this report authored by O. Ayisit a 3D­model was setup in AUTODYN where different asymmetries were simulated with a similar SC with a trumpet copper liner. In this paper, asymmetries such as off­center detonation where trialed and compared to air bubbles in the highly explosive among other asymmetries.

However, the asymmetry of interest is off­center detonation as this is going to be trialed in this thesis too. The conclusions the author seems to draw is that the greater the distance between the center­axis and detonation point the more the jet drifts. The author does not try to create some link as if the drift is linear or a polynomial to the off­axis distance but does acknowledge that as the distance increases so does the negative affects on the jet formation. [1]

2.1.2 Experimental Study of Shaped Charges with Built­in Asymmetry In contrast to the prior study this one was conducted experimentally where the off­

axis detonations where examined with the help of radiographs. This experiment

solely focused on off­center detonations and here too the authors made the observation that a greater off­set led to a greater off­axis velocity. They tried to create a relationship between the off­axis distance and jet angle, however, the authors are quick to realize that the relationship is fraud and merely mentionable in an ideal scenario. [2]

3 Technical Background

3.1 Model Setup

The model setup in IMPETUS begins with defining the geometries, this is done with the help of the STL­parts imported from ANSYS. The assembly of the simulation can be seen in figure 3.1. The green box is the sensor box which records

• Position X,Y,Z

• Velocity X,Y,Z

of the jet formation in figure 1.3.

Figure 3.1: Assembly of simulation

3.1.1 Material Parameters

Further on, material parameters are established for the copper­liner and the highly explosive material. To define the material parameters for the liner a Johnson­Cook flow stress constitutive model is built. However, such a model is purely empirical and defines the flow stress as the liner collapses. The Johnson­

Cook model is a well used model even though it still has its short comings. It is renowned for having a small­strain rate dependence at high temperatures. See appendix A.2 for complete list of material parameters.

3.1.2 Equation of State

In simulations with high strain rates and propagating shock waves in combination with pressures that exceed material yield strengths an equation of state (EOS) has to be set. An EOS determines the pressure as a function of energy/temperature.

There are of course different EOS’s and in this thesis two different equation of states were created, one for the highly explosive and one for the liner.

The relationship between pressure and volume for the highly explosive material is established with the help of a Jones Wilkins­ Lee EOS which is commonly used when modeling explosives.

To model the pressure wave in the copper liner the Mie–Grüneisen equation of state was used. This EOS is regularly used when modeling the collapse of the liner in a shaped charge.

3.1.3 Smoothed­Particle Hydrodynamics

Smoothed­particle hydrodynamics or SPH are commonly used in the fields solid mechanics and fluid mechanics to numerically model the mechanical behaviour of continuum mediums. SPH utilizes the Lagrangian method, therefore, instead of discretizing the control volume we discretize the mass of each particle. Due to this property among others SPH is a suitable method to model high deformation scenarios which is the case with ballistics.[4]

3.2 Theory

3.2.1 The Hydrodynamic Sequence

As was mentioned in the introduction the shock wave that propagates through the high explosive forces the liner to collapse towards the symmetry axis of the SC. However, the velocity of this collapse depends on the mass ratio between the liner and the high explosive; hence, this velocity varies along the symmetry axis.

Due to this difference in collapse velocity the jet formation in figure 1.3 has a velocity gradient from the top to the bottom which stretches the jet. Therefore, as a result of the velocity of the jet the initial impact with the target can be seen

as a hydrodynamic process where the target, usually armour and the jet behave as fluids.

3.2.2 Mechanics of Simulation

Figure 3.2: Top view of Simulation Assembly

As can be seen in figure 3.2 the velocities form a composant which has a 45 degree angle with the initial velocity direction. Thus, to compensate for this velocity composant the jet direction, which is the only velocity that can be altered is tilted towards the second quadrant. This is done in numerous ways by affecting the symmetry of the liner and high explosive assembly. One obvious method would be to angle the liner into the second quadrant which is one among many asymmetries which has been trailed, see figures in A.1 for a clearer understanding.

3.2.3 Wave Shaper

The purpose of the wave shaper is to to increase the jet velocity in figure 1.3 and this is achieved by having a perpendicular pressure wave to the liner, see figure 3.3. Thus, with the inclusion of the wave shaper as can be seen in appendix A.1 the detonation wave has to move around a rigid wave shaper and reach the liner perpendicularly. With a successful application one can increase the jet tip velocity by up to 2 km/s hence increasing the penetration ability of the SC. [3]

Figure 3.3: Pressure wave with wave shaper

4 Methodology

These asymmetries will be simulated in the 3D­model simulation program IMPETUS from a STL­model created in ANSYS Spaceclaim. The asymmetries that will be implemented are

• Angled Liner I. 0.5 degrees II. 1 degrees

• Off­center Detonation I. 1 mm

II. 2 mm

• Wave shaper

• Wave shaper displaced 1 mm

• Wave shaper displaced 1 mm & Angled Liner 0.5 deg.

For a better understanding of these asymmetries see appendix A.1.

4.1 Workflow

The general workflow of one simulation is presented in figure 4.1.

Figure 4.1: Flowchart of the work process; black ­ step in flow,green ­ software used.

4.1.1 Model Design

By far the most time­consuming step of the process was the modelling and assembly of the liner and casing of highly explosive seen in figure 1.4. This CAD model was first designed in Siemens NX with the help of an already existing model and then exported to Ansys SpaceClaim where the liner and highly explosive casing was assembled and the different asymmetries were built­in.

4.1.2 Pre­Processing

The pre­processing was the most crucial part of the simulation as this is where the model was set up and most of the debugging occurred. A lot of time was spent on this step to insure that the model was accurate but at the same time efficient. To insure this tools such as symmetry planes and simulation termination were used throughout the simulations. Nonetheless, to obtain accurate results on asymmetries that were not symmetrical in any plane the symmetry planes had to be scrapped which resulted in drastically longer simulation time.

In most traditional simulation programs a mesh is usually set­up with the help of a 3D geometry, but in IMPETUS the 3D­geometry is instead defined with the help of smoothed­particle hydrodynamics or SPH. For a deeper understanding, see section 3.1.3.

Some general and important model parameters are presented in table 4.1. Stand off is the distance between the sensor and the SC in figure 3.1. The simulation length was set after the particle speed through the sensor; as the jet looses speed it looses it’s ability to penetrate and speeds lower then 2 km/s are not interesting to study. Hence, the simulation was set to last until the jet speed was approximately 2 km/s.

Parameters Value

SPH Particle Distribution Distance 0.3 mm

Detonation Velocity 8800 m/s

Stand­off 0.5 m

Initial Velocity 300 m/s

Simulation Time 0.26775 ms

Table 4.1: Model Parameters

4.1.3 Post­Processing

Post­Processing was of course a vital part of the simulation. First and foremost figures akin 1.3 were examined to verify that the model was accurate and did not behave in an unexpected manner were the jet splits or behaves in a asymmetric fashion. Furthermore, the post­processing tool was used to visualize the results,

for example the pressure wave in figure 3.3 was incorrect at first try but with the help of this visualization tool this could be quickly corrected.

4.1.4 Analysis

All the data for the analysis was collected with the help of the sensor(green rectangle) in figure 3.1. The data it collected was exported into an Excel­sheet were it later formed the basis of the analyses. As can be seen in section 3.1 the number of output data is not a lot, nonetheless it is enough to form an understanding regarding which asymmetries help compensate the velocity. The data was then imported intoPython with the help of Pandas, manipulated with the help of NumPy and visualised with the help ofmatplotlib.

4.2 Analysis Parameters

Usually the performance of a SC is measured through it’s penetration capability, P = L^√_ρ^ρjet

target where L is the length of the jet and ρ is the density [5]. But, in this thesis it’s not the penetration capability that is going to be compared between the different asymmetries even though one could draw conclusions about it from the results that are going to be presented. In this thesis the velocities of the jet in the x­direction(the direction the of the projectile) are going to be compared to understand which asymmetry drifts the most. The discussion is going to be further nuanced by comparing the distance δ from the first point of impact to all the points of impact that follow to better understand how far from the original impact the jet travels. This is described in equation 1.

δ =|x1− xi| for i = 2, 3... (1)

Were x₁is the first particle to penetrate the sensor and x_iare all the particles that follow.

As can be seen in section 3.1 some asymmetries are the same but with different angles or distances; to better compare these domestic asymmetries a histogram of their distance to the first penetration, equation 1 is going to be plotted. To understand how everything was calculated see appendix B.2.

5 Results

5.1 Reference

Reference, that is without any asymmetries.

0.000075 0.000100 0.000125 0.000150 0.000175 0.000200 0.000225 0.000250 0.000275 Time [s]

0.00 0.01 0.02 0.03 0.04 0.05 0.06

|x| [m]

Reference

Figure 5.1: Distance to First Point of Impact

Figure 5.2: Jet Formation of the Reference Simulation

5.2 Angled Liner

0.00010 0.00015 0.00020 0.00025

Time [s]

0.000 0.005 0.010 0.015 0.020

|x| [m ]

Angled Cone 1 deg Angled Cone 0.5 deg

Figure 5.3: Distance to First Point of Impact

0.000 0.005 0.010 0.015 0.020 Distance to first point of impact [m]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Pro ba bil ty

1 deg 0.5 deg

Figure 5.4: Angled Liner, histogram of the distance to the first point of impact

Figure 5.5: Jet Formation of Angled Liner 0.5 deg

Figure 5.6: Jet Formation of Angled Liner 1 deg

5.3 Off­center Detonation

0.00010 0.00015 0.00020 0.00025

Time [s]

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

|x| [m ]

Off-axis Det. 1 mm Off-axis Det. 2 mm

Figure 5.7: Distance to First Point of Impact

Figure 5.8: Jet Formation of Off­center Detonation 1 mm

Figure 5.9: Jet Formation of Off­center Detonation 2 mm

5.4 Wave Shaper Displaced 1 mm

0.00005 0.00010 0.00015 0.00020 0.00025

Time [s]

0.00 0.02 0.04 0.06 0.08

|x| [m ]

Wave Shaper, displaced 1 mm

Figure 5.10: Distance to First Point of Impact

Figure 5.11: Jet Formation of Displaced Wave Shaper 1 mm

5.5 Wave Shaper & Angled Liner

0.00005 0.00010 0.00015 0.00020 0.00025

Time [s]

0.00 0.01 0.02 0.03 0.04

|x| [m ]

Wave Shaper, displaced 1 mm & Angled Cone 0.5 deg

Figure 5.12: Distance to First Point of Impact

Figure 5.13: Jet Formation of Displaced Wave Shaper 1 mm & Angled Liner 0.5 deg

5.6 Compiled Results

0.00000 0.00005 0.00010 0.00015 0.00020 0.00025

Time [s]

−200

−100 0 100 200 300

Vel city [m/s]

Reference Off-axis Det. 1 mm Off-axis Det. 2 mm Angled Cone 1 deg Angled Cone 0.5 deg Displaced Wave Shaper 1 mm

Displaced Wave Shaper 1 mm & Angled Cone 0.5 deg

Figure 5.14: Velocity in the X­direction

0.00005 0.00010 0.00015 0.00020 0.00025 Time [s]

0.00 0.02 0.04 0.06 0.08

|x| [m]

Reference Off-axis Det. 1 mm Off-axis Det. 2 mm Angled Cone 1 deg Angled Cone 0.5 deg

Wave Shaper, displaced 1 mm

Wave Shaper, displaced 1 mm & Angled Cone 0.5 deg

Figure 5.15: Distance to First Point of Impact

Asymmetry RMS Value [m/s] Percentage of Reference

Reference 255.25 100%

Off­axis 1 mm 238.81 93.56%

Off­axis 2 mm 218.69 85.68%

Angled Liner 0.5 deg 120.60 27.61%

Angled Liner 1 deg 70.47 47.25%

Waveshaper 1 mm 252.72 99.00%

WS 1 mm & AL 0.5 deg 105.43 41.3%

Table 5.1: Values over RMS velocity in the X­direction

6 Conclusions

6.1 Discussion

6.1.1 Angled Liner

At first glance when analysing the jet formation in figure 5.6 it was noted that the velocity component probably dipped into the negative digits and this was confirmed in figure 5.14. This was at first thought to be a negative trait but when figure 5.3 was compiled and analyzed the conclusion that this trait should be utilized but to a lesser degree was done. Hence, a simulation with angled liner 0.5 degree was conducted.

Comparing figure 5.5 and 5.6 one can see that the bow of the jet formation is less aggressive for the 0.5 degree liner, as expected. The 0.5 deg configuration did not lead to a shorter distance to first point of impact as one can see in figure 5.3.

Nonetheless, as can be seen in figure 5.4 the probability of the particles hitting closer to the first point of impact is higher for angled liner 0.5 deg, hence more energy is concentrated around one point increasing penetration potential. The opposite can be said about angled liner 1 deg where the probability density of the particles in the jet seem to be concentrated around the point furthest away from first point of impact.

6.1.2 Off­center Detonation

Both off­center detonations seemed to have decent velocity compensation at the tip of the jet as can be seen in figure 5.7. However, as can be seen in figure 5.15 the rest of the jet shows the same linear behaviour as the reference but the final point of impact is greater then the reference which is not promising.

When comparing the jet tips of off­center detonation in figure 5.8 with the jet tip of for example angled liner in figure 5.5 one can see that the spread of the SPH particles is higher in the case of the off­center detonation. This can potentially be an indication of a poor simulation where the SPH particle distribution distance in table 4.1 needs to be lowered.

6.1.3 Wave Shaper Displaced 1 mm

The displaced wave shaper seems to behave in a similar fashion as the off­center detonation were the tip of the jet formation has decent velocity compensating abilities but the rest of the jet has the same linear|x(t)| behaviour as the reference.

Looking at the velocity graph in figure 5.14 the displaced wave shaper shows some potential as the curve behaves as a log function; however, it reaches the initial velocity of 300 m/s so quickly that it’s RMS value is still 99% of the reference as can be seen in table 5.1.

6.1.4 Wave Shaper & Angled Liner

The combination of a wave shaper and angled liner was tested because of the wave shaper’s ability to velocity compensate at the tip of the jet and the angled liner’s ability to velocity compensate through the rest of the jet.

As can be seen in figure 5.14 the combination of WS & AL has almost superpositioned with wave shaper when looking at the beginning of the impact but as the time progresses the x­velocity converges towards angled liner 0.5 deg.

Hence, the conclusion that when t is small(beginning of simulation) the WS &

AL behaves as a wave shaper but as the simulation goes on it behaves as the angled liner. The same however, cannot be said about the distance to first point of impact.

When analysing figure 5.15 one can see that WS & AC follows wave shaper for small t and as the simulation progresses it follows angled cone 0.5 deg but half way through the simulation it diverges and shows a linear pattern as the reference or the wave shaper.

Hence, the result was promising from a velocity point of view but disappointing from a impact radius point of view.

6.2 Compiled Conclusion

When comparing the different asymmetries trialed in this thesis it is quite blunt that the asymmetry that shows most potential is the angled liner, more specifically angled liner 0.5 deg. This can be seen when one compares the velocity data in

figure 5.14 & table 5.1; this is also consolidated by the data in figure 5.15. Hence, the facts are conclusive.

6.3 Future Work

To complete this work one could actually conduct real world experiments to validate these results. This could be done in similar fashion to the report [2]

were one uses radiography to understand the X­direction velocity of the jet.

Nonetheless, with this method all the validation would have to occur through the comparison of velocity and not with figure 5.15. On the other hand one could with the data in this report generate for example the angle between the first point of impact and compare that to the same angle in the real life experiments.

References

[1] Ayisit, O. “The influence of asymmetries in shaped charge performance”.

In: International Journal of Impact Engineering (2008). DOI: https : / / doi . org / 10 . 1016 / j . ijimpeng . 2008 . 07 . 027. URL: https : / / www . sciencedirect.com/science/article/abs/pii/S0734743X0800184X.

[2] John Brown I.D Softley, P.Edwards. “Experimental Study of Shaped Charges with Built­in Asymmetries”. In: (1993).

[3] Kurt Andersson Stefan Axberg, etc. Lärobok i Militär, vol 4: Verkan och Skydd. Försvarshögskolan, 2009. ISBN: 9789189683082.

[4] Stanic, Milos. The Smoothed Particle Hydrodynamics Method vs. Finite Volume Numerical Methods. URL: https://www.altair.com/newsroom/

articles/the- smoothed- particle- hydrodynamics- method- vs- finite- volume-numerical-methods/. (accessed: 2016.05.04).

[5] W.P Walters, J.A Zukas. Fundamentals of Shaped Charges. A Wiley­

Interscience Publication, 1989. ISBN: 9780471621720.

Appendices

Appendix ­ Contents

A First Appendix 30

A.1 Asymmetries . . . 30 A.2 Material parameters . . . 32

B Second Appendix 33

B.1 Pre­Processing Code . . . 33 B.2 Python Analyses code . . . 36

A First Appendix

A.1 Asymmetries

Figure A.1: Angled Liner

Figure A.2: Off­Center Detonation (Not to scale)

Figure A.3: Displaced Wave Shaper

A.2 Material parameters

LX­14 Value Unit

Density 1821 kg/m³

Linear Bulk Modulus 2.2 GP a

Viscosity 0 P as

Pressure cut­off −1 f P a

Copper Value Unit

Density 8940 kg/m³

Young’s Modulus 120 GP a Poission’s ratio 0.345 −

B Second Appendix

B.1 Pre­Processing Code

# V e l o c i t y Compensation

#

# R e f e r e n c e s i m u l a t i o n WITHOUT asymmetri

#

#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

*

^PARAMETER

dx = 0.3 e−3 # P a r t i c l e d i s t r i b u t i o n D = 8800.0 #Detonation v e l o c i t y

y d e t = 0.0 #Detonation pos . z d e t = 0.0 #Detonation pos . s t a n d _ o f f = 0.5 #m

v _ i n t = 300 #m/ s #Shaped charge i n t . v e l

t_end = [% s t a n d _ o f f ]/2000 + 1 . 7 7 5 e−05 #2000 m/ s i s t he end o f the j e t , 1 . 7 7 5 e−5 s i s approx the time i t t a k e s f o r the j e t t o form #S i m u l a t i o n d u r a t i o n

t_death = 2 . 1 e−05 #When t he LX−14 i s t e r m i n a t e d from the s i m u l a t i o n

#

#

#

*

UNIT_SYSTEM SI

*

INCLUDE #I n c l u d e s mesh o f shaped charge viper_shaped_charge . k

1 . 0 e −2 , 1 . 0 e −2 , 1 . 0 e−2

*

^TIME

[%t_end ]

*

^PART

”HE v o l ” #D e f i n e s HE as r i g i d 1 , 10

” L i n e r Vol ” # D e f i n i e s the l i n e r as r i g i d 2 , 10

” Lx 1 4 ” #D e f i n e s the p a r t i c l e s and t h e r e p a r t i d 1 1 , 1 1 , 1 1

” Copper L i n e r ” 1 2 , 1 2 , 12

#

#

*

MAT_FLUID #Def . Lx−14 mat . parametrar

” Lx −14”

1 1 , 1 8 2 1 . 0 , 2 . 2 e9 , 0 . 0 , −1.0 e15

*

MAT_JC #Def . koppar−mat . parametrar

” copper ”

1 2 , 8940.0 , 120e9 , 0.345 350e6

*

EOS_JWL #Equation o f s t a t e − Jones Wilkins − Lee f o r e x p l o s i v e s

1 1 , 826.1 e9 , 1 7 . 2 4 e9 , 4 . 5 5 , 1 . 3 2 , 0 . 3 8 , 1 0 . 2 e9

#

#

*

DETONATION_POINT_COORDINATE #P o i n t o f d e t o n a t i o n 0 . 0 , %ydet , %zdet , 0 . 0 , %D

#

#

*

EOS_GRUNEISEN #Equation o f s t a t e , r e l a t e s p r e s s u r e t o volume t o desc . the copper l i n e r

1 2 , 1 . 4 8 9 , 1 . 9 9

#

#

*

MAT_RIGID #D e f i n e s t he l i n e r & and HE as a s o l i d 10 , 7800

*

SPH #D e f i n e s the p a r t i c l e s i n t he l i n e r and th e HE

1 1 , 1 1 , 1.0

*

^%dx

1 2 , 1 2 , 1.0

*

^%dx

#

#

*

ACTIVATE_ELEMENTS #Hides th e HE a f t e r a s e t amount o f time 1 , G, 1 1 , 0 . 0 , [%t_death ]

#

#

*

GEOMETRY_PART # Read desc . 1 1

1

*

GEOMETRY_PART 12

2

#

#

*

SET_PART #C r e a t i n g a s e t p a r t so t h a t th e shaped charge can be g i v e n a v _ i n t

” Combining l i n e r & HE”

1122 1 , 2

*

INITIAL_VELOCITY #The shaped charge i s g i v e n an i n t i a l v e l .

PS , 1 1 2 2 , [ % v _ i n t ]

*

INITIAL_VELOCITY G, 1 1 , [ % v _ i n t ]

*

INITIAL_VELOCITY G, 1 2 , [ % v _ i n t ]

#

#

*

GEOMETRY_BOX #Drawn i n assemble

” Sensor Box ” 99

−0.05 , [% s t a n d _ o f f ] , −0.05 , 0.25 ,[% s t a n d _ o f f ] + 0 . 0 1 , 0.05

*

SPH_SENSOR_STATE #Sensor output 1 , 9 9 , 0

*

^END

B.2 Python Analyses code

B.2.1 Analysis of Velocity

import m a t p l o t l i b . p y p l o t as p l t import numpy as np

import pandas as pd

r e f = pd . r e a d _ e x c e l ( ’ R e f e r e n c e . x l s x ’ , )

off_ax_1mm = pd . r e a d _ e x c e l ( ’ Off − a x i s Detonation 1 mm. x l s x ’ ) off_ax_2mm = pd . r e a d _ e x c e l ( ’ Off − a x i s Detonation 2 mm. x l s x ’ )

angl_1 = pd . r e a d _ e x c e l ( ’ Angled Cone 1 deg . x l s x ’ ) angl_05 = pd . r e a d _ e x c e l ( ’ Angled Cone 0.5 deg . x l s x ’ )

disp_1mm = pd . r e a d _ e x c e l ( ’ D i s p l a c e d Cone 1 mm. x l s x ’ ) disp_05mm = pd . r e a d _ e x c e l ( ’ D i s p l a c e d Cone 0.5 mm. x l s x ’ )

WS_disp1 = pd . r e a d _ e x c e l ( ’ WS_disp1 . x l s x ’ )

WSAC = pd . r e a d _ e x c e l ( ’WS+AC . x l s x ’ )

RMS_refX = np . s q r t ( np . mean ( np . power ( r e f [ ’ V e l o c i t y X ’ ] , 2 ) ) )

RMS_off_ax1mm_X = np . s q r t ( np . mean ( np . power ( off_ax_1mm [ ’

RMS_off_ax2mm_X = np . s q r t ( np . mean ( np . power ( off_ax_2mm [ ’ V e l o c i t y X ’ ] , 2 ) ) )

RMS_angl1 = np . s q r t ( np . mean ( np . power ( angl_1 [ ’ V e l o c i t y X ’ ] , 2 ) ) )

RMS_angl05 = np . s q r t ( np . mean ( np . power ( angl_05 [ ’ V e l o c i t y X ’ ] , 2 ) ) )

RMS_disp1mm = np . s q r t ( np . mean ( np . power ( disp_1mm [ ’ V e l o c i t y X ’ ] , 2 ) ) )

RMS_disp05mm = np . s q r t ( np . mean ( np . power ( disp_05mm [ ’ V e l o c i t y X ’ ] , 2 ) ) )

RMS_WSdisp1mm = np . s q r t ( np . mean ( np . power ( WS_disp1 [ ’ V e l o c i t y X ’ ] , 2 ) ) )

RMS_WSAC = np . s q r t ( np . mean ( np . power (WSAC[ ’ V e l o c i t y X ’ ] , 2 ) ) )

print ( RMS_off_ax1mm_X/RMS_refX ) print (RMS_off_ax2mm_X/RMS_refX )

print ( RMS_angl1/RMS_refX ) print ( RMS_angl05/RMS_refX )

print (RMS_disp1mm/RMS_refX ) print (RMS_disp05mm/RMS_refX )

print (RMS_WSdisp1mm/RMS_refX )

print (RMS_WSAC/RMS_refX )

p l t . p l o t ( r e f [ ’ Time ’ ] , r e f [ ’ V e l o c i t y X ’ ] , l a b e l = ” R e f e r e n c e

” )

p l t . p l o t ( off_ax_1mm [ ’ Time ’ ] , off_ax_1mm [ ’ V e l o c i t y X ’ ] , l a b e l = ” Off − a x i s Det . 1 mm” )

p l t . p l o t ( off_ax_2mm [ ’ Time ’ ] , off_ax_2mm [ ’ V e l o c i t y X ’ ] , l a b e l = ” Off − a x i s Det . 2 mm” )

p l t . p l o t ( angl_1 [ ’ Time ’ ] , angl_1 [ ’ V e l o c i t y X ’ ] , l a b e l = ” Angled Cone 1 deg ” )

p l t . p l o t ( angl_05 [ ’ Time ’ ] , angl_05 [ ’ V e l o c i t y X ’ ] , l a b e l = ” Angled Cone 0.5 deg ” )

p l t . p l o t ( disp_1mm [ ’ Time ’ ] , disp_1mm [ ’ V e l o c i t y X ’ ] , l a b e l =

” D i s p l a c e d Cone 1 mm” )

p l t . p l o t ( disp_05mm [ ’ Time ’ ] , disp_05mm [ ’ V e l o c i t y X ’ ] , l a b e l

= ” D i s p l a c e d Cone 0.5 mm” )

p l t . p l o t ( WS_disp1 [ ’ Time ’ ] , WS_disp1 [ ’ V e l o c i t y X ’ ] , l a b e l =

” D i s p l a c e d Wave Shaper 1 mm” )

p l t . p l o t (WSAC[ ’ Time ’ ] , WSAC[ ’ V e l o c i t y X ’ ] , l a b e l = ” D i s p l a c e d Wave Shaper 1 mm & Angled Cone 0.5 deg ” )

p l t . x l a b e l ( ’ Tid [ s ] ’ )

p l t . y l a b e l ( ’ H a s t i g h e t [m/ s ] ’ ) p l t . t i t l e ( ’X− r i k t n i n g ’ )

p l t . legend ( ) p l t . show ( )

B.2.2 Analysis of Position

import m a t p l o t l i b . p y p l o t as p l t import numpy as np

import pandas as pd

r e f = pd . r e a d _ e x c e l ( ’ R e f e r e n c e . x l s x ’ , )

off_ax_1mm = pd . r e a d _ e x c e l ( ’ Off − a x i s Detonation 1 mm. x l s x ’ ) off_ax_2mm = pd . r e a d _ e x c e l ( ’ Off − a x i s Detonation 2 mm. x l s x ’ )

angl_1 = pd . r e a d _ e x c e l ( ’ Angled Cone 1 deg . x l s x ’ ) angl_05 = pd . r e a d _ e x c e l ( ’ Angled Cone 0.5 deg . x l s x ’ )

disp_1mm = pd . r e a d _ e x c e l ( ’ D i s p l a c e d Cone 1 mm. x l s x ’ ) disp_05mm = pd . r e a d _ e x c e l ( ’ D i s p l a c e d Cone 0.5 mm. x l s x ’ )

WS_disp1 = pd . r e a d _ e x c e l ( ’ WS_disp1 . x l s x ’ )

WSAC = pd . r e a d _ e x c e l ( ’WS+AC . x l s x ’ )

ref_nonz = np . a r r a y ( np . where ( r e f [ ’ P o s i t i o n X ’ ]!=0 ) ) # Indexes o f non−zero v a l u e s

offax1_nonz = np . a r r a y ( np . where ( off_ax_1mm [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

offax2_nonz = np . a r r a y ( np . where ( off_ax_2mm [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

angl1_nonz = np . a r r a y ( np . where ( angl_1 [ ’ P o s i t i o n X ’ ]!=0 ) )

#Indexes o f non−zero v a l u e s

angl05_nonz = np . a r r a y ( np . where ( angl_05 [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

disp1_nonz = np . a r r a y ( np . where ( disp_1mm [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

disp05_nonz = np . a r r a y ( np . where ( disp_05mm [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

WSdisp1_nonz = np . a r r a y ( np . where ( WS_disp1 [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

WSAC_nonz = np . a r r a y ( np . where ( WSAC[ ’ P o s i t i o n X ’ ]!=0 ) ) # Indexes o f non−zero v a l u e s

# C a l c u l a t i n g t h e d i s t a n c e between each p a r t i c l e t o t h e f i r s t p a r t i c l e i n t h e x− d i r e c t i o n

d i f f _ r e f = np . a r r a y ( np . a b s o l u t e ( r e f . i l o c [ ref_nonz [ 0 , 0 ] ] [ 1 ]

− r e f . i l o c [ ref_nonz [ 0 , 0 ] : ref_nonz [ 0 , − 1 ] , 1 ] ) )

d i f f _ o f f a x 1 = np . a r r a y ( np . a b s o l u t e ( off_ax_1mm . i l o c [ offax1_nonz [ 0 , 0 ] ] [ 1 ] − off_ax_1mm . i l o c [ offax1_nonz [ 0 , 0 ] : offax1_nonz [ 0 , − 1 ] , 1 ] ) )

d i f f _ o f f a x 2 = np . a r r a y ( np . a b s o l u t e ( off_ax_2mm . i l o c [ offax2_nonz [ 0 , 0 ] ] [ 1 ] − off_ax_2mm . i l o c [ offax2_nonz [ 0 , 0 ] : offax2_nonz [ 0 , − 1 ] , 1 ] ) )

d i f f _ a n g l 1 = np . a r r a y ( np . a b s o l u t e ( angl_1 . i l o c [ angl1_nonz [ 0 , 0 ] ] [ 1 ] − angl_1 . i l o c [ angl1_nonz [ 0 , 0 ] : angl1_nonz [ 0 , − 1 ] , 1 ] ) )

d i f f _ a n g l 0 5 = np . a r r a y ( np . a b s o l u t e ( angl_05 . i l o c [ angl05_nonz [ 0 , 0 ] ] [ 1 ] − angl_05 . i l o c [ angl05_nonz [ 0 , 0 ] : angl05_nonz [ 0 , − 1 ] , 1 ] ) )

d i f f _ d i s p 1 = np . a r r a y ( np . a b s o l u t e ( disp_1mm . i l o c [ disp1_nonz [ 0 , 0 ] ] [ 1 ] − disp_1mm . i l o c [ disp1_nonz [ 0 , 0 ] : disp1_nonz

[ 0 , − 1 ] , 1 ] ) )

d i f f _ d i s p 0 5 = np . a r r a y ( np . a b s o l u t e ( disp_05mm . i l o c [

disp05_nonz [ 0 , 0 ] ] [ 1 ] − disp_05mm . i l o c [ disp05_nonz [ 0 , 0 ] : disp05_nonz [ 0 , − 1 ] , 1 ] ) )

diff_WSdisp1 = np . a r r a y ( np . a b s o l u t e ( WS_disp1 . i l o c [ WSdisp1_nonz [ 0 , 0 ] ] [ 1 ] − WS_disp1 . i l o c [ WSdisp1_nonz

[ 0 , 0 ] : WSdisp1_nonz [ 0 , − 1 ] , 1 ] ) )

diff_WSAC = np . a r r a y ( np . a b s o l u t e (WSAC. i l o c [WSAC_nonz[ 0 , 0 ] ] [ 1 ] − WSAC. i l o c [WSAC_nonz [ 0 , 0 ] : WSAC_nonz [ 0 , − 1 ] , 1 ] ) )

# P l o t t n i n g t h e time vs t h e d i s t a n c e p l t . f i g u r e ( 1 )

p l t . p l o t ( r e f . i l o c [ ref_nonz [ 0 , 0 ] : ref_nonz [ 0 , − 1 ] , 0 ] , d i f f _ r e f , l a b e l = ’ R e f e r e n c e ’ )

p l t . p l o t ( off_ax_1mm . i l o c [ offax1_nonz [ 0 , 0 ] : offax1_nonz [ 0 , − 1 ] , 0 ] , d i f f _ o f f a x 1 , l a b e l = ’ Off − a x i s Det . 1 mm’ ) p l t . p l o t ( off_ax_2mm . i l o c [ offax2_nonz [ 0 , 0 ] : offax2_nonz

[ 0 , − 1 ] , 0 ] , d i f f _ o f f a x 2 , l a b e l = ’ Off − a x i s Det . 2 mm’ )

p l t . p l o t ( angl_1 . i l o c [ angl1_nonz [ 0 , 0 ] : angl1_nonz [ 0 , − 1 ] , 0 ] , d i f f _ a n g l 1 , l a b e l = ’ Angled Cone 1 deg ’ )

p l t . p l o t ( angl_05 . i l o c [ angl05_nonz [ 0 , 0 ] : angl05_nonz

[ 0 , − 1 ] , 0 ] , d i f f _ a n g l 0 5 , l a b e l = ’ Angled Cone 0.5 deg ’ )

p l t . p l o t ( disp_1mm . i l o c [ disp1_nonz [ 0 , 0 ] : disp1_nonz

[ 0 , − 1 ] , 0 ] , d i f f _ d i s p 1 , l a b e l = ’ D i s p l a c e d Cone 1 mm’ ) p l t . p l o t ( disp_05mm . i l o c [ disp05_nonz [ 0 , 0 ] : disp05_nonz

[ 0 , − 1 ] , 0 ] , d i f f _ d i s p 0 5 , l a b e l = ’ D i s p l a c e d Cone 0.5 mm’ )

p l t . p l o t ( WS_disp1 . i l o c [ WSdisp1_nonz [ 0 , 0 ] : WSdisp1_nonz [ 0 , − 1 ] , 0 ] , diff_WSdisp1 , l a b e l = ’ Wave Shaper , d i s p l a c e d

1 mm’ )

p l t . p l o t ( WSAC. i l o c [WSAC_nonz [ 0 , 0 ] : WSAC_nonz[ 0 , − 1 ] , 0 ] ,

diff_WSAC , l a b e l = ’ Wave Shaper , d i s p l a c e d 1 mm & Angled Cone 0. 5 deg ’ )

p l t . legend ( )

p l t . x l a b e l ( ’ Tid [ s ] ’ ) p l t . y l a b e l ( ’ | x | [m] ’ )

p l t . t i t l e ( ’ Avstånd t i l l f ö r s t a t r ä f f p u n k t e n ’ )

# P l o t t i n g t h e X− d i r e c t i o n p l t . f i g u r e ( 2 )

p l t . p l o t ( r e f [ ’ Time ’ ] , r e f [ ’ P o s i t i o n X ’ ] , l a b e l = ’ R e f e r e n c e

’ )

p l t . p l o t ( off_ax_1mm [ ’ Time ’ ] , off_ax_1mm [ ’ P o s i t i o n X ’ ] , l a b e l = ’ Off − a x i s Det . 1 mm’ )

p l t . p l o t ( off_ax_2mm [ ’ Time ’ ] , off_ax_2mm [ ’ P o s i t i o n X ’ ] , l a b e l = ’ Off − a x i s Det . 2 mm’ )

p l t . p l o t ( angl_1 [ ’ Time ’ ] , angl_1 [ ’ P o s i t i o n X ’ ] , l a b e l = ’ Angled Cone 1 deg ’ )

p l t . p l o t ( angl_05 [ ’ Time ’ ] , angl_05 [ ’ P o s i t i o n X ’ ] , l a b e l = ’ Angled Cone 0.5 deg ’ )

p l t . p l o t ( disp_1mm [ ’ Time ’ ] , disp_1mm [ ’ P o s i t i o n X ’ ] , l a b e l =

’ D i s p l a c e d Cone 1 mm’ )

p l t . p l o t ( disp_05mm [ ’ Time ’ ] , disp_05mm [ ’ P o s i t i o n X ’ ] , l a b e l

= ’ D i s p l a c e d Cone 0.5 mm’ )

p l t . p l o t ( WS_disp1 [ ’ Time ’ ] , WS_disp1 [ ’ P o s i t i o n X ’ ] , l a b e l =

’ Wave Shaper , 1 mm d i s p l a c e d ’ )

p l t . p l o t (WSAC[ ’ Time ’ ] , WSAC[ ’ P o s i t i o n X ’ ] , l a b e l = ’ Wave Shaper , 1 mm d i s p l a c e d & Angled Cone 0.5 deg ’ )

p l t . legend ( )

p l t . x l a b e l ( ’ Tid [ s ] ’ ) p l t . y l a b e l ( ’ [m] ’ )

p l t . t i t l e ( ’X− D i r e c t i o n ’ )

p l t . show ( )

B.2.3 Analysis with histogram

import m a t p l o t l i b . p y p l o t as p l t import numpy as np

import pandas as pd

angl_1 = pd . r e a d _ e x c e l ( ’ Angled Cone 1 deg . x l s x ’ ) angl_05 = pd . r e a d _ e x c e l ( ’ Angled Cone 0.5 deg . x l s x ’ ) disp_1mm = pd . r e a d _ e x c e l ( ’ D i s p l a c e d Cone 1 mm. x l s x ’ ) disp_05mm = pd . r e a d _ e x c e l ( ’ D i s p l a c e d Cone 0.5 mm. x l s x ’ )

angl1_nonz = np . a r r a y ( np . where ( angl_1 [ ’ P o s i t i o n X ’ ]!=0 ) )

#Indexes o f non−zero v a l u e s

angl05_nonz = np . a r r a y ( np . where ( angl_05 [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

disp1_nonz = np . a r r a y ( np . where ( disp_1mm [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

disp05_nonz = np . a r r a y ( np . where ( disp_05mm [ ’ P o s i t i o n X ’ ]!=0 ) ) #Indexes o f non−z ero v a l u e s

# C a l c u l a t i n g t h e d i s t a n c e between each p a r t i c l e t o t h e f i r s t p a r t i c l e i n t h e x− d i r e c t i o n

d i f f _ a n g l 1 = np . a r r a y ( np . a b s o l u t e ( angl_1 . i l o c [ angl1_nonz [ 0 , 0 ] ] [ 1 ] − angl_1 . i l o c [ angl1_nonz [ 0 , 0 ] : angl1_nonz [ 0 , − 1 ] + 1 , 1 ] ) )

d i f f _ a n g l 0 5 = np . a r r a y ( np . a b s o l u t e ( angl_05 . i l o c [ angl05_nonz [ 0 , 0 ] ] [ 1 ] − angl_05 . i l o c [ angl05_nonz [ 0 , 0 ] : angl05_nonz [ 0 , − 1 ] + 1 , 1 ] ) )

d i f f _ d i s p 1 = np . a r r a y ( np . a b s o l u t e ( disp_1mm . i l o c [ disp1_nonz [ 0 , 0 ] ] [ 1 ] − disp_1mm . i l o c [ disp1_nonz [ 0 , 0 ] : disp1_nonz [ 0 , − 1 ] + 1 , 1 ] ) )

d i f f _ d i s p 0 5 = np . a r r a y ( np . a b s o l u t e ( disp_05mm . i l o c [

disp05_nonz [ 0 , 0 ] ] [ 1 ] − disp_05mm . i l o c [ disp05_nonz [ 0 , 0 ] : disp05_nonz [ 0 , − 1 ] + 1 , 1 ] ) )

# S t a t i s t i c a l Histogram p l o t s p l t . f i g u r e ( 1 )

h i s t _ a n g l 1 , b i n _ a n g l 1 = np . histogram ( d i f f _ a n g l 1 , d e n s i t y = True )

binWidth_angl1 = b i n _ a n g l 1 [ 1 ] − b i n _ a n g l 1 [ 0 ]

p l t . bar ( b i n _ a n g l 1 [ : − 1 ] , h i s t _ a n g l 1

*

binWidth_angl1 , binWidth_angl1 , a l i g n = ’ edge ’ , l a b e l = ’ 1 deg ’ )

p l t . t i t l e ( ’ Vinklad kon , histogram över a v s t å n d e t t i l l f ö r s t a t r ä f f p u n k t e n ’ )

p l t . x l a b e l ( ’ Avstånd t i l l f ö r s t a t r ä f f p u n k t e n [m] ’ ) p l t . y l a b e l ( ’ S a n n o l i k h e t ’ )

hi s t _ a n gl 0 5 , bin_angl05 = np . histogram ( d i f f _ a n g l 0 5 , d e n s i t y = True )

binWidth_angl05 = bin_angl05 [ 1 ] − bin_angl05 [ 0 ]

p l t . bar ( bin_angl05 [ : − 1 ] , h i s t _ a n g l 0 5

*

binWidth_angl05 , binWidth_angl05 , a l i g n = ’ edge ’ , l a b e l = ’ 0 .5 deg ’ )

p l t . legend ( )

p l t . f i g u r e ( 3 )

h i s t _ d i s p 1 , b i n _ d i s p 1 = np . histogram ( d i f f _ d i s p 1 , d e n s i t y = True )

binWidth_disp1 = b i n _ d i s p 1 [ 1 ] − b i n _ d i s p 1 [ 0 ]

p l t . bar ( b i n _ d i s p 1 [ : − 1 ] , h i s t _ d i s p 1

*

binWidth_disp1 , binWidth_disp1 , a l i g n = ’ edge ’ , l a b e l = ’ 1 mm’ )

h i s t _ d i s p 0 5 , bin_disp05 = np . histogram ( d i f f _ d i s p 0 5 , d e n s i t y = True )

binWidth_disp05 = bin_disp05 [ 1 ] − bin_disp05 [ 0 ]

p l t . bar ( bin_disp05 [ : − 1 ] , h i s t _ d i s p 0 5

*

binWidth_disp05 , binWidth_disp05 , a l i g n = ’ edge ’ , l a b e l = ’ 0.5 mm’ )

p l t . legend ( )

p l t . t i t l e ( ’ F ö r f l y t t a d kon , histogram över a v s t å n d e t t i l l

f ö r s t a t r ä f f p u n k t e n ’ )

p l t . x l a b e l ( ’ Avstånd t i l l f ö r s t a t r ä f f p u n k t e n [m] ’ ) p l t . y l a b e l ( ’ S a n n o l i k h e t ’ )

p l t . show ( )