Asymmetries in Shaped Charges

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DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM, SWEDEN JUNE 2021

Asymmetries in Shaped Charges

KTH Bachelor Thesis Report

Gustav Boman

KTH ROYAL INSTITUTE OF TECHNOLOGY

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Abstract

This thesis concerns the effects of a select few asymmetries on a jet forming shaped charge. A Shaped charge(SC) is a formed explosive with metal liner that collapses under the detonation. The resulting shockcompression results in a jet of metal being shot forward in up 10km/s. The asymmetries that were modelled are bubbles in the LX14 high explosive, rust on the copper liner and offcenter detonation. To simulate the SC the software IMPETUS afea is used. By replacing the geometry with SPH particles instead of a mesh, a simulation with broad boundaries is possible which helps in the simulation of explosives. The results showed that the offcenter detonation caused the biggest deviation in the jet. The rust caused fragmentation in the jet but a smaller deviation and the bubbles caused very little damage to the jet formation.

Keywords

Shaped charge, Thesis, Jet deviation, asymmetry

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Abstract

Detta examensarbete handlar om ett antal valda asymmetriers påverkan på en strålbildande riktad sprängladdning. En riktad sprängladdning (RSV) är en formad sprängladdning med en liner av metall som kollapsar på sig själv under explosionens forlopp. The resulterande shockkompresssionen resulterar i att metallen skjuts i väg i en stråle som rör sig i upp till 10km/s. Asymmetrierna som simulerades var sned tändning, luftbubblor i det högexplosiva ämnet LX

14 och rost på kopparlinern. För att simulera en RSV används programmet IMPETUS afea. Genom att använda SPH partiklar istället för en mesh kan simuleringen ha stora ränder och gränser vilket är användbart i simuleringen av explosioner. Resultaten visade att snedtändningen skapade störst avvikelse från yaxeln. Därefter kom rost på linern och sist bubblorna som hade ytterst lite inverkan på strålbildningen.

Nyckelord

Kandidatexamensarbete, Riktad sprängladdning, Stål avvikelse, asymmetri

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Acknowledgements

Thank you to SAAB dynamics in Karlskoga for allowing me, Kristoffer Seidel and Anmol Buhllar to write our theses with their help and expertise. Thank you to SAAB dynamics in Thun, Swizerland who initiated the project and then passed it on to Sweden when out exchange to ETH was cancelled.

Thank you to our supervisor at KTH, Ricardo Vinuesa

Thank you to Kristoffer Seidel and Anmol Buhllar who did parallel bachelor’s thesis work at SAAB and were a huge help during this project.

And finally a huge thank you to our supervisor at SAAB, Victor Björkgren who has been a huge support and whose technical expertise we could rely on throughout the whole project.

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Author

Gustav Boman <gboman@kth.se>

Information and Communication Technology KTH Royal Institute of Technology

Place for Project

Stockholm, Sweden Karlskoga, Sweden

Examiner

Gunnar Tibert

KTH Royal Institute of Technology

Supervisors

Ricardo Vinuesa

KTH Royal Institute of Technology

Victor Björkgren, Specialist in explosive modelling SAAB Dynamics, Karlskoga, Sweden

Cristian Herren

SAAB Dynamics, Thun, Switzerland

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1 Introduction

Shaped Charges have been in use for decades as a means of an explosion for military use but also for more civilian use as in mining. General and fundamental information regarding shaped charges, disregarding of use is available to the general public. However, there are very few specialized and detailed reports done with modern softwares. In this paper the effects of asymmetries in shaped charges will be investigated and discussed with the help of modern softwares.

Figure 1.1: High explosive antitank artillery shell, HEAT (Fundamental of shaped charges [4])

1.1 Background

The purpose of a shaped charge is to concentrate the explosive energy into one specific point. A shaped charge is created by surrounding a cone or sphere called a liner [often made in metal] with explosives. The following detonation will cause the liner to collapse in on itself under immense shock compression.

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(a) Illustration of the elapsing detonation from The fundamentals of shaped charges

(b) The elapse

of the detonation simulated in Impetus

Figure 1.2: Illustration of the collapse of the liner as the detonation is elapsing.

Illustration based on flash xray photography vs simulated elapse ( Fundamental of shaped charges [4])

Placed against a steel barrier, the crater depth will be half the charge diameter, figure 1.3a. Installing a liner, most commonly copper, the craterdepth will dramatically increase, figure 1.3b. Placing the charge a certain distance from the steel plate, the crater depth will grow dramatically as the jet now has the possibility the properly form, figure 1.3c. This optimal distance is known as Standoff.[4]

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Figure 1.3: Illustration of crater depth by SC’s without liner, with liner and with liner at a proper standoff. (Fundamental of shaped charges [4])

When allowed to properly form, the tip of the jet of copper can reach speeds from 5km/s up to 10km/s. The jet’s penetrating capabilities depend on a variety of factors. A simplified model of the penetration depth [P] is described as

P = L

√ ρ_jet ρtarget

where L is the length of the jet. In reality the penetration is also affected by the jets straightness, mass and velocity distribution, fragmented or nonfragmented, the fragments shape, fragments tumble and the jets angle of attack towards the target.

But when the jet is properly formed its penetration capabilities are astonishing. To get an idea of the penetration look at figure 1.4. A solid cubic decimeter of steel is easily punched through with the majority of the jet still being intact afterwards.

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Figure 1.4: A fully formed jet penetrating a solid steel block (10 · 10 · 10cm) in IMPETUS afea

The explosive used in high explosive antitank shaped charge rounds (common acronym HEAT) is most commonly octogen combined with a plastic binder to stabilise the compound. One variation of this compound, LX14 is very common in this application and is also what this thesis makes use in its model. The charge is initiated by an ignition source, usually activated by the warheads strikingcap when making contact with the target. It is important that the shock wave that propagates through the explosive is uniform and straight so that the liner collapses symmetrically. If the wave would strike the liner at an offset it could result in a malformed jet.

A so called waveshaper can be used to direct the shock wave but it is not a necessity. When the liner collapses, the material is rapidly heated up, stretched and deformed. This leads to the requirement that the liner is of a very malleable material that can withstand rapid deformation without hardening under the stress. Pure copper has proven to be a good material for this application.

When the jet is formed, the velocity is distributed in such a manner that the tip of the jet can be up to 10 times the velocity of the tail. The majority of the penetrative tip originates from the inner and lower part of the liner. The rest of the liner forms the base of the jet and the part that is often referred to as a mushroom due to its shape. This can be observed in figure 1.5.

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Figure 1.5: A fully formed jet from a shaped charge in IMPETUS afea

The fast tip is what causes the majority of the penetration while the slower tail is what causes the damage inside the target (in a military application).

The jet that is formed is extremely sensitive to asymmetries in the liner and places a very high demand on the rotational asymmetry. [1]

1.2 Problem

As mentioned in 1.1 the penetration of the jet is dependant on many different characteristics. These characteristics are extremely sensitive to the liner’s rotational symmetry, local material thickness on the micrometer as well as the explosive’s shock wave propagation.[1] The sensitivity of the system requires a high precision in manufacturing and the proper storage of SC’s to prevent damage.

However, production faults and hostile environments are impossible to nullify.

Hence, it is very desirable to know how asymmetries in form of production faults and damages effect the jet.

1.3 Purpose

Historically it has been very hard theoretically predict a SC. To verify the warhead one would need to do a test fire and photograph the jet with flash xray photography. Softwares in the last decades have progressed enough to accurately be able to predict asymmetries effect on the jet formation. However, these simulations have been limited to 2D with extremely long computational time. However these last few years better softwares combined with increasingly powerful GPU’s has given rise to the possibility of calculating these simulations in 3D with millions of particles within a relatively short time span. With the

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use of IMPETUS afea, a nonlinear finite element analysis tool and the proper damage models, it is possible to accurately simulate the elapse of a shaped charge detonation.

This thesis aims to evaluate different asymmetries and present their effect on the jet of a shaped charge. Their effect on the penetration capabilities of the jet will not be discussed or examined to a large extent.

1.4 Goal

The goal of this project is to find the effect of selected asymmetries on a viper shaped charge. This is done by investigating the jets formation, its characteristics and divergence when the asymmetries are applied.

1.5 Methodology

The asymmetries will be evaluated by simulating them in IMPETUS afea. These asymmetries are modelled in ANSYS spaceclaim and then exported to impetus in stl. format. The shaped charge that is being tested is a Viper charge. Which is an ’open source’ SC defined by a cone angle of 22.5^◦ and a base diameter of 65mm.

The asymmetries that will be examined are the following.

§ Displaced detonation point [0.5mm and 1mm]

• Individual airbubbles in the LX14 high explosive material.

• Clusters of airbubbles in the LX14 high explosive material.

• Rust/Oxidation on the copper liner

1.6 Delimitations

As mentioned in 1.4 the goal is to investigate the effects of the asymmetries on a shaped charge. However this is done by investigating the divergence of jet over time as it hits a sensor. To truly find the effect on the warhead’s performance, one would simulate the penetration in its entirety. However, a fullscale simulation

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with an adequately large target would require too much GPU memory and would either be impossible to run or take too long time with the computers we have access to.

1.7 Outline

In chapter 2 Theoretical Background the sources used for this thesis are presented and discussed. In chapter 3 Method The model used to simulate the shaped charge is presented together with some theory on the solutionmethod the software uses. Chapter 4, 5 and 6 present the work done, results and discussion on the results respectively.

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2 Theoretical Background

Information regarding shaped charges can be difficult to find. Our supervisor at SAAB, Victor gave us some tips on useful papers/books to use. One of these was Fundamentals of Shaped Charges by W. P. Walters, Jonas A. Zukas.

Otherwise we were able to find some published works regarding shaped charges and asymmetries.

2.1 The Fundamentals of Shaped Charges

The covers the shaped charge’s history, application, different models, formation of the jet and much more. It was a good starting point to gain understanding of the subject. Although one has to be careful with various physical explanations given in the book as it was written long before accurate software and other modern tools. As such a few theories in the book are speculative. For example how the jet actually penetrates. The author discusses possible theories such as the jet pushing the material to the side or simply plastically deforming the material into dust. But besides that it gives a very good overview and the results presented from test

detonations are still valid. [4]

2.2 The influence of asymmetries in shaped charge performance

This paper evaluates different asymmetries in shaped charges in AUTODYN with a 2D model. It’s methods are very similar to what this thesis presents and is a good reference on what to expect. This paper gives this thesis a different model to compare against. This report is a decade old and computational power has come a long way and the reader should keep that in mind. None the less it is a good paper to have as a reference but not as a source. [2]

2.3 Lärarbok i Militärteknik

This book is published by Försvarshögskolan and concerns warheads, their delivery and protection against these. The book is written by researcher at Försvarets Forskningsinstitut, FOI. Hence, a lot of information regarding shaped

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charges was sourced from this report as FOI is highly reputable and uses many sources including real life testing to advanced softwares [1]

2.4 Smoothed Particle Hydrodynamics

This is a published work by the developers of the SPH method. The paper uses the SPH method to analyse a known distribution of fluid particles with Newtonian equations. The paper first presents a nonaxisymmetric distribution of 80 particles in three dimensions. The authors state in the conclusion that this method can easily be upscaled. The same principle method is used in Impetus afea but with millions of particles instead. [3]

2.5 Mei Grüneisen Equation of State (EOS)

This wikipedia article has summarized various references on the Mei Grüneisen EOS. Most importantly Gustav Mei and Eduard Grüneisen, the developers own work. The reason for not referencing the original work directly is that the articles are published in german. One has to be wary of Wikipedia as anyone can edit the article. Hence, for deeper knowledge one should follow the references cited in the article. [5]

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3 Model and Simulation Setup

3.1 Model setup

The model setup is the viper charge placed with the detonationpoint placed in the origin, which can be seen on the right in figure 3.1a. A sensor is placed 30cm from base of the cone. In figure 3.1a the sensor is placed 5 cm in front of the base for visualization purposes.

(a) Viper Geometry with a sensor

(b) Geometry filled with SPH particles

Figure 3.1: The geometry (a) which is then filled with SPH particles

In figure 3.1b the gray and dark brown is the high explosive Lx14 and the yellow and beige is the copper liner.

The code for the Impetus model can be found in Appendix A

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3.1.1 Material Data

The material data that was used came from Ansys’ Material Library. However for the material data regarding the rust on the liner we used two different methods to work around the lack of data. The first method was to simply lower the density from 8900_m^kg3 to 8000_m^kg3 as well as lowering the initial yield strength from 350e6P a to 200e6P a. The more scientific solution was to use unique a failure criteria for the oxidized metal. The model was provided by one of the engineers at SAAB and is illustrated in figure 3.2. The model attempts to simulate the brittleness of an oxidized material by assigning the SPH particles with a linear elastic deformation limit, that when crossed results in the internal stress reducing linearly with the plastic deformation.

Figure 3.2: Illustration of the damage model. The blue is elastic deformation and the yellow is plastic deformation (D).

3.1.2 Equation of state

Impetus also uses Grüneisen equation of state, which relates pressure, volume and temperature in a solid. It is used to determine the pressure in a shock

compressedsolid and is necessary for solving how the copper liner behaves under the detonation. The equation was developed by Gustav Mei in 1903 and extended in 1912 by Eduard Grüneisen. Impetus provides a manual which is available to everyone at (https://www.impetusafea.com/support/manual/) which describes

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the formula implemented in the program. Here we can see how the EOS is calculated and what inputs are required.[5]

(a) The equation of state

(b) The input parameters

Figure 3.3: Screenshot from IMPETUS manual on Equation of State

3.1.3 SPH particles

The simulations in IMPETUS were calculated with SPH (SmoothedParticle hydrodynamics). The method calculates fluid dynamics with a Mesh free lagrangian method. Each particle has its own coordinated system that follows the particle as it travels. A useful advantage in this application is that the method allows for big displacements in the boundary. The simulations performed had very few boundaries and particles were free to travel endlessly until a lifespan command was used to kill the particles or stopping the simulation.

To simulate the explosion, the detonation point initiates a shockwave within a 2mm diameter sphere at the bottom of the explosives. When a LX14 SPH particle is ignited the program follows the JWL equation of state (EOS) for LX14 which is a form of meigrüneisen EOS. [3]

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4 Asymmetry modeling and Post processing

The base model for the viper charge was already provided by Saab. As such the work consisted of modeling the asymmetries and postprocessing.

4.1 Offcenter ignition

This asymmetry was very easy to implement. The ignition point could be altered by changing one simple parameter in the code called D_x. D_x was first changed to 1mm then 0.5mm. The results were saved and treated in the postprocessing.

Figure 4.1: The detonation sphere on the base of the HE

4.2 Air bubbles

To model a single airbubble the geometry of the high explosive was exported to Ansys Spaceclaim in .stl format. Here the part was converted to an Ansys solid (.scdoc) so that the part could be modified. Using the sphere tool in Space Claim a sphere of 2mm in diameter was placed on 4 different places, and saved as 4

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separate HEparts in .stl format. After they were simulated the decision to ad more cavities was made from the post processing result.

(a) Bubbles 1 (b) Bubbles 2

(c) Bubbles 3 (d) Bubbles 4

Figure 4.2: The four different asymmetries with a single bubble

To get a more realistic representation of cavities in the HE, some xray photographs of cast high explosives were analysed. These were provided by SAAB but cannot be included in this report due to it being sensitive material. Two new HE models were made where there were two different clusters of bubbles with both 2mm and 1mm in diameter near the base of the liner and at the top.

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(a) Bubbles 5 (b) Bubbles 6

Figure 4.3: The two different asymmetries with multiple bubbles

4.3 Rust

Modeling the rust in the liner was also done in spaceclaim. By splitting the liner model in two a rectangular sketch was drawn along the profile of the liner. Around 5mm long and 0.6mm thick. This was to simulate a small fingerprint that had been made in production and left to rust in storage for a long time. Three different positions were simulated with two different models as mentioned in section 3.1.1.

Firstly the rust at the top of the liner was examined. Looking at the results it was decided to move the patch downward along the liner as the top of the liner only generates a very small segment of the jet.

(a) Rust base (b) Rust Mid (c) Rust top

Figure 4.4: The three different placements for the rust patch

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4.4 Post Processing

The postprocessing started by inspecting the jet to get a general feel for what had happened and to control that the solution didn’t diverge and produce chaos. After that, the information gathered by the sensor placed at 30cm from the base was examined. To present the divergence of the jet the information on the particles x and z position was exported with a .csv format to matlab. In matlab the data was read and converted into vectors for both x and z position. The total divergence from the yaxis was then derived with Pythagoras. The respective asymmetries divergence was then plotted against time.

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5 Result

Each asymmetry’s divergence is plotted individually and together. A few images of the jet are also presented.

Figure 5.1: The Jet from the viper charge without any asymmetries

5.1 Offcenter detonation

Figure 5.2: The divergence from the y axis for off centre detonation

In figure Figure ?? see that the tip of the jet experiences a large divergence from the yaxis that increases with the detonation point’s distance from the origin. The abrupt spike in the divergence is the average of the particles that are almost flying

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away. These can be see between the grid coordinates 0.55 m and 0.5 m in figure Figure 5.3

Figure 5.3: Jet from 1mm offcenter detonation at 7.2 µ s

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5.2 Airbubbles in the HE

(a) Zoomed out (b) Figure to clarify plot

index

(c) Zoomed in

Figure 5.4: The Divergence of the 6 different bubblescenarios

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All the scenarios are very similar in divergence. The Ones that have a bigger divergence are line 5 and 6 as these are the ones with multiple cavities. This implies that that the tip of the jet is very messy without much structure. The rest of the jet does however retain a good quality.

(a) Bubbles 4

(b) Bubbles 5

(c) Bubbles 6

Figure 5.5: The jet from Bubbles 4 to 6 at 7.2µs

In figure 5.5 we can see the jets from bubbles 4, 5 and six. We can see the messy tip of the jet from the viper charge with multiple cavities. Bubbles 1, 2 and 3 were not plotted as they were visually identical to 4.

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5.3 Rust on the liner

(a) Zoomed out

(b) Zoomed in with new line colors

Figure 5.6: The divergence from the yaxis for the 4 different cases of rust on the liner

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Table 5.1: Indexes for the rust asymmetry

Index Asymmetry

Rust 1 Rust on the top of the liner with improvised data Rust 2 Rust on the base of the liner with improvised data Rust 3 Rust on the base of the liner with damage model Rust 4 Rust on the middle of the liner with damage model

From the zoomed out graph, 5.6a it is visible that all the lines have some form of major displacement at the tip of the jet. However only line 4 is observed to diverge after the initial displacement. This can also be observed in figure 5.9.

The jet with the rust patch at the bottom with the deformation model

The jet with the rust patch at the bottom with the improvised material data Figure 5.7: Two jets formed from the same asymmetry. Rust at the bottom as seen in figure 4.4c but with a different material model for the rust

Both of these images are taken at 7.2µs from detonation. Both behave very similarly. A crooked tip but a straight and uniform jet without to many different segments.

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Figure 5.8: the base of the jet with the rust patch at the top as seen i figure 5.6a

This is the base of the asymmetry with rust at the bottom taken at 7.2µs after detonation. The damage is only present at the base of the jet.

Figure 5.9: The Jet from the liner with the rust patch in the middle as seen in figure 4.4b. Taken at 7.2µs after detonation.

The damage at the middle of the liner resulted in a crooked jet that displays elements of segmentation as well.

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5.4 Compiled Results

(a) Zoomed out

(b) Zoomed in

Figure 5.10: All the asymmetries divergence gathered in one single plot.

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It is apparent that the offaxis detonation results in the largest deviation.

Furthermore, all asymmetries seem to experience a larger deviation at the same place.

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6 Conclusions

Evaluating the result is hard as this report doesn’t not discuss what the effect of the jets divergence has on the viper’s penetrative capabilities. What can be discussed and evaluated are the jets properties and its resulting deviation. The one conclusion that can be made with confidence is that all of the asymmetries are unwanted.

6.1 Discussion

From looking at the plots in section 5 we can see that the off centre detonation causes the largest divergence from the yaxis. Not only is a big divergence on the first hit but the divergence slowly converges towards the y axis. This will most likely result in the jet being evenly spread out along the target and thus loosing a lot of its penetrative power.

Following the order of the the results the bubbles appear to only cause minor disturbances by increasing the divergence of these rogue particles that can even be observed in the reference detonation, figure 5.1 in section 5. these rogue particles skew the average position of the jet, hence why one should cross reference with the actual simulation image of the jet.

Increasing the number of bubbles barely made any impact. This could be because the bubbles don’t cause any real hindrance to the incoming detonationwave and only remove a minute amount of explosive mass from the LX14. It is however important to note that larger or other kinds of cavities such as cracks and porosity could have a more drastic effect with almost endless permutations of defects.

The rust had a very interesting impact but somewhat predictable result. As mentioned in section 1.1 the jet is mostly formed by the inner middle and lower part of the liner. As such placing the damage in this region causes the frontmost part of the jet to bend and fragment.

Stacking these asymmetries up against each other, the offcenter detonation

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obviously had the biggest divergence, with the rust on the liners middle part coming in at a second place. However realistically the problem of offcenter detonation can be somewhat circumvented with a waveshaper. Where as the problem of oxidation is something you cant circumvent in the design. As such it would be reasonable to assume that the rust has more potential to cause damage.

Especially when countries such as Sweden keep many weapons in storage under a long tome in case of an arising conflict. This damage also has the potential to be worse as the one modeled in this report was roughly 5x5mm. A single thumbprint can be much larger than that. The rust does also raise the question of jet control via composite liner material.

6.2 Future Work

This report was in many ways very shallow and is only meant to give an overview of the effect of these asymmetries. Each asymmetry can warrant its own report with an extensive penetration simulation as well to clarify how much the divergence affects the penetration.

As mentioned it would also be interesting to see the effects of a composite liner.

6.3 Final Words

The work was fun and a great learning experience of actually working as an engineer. As such, a huge thank you to SAAB for making this thesis possible.

The workflow was constantly met with problem solving in CAD modelling and software errors with IMPETUS. This did however result in it taking more time than expected. Ideally I would also have liked to implement a greater amount of asymmetries such as a displaced cone. But overall I am happy with the amount of results and I hope this report comes in use in giving pointers to which direction to pursue for future investigations regarding this subject.

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References

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

[2] O, Ayisit. “The influence of asymmetries in shaped charges performance”.

In: (2008).

[3] R. A. Gingold, J. J. Monaghan. Smoothed Particle Hydrodynamics. Monthly Notices of the Royal Astronomical Society, Volume 181, Issue 3, December 1977. ISBN: Non existant. URL: https : / / academic . oup . com / mnras / article/181/3/375/988212. 01 December 1977.

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

Interscience Publication, 1989. ISBN: 9780471621720.

[5] wikipedia. Mie–Grüneisen equation of state. ISBN: Non existant. URL:

https : / / en . wikipedia . org / wiki / Mie % E2 % 80 % 93Gr % C3 % BCneisen _ equation_of_state.

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Appendices

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Appendix Contents

A Impetus Viper code 31

B Matlab code for postprocessing 33

C Impetus manual 42

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A Impetus Viper code

The source code for the Viper shaped charge

Viper-main.txt

#--- Verifier ---

# Ref :

# Criterion : at tend = 0.04e-3 s

# Copper Maximum Velocity = 8853.81 +- 10%

#---

dx = 0.3e-3

D = 8800.0

ydet = 0.0 zdet = 0.0 yref = 0.123439 ydist = 250e-3

Rtarget = 1.5Ltarget = 450e-3 SI

viper_shaped_charge.k 1.0e-2, 1.0e-2, 1.0e-2 100.0e-6

"HE vol"

1, 10

"Liner Vol"

2, 10

"Lx 14"

11, 11, 11

"Copper Liner"

12, 12, 12

#"Target"

#15, 15, 15

11, 1821.0, 2.2e9, 0.0, -1.0e15

##15, 7830, 1.59e11, 0.0

#4e7, 400, 0.27

# id, A, B, R1, R2, Omega, E0

11, 826.1e9, 17.24e9, 4.55, 1.32, 0.38, 10.2e9

# x, y, z, t0, D0

0.0, %ydet, %zdet, 0.0, %D

"deto HE"

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14

0.0, %ydet, %zdet, 3.0"copper"

12, 8940.0, 120e9, 0.345 350e6

12, 1.489, 1.99

##15, 1.25, 1.97 10, 7800

11, 11, 2.012, 12, %dx

#15, 15, 21, G, 11, 0.0, 4.0e-5 11

1 12 2

"sensor1"

13

0.0, 0.28, 0.0,0.0, 0.29, 0.0, 0.05

"sensor2"

16

0.0, 0.18, 0.0,0.0, 0.19, 0.0, 0.05

"sensor3"

17

0.0, 0.38, 0.0,0.0, 0.39, 0.0, 0.05

"sensor3"

18

0.0, 0.48, 0.0,0.0, 0.49, 0.0, 0.05

##"Target"

#15

#0.05,0.24,0.05,-0.05,0.34,-0.05 1, 13, 0

2, 16, 0 3, 17, 0 4, 18, 0 X

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B Matlab code for postprocessing

this is the information

1 c l c

2 c l e a r a l l

3 c l o s e a l l

4 %Snedtändning FIXA 0.5MM OCKSÅ

5 x_1mm= r e a d t a b l e ( ’ x_pos_1mm . c s v ’) ; %l ä s e r e x c e l

6 z_1mm= r e a d t a b l e ( ’z_1mm . c s v ’) ;

7 x_05mm= r e a d t a b l e ( ’ 0.5mm_x. c s v ’) ;

8 z_05mm= r e a d t a b l e ( ’ 0.5mm_z. c s v ’) ;

9 y_v= r e a d t a b l e ( ’ y_v . c s v ’) ;

10 11

12 %%%%%%%%%%%%

13 %%%%Rost%%%%

14 %%%%%%%%%%%%

15

16 %Improv top

17 s i t _ x = r e a d t a b l e ( ’ s i t _ x . c s v ’) ;

18 s i t _ z = r e a d t a b l e ( ’ s i t _ z . c s v ’) ;

19 time= t a b l e 2 a r r a y ( s i t _ x ( : , 1 ) ) ;

20 s i t _ x = t a b l e 2 a r r a y ( s i t _ x ( : , 2 ) ) ;

21 s i t _ z = t a b l e 2 a r r a y ( s i t _ z ( : , 2 ) ) ;

22 s i t _ x = s i t _ x ( s i t _ x ~=0) ;

23 s i t _ z = s i t _ z ( s i t _ z ~=0) ;

24 s i t =1000

*

^{s q r t}( s i t _ x .^2+ s i t _ z . ^ 2 ) ;

25 s i t = s i t ( 1 :end−1) ;

26 s i t _ d = s i t ( 1 ) − s i t ;

27 28

29 time =10^6

*

time ( 1 : 5 6 3 ) ;

30 %Improv bas

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31 sib_x= r e a d t a b l e ( ’ sib_x . c s v ’) ;

32 s i b _ z= r e a d t a b l e ( ’ s i b _ z . c s v ’) ;

33 sib_x= t a b l e 2 a r r a y ( sib_x ( : , 2 ) ) ;

34 s i b _ z= t a b l e 2 a r r a y ( s i b _ z ( : , 2 ) ) ;

35 sib_x=sib_x ( sib_x ~=0) ;

36 s i b _ z=s i b _ z ( s i b _ z ~=0) ;

37 s i b =1000

*

^{s q r t}( sib_x .^2+ s i b _ z . ^ 2 ) ;

38 sib_d= s i b ( 1 ) − s i b ;

39

40 %Dmg bas

41 sbd_x= r e a d t a b l e ( ’ sdb_x . c s v ’) ;

42 sbd_z= r e a d t a b l e ( ’ sdb_z . c s v ’) ;

43 sbd_x= t a b l e 2 a r r a y ( sbd_x ( : , 2 ) ) ;

44 sbd_z= t a b l e 2 a r r a y ( sbd_z ( : , 2 ) ) ;

45 sbd_x=sbd_x ( sbd_x~=0) ;

46 sbd_z=sbd_z ( sbd_z~=0) ;

47 sbd=1000

*

^{s q r t}( sbd_x .^2+ sbd_z . ^ 2 ) ;

48 sbd_d=sbd ( 1 ) −sbd ;

49

50 %Dmg Mid

51 smd_x= r e a d t a b l e ( ’ smd_x . c s v ’) ;

52 smd_z= r e a d t a b l e ( ’ sdm_z . c s v ’) ;

53 smd_x= t a b l e 2 a r r a y ( smd_x ( : , 2 ) ) ;

54 smd_z= t a b l e 2 a r r a y ( smd_z ( : , 2 ) ) ;

55 smd_x=smd_x ( smd_x~=0) ;

56 smd_z=smd_z ( smd_z~=0) ;

57 smd=1000

*

^{s q r t}( smd_x.^2+smd_z . ^ 2 ) ;

58 smd=smd ( 1 :end−1) ;

59 smd_d=smd ( 1 ) −smd ;

60 61 62

63

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64 %Bubbla x v e l o c i t y

65 bx_1= r e a d t a b l e ( ’ Bubbla1_xpos . c s v ’) ;

69 bx_5= r e a d t a b l e ( ’ bmulti_x . c s v ’) ;

70 bx_6= r e a d t a b l e ( ’ bmt_x . c s v ’) ;

71 %l ä g g t i l l m u l t i

72 bx_1= t a b l e 2 a r r a y ( bx_1 ( : , 2 ) ) ;

73 bx_2= t a b l e 2 a r r a y ( bx_2 ( : , 2 ) ) ;

74 bx_3= t a b l e 2 a r r a y ( bx_3 ( : , 2 ) ) ;

75 bx_4= t a b l e 2 a r r a y ( bx_4 ( : , 2 ) ) ;

76 bx_5= t a b l e 2 a r r a y ( bx_5 ( : , 2 ) ) ;

77 bx_6= t a b l e 2 a r r a y ( bx_6 ( : , 2 ) ) ;

78

79 bx_1=bx_1 ( bx_1~=0) ;

80 bx_2=bx_2 ( bx_2~=0) ;

81 bx_3=bx_3 ( bx_3~=0) ;

82 bx_4=bx_4 ( bx_4~=0) ;

83 bx_5=bx_5 ( bx_5~=0) ;

84 bx_6=bx_6 ( bx_6~=0) ;

85

86 %Bubbla z v e l o c i t y

87 bz_1= r e a d t a b l e ( ’ Bubbla1_zpos . c s v ’) ;

91 bz_5= r e a d t a b l e ( ’ bmulti_z . c s v ’) ;

92 bz_6= r e a d t a b l e ( ’ bmt_z . c s v ’) ;

93

94 bz_1= t a b l e 2 a r r a y ( bz_1 ( : , 2 ) ) ;

95 bz_2= t a b l e 2 a r r a y ( bz_2 ( : , 2 ) ) ;

96 bz_3= t a b l e 2 a r r a y ( bz_3 ( : , 2 ) ) ;

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97 bz_4= t a b l e 2 a r r a y ( bz_4 ( : , 2 ) ) ;

98 bz_5= t a b l e 2 a r r a y ( bz_5 ( : , 2 ) ) ;

99 bz_6= t a b l e 2 a r r a y ( bz_6 ( : , 2 ) ) ;

100

101 bz_1=bz_1 ( bz_1~=0) ;

102 bz_2=bz_2 ( bz_2~=0) ;

103 bz_3=bz_3 ( bz_3~=0) ;

104 bz_4=bz_4 ( bz_4~=0) ;

105 bz_5=bz_5 ( bz_5~=0) ;

106 bz_6=bz_6 ( bz_6~=0) ;

107

108 %Totpos

109 TotPos1 =1000

*

^{s q r t}( bx_1 .^2+ bz_1 . ^ 2 ) ;

110 TotPos2=1000

*

^{s q r t}( bx_2.^2+bz_2 . ^ 2 ) ;

111 TotPos3=1000

*

^{s q r t}( bx_3.^2+bz_3 . ^ 2 ) ;

112 TotPos4=1000

*

^{s q r t}( bx_4.^2+bz_4 . ^ 2 ) ;

113 TotPos5=1000

*

^{s q r t}( bx_5.^2+ bz_5 . ^ 2 ) ;

114 TotPos6=1000

*

^{s q r t}( bx_6.^2+bz_6 . ^ 2 ) ;

115

116 %Delar upp t i d e n och x−pos samt k o v e r t e r a r c e l l format t i l l v e k t o r

117 t_1mm= t a b l e 2 a r r a y (x_1mm ( : , 1 ) ) ;

118 x_1mm= t a b l e 2 a r r a y (x_1mm ( : , 2 ) ) ;

119 z_1mm= t a b l e 2 a r r a y (z_1mm ( : , 2 ) ) ;

120 x_05mm= t a b l e 2 a r r a y (x_05mm( : , 2 ) ) ;

121 z_05mm= t a b l e 2 a r r a y (z_05mm ( : , 2 ) ) ;

122 yv= t a b l e 2 a r r a y ( y_v ( : , 2 ) ) ;

123 %Tar b o r t n o l l v ä r d e n

124 x_1mm=x_1mm(x_1mm~=0) ;

125 x_05mm=x_05mm(x_05mm~=0) ;

126 %x_1mm=x_1mm( 2 : end ) ;

127 x _ r e f=z e r o s(s i z e(x_1mm) ) ;

128

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129

130 z_1mm=z_1mm(z_1mm~=0) ;

131 z_05mm=z_05mm(z_05mm~=0) ;

132 %z_1mm=z_1mm( 2 : end ) ;

133

134 yv=yv ( yv~=0) ;

135 yv=yv ( 3 :end) ;

136

137 %Beräknar t o t a l a d i v e r g e n s e n f r å n centrum− a x i s

138 %samt k o v e r t e r a r t i l l mm

139 TotPos=1000

*

^{s q r t}(x_1mm.^2+z_1mm. ^ 2 ) ;

140 TotPos_05mm=1000

*

^{s q r t}(x_05mm.^2+z_05mm. ^ 2 ) ;

141 TotDiv=TotPos ( 1 ) −TotPos ( 2 :end) ;

142

143 %s e n s o r v i d 38.39 cm . Verkansdel s l u t a r 12 b o r t f r å n o r i g o

144 %h a s t i g h e t k o n s t a n t . Därför bör man kunna i n t e r p o l e r a d i v e r g e n s e n med v i n k e l

145 %v i n k e l n tan ( d i v / y d i s t )

146

147 v i n k e l =tan( TotDiv ./(385 −120) ) ;

148

149 f i g u r e

150 p l o t( time ( 1 :l e n g t h( TotPos_05mm ) ) , TotPos ( 1 : l e n g t h(

TotPos_05mm ) ) , time ( 1 :l e n g t h( TotPos_05mm ) ) , TotPos_05mm )

151 t i t l e ( ’ Off − c e n t r e d e t o n a t i o n ’ , ’ F o n t S i z e ’ , 1 4 )

152 legend( ’ 1mm’ , ’ 0.5mm’ , ’ F o n t S i z e ’ , 1 4 )

153 x l a b e l( ’ { \mu} s ’ , ’ F o n t S i z e ’ , 1 4 )

154 y l a b e l( ’ Divergence from the c e n t r e a x i s (mm) ’ , ’ F o n t S i z e ’ , 1 4 )

155 156

157 f i g u r e

158 p l o t( TotPos1 )

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159 hold on

161 hold on

163 hold on

165 hold on

167 hold on

169 legend( ’ Bubble 1 ’ , ’ Bubble 2 ’ , ’ Bubble 3 ’ , ’ Bubble 4 ’ , ’ Bubble 5 ’ , ’ Bubble 6 ’ , ’ F o n t S i z e ’ , 1 2 )

172 t i t l e ( ’ Bubbles i n LX−14 ’ , ’ F o n t S i z e ’ , 1 4 ) ;

173

174 y=120

175 f i g u r e

176 p l o t( TotPos1 ( 1 : y ) )

177 hold on

178 p l o t( TotPos2 ( 1 : y ) )

179 hold on

180 p l o t( TotPos3 ( 1 : y ) )

181 hold on

182 p l o t( TotPos4 ( 1 : y ) )

183 hold on

184 p l o t( TotPos5 ( 1 : y ) )

185 hold on

186 p l o t( TotPos6 ( 1 : y ) )

187 legend( ’ Bubble 1 ’ , ’ Bubble 2 ’ , ’ Bubble 3 ’ , ’ Bubble 4 ’ , ’ Bubble 5 ’ , ’ Bubble 6 ’ , ’ F o n t S i z e ’ , 1 4 )

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190 t i t l e ( ’ Bubbles i n LX−14 ’ , ’ F o n t S i z e ’ , 1 4 ) ;

191

192 % f i g u r e

193 % p l o t ( yv , TotDiv )

194 % x l a b e l ( ’m/ s ’ )

195 % y l a b e l ( ’mm’ )

196 % t i t l e ( ’ Divergence from f i r s t h i t a t 260mm’ )

197 198

199 % f i g u r e

200 % TotDiv_30= s i n ( v i n k e l )

*

^300;

201 % p l o t ( yv , TotDiv_30 )

202 % h l i n e = r e f l i n e ( 0 , 3 ) ;

203 % h l i n e . Col or = ’ r ’ ;

204 % x l a b e l ( ’m/ s ’ )

205 % y l a b e l ( ’mm’ )

207

208 209

210 %%

211 %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

212 % f i g u r e %r o s t

213 % p l o t ( time , s i t _ d , time , sib_d , ’ r ’ , time , sbd_d , ’ − − ’ , time , smd_d , ’ g ’ )

214 % y l a b e l ( ’mm’ )

216 % y l i n e ( 0 , ’ − − ’ )

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217 % legend ( ’ Top −Improvised ’ , ’ Base − Improvised ’ , ’ Base − Dmg model ’ , ’ Mid − Dmg Model ’ )

218

219 f i g u r e %r o s t

220 p l o t( time , s i t , time , s i b , ’ r ’ , time , sbd , ’−− ’ , time , smd , ’ g ’)

223 t i t l e ( ’ Rust on the l i n e r ’ , ’ F o n t S i z e ’ , 1 4 )

224 legend( ’ Rust 1 ’ , ’ Rust 2 ’ , ’ Rust 3 ’ , ’ Rust 4 ’ , ’ F o n t S i z e ’ , 1 4 )

225

226

227 % f i g u r e %r o s t

228 % p l o t ( s i t _ d ( 1 : 6 0 ) )

229 % hold on

230 % p l o t ( sib_d ( 1 : 6 0 ) )

231 % hold on

232 % p l o t ( sbd_d ( 1 : 6 0 ) )

233 % hold on

234 % p l o t (smd_d ( 1 : 6 0 ) )

235 % y l a b e l ( ’mm’ )

238

239 % f i g u r e %r o s t

240 % p l o t ( time ( 1 : 6 0 ) , s i t ( 1 : 6 0 ) , time ( 1 : 6 0 ) , s i b ( 1 : 6 0 ) , ’ r −o ’ , time ( 1 : 6 0 ) , sbd ( 1 : 6 0 ) , ’ − − ’ , time ( 1 : 6 0 ) ,smd ( 1 : 6 0 ) , ’ −

*

^{’ )}

241 % y l a b e l ( ’mm’ )

242 % x l a b e l ( ’ time ( s ) ’ )

243 % y l i n e ( 0 , ’ − − ’ )

244 % t i t l e ( ’ Divergence c e n t r e a x i s a t 300mm’ )

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246 t =130

247

248 f i g u r e %r o s t

249 p l o t( time ( 1 : t ) , s i t ( 1 : t ) , time ( 1 : t ) , s i b ( 1 : t ) , ’ r −o ’ , time ( 1 : t ) , sbd ( 1 : t ) , ’−− ’ , time ( 1 : t ) ,smd ( 1 : t ) , ’ −

*

^’⁾

252 y l i n e ( 0 , ’−− ’)

253 t i t l e ( ’ Rust on the l i n e r ’ , ’ F o n t S i z e ’ , 1 4 )

254 legend( ’ Rust 1 ’ , ’ Rust 2 ’ , ’ Rust 3 ’ , ’ Rust 4 ’ , ’ F o n t S i z e ’ , 1 4 )

255

256 f i g u r e

257 p l o t( time ( 1 : t ) , s i t ( 1 : t ) , ’−− ’ , time ( 1 : t ) , s i b ( 1 : t ) , ’−− ’ , time ( 1 : t ) , sbd ( 1 : t ) , ’−− ’ , time ( 1 : t ) ,smd ( 1 : t ) , ’−− ’ , time ( 1 : t ) , TotPos1 ( 1 : t ) , time ( 1 : t ) , TotPos2 ( 1 : t ) , time ( 1 : t ) , TotPos3 ( 1 : t ) , time ( 1 : t ) , TotPos4 ( 1 : t ) , time ( 1 : t ) , TotPos5 ( 1 : t ) , time ( 1 : t ) , TotPos6 ( 1 : t ) , time ( 1 : t ) , TotPos ( 1 : t ) , ’ −

*

^’ , time ( 1 : t ) , TotPos_05mm ( 1 : t ) , ’ −

*

^’⁾

258 legend( ’ Rust 1 ’ , ’ Rust 2 ’ , ’ Rust 3 ’ , ’ Rust 4 ’ , ’ Bubble 1 ’ , ’ Bubble 2 ’ , ’ Bubble 3 ’ , ’ Bubble 4 ’ , ’ Bubble 5 ’ , ’ Bubble 6 ’ , ’ 1 mm’ , ’ 0.5mm’ , ’ F o n t S i z e ’ , 1 4 )

259 t i t l e ( ’ A l l the A s s y m e t r i e s ’ , ’ F o n t S i z e ’ , 1 4 )

262

263

264 t =250

265 f i g u r e

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266 p l o t( time ( 1 : t ) , s i t ( 1 : t ) , ’−− ’ , time ( 1 : t ) , s i b ( 1 : t ) , ’−− ’ , time ( 1 : t ) , sbd ( 1 : t ) , ’−− ’ , time ( 1 : t ) ,smd ( 1 : t ) , ’−− ’ , time ( 1 : t ) , TotPos1 ( 1 : t ) , time ( 1 : t ) , TotPos2 ( 1 : t ) , time ( 1 : t ) , TotPos3 ( 1 : t ) , time ( 1 : t ) , TotPos4 ( 1 : t ) , time ( 1 : t ) , TotPos5 ( 1 : t ) , time ( 1 : t ) , TotPos6 ( 1 : t ) , time ( 1 : t ) , TotPos ( 1 : t ) , ’ −

*

^’ , time ( 1 : t ) , TotPos_05mm ( 1 : t ) , ’ −

*

^’⁾

267 legend( ’ Rust 1 ’ , ’ Rust 2 ’ , ’ Rust 3 ’ , ’ Rust 4 ’ , ’ Bubble 1 ’ , ’ Bubble 2 ’ , ’ Bubble 3 ’ , ’ Bubble 4 ’ , ’ Bubble 5 ’ , ’ Bubble 6 ’ , ’ 1 mm’ , ’ 0.5mm’ , ’ F o n t S i z e ’ , 1 4 )

268 t i t l e ( ’ A l l the A s s y m e t r i e s ’ , ’ F o n t S i z e ’ , 1 4 )

C Impetus manual

https://www.impetus.no/support/manual/

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Asymmetries in Shaped Charges