The Development of a Semi-empirical Barrel Engraving ...

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The Development of a Semi-empirical Barrel Engraving Resistance Method for

Internal Ballistic Simulation Codes

Nawaf S Alanazi

orcid.org/0000-0002-4369-8037

Dissertation accepted in partial fulfilment of the requirements for the degree Master of Science in Mechanical Engineering at

the North-West University

Supervisor: Prof. Willem den Heijer

Co-Supervisor: Mr Victor Schabort

Graduation: June 2021

Student number: 27359875

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ACKNOWLEDGEMENT

First and foremost, I wish to express my sincerest gratitude to Rheinmetall Denel Munitions that

gave me the opportunity to conduct this research. I wish to convey my appreciation to the

research supervisor, Professor Willem den Heijer, the technical research supervisor, Mr Victor

Schabort, and Mr Hans Heunis for their valuable assistance and guidance.

I dedicate this dissertation to my parents that inspired and supported me; to my wife that sacrificed

her studies to relocate to South Africa with me whilst patiently giving me all the support and

encouragement I dearly needed; and lastly to my brothers that looked after my children while we

were abroad.

I wish to thank the Saudi Arabian Government, particularly the Military Industries Corporation,

who provided funding and gave me the opportunity to develop myself to serve my country.

I also wish to pass my sincerest appreciation to the beautiful country and people of South Africa.

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ABSTRACT

This research focused on the implementation of a semi-empirical projectile barrel engraving

resistance model for internal ballistic simulation codes. An alternative method is provided to

determine the engraving resistance profile by the direct measurement of projectile movement and

subsequent creation of the resistance curve. The identified research problem, subsequent

literature survey, and the development of a semi-empirical projectile barrel engraving resistance

model in internal ballistic simulation codes have not been previously investigated based on the

literature survey conducted.

The advantage of the developed approach is its relative simplicity and the elimination of the need

for detailed material and structural analyses, as well as measurements of the projectile driving

band material interaction with the gun barrel.

The developed approach greatly improves the confidence over the traditional method in defining

the shape of resistance curves for the simulation model of gun systems. This approach is based

on intermediate experimental data, whereas the traditional method estimates the resistance curve

definition only based on the final experimental velocity and maximum pressure.

KEYWORDS: Semi-empirical, barrel engraving, internal ballistic simulation, strain gauge,

resistive pressure, STANAG 4367.

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ABBREVIATIONS AND VARIABLES

𝐹𝑖. 𝑖 𝑡ℎ propellant force per unit mass

𝑇0𝑖 𝑖𝑡ℎ propellant adiabatic flame temperature

𝐹𝑖′ 𝑖𝑡ℎ propellant adjusted force per unit mass

𝜌𝑖 𝑖𝑡ℎ propellant density

𝛾𝑖 𝑖𝑡ℎ propellant specific heat ratio

𝑃′𝑅 Adjusted resistive pressure because of engraving and friction

𝑃𝑎 Ambient air pressure

Aw Area of the chamber wall and gun tube wall area

BD Band diameter

BL Band length

𝑃𝑅 Bore resistive pressure due to engraving and friction

𝐴𝑏𝑟 Breech face area

Z Burning rate of the mass fraction of the propellant grain

𝑉𝑐 − 𝑉0 Chamber volume enlargement due to projectile displacement

Tc Chamber wall temperature

COR Commencement of rifling

FCurve Curvature force

D Diameter of the propellant grain

d Distance

𝑥 Displacement of the projectile

𝑓𝑅 Down-tube resistance factor

𝑉0 Empty gun chamber volume

𝐸𝑑 Energy loss due to air resistance

𝐸ℎ Energy loss due to barrel walls and heat transfer to the chamber

𝐸𝑏𝑟 Energy loss due to engraving and friction of driving band

𝐸𝑟 Energy loss due to recoil

EB Exterior ballistics

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FExt Extrusion force

𝑓𝛽 Factor of burn rate

𝑓𝛽𝑇 Factor of burning rate temperature

𝑓𝑟 Factor of down-tube resistance

𝑓𝐹𝑇 Factor of force temperature

𝜃 Forcing cone included angle

Ffriction Friction force

DL Groove diameter

G/L Groove to land width ratio

�� Gun chamber space-mean pressure

𝑥𝑟𝑝 Gun Displacement due to recoil

𝑇0𝐼 Igniter adiabatic flame temperature

𝑏𝐼 Igniter covolume

𝐶𝐼 Igniter initial mass

𝐹𝐼 Igniter propellant force per unit mass

𝛾𝐼 Igniter specific heat ratio

FInc Incision force

𝑉0 Initial chamber volume

IDE Integrated development environment

IB Interior ballistics

NB Intermediate ballistics

IBS Internal ballistics simulation

DG Land diameter

LPM Lumped parameter model

𝜀 Mass ratio of ballistic

FF Metrical code

MIC Military Industries Corporation

MC Mono Cast

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MV Muzzle Velocity

NC Nitrocellulose

NG Nitroglycerine

NQ Nitroguanidine

PrN Normalised resistance pressure

NATO North Atlantic Treaty Organisation

𝑛 Number of propellants

𝑉𝑃𝑖 Parasitics associated with the 𝑖𝑡ℎ propellant volume

PDV Photonic Doppler velocimeter

KNO3 Potassium nitrate

𝑃0 Pressure of the breech

A Projectile base area including rotating band

𝑃𝑏 Projectile base pressure

𝑃𝑔 Projectile gas pressure

𝑚𝑝 Projectile mass

vp Projectile velocity

𝑉𝑐 Projectile volume available for propellant gas

𝑟 Propellant burning rate

β and α Propellant chemical composition parameters

C_gas Propellant gas mass

Vg Propellant grain initial volume

C Propellant initial total mass

CT Propellants and igniter total mass

Pr Resistance pressure

𝑃𝑅 Resistive pressure

RDM Rheinmetall Denel Munition

FRif Rifling force

TW Rifling twist

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RDX Royal Demolition Explosive

k Small residual constant

C_solid Solid propellant mass

STANAG Standard Agreement of North Atlantic Treaty Organisation

SG Strain gauge

S Sulfur

S Surface area of the propellant grain

TB Terminal ballistics

t Time

𝑡𝑟 Time of recoil

𝑉𝑐 Total chamber volume after projectile displacement

F Total resistance of force

TNT Trinitrotoluene

𝑣𝑟𝑝 Velocity of the gun recoil

V Volume of the propellant grain

𝐺𝐿𝑅 Width ratio of land and groove

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TABLE OF CONTENTS

1.1 Background ...................................................................................................... 1

1.2 Ignition System ................................................................................................. 2

1.3 Gun System ...................................................................................................... 3

1.4 Propelling Charge System ............................................................................... 3

1.5 Projectile Motion ............................................................................................... 4

1.6 Interior Ballistic Cycle and Simulation Models (IBS) ..................................... 5

1.7 Problem Statement ........................................................................................... 7

1.8 Aim .................................................................................................................... 8

1.9 Research Objectives: ....................................................................................... 8

1.10 Methodology ..................................................................................................... 8

1.11 Study Limitations ............................................................................................. 9

1.12 Contributions of the Study ............................................................................... 9

1.13 Chapter Division ............................................................................................... 9

1.14 Summary ......................................................................................................... 10

2.1 Introduction of Lumped-Parameter Thermodynamic Simulation ................ 11

2.1.1 Equation of Motion ............................................................................................ 12

2.1.2 Pressure Gradient Model .................................................................................. 12

2.1.3 Combustion Models .......................................................................................... 13

2.1.4 Equation of State .............................................................................................. 15

2.1.5 Energy Model ................................................................................................... 16

2.2 Engraving and Bore Resistance Cycle .......................................................... 18

2.2.1 History of Engraving and Bore Resistance Cycle .............................................. 18

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2.2.2 Simulation on the engraving process of the projectile rotating band .................. 25

2.2.3 Concept of Engraving and Bore Resistance Cycle ............................................ 29

2.2.4 Resistive Pressure ............................................................................................ 30

2.3 Conclusion ...................................................................................................... 31

3.1 Background .................................................................................................... 32

3.2 Weapon System and Setup ............................................................................ 32

3.3 Gun preparation .............................................................................................. 33

3.4 Strain Gauge Installation Methodology ......................................................... 34

3.4.1 Employment of Strain Gauges, ......................................................................... 34

3.4.2 Strain Gauge Sensor ........................................................................................ 35

3.5 Test Setup ....................................................................................................... 37

3.6 Firing Trail ....................................................................................................... 39

3.7 Practical Test Instrumentation Limitation ..................................................... 40

3.8 Chapter Summary ........................................................................................... 41

4.1 Introduction .................................................................................................... 42

4.2 Test Setup ....................................................................................................... 42

4.3 Tests Results .................................................................................................. 42

4.3.1 Shot Number 1 Results ..................................................................................... 43


4.3.3 Shot Number 3 Results: .................................................................................... 45



4.4 Test summary ................................................................................................. 48

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4.5 Chapter summary ........................................................................................... 49

5.1 Introduction .................................................................................................... 50

5.2 Data Analysis .................................................................................................. 50

5.3 Data Summary ................................................................................................ 52

5.4 Model Construction ........................................................................................ 53

5.5 Model Verification ........................................................................................... 57

5.6 Results of the Semi-empirical Resistive Pressure Curve of 35mm

Cannon ............................................................................................................ 57

5.7 Implementation of the Semi-Empirical Resistive Pressure Curve of

35mm Cannon into the Developed Model ..................................................... 59

5.8 Limitations of the experimental data ............................................................. 60

5.9 Chapter summary ........................................................................................... 60

6.1 Research Overview......................................................................................... 61

6.2 Conclusion ...................................................................................................... 61

6.3 Future Work .................................................................................................... 62

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LIST OF TABLES

Table 2-1 The resistive pressure inside a gun system ............................................................. 19

Table 2-2 Material properties that affect the stability of the projectile ....................................... 24

Table 2-3 Comparison of simulation results ............................................................................. 28

Table 3-1 Locations of the strain gauges on the barrel ............................................................ 37

Table 3-2 Propellant mass in each shot ................................................................................... 39

Table 4-1 Results of the measurements at different shots ....................................................... 48

Table 5-1 RDM IB model vs Developed Model ........................................................................ 57

Table 5-2 The semi-empirical resistive pressure at different time intervals .............................. 59

Table 5-3 The effect of the resistance pressure sub-model on IB results ................................. 60

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LIST OF FIGURES

Figure 1-1 Sub-systems of a typical ballistic system .................................................................. 2

Figure 1-2 Ignition sub-system in the ballistic system ................................................................ 2

Figure 1-3 Rifled-bore and smooth-bore gun sub-systems in a ballistic system ......................... 3

Figure 1-4 Difference between the smooth-bore and rifled barrels ............................................ 5

Figure 1-5 Effect of the bore on projectile motion after exiting the gun muzzle (Koffler,

2015).............................................................................................................. 5

Figure 1-6 Internal layout of the loaded gun system .................................................................. 6

Figure 1-7 Chamber pressure profile for the gun system ........................................................... 6

Figure 2-1 Illustrating Flowchart of LPM of interior ballistics .................................................... 11

Figure 2-2 Pressure gradient profile ........................................................................................ 13

Figure 2-3 Schematic of a typical PDV system (Sättler, 2013) ................................................. 21

Figure 2-4 Rack with optical components (left) and oscilloscope (right) (Sättler, 2013)............ 21

Figure 2-5 Working principle of the PDV (Sättler, 2013) .......................................................... 22

Figure 2-6 Pressure and velocity as a function of time ............................................................ 22

Figure 2-7 Acceleration and velocity as a function of displacement ("Planar Doppler

Velocimetry,") ............................................................................................... 23

Figure 2-8 Partially enlarged drawing of touch between band and barrel ................................. 23

Figure 2-9 Dynamic response of projectile centre of gravity in the system with different

band materials during the engraving process ............................................... 25

Figure 2-10 Schematic drawing of rotating band engraving process (Quanzhao, 2017) ......... 26

Figure 2-11 Evolution of Von Mises stress for rotating band (Quanzhao, 2017) ..................... 27

Figure 2-12 Engraving resistance (Quanzhao, 2017) ............................................................. 28

Figure 2-13 Inside of the gun system ....................................................................................... 29

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Figure 2-14 Total resistive forces acting on projectile motion (Crowley, 1999, November) ...... 30

Figure 2-15 Variation of the resistance force due to friction with projectile motion (Crowley) ... 31

Figure 3-1 experimental weapon setup .................................................................................... 33

Figure 3-2 The preparations of the gun ................................................................................... 33

Figure 3-3 Definition of strain................................................................................................... 35

Figure 3-4 Construction and the principle of the work of a strain gauge ................................... 36

Figure 3-5 Full-bridge configuration of strain gauges ............................................................... 36

Figure 3-6 Locations of the strain gauges on the barrel ........................................................... 37

Figure 3-7 Bridge of strain gauges on the barrel of the 35mm ................................................. 38

Figure 3-8 Test methodology and setup .................................................................................. 38

Figure 4-1 Shot Number 1 results ............................................................................................ 43





Figure 4-6 Breech Pressure and SGs vs time for shot number 4 ............................................. 48

Figure 5-2 Determination of shot exit pressure ........................................................................ 51

Figure 5-3 Iterative procedure for measuring the actual resistance pressures ........................ 52

Figure 5-4 Flow chart plan for the IB model ............................................................................. 56

Figure 5-5 Showing the semi-empirical resistive pressure curve of 35mm cannon .................. 58

Figure 5-6 The semi-empirical resistive curve of the 35mm cannon ........................................ 59

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CHAPTER 1 INTRODUCTION

1.1 Background

The attention to ballistics began with the use of traditional tools such as spears, arrows, bows,

and slingshots. In the early 9th century, the revolution of the ballistics field was developed by the

discovery of gunpowder, which is an essential part of the ballistic systems. Chinese scientists

accidentally encountered gunpowder from the reaction of charcoal, potassium nitrates, and

sulphur, which recrystallised from boiling water. The generation of these materials, described as

extremely explosive, led to the development of the ballistics field. After four centuries, gunpowder

has been used as a chemical propelling charge, in which it is ignited and to push and accelerate

a projectile through the barrel of a gun. The extensive progress in the field opens the gates for a

new field of ballistics (Gray, E., Marsh, H., & McLaren, M., 1982) (Gunpowder, 2019) (Kelly,

2004).

Ballistics can be defined as the scientific investigation of projectile motion and classified into four

disciplines, namely interior, intermediate, exterior, and terminal ballistics. The principles of these

disciplines are described as follows:

Interior ballistics (IB) deals with the interaction on the gun behaviour, propelling charge,

and projectile from the ignition of the propelling charge to the point in time when the

projectile exits the gun muzzle.

Intermediate ballistics deals with the period just after the projectile exits the barrel where the combustion gasses influence the orientation of the projectile.

Exterior ballistics (EB) covers the projectile behaviour from the time that the projectile exits

the gun muzzle until it reaches the target.

Terminal ballistics (TB) encompasses the comprehensive data of all the aspects when the

projectile impacts the target. (Carlucci & Jacobson, 2018) (Okunev, 1943)

A ballistics system comprises four sub-systems as shown in Figure 1-1, namely the projectile, the

propelling charge, the gun, and the igniter,.

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Figure 1-1 Sub-systems of a typical ballistic system

In the following sections, more details will be provided regarding these sub-systems.

1.2 Ignition System

The ballistics process starts when the projectile is loaded from the back of the barrel. Ignition

commences when the firing pin crushes the primary cartridge, which produces a flame that is

transferred through the flash hole to the propellant grain (see Figure 1-2). The main function of

the primary cartridge is to ignite the propellant grain.

The two types of primary cartridges are boxer and Berdan cartridge primers. Both types have

same role but different mechanism that boxer has one hole to transfer the ignition to the main

cartridge while Berdan has two holes between the primary and main cartridges.

(Worsfold, Townshend, Poole, & Miro, 2019.)

Figure 1-2 Ignition sub-system in the ballistic system

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1.3 Gun System

The gun system in medium calibre applications contains the following components as shown in

Figure 1-3, namely the breech, the barrel and the chamber. A gun barrel is made from a high-

strength metallic tube that allows the projectile to travel. The front end of the barrel is called the

muzzle in which the projectile comes out from. The interior of the barrel, known as the bore, can

be classified into two types , as shown in Figure 1-3 (Deng, Sun, Chiu, & Chen, 2014).. The first

type is smooth bore which has flat surface throughout the barrel. The second type is rifled bore

which has circumferential grooves to spin the projectile throughout the barrel.

The barrel has a smaller diameter than the chamber to accommodate the propelling charge. Thus,

the diameter decreases from the chamber (chambrage) through the forced cone to the bore. This

is shown in Figure 1-3.

The purpose of the chambrage is to direct the projectile into the bore. The back-end of the

chamber is sealed by a breechblock. Interestingly, some guns have similar diameter sizes

between the chamber and bore with the absence of a forcing cone.

Figure 1-3 Rifled-bore and smooth-bore gun sub-systems in a ballistic system

1.4 Propelling Charge System

In recent years, a new type of propelling charge, known as gun propellant, has been discovered.

Gun propellant has several advantages, such as clean and smokeless combustion, hydrophobic

properties. Also, gun propellant has more applications than gunpowder owing to its ability to

produce high temperatures and a large amount of energy. (Kubota & VCH-Verlagsgesellschaft.,

2015)

The propelling charge is a granular chemical substance that is ignited by the flame transferred

from the igniter. The resulting rapid combustion of the propelling charge generates hot expanding

gases with high energy to deliver significant pressure build-up. This chemical reaction is called

deflagration which has shockwave propagation lower than sonic. This pressure is responsible for

propelling the projectile forward within the gun barrel, the path of least resistance for the pressure.

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The explosive materials and propelling charges are produced from the same family of energetic

materials. The primary difference is that the explosive materials detonate from shockwaves,

producing uncontrollable explosions, while the propelling charges have subsonic burn rates that

are more controllable, (Kubota & VCH-Verlagsgesellschaft., 2015).

There are three types of propellants:

single-based propellant, consisting primarily of nitrocellulose (NC) as the energetic

material

double-based propellant, containing primarily nitroglycerine (NG) and NC

triple-based propellant, consisting primarily of NG, NC and Nitroguanidine (NQ)

The selection of which propellant to use is based on the energy requirement, processing

techniques available, toxicity limitations, rate of combustion, and weapon system requirements

(Kirchner, 1993).

The rate of gas generation depends on the shape of the propellant grain. There are three types

of propellant grains, classified based on their geometry:

digressive burning grains;

neutral burning grains; and

progressive burning grains.

Once a digressive burning grain begins to burn, its surface area decreases over time; thus, gas

generation decreases over time. A neutral burning grain’s surface area remains constant; thus,

gas generation remains constant. A progressive burning grain increases its total surface area

over time, subsequently, gas generation increases.

1.5 Projectile Motion

The life cycle of projectile motion inside the barrel depends on the barrel type. As mentioned

above in the gun system section, there are smooth-bore and rifled barrels. The smooth barrel is

a smooth metallic tube, and the smooth bore affects the projectile motion by not forcing the

projectile to spin after exiting the gun muzzle, reducing the stability of the projectile. The rifled

barrel has lands and grooves, which force the projectile to spin after exiting the gun muzzle. The

induced spin increases the balance, range, and stability of the projectile (Koffler, 2015). Figure 1-4

and Figure 1-5 show the differences between the smooth-bore and rifled barrels.

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Figure 1-4 Difference between the smooth-bore and rifled barrels

Figure 1-5 Effect of the bore on projectile motion after exiting the gun muzzle (Koffler, 2015)

This research project focusses only on the life cycle of projectile motion inside the rifled barrel,

particularly during the internal ballistic cycle.

1.6 Interior Ballistic Cycle and Simulation Models (IBS)

The internal layout of the loaded ballistic system is shown in Figure 1-6. The IB cycle occurs in

several steps:

1. The primary cartridge located behind the propellant is ignited by percussion to provide a

source of heat.

2. This heat ignites the propellant causing a pressure build-up.

3. Pressure builds in the projectile’s casing.

4. At the shot start pressure, the projectile begins to move inside the barrel due to the high

pressure resulting from the propellant combustion (Schabort., 2016).

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Figure 1-6 Internal layout of the loaded gun system

The engraving of the driving band into the circumferential grooves begins at this point. The

projectile encounters resistance encountered from friction and engraving forces during its

acceleration down the barrel, which is the main focus of this research. The friction force between

the rotational band and the bore wall increases before containing the entire rotational bond. After

that, the friction force between the two surfaces significantly reduces, causing an acceleration of

the projectile. As the projectile passes through the barrel, the volume of the chamber begins to

increase, leading to a pressure reduction. The projectile continues to travel towards the muzzle,

increasing the confined volume for the combustion gasses. When the projectile leaves the muzzle

completely, a blast is observed owing to the pressure difference between the propellant gas and

ambient pressures. Figure 1-7 shows the chamber pressure of the gun system during firing

(Schabort., 2016).

Figure 1-7 Chamber pressure profile for the gun system

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Figure 1-7, the pressure increases significantly and rapidly to the point of maximum pressure

when the projectile is displaced. The pressure then decreases dramatically from the movement

of the projectile through the barrel that leads to an increase in the volume behind the projectile.

Scientists in ballistic fields are studying the behaviour of the projectile to determine the optimum

parameters and predict behaviour with mathematical code that describes the IB cycle. The goal

is to fill the knowledge gap of aspects within the internal ballistic system. The new simulation

pathway identified as internal ballistics simulation (IBS) models that can be categorised as follows:

zero-dimensional internal ballistic (0D-IB), one-dimensional internal ballistic (1D-IB), and two-

dimensional internal ballistic (2D-IB). These categories are used to calculate variable parameters,

including density, velocity, mass, and pressure (Schabort., 2016).

1.7 Problem Statement

IBS models make use of, amongst others, a projectile resistance model to simulate the internal

ballistic cycle. Several approaches have been proposed in the internal ballistics field, ranging from

an empirical approach with the simple fit of internal factors to a semi-empirical approach in which

projectile resistance is calculated based on theoretical modelling using prior experimental

investigations. A problem that internal ballisticians encounter is that the pure empirical, fitting

factor approach is easy to implement but has shortcomings in terms of prediction capabilities.

However, to improve general ballistic prediction capabilities, a fundamental analytical approach,

pose modelling, and experimental investigations require dedicated research and analysis

infrastructure.

The resources for such ventures are generally not available in the industry. An intermediate

approach is proposed as a solution, which is likely more reachable within the typical industry

organisation’s capabilities.

The existing models used by internal ballisticians, which comply with the Standardization

Agreement of NATO (STANAG 4367, 2012), are lumped-parameter thermodynamic simulation

(LPM) packages. These models predict propellant combustion, gas pressure, pressure gradient,

heat loss, friction resistance, projectile movement, and related internal ballistic parameters.

However, in this dissertation, an experimental approach is considered to obtain empirical

resistance values from direct measurements of projectile travel at various selected displacements

inside the barrel. These empirical values are then utilised to construct a semi-empirical resistance

curve as a function of travel for the specific gun system.

Therefore, the problem can be solved by empirically determining the resistance as a function of

projectile travel during the ballistic cycle. The experiments performed in this study use a 35mm

anti-aircraft weapon as the test platform.

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1.8 Aim

This study offers an alternative approach to derive the resistance parameters from the empirical

measurement of the movement of the projectile down the barrel during the firing cycle. This

method could improve the quantitative and qualitative characteristics of the empirical approach

based on simple fitting factors. The advantages of the proposed approach are its relative simplicity

and operation without detailed material/structural analysis and related measurements of the

projectile driving band material interaction within the gun barrel. The method establishes nominal

resistance curves of representative ballistic systems and enhances the modelling of the

resistance curve with an improved empirical model.

1.9 Research Objectives:

The specific objectives of this study are as follows:

- Establish a test platform for measuring projectile travel and relevant ballistic parameters

during live firings.

- Develop a method and related software tools to analyse the measured data and derive a

resistance curve for the ballistic system.

- Obtain suitable data from live firings conducted from a series of experiments, which then

could be utilised to derive resistance curves to be implemented in an internal ballistic

simulation model.

- Demonstrate the implementation of such a resistance model in comparison to the

traditional empirical model.

1.10 Methodology

This research will establish a reference for the IBS model based on the STANAG standard and

its related resistance model. A complete literature study on all related topics will be performed

with complete background knowledge on all involved disciplines. The test layout for a ballistic

system of choice will be designed that will employ qualitative and quantitative methods. The

research methodology will involve the following detailed activities:

- Literature study on current models and methods;

- Definition, description, and formulation of the analysis method to be used to derive the

resistance curve from the measured data;

- Ballistics tests and data acquisition;

- Implementation and verification of an internal ballistic code based on the STANAG

standard (using the MATLAB software platform);

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- Analysis of the data with the data reduction methods defined above to derive the

characteristic resistance curves;

- Implementation of the resistance curves into the verified IBS model; and

- Verification of the method by comparing the result to the measured data for the ballistic

configuration deployed.

1.11 Study Limitations

This study only focusses on semi-empirical barrel engraving and not on other components of the

IBS model. The data derived from this study will apply specifically to the selected testing platform

(i.e. 35mm cannon gun, rifled barrel).

1.12 Contributions of the Study

The results of this study will contribute to the fidelity of the internal ballistic model. The physics

will be more closely represented when the resistance curve is neither estimated with an educated

guess nor as a correction factor to match the muzzle velocity of the simulation with the

experimental result.

Also, the advantage of this approach is its relative simplicity and the elimination of the need for

detailed material/structural analysis and related measurements of the projectile driving band

material interaction with the gun barrel. It also may provide a more realistic initial estimation of

the resistance pressure based on experimental data for characteristic ballistic systems.

1.13 Chapter Division

The remainder of the present dissertation is structured as follows:

Chapter 2 will present a review of existing literature and the latest developments concerning

resistance models of internal ballistics systems. It will highlight available scholarly work in the field

of internal ballistics and present the various concepts necessary to explain the internal ballistic

system.

Chapter 3 deals with the methodology of the experiment, safety precaution, weapon system and

setup, gun preparation, strain gauge installation, firing trail, practical test instrumentation

limitation, and test layout for the experiment.

Chapter 4 describes the results obtained from the experimental tests, including the relationship

between pressure, time, and strain gauge (SG) variables at different shots.

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Chapter 5 describes the data analysis and processing of the IBS model simulated by MATLAB

software. The data summary and the results of the semi-empirical resistive pressure curve of

35mm cannon are presented. The results are also verified by comparing the Rheinmetall Denel

Munition (RDM) IBS model with the developed model.

Chapter 6 concludes the research by recalling all the previous chapters and gives an overview,

the final conclusion shall be drawn. Moreover, this chapter will suggest fields and areas of future

studies that shown a gab needed to be filled and taken a place in future research and efforts.

1.14 Summary

This chapter presented an introduction to the models used and the limitations within the internal

ballistics body of work. Further information on types of gun systems and the classification of gun

barrels was given. The chapter then noted stages in which the projectile passes inside and outside

the barrel and the energy responsible for the movement of the projectile. Finally, the research

problem, aim of the research, limitation, and contributions were listed.

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CHAPTER 2 BACKGROUND STUDY

2.1 Introduction of Lumped-Parameter Thermodynamic Simulation

The STANAG on the specifications and standards of more than 1,300 munitions, weapons,

communications, and maps is designed to facilitate the exchange of information around military

specifications. Ballisticians consider the STANAG 4367 LPM model as a critical tool in the ballistic

field because of its ability to predict the maximum pressure and the muzzle velocity of the weapon

system. (Schabort., 2016)

The STANAG 4367 LPM model comprises the following: energy model, pressure gradient model,

combustion model, equation of motion, and equation of state, as displayed in Figure 2-1.

Ballisticians generate significant predictions across the entire internal ballistic cycle, including the

maximum projectile acceleration and spin rate, maximum chamber pressure, maximum base

pressure, muzzle velocity of the projectile, barrel exit pressure, efficiency of the ballistic system,

propellant’s burn time, gun backward velocity caused by recoil, bore maximum wall temperature,

maximum gas rate created from propellants, time at the maximum pressure, maximum kinetic

energy of the projectile, exit time of the barrel, and the interior ballistic cycle duration time

(Dyckmans, 2015).

Figure 2-1 Illustrating Flowchart of LPM of interior ballistics

The energy model, pressure gradient model, combustion model, equation of motion, and equation

of state shown in Figure 2-1 are described in detail in the next sections.

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2.1.1 Equation of Motion

The equation of motion model investigates the velocities of the projectile and the recoiling parts.

The concept of the model can be attributed to the pressure increase generated from the ignition

of the propellant causing a large force on the base of projectile and breech, leads to an

acceleration in forwards and backwards. The velocity of the projectile 𝑣𝑟𝑝 can be calculated in the

next equation:

𝑣𝑟𝑝 =

𝑑(𝑣𝑝)

𝑑𝑡=

𝐴(𝑃𝑏 − 𝑓𝑅𝑃𝑅)

𝑚𝑝,

(1)

where t is time, the projectile mass is denoted as 𝑚𝑝, the projectile pressure on the base is

denoted as 𝑃𝑏, A is the projectile area on the base, including the rotating band, and 𝑓𝑅 is the

down-tube resistance factor. The bore resistance 𝑃𝑅 is due to the engraving and friction. Now, A

can be determined by Equation 2:

𝐴 =𝜋

4𝐷𝑏

2, (2)

where 𝐷𝑏 is bore diameter, which depends on the nature of the bore as smooth or rifled. The

differences between a smooth and rifled barrel were explained previously and shown in

Figure 1-4. Hence, the rifled bore diameter can be determined using Equation 3:

𝐷𝑏

2 = [𝐺𝐿𝑅 ∗ 𝐷𝐺2

𝐺𝐿𝑅 + 1+

𝐷𝐿2

𝐺𝐿𝑅 + 1],

(3)

where GLR is the width ratio of land and groove, DL is the groove diameter, and DG is the land

diameter. In the case of the velocity of recoiling parts 𝑣𝑟𝑝 can be determined as

𝑣𝑟𝑝 = −

𝐴𝑏𝑟𝑃0 − 𝑅𝑅 − 𝐴𝑓𝑅 𝑃𝑅

𝑚𝑟𝑝,

(4)

where 𝐴𝑏𝑟 is the breech face area, 𝑃0 is the breech pressure, 𝑚𝑟𝑝 is the mass of the recoil

parts, and RR is the recoil resistance force (Moreno, 2009).

2.1.2 Pressure Gradient Model

When the acceleration of the projectile begins to grow, the expansion wave moves toward the

breech resulting in a high rarefaction wave impact. At this point, the pressures in the base and

breech are equal. After the rarefaction wave contacts the projectile, the pressure ratio drops

dramatically leading to a decrease in the kinetic energy and propellant acceleration of gases. The

13

breech pressure is relatively higher than the projectile’s base pressure, as shown in Figure 2-2.

(Hansen & Heiney)

Figure 2-2 Pressure gradient profile

The Lagrange gradient model is vital in the investigation of the pressure gradient phenomenon,

connecting the breech and base pressure (Morrison, Wren, Oberle, & Richardson, 1993). The

projectile pressure on the base 𝑃𝑏 can be calculated using the following equation:

𝑃𝑏 =

[�� +𝐶𝑇(𝑓𝑅𝑃𝑅)

3𝑚𝑝]

[1 +𝐶𝑇

3𝑚𝑝]

,

(5)

where �� is the space-mean pressure, 𝐶𝑇 is the total mass of propellants and igniter. Now, the

pressure of the breech 𝑃0 can be investigated as follows (Moreno, 2009):

𝑃0 = 𝑃𝑏 +

𝐶𝑇

2𝑚𝑝

(𝑃𝑏 − 𝑓𝑅𝑃𝑅) .

(6)

2.1.3 Combustion Models

The main role of the combustion model is to determine the gas produced by propellant during the

ignition process. The ignition of propellant depends on a) the nature of the chemical composition,

because high energetic materials burn faster than the low energetic materials, and b) the

surrounding pressure because higher surrounding pressure burns the propellant faster than the

lower surrounding pressure. The fraction of mass burned of ith propellant 𝑍𝑖 can be defined as

follows:

14

𝑑𝑍𝑖

𝑑𝑡=

𝑆𝑖𝑟𝑖

𝑉𝑔𝑖 ,

(7)

where Si is the surface area of the propellant grain, and Vgi is the propellant grain initial volume.

Si and Vgi are calculated using algebraic equations based on the shape of the propellant grain.

The adjusted propellant burning rate 𝑟𝑖 can be described as follows:

𝑟𝑖 = 𝑓𝐵𝑓𝐵𝑇𝛽𝑖(��)𝑎𝑖 + 𝑘𝑣𝑣𝑝 , (8)

where 𝛽 and 𝛼 are parameters describing the chemical composition of the propellant, 𝑓𝐵 is the

burning rate factor, 𝑓𝐵𝑇 is the burning rate temperature factor, and 𝑘𝑣 is the erosive burning

coefficient. The value of 𝑘𝑣 mainly relies on the shape of the propellant grain and ��.

As mentioned previously, the surface area S and volume V of the propellant grain are dependent

on its geometry, exemplified by the following types of grains: spherical, slab, single-perforation,

seven-perforated, and slotted stick. For a spherical grain, the value of V and S are calculated

based on the following equations:

V =1

6 π(D)3 and (9)

S = π(D)2, (10)

where D is the diameter of propellant grain. For a slab grain, V and S can be described by the

following:

𝑉 = 𝐿(𝑊𝑇)(𝑊𝑆) and (11)

𝑆 = 2[𝐿(𝑊𝑆) + 𝐿(𝑊𝑇) + (𝑊𝑆)(𝑊𝑇)], (12)

where WS is the slot width, WT represents slab thickness, and L is the length of the propellant

grain. For single-perforation grain, V and S are calculated as follows:

𝑉 =𝜋

4 (𝐿 − 2𝑥)[(𝐷 − 2𝑥)2 − 7(𝑃 − 2𝑥)2] and (13)

𝑆 =

2𝑉

(𝐿 − 2𝑋)+ 𝜋(𝐿 − 2𝑥)(𝐷 + 7𝑃 + 12𝑥),

(14)

where DL is the diameter of rifling lands. For a seven-perforated grain, V and S are determined

using the following equations:

15

𝑉 =1

4 𝜋𝐿(D2

− P2 ) and (15)

𝑆 = 𝜋 (𝐷𝐿 + 𝑃𝐿 +

D2

2−

P2

2).

(16)

Finally, for a slotted stick propellant grain, V and S can be determined using the following

equations (Moreno, 2009);

𝑉 = 2𝐿[(𝜋 − 𝜃)R2 − (𝜋 − 𝑎)r2 − (1

2 SW R cos θ −

1

2 SW r cos 𝑎)] and (17)

𝑆 = 2𝐿[(𝜋 − 𝑎)𝑟 + 𝑅 𝑐𝑜𝑠 𝜃 − 𝑟 𝑐𝑜𝑠 𝑎]. (18)

2.1.4 Equation of State

The equation of state is used to calculate the pressure of the space-mean chamber �� of the

ballistic system after determining the energy of propellants. The pressure of the space-mean

chamber can be determined using the following equation:

�� =

𝑇

𝑉𝑐 [∑

𝐹𝑖′𝐶𝑖𝑍𝑖

𝑇0𝑖+

𝐹𝐼𝐶𝐼

𝑇0𝐼

𝑛

𝑖=1

] .

(19)

Then, the projectile volume available for propellant gas 𝑉𝑐 can be determined using the equation

below:

𝑉𝑐 = 𝑉0 − ∑ 𝑉𝑃𝑖

+ 𝐴𝑥 −

𝑛

𝑖=1

∑𝐶𝑖

𝜌𝑖

𝑛

𝑖=0

(1 − 𝑍𝑖) − ∑ 𝐶𝑖𝑏𝑖𝑍𝑖 − 𝐶𝐼𝑏𝐼

𝑛

𝑖=0

,

(20)

where 𝑇 is the temperature of the gases, 𝐹𝑖′ is the adjusted force per mass of ith propellant, 𝐹𝐼 is

the force per mass of ith propellant, 𝐶𝑖 is the initial mass of ith propellant, 𝐶𝐼 is the initial mass of

igniter, 𝑇0𝑖 is the adiabatic flame temperature of ith propellant, 𝑇0𝐼 is the adiabatic flame

temperature of igniter propellant, 𝑉0 is the volume of empty cannon chamber, 𝑉𝑃i is the volume of

the parasitics associated with ith propellant, 𝜌𝑖) is the density of ith propellant, and the 𝑏𝐼 is the

igniter covolume. From Equation 20, the increasing volume of the chamber as the projectile moves

can be determined. Similarly, the chamber volume builds up as a propellant charge reduces

because of burning (Moreno, 2009).

16

2.1.5 Energy Model

Energy cannot be created or destroyed, but it can transfer from one phase to another. The

reaction of the propellant is exothermic, which means heat and work are generated during the

ignition process. The available energy in the propellant gases can be determined using the

temperature-of -the-gases equation, as shown below:

𝑇 =∑

𝐹𝑖′ 𝐶𝑖𝑍𝑖

𝛾𝑖 − 1 +𝐹𝐼𝐶𝐼

𝛾𝐼 − 1𝑛𝑖=1 − 𝐸𝑝𝑡 − 𝐸𝑝𝑟 − 𝐸𝑝 − 𝐸𝑏𝑟 − 𝐸𝑟 − 𝐸ℎ

∑𝐹𝑖

′ 𝐶𝑖𝑍𝑖

(𝛾𝑖 − 1)𝑇0𝑖+

𝐹𝐼𝐶𝐼

(𝛾𝐼 − 1) 𝑇0𝐼

𝑛𝑖=1

,

(21)

where the variables are defined as follows:

- 𝑛 is the number of propellants.

- 𝑇0𝐼 is the igniter adiabatic flame temperature.

- 𝐹𝑖′ is the 𝑖𝑡ℎ propellant adjusted force per unit mass.

- 𝑇0𝑖 is the 𝑖𝑡ℎ propellant adiabatic flame temperature.

- 𝐸ℎ is the energy loss because of barrel walls and heat transfer to the chamber.

- 𝐹𝐼 is the igniter propellant force per unit mass.

- 𝐶𝐼 is the initial igniter mass.

- 𝐸𝑟 is the energy loss due to recoil.

- 𝛾𝑖 is the 𝑖𝑡ℎ propellant specific heats ratio.

- 𝐸𝑏𝑟 is the energy loss because of engraving and friction of driving band.

- 𝛾𝐼 is the igniter specific heats ratio.

- 𝐸𝑝𝑡 is the energy consumed because of projectile translation.

- 𝐸𝑝 is the energy loss because of unburned propellant motion and propellant gas.

- 𝐸𝑝𝑟 is the energy loss because of projectile rotation.

The kinetic energy 𝐸𝑝𝑡 obtained from the total gas energy can be determined using the following

equation:

𝐸𝑝𝑡 =𝑚𝑝𝑣𝑝

2

2 ,

(22)

The energy in the projectile 𝐸𝑝𝑟 is the sum of translational kinetic energy and the rotational kinetic

energy, a relatively small value, and can be determined as follows:

17

𝐸𝑝𝑟 =𝜋2𝑚𝑝𝑣𝑝

2

4𝑇𝑊2 ,

(23)

where 𝑇𝑊 is the rifling twist. 𝐸𝑝 can be determined by the following equation:

The energy loss due to the engraving and friction of the driving band 𝐸𝑏𝑟 is determined in the

following equation:

where A is the projectile base area, including a suitable portion of the driving band, and 𝑓𝑅 is the

bore adjusted resistance during the engraving process (x < xe). More details about the engraving

and bore resistance cycle are displayed in Engraving and Bore Resistance Cycle section below.

The energy loss due to the recoil process 𝐸𝑟 can be determined using the following equation:

𝐸𝑟 =

𝑚𝑟𝑝𝑣𝑟𝑝2

2 .

(26)

𝐸ℎ can be determined using the following equation:

𝐸ℎ = ∫ 𝐴𝑤ℎ(𝑇 − 𝑇𝑐)

𝑡

0

𝑑𝑡 ,

(27)

where Tc is the chamber wall temperature, and ℎ is the Nordheim, Soodak, and Nordheim heat

transfer coefficient. The area of the chamber wall and gun tube wall area Aw can be determined

using the following equation (Moreno, 2009):

The STANAG 4367 LPM model requires assumptions to process the model owing to the limited

testing and theoretical considerations, including the following: .

All ignited materials fully burn at the same rate at the beginning of the simulation.

All propellant grains are made from the same material and, hence, have the same chemical and physical properties.

𝐸𝑝 =𝐶𝑇𝑣𝑝

2

6 ,

(24)

𝐸𝑏𝑟 = 𝐴 ∫ 𝑓𝑅

𝑥

0

𝑃𝑅𝑑𝑥 , (25)

𝐴𝑤 =𝑉0

A𝜋 𝐷𝑏 + 2A + 𝜋 𝐷𝑏(𝑥).

(28)

18

The mass of the projectile remains constant, along with the interior ballistic system

(Bougamra & Lu, 2014).

The movement of the projectile is restricted by the presence of shot-start pressure. The

pressure inside the rifled barrels is estimated to be within the range of 15 to 20 MPa

(Moreno, 2009).

There is a direct relationship between the distance and the engraving resistance inside

the barrel. These values can be estimated using the geometry interference between

forcing cone and driving band.

Several assumptions used as inputs for the Lagrange gradient model include the following:

There is uniform gas density from breech to the projectile’s base.

The diameters of the ballistic system chamber and the barrel are equal.

Propellant ignition occurs before the motion of the projectile. These assumptions

contribute to minor errors in the LPM (STANAG 4367, 2012) model (Moreno, 2009).

The computational calculations for the velocity and pressure of the STANAG 4367 LPM calibration

system introduce some error relative to the experimental calculations. Calibration factors, ranging

from zero to one, are applied to the computational equations for correction. The STANAG 4367

LPM has calibration factors that include the following: the burn rate factor 𝑓𝛽, which is used to

adjust the burn rate coefficient and exponent, the burning rate temperature factor 𝑓𝛽𝑇, which

corrects the propellant charge temperature; the force temperature factor 𝑓𝐹𝑇, which is applied to

the propellant force constant; and the down-tube resistance factor 𝑓𝑅, which adjusts the value of

the down-tube resistance (Moreno, 2009). The next section will focus on the energy loss due to

the engraving and driving band into the interior ballistic system.

2.2 Engraving and Bore Resistance Cycle

2.2.1 History of Engraving and Bore Resistance Cycle

One of the earliest studies of the engraving and bore resistance cycle was conducted by Wolf and

Cochran in 1979. They performed an empirical investigation to propose a model that accurately

calculates the resistive pressure inside a gun system for small and medium calibres. The exact

19

resistive pressure was measured at particular points along the barrel, contributing to the

mathematical model that is widely used among ballisticians. Table 2-1 shows the normalised

resistive pressures PrN inside a gun system. Equation 36

Table 2-1 The resistive pressure inside a gun system

PrN Travel band lengths

(MPa) (psi) (m)

13.8 2 000 0.0

103.4 15 000 0.2

103.4 15 000 0.8

69.0 10 000 1.0

48.3 7 000 1.5

27.6 4 000 4.0

17.2 2 500 10.0

13.8 2 000 30.0

10.3 1 500 60.0

10.3 1 500 2 000.0

Equation 35 shows the calculation for resistance pressure Pr from the normalised values:

𝑃r =𝑃𝑟𝑁(𝐵𝐷 − 1.0)(𝐵𝐿) (

1.92𝐺/𝐿

) 𝐹𝐹

𝐶𝑜𝑠𝜃2

+ 𝑘,

(35)

where Pr is resistance pressure, PrN is normalised resistance pressure, BD is band diameter,

calibres, BL is band length, calibres, FF is the metrical code (1.0 for copper or iron, 0.2 for plastic),

𝜃 is forcing cone included angle, G/L is the ratio of groove to land width, and k is small residual

constant.

More recently, the Sättler method has been considered a critical study in this field (Sättler, 2013).

The study incorporates modern techniques to investigate the projectile movement inside a gun

tube and produce near-accurate results. The results have not been verified yet; however, Sättler

published his methodology, allowing the approach to be applied in this research.

In the Sättler method, speed, time, and displacement are three essential factors that determine

the overall performance of a gun or projectile. Sättler describes speed as the strongest factor

affecting the science of internal ballistics and the life of the projectile. In 2013, Sättler et al., in

20

cooperation with the Fraunhofer Institute for High-Speed Dynamics, conducted an experiment of

the projectile movement which focused on the following:

Identification and evaluation of the performance of the gun and projectile system

Validation of mathematical models and numerical simulations for essential information,

such as initial movement, speed, and displacement

The simple concept of a photonic Doppler velocimeter (PDV) system is attributed to light scattered

by particles seeded in the flow that is Doppler-shifted in frequency. The magnitude of the shift is

dependent on both the propagating direction of the laser and the view of the region under

investigation. The velocity of the particle is associated with the Doppler shift in the frequency of

the laser light by the Doppler formula (Sättler, 2013).

A system developed two years before the experiment was published in cooperation with the

Fraunhofer Institute for High-Speed Dynamics. Different speed gauges were applied to the

experiment, based on the Doppler of electromagnetic waves that reflect off a moving object, which

incorporated shock wave research and detonation physics. A PDV uses a sound gauge to

measure the movement of the projectile inside a tube. This measurement is performed via a laser,

microwaves, or radar waves in the form of a cohesive or directed beam.

A PDV can also be used in a narrow infrared spectral laser system with a wavelength of 1550 nm,

the standard wavelength in communication technology, as shown in Figure 2-3. This research

used this measurement system of PDV light. Sättler et al. state that results obtained with this

approach are satisfactory and can be widely applied. They have been contacted through the

University of Potchefstroom to exchange knowledge and compare the results from this research.

However, no response was obtained.

21

Figure 2-3 Schematic of a typical PDV system (Sättler, 2013)

Some requirements for the PDV system include laser light, micro or radar wave, a rack with optical

components, and an oscilloscope, as shown in Figure 2-4. The calibre used in the experiment

was a 45 mm projectile with a weight of 700 g on a smooth barrel that is fired by a conventional

400 g propellant. Results indicate that it is possible to determine acceleration, velocity, and

displacement by a differential and integration method.

Figure 2-4 Rack with optical components (left) and oscilloscope (right) (Sättler, 2013)

22

A coherent beam is directed down the barrel towards the projectile, from the direction of the

muzzle through the use of a mirror. The coherent beam suffers a Doppler shift upon its reflection

by the projectile. Figure 2-5 shows the working principle of the PDV.

Figure 2-5 Working principle of the PDV (Sättler, 2013)

Figure 2-6 shows the time histories for the pressures p1 at the breech (red), p3 close to the initial

position of the projectile (violet), and the projectile velocity v(t) (blue). All three measurements

were triggered on the ignition pulse for the igniter cab at t = 0 ms. After 5.7 ms, the projectile

started moving. The initial movement of the projectile reflected the impact of the pressure waves.

The projectile then moved smoothly until it reached the muzzle at t = 11.9 ms with v0 = 1123 m/s.

After muzzle exit, the velocity slowly decreased owing to air drag.

Figure 2-6 Pressure and velocity as a function of time

Figure 2-7 shows the projectile acceleration a(x) (green) and its velocity as a function of

displacement v(x) (blue). The maximum acceleration of about 39 000 g was reached at x = 0.5

m. Here the projectile had a velocity of 535 m/s after travelling 3.6 m, which is the barrel length,

when the projectile reached the muzzle.

23

Figure 2-7 Acceleration and velocity as a function of displacement ("Planar Doppler Velocimetry,")

Figure 2-8 Partially enlarged drawing of touch between band and barrel

One study by Xue et al. explains that the engraving process plays a key role in determining

ballistic stability and performance in the projectile. Additionally, the band material type is a primary

factor in the stabilisation process. Therefore, the authors focused their research on two areas:

effects caused by the different band geometry.

effects caused by different band materials.

Xue et al. state that the friction between the band and barrel when the projectile is launched is

due to the gas propulsion caused by the combustion of gunpowder. The engraving occurs in only

a few milliseconds, and resistance R(t) reaches 123 kN and decreases at the end of the engraving

process (see Figure 2-8). Xue et al. also state that significant fluctuations occur at the beginning

of the projectile launch, as there are several factors that can affect the engraving, such as force

cone angle and rifling (Xue, Zhang, & Cui, 2014).

24

After determining the factors that affect the behaviour of the band, Xue et al. chose three types

of materials: Mono Cast (MC) nylon, copper, and steel. The materials selected have different

properties (ranging from low to high toughness) to assess the various effects of the different

materials on the band (Xue, Zhang, & Cui, 2014).

After applying the test, the results confirm that the type and properties of the material significantly

affect the stability of the projectile (see Table 2-2 and Figure 2-9).

Table 2-2 Material properties that affect the stability of the projectile

Material Density (g cm-3)

Modulus of elasticity

(GPa)

Poisson ratio Yield strength (GPa)

MC nylon 1.40 8.64 0.29 0.092

Copper 8.93 118 0.35 0.184

Steel 7.85 207 0.29 0.350

25

Figure 2-9 Dynamic response of projectile centre of gravity in the system with different band materials during the engraving process

2.2.2 Simulation on the engraving process of the projectile rotating band

A recent study contains the simulation of the deformation process on the band, simulation of the

projectile movement and resistance on the band engraving, and average movement pressure and

the pressure affecting the engraving process (Deng, Sun, Chiu, & Chen, 2014). Deng et al.

describe the default measurement in existing simulation models as inaccurate. Also, the

significant development observed in shooting operations, advanced firing rate, and high precision

require a dynamic analysis of the engraving processes. However, a dynamic analysis is a

challenge due to the distortion, high speed, high heat, and spin that occur to the band. Hence,

Deng et al. offer a logical analysis of the engraving inside the barrel, which is divided into three

26

parts. Figure 2-10 is a schematic drawing of the rotating band engraving process: (a) initial

configuration of the projectile and barrel, (b) rotating band engraving into the rifling, and (c)

rotating band completely engraved by the rifling.

Figure 2-10 Schematic drawing of rotating band engraving process (Quanzhao, 2017)

However, Sun et al. ignored the dynamic balance of the projectile and neglected the subsequent

recoil. The pressure gradient was also not taken into consideration, as the study only measured

the pressure of the chamber. Additionally, the study ignored the thermal stress and material

property of the band, which play fundamental and decisive roles in the behaviour of the engraving

process.

(Quanzhao, 2017) applied the experiment on a 155-mm (39 calibre) for all charge configurations

(charge 1 to charge 5). The experiment examines five cases with increasing charge mass ratios

from Case 1 to Case 5. The five curves are represented in Figure 2-12 from the firing of live

ammunition under different situations with an increase in the charge mass from Case 1 to Case

2. Case 1 appears in a slightly different curve owing to the charge position. Their experiment

results indicate that the engraving process on the band was completed at 7.5 ms for Case 5.

Figure 2-11 (I) shows the levels of engraving and stress as functions of time.

Figure 2-11 (I) also illustrates the evolution of Von Mises stress for the rotating band and depicts

the stress of the engraving processes on the band and how it affects the band shape. The band

had varying distortions at each time increment. Figure 2-11 (II) illustrates projectile base pressure

on the middle, and Figure 2-11 (III) illustrates projectile motion curves ((a) Displacement–time

curves of the projectile, (b) velocity–displacement curves of the projectile, (c) acceleration–

displacement curves of the projectile).

27

Figure 2-11 Evolution of Von Mises stress for rotating band (Quanzhao, 2017)

There is a slight difference in displacement with the engraving process in the five cases. Another

noticeable difference, which is normal and expected, is the increase in speed as the gas pressure

increases. The engraving process was completed quickly in Case 5, while the process was slower

(I) (II)

(III)

28

in Cases 1 and 2. All cases, except Case 5, indicate that the acceleration of the projectile

stabilises but with some vibration.

Figure 2-12 Engraving resistance (Quanzhao, 2017)

It can be noted that the engraving resistance increases in depth as it maintains a constant level

with limited vibration. All the engraving resistance curves in the five cases exhibit similar trends,

having two distinct peaks with the first peak being higher than the second, excluding Case 4.

Table 2-3 lists the differences between the five cases with respect to time, speed, displacement,

etc. Values in parentheses correspond to the first peak in Case 4.

Table 2-3 Comparison of simulation results

Case 1 2 3 4 5

Time at engraving completion point (ms) 19.8 18.9 15.3 12.7 7.5

Velocity at engraving completion point (m/s)

34.9 50.2 68.7 87.8 108.4

Displacement at engraving completion point (mm)

108.9 114.3 116.3 117.3 116.2

Projectile base pressure at engraving completion point (MPa)

58.5 78.0 116.1 161.1 225.9

Average work pressure (MPa) 16.2 14.2 18.8 24.7 45.2

Time at max engraving resistance point (ms)

16.1 16.8 13.9 12.1 6.7

Max engraving resistance value (N) 7.1×105 7.7×105 8.1×105 8.9×105 1.0×105

Velocity at max engraving resistance point (m/s)

9.9 21.2 33.2 59.8 58.8

Engraving pressure value (MPa) 34.5 50.3 73.0 135.9 170.6

29

2.2.3 Concept of Engraving and Bore Resistance Cycle

The projectile diameter is slightly larger than barrel diameter, and, as the propellant is ignited, it

propels the projectile causing an engraving process inside the barrel. The diameter of the rotating

band attached to the projectile is slightly larger than the projectile diameter. Therefore, as the

projectile travels through the barrel, there is a substantial amount of energy that is lost from friction

and rotating band engraving. Sättler (2013) describes the engraving and bore resistance cycle in

a large calibre. When the propellant charge is ignited, the pressure starts to build up inside the

gun chamber. The projectile remains still until the pressure inside the chamber rises to the pre-

defined shot start pressure, after which the driving band material starts to fill the forcing cone. As

pressure continues building up, the driving band material starts to penetrate through barrel

grooves until no driving band material remains in the forcing cone. After all the driving band

material is contained within barrel grooves, the projectile begins travelling along the barrel to the

point where the projectile and all driving band material are expelled from the barrel. Figure 2-13

illustrates the inside of the gun system.

Figure 2-13 Inside of the gun system

30

Figure 2-14 Total resistive forces acting on projectile motion (Crowley, 1999, November)

Crowley (1999) explains the total forces acting upon the projectile and the driving band. From

Figure 2-14, there are five different forces, and each force has a different magnitude. The

extrusion force stresses the driving band material to its limit point. The second force is the incision

force, the force required to cut into the driving band material. There is a force required to spin up

the projectile. As mentioned previously, the projectile diameter is slightly larger than the barrel.

Thus, there is a loss in force due to friction. Finally, there are the force losses due to the curvature

of the barrel. The equilibration equation is determined in the following equation (Crowley, 1999,

November):

F = FExt + FInc + FRif + FFric + FCurv (36)

where F is the total resistance of force, FExt is the extrusion force, FInc is the incision force, FRif is

the rifling force, Ffric is the friction force, and FCurve is the curvature force.

2.2.4 Resistive Pressure

A large force is required for the projectile to overcome the engraving and bore friction. Interior

ballistics determine total engraving and bore friction forces by calculating resistive pressure.

Resistive pressure has noticeable effects on muzzle velocity and base pressure. However, the

extremely short duration of the interior ballistic cycle is considered a challenge for interior

ballisticians to determine resistive pressures along the barrel. Many ballisticians have developed

an accurate model to calculate resistive pressure. There are different approaches to formulate

resistive pressure models, including approximation, semi-empirical, and empirical approaches.

31

In the case of STANAG (STANAG 4367, 2012), the thermodynamic interior ballistic model is

considered. In this model, the energy loss due to the engraving process between the driving band

and a gun’s barrel is explained. However, research on the physical processes and internal forces

is not yet comprehensive. There are limited testing and scientific consideration on this energy

loss. Current research states that a projectile moves at its launch between 15 and 20 MPa. This

is based on a limited theoretical framework and not real measurements, mostly reliant on

assumptions. This body of work is mainly based on Gabot’s model, as discussed below.

Figure 2-15 Variation of the resistance force due to friction with projectile motion (Crowley)

Gabot’s approximation model is used to calculate the maximum engraving pressure during the

interior ballistic cycle. This research is based on Gabot’s approximation and empirical evidence.

The main parameters of the model are start pressure and engraving pressure. Engraving pressure

is the peak pressure, while the sliding pressure focuses on the projectile after its fully engraved,

as shown in Figure 2-15.

2.3 Conclusion

To conclude, the model of STANAG 4367 LPM, in terms of the energy model, pressure gradient

model, combustion model, equation of motion, and equation of state was discussed in detail.

Secondly, several assumptions related to the LPM of interior ballistics were determined. Finally,

the calibration factors were described. These factors are applied to the computational equations

to adjust the errors in the velocity and pressure of the interior ballistics system.

32

CHAPTER 3 TEST METHODOLOGY

3.1 Background

Accurate analyses of energy losses within the barrel, based on empirical data, help to understand

projectile performance and improve efficiency. In ballistics, testing is an integral part of the

development. This paper discusses the search for a non-complex sensor or device through which

the projectile movement and the engraving process inside the barrel can be evaluated. The

extreme speed in which the internal ballistic process occurs prompts the researcher to find an

adequate device, capable high-speed measurements. These measurements can then be

represented numerically, particularly considering the progress made since the computer was

introduced in the world of ballistics.

This chapter describes the experimental procedure used in this study. The experiment and its

various aspects will be discussed with elaboration on the following:

Safety precaution

Description of weapon used (35mm) and gun preparation

Test setup

Strain gauge installation

Firing trail

3.2 Weapon System and Setup

The 35mm artillery that was used in this experiment is a weapon originally for anti-aircraft,

renamed as Rheinmetall Air Defence Ag, following the merger with Rheinmetall in 2009. The anti-

aircraft weapon was customised to be used against ground targets. Many improvements have

followed over the years, such as the addition of an independent vision system and other

enhancements in performance, resulting in several variants, including GDF-001 in 1959, the

second stage GDF-002. The final version, GDF-005 in 1985, was the most comprehensive;

Figure 3-1 shows the experimental weapon setup.

33

Figure 3-1 experimental weapon setup

3.3 Gun preparation

Figure 3-2 The preparations of the gun

With reference to Figure 3-2, the following steps were required prior to the firing of the rounds

from the 35mm Oerlikon test weapon, as per RDM document 03986-N25-604001 and the

Oerlikon Operating Manual:

1. Fasten the gun mount (4) to the floor of the shooting range.

2. Check that the recoil mechanism and cradle (3) is securely fixed to the gun mount.

3. Check that the pressure barrel jacket with the breech block (2) is able to slide freely in the recoil mechanism (3) and that the locking lever is in the correct position.

4. Check that the pressure barrel with a muzzle brake (1) is inserted into the pressure barrel jacket with the breech block (2) and that the barrel pressure ports are accessible.

Muzzle Chamber Barrel

34

5. Test the firing mechanism for smooth operation.

6. Test the wedge breech block for smooth operation.

3.4 Strain Gauge Installation Methodology

The following process was used to install the strain gauges:

1. Measure barrel and assign locations for strain gauges.

2. Prepare locations with sanding paper.

3. Clean locations with acetone and gauze.

4. Remove acetone and other residues with acid and neutralise with base afterwards. Be careful about cross-contamination.

5. Place strain gauges on a clean, smooth surface (glass) and stick them down with adhesive tape.

6. Carefully remove strain gauges from the surface with tape and place on location.

7. Peel back the adhesive tape and place a small amount of cyanoacrylate underneath strain gauge.

8. Roll down the tape and squeeze excess cyanoacrylate from beneath strain gauge. Keep applying pressure to ensure strain gauge adheres to barrel.

9. Carefully remove the adhesive tape.

10. Apply catalyst to solder tabs on strain gauge and solder wires to strain gauges with soldering iron and solder.

11. Measure continuity and resistance with the multi-meter to ensure a good solder connection was made.

12. Apply protective coating to the strain gauges and connections.

13. Route wires and secure with duct tape.

14. Solder plugs and checks for continuity as well as resistance between connections/legs of the bridge.

3.4.1 Employment of Strain Gauges,

The goal of the test is to detect any corresponding pressure changes while the projectile is moving

along the barrel. A simple way to quantify the pressure is to measure the strain behaviour when

the projectile reaches a specific location within the barrel; therefore, allocating various sensors

will indirectly determine the pressure.

35

Figure 3-3 Definition of strain

Strain in a material represents the change in length caused by a force applied the material. The

internal stress of the material creates this change in length which is the result of a force acting

over a specific area. The strain is defined as the difference in length over the original length.

Figure 3-3 shows the relationship between the strain value to the change in the length of the

material.

During the engraving process, the resistance curve is subjected to the stress of the driving band

material, resulting in corresponding shear stress over a specific area.

By connecting strain measurements across the length of the barrel, the test output can produce

a curve of pressure over time. Consequently, the area of peak pressure determines the resistance

curve of the gun; subsequently, the results can be implemented in the simulation of the IB cycle.

3.4.2 Strain Gauge Sensor

A sensor is a device or subsystem designed to measure absolute or specific variables based on

the type and purpose of the sensor. It helps detect events or changes in its environment and

sends the information to electronics, frequently a computer processor. The variables can be as

simple as the temperature of a body or as complex as the computer. A paper was published that

the strain gauge, invented by Ruge (US Patent No. 2,350,97, 1944), can be the most suitable

sensor/device to measure the movement of the projectile inside the barrel.

The strain gauge sensor was chosen for this research for several reasons. It is a high-precision

sensor used to measure various mechanical quantities, including changes in physical dimensions,

by converting tension, force, and pressure into an electrical signal. Additional benefits include

large-scale and wide-range applicability, low cost, compact size, and reusability. Figure 3-4

illustrates the construction and the principle of the work of a strain gauge.

36

Figure 3-4 Construction and the principle of the work of a strain gauge

A strain gauge consists of a network of thin metal wires, 0.001 mm thick, connected directly to

the strained surface by a thin layer of epoxy resin. When a load is applied to the surface of the

strain gauge, the subsequent transformation in surface length is signalled to the resistor, and the

matching strain is measured in terms of the electrical resistance of the foil wire, which varies

linearly with strain. The external load applied to a stationary body is the result of stress and

pressure. Stress is defined as the strength of internal resistance as well as displacement and

distortion that occur; deriving the value of the measured stress gives the value of influence

(Dyckmans, 2015).

To record pressure, the strain gauge should be connected to an electric circuit that can observe

the change in resistance due to the change in strain. There are various strain gauge

configurations, depending on the application of the measurement, such as quarter-, half-, and full-

bridge configurations. Figure 3-5 shows the scheme of the full-bridge configuration of strain

gauges. This configuration allows the circuit to detect the signal of four strain gauges

simultaneously.

Figure 3-5 Full-bridge configuration of strain gauges

37

3.5 Test Setup

The test weapon was an Oerlikon 35mm anti-aircraft weapon. This weapon was adapted to fit a

base and only fire single shots. The barrel was instrumented with SGs at set intervals. The

physical value of the strain was not the focus; each SG configuration was used to determine when

the projectile passed its location. The projectile velocity could then be calculated based on the

distance between the SG bridges and the time between the strain occurrences. The first attempts

involved four full-bridge configurations (12 strain gauges), but the signal was too noisy to obtain

reasonable data. Better results were obtained when the strain circuit was changed, and four of

the strain gauges were rewired to a quarter-bridge configuration with bridge completion by the

bridge amplifier. The locations of the SGs are indicated in Table 3-1 and Figure 3-6:

Table 3-1 Locations of the strain gauges on the barrel

SG Location (m)

1 0.788

2 1.575

3 2.363

4 2.931

Figure 3-6 Locations of the strain gauges on the barrel

38

Figure 3-7 Bridge of strain gauges on the barrel of the 35mm

Figure 3-7 shows the SG configurations mounted on top of the barrel to measure the shear stress

on the barrel due to the movement of the projectile. The force acting on the barrel was proportional

to the amount of the pressure over that specific area where the strain gauge was located. By

using a sufficient number of strain gauges along the barrel, the time at which the projectile passed

each SG was recorded throughout the IB cycle.

There was a piezoelectric pressure sensor that measured the pressure inside the barrel to create

a pressure-time curve. All signals coming out of the strain gauges, pressure sensor, and

acceleration sensor were moved through a moving average filter to remove any noise. The

moving average filter defines sequential range of sample sets and take the average of them for

better filtration. Then, the clear signals were fed through amplifiers. Next, the stand-alone signals

were synchronised over the same period using a signal conditioner to build the full pressure curve.

Lastly, the full signal was input to a computer to evaluate and record the measurements. The data

measured from each firing, therefore, comprised two useful time-dependent datasets: projectile

position and chamber pressure.

Figure 3-8 Test methodology and setup

39

Figure 3-8 shows both the accelerometer and strain gauges connected to bridge amplifiers for

signal conditioning. The data from the accelerometer was not used for any analysis purposes.

The pressure transducer that was used to measure chamber pressure required a charge amplifier

for signal processing. The pressure transducer was mounted directly into the barrel, and the cases

were drilled to allow the pressure to be measured.

The test weapon had a contact that closed once the trigger was pulled; this contact was used to

measure T0. Radar was used to measure the muzzle velocity of the projectile. A calibrator was

used to check the calibration settings for the charge amplifier and data acquisition system. Data

acquisition of the signals was accomplished with a PicoScope 4424 data acquisition system

connected to a laptop computer. Pico3424 software on the laptop captures and stored the

unfiltered data at various sampling rates. MATLAB was used to further analyse the data.

The following data were recorded for each of the firings:

Muzzle velocity of the projectile – measured by the radar placed next to the weapon

Projectile velocity/displacement – measured by using data from strain gauges

Acceleration of the barrel – measured but not used

Pressure – measured by the piezoelectric transducer

3.6 Firing Trail

The firing trial was performed under the supervision of a specialist, and all the approval

documentation was obtained by the department of shooting range area. All safety aspects were

taken into consideration before and during the experiment. In the case of the firing trail, a projectile

was fired from the 35mm cannon. The tests were conducted at a proofing test range at

Rheinmetall Denel Munition (Somerset West). Five shots were fired during the test with different

propellant masses. The table below shows the particular propellant mass for each shot.

Table 3-2 Propellant mass in each shot

Shot Number Mass Charge (g)

1 270

2 270

3 290

4 315

5 250

40

The used sampling rate was 20 MHz (50 ns sampling time) as this was the highest sampling rate

could be accomplished with the channel count. In order to check that it was within the Nyquist

frequency, the expected time that would take a projectile to travel the shortest distance between

strain gauges has been calculated based on a muzzle velocity (MV) of 1200 m/s:

𝑑3−4 = 0.568𝑚

𝑡3−4 = 𝑑3−4

𝑀𝑉

𝑡3−4 = 473𝑚𝑠

where; d= distance, t= time, MV= Muzzle Velocity

Therefore, the sampling was almost 9500 times faster than this time and were confident that it

would be able to accurately capture the event. Also, capturing strains that resulted from the

excitation of the barrel as these would be orders of magnitude smaller that the strain caused by

the bullet was not the concern.

3.7 Practical Test Instrumentation Limitation

Below is the summary of the instrumentation limitation of the practical test:

The distance between the strain gauges was taken into consideration because of the

attenuations that can be produced by signal interference. The projectile moved too fast

between two strain gauges, leaving insufficient time for the next strain gauge to detect the

signal.

The locations where the strain gauges could be mounted were limited because there are

positions along the gun where there was no possible direct contact between the barrel

and a sensor.

Because of the nature of the projectile resistance curve, the most significant interaction

was during the first 10% of movement; however, that where it is most difficult to access

strain.

The number of available data capturing channels was limited.

41

3.8 Chapter Summary

Chapter 3 provided insights to understand and overcome problems in the current modelling

system. Projectile behaviour, methods of measuring projectile motion, the strain gauge sensor

system, and test limitations were discussed. The test layout for the experiment was presented,

along with rationality for the choice in sensor type. Finally, this chapter provided the sensor

features and elaborated on the expected results.

42

CHAPTER 4 RESULTS

4.1 Introduction

The purpose of this chapter is to presents the results obtained from the experimental tests. The

objective of the tests was to investigate the relationship between pressure, time, and SG

measurements at different shots. The experiments were performed successfully, and the results

are presented in this chapter in detail.

4.2 Test Setup

The experiments were accomplished under the following conditions:

Weighing and loading of 35mm ammunition

Propellant – S350 Final Lot 059/B/2018

Bulk density – 983 g/dm3

Receiver type – OERLIKON KV353 Nr.23 Barrel Nr. Oerlikon-020

Barrel length – 3250 mm

Top charge – 315 g

Middle charge – 290 g

Low charge – 270 g

4.3 Tests Results

The behaviour and the movement of the projectile and the challenges that face the projectile

inside the barrel were investigated during the experiment. In all shots, the pressure and time were

measured at different SG locations. Maximum pressure for each shot was recorded at SG1. Then,

the pressure slightly decreased at SG2 and continued decreasing to the lowest value at SG4.

It was observed that the projectile was affected by the friction and engraving processes due to

the contact between the projectile and the groove and lands of the rifled barrel. The influence of

the friction at the initial movement of the projectile was significant. As the projectile travelled

through the barrel, the effect of the friction decreased and, therefore, the pressure decreased.

The experiment was successful and matched the expected result. The friction and engraving

processes influenced the pressure measurements, as predicted.

43

4.3.1 Shot Number 1 Results

Figure 4-1 illustrates the measurements of the strain gauges and pressure transducer for shot

number 1. The pressure curve represents the breech pressure. The spikes in the strain gauge

curves indicate the time when the projectile passes the corresponding strain gauge.

Figure 4-1 Shot Number 1 results

44


Figure 4-2 illustrates the measurements of the strain gauges and the pressure transducer for shot

number 2. The pressure curve represents the breech pressure. The spike of the strain gauges

indicates the time when the projectile passes them.


45

4.3.3 Shot Number 3 Results:

Figure 4-3 illustrates the measurements of the sensors, which are the strain gauges as well as

the pressure transducer for shot number 3. The pressure curve represents the breech pressure.

The spike of the strain gauges indicates the time when the projectile passes them.


46






47






48

Measuring data within such a short duration is difficult. Therefore, multiple attempts were

conducted to obtain accurate measurements. All attempts yielded suitable data since there were

no discontinuities or meaningless oscillations. However, the most accurate data was considered

to be from shot number 4, which exhibited the least amount of noise in the signals. Therefore,

data from shot number 4 were analysed, as shown in Figure 4-6. The corresponding results

algorithm is displayed in the appendix.

Figure 4-6 Breech Pressure and SGs vs time for shot number 4

Figure 4-6 shows the time the projectile passes each strain gauge and the value of the breech

pressure at that moment. This information was the core of this testing since these measurements

were then processed with IB simulation model to obtain the proper resistive curve to calibrate the

IB model.

4.4 Test summary

Table 4-1 presents the results of the measurements at different shots which shows different time

periods of the SGs for different charge masses.

Table 4-1 Results of the measurements at different shots

Shot

number

Muzzle velocity (m/s)

Pressure

(MPa)

Acceleration (g)

Delta t

SG1–SG2

(ms)

Delta t

SG2–SG3 (ms)

Delta t

(SG3–SG4 (ms)

Mass of Charge (g)

1 1025.69 247 2000 1,236 0,724 0,750 270

2 1035.00 257 1140 1,231 0,730 0,730 270

3 1089.29 305 2000 1,147 0,687 0,688 290

4 1171.12 397 400 1,044 0,638 0,627 315

5 1016.18 263 2000 1,222 0,744 0,747 250

49

4.5 Chapter summary

Chapter 4 provided insights into the results obtained from the experimental tests. The projectile

behaviour and the effect of the friction at the initial and final movements through the barrel were

analysed. The maximum pressure was observed at the SG1 for each test. Then, the pressure

value reduces gradually recording the lowest value at SG4. It was observed that the projectile is

affected by the friction and engraving processes due to the contact between the projectile and the

groove and lands of the rifled barrel. Finally, it was determined that shot number 4 data was the

most accurate data and can then be used in the model.

50

CHAPTER 5 DATA ANALYSIS AND PROCESSING

5.1 Introduction

The tests were conducted at the proofing test range at Rheinmetall Denel Munition, and the

results presented in the previous chapter. A 35mm artillery system was used in the experiment,

which during which the behaviour of the projectile inside the barrel due to driving band material

and the engraving process was studied. The test results were analysed using a MATLAB program

to process real data, simulate the result, and investigate variable parameters.

The test results were evaluated to derive a relationship that defines the characteristic resistance

curve for the specific gun system based on the following:

Gun barrel movement at shot start

Pressure of the gun chamber as a function of time

Position of the projectile’s drive band as a function of time

Muzzle velocity (projectile exit velocity)

The above data were utilised to derive the actual projectile resistance as a function of position in

the gun’s barrel. Testing devices and measuring tools used were carefully selected and calibrated

with respect to international standards to avoid any errors during the testing process and to limit

any discrepancies in the results.

Pico3424 software was used to capture and store the unfiltered data at various sampling rates.

MATLAB was used to further analyse the data. As mentioned previously, the shot 4 dataset was

chosen for full data analysis since it had the least noise in the signals.

In this section, experimental findings are discussed and compared to what was expected. The

data, processing, and iteration principles are discussed in detail in this section. Finally, the

limitation of the experimental data and improvements for future work are stated.

5.2 Data Analysis

All the data and relevant variables were initialised for the shot being processed. The shot start

pressure from the accelerometer measurement data was determined, as shown in Figure 5-1, by

inspecting the pressure and integrated acceleration curves. All datasets and relevant variables

were isolated for the shot being processed. The exit pressure can also be investigated by

inspecting the integrated acceleration. As soon as maximum recoil velocity is reached, this implies

barrel exit (~50 MPa), as displayed in Figure 5-1.

51

Figure 5-1 Determination of shot exit pressure

The integrated value of acceleration was plotted with the pressure, and pressure at the maximum

recoil velocity was determined, as shown in Figure 5-1. The data after the exit pressure was

discarded. The strain gauge times were used in the initial calculations of projectile velocity and

translation, which were expected to be higher than the actual values since the resistive pressures

were not yet included. This initial calculation provided an indication of whether the data reduction

process so far was valid.

The resistive pressures and the gas pressure in front of the projectile were considered. The actual

resistance pressures at the SG positions, based on the time the projectile reaches the SG

position, was determined through a multistep iterative procedure, i.e. an iterative calculation for

each strain gauge, as shown in Figure 5-2.

52

Figure 5-2 Iterative procedure for measuring the actual resistance pressures

The first three resistance pressure values determined the initial conditions PR2, PR3, and PR4 at

the SG1 position (PR1 was already calculated and considered). PR2 was considered as the

maximum resistance.

5.3 Data Summary

The result from the data analysis provided the input data to be implemented in MATLAB to

simulate results and investigate variable parameters. MATLAB is a comprehensive integrated

development environment (IDE). MATLAB software has a multi-paradigm numerical computing

environment and is classified as a programming language that was established by MathWorks.

The MATLAB software functionality used in this research is listed in the appendix.

The capabilities of the IDE simulated environment allow the program to be used for many

applications. The IDE was essential to develop the model and achieve the fundamental target of

this research. The IDE simulation yielded a more accurate result than the engraving and friction

model cycle of STANAG 4367.

The data used in the software are the following:

Barrel length from the commencement of rifling (COR) to front face – 2931 mm

Relative position of SG1 from COR – 788 mm


53



Mass of propellant – 315 g

Mass of projectile – 550 g

Pressure (Max) – 397 MPa

Velocity – 1171 m/s

Barrel groove diameter – 35.1 mm

Land diameter – 35 mm

Land-to-groove ratio – 1.1

5.4 Model Construction

Scientists in ballistic fields use data analysis algorithms to analyse the behaviour of the projectile,

determine the optimum parameters, and make predictions by applying mathematical code

describing the interior ballistic cycle. These algorithms also help address the lack of information

within the internal ballistic system.

The model data analysis algorithms are used to solve a specific problem using mathematical

codes. STANAG equations for pressures and projectile travel were used to build a model that

calculates the position of the projectile at a given time for each strain gauge (from strain gauge

measurements). Applying the equations listed in Chapter 2, MATLAB software was used to

compile the data analysis code. The general sequence of the MATLAB code was as follows:

1. Input Data – All input data, including that of the igniter and propellant, were analysed using

various parameters]. These data are necessary to determine the gas constant, heat

capacities, and others.

2. Initialisation – The input data were initialised to set values, e.g. derivatives and state vector

at zero time t0.

3. Calculation of IB cycle – Integrated parameters were assigned to obtain suitable variables,

such as distance, recoil distance, velocity, the energy loss of air displacement, the energy

of heat transfer, propellant distance, the mass fraction of propellant gas based on burn

distance, total burn, the average remaining propellant volume and gas properties, kinetic

energy of projectile and gas/propellant, projectile rotational velocity and energy, F\friction

energy, gas temperature, an average of the chamber pressure, breech and base

pressures, burn rates of propellant, projectile acceleration based on base pressure, and

recoil acceleration based on breech pressure. All the parameters were integrated over

54

time to the endpoint where the projectile reached the barrel. IB model calculations were

then terminated.

4. Output array: distance, acceleration, time, velocity, and average pressure (Brinson &

Brinson, 2015).

During the analysis, the sequence of MATLAB code was as follows:

1. Initialise all data sets and relevant variables for the shot being processed. The

experimental data was imported and allocated to variables for processing.

2. Determine the shot start pressure from the accelerometer measurement data. The plot is

used to determine shot start pressure by inspecting the pressure and integrated

acceleration curves.

3. Process the data. Only relevant data was prepared for display.

The shot start pressure was estimated based on the plot of the integrated acceleration curve. The

relevant datasets from the shot start onwards were isolated. The exit pressure was also

determined by inspecting the integrated acceleration. The maximum recoil velocity implies the

moment of barrel exit (~50 MPa). The integrated value of acceleration (i.e. velocity) was plotted

with the pressure, and the pressure at the maximum recoil velocity was determined. All data after

the exit pressure was discarded.

The strain gauge times were determined in the following step. The boundary effect was then

removed, and all the other datasets were followed the same procedure. The data was isolated

further and discarded all other data after the projectile. Now, the pressure data was filtered, and

the time and pressure index of strain gauges were determined where projectile passes. Then,

time and pressure for each strain gauge were recorded. Now, the time and pressure were

extracted when each strain gauge reflects the projectile moving passed.

The base pressure based on the Lagrange gradient model was then calculated, as introduced in

Chapter 2. The raw, base, and filtered pressures were plotted. Also, amplified strain gauge signals

were plotted on the same graph for visualisation. The position on the pressure curve for each

strain gauge activation by a projectile was marked. The initial nominal calculated velocity was

plotted on the secondary axis. The next step was to consider the resistive pressures and the gas

pressure in front of the projectile, as mentioned in Chapter 2.

The actual resistance pressures PR at the SG positions were determined based on the time the

projectile reached the SG, which required a multistep iterative procedure for each strain gauge.

The first three resistance values were estimated as initial conditions, which were PR2, PR3 and

PR4 (SG1 position), where PR1 was already calculated.

55

For procedure step 1, the value and iterate PR4 (SG1 position) resistance value was estimated.

Then, the value and iterate until PR2-4 were calculated. Finally, the value and iterate until SG1

time were to matched calculation.

In procedure step 2, the PR5 (SG2 position) resistance value was determined and adjusted in an

iteration. The iterate until SG2 time was matched to the calculation.

In the case of procedure step 3, PR6 (SG3 position) resistance value was determined and

adjusted in iterate until SG3 time was matched to the calculation.

For procedure step 4, PR7 (SG4 position) resistance value was determined and adjusted in

iteration until SG4 time was matched to the calculation. At this point, all the PR values are now

solved.

In procedure step 5, the calculation with a complete cycle integration using the calculated PR

values was verified. Therefore, the calculated velocity is measured as additional have more

verification.

Procedure step 6 used the simulation and calibration to verify the results. All the equations are

provided in the appendix.

56

Figure 5-3 Flow chart plan for the IB model

From Figure 5-3, the input data are assigned and then initialised to set different variables at zero

time t0. There are eight points along the barrel assessed in the calculation, including the four strain

gauge locations. All the integrated data of the IB model for SG1 were assigned to obtain suitable

variables. If the projectile transition aligned with the measured data, the integrated data were

submitted to the IB model for SG2. Otherwise, the Pr value was adjusted, and then the IB model

was rerun for SG1. Similar procedures were followed to run the IB model for SG2, SG3, and SG4.

Finally, the output data were posted, and the IB model was terminated.

57

5.5 Model Verification

After developing the interior ballistic model from this research, it required verification against the

interior ballistic simulation program of RDM, which is STANAG 4367 compliant. The RDM IB

model has been in service for almost 20 years. Throughout these years, the interior ballistic

simulation program of RDM has been continuously improved, verified, and validated and is

extensively utilised in product development. It has been tested against hundreds of ballistic

systems from small, medium, and large calibre systems (used for 35mm calibre simulation).

Table 5-1 shows the outputs generated by the developed IB model versus that of the RDM IB

model using the same inputs. Despite the relatively large deviation between the outputs of the

two models, the results of the developed model were closer to the measured data. As a result,

the developed IB model was verified.

Table 5-1 RDM IB model vs Developed Model

35mm System RDM Developed program

Maximum Pressure (MPa) 378.3 331.3

Muzzle Velocity (m/s) 917.1 969.7

5.6 Results of the Semi-empirical Resistive Pressure Curve of 35mm Cannon

By following the data analysis principles, the method worked successfully. Ballistic verification

aligned with the measured values. The factor for adjusting the base pressure was introduced.

58

Figure 5-4 Showing the semi-empirical resistive pressure curve of 35mm cannon

The semi-empirical resistive pressure curve of the 35mm cannon is shown in Figure 5-4. The

friction model was used to adjust the data to realistic values and calculate the maximum engraving

pressure and velocity during the interior ballistic cycle.

The graph contains eight points in the calculation, including four strain gauges as follows:

Point one (P1) is shot start pressure point and shows the resistance pressure before the

movement of the projectile.

Points two and three (P2 and P3) are assigned when the projectile begins to travel through

the barrel. The two points between the shot start pressure point and point four (SG1

position) are calculated during the solution from shot start pressure point to SG1 position.

The two points are used to predict the resistance values between the shot start pressure

point and the SG1 position. This is the area where the pressure peaks owing to the

engraving and friction processes.

The following points five (P5), six (P6), and seven (P7) correspond to the SG2, SG3, and SG4 positions, respectively.

Finally, Point eight (P8) represents the barrel exit.

Barrel Length (m)

59

The results are summarised in Table 5-2.

Table 5-2 The semi-empirical resistive pressure at different time intervals

Points 1 2 3 4 5 6 7 8

PR (MPa) 0.043 15 66 33 2.5 1.3 8.4 0.68

Projectile displacement (m)

0.000 0.010 0.020 0.788 1.575 2.363 2.931 3.330

5.7 Implementation of the Semi-Empirical Resistive Pressure Curve of 35mm Cannon

into the Developed Model

The following Figure 5-5 shows the semi-empirical resistive pressure curve, which was applied in

the developed IB model. The semi-empirical resistive curve of the cannon consists of eight points,

unlike the classical one, which consists of four points. In other words, the more points to measure

the pressure the more accurate the results are.

Figure 5-5 The semi-empirical resistive curve of the 35mm cannon

Table 5-3 shows the result of the developed IB model before and after implementing the resistive

semi-empirical resistive pressure curve of 35mm cannon. There is no difference in maximum

pressure predicted. The muzzle velocity only shows a slight increase of 0.15 m/s.

60

Table 5-3 The effect of the resistance pressure sub-model on IB results

35mm System Before Resistance

Model

After Implementing the Resistance

Model

Measured data

Maximum Pressure (MPa) 331.30 331.30 397.00

Muzzle Velocity (m/s) 969.70 969.85 1171.12

5.8 Limitations of the experimental data

There were limitations of the experimental data to be considered:

No strain data was available close to the breech where maximum pressure was expected. The configuration test weapon did not allow for the installation of strain gauges at this location.

Uncertainty regarding the shot start pressure was present. The exact time when the

projectile starts moving was immeasurable, but it could be derived with reasonable

accuracy from the acceleration data of the weapon.

Uncertainty regarding barrel exit time was present. This was not measured directly but

derived from the accelerometer data of the weapon.

The actual times of the strain measurements when the projectile passes should be

carefully evaluated. Since the exact shot start is not known, the strain measurements and,

hence, time/distance of the projectile are effected.

The number of strain measurements was limited. Because of the constraint of available

measuring channels, there were only eight points in the calculation, including the four

strain gauge locations. With additional channels, the distances between strain

measurements could have been closer.

5.9 Chapter summary

The test results from Chapter 4 have been analysed in this chapter. The data were processed

using MATLAB to generate the semi-empirical resistance pressure curve, which was used to

improve the confidence over the traditional method in defining the shape of resistance curves for

the simulation model of gun systems since it is based on intermediate experimental data. The

results showed the accuracy of the IB model with resistive pressure considering.

61

CHAPTER 6 CONCLUSION

6.1 Research Overview

This dissertation examined the implementation of the resistive pressure sub-model to enhance

the accuracy of the overall IB model. The approach was to develop a test setup to gather

information necessary to realise the resistance curve that is characterised by different firing

conditions.

Chapter 1 – This chapter gave a detailed introduction to the gun system. Current problems facing

internal ballisticians were presented. The research methodology was also outlined.

Chapter 2 – This chapter presented a review of existing literature and the latest developments

concerning resistance models of internal ballistics systems. Available scholarly work in the field

of internal ballistics was highlighted, and various concepts explaining the internal ballistic system

were presented.

Chapter 3 – This chapter dealt with the methodology of the experiment, safety precaution,

weapon system and setup, gun preparation, strain gauge installation, firing trail, practical test

instrumentation limitation, and test layout for the experiment.

Chapter 4 – This chapter described the result obtained from the experimental tests. The result of

the experiments showed the relationship between pressure, time, and variable SGs at different

shots. The results were then displayed.

Chapter 5 – This chapter described the data analysis and processing of the IB model simulated

by MATLAB software. The data summary and the results of the semi-empirical resistive pressure

curve were presented showing the adequacy of resistive curve implementation to the IB model.

6.2 Conclusion

This study developed a semi-empirical barrel engraving resistance test method for internal

ballistic simulation codes. The aim of this study was to determine the characteristic pressure

resistance curve via the stated intermediate approach through internal ballistic simulation using

experimental data from live firings. The objectives in this study were to a) establish a test platform

for measuring projectile travel and relevant ballistic parameters during live firings, b) develop a

method and related software tools to analyse the measured data and derive a resistance curve

for the ballistic system, c) obtain suitable data from live firing conducted from a series of

experiments, which then could be utilised to derive resistance curves to be implemented in an

62

internal ballistic simulation model, and d) demonstrate the implementation of such a resistance

model in comparison to the traditional empirical model.

The experiments were conducted successfully, and the required results were obtained, as

presented in the previous chapter. The measured data was processed and modelled to derive the

semi-empirical resistive curve of the 35mm cannon. After that, the IB model is developed and

verified against RDM’s IB model. Finally, the semi-empirical resistance curve was implemented

into the developed IB model, and the result was compared to the result of the IB model before

implementing the semi-empirical resistance curve with the measured data.

The difference between the simulation result before and after implementing the semi-empirical

resistance curve did not significantly vary. The maximum pressure showed no changes whereas

the muzzle velocity showed a slight difference by 0.15 m/s. This unchanging model results is

because of the following reason. The strain gauges were placed after the real engraving

phenomenon, which occurs at the beginning of the barrel. In this study’s strain gauge placement,

the driving band was already worn out by the time the projectile passes the strain gauges. As a

result, the friction that the projectile is subjected to is minimal, which might have minimal effect

on the ballistics of the projectile.

6.3 Future Work

After investigating the interior ballistic cycle and obtaining the results mentioned above, some

improvements are proposed for future work:

Shot start pressure measurement by an alternative method. The exact time the projectile

starts moving needs to be measured to accurately determine the pressure at which this

occurs. Possible solutions include a broken wire or contact screen in the barrel.

Accurate measurement of exit time to determine exit pressure. A contact screen, broken

wire, or pressure gauge could be used to measure the exact time of barrel exit.

Additional strain gauges between the shot start and maximum pressure, if possible (100–

200mm from COR). This would provide essential information to the pressure resistance

model.

Continuous displacement measurements. A promising solution could be to use a wire

fitted to the front of the projectile and isolating it from the barrel. The resistance of this

circuit can then be measured as it changes as the projectile moves through the barrel.

This measurement can address several problems, including the challenges at the shot

start and barrel exit.

63

BIBLIOGRAPHY

Bougamra, A., & Lu, H. (2014). Multiphase CFD simulation of solid propellant combustion in a

small gun chamber. International Journal of Chemical Engineering, v2014 (2014 01 01):.

Brinson, H. F., & Brinson, L. C. (2015). Polymer Engineering Science and Viscoelasticity : an

Introduction. New York: Springer.

Carlucci, D. E., & Jacobson, S. S. (2018). Ballistics : Theory and Design of Guns and

Ammunition, Third Edition. Boca Raton, FL : CRC Press.

Crowley, A. B. (1999, November). Effect on barrel wear on weapon performance. the 21st

International Symposium on Ballistics , 228-235.

Deng, S., Sun, H. K., Chiu, C.-J., & Chen, K.-C. (2014). Transient finite element for in-bore

analysis of 9 mm pistols. Applied Mathematical Modelling, v38 n9-10 (201405): 2673-

2688.

Dyckmans, G. (2015). Energy Balance. e-ballistics. Retrieved from Retrieved from https://e-

ballistics.com/ebook/internal ballistics - energy balance.htm#GasInertia

Gray, E., Marsh, H., & McLaren, M. (1982). A short history of gunpowder and the role of

charcoal in its manufacture. Journal of Materials Science, 17(12), 3385-3400.

Gunpowder. (2019, 03 1). Retrieved from Wikipedia, the free encyclopedia. Retrieved from:

https://en.wikipedia.org/wiki/Gunpowder

Hansen, E. C., & Heiney, O. K. (n.d.). Pressure and gas flow gradients behind the projectile

during the interior ballistic cycle. ; University of Florida. Department of Mechanial

Engineering: Canoga Park, CA : Rockwell Intl Corp., 1987.

Kelly, J. (2004). Gunpowder : Alchemy, Bombards, and Pyrotechnics - The History of the

Explosive That Changed the World. New York : Basic Books, 2009.

Kirchner, D. B. (1993). Combustion products of propellants and ammunition. Occupational

health–the soldier and the industrial bas. Falls Church, Va. : Office of the Surgeon

General, U.S. Dept. of the Army ; Aberdeen Proving Grounds, Md. : U.S. Army

Environmental Hygiene Agency ; Bethesda, Md. : Uniformed Services University of the

Health Sciences ; Washington, DC : For sale by the S.

64

Koffler. (2015). Rifled Barrel vs Smooth Barrel. Retrieved from Retrieved from

https://rivervalleyarms.com/2015/11/07/slug-country-debates-rifled-barrel-versus-

smooth-bore/

Kubota, N., & VCH-Verlagsgesellschaft. (2015). Propellants and explosives : thermochemical

aspects of combustion. John Wiley & Sons.

Moreno, J. A. (2009). Thermodynamic Interior Ballistic Model With Global Parameters.

Retrieved from. NATO.

Morrison, W. F., Wren, G. P., Oberle, W. F., & Richardson, S. L. (1993). The Application of

Lagrange and Pidduck-Kent Gradient Models to Guns Using Low Molecular Weight

Gases. Ft. Belvoir Defense Technical Information Center FEB 1993.

Okunev, B. O. (1943). ballistiki [Fundamentals of Ballistics]. Voenizdat, . . Moscow.

Quanzhao, S. G. (2017). Modeling and simulation on engraving process of projectile rotating

band under different charge cases. Journal of Vibration and Control, 23, 6,, 1044-1054.

Ruge, A. (1944). US Patent No. 2,350,97.

Sättler, A. P. (2013). Investigation of the Projectile Movement Inside a Gun Tube Using a

Photonic Doppler Velocimeter. (pp. 50-58).

Schabort. (2016). RDM Internal Ballistic Course. RDM. Somerset West.: 2016.

STANAG 4367. (2012). THERMODYNAMIC INTERIOR BALLISTIC MODEL WITH GLOBAL

PARAMETERS 4367. NATO.

Worsfold, P., Townshend, A., Poole, C. F., & Miro, M. (2019.). Encyclopedia of analytical

science. Amsterdam : : Elsevier,.

Xue, T., Zhang, X.-b., & Cui, D.-h. (2014). Dynamic analysis of a guided projectile during

engraving process. Defence Technology, v10 n2 (201406):, 111-118.

65

ANNEXURES

Code - Data analysis

clear all;

close all;

clc;

% I WILL DO SOME AUTOMATION HERE - FRAMEWORK TO BE ESTABLISHED

%Step1

%Step2

%Step3

%Iteration Step1

%Iteration Step2

%Iteration Step3

%Iteration Step4

%Verification

%Simulation

%% Shot 4 Summary:

% Barrel length from COR (commencement of rifling) to front face: 2931 mm ??

% Relative Position of SG1 (Strain guage 1) from COR: 788 mm




% Mass of propellant = 315 g

% Mass of projectile = 550 g

% Pressure (Max) = 1171 [m/s]

% Velocity = 397 [MPa]

% Barrel Groove Diameter = 35.1 mm

% Land Diameter = 35 mm

% Land to Groove ratio = 1.1

66

%

%% First Step is to Initialize all data sets and relevant variables for the

Shot being processed

%

%

%

dt = 5e-08; % Time step size [s] from Sampling rate

WindowSize = 500; % Filter window size

X = 1:WindowSize; % create filter function

Filter = 1/sum(sin(pi*X/WindowSize))*sin(pi*X/WindowSize); % Moving average

filter window based on die function

LandDiam = 0.035; % mm

GrooveDiam = 0.0351; % mm

LndGrvratio = 1.1;

ProjMass = 0.550; % Projectile Mass [kg]

PropellantMass = 0.315; % Propellant Mass [kg]

BaseArea = 0.25*pi*GrooveDiam^2; %

Eps = PropellantMass/ProjMass; % See Course notes (Lagrange ratio)

% Import experimental data and allocate to variables for processing

data = importdata('Shot_4.txt');

P = data (:,1); % Pressure

x1 = data (:,2); % ?

SG1 = data (:,3); % Strain Guage 1

SG2 = data(:,4); % Strain Guage 2

x2 = data(:,5); % ?



67

Acc = data(:,8); % Accellerometer

Acc = filter(Filter',1,Acc); % Filtered data

AccInt = cumtrapz(Acc); % Integrate Accelleration to get

velocity (trapezium rule)

%% Next step is to determine the Shot Start Pressure form the accellerometer

measurement data

%

% Use this plot to determine shot start pressure by inspecting the P and

% integrated acceleration curves

[MaxP,MaxIndex] = max(P); % Maximum pressure index

Time = dt*(0:1:(length(P)-1))'; % Time series based on

time step value

figure(1);

plot(Time,P); % Plot pressure time

curve

hold on;

AccInt(MaxIndex+100000:length(AccInt),:) = 0; % Ignore Acc data

after max pressure (we do not need this)

Acc(MaxIndex+100000:length(AccInt),:) = 0;

ylabel("Pressure");

yyaxis right;

plot(Time,Acc*1000,'b:'); % Plot accelleration as

well as velocity vs time

plot(Time,AccInt,'r-');

grid on;

title("Plot to determine Shot Start Presure");

xlabel("Time [s]");

ylabel("Barrel Velocity");

legend('Pressure','Barrel Acc','Barrel V (Integrated Acc)');

%return; % Un-comment this stement

and first confirm the Shot Start Pressure

68

% Shot Start pressure was determined - so we can move on - comment out the

return statement

%% Next Step - Process the data - Isolate only relevant data and prepare for

display on Figure 2

%

% Please follow comments to inspect process

% You can descrive this process in you dissertation

%

ShotStartP = 35.0; % Estimated Shot Start Pressure based on plot 1 -

integrated accelleration curve

for i = 1:(length(P)) % Find the index of the shot start pressure

if P(i) >= ShotStartP % Pressure on the way up

ShotStartIndex = i;

break;

end

end

% Now Isolate relevant data from shot start onwards

P(1:ShotStartIndex,:) = []; % Discard data up to shot start

SG1(1:ShotStartIndex,:) = [];




Acc(1:ShotStartIndex,:) = [];

AccInt(1:ShotStartIndex,:) = [];

Size = length(P);

69

%% Exit pressure can also be determined by inspecting the integrated Acc

% As soon as maximum recoil velocity is reached, this implies barrel exit

% (this is around 50 MPa - you can also include this in ypou dissertation!)

% Simply Plot the integrated value against the pressure, read of the

% pressure against the maximum recoil velocity - This is done in the

% following plot

figure(2);

Time = dt*(0:1:(length(P)-1))'; % Create new Time series

plot(Time,P);

hold on;

plot(Time,AccInt/10000); % You can inspect this graph and

derive exit pressure/time

title('Graph to determine shot exit pressure by inspection');

xlim([0 10/1000]); % Zoom to first 10 milliseconds

[MaxAcc,MaxAccIndex] = max(AccInt);

scatter(Time(MaxAccIndex),P(MaxAccIndex),150,300,'filled');

ExitPressure = P(MaxAccIndex); % This should be adjusted according

to inspection of data (can also be verified by simple velocity calulations)

%return; % Commnet this statment before

moving to the next step

%% Discard of all data after exit pressure

% Get the starin guage times to be used for the next step

%

for i = 10000:(length(P)) % Find the index of the shot exit pressure

if P(i) < ExitPressure % Pressure on the way down

ExitIndex = i;

break;

end

70

end

% Isolate data further

P( ExitIndex:length(P),:) = []; % Discard all data after projectile

exit

SG1(ExitIndex:length(SG1),:) = [];




Acc(ExitIndex:length(Acc),:) = [];

% Now Filter the Pressure Data

PFilt = filter(Filter',1,P); % Filtered pressure

P(1:WindowSize,:) = []; % Discard filter boundary effect

PFilt(1:WindowSize,:) = []; % Remove boundary effect

SG1(1:WindowSize,:) = []; % Do the same for all the other

data sets

SG2(1:WindowSize,:) = [];



Acc(1:WindowSize,:) = [];

Time = dt*(0:1:(length(P)-1))'; % Create new Time series

% Find the time and pressure index of strain gauges where projectile passes

for i=1:length(P)

if(SG1(i)*60 > P(i))

I1 = i;

break;

end

end

for i=1:length(P)

if(SG2(i)*60 > P(i))

I2 = i;

71

break;

end

end

for i=1:length(P)

if(SG3(i)*60 > P(i))

I3 = i;

break;

end

end

for i=1:length(P)

if(SG4(i)*60 > P(i))

I4 = i;

break;

end

end

% Record Time and Pressure for each strain guage

tSG1 = Time(I1); PSG1 = P(I1);



tSG4 = Time(I4); PSG4 = P(I4); % This was extracted by inspection of the

data

tSG = [tSG1 tSG2 tSG3 tSG4]; % Extract the time and Pressure when each

strain gauge reflects the projectile moving passed

PSG = [PSG1 PSG2 PSG3 PSG4]; % And the related pressure

%% Initial calculation of Projectile velocity and translation

% These initial values should be higher than the actual values, since the

% resistive pressures have not been considered yet

% This will provide a good indication whether data reduction process so far

was

72

% sufficiently correct

Pbase = P/(1+Eps/2); % Calculate Base pressure from

from Lagrange

ProjA = (Pbase*1000000)*BaseArea/ProjMass; % Air and Barrel Resistance not

yet considered- First Order calculation to check the data processing

ProjV = cumtrapz(ProjA)*dt;

ProjX = cumtrapz(ProjV)*dt;

% Now create a new Figure and plot all relevant data for visualization and

inspection

figure(3);

plot(Time,P); % Plot raw and filtered pressure and Base pressure

hold on;

plot(Time,PFilt);

plot(Time,Pbase);

plot(Time,SG1*60); % Plot Strain gauge signals on same axis with some

amplification just for visualization

plot(Time,SG2*60);

plot(Time,SG3*60);

plot(Time,SG4*60);

sz = 120;

scatter(tSG1,PSG1,sz,300,'filled'); % Mark position on pressure curve for

each strain guage activation by projectile

scatter(tSG2,PSG2,sz,300,'filled');



xlabel('Time [s]');

ylabel('Pressure [MPa] and SG signals');

yyaxis right; % Plot initial nominal calculated

velocity on the secondary axis

plot(Time,ProjV);

ylabel('Velocity [m/s]');

grid on;

73

title('Resistive Pressure Data analysis - Shot 4');

legend('Raw Pressure','Filtered Pressure','Base Pressure','SG1 signal',...

'SG2 signal','SG3 signal','SG4 signal','SG1 time','SG2 time','SG3

time','SG4 time','Projectile Velocity');

%%

%% The next step is to consider the resistive pressures also as well as the

gas pressure in front of projectile

% The objective is now to calculate the actual resistance pressures at the

% SG positions, based the time the projectile reaches the SG position.

% This will require a multistep iterative procedure -

% i.e. an iterative calculation for each strain guage

% The first three resistance values are guessed as an initial condition -

% That is PR2 PR3 and PR4(SG1 position), PR1 was already calculated

% So:

% Procedure Step 1

% PR1 - Shot Start Pressure (already determined)

% PR2 - Maximum resistance - Iterative calculation 1 - Guess value

and iterate

% PR3 - Second Resistance Pressure - Iterative calculation 1 - Guess value

and iterate

% PR4 - SG1 resistance value - Iterative calculation 1 - Guess value

and iterate - Iterate until PR2-4 is % calculated

% Iterate until SG1 time matched calculation

% Procedure Step 2:

% PR4 - SG2 resistance value - Iterative calculation 2 - Adjust this

value in iteration


74

% Procedure Step 3:


value in


% Procedure Step 4:


value in iteration


% All the PR value are now solved

% Procedure Step 5: verify the calculation with a complete cycle

% integration using the calculated PR values and verify that calculated

velocity = measured velocity

% Working on this - to follow!!!

% Procedure Step 6: Verify by simulation and calibration

% To follow !!!

%% Save data

% Write extracted Pressure, Time and SG Times to files

fid = fopen('Pressure.txt','wt');

fprintf(fid,'%f\n',P);

fclose(fid);

fid = fopen('SGData.txt','wt');

fprintf(fid,'%f\n',[tSG PSG]);

fclose(fid);

75

Code - Resistance Pressure model

clear all;

%%% data processing

data = importdata('pt.txt');

data1 = importdata('ts.txt');

PRdata = importdata('PR.txt'); % in Pa;

Pbr = data*1000000;

ts = data1;

dt = 5e-08;

t = dt*(0:1:(length(Pbr)-1))';

%%%

global PropGeom; % Gas properties (Input)

PropGeom =

struct('Type',0,'a',0.0,'b',0.0,'c',0.0,'L',0.0,'D',0.0,'P1',0.0,'P2',0.0,'W1

',0.0,'W2',0.0,'SlabW',0.0,'SlabL',...

0.0,'SlabT',0.0,'SlotW',0.0);

global PropChem; % Gas properties (Input)

PropChem =

struct('R',0.0,'Cv',0.0,'Cp',0.0,'Fp',0.0,'CoVol',0.0,'FlamTemp',0.0,'Rho',0.

0,'Gamma',...

0.0,'Mass',0.0,'BurnCoeff',0.0,'BurnExp',0.0);

global XR;

global PR;

%%% initial condition

% Gun Input (35 mm)

ChVolume = 550.0; % Chamber Volume [cc]

ChVolume = ChVolume/1000000.0; % to m3

76

ProjMass = 0.550; % ProjMass[kg]

GrooveLength = 2.931; % [m]

RiflingTwist = 20; % 1x360deg (2pi radians) Rotation in this

number of calibres travel [rad/m]

RecoilMass = 0.0; % Total mass of recoil - in this case the

rifle - free recoil - if mass = 0, recoil is ignored

RecoilResistance0 = 0.0; % Initial threshold recoil resistance [N]

RecoilResistance = 0.0; % Cycle Recoil resistance [N]

GrooveDiam = 0.0351; % [m]

LandDiam = 0.035; % [m]

GLRatio = 1.1; % Ratio of groove width to land width

BarrelDiamSq = LandDiam*GrooveDiam;

ProjBaseArea = 0.25*pi*BarrelDiamSq;

% Base Area according to STANAG

Fft = 1;

% Propellant 1 Input

% Thermochemical and burn rate

PropChem.Mass(1) = 0.275; % kg

PropChem.Fp(1) = 972.9*1000.0; % J/kg

PropChem.CoVol(1) = 1.043; % cc/g

PropChem.FlameTemp(1) = 2716.0; % K

PropChem.Gamma(1) = 1.2530; %

PropChem.Rho(1) = 1.589*1000.0; % kg/m3

PropChem.BurnCoeff(1) = 1.3422; %

PropChem.BurnExp(1) = 0.8950; %

PropChem.Cv(1) = Fft*PropChem.Fp(1)/((PropChem.Gamma(1) -

1)*PropChem.FlameTemp(1)); % J / (kg K)

PropChem.Cp(1) = Fft*PropChem.Fp(1)*PropChem.Gamma(1)/((PropChem.Gamma(1) -

1)*PropChem.FlameTemp(1));

PropChem.R(1) = Fft*PropChem.Fp(1)/PropChem.FlameTemp(1);

% Geometry

PropGeom.Type(1) = 2; % Single perforation - See Form.m for key

77

PropGeom.L(1) = 2.4; % Length [mm]

PropGeom.P1(1) = 0.2; % Perf diameter [mm]

PropGeom.D(1) = 2.0; % Tube diameter [mm]

% Resistance Model

XR(1) = 0.0; % [m]

XR(2) = 0.030; % Half of drive band width

XR(3) = 0.060; % Full Drive band width

XR(4) = 787.5/1000;

XR(5) = 1575/1000;

XR(6) = 2362.5/1000;

XR(7) = GrooveLength;

% PR(1) = PRdata(1); % Initial Resistance Pressure - Shot

Start [MPa]

% PR(2) = PRdata(2); % Further resistance values [MPa]

% PR(3) = PRdata(3);





k = 66*0.03;

PR(1) = 15; % Initial Resistance Pressure - Shot Start

[MPa]

PR(2) = k /(XR(2)); % Further resistance values [MPa]

PR(3) = k /(XR(3));

PR(4) = k /(XR(4));

PR(5) = k /(XR(5));

PR(6) = k /(XR(6));

PR(7) = k /(XR(7));

PR = PR *1000000;

78

% Modelling

A(1) = 0.0;

V(1) = 0.0;

X(1) = 0.0;

Pba(1) = Pbr(1);

Pba(1) = Pbr(1);

eps = PropChem.Mass(1)/ProjMass;

Pg = 0.101;

Pr(1) = 0.0;

basepressure = 1;

time(1) = 0.0;

time1(1) = 0.0;

for i = 2:length(Pbr)

if Pbr (i) >= PR(1) || X(i-1) > 0.0

time1(i) = time1(i-1)+dt;

switch basepressure

case 1

Pba(i) = Pbr(i)/(1+0.5*(eps)-

(1/(24*PropChem.Gamma(1)))*((eps)^2)+((1/(80*PropChem.Gamma(1)))+(1/(360*((Pr

opChem.Gamma(1))^2))))*(eps)^3);

case 2 %STANAG

Pba(i) = (Pbr(i)+0.5*eps*Pr(i-

1)+0.5*eps*Pg*1000000)/(1+0.5*eps); % Breech Pressure according to STANAG

[Mpa]

end

k = 1;

if X(i-1) < XR(3)

79

A(i) = (((Pba(i) - Pr(i-1)-

Pg*1000000)*ProjBaseArea))/ProjMass; % Unit in [Pa]

elseif X(i-1) >= XR(3)&& X(i-1) <= 2*XR(3)

x = 2*XR(3) - X(i-1);

if x < 0.0

x = 0.0;

end

A(i) = (((Pba(i) - Pr(i-1)-

Pg*1000000)*ProjBaseArea)+(x*k))/ProjMass; % Unit in [Pa]

else

A(i) = (((Pba(i) - Pr(i-1)-

Pg*1000000)*ProjBaseArea))/ProjMass; % Unit in [Pa]

end

V(i) = A(i)*dt+V(i-1);

% Air resistance pressure in front of projectile

VSoundAir = 343.0; %

[m/s]

Air_Gamma = 1.402;

if V(i) > 0.0

MachNo = V(i)/VSoundAir; %

MachNo of projectile

Pg = 0.101*(1 + Air_Gamma*MachNo*MachNo*... % Gas

pressure in front of projectile [MPa]

(((1+Air_Gamma^2)/4)+...

sqrt(((1+Air_Gamma)*(1+Air_Gamma(1))/4) +

1/(MachNo*MachNo))));

else

Pg = 0.101;

end

X(i) = V(i)*dt+X(i-1);

Pr(i) = CalcResistance(X(i));

else

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Pba(i) = Pbr(i);

A(i) = 0.0;

V(i) = 0.0;

X(i) = 0.0;

Pr(i) = 0.0;

time1(i) = 0.0;

end

time(i) = time(i-1)+dt;

Pbrr (i) = Pbr (i);

if X(i) > GrooveLength

break

end

end

XR1 = 0;

XR2 = 0;

XR3 = 0;

XR4 = 0;

for i = 1:length(X)

if X(i) >= XR(4) && XR1 == 0

tAtXR1 = time1(i);

tss(1,1) = tAtXR1;

XR1 = 1;

elseif X(i) >= XR(5)&& XR2 == 0

tAtXR2 = time1(i);

tss(2,1) = tAtXR2;

XR2 = 1;

elseif X(i) >= XR(6)&& XR3 == 0

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tAtXR3 = time1(i);

tss(3,1) = tAtXR3;

XR3 = 1;

elseif X(i) >= XR(7)&& XR4 == 0

tAtXR4 = time1(i);

tss(4,1) = tAtXR4;

XR4 = 1;

end

end

display(ts)

display(tss)

V(i)

subplot(2,1,1); plot(XR,PR)

subplot(2,1,2); plot(X,Pr)

function Pr = CalcResistance(x)

% Function to determine resistance pressure as function of x

%

%

global XR;

global PR;

Pr = 0.0;

if x <= XR(2)

Pr = (PR(2)-PR(1))/(XR(2)-XR(1))*x + PR(1); % PR(1) +

Interpolate from X(1) to X(2)

end

if x > XR(2) && x <= XR(3)

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Pr = (PR(3)-PR(2))/(XR(3)-XR(2))*(x-XR(2)) + PR(2); % PR(2) +


end

if x > XR(3) && x <= XR(4)

Pr = (PR(4)-PR(3))/(XR(4)-XR(3))*(x-XR(3)) + PR(3); % P((3) +


end

if x > XR(4) && x <= XR(5)



end

if x > XR(5) && x <= XR(6)



end

if x > XR(6) && x <= XR(7)



end

if x > XR(7) % Beyond X(4)

Pr = PR(7);

end

end

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PR Function

function Pr = CalcResistance(x)

% Function to determine resistance pressure as function of x

%

%

global XR;

global PR;

if x <= XR(2)

Pr = (PR(2)-PR(1))/(XR(2)-XR(1))*x + PR(1); % PR(1) +


end

if x > XR(2) && x <= XR(3)

Pr = (PR(3)-PR(2))/(XR(3)-XR(2))*(x-XR(2)) + PR(2); % PR(2) +


end

if x > XR(3) && x <= XR(4)



end

if x > XR(4) && x <= XR(5)



end

if x > XR(5) && x <= XR(6)

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end

if x > XR(6) && x <= XR(7)



end

if x > XR(7) && x <= XR(8)



end

if x > XR(8) % Beyond X(8)

Pr = PR(8);

end

end

The Development of a Semi-empirical Barrel Engraving ...

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