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Course of study: Defence Engineering (MS in Vehicle and Weapon Engineering)
Name of designated person authorising scanning: Dr A Hameed
Title: Military Ballistics
Name of author: G.M. Moss, D.W. Leeming & C.L. Farrar
Name of publisher: Brassey’s
Name of visual creator (as appropriate):
Land Warfare: Brassey’s New Battlefield
Weapons and Technology Series
into the 21st Century
Volume 1
MILITARY
BALLISTICS
A Basic Manual
13569_int.indd 1 11/15/11 11:20:14 AM
Land Warfare: Brassey’s New Battlefield Weapons Systems and
Technology Series into the 21st Century
Executive Editor: Colonel R G Lee OBE, Former Military Director of Studies,
Royal Military College of Science, Shrivenham, UK.
Editor-in-Chief: Professor Frank Hartley, Vice Chancellor, Cranfield
University, UK.
The success of the first and second series on Battlefield Weapons Systems and
Technology and the pace of advances in military technology have prompted
Brassey’s to produce a new Land Warfare series. This series updates subjects
covered in the original series and also covers completely new areas. The new
books are written for military personnel who wish to advance their professional
knowledge. In addition, they are intended to aid anyone who is interested in the
design, development and production of military equipment. Series 2 consisted of:
Volume 1 Guided Weapons – R G Lee
Volume 2 Explosives, Propellants and Pyrotechnics – A Bailey and
S G Murray
Volume 3 Noise in the Military Environment – R F Powell and
M R Forrest
Volume 4 Ammunition for the Land Battle – P R Courtney-Green
Volume 5 Communications and Information Systems for Battlefield
Command and Control – M A Rice and A J Sammes
Volume 6 Military Helicopters – E J Everett-Heath, G M Moss,
A W Mowat and K E Reid
Volume 7 Fighting Vehicles – Col. T W Terry, S Jackson,
Col. C E S Ryley, B E Jones, and P J H Wormell
Volume 8 Surveillance and Target Acquisition Systems –
Col. A Figgures, Lt-Col. M McPherson, Prof. A L A Rogers,
Dr P S Hall, T K Garland-Collins and J A Gould
Volume 9 Radar – Dr P S Hall
Volume 10 Nuclear Weapons: Principles, Effects and Survivability –
Charles C Grace
Volume 11 Powering War – Modern Land-Force Logistics – P D Foxton
Volume 12 Command and Control Support Systems in the Gulf War –
M A Rice and A J Sammes
MILITARY
BALLISTICS
A basic manual
G M Moss, D W Leeming & C L Farrar
Royal Military College of Science, Shrivenham, UK
Brassey’s
London • Washington
13569_int.indd 2-3 11/15/11 11:20:16 AM
Preface Copyright © 1995 Brassey’s (UK) Ltd
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted in any
form or by any means: electronic, electrostatic, magnetic tape,
mechanical, photocopying, recording or otherwise, without
permission in writing from the publishers.
First English edition 1983
Revised Edition 1995
UK editorial offices: Brassey’s Ltd, 33 John Street London
WC1N 2AT
UK Orders: Marston Book Services, PO Box 87, Oxford OX2
ODT
North American Orders: Brassey’s Inc, PO Box 960, Herndon,
VA 22070
Messrs Moss, Leeming and Farrar have asserted their moral
right to be identified as the authors of this work.
Library of Congress Cataloging in Publication Data
Available
British Library Cataloging in Publication Data
A catalogue record for this book is available from the British
Library
ISBN 1 85753 0845 Flexicover
Typeset by Florencetype Ltd, Stoodleigh, Devon
Printed in Great Britain by
Butler and Tanner Ltd, Frome and London
This series of books is written for those who wish to improve their knowledge of
military weapons and equipment. It is equally relevant to professional soldiers,
those involved in developing or producing military weapons or indeed anyone
interested in the art of modern warfare.
All the texts are written in a way which assumes no mathematical knowledge
and no more technical depth than would be gleaned from school days. It is
intended that the books should be of particular interest to army officers who are
studying for promotion examinations, furthering their knowledge at specialist
arms schools or attending command and staff schools.
The authors of the books are all members of the staff of the Royal Military
College of Science, Shrivenham, which is comprised of a unique blend of academic
and military experts. They are not only leaders in the technology of their
subjects, but are aware of what the military practitioner needs to know. It is
difficult to imagine any group of persons more fitted to write about the application of technology to the battlefield.
This Volume
There are many textbooks on ballistics written primarily for the specialist.
However, none of these offers a simple introduction to this complex subject. This
volume was very successful when first published in 1984. It has been revised to
take note of recent developments but still concentrates on the principles of
ballistics, illustrated by reference to military applications. The subject os broadly
divided into its components of internal, intermediate, external and terminal
ballistics. As the book is intended for use by both the army officer and scientist,
some of the chapters are divided into two sections. The first part is largely the
qualitative while the second part provides a mathematical background for
further study. This treatment of the subject has proved to be useful and popular.
Shrivenham, February 1995 Geoffrey Lee
13569_int.indd 4-5 11/15/11 11:20:16 AM
Acknowledgements Contents
List of Illustrations viii
List of Tables xiii
Chapter 1 History of Ballistics 1
Chapter 2 Internal Ballistics – Part I 9
Internal Ballistics – Part II 37
Chapter 3 Intermediate Ballistics 51
Chapter 4 External Ballistics – Part I 67
External Ballistics – Part II 112
Chapter 5 Terminal Ballistics – Part I 147
Terminal Ballistics – Part II 162
Chapter 6 Wound Ballistics 167
Chapter 7 Ballistics Instrumentation 177
Answers to Self Test Questions 203
Bibliography 210
Index 211
The authors are grateful to all their colleagues at the Royal Military College of
Science, Shrivenham for their advice and help in writing this book.
13569_int.indd 6-7 11/15/11 11:20:16 AM
viii
List of Illustrations ix
List of Illustrations
Chapter 1
Figure 1.1 Ballista 1
Chapter 2 – Part I
Figure 2.1 The main ballistic components of the loaded gun 10
Figure 2.2 105 mm shell featuring a single driving band 11
Figure 2.3 7.62  51 mm cartridge together with a fired boattailed
bullet and an unfired flat-based bullet 12
Figure 2.4 105 mm calibre APFSDS (Armour Piercing Fin-Stabilised
Discarding Sabot) with sabot attached 12
Figure 2.5 Cross-section of a shotgun cartridge 14
Figure 2.6 Common shapes of propellant granules 14
Figure 2.7 Preparation of smokeless propellants 14
Figure 2.8 Three stages in the burning of a cylindrical propellant
granule 15
Figure 2.9 Ignition temperature of smokeless propellants 15
Figure 2.10 Typical burning rate constants 16
Figure 2.11 Burning rate against pressure for a typical propellant 16
Figure 2.12 Pressure within a closed-vessel plotted against time 17
Figure 2.13 Typical adiabatic (closed-vessel) flame temperatures 18
Figure 2.14 Examples of ballistic size 18
Figure 2.15 Types of progressive granules 19
Figure 2.16 Cross-sections of 7 hole multi-tube granule 20
Figure 2.17 Examples of granule shapes and their form function
coefficients 20
Figure 2.18 Configuration of igniter and propellant charge in typical
ammunition 21
Figure 2.19 Typical pressure/time, velocity/time and travel/time curves 23
Figure 2.20 Typical pressure/space and velocity/space curves 23
Figure 2.21 The four factors which determine projectile acceleration 24
Figure 2.22 Typical frictional force during firing 25
Figure 2.23 The approximate distribution of liberated energy 26
Figure 2.24 Approximate recoil energies relative to total available
energy 26
Figure 2.25 Examples of peak pressure 26
Figure 2.26 Pressure/space curves for three full charges of propellant 28
Figure 2.27 Erosion of the gun 29
Figure 2.28 An eroded gun sectioned to show enlargement of the bore
and worn rifling 30
Figure 2.29 Cross-section of the bore and relative rates of erosion 31
Figure 2.30 Rifling depth in a Probertised barrel 32
Figure 2.31 Examples of practical barrel life 32
Figure 2.32 Chieftain tank fitted with 120 mm lagged gun 33
Figure 2.33 The principle of the recoilless jet reaction gun 33
Figure 2.34 LAW 80 (early model) anti-tank recoilless rocket launcher
(Photo by courtesy of Hunting Engineering Ltd) 34
Figure 2.35 120 mm Conbat recoilless jet reaction gun 34
Chapter 2 – Part II
Figure 2.36 Flow diagram for a lumped parameter computer model of
internal ballistics, with description given on left hand side 47
Chapter 3
Figure 3.1 Shock waves formed by the release of high pressure gas
from a muzzle 53
Figure 3.2 Shock wave formation before projectile exit 53
Figure 3.3 The precursor blast field of a 5.56 mm calibre rifle 54
Figure 3.4 The initial formation of shock waves shortly after projectile
exit 55
Figure 3.5 Expansion of the blast field 55
Figure 3.6 Final phase of the blast field before contraction of the bottle
shock and Mach disc 56
Figure 3.7 Preflash 57
Figure 3.8 Flash 57
Figure 3.9 The intense secondary flash of a 120 mm calibre Chieftain
tank gun 58
Figure 3.10 The intense secondary flash from the breech nozzle of a 120
mm calibre Wombat recoilless gun 58
Figure 3.11 The primary and intermediate flash of a 7.62 mm calibre
rifle showing the total absence of secondary flash 59
Figure 3.12 Conical, slotted tube and bar type flash suppressors on 5.56
mm calibre rifles 59
Figure 3.13 Cross-section of an idealised shock wave 61
Figure 3.14 Table of overpressures in atmospheres against intensity of
sound in decibels (reference pressure 2  10–5 Pascals) 62
Figure 3.15 Methods of blast suppression 63
Figure 3.16 9 mm calibre sten sub machine gun fitted with silencer 64
Figure 3.17 The principle of the muzzle brake 65
Figure 3.18 Muzzle brake fitted to British 105 mm calibre light gun 65
13569_int.indd 8-9 11/15/11 11:20:16 AM
x List of Illustrations List of Illustrations xi
Chapter 4 – Part I
Figure 4.1 Steady streamline flow 71
Figure 4.2 Real and in vacuo trajectories 72
Figure 4.3 Flow past progressively more streamlined bodies 74
Figure 4.4 (a) Pressure waves from a source moving at half the
speed of sound 75
(b) Pressure waves from a source moving at twice
the speed of sound 75
Figure 4.5 Shock angle for wedges in supersonic flow 77
Figure 4.6 Shadowgraph of 7.62 mm bullet 78
Figure 4.7 Shadowgraph of high drag training round 78
Figure 4.8 Components of drag 79
Figure 4.9 Drag coefficient of 7.62 mm bullet 80
Figure 4.10 Drag of 7.62 mm bullet 81
Figure 4.11 Nose drag at subsonic speeds 82
Figure 4.12 Tangent ogive nose (calibre radius head) 82
Figure 4.13 Secant ogive nose (fractional calibre radius head) 83
Figure 4.14 Nose drag at supersonic speeds 84
Figure 4.15 Boat-tail drag at subsonic speeds 85
Figure 4.16 Boat-tail drag at supersonic speeds 85
Figure 4.17 A typical base bleed projectile 86
Figure 4.18 The effect of base bleed on range 87
Figure 4.19 Components of the force acting on a body in flight 88
Figure 4.20 Body sideforce distribution due to yaw 89
Figure 4.21 Body normal force divided by Mach number squared 89
Figure 4.22 (a) Cone on base – stable; (b) Cone on side – neutral; (c)
Cone on apex – unstable 90
Figure 4.23 Drag variation with yaw angle at M  2.5 91
Figure 4.24 Effect of after-body on centre of pressure position 92
Figure 4.25 Fin stabilisation 93
Figure 4.26 Comparison of lift on wings and a body 93
Figure 4.27 Relative lifting efficiency of different wing planforms 94
Figure 4.28 The gyroscope 95
Figure 4.29 Shell gyroscopic stability 96
Figure 4.30 Shell motion following a disturbance in yaw 96
Figure 4.31 What is required for gyroscopic stability 97
Figure 4.32 Gunnery angles 99
Figure 4.33 Variation of trajectories with launch angle (QE) 101
Figure 4.34 Equilibrium yaw 102
Figure 4.35 Flow about a spinning cylinder 102
Figure 4.36 Rigidity of trajectory 106
Figure 4.37 Effects of rotation of the earth (1) 106
Figure 4.38 Effects of rotation of the earth (2) 107
Figure 4.39 Effects of cross-wind on drift 108
Figure 4.40 Effect of delayed opening fins for artillery rockets 110
Chapter 4 – Part II
Figure 4.41 Boundary layer velocity profiles 116
Figure 4.42 The boundary layer on a flat plate 117
Figure 4.43 (a) Inviscid dlow about circular cylinder;
(b) Pressure distribution 118
Figure 4.44 Flow in the neighbourhood of separation 118
Figure 4.45 (a) Real flow about circular cylinder;
(b) Pressure distribution 119
Figure 4.46 Flow about circular cylinder for laminar and turbulent
separation 120
Figure 4.47 Variation of drag coefficient with Reynolds number for a
sphere 120
Figure 4.48 The effect of surface roughness just below the critical
Reynolds number 121
Figure 4.49 Laminar and turbulent flow behind a truncated body at
supersonic speeds 122
Figure 4.50 The flow through a shock wave 122
Figure 4.51 Supersonic flow past a cone 122
Figure 4.52 Maximum turning angles for supersonic downstream flow 123
Figure 4.53 Body upwash effect on wing local incidence angle 126
Figure 4.54 Wing-body interference effects 127
Figure 4.55 (a) A spinning-top; (b) Components of  in x, y axes 128
Figure 4.56 Components of angular momentum 129
Figure 4.57 Variations of yawing moment characteristics 131
Figure 4.58 Convention system 132
Figure 4.59 The stability diagram for small perturbations 135
Figure 4.60 Motion in vacuo 135
Figure 4.61 The conservation of linear momentum 139
Figure 4.62 Rocket motion 141
Figure 4.63 Equilibrium of a spinning projectile 142
Figure 4.64 Projectile subjected to control moment 143
Figure 4.65 Resultant equilibrium state 144
Figure 4.66 Effect of impulse control 145
Chapter 5
Figure 5.1 Angle of attack 148
Figure 5.2 Impact of a long rod penetrator 151
Figure 5.3 Perforation mechanisms 152
Figure 5.4 Basic components of an APDS projectile 154
Figure 5.5 Basic components of an APFSDS projectile 155
Figure 5.6 Basic components of a HE fragmentation shell 155
Figure 5.7 The shaped charge principle 157
Figure 5.8 Penetrative performance of a conical liner 157
Figure 5.9 Basic components of a HEAT projectile 158
Figure 5.10 HEAT effect 159
Figure 5.11 Basic components for a HESH projectile 159
Figure 5.12 HESH effect 160
13569_int.indd 10-11 11/15/11 11:20:16 AM
xii List of Illustrations List of Illustrations xiii
List of Tables
Chapter 6
Figure 6.1 Wound track of subsonic projectile 169
Figure 6.2 Stress wave formed by high velocity projectile 169
Figure 6.3 Temporary cavity formed by high velocity projectile 169
Figure 6.4 Selected stills from a high speed cine film showing the
temporary cavitation
(Photo by courtesy of CDE, Porton Down) 171
Figure 6.5 The respective wounding effects of a soft bullet, unstable
jacketed bullet and stable jacketed bullet 171
Figure 6.6 The permanent record of cavitation effects as shown in
standard soap blocks
(Photo by courtesy of CDE, Porton Down) 172
Figure 6.7 Possible cavitation effects with metallic armour 173
Figure 6.8 The non-penetrating impact of a 9 mm bullet travelling at
330 m/s against 16 plys of Kevlar body armour
(Photo by courtesy of CDE, Porton Down) 174
Chapter 7
Figure 7.1 Time scale for various illuminating sources 178
Figure 7.2 A microflash photograph of a 105 mm shell in flight
(Photo by courtesy of Cranfield University) 178
Figure 7.3 A microflash photograph of 120 mm APDS in flight
(Photo by courtesy of P. Fuller) 179
Figure 7.4 Experimental arrangement for spark photography 180
Figure 7.5 Diagrammatic representation of the shadowgraph 180
Figure 7.6 Multiple spark photograph
(Photo by courtesy of P. Fuller) 181
Figure 7.7 Schlieren photograph of sphere
(Photo by courtesy of P. Fuller) 181
Figure 7.8 The Schlieren system 182
Figure 7.9 Rocket launching
(Photo by courtesy of P. Fuller) 182
Figure 7.10 Compensation for film movement by rotating glass block 183
Figure 7.11 Ballistic synchro technique
(Diagram by courtesy of P. Fuller) 184
Figure 7.12 A ballistic synchro record of 120 mm APDS
(Photo by courtesy of P. Fuller) 185
Figure 7.13 The rotating mirror streak camera
(Diagram by courtesy ofP. Fuller) 185
Figure 7.14 The rotating mirror framing camera
(Diagram by courtesy of P. Fuller) 186
Figure 7.15 Image converter camera schematic 186
Figure 7.16 Lead pellet impacting on hard target
(Photo by courtesy of John Hadland Photographic
Instrumentation Ltd., Herts, UK) 187
Figure 7.17 Streak photograph of 7.62 mm bullet travelling at approximately 840 m/s
(Photo by courtesy of Cranfield University) 187
Figure 7.18 X-ray shadowgraph of 0. 45 automatic during firing
(Photo by courtesy of Hewlett Packard Ltd., Berks, UK) 188
Figure 7.19 PCC equipment 189
Figure 7.20 Principle of radio-doppler 190
Figure 7.21 Principle of yaw sonde 192
Figure 7.22 Principle of the ‘V’ cell 192
Figure 7.23 Interpretation of pulse record 193
Figure 7.24 Results from yaw sonde 193
Figure 7.25 Shot position indicator
(Photo by courtesy of M. S. Instruments Ltd., Kent, UK) 194
Figure 7.26 Position of shock wave front 195
Figure 7.27 Cartesian co-ordinate system 195
Figure 7.28 Crusher gauge 196
Figure 7.29 Basic interferometer system 198
Figure 7.30 Direct optical method 199
Figure 7.31 Bore-wire resistance method 199
Figure 7.32 Linear displacement transducers 200
Table 4.1 The Standard Atmosphere (ICAO) 69
Table 4.2 Drah breakdown for a typical shell shape 85
Table 4.3 Drag breakdown for typical APFSDS round 86
Table 4.4 Spin rates for normal stability for different calibre
projectiles 97
Table 4.5 The effect of air resistance on range 100
Table 4.6 Input requirements of trajectory models and comparative
computing requirements 103
Table 4.7 Typical values for specific impulse 140
13569_int.indd 12-13 11/15/11 11:20:17 AM
1
1.
History of Ballistics
FIG. 1.1 Ballista
External Ballistics
Pre-Newton Era
The first stone hurled by prehistoric man was probably the earliest example of
external ballistics. The advantages of being able to throw farther and with more
power led to devices such as slings and spears. Next came the bow, and an
extension of it called the ‘ballista’ from which ballistics derives its name. In turn,
the word ballista owes its origin to a Greek word ballein, meaning ‘to throw’.
The ballista was a fairly complicated device used for propelling large arrows.
(See Fig. 1.1).
It was the work of Leonardo da Vinci (1452–1519) which led to the early
development of modern ordnance engineering. He designed many kinds of
13569_int.indd 14-1 11/15/11 11:20:17 AM
2 Military Ballistics History of Ballistics 3
weapons, both offensive and defensive, ranging from cannon balls, mortars, rifled
firearms, up to primitive versions of the tank and submarine. Da Vinci was also
the first to provide a theoretical basis for the phenomena of aerodynamics: for
example he conceived the idea of a centre of pressure whilst studying the flight of
birds. External ballistics was founded as an exact science with the work of
Galileo Galilei of Italy (1562–1642).
Galileo destroyed the Aristotelian theory of motion* and succeeded in laying
the foundation for an accurate scientific study of motion. He deduced a parabolic
path for a projectile in vacuo. His pupil Evangelista Torrecelli formulated the
equation for the range of a projectile, constructed the parabola and studied its
various properties. The quadrant angles of elevation of cannon had been accurately measured from the time of Niccolo Tartaglia in 1537, but crude methods of
measuring muzzle velocity were first found in the century following Galileo’s
death. It was then learned that the ranges actually attained by projectiles were
very much shorter than those predicted by Galileo’s parabolic trajectory.
However, Galileo had known that the atmosphere resisted the motion of projectiles and thus he stated that his demonstrations were accurate ‘in the case of no
resistance’. Galileo argued that the retardation or acceleration due to the drag of
the air on a moving body was a function ‘of weight, of velocity, and also of form’.
He stated that this resistance decreased with the projectile’s density, increased
with its speed and varied greatly with its shape. Galileo’s work paved the way for
Sir Isaac Newton (1642–1727) who was probably the greatest of the modern
founders of ballistics. Newton’s work on dynamics appeared in the ‘Philosophiae
Naturalis Principia Mathematica’.
The Principia contained two volumes: they were concerned with the motion of
rigid bodies and the motion of fluids. Both subjects are of prime interest in
modern ballistics. Newton began his argument on gravitation by considering the
motion of a projectile fired horizontally from a mountain top in vacuo. He showed
that, by continually increasing the initial speed of the projectile a speed would
eventually be obtained which would result in the projectile moving completely
around the earth and returning to pass through the firing position.
Post-Newton Era
The most important early successor of Newton in ballistics was Leonhard Euler
of Switzerland (1707–1783). He analysed the results of experimental range
firings in order to determine the drag on cannon balls. It is important to note that
he was the first to work in analytical rather than geometrical terms. †
In 1742
Benjamin Robins invented the ballistic pendulum and with it determined the
muzzle velocity of musket balls. He was able to measure velocities up to 518 m/s
at distances up to 76m from the gun. He investigated drag at low and high
velocities, and found that Newton’s velocity squared law for drag held quite well
up to velocities of 244 m/s but that with velocities of 336 m/s or more the
resistance was very much greater.†
The importance of accurate experimental methods for determining the drag of
projectiles increased steadily during the 19th century because of developments in
internal ballistics and ordnance which had increased both the magnitude and
consistency of muzzle velocities. Progress in accurate timing of shell motion was
made by Sir Charles Wheatstone (1802–1875) who used a screen through which
the projectile passed, thus breaking an electrical circuit. This led to the work of
Francis Bashforth; (1865–1870, 1878–1880) using an electrical chronograph,
which he had devised for determining the drag of artillery projectiles.
Experimental work continued throughout Europe to provide a drag law, involving the velocity, for a standard shell. It was during the 19th century that a
greater understanding of aerodynamics occurred. Drag as a function of the
properties of air was recognised and this gradually led to the shape of the
projectile now in common use. The smoothbore guns of the 18th century produced
low muzzle velocities and were inaccurate. This led to the appearance of the rifled
gun in European warfare during the early part of the 19th century. Cannon used
by European armies in the 18th century were effective only at short distances
because of their low muzzle velocity and internal clearances. This led to the
return of the artillery rocket.
Incendiary rockets had been extensively used in Italy and Germany during the
14th century, but they were gradually abandoned in European land warfare after
1450, largely because of their tendency to explode during manufacture or upon
firing. Rockets continued to be used in the Middle East however. The Indian
rockets were inaccurate and consisted basically of iron tubes weighing three to
five kilograms. However, they were effective enough to alarm the British army
campaigning in India, and they interested William Congreve of England (1772–
1828) who developed incendiary rockets which could achieve ranges of about
three kilometres. As a consequence, by the time of Napoleon every European
nation had a Rocket Corps in its army.
The spin-stabilised rocket was invented by the American, William Hale, in
1855. With the adoption of rifled artillery came the first need to treat the
projectile as a body subject to aerodynamic forces other than gravity and drag.
Ballisticians now had to explain other phenomena such as projectile drift. During
the 20th century a thorough mathematical basis has been established to describe
all the aerodynamic forces acting on a projectile in flight. In recent years the
elaborate ballistic tables for calculating trajectories have been largely superseded by computers. Of course the drag law for individual projectiles is still
required and may sometimes be deduced from theoretical considerations coupled
with wind tunnel tests and live firings in a laboratory setting rather than the
extensive range firings formerly undertaken. * Aristotle reasoned that no inanimate body could move without a motive force; unless it belonged to
four natural elements from which the Universe was made up; fire, air, water and earth. Air and fire
could only move upwards, water and earth downwards, their natural homes. This could not explain
the flight of an arrow, for example, after the motive force of man had been removed. To overcome this
he introduced the medium in which motion takes place. This medium assisted all ‘violent’ motions.

The letter e was first used to denote the base of the natural system of logarithms in one of Euler’s
papers on ballistics.

We now recognise that this rise in resistance at approximately 340 m/s is due to the transition from
subsonic to supersonic flight.
13569_int.indd 2-3 11/15/11 11:20:17 AM
4 Military Ballistics History of Ballistics 5
Internal Ballistics
The Rise of Gunpowder
The history of internal ballistics begins with the use of gunpowder and,
although the actual date of its first use as a propellant has never been accurately
determined, it dates back to the early 14th century and was certainly used in the
battle between the English and French at Crecy in 1346. By the end of the 18th
century the composition of gunpowder was fairly well standardised at 75 per cent
saltpetre, 15 per cent charcoal and 10 per cent sulphur. The first recorded attempt
to test powder was made by Bourne in 1578. He fired the powder in a small metal
cylinder and the extent to which the lid rose gave an indication of the ‘strength’ of
the powder.
Early Measurements
The earliest attempts to measure the ballistics of the powder were made by
Luys Collado in Italy and William Eldred and Nathanial Nye in England, during
the 17th century. These attempts consisted of firing shots at a series of elevations
and measuring the range. Benjamin Robins with his invention of the ballistic
pendulum in 1742 measured the muzzle velocity of musket balls and deduced the
pressure of the propellant gases. His book ‘New Principles of Gunnery’ published
in 1742 dealt with the fundamental problem of internal ballistics namely, achieving a given velocity within the pressure limits. The first attempt to measure the
pressure of propellant gases directly was made by Count Rumford in America in
1792. His experiments led him to deduce a relation between pressure and density
of the gases. At the end of the 18th century ballisticians were able to calculate the
relation between pressure and shot travel, using Rumford’s pressure density
relation and assuming that the charge was completely burnt before the shot
started to move. Integrating the pressure-shot travel curve enabled them to
calculate the muzzle velocity and so to compare their results with experimental
findings.
Piobert of France announced his laws of burning in 1839. Although these laws
relate to black powder, one of them, namely, that the burning of each individual
granule takes place in parallel layers, has been found to be applicable to modern
propellants. Piobert also gave an approximate solution to the problem of the
motion of the gases in the bore, originally treated by Lagrange during the period
of the French revolution. The approximate relation between the pressures on the
breech and on the base of the shot was also worked out by Piobert. In 1857,
General Rodman of America invented an ‘Indentation’ pressure gauge for
measuring the pressure of the propellant gases. The pressure was determined by
the indentation made in a copper or lead plate by a wedged piston in contact with
the gases. With this gauge he was able to measure the maximum pressure in
guns and also deduced a pressure-density relation in a closed vessel. Rodman also
showed how the shape of the granule affects the rate of burning.
The Basis of Modern Measuring Systems
A more accurate pressure measuring device called the ‘Crusher gauge’ was
invented by Andrew Noble in 1860. With this, he and Frederick Abel deduced the
law relating to pressure and density at constant volume. The energy equation for
burning propellants was given by Resal in 1864, thus laying the foundation of
internal ballistics on thermodynamic principles.
By the end of World War II complex mathematical models existed. However
these were generally inapplicable due to various simplifications. The advent of
computers has enabled a more accurate determination of real gun systems and
modern programs may have as many as fifty variables.
Modern Propellants
Modern propellants date from 1845 when a German chemist Christian
Schonbein, discovered nitrocellulose, which burns completely leaving no solid
residues. Gunpowder by comparison produces over half its weight as solid residue. A satisfactory propellant in the form of cakes was first produced in 1884 by a
French physicist Paul Vieille by gelatinising nitrocellulose with an ether-alcohol
mixture. This was used by the French army under the name of Poudre B. Alfred
Nobel produced a similar propellant by using nitro-glycerine instead of etheralcohol. Abel in Britain gelatinised nitrocellulose by acetone using a mixture of
nitroglycerine and vaseline. It was known as cordite due to its shape and was
adopted by the British army in 1891 and is still used.
Since the firing of the first liquid propellant rocket by Robert Goddard in 1926,
tremendous progress has been achieved in the development of liquid and hybrid
propellants capable of reaching very high impulses. Prior to World War II, the
use of solid propellant was generally restricted to armament rockets and boosters. Progress in this field has made its use in rockets and missiles widespread.
Terminal Ballistics
Armour Piercing Developments
The scientific study of the effects of a projectile striking a target is comparatively new. The early efforts to increase the efficiency of a weapon simply
consisted of making the shell larger. Introduction of armour and aircraft into
warfare forced the development of armour-piercing devices. However, it is only
the recent advances in metallurgy, strength of materials, and the development of
sophisticated ballistic instruments capable of measuring very high pressure and
phenomena taking place in the range of milliseconds that have enabled a proper
study of terminal ballistics. Since the beginning of the Second World War
considerable progress has been achieved in making high explosive shells and
shots capable of causing heavy damage by blast, scabbing, fragmentation, and
penetration. The study of the terminal effects of tactical nuclear weapons in
recent years has extended the boundaries of terminal ballistics.
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6 Military Ballistics History of Ballistics 7
Forensic and Wound Ballistics
The use of handguns in civilian life has recently brought medicine and law into
the realms of ballistics under the heading offorensic ballistics. In parallel with it
the damage caused by ammunition to the human body has given rise to a new
specialisation called wound ballistics.
QUESTION 9 Until recently, what practical factors limited an extensive
scientific study of the subject of terminal ballistics?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 10 What are the two most recent applications of ballistics?
Answer ……………………………………………………………………………….
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Self Test Questions
QUESTION 1 Who pioneered the early development of modern ordnance
engineering?
Answer ……………………………………………………………………………….
QUESTION 2 Who succeeded in laying the foundation for an accurate scientific study of motion?
Answer ……………………………………………………………………………….
QUESTION 3 What two important subjects of prime interest to modern ballistics were contained in Newton’s Principia?
Answer ……………………………………………………………………………….
………………………………………………………………………………………….
QUESTION 4 What important invention was created in 1742 by Benjamin
Robins to measure muzzle velocities of musket balls?
Answer ……………………………………………………………………………….
QUESTION 5 What important factor led to the shape of the projectile now in
common use?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
QUESTION 6 Why was rifled artillery adopted?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 7 How did Bourne in 1578 attempt to test gunpowder?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 8 What important law did the Crusher gauge help to determine?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
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9
2.
Internal Ballistics – Part I
Definition of the Gun and Internal Ballistics
The gun can be viewed as a mechanical device in which heat, liberated by a
burning propellant, is converted into the useful kinetic energy of a projectile, and
its function is to propel projectiles toward specified targets. Internal ballistics is
the scientific study of the operating processes within the gun from the moment
that the burning of the propellant is initiated.
The Ballistic Components of the Gun
There is a wide variety of firearms to be found today; in practice the various
types, from handguns through to heavy artillery, show distinct differences, but
from a ballistic standpoint they are all similar. The majority of firearms work on
the principle of the gun, and it is this majority which will be considered in this
chapter. The few unconventional firearms, notably recoilless guns, will be just
briefly described: they are in effect rocket powered and so do not conform to the
traditional concept of the gun.
Firearms are devices for propelling projectiles toward specified targets: their
common component is a tube in which both motion and direction are imparted to
the projectiles fired. To avoid confusion, the gun will be defined here as having
one closed end, whereas the recoilless gun is effectively open at both ends. The
closed end of a gun forms a chamber which, when loaded, is filled with propellant
and has a projectile positioned ahead of it. When the propellant is ignited it
rapidly produces gases which in turn push the projectile along the barrel that
forms the remaining part of the tube.
To fire a gun the propellant must be ignited, so an ignition device is included in
the propellant chamber. Once ignited the propellant is said to burn, however it is
unlike more familiar fuels as its combustion does not require oxygen from the air:
only the chemicals within the propellant react to produce the resultant hot gases.
Many guns fire projectiles which need to be spun to maintain a stable flight. In
such a case a gun has a rifled bore to its barrel, which causes the projectiles to
rotate as they pass along the barrel.
The internal diameter of the barrel is called the calibre; if the barrel is rifled
the calibre is the diameter of the bore prior to the cutting of rifling grooves in the
bore surface.
For ease of use the propellant and ignition device are often assembled as a unit
13569_int.indd 8-9 11/15/11 11:20:18 AM
10 Military Ballistics Internal Ballistics – Part I 11
FIG. 2.1 The main ballistic components of the loaded gun
in a combustible bag, or within a cartridge. Once inserted, a cartridge case can be
considered as part of the chamber wall.
The components that are renewed for each firing of the gun are the projectile,
the propellant charge, the primer and, where applicable, the cartridge case: they
are collectively known as the ammunition. The dimensions and characteristics of
these components together determine the loading conditions of the gun.
The Practical Gun
The fundamental differences between the internal designs of guns are usually
only differences in the size and the shape of the ballistic components. However,
the definition so far is only sufficient to describe a simple muzzle-loaded cannon,
so additional parts must be incorporated in the design to make the gun a practical
and effective weapon. Four mechanisms must be included to enable mounting,
loading, aiming and firing of the gun.
The simplest mounting is used for handguns and most rifles: they are simply
hand held. Heavier guns of course need mechanical support and this is usually
provided by a pivoting block at the chamber end which allows the gun to be
turned for aiming.
Normally the back of the chamber can be opened so that a new projectile,
propellant charge and primer can be loaded into the breech: this is achieved
either manually or by an automatic mechanism. The designs of breech mechanisms vary significantly between the different types of guns, though their function
is the same: it is to permit reloading of the gun for the next firing sequence.
Accurate aiming is achieved either visually by the use of sights or by calculated alignment of the gun according to data on the relative position of the target.
An aiming device is of little concern in the study of internal ballistics, though the
accuracy of both the aiming device and the internal ballistics are together vital to
the gun’s ability to deliver projectiles to the point of aim.
The final mechanism necessary to the operation of the gun is an ignition
device. Usually this consists of a primer, which is a small but powerful propellant
charge which on firing discharges hot gases and particles into the main propellant charge. The firing is initiated by electrical heating or mechanical crushing of
the primer.
The internal ballistics of the gun that will be described in this chapter is only
an idealised behaviour of the gun. It roughly describes the components, firing
sequence and problems encountered in most guns. To model accurately the
internal ballistics and response of specific guns, the effect of their added practical
components must also be considered. The structure of a gun and its condition on
firing have a most significant effect on the recoil of the gun and the flexure of its
individual components. Internal ballistics alone cannot fully describe such
effects, but it can provide explanations and predictions of the dominating forces
and effects observed in most guns.
The Projectile
The projectile component of ammunition can also be referred to as the shot.
There are three common types of projectile.
Bore-calibre Projectiles
Bore-calibre projectiles such as the conventional shell and bullet are of the
same diameter as the bore of the gun and so their sides bear directly against the
sides of the bore. This type also includes multiple-projectiles in which a number
of projectiles are encased within a solid jacket until they have left the gun. Most
large bore-calibre projectiles are fitted with a driving band or bands, to allow
location, obturation (sealing) and spin of the projectiles within the barrel. To
achieve the same functions, projectiles without driving bands rely on the high gas
pressures on firing to expand the projectile base and the swaging action of the
barrel to crush the projectile to the shape of the rifling.
Sabotted Projectiles
There are considerable external ballistic and terminal ballistic advantages to
be gained by using sub-calibre projectiles supported by lightweight sabots while
in the gun. The sabot will support any shape of projectile, notably including the
fin-stabilised projectile, during its motion along the barrel: the sabot is discarded
shortly after leaving the gun. There are two basic types of sabot, the axially and
the radially discarding sabots. The axially discarding sabot, known as a pot
FIG. 2.2 105 mm shell featuring a single driving band
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12 Military Ballistics Internal Ballistics – Part I 13
* The individual pieces of propellant are often known as grains; to avoid confusion with the grain (a
unit of weight  1/7000 pound) sometimes used in ballistics, the propellant pieces are here termed
granules.
sabot, is pulled off the back of the projectile by the excessive drag generated by its
poor aerodynamic shape. Radially discarding sabots consist of a number of petals
which are either spun off or blown off by the gas flow after muzzle exit.
A sabot is fitted with a driving band, though when a fin-stabilised projectile is
used the driving band is often allowed to rotate freely around the sabot to avoid
detrimental spin of the projectile if fired from a rifled gun. Whilst in the gun, the
projectile and sabot can together be considered as a single projectile.
Shot
The term shot can be applied to loosely supported lead shot (ie, lead spheres) or
a number offlechettes positioned ahead of a gas sealing wad, but its use is usually
limited to shotguns and warhead fillings. The internal ballistics of guns and
shotguns are similar since the behaviour of loose shot driven ahead of a wad is
similar to that of other projectiles.
The Propellant
Basic Characteristics
Once ignited the components of the propellant, depending on composition,
rapidly decompose or react together to produce energetic gases which generate
high pressures and temperatures within the gun. The high pressure acts on the
base of the projectile and so pushes it along the barrel.
Propellants are available in a vast number of compositions, shapes and sizes.
Though some types, especially those used in large calibre gun charges, are
formed into long cords, ribbons or slotted tubes, all individual pieces of propellant
are referred to here as propellant granules.*
For each gun and projectile combination the propellant type, granule design
and quantity used is selected primarily to produce suitable muzzle velocities
without exceeding the pressure limitations of the gun. The burning characteristics of a propellant composition are the burning rate constant, the pressure
index, the force constant and the co-volume. The burning characteristics of a
propellant granule design are the ballistic size and the form function.
Propellant Compositions
Black powder, often and ambiguously known as gunpowder, is the modern
formulation of the earliest propellant; currently it is used in some gun igniters,
low velocity guns, and rockets. It consists of a mixture of saltpetre, charcoal and
sulphur roughly in proportions of 75:15:10. All other propellants for guns are
known as smokeless powders although they are neither completely smokeless nor
powders. The three basic types are derived from explosives whose explosive
qualities are moderated in the processes which convert them to propellants.
Double-base propellants are more powerful than single-base propellants,
however they suffer from high propellant gas temperatures which can cause
excessive barrel erosion and muzzle flash. Triple-base propellants are similarly
powerful, but the addition of cool burning nitroguanidine reduces the temperature of the gas to near that of the single-base propellants. Other ingredients
added to smokeless propellants are used primarily to control burning rate and
suppress decomposition during storage.
Piobert’s Law
Before considering what effect the composition and granule design have on the
performance of a burning propellant, the mechanical process by which a granule
burns must be assumed. It is assumed that on firing, the entire surface of each
FIG.2.3 7.62  51 mm cartridge together with a fired boattailed bullet and
an unfired flat-based bullet
FIG. 2.4 105 mm calibre APFSDS (Armour Piercing Fin-Stabilised Discarding Sabot)
with sabot attached
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14 Military Ballistics Internal Ballistics – Part I 15
Propellant type Basic preparation
Single-base Nitrocelulose dissolved in ether and alcohol.
Double-base Nitrocellulose dissolved in nitroglycerine.
Triple-base Nitrocelulose dissolved in nitroglycerine,
nitroguanidine added.
Each preparation is then pressed to shape and dried.
propellant granule is ignited almost simultaneously and all the surfaces then
recede at an identical rate: the granules are said to burn in parallel layers. For
example, if the initial granule is cylindrical, it will retain its cylindrical shape
throughout the burning process as the diameter and the length of the cylinder
will reduce at the same rate. The diameter, being less than the length, will be
exhausted first.
The assumption that propellant granules burn in parallel layers is known as
Piobert’s Law, and is strongly supported by experimental evidence.
Burning Rate
As a granule of propellant burns, its surface undergoes reactions which convert
it to energetic gases. Most of the liberated energy generates high pressures and
temperatures of the gas. A small proportion of the liberated energy is conducted
into the granule, raising the temperature of successive layers of propellant until
they too are ignited. Examples of the ignition temperatures are:
FIG. 2.9 Ignition temperature of smokeless propellants
FIG. 2.5 Cross-section of a shotgun cartridge
Propellant type Ignition temperature
Single-base 315°C approx.
Double and Triple-base 150° -170°C
FIG.2.6 Common shapes of propellant granules
FIG. 2.7 Preparation of smokeless propellants
FIG. 2.8 Three stages in the burning of a cylindrical propellant granule
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16 Military Ballistics Internal Ballistics – Part I 17
The ignition process is thus sustained until all of the propellant has been
consumed. As burning occurs on all sides, the rate at which a granule reduces in
size, known as the burning rate, is equal to twice the rate at which the ignition
process spreads through the granule. It is calibrated for a pressure of 1 MPa
(approximately 10 atmospheres*): this value is known as the burning rate constant. For example:
From this table it can be seen that a cylindrical stick of typical propellant
1.1 mm in diameter will take over 7 seconds to burn in free air, whereas at the
high pressure of 380 MPa, such as may occur in a high velocity gun, the same
stick would be completely burnt in less than 2 ms.
Force Constant, Co-volume and Flame Temperature
The amount of energy released by a certain mass of propellant is related to
its force constant; thus the powerful double and triple-base propellants have
higher force constants than the less powerful single-base propellants. The value
of the force constant is found by burning a measured mass of propellant in a
strong airtight chamber known as a closed-vessel. The propellant is electrically
ignited and the pressure developed within the closed-vessel is measured by a
pressure gauge.
Maximum pressure is reached once all the propellant has burnt; the pressure
then begins to fall as the hot gases are cooled by contact with the cold sides of the
closed-vessel.
Before the force constant can be calculated, the volume of the closed-vessel
must be known. The effective volume of the closed-vessel is less than its actual
volume owing to the volume occupied by the propellant molecules. The volume
occupied by the molecules of a kilogram of propellant is called its co-volume; this
can be determined during a number of closed-vessel firings using a variety of
masses of the propellant under test.
Effective volume of  actual volume – co-volume of  mass of
closed-vessel of closed-vessel propellant propellant
The force constant can then be calculated:
force constant  maximum  effective volume ÷ mass of
pressure of closed-vessel propellant
Each type of propellant burns with a characteristic flame temperature. This
In some texts, burning rate is taken to be equal to the ignition spread rate, so
the burning rate constant appears to be half the expected value. The burning rate
constant is affected by changes in the initial temperature of the propellant: an
increase in the initial temperature will increase the burning rate.
Pressure Index
The burning rate also increases as the pressure increases, though it does not
necessarily increase by the same proportion. The coefficient which relates
changes in burning rate to changes in pressure is the pressure index. If the value
of the pressure index is 1, the burning rate and pressure rise in direct proportion,
so a doubling of the pressure will double the burning rate. The greater the value
of the pressure index, the quicker will be the rise in burning rate as pressure
increases. For modern propellants the pressure index is usually close to 1 and the
burning rate constant is typically 1.5 mm/second/MPa. The increase in burning
rate against increasing pressure can be seen in the table below.
Propellant type Typical burning rate constant
Black powder 18 mm/second/MPa approx.
Smokeless powders 0. 5-3 mm/second/MPa
FIG. 2.10 Typical burning rate constants
FIG. 2.11 Burning rate against pressure for a typical propellant
* A pressure of one atmosphere is the natural pressure of the air at sea level.
FIG. 2.12 Pressure within a closed-vessel plotted against time
Pressure Burning Rate
MPa Atmospheres (approx.) mm/second
0.1 1 0.15
1 10 1.5
50 500 75
380 3800 570
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18 Military Ballistics Internal Ballistics – Part I 19
temperature is only effectively achieved in a closed-vessel; in guns the temperature of the gases evolved within the flames is simultaneously cooled by their own
expansion. The volume of a closed-vessel is fixed and the rate at which heat is lost
by conduction to the vessel sides is slow, so the associated flame temperature is
known as the adiabatic* or isochoric†
flame temperature.
Form Function
The configuration of the propellant, or form function, as it is known, is important. It is interesting to take the case of numerous long cylindrical granules of
propellant. At ignition they possess a relatively large total surface area. The rate
at which propellant gas is produced is directly proportional to the total surface
area of all the granules so there is a relatively fast initial production of gas. As
the burning proceeds, the surface area reduces and results in a corresponding
reduction in the relative rate at which gas can be produced. Each granule is
simultaneously being subject to variations in pressure which affect the burning
rate and so also affect the rate at which gas is produced. Finally, the surface area
of the propellant granules falls to zero and the production of gas stops. Such a
granule shape, in which the surface area diminishes as the burning progresses, is
said to be degressive.
For each granule shape, a geometric relationship exists between the fraction of
propellant burnt and the fraction of the ballistic size remaining at any moment.
This relationship is called the form function. The degressiveness of a granule is
described mathematically by the form function coefficient that appears in the
form function. Degressive granules, such as long cylinders, have positive coefficients. By comparison, a thin disc shaped granule maintains almost the same
surface area throughout the burning process, and so has a coefficient very close to
zero.
Tubular granules are frequently used as they are less degressive than cylindrical granules. When long tubes are used, they are slotted along one side to avoid
excessive pressures within the tube that would otherwise cause the tubes to
disintegrate.
Some propellant granules, on the other hand, are designed to generate most
gas near the end of the burning process. This is achieved by either forming the
propellant into tubes, or by doping the degressive outer layers of the granules
with burn suppressants or ignition inhibitors. Such granules have negative form
function coefficients and are said to be progressive. The initial production of gas
is slowed by suppressants in the outer layers of solid suppressed burn granules.
Once the suppressed layers have burned, the burning proceeds at the normal
degressive rate
In the examples above it is interesting to note that the single-base propellant
burns with the lowest flame temperature, and that the triple-base propellant
burns at a considerably lower temperature than the double-base propellant even
though the two are similarly powerful (ie, have similarly high force constants).
Ballistic Size
When, say, a long cylindrical granule of propellant burns, its length and
diameter recede at the same rate in accordance with Piobert’s Law of burning. As
the diameter is less than the length, it completes burning when the diameter,
rather than the length, reduces to zero. From this it can be seen that the diameter
is the significant dimension or ballistic size of a long cylinder.
The ballistic size is usually the shortest distance between any two opposing
surfaces of a granule. One exception to this is the multi-tube granule which, as
will be explained later, continues to burn after the ballistic size has reduced to
zero: in this case the significant dimension is called the web size.
U. S. standard propellant Propellant type Adiabatic flame temperature
M6 Single-base 2570 K (2297°C)
M2 Double-base 3319 K (3046°C)
M30 Triple-base 3040 K (2767°C)
FIG. 2.13 Typical adiabatic (closed-vessel) flame temperatures
FIG. 2.14 Examples of ballistic size
* Adiabatic  without transference of heat.
† Isochoric  at constant volume. FIG. 2.15 Types of progressive granules
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20 Military Ballistics Internal Ballistics – Part I 21
Whilst the outer surface of a tube burns degressively, its progressive inner
surface increases in area as it burns; but if the outer surface is completely
inhibited then the granule will burn progressively.
Alternatively, the degressive outer surface can be compensated by the progressive burning of a number of holes through the granule; this is the principle
applied to multi-tube granules to produce neutral or slightly progressive burning. Once the web size of a multi-tube granule has reduced to zero, it breaks into
so-called slivers which continue to burn until all the propellant has been
exhausted.
The Primer
A primer initiates the burning of a propellant by releasing hot gases and
particles into the charge. The surface temperature of granules in the flow of hot
primer gases rises until they are ignited. The flow of hot primer and propellant
gases then spreads throughout the chamber perpetuating the flame spread like a
wave.
The wave form flame spread is accompanied by a pressure wave generated as a
result of the gas flow and enhanced production of propellant gases in regions of
high relative pressure. The pressure waves are reflected at the chamber sides and
so pass back and forth through the propellant. This behaviour is normally
detected as rapid fluctuations in the pressure measured at the base and projectile
ends of the chamber. The oscillations fade once the projectile begins to accelerate
along the bore. Such oscillations can lead to inconsistent burning and excessive
chamber pressures: this serious problem is frequently encountered with modern
guns using very slow burning propellants in relatively large chambers.
The importance of the primer and propellant charge interaction cannot be
overstressed. Any inconsistencies in ignition will manifest themselves on the
entire ballistic sequence: consequential variations in muzzle velocity and recoil
effects will reduce accuracy.
The Firing Sequence
The firing sequence is usually initiated by the ignition of the primer. This
causes the primer charge combustion products, consisting of hot gases and
FIG. 2.16 Cross-sections of 7 hole multi-tube granule
Granule shape Approximate form
function coefficient
Random chips positive Most degressive
Spherical positive
Cylindrical / Cord positive
Disc positive, near zero
Tube / Slotted tube positive, near zero
Ribbon positive, near zero
Solid, suppressed burn of outer layers near zero Neutral
Multi-tube near zero
Tube, inhibited ignition of outer surface negative Most degressive
FIG. 2.17 Examples of granule shapes and their form function coefficients
FIG. 2.18 Configuration of igniter and propellant charge in typical ammunition
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22 Military Ballistics Internal Ballistics – Part I 23
incandescent particles, to be injected into the propellant. The hot gases flowing
between the propellant granules ignite the granule surfaces; the primer and
propellant combustion products then act together, perpetuating the flame spread
until all the propellant granules are ignited.
At first the chamber is virtually sealed by the projectile, so the gases and
energy liberated by the primer and propellant cannot escape: there is a resultant
dramatic increase in the pressure and temperature within the chamber. The
burning rate of the propellant is roughly proportional to the pressure, so the
increase in pressure is accompanied by an increase in the rate at which further
gas is produced. Without any means to check the accelerating production of gas,
the gun would explode.
The rising pressure is moderated by the motion of the projectile along the
barrel. The pressure at which this motion begins is the shot-start pressure. The
projectile will then almost immediately encounter the rifling and the projectile
will slow or stop again until the pressure has increased sufficiently to push it into
the constricting bore. The driving bands or the surface of the projectile itself,
depending on design, will be engraved to the shape of the rifling if the bore is
rifled. The resistance then falls, allowing the rapidly increasing pressure to
accelerate the projectile.
As the projectile moves forward it leaves behind it an increasing volume to be
filled by the high pressure propellant gases. The propellant is still burning,
producing high pressure gas so rapidly that the motion of the projectile cannot
fully compensate: as a result, the pressure continues to rise until the peak
pressure is reached. The peak pressure is attained when the projectile has
travelled about one tenth of the total length of a full length gun barrel.
The extra space being created behind the rapidly accelerating projectile then
exceeds the rate at which high pressure gas is being produced, and so the
pressure begins to fall. The next stage is the all-burnt position at which the
burning of the propellant is completed. However, there is still a considerable
pressure in the gun so, for the remaining motion along the bore, the projectile
continues to accelerate: as it approaches the muzzle the propellant gases expand,
the pressure falls, and so the acceleration lessens. At the moment the projectile
leaves the gun the pressure will have reduced to about one sixth of the peak
pressure.
The flow of gases following the projectile out of the muzzle provides additional
acceleration for a short distance so that the full muzzle velocity is not reached
until the projectile is some distance beyond the muzzle. After this, the projectile
is soon lost from the gun’s influence and begins its fast, independent flight.
This entire sequence, from primer firing to muzzle exit, typically occurs within
15 milliseconds. For a handgun the sequence may take less than 1 millisecond,
but perhaps as much as 25 milliseconds for a large artillery gun.
Time and Space Curves
Pressure gauges fixed in the side of the chamber can measure the propellant
gas pressure during the firing sequence, whilst velocity and displacement transducers can measure the velocity and travel of the projectile along the bore. The
measurements taken can be automatically recorded against time, so the plotted
data produces graphs which are known as the pressure/time, velocity/time and
travel! time curves and are illustrated in Fig. 2.19.
The pressure and velocity data can also be plotted against the simultaneous
displacement of the projectile; these graphs are known as the pressure/space and
velocity/space curves.
FIG. 2.19 Typical pressure/time, velocity/time and travel/time curves
FIG. 2.20 Typical pressure/space and velocity/space curves
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24 Military Ballistics Internal Ballistics – Part I 25
Forces Acting on the Projectile
There are two opposing forces acting on a projectile within the gun: the
propelling force is due to the high pressure propellant gases pushing on the base
of the projectile, whilst the frictional force between the projectile and bore, which
includes the high resistance during the engraving process, opposes the motion of
the projectile. Additionally, the reaction between the projectile and the rifling of
a rifled gun translates a small part of the propelling force into a torque which
causes the projectile to rotate. So we can derive the following equations:
Propelling force  Propellant gas pressure  Area of projectile base.
Ignoring the small energy translated into the rotation of the projectile within a
rifled gun:
Total force on projectile  Propelling force  Frictional force.
If the gun is elevated, the frictional force will include a slight additional force due
to the weight of the projectile. If the total force and projectile mass are known, it
is possible to calculate the acceleration of the projectile:
The pressure and frictional force vary continuously during the firing sequence,
and so the acceleration varies also.
In practice, the projectile mass and base area will be known, the gas pressure
can be measured and recorded during the firing sequence, and the acceleration
deduced from the observed motion of the projectile. Consequently it is possible to
calculate the frictional force resisting the motion of the projectile.
Distribution of Energy
When a propellant burns it releases a large amount of energy in the form of hot
propellant gases. As the firing sequence progresses, a large proportion of this
energy is converted into other forms. The approximate distribution of energy at
the end of the firing sequence is given in Fig. 2.23.
This table reflects the basic function of the gun, which is to convert the heat of
propellant gases into useful kinetic energy of the projectile. In this case, the gun
has achieved its function with 32% efficiency on the basis of:
Muzzle energy of projectile Percentage propulsive efficiency   100 Total heat of propellant
Unless the propulsive efficiency varies from shot to shot it is of no significance
to the gun’s efficiency to deliver a projectile to a specified target.
There are some extra minor effects on the efficiency of a gun. For example, if
the gun is rifled, the projectile will exit the gun with a rotational energy which,
for a medium calibre, will amount to only 0.15% of the total energy liberated by
the propellant. Then some energy is lost to the recoil motion of the gun; this will
account for between 0.02% and 0.5% of the total available energy. In simple
ballistic studies recoil energy is negligibly small: to illustrate this, some typical
recoil energies, given as a percentage of total available energy in each case, are
shown in Fig. 2.24.
An inefficient gun/ammunition design may fail to reach all-burnt. In this case
unburnt propellant will be ejected through the muzzle: this unburnt propellant
can be a significant cause of energy loss, but should not normally happen.
Pressure of Propellant Gases
There are many complex factors contributing to the pressure at any moment
within the gun. However, as a simple guide, the propellant gas pressure is
roughly proportional to heat retained by propellant gases ÷ volume of propellant
gases. Shortly after ignition, large pressure fluctuations may occur owing to
oscillations of the gas within the chamber. Later, when the projectile is moving
rapidly, there will be a fall in pressure towards the base of the projectile as
friction between the propellant gases and the bore resists the motion of the gases
along the bore.
Peak Pressure
There are two main factors which contribute to a high peak pressure. The first
is the rapid liberation of energetic gases during the early stages of the firing
sequence and the second is the high projectile mass.
FIG. 2.21 The four factors which determine projectile acceleration
FIG. 2.22 Typical frictional force during firing
13569_int.indd 24-25 11/15/11 11:20:23 AM
26 Military Ballistics Internal Ballistics – Part I 27
A rapid liberation of energy requires either a large total surface area of the
propellant granules, a high value of force constant or a fast burning rate. The
size, shape and number of granules will determine the total surface area, whilst
the force constant and burning rate are dependent on the composition of the
propellant.
A projectile of high mass will tend to resist acceleration, so restricting the
volume into which the gases can expand and so increases the peak pressure.
Conversely, if the early rate of energy liberation or the projectile mass are
reduced, the peak pressure will be less.
Motion of projectile 32%
Motion of propellant gases 3%
Frictional losses 3%
Heat loss to gun and projectile 20%
Heat retained by propellant gases 42%
Total energy liberated by propellant 100%
FIG. 2.23 The approximate distribution of liberated energy
Weapon Relative recoil energy
5. 56 mm Rifle 0.1%
120 mm Tank gun 0.2%
. 44 Magnum pistol 0.5%
FIG. 2.24 Approximate recoil energies relative to total available energy
FIG. 2.25 Examples of peak pressure
Gun / ammunition Typical peak pressure
5. 56 x 45 mm (Rem. 223) 354 MPa
120 mm APDS 425 MPa
When progressive granules are used, the initial liberation of energy is limited,
thereby limiting the peak pressure to the safe limits of pressure and erosion that
may be tolerated by the gun.
All-Burnt
All-burnt refers to the moment at which all of the propellant has been burned;
this happens almost simultaneously for all the granules though the exact instant
for each granule will be dependent on its local conditions within the gun during
the firing sequence.
The position of the projectile at all-burnt is largely dependent on the peak
pressure and the form function of the propellant granules. A high peak pressure
implies that the pressure is relatively high throughout the firing sequence and,
as the burning rate is roughly proportional to pressure, all-burnt will be reached
very rapidly. Similarly, a low peak pressure implies a late all-burnt.
Progressive propellant charges, which are discussed more fully later in the
chapter, usually produce late all-burnt positions due to their moderated peak
pressure. Indeed, when multi-tube granules are used, all-burnt is frequently
never reached and some of the remaining slivers of propellant are ejected in an
intense muzzle flash. After peak pressure has occurred, the fall in pressure, and
consequent burning rate, are dependent on the form function. Progressive granules tend to sustain the pressure slightly so that at all-burnt the pressure is
higher than with degressive granules.
If all-burnt occurs early, when the projectile has travelled only a short distance, the results will be an increased propulsive efficiency, a reduced muzzle
blast and flash and an increased consistency of muzzle velocity from shot to shot.
After all-burnt, all the available propellant gases are able to contribute to the
propulsion of the projectile for its remaining travel along the bore. If all-burnt
occurs early, all the available gases will be able to act on the projectile over an
increased distance. The greater the distance travelled by the projectile after allburnt, the greater the expansion of the propellant gases. Thus the propulsive
efficiency is increased and, for a given projectile mass, the muzzle velocity is
increased. As expansion of the gases reduces their pressure and temperature, an
early all-burnt will also result in reduced blast and flash at the muzzle. The use of
a longer barrel will have a similar effect to an early all-burnt position unless the
barrel is excessively long, in which case the sustained projectile/bore friction will
exceed the propulsive force and reduce the muzzle velocity.
In practice the greatest variation of projectile velocities observed from shot to
shot occurs at the position of all-burnt; but the variation of velocities lessens
towards the muzzle, so the longer that projectiles travel after all-burnt the less
the variation of muzzle velocities, which again is an argument for having an
early all-burnt position.
Progressive Propellant Granules
Gun systems usually use the largest propellant charge that may be inserted
into the chamber to achieve the highest practical peak pressure that the gun can
withstand. The question then arises, how is it possible to increase the muzzle
velocity if the projectile mass remains the same?
The burning rate, form function, ballistic size and number of granules can be
adjusted to give an earlier all-burnt; the resultant increase in propulsive
efficiency will probably be small, and the peak pressure may exceed the limitations of the gun.
A larger increase in the muzzle velocity implies that the amount of energy
supplied by the propellant must be increased, which can only be achieved by
13569_int.indd 26-27 11/15/11 11:20:23 AM
28 Military Ballistics Internal Ballistics – Part I 29
using a propellant with a higher force constant. Providing the gun can withstand
the very high pressures developed, the muzzle velocity will be greatly improved.
However, the peak pressure can be moderated by forming the propellant into
neutral or progressive granules rather than the more usual degressive granules.
The majority of the energy will then be liberated after the peak pressure has
occurred, so sustaining the pressure at high levels until all-burnt is reached. This
delayed liberation of energy will result in a reduced propulsive efficiency; nonetheless, the losses due to inefficiency will be small compared to the large increase
in available energy. The low efficiency will be reflected by an increase in muzzle
blast and flash. The delayed liberation of energy will also tend to give inconsistent muzzle velocities: in consequence progressive propellant granules are usually
limited to guns where high muzzle velocity is more important than accuracy or
low blast and flash.
Barrel Life
The life of an in-service gun, usually stated in terms of the number of times it
can be fired accurately, is limited by the three prime causes of internal barrel
wear, which are corrosion, abrasion and erosion.
Erosion is the most significant as is explained later.
Corrosion
Corrosion is normally the most insignificant limit to barrel life as modern
propellant combustion products are chemically non-corrosive. However, gunpowder and other primer compounds may cause corrosion; and as most barrels
are made of steel, they may rust in damp climates. Such causes of corrosion can
easily be avoided by suitable treatment of the bore after use, and by using selfcleaning and lubricating projectiles.
Abrasion
The friction of the projectile against the bore will cause some wear of the bore.
The extent of this abrasion varies with the type of projectile used, though in most
cases the amount of purely abrasive wear is relatively slight. In large calibre
guns the use of plastic driving bands reduces abrasion, but small calibre guns, in
which separate driving bands are not used, rely on the choice of projectile surface
material and projectile lubrication to control abrasion. The choice of projectile
surface material is not simple; indeed, hard armour-piercing bullets have on
occasion produced less wear than did standard bullets.
In some circumstances, foreign material such as sand may repeatedly enter the
bore between shots; the result would be a serious shortening of the gun’s life.
Large quantities of material obstructing the bore would cause local expansion of
the barrel as the projectile passes, possibly resulting in the muzzle end of the gun
being blown off.
Conversely, the abrasion may wear the projectile rather than the bore: the
accumulation of material lost from a number of projectiles causes so-called
leading of the bore, and the gun will consequently lose accuracy. Leading can be
removed by scrubbing of the bore, so it has no effect on the life of the gun. Also,
lubrication of the projectile can often be used to suppress both leading and
abrasion of the bore.
Erosion
Erosion, the major cause of barrel wear, is caused by the transfer of heat from
the energetic propellant gases to the bore. The region of greatest erosion is near
the start of the rifling, or at the same relative position in a smooth-bore gun. The
collision of energetic gases against the bore surface produces rapid local heating.
The hot steel reacts with the propellant gases to produce a weak layer of brittle
compounds that is removed during subsequent firings by the abrasive action of
the propellant gases, propellant granules, and projectiles. The removed material
is carried out with the propellant gases through the muzzle.
FIG. 2.26 Pressure/space curves for three full charges of propellant
FIG. 2.27 Erosion of the gun
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30 Military Ballistics Internal Ballistics – Part I 31
Although the temperature of the entire barrel rises slightly as a result of
firing, it is the high temperature of the bore surface during the firing sequence
that determines the rate of wear. The higher the bore surface temperature during
firing, the greater the rate of wear.
The rate of erosion is dependent on seven factors:
1. propellant
2. wear additives
3. rate of fire
4. barrel cooling
5. calibre
6. barrel material
7. gun design.
The primary features of a propellant charge which control the rate of erosion are
the force constant and form function. A high force constant implies high temperatures of the propellant gas and a consequent high rate of erosion. The erosion rate
is very sensitive to even small changes in gas temperature, so it is important to
avoid excessively high force constants. However, relatively high force constants
may be used if the peak pressures are moderated by the use of progressive
granules, because the very high rates of heat transfer that accompany very high
pressures are avoided. Low burning rates and pressure indices also reduce
erosion, but at the cost of reduced muzzle velocity.
The rate of erosion can be reduced by the addition of small quantities of
chemically inert wear additives to the propellant. On firing, the wear additives
will tend to settle on the bore surface; the insulative layer formed will resist heat
transfer from the propellant gas to the barrel, thus reducing the rate of erosion.
Successful additives include titanium dioxide/wax and talc/wax mixtures.
During rapid fire, when there is little chance for cooling between shots, the
overall temperature of the barrel will rise, and so the bore surface temperature
during firing will also rise. The consequence is a shortened barrel life unless a
cooling system is used. Coolant can be pumped through tubes in the barrel and
then passed through ventilated radiators. Aircraft guns simply use forced airflow
during flight to maintain cooling.
Erosion is most severe in guns in which the circumference of the bore is large
compared with the cross-sectional area of the bore (see Fig. 2.29). This describes
small arms and the relatively small large calibre guns. This problem is conveniently overcome when discarding sabot projectiles are used.
Barrel steels containing alloys, with such materials as tungsten and chromium, do much to increase the erosion resistance, and chromium electroplating
of the bore can almost completely eliminate wear by erosion. However, they are
expensive and difficult to produce.
Erosion can, however, be further minimised by careful design of the gun.
Probertised rifling is an excellent example of this. As erosion is accentuated by
rifling, the gun is made smooth-bore at the shot-start position where erosion is
greatest. The depth of rifling increases towards the muzzle, cutting into the
driving bands during its motion along the barrel. Then the bore narrows and the
rifling decreases again near the end of the bore, squashing the driving bands flat
to improve the projectile’s flight characteristics.
Probertised rifling is in fact smooth-bored at the muzzle. The result is that the
life of a Probertised gun is up to 6 times longer than that of a similar conventional gun, though both the gun and its ammunition are expensive to produce.
Practical Barrel Life
A barrel can be considered worn out when the gun cannot achieve its required
accuracy in normal use. The point at which this occurs varies, depending on the
type of gun, ammunition used, and minimum acceptable accuracy. The amount of
wear is measured at the commencement of rifling, and is the percentage increase
in diameter of the bore. Whilst barrel life is defined as the number of shots
FIG. 2.28 An eroded gun sectioned to show enlargement of the bore and worn rifling
FIG. 2.29 Cross-section of the bore and relative rates of erosion
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32 Military Ballistics Internal Ballistics – Part I 33
required to increase the bore diameter by 5%, the practical increase in diameter
that can be tolerated is usually between 1% and 5%.
The muzzle velocity of a gun is determined by factors which also affect its rate
of wear: so it is possible to relate barrel life to muzzle velocity, though the
relationship is not simple. Generally, a small increase in muzzle velocity will
considerably shorten the life of a gun.
Most guns that suffer short barrel life do so as a result of high rates of erosion
caused by high muzzle velocities or rapid rates offire. Such guns include antitank
guns, anti-aircraft guns, high velocity rifles and machine guns.
Gun / ammunition Approx. muzzle Typical practical
velocity barrel life
.22 (5.5 mm) LR 320 m/s 1,000,000 rounds
5.56 x 45 mm (Rem. 223) 990 m/s 10,000 rounds
88 mm anti-aircraft 1000 m/s 100 rounds
120 mm anti-tank (sabotted) 1370 m/s 100 rounds
.50 inch Browning, Rapid Fire 890 m/s 500 rounds
FIG. 2.32 Chieftain tank fitted with 120 mm lagged gun
FIG. 2.33 The principle of the recoilless jet reaction gun
FIG.2.31 Examples of practical barrel life
Note that the .50 inch Browning machine gun firing at 1000 rounds per minute
has a barrel life of just 30 seconds.
Barrel Distortion
A secondary effect, due to the transfer of heat to and from the barrel, is barrel
distortion. Heating by the propellant gas, and uneven cooling by, say, wind will
cause the barrel to bend slightly, so affecting the accuracy. This can be a serious
problem for some modern thin-barrelled guns. Fortunately, it can be controlled
by the choice of propellant and wear additives, insulation of the barrel by lagging
on the external surfaces and, where necessary, an adequate cooling system.
Recoilless Guns
A recoilless gun consists of a tube open at both ends. The projectile is propelled
through a conventional muzzle, and the propellant gases are ejected through the
open breech of the gun. The forward momentum imparted to the projectile equals
the backward momentum imparted to the propellant gases; consequently the gun
does not move, and it is therefore recoilless. In some alternative recoilless
systems, the backward momentum is provided by the ejection of a counter-mass
from the rear of the tube. In its simplest form, the recoilless gun is no more than a
rocket launcher tube, but the term is usually applied to the more efficient jet
reaction gun. (See Fig. 2.33).
FIG. 2.30 Rifling depth in a Probertised barrel
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34 Military Ballistics Internal Ballistics – Part I 35
During the firing sequence, the pressure within the recoilless gun increases,
thus efficiently driving the projectile forward. Simultaneously, the pressure acts
on the breech, tending to drive the gun backwards. This force is balanced by a
high velocity jet of propellant gases ejected through a nozzle, or nozzles, which
generates an equal forward reaction force.
The advantages of the recoilless gun are that it is lightweight, of simple
construction and has a long barrel life. Its disadvantages are that it requires a
charge some 3 to 4 times greater than that of a conventional gun, it has a low
maximum achievable muzzle velocity and its accuracy is poor.
So far in this first section on internal ballistics, we have looked, in a descriptive
way, at the various factors which affect internal ballistics. Some readers, who do
not have a mathematical background, will find it sufficient. Others, with a
mathematical turn of mind, will be interested in the development of the theme in
a more theoretical way in Section II.
Self Test Questions
QUESTION 1 How does the combustion of propellants differ from the combustion of conventional fuels?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
QUESTION 2 What are the purposes of driving bands?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
QUESTION 3 Which burning characteristics are dependent on the compo-sition of a propellant, and which are dependent on the shape of
the granules?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 4 What is the order of events of a typical firing sequence?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
QUESTION 5 What are the characteristics upon firing of guns using multitube propellant granules?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 6 Which configuration of chamber size and propellant burning
rate promotes pressure waves in the chamber?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 7 In what fashion does the surface area of a progressive granule of
propellant change during burning, providing that it is not treated with burn suppressants or inhibitors?
Answer ……………………………………………………………………………….
QUESTION 8 What are the arguments for and against barrel lagging?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
FIG. 2.34 LAW 80 (early model) anti-tank recoilless rocket launcher
FIG. 2.35 120 mm Conbat recoilless jet reaction gun
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36 Military Ballistics
37
Internal Ballistics – Part II
QUESTION 9 In the case of a typical spin-stabilised projectile, what proportion of its kinetic energy at muzzle exit is due to rotation?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 10 Why is the temperature of the propellant gases within a gun
lower than the flame temperature?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
………………………………………………………………………………………….. Scope
To produce a mathematical model of the internal ballistics of the gun as a
whole, it is necessary to model the dynamics of each of the components. The
treatment given here is a simplification of both the real gun and typical models of
the gun as only the main internal ballistic components are considered. These
components are the propellant, the propellant gases and the projectile. In the
following pages five equations will be derived that model the burning of the
propellant, the behaviour of the gases and the resultant motion of the projectile.
In deriving these equations, well known principles of physics will be applied,
together with explanation of many of the processes particular to the gun.
Although referred to, the equations governing some of the more complex processes are not given; these can be found in various reference books which deal
with internal ballistics more fully. Both classical and contemporary applications
of the five equations are discussed, and an outline is given of the two main types
of computer gun simulation.
Burning Rate Law
Looking first at an individual granule or propellant, we need some idea of the
way in which it burns. In obeyance with Piobert’s law of burning the entire
surface of a burning propellant granule regresses at the same rate, and so recedes
in parallel layers. As a granule burns from all sides, the rate at which it reduces
in size is twice the rate of regression of just one side. This rate of reduction in size,
known as the burning rate*, varies with the pressure of the surrounding gases in
accordance with the following equation:
in which s is the reduction in size of the granule;
t, time in seconds;
P, gas pressure in the region of the granule:
a, the pressure index of the propellant composition
and B is the burning rate constant of the propellant composition
* In some references, notably American, burning rate is taken to be equal to the rate of regression,
and so the burning rate constant appears to be half the expected value.
ds
burning rate, —  BPa
dt
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38 Military Ballistics Internal Ballistics – Part II 39
The ballistic size, the initial size, of the granule is represented by D. Now, if at
some instant after ignition when the size of the granule has reduced by a distance
s, and the remaining size of the granule is a fraction f of the original size D, then:
At ignition f equals one, decreases in value as the burning proceeds and equals
zero when all burnt is reached. This relationship between sand f, differentiated
with respect to time gives
and by substitution for –– in the burning rate equation leads to:
This function can now be integrated with respect to time, so the fraction f of the
ballistic size burnt at time t can be equated with the variations of pressure during
firing:
The value of –– is known as the propellant vivacity. For a propellant of given
pressure index and granule shape, it is possible to vary the values of Band D
without affecting the internal ballistics of the gun in which it is used, providing
that the vivacity remains the same. Of course, the extremes of ballistic size would
cause problems in practice.
Form Function
We next need to relate the proportion of propellant burnt with the remaining
fraction f of the ballistic size. The relationship is purely geometrical, and depends
only on the shape of the granules used. Writing z as the fraction of propellant
volume burnt at some instant t, then
Now, if the granules have a cross-sectional area Ag and a length L then
As the length L of the granules is generally much greater than the ballistic size
D, the relative change in granule length during burning is negligibly small. It is
therefore possible to make the approximation:
Taking a long cylindrical granule as an example,
and
Putting these into the equation for z gives
z  1  f
2
,
and this can be expanded to give:
z  (1  f) (1  kf). (Equation 2)
This equation relating z and f is known as the form function and here, in the case
of the long cylindrical granule, the form function coefficient k equals one.
Different values of k allow the form function to be used for a variety of propellant
shapes. For example, for long tube or slotted tube, k  0; and for long ribbon of
thickness D and width wD, k  . In the case of multi tube granules, such as the
7 hole cylinder, an empirical form function is used: the ballistic size is taken to be
1.28 times the web size, and k  0.12.
The Closed-Vessel
As was seen in Part 1, the closed-vessel is the primary tool in the calibration of
propellants. By extending the principles governing the behaviour of propellant
gases in the restricted conditions of a closed-vessel, it is possible to grasp an
understanding of the complex gas behaviour during the firing sequence of a gun.
Equation of State
The starting point for the underlying theory is Van der Waals’ equation of state
for an ideal gas:
PV  nRuT
where P is gas pressure,
V is gas volume,
n is the number of moles* of gas,
* A mole is the value of the molecular weight of a given substance measured in grams. Example:
carbon monoxide, formula CO; molecular weight  12  16  30, therefore one mole of carbon
monoxide weighs 30 grams.
D  s  fD.
ds df
– —  D — ,
dt dt
df B
—   —Pa
. dt D
B t
f   — Pa
dt. (Equation 1)
D O
ds
dt
B
D
original volume  volume at time t
z  original volume
volume at time t
z  1 –
original volume
AgL (at time t)
z  1 –
AgL (original)
AgL (at time t)
z  1 –
AgL (original)
D2
Ag (original)  4
 (fD)2
Ag (at time t)  4
L (at time t)  L (original), and so
1
w
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40 Military Ballistics Internal Ballistics – Part II 41
Ru, the universal gas constant
and T is the absolute gas temperature.
This equation describes the relationship between P, V, n and T for a gas
consisting of very simple molecules. However the real gases evolved by a burning
propellant are relatively complicated and so do not behave exactly as Van der
Waal’s equation predicts. The most significant cause of discrepancy between
theory and practice is the coalescence of gas molecules as they are forced together
at the high pressures achieved. This effect is known as condensation as it is, in
effect, similar to the condensation of a mist of water droplets from humid air. As a
result, the volume of the gas is reduced by the condensation of apparently nongaseous particles from the high pressure gases. To compensate for the reduction
in volume, the equation of state is assumed to be of the form
where m is the mass of propellant burnt
and c is the co-volume of the propellant.
This, the Noble-Abel equation, is used as the equation of state for the gases of
both the closed-vessel and the internal ballistics of the gun. Though the covolume is an assumed characteristic of propellant gases, it can be evaluated for
each propellant composition by closed-vessel testing, as will be shown later. The
Noble-Abel equation is a simplified treatment of the complex interaction of gases
at high pressure; nevertheless, it is respected for its close agreement with
experimental results.
Force Constant
When propellant is burned, it evolves gases at its characteristic flame temperature To. If these gases are restrained within the solid walls of a closed-vessel, they
cannot cool by expansion. Also, as the burning occurs over just a short period of
time, the initial heat loss to the vessel walls will be negligible. Thus, at the
instant the burning has completed, the gas temperature is still To, and the
pressure simultaneously reaches a maximum value Pmax. The equation of state is
then
where Vcv is the volume of the closed-vessel
and m, the mass of propellant.
The value of is a characteristic of each propellant, so the equation can be
written
where F is the force constant of the propellant composition.
Measurement of Co-volume and Force Constant
Now, if the closed-vessel is fired with a mass m1 of a propellant, it will produce
a peak pressure Pmax1
, so
A second firing with a different mass, m2 gives
The co-volume and force constant for the propellant can then be found by solving
the two equations above.
Energy Liberated
The gases evolved by a burning smokeless propellant carry with them the
energy liberated by the burning process. The amount of energy liberated is given
by
where m is the mass of propellant burnt,
Cv is the specific heat capacity per unit mass of gas at constant volume
and To is the absolute flame temperature.
Using the two fundamental gas equations which relate Cv above to Cp, the
specific heat capacity per unit mass of gas at constant pressure:
and
we get
where G is the ratio of specific heat capacities. Putting this into the energy
equation gives:
and as nRuT0
equals mF,
P(V  mc)  nRuT
Pmax (VCV  mc)  nRuTo
Pmax1
(VCV  m1
c)  muF
energy, E  m Cv To
Pmax2
(VCV  m2
c)  m2F
Pmax (VCV  mc)  mF
nRuT0
m
Vcv Pmax2 Pmax1
Co-volume, c   –  Pmax2
 Pmax1 m2 m1
1
1 1 1
Force constant, F  Vcv      –  m1 m2 Pmax1
Pmax2
nRu
Cp  Cv  m
Cp
 G , Cv
nRu
Cv  m(G  1)
nRuT0
E  G  1
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42 Military Ballistics Internal Ballistics – Part II 43
If, at some instant t during the firing sequence of a gun, a fraction z of an initial
propellant mass C has burnt, the energy liberated will be
Energy Balance
As with all physical systems, the gun obeys the principle of conservation of
energy, so although the energy liberated in the form of propellant gases at the
flame temperature is converted into other forms, the total quantity of energy
remains the same. Consequently the sum of the converted forms of energy equals
the liberated energy throughout the internal ballistic sequence.
Et  Ep  Eg  Eu  Er  Eb  Eh  Es  Ef
where Ep is the kinetic energy due to the projectile motion,
Eg is the kinetic energy of the propellant gas motion,
Eu is the kinetic energy of the unburnt propellant motion,
Er is the kinetic energy due to recoil of the gun,
Eb is the heat energy lost to the barrel,
Eh is the residual heat energy in the propellant gases,
Es is the strain energy used in expanding the barrel,
and Ef
is the energy lost in engraving the driving band and overcoming
friction in the bore.
Although mathematical models do exist for the energy losses Eg, Eu, Er Eb, Es,
and Ef
, they will for simplicity be summed together here as E1, so the energy
balance equation appears:
Et  Ep  Eh  E1
Now, ignoring the slight rotational energy of the projectile, Ep is simply given by
the basic equation for the kinetic energy of a moving mass:
Ep  Mv2
where M is the mass of the projectile
and v is the velocity of the projectile.
However, evaluating the residual heat energy Eh of the propellant gases is more
difficult. The propellant gases evolved at the flame temperature cool by expansion as they do work in driving the projectile along the barrel, and are further
cooled by contact with the barrel. The average temperature of the gases is some
value T, so
Eh  Cz CvT .
As before
so
Substituting the Noble-Abel equation
gives
where P is the average propellant gas pressure and V, the volume occupied by the
propellant gases is given by:
in which Va is the initial air space in the chamber,
A is the cross-sectional area of the bore,
x is the distance moved by the projectile,
and Vi is the increase in volume due to the diminishing size of the
propellant granules as they burn.
As Vi  , where d is the density of the propellant, the residual heat of the gases
can now be written
By substituting the equations for Et, Ep and Eh, the energy balance equation can
now he written
For convenience, this is now rearranged to the final form:
(Equation 3)
Assuming that E1 has been substituted by suitable equations, all the parameters
in this rearranged energy balance equation are known except for P, z, v and x.
Equations of Motion
Moving on now, we require equations which govern the motion of the projectile.
One of Newton’s equations of motion states
Ma F0
mF
E  G  1
nRu
Cv  , Cz(G  1)
P
Eh  (V  Czc) , G  1
nRuT
Eh  . G  1
CzF  (G  1) (½Mv2
 E1) P  Va  Ax  Cz ((1/d)  c)
CzF
Et  G  1
P(V  Cz)  nRuT
V  Va  Ax  Vi
Cz
d
P 1
Eh   Va  Ax  Cz  – c   G  1 d
CzF 1 P 1  Mv2
  Va  Ax  Cz   c  
 E1 . G  1 2 G  1 d
1–
2
13569_int.indd 42-43 11/15/11 11:20:26 AM
44 Military Ballistics Internal Ballistics – Part II 45
That is to say, mass multiplied by acceleration equals force. Within the gun, the
force acting on the projectile,
FO  A(Pp  Pa)  Fr
where A is the cross-sectional area of the bore,
Pp is the pressure acting on the base of the projectile,
Pa is the atmospheric pressure,
and Fr is the frictional force between the projectile and bore.
Owing to both the inertia of the gases and unburnt propellant, and their friction
with the bore surface, the projectile base pressure Pp is somewhat less than the
average pressure P. An expression exists which takes this into account and so
allows Pp to be calculated once P is known.*
The frictional force Fr is owed to the unrelieved stresses acquired as the driving
band engraves into the rifling. Simple mathematical models of the frictional force
impose a shot-start pressure which has to he exceeded before the projectile is
allowed to move, and apply a constant resistance force for the remainder of the
firing sequence. Sophisticated models calculate the energies required to stress
and yield the driving band during the engraving process, and use the residual
stress once the engraving has been completed to find the sliding frictional force.
Writing acceleration as the change of velocity with time t,
then
where M is the mass of the projectile.
Integrating this equation with respect to time gives
(Equation 4)
This equation will be used later in finding the velocity of the projectile throughout the firing sequence. Similarly, the equation
which defines velocity as the change of distance with time, can be integrated to
give
(Equation 5)
and this will be used to find the position of the projectile.
Summary of the Internal Ballistics Equations
Classical Solutions
Until the advent of computers, these equations were solved by mathematical
analysis. However, the continuous natures of mathematical functions are poorly
able to model the discontinuous processes of internal ballistics, so the five
equations were simplified to suit the mathematics. Typical simplifications included taking a  1, imposing a shot-start pressure instead of modelling the
engraving process, and taking a constant resistance force to mimic the real
frictional resistance force. Further simplifications were made to the equations
that constitute E1, though this was often done because the underlying physical
processes were not fully understood. Analytical solutions, such as ‘A System of
Internal Ballistics 1945’ by Hunt, Hinds et al. of RMCS, were capable of predicting the performance of the contemporary guns with considerable accuracy.
However, as the performance of modern guns has increased, the effect of variations in a, Fr and E1 has become highly significant. It is here that the computer,
with its ability to model complex processes, has in recent times ousted the
classical analytical models.
Computer Modelling
A working computer model of the gun has three advantages over the analytical
models. It allows realistic data and functions to be included that model discontinuous processes such as ignition and engraving. Its flexibility allows the design
of the modelled gun system to be changed at will. And its fast computing speed
enables rapid predictions of gun performance. Clearly, the ability to experiment
with gun system designs without having to fire a shot has made computer
simulation one of the gun designer’s most important tools.
The Lumped Parameter Model
The two main types of computer methods used for internal ballistic modelling
* Internal Ballistics, HMSO 1951. are the lumped parameter and the finite difference methods.
dv
a  dt ’
B
1. f  – 0
t
Pa
dt D
2. z  (1  f) (1  kf)
CzF  (G  1) (½Mv2
 E1)
3. P  Va  Ax  Cz((1/d)  c)
1
4. v  v  0
t

A(Pp  Pa)  Fr 
dt
M
5. x  0
t
v dt
dx  v ,
dt
dv
M  A (Pp  Pa)  Fr
dt
1
v  0
t

A(Pp  Pa)  Fr  dt
M
x  0
t
v dt.
13569_int.indd 44-45 11/15/11 11:20:27 AM
46 Military Ballistics Internal Ballistics – Part II 47
Lumped parameter models are in effect numerical solutions to the equations
that used to be solved by analytical methods. They are so named because the
equations that describe the behaviour of the propellant gases treat the gases as a
homogeneous fluid of arbitrary shape. The only term that accounts for any
variation in state within the gases at any given moment is the equation for Pp,
which estimates the fall in pressure from the gases as a whole to the projectile
base. A more accurate model would determine the state of the gases throughout
the gun, which in fact is what finite difference models do, though these will be
described later.
An essential requirement when modelling such systems as the gun by use of
digital computer is that the action described is resolved into discrete steps. The
fundamental step that is usually used is a small increment in time, t. This in
turn requires that integral relationships such as equation 1,
have to be written as summations, so
Similarly, equations 4 and 5 are written
and
To ensure that the summations closely approximate the theoretical equations the
size of t must be kept small. Typically, t is taken to be one-twentieth of a
millisecond.
The flow diagram of a computer program into which the five equations fit is
shown in Fig. 2.36. The computations iterate about the program, incrementing by
t each time, until the projectile either exits from the barrel or stops owing to
insufficient propellant.
For accurate modelling it is necessary to make the discrepancy between the
summations and theoretical equations negligibly small. One method of achieving
this would be to make t extremely small, however the program would then take
an unacceptably long time to run. The practical alternative is to apply a
predictor-corrector technique,* such as the Runge-Kutta procedure, to estimate
the discrepancy and so correct each computation.
The Finite Difference Model
The aim of finite difference models is to compute the pressure and flow of the
propellant gases throughout the gun. In practice, large pressure differentials are
FIG.2.36 Flow diagram for a lumped parameter computer model of internal ballistics,
* Numerical Methods for Scientists and Engineers, McGraw-Hill, 1962. with description given on left hand side
B
f  – 0
t
Pa
dt D
B
f  –
0
t
Pa t D ∑
1
v  –
0
t

A (Pp  Pa)  Fr  t M ∑
x 
0
t
v t ∑
13569_int.indd 46-47 11/15/11 11:20:28 AM
48 Military Ballistics Internal Ballistics – Part II 49
observed along the length of chambers and bores, whereas the pressure differentials across their diameters tend to be slight. This is owed to the geometry of
chambers: the length is generally greater than twice the diameter; and the
longitudinal asymmetry, with the projectile at one end and the primer often at
the other, promotes asymmetry of pressure.
Finite difference models employ a second fundamental step, this being a small
distance Ax separating points along the axis of the gun. The gas in the region of
each point is individually modelled by a lumped parameter type model which
differs from the previous model in that the influence of surrounding regions of gas
have to be taken into account. If required, the non-axial regions can be modelled
by a two-dimensional array of points lying on a plane that includes the axis. The
slight gain in realism afforded by two-dimensional modelling is often outweighed
by a large increase in both model complexity and computing requirements.
The finite difference method is able to model the pressure oscillations that
occur between the breech and projectile base, and can be adapted to model the
process of flame spread during ignition. Both pressure fluctuation and inconsistent ignition are serious problems in modern high performance guns, so this
method has considerable advantages over lumped parameter models.
Conclusion
This introduction to theoretical internal ballistics has concentrated on the
underlying principles of the basic gun and its modelling by computer. Although
the vast spectrum of research and modelling is not covered here, there are
numerous reference books and papers to which the reader can turn for further
study.
There is now a considerable research effort in internal ballistics; it is one of the
most significant areas of advancement in ballistics, and its importance is growing
with the search for ever increasing gun performance.
Self Test Questions
QUESTION 1 What is the relationship between burning rate and rate of
regression?
Answer ……………………………………………………………………………….
………………………………………………………………………………………….
QUESTION 2 How can the internal ballistic performance of a gun be maintained if the propellant burning rate constant is halved?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 3 If C is the total mass of propellant and z is the fraction of
propellant volume burnt, what is the mass of gas liberated?
Answer ……………………………………………………………………………….
QUESTION 4 When the frictional force acting on the projectile exceeds the
propelling force, what does the projectile do if it is a) stationary,
b) moving?
Answer a)…………………………………………………………………………..
b)…………………………………………………………………………..
QUESTION 5 What are the advantages of finite difference methods in computer modelling of internal ballistics?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
QUESTION 6 The Noble-Abel equation alone cannot completely describe the
state of the gas within a gun; why is this so?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
13569_int.indd 48-49 11/15/11 11:20:28 AM
51
3.
Intermediate Ballistics
Definition of Intermediate Ballistics
Intermediate ballistics is defined as the study of the transition from internal to
external ballistics, which occurs in the vicinity of the gun muzzle. In the case of
recoilless guns, the study is extended to the region of the recoil jet nozzles.
The distribution of energy at muzzle exit can be simplified down to:
Motion of projectile 30
% 75
% Released from gun Energy of propellant gases 45%
Heat retained by gun 25
%
Roughly three-quarters of the available energy passes through the muzzle; the
majority of it is carried by the propellant gases in the form of heat, pressure and
motion. After muzzle exit, the behaviour of these gases has considerable influ

ence on the projectile and gun motions; they also give rise to the effects known as
blast and flash. At this stage the gas flow takes a distinctive form: it is worth
describing the features of it before considering the additional effects that a
projectile has as it exits from the muzzle and passes through the gas flow field.
The Gas Flow Field Near a Muzzle
The release of high pressure gas from a muzzle causes turbulence as it mixes
with the ambient air. The resultant pressure waves radiate at the speed of sound
as noise. The speed of sound through the muzzle gas flow and surrounding air
varies according to the types and state of the gases present. It is roughly given by
the equation:
where
G  Ratio of specific heats of the gas mixture R  The gas constant, 287 J/kg K T  Absolute temperature of the gas.
For air under normal conditions
G  1. 404
and
T  288K (15
°C),
so the speed of sound in air is 340 m/s.
Speed of sound in gas, a
g   (GRT)

13569_int.indd 50-51 11/15/11 11:20:28 AM
52 Military Ballistics Intermediate Ballistics 53
By substituting the gas equation RT  P/d into the speed of sound equation gives
where P  gas pressure
and d  gas density
In the vicinity of a gun muzzle during firing, the gas temperature, pressure and
density vary considerably, and so the speed of sound also varies. In addition the
mixture of gases produced by the burning propellant is very much different from
that of air: the resultant differing value of G produces further variations in the
speed of sound. Although a shock wave is simply an intense sound wave, the
momentary rise of the gas temperature as a shock wave passes induces an
increase in the speed of sound; consequently shock waves travel faster than
sounds of lower intensity. Conversely, a shock wave can be defined as a sound
wave of sufficient intensity to self-induce a velocity significantly greater than
that predicted by the speed of sound equation.
Noise generated by the turbulent mixing of gases near the muzzle travels both
away from the muzzle and towards it. If high pressure gas is suddenly released
from the muzzle, which occurs when a projectile exits from a gun, the outgoing
noise primarily takes the form of an abrupt increase in pressure known as a blast
shock wave. This wave travels away from the gun at speeds slightly greater than
the speed of sound and is heard as a sonic bang. The ingoing noise forms a shock
wave which travels towards the muzzle against the flow of the gas. Near the
muzzle, the speed of the shock wave may equal the speed of the gas flow; when
this happens the ingoing shock wave makes no headway and so forms a quasistatic shock wave. This shock wave is bottle shaped, and is sometimes referred to
as a ‘bottle shock’. The curved sides of the bottle shock that extend from the
muzzle are called the barrel shock, and the almost flat base to the bottle shock is
called a Mach disc. The size of the bottle shock increases as the outflowing gas
velocity increases. As the gas velocity falls the bottle shock shrinks and eventually disappears into the muzzle.
Muzzle Gas Flow Field During Firing
The muzzle gas flow field during firing consists of two phases: the precursor
blast field that precedes the projectile exit from the muzzle, and the main blast
field that follows as the high pressure propellant gases are ejected into the air. In
recoilless guns, the blast fields are dominated by the rearward jet of propellant
gases from the breech; the flash and blast produced by the jet is similar to that
produced by the main blast field of a conventional large calibre gun.
Before Projectile Exit
As the projectile accelerates along the bore, it pushes ahead of it a column of air
augmented by any leakage of propellant gases past the projectile. A shock wave
forms just ahead of the projectile, travels along the bore, and is released as a
near-spherical precursor blast shock at the muzzle. Once the outflowing air
velocity is sufficient, a small bottle shock will form about the muzzle, growing in
size as the flow velocity increases.
FIG. 3.1 Shock waves formed by the release of high pressure gas from a muzzle
FIG. 3.2 Shock wave formation before projectile exit
GP
Speed of sound in gas, ag    –—  d
13569_int.indd 52-53 11/15/11 11:20:29 AM
54 Military Ballistics Intermediate Ballistics 55
After Projectile Exit
The projectile will then emerge, and once the projectile gas seal has passed the
muzzle, the high pressure propellant gases will be released into the atmosphere,
so generating a powerful blast shock. Initially the blast shock is highly nonspherical as it is distorted by the presence of the projectile and the high velocity
flow of the propellant gases. The propellant gases rapidly expand, accelerating to
velocities much greater than that of the projectile, so that shock waves form
around the base of the projectile, rather as though the projectile is moving
backwards. This apparent reverse gas flow provides slight additional acceleration
of the projectile for several calibres distance beyond the muzzle. Also, the muzzle
gas flow can have an adverse effect on the accuracy of the gun by causing
abnormal yawing of the projectile.
A new large barrel shock and Mach disc then forms around the muzzle. As the
velocity of the outflowing gases slows, the barrel shock and Mach disc shrink in
size; then the remaining Mach disc enters the muzzle and becomes a rarefaction
wave that travels back along the bore. Providing the projectile is fully supersonic
it will in the meantime pass through the blast shock. Owing to its high intensity,
the blast shock travels faster than the speed of sound and so tends to catch up
with the less intense precursor blast shock.
The figures below show three phases in the development of the blast field after
projectile exit.
Flash
Flash is the light emitted in the vicinity of the muzzle by the hot propellant
gases and the chemical reactions that follow as the propellant gases mix with the
surrounding air.
Before projectile exit, a slight preflash may occur owing to hot gases and
particles that have leaked past the projectile. Following muzzle exit, the temFIG. 3.3 The precursor blast field of a 5. 56 mm calibre rifle
FIG. 3.4 The initial formation of shock waves shortly after projectile exit
FIG. 3.5 Expansion of the blast field
13569_int.indd 54-55 11/15/11 11:20:30 AM
56 Military Ballistics Intermediate Ballistics 57
perature of the propellant gases is generally sufficiently high to emit visible
radiation known as primary flash. As the gases rapidly expand and cool they
continue to emit a relatively faint muzzle glow; they are then recompressed as
they pass through the Mach disc, and the consequent high temperature produces
intermediate flash. Ignition of the hot combustible gases, mainly hydrogen and
carbon monoxide, may then follow as they mix with oxygen in the surrounding
air; secondary flash, the brightest form of flash, is produced by the ensuing large
flame. In small calibre weapons, both the temperature and density of combustible
gases in the region of the intermediate flash is usually insufficient to allow
ignition, so secondary flash does not occur. Finally, hot particles and remnants of
burning propellant may appear as a long streak of light in the wake of the
projectile.
For military purposes flash, and especially secondary flash, is undesirable at
night as it will indicate the positions of guns to the enemy and will also
temporarily blind the gun crews. The three methods of flash suppression are
muzzle devices, choice of propellant and propellant additives.
Flash suppression devices are usually designed to suppress the intermediate
flash. The reduction of intermediate flash can also suppress the ignition of the
secondary flash that occurs in the case of large calibre guns.
The simplest form of suppressor is the flash hider: this is a device which
surrounds the primary flash and hides it from all directions except the line of fire.
However, as primary flash tends to be insignificant, the original intended purpose of flash hiders is usually ineffective. Modern flash suppressor devices are
often referred to as flash hiders, but their real purpose is to disperse or break up
FIG. 3.6 Final phase of the blast field before contraction of the bottle shock and Mach disc
FIG. 3.7 Preflash
FIG. 3.8 Flash
13569_int.indd 56-57 11/15/11 11:20:31 AM
58 Military Ballistics Intermediate Ballistics 59
FIG. 3.11 The primary and intermediate flash of a 7.62 mm calibre rifle showing
the total absence of secondary flash
the barrel shock and Mach disc so that the intermediate flash is reduced. The
three types of flash suppressor devices commonly used are the conical tube,
slotted tube, and bar type. The slotted tube and bar type employ a number of slots
or prongs protruding forward around the muzzle: an odd number is usually
chosen to avoid oscillations of the muzzle gas flow that can interfere with the
motion of the projectile. (See Fig. 3.12).
In large calibre guns, the most effective method of secondary flash reduction is
to use a propellant which evolves a large proportion of inert nitrogen gas at
FIG. 3.9 The intense secondary flash of a 120 mm calibre Chieftain tank gun
FIG. 3.10 The intense secondary flash from the breech nozzle of a 120 mm calibre
Wombat recoilless gun
FIG. 3.12 Conical, slotted tube and bar type flash suppressors on 5.56 mm calibre rifles
13569_int.indd 58-59 11/15/11 11:20:32 AM
60 Military Ballistics Intermediate Ballistics 61
relatively low temperatures. Such a technique will dilute the combustible fraction of the propellant gases and reduce the overall temperature. It is achieved by
triple-base propellants owing to their large proportion of nitrogen rich
nitroguanidine.
Flash can also be reduced by the addition of potassium and sodium salts to the
propellant, especially potassium sulphate or nitrate, potassium cryolite and
sodium cryolite. Their action is not fully understood, but is known to inhibit the
formation and burning of hydrogen gas. However these additives are generally
not used because they increase the amount of smoke produced which provides a
strong unwelcome firing signature in daytime.
Blast
The term blast encompasses the effects produced by gas pressure waves in both
intermediate and terminal ballistics. The most familiar feature of blast from a
gun is the noise produced by the release of high pressure propellant gases into the
atmosphere when the gun is fired. Close to the gun, the blast can be sufficiently
intense to cause hearing damage and, in extreme cases, injury to lungs and other
soft tissues. The terminal ballistic blast effects produced by explosive projectiles
is discussed in Chapter 5.
Problems due to muzzle blast are normally restricted to military weapons and
usage. Gun crews are subjected to intense blast noise, often in excess of 140
decibels, which can cause hearing damage. Some form of hearing protection is
normally used to avoid temporary hearing loss, or permanent damage in cases of
extreme blast or repetitive firings. Excessive blast noise is a particularly serious
problem with hand-held anti-tank recoilless guns as the blast from the rear of the
tube is severe, and the firer’s head has to be held close to the barrel to allow
aiming of the weapon. In such cases the blast can not only be heard, but also felt
as a blow when it strikes the body. Anticipation of the violent blast the soldier
will suffer on firing can markedly reduce his ability to use the gun efficiently if he
is not thoroughly trained to it.
Blast noise is further undesirable in military weapons as it indicates gun
positions to the enemy. Most guns fire supersonic projectiles so, even if a gun is
virtually silent in operation, the shots are still likely to be heard owing to the
shock waves generated by the projectile during flight. However, any reduction in
muzzle blast will help to avoid identification of the shot’s source.
There are two main sources of muzzle blast: blast shock waves and flash blast.
The sudden release of high pressure propellant gases that follows the projectile
muzzle exit generates an inevitable blast shock wave. Flash blast is produced by
the rapid heating and consequent expansion of gases within the secondary flash;
it is most prominent in large calibre guns, and can contribute up to half of the
total noise on firing. Both of these types of blast cause a rise in the gas and air
pressure in the vicinity of the muzzle. In the case of blast shock this overpressure
spreads outward as a shock wave, producing an abrupt increase in pressure as it
travels. Flash blast resembles an intense noise, accompanied by an overall
increase in pressure. Close to the muzzle the overpressure is extremely high, and
it rapidly falls to safe levels further away. Gun crews and small arms firers often
wear ear defenders to guard against overpressures in excess of 0.2% of atmospheric pressure as they are likely to cause hearing damage.
The motion of the projectile and the muzzle gas flow generate a succession of
shock waves together with further oscillations of pressure due to turbulence and
flash. These variations in pressure can be monitored during firing by fast
response pressure gauges, though care must be taken to ensure that their design
and orientation do not distort the gas flow, otherwise false measurements will be
produced.
The intensity of blast, sound and noise is measured in decibels, a quantity
which relates the overpressure produced by a sound to a reference pressure of
2  10–5 Pascals, where 1 Pascal equals 1 N/m2‚
Intensity in decibels is given by:
where P0
is the reference pressure.
For example: find the approximate intensity of an overpressure of 0.2 atmospheres.
As 1 atmosphere roughly equals 1  105 Pascals
then 0.2 atmosphere roughly equals 2  104 Pascals
The blast shock from small-calibre guns can be suppressed by so-called
silencers attached to the muzzle. A silencer never totally silences a weapon, it
just reduces the intensity of the muzzle blast; consequently silencers are often
FIG. 3.13 Cross-section of an idealised shock wave
dB  20 log10
Intensity  20 log10
Intensity  20 log10 109
Intensity  180 dB
P1
P0
2  104
2  10–5
13569_int.indd 60-61 11/15/11 11:20:32 AM
62 Military Ballistics Intermediate Ballistics 63
known as moderators. The three methods of blast suppression commonly
employed in silencers are shown below. A silencer may incorporate a combination
of these blast suppression methods in its design.
In large calibre weapons, the size of silencer required to suppress the blast
shock would be impracticably large. Though blast shock is unavoidable, the flash
blast from large guns can be suppressed by fitting a flash suppressor, or by adding
a flash suppressant to the propellant. Ordinarily, the majority of the blast is
directed forwards by the motion of the propellant gases, so the blast that the
firers suffer is relatively small. If a muzzle brake has been fitted, some of the
propellant gases and the blast they produce are deflected sideways and backwards, so the blast received by the firers is stronger.
There are no clearly defined rules governing the levels above which hearing
can be impaired by gun blast. Although recommendations have been made, the
suggested limits of safe hearing vary according to the opinions of the recommending council. When assessing the likelihood of hearing damage from gun blast,
there are six main factors to be considered:
1. Peak overpressµre at the ear.
2. Effective duration of the intense blast.
3. Frequency range of the blast noise.
4. Ear protection used.
5. Repetition of exposure to blast.
6. Personal susceptibility to blast.
The effective duration of the blast can be defined in a number of ways. For
example, the blast duration can be defined as the time taken from the initial rise
of pressure to the moment that the overpressure falls to a level permanently
below one tenth of the peak pressure. The sensitive frequency range of the ear is
roughly from 1000 Hz to 6000 Hz; intense noise in this range tends to cause
damage and the consequent impairment of hearing will reduce sensitivity in this
range first. Mild loss of hearing in the sensitive range often goes unnoticed as the
reduction in clarity of most sounds is disguised by the continued ability to hear
sounds at other frequencies. Proper use of ear defenders can typically reduce the
effective intensity of the blast noise at the ear drum by up to 35 dB. This alone
will protect hearing in most cases: the main exceptions are recoilless and tank
guns in which the combined effect of intense blast and unavoidably close proximity of the crews can sometimes lead to blast levels in excess of 185 dB; in such
cases extra hearing protection should be used. Unless properly fitted, ear
defenders and ear plugs are generally far less effective, giving protection of 15 to
25 dB. The ear is usually capable of withstanding short single blasts in excess of
150 dB, but repetitive exposure to even low levels of blast can lead to permanent
hearing damage. The restriction on the number of repeat firings and the period of
firing depends on several factors, complicated by the circumstances in which guns
are used. For example, firing an anti-tank recoilless rocket launcher without ear
protection more than once every day would soon cause hearing damage, but a
soldier in battle could fire the same gun many times in one day yet not suffer FIG. 3.14 Table of overpressures in atmospheres against intensity of sound in decibels
(reference pressure 2  10–5 Pascals)
FIG. 3.15 Methods of blast suppression
Intensity in dB Overpressure in atmospheres
100 0.00002 Approx
120 0.0002
140 0.02
160 0.02
180 0.2
200 2
13569_int.indd 62-63 11/15/11 11:20:33 AM
64 Military Ballistics Intermediate Ballistics 65
permanent hearing damage as he may never need to fire such a gun again. The
ability to tolerate blast noise varies considerably from person to person and so
care should be taken to ensure that adequate protection is used in every case to
safeguard the hearing of those most susceptible to damage.
Recoil
Recoil is the rearward motion of the gun in reaction to the forward motion
imparted to the projectile and propellant gases. The forward momentum gained
by the projectile and gases is accompanied by an identical gain in rearward
momentum of the gun. The speed of the recoiling gun is small in comparison to
the muzzle velocity, but the gun is relatively massive and so requires a considerable braking effect to halt its motion. The recoil of small arms is usually
acceptable, but for larger weapons recoil can be a serious problem.
Typically, a gun will have attained about half of its final recoil energy when
the projectile leaves the muzzle; the remaining half of the recoil energy is gained
by reaction to the rapid outflow of gases at the muzzle. Alternatively, a large
proportion of the outflowing gases can be deflected backwards by a muzzle brake,
and so generate a forward thrust that partially counteracts the recoiling motion
of the gun.
An efficient muzzle brake can reduce recoil by over 50%, though such muzzle
brakes are complex, costly and, as has been indicated, can damage hearing.
Practical muzzle brakes reduce recoil by about 25% by deflecting some of the gas
flow sideways rather than backwards.
Muzzle brakes can be designed to deflect gas mainly upwards, for example, to
control the upward muzzle jump of submachine guns. The prime disadvantage of
the muzzle brake is that it increases the blast noise suffered by the firer. It can
also subject the fore-end of the barrel to excessive stress. Generally sabotted
ammunition cannot be used in guns fitted with muzzle brakes as the complex gas
flow at the muzzle interferes with the sabot discarding process, thereby reducing
the accuracy of fire. Besides, the sabot is very likely to collide with the muzzle
brake.
The Dilemma of Intermediate Ballistics
There are so many variables which complicate the objective study of intermediate ballistics that it is by no means an exact science. Indeed it is an area which
can cause gun and ammunition designers considerable problems. This chapter
has covered the areas which theory and empirical methods have isolated
as affecting design. However there remain no precise methods to predict intermediate ballistic performance, so practical trials are still necessary to assess
every new design.
FIG. 3.16 9 mm calibre Sten sub machine gun fitted with silencer
FIG. 3.17 The principle of the muzzle brake
FIG. 3.18 Muzzle brake fitted to British 105 mm calibre light gun
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66 Military Ballistics
67
4.
External Ballistics – Part I
Self Test Questions
QUESTION 1 Name the shock waves produced by the firing of a gun.
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 2 What effects does the muzzle gas flow have upon the projectile?
Answer………………………………………………………………………………..
QUESTION 3 What is the minimum speed of a shock wave?
Answer ……………………………………………………………………………….
QUESTION 4 Which types of flash would you expect from a rifle?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
QUESTION 5 Which type of blast is not produced by small calibre guns?
Answer ……………………………………………………………………………….
QUESTION 6 If an overpressure of 0.08 atmospheres is recorded near a gun,
what is the intensity of the blast at that point relative to a
reference pressure of 2  10-5 Pascals?
Answer ……………………………………………………………………………….
QUESTION 7 Why is the recoil momentum of a gun greater than the forward
momentum imparted to the projectile?
Answer ……………………………………………………………………………….
…………………………………………………………………………………………..
…………………………………………………………………………………………..
Basics
Once the projectile has left the gun and the influence of the emerging gases, the
part of the flight known as external ballistics begins. There are a number of
factors which affect the motion of a projectile, some associated with the projectile
itself and others with the atmosphere through which the projectile is moving. The
properties of the projectile which enter into the problem are its mass, calibre
(that is its maximum diameter), shape and axial spin rate. The relevant properties of the atmosphere are air density, temperature, static pressure, viscosity and
wind speed and direction. The effects of these are made manifest through the
projectile properties introduced above.
This chapter will deal first of all with the atmosphere and its properties,
followed by a conceptual introduction to the important properties of the air from
which it is composed. The influence of these properties on the flight of a projectile
will then be dealt with in some detail.
The Atmosphere
The atmosphere up to about 20 km altitude is basically composed of 78%
nitrogen, 21% oxygen with the remaining 1% comprising water vapour, carbon
dioxide and other gases. At very high altitude, separation of the gases occurs. For
example at about 100 km atomic oxygen separates and the lighter gases diffuse
upwards and separate out into layers.
Air pressure, temperature, density and viscosity all vary with altitude. These
changes in the physical properties of the atmosphere affect the resistance of the
air to the passage of a projectile and hence its range. Since the trajectories of
small arms ammunition, artillery projectiles and ballistic missiles normally have
peak altitudes of 50 m, 20 km and 600 km respectively, projectiles in these
different roles will experience significantly different environments in flight. For
example, the density of air decreases as the height above sea level increases and
consequentially the range of a projectile increases as the trajectory gets higher
and higher. This is exploited by intercontinental ballistic missiles which travel
for most of their flight in the upper atmosphere where there is practically no air
resistance at all.
Atmospheric conditions vary from place to place and from time to time.
For comparative performance assessment, a standard atmosphere is required.
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68 Military Ballistics External Ballistics – Part I 69
This requirement is also common to aeronautics. The standard atmosphere most
commonly used today is that proposed by the International Civil Aviation
Organisation (ICAO). It approximates to the average atmospheric conditions in
Continental Europe and North America. Typical values are given in Table 4.1. In
these regions, it has been observed that, on average, at heights up to the
tropopause at 11 km, the temperature falls linearly with height at a constant rate
of 6.5°C per km – this is known as the temperature lapse rate. This region is the
troposphere. Also it has been found that the average temperature at Sea Level
(SL) is 15°C, ie 288.15K. The region above the troposphere, up to a height of 20
km, is called the stratosphere. In the stratosphere, the temperature remains
constant. The important characteristics of the atmosphere are:
1. Temperature (T). This is given in Table 4.1 in the Kelvin temperature scale
in which the temperature is zero at absolute zero, ie 0°C  273.15K with
Celsius degree intervals. It is a measure of the kinetic energy of the molecules of the gas. In American literature, the absolute scale of temperature is
in degrees Rankine (°R) and this scale has Farenheit intervals. Hence:
2. Pressure (p). The third column in the Table shows how the local static
pressure varies with height. This is the pressure which would be recorded by
a barometer at the same height.
3. Density (). Column 4 shows the equivalent variation in mass density.
Pressure, density and temperature are related in a gas by the equation of
state:
p  RT
in which R is the gas constant. Since vertical equilibrium is assumed, once
the temperature variation is specified, the variation of density and pressure
with height are prescribed. Column 5 gives the variation in speed of sound;
this is dependent only on temperature and it therefore becomes constant in
the stratosphere where the temperature itself is constant.
4. Viscosity (µ). The (dynamic) viscosity of a fluid is a measure of its resistance
to shearing motion. Examples of very viscous fluids in every-day use are
syrup and heavy oil. The viscosity of liquids reduces as temperature increases; for gases it increases as temperature rises. The word ‘viscosity’
usually refers to this quantity.
The final column in Table 4.1 is the coefficient of kinematic viscosity,
usually written . This is the (dynamic) viscosity of the fluid divided by its
mass density, ie µ/. Since air has a very small viscosity, this number is also
very small.
In practice, the atmosphere varies markedly from this standard. In trials use is
often made of two different altitudes. One is pressure altitude. This is the altitude
at which the observed pressure is equal to the pressure in the standard atmosphere; for example if the pressure altitude were quoted to be 5000m this would
mean that at the height of interest, the static pressure would be 54020 N/m2. This
is not unreasonable since the altimeter used in aircraft is really a barometer
which therefore measures static pressure.
The second altitude often used is density altitude. This is similarly the altitude at
which the density would occur in the standard atmosphere. Except on rare occasions,
neither altitude corresponds to the true, geometric height above sea level.
Subsonic, Transonic and Supersonic Projectiles
A division in the types of projectile will be made in terms of their speed of
flight. Because of the basic change in the way in which the fluid behaves when
H T p r a 
m K N/m2 kg/m3 m/s m2
/s  10–5

0 288.15 101325 1.2250 340.29 1.461
500 284.90 95461 1.1673 338.27 1.519
1000 281.65 89875 1.1116 336.43 1.581
1500 278.40 84556 1.0581 334.49 1.646
2000 275.15 79495 1.0065 332.53 1.715
2500 271.90 74683 0.9569 330.56 1.787
3000 268.65 70109 0.9091 328.58 1.863
3500 265.40 65764 0.8623 326.58 1.943
4000 262.15 61640 0.8191 324.58 2.027
4500 258.90 57728 0.7768 322.56 2.117
5000 255.65 54020 0.7361 320.53 2.212
5500 252.40 50507 0.6971 318.49 2.311
6000 249.15 47181 0.6597 316.43 2.417
6500 245.90 44035 0.6238 314.36 2.529
7000 242.65 41061 0.5895 312.27 2.649
7500 239.40 38251 0.5566 310.18 2.774
8000 236.15 35600 0.5252 308.06 2.906
8500 232.90 33099 0.4951 305.94 3.050
9000 229.65 30742 0.4564 303.79 3.200
9500 226.40 28524 0.4389 301.64 3.360
10000 223.15 26436 0.4127 299.46 3.531
10500 219.90 24474 0.3877 297.27 3.713
11000 216.65 22632 0.3639 295.07 3.907
15000 216.65 12045 0.1937 295.07 7.340
20000 216.65 5475 0.0880 295.07 16.151
25000 221.65 2511 0.0395 298.45 36.71
30000 226.65 1172 0.0180 301.80 81.94
List of symbols: H  height above SL
p  static pressure
a  speed of sound
T  arnbient temperature
  mass density for dry air
  coefficient of kinematic viscosity
TABLE 4.1
The Standard Atmosphere (ICAO)
°R  °F  460
5
K  °R
9
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70 Military Ballistics External Ballistics – Part I 71
objects which are within it are travelling at speeds which are below or above the
speed of sound, the ratio of the speed of the projectile to the speed of sound in the
fluid will form the basis of this division.
This ratio is known as Mach number and is defined as follows:
The speed of the projectile Mach number (M)  The ambient speed of sound
The speed of sound in a fluid is very dependent on a combination of the
compressibility and the mass density of the fluid. For example, in air at sea level
on an average day in Continental Europe, it is about 340 m/s and it varies as the
square root of the absolute temperature. In fresh water, the corresponding speed
is about 1450 m/s.
Projectiles for which the flight Mach number is less than about 0.8, are known
as subsonic projectiles and those for which the flight Mach number remains above
about Mach 1.2 are called supersonic projectiles. The regime of flight which
occupies the Mach number range of about 0.8 to 1.2 is known as the transonic
region. If the flight Mach number is above about Mach 5, the projectile is
hypersonic. (The prefix hyper- is now somewhat uncritically applied to many
high speed, ‘hyper-velocity’ projectiles with speeds from about 1000 m/s
upwards.)
Any given projectile may be supersonic, transonic or subsonic at different parts
of its flight depending on the Mach number at the time. For example, even a
projectile which was highly supersonic in air would become subsonic on entering
water (or body muscle tissue in which the speed of sound is very similar to that in
water).
Reference Frames
We are really concerned with the motion of a projectile through an essentially
stationary atmosphere, but it is often convenient to consider the projectile to be
stationary and to examine the air flowing past it, as in a wind tunnel. We will use
both references, frequently changing from one to the other. In most cases these
two reference frames are equivalent: where the differences are significant, they
will be noted.
Basic Equations of Fluid Flow
When analysing fluid flows we need some way of depicting the flow in detail.
This is done by the use of streamlines. These are lines drawn in the fluid in the
direction of the local fluid flow. It follows therefore that fluid cannot cross a
streamline and also that, in steady flow, two streamlines cannot cross. Because of
this, any streamline can notionally be replaced by a solid boundary. In steady
flow, the streamlines are fixed in position.
It is also helpful when attempting to gain an understanding of the behaviour of
fluids in motion to have some knowledge of the basic equations which the fluid
obeys. The first of these is known as the continuity equation and is the fluid
mechanics equivalent of the law of conservation of mass.
Let us look at the flow between the two streamlines shown in Fig. 4.1. The fluid
density is denoted by , the local cross sectional area by A and the speed of the
fluid by V. The equation of continuity then states that:
AV  constant
For low speed flight, in which density is essentially constant, where the flow
speeds up, the streamlines come closer together. Where the flow slows down, they
are further apart.
At higher speeds, the density starts to change at an increasingly faster rate as
the speed increases. Above Mach 1, this change is so rapid that it reverses the
behaviour described previously for low speeds: where the flow slows down, the
streamlines come closer together, and where the flow speeds up, the streamlines
get further apart.
Additionally, provided that the effects of viscosity are not large, then where the
fluid speed increases the static pressure falls and vice versa. This is very useful in
gaining a qualitative idea of the forces consequent upon fluid motion.
Aerodynamic Drag
In a vacuum, the only force acting on a projectile in flight is that due to
gravitational acceleration. In air, there will be an additional force opposing the
forward motion of the projectile caused by the resistance of the air; this is called
drag. In general terms, over the speed range of interest here, this force depends
on the forward speed of the projectile. For projectiles which fly at high speeds, it is
the dominant force on the projectile, and substantially modifies the trajectory.
During the rising part of the trajectory, drag acts in addition to gravity, and the
vertical component of the launch velocity is eroded more quickly. The horizontal
velocity component is also reduced by drag. The net result is sketched in Fig. 4.2.
Compared with the in-vacuo trajectory range is reduced and the angle of descent
steepened.
For example, if an L15 155mm artillery shell were fired at its current maximum muzzle velocity, the aerodynamic drag would be about 5 times the weight of
the shell at launch. This reduces the in vacuo maximum range by a factor of
nearly 3 from 67km to 24km. For small projectiles, which have a much greater
surface area for their size and weight, the importance of aerodynamic drag
becomes more pronounced. Fired in similar conditions to the L15 shell, a NATO
standard 7.62mm bullet would initially have an aerodynamic drag of 60 times its
weight. The reduction of the in-vacuo maximum range in this case is from 72 km
to 4 km.
FIG. 4.1 Steady streamline flow
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72 Military Ballistics External Ballistics – Part I 73
In view of the very large effects of drag on range, it is worth spending a little
time exploring its origins.
Types of Drag
Essentially, there are three separate contributions to the drag force on an
object moving through a fluid medium. These may be summarised as:
1. Skin friction
2. Pressure drag and
3. Yaw-dependent drag
Skin Friction
Skin friction is produced directly by the viscosity of the fluid. Viscosity is a
measure of the resistance to shearing motion of the fluid. When an object is in
motion through the fluid, the molecules immediately adjacent to the surface of
the object, stick firmly to the surface. Molecules a little further away from the
surface move parallel to it. There is then a thin layer of fluid close to the surface
of the body in which shearing is taking place; this is known as the boundary
layer.
The viscosity of air is small. Nevertheless, it can still make a substantial
contribution to the drag of a projectile because of its effect on pressure drag.
Pressure Drag
In general, as a consequence of viscosity, the average static pressure at the
front of the moving body is greater then the average static pressure at the rear of
the body. Consequently a force acts against its forward motion. This is called
pressure drag.
At subsonic speeds, smooth shapes with rounded noses and gently tapering
rear sections with pointed bases are required if pressure drag is to be minimised.
If sharp corners are present elsewhere on the object or if it changes its cross
sectional shape too quickly, the boundary layer may separate from the surface.
This causes the formation of a region of fluid behind the body which tends to move
along with it. This region is known as the wake. In it the static pressure is lower
than it would otherwise be and therefore the drag is increased. This form of drag
is usually called ‘base drag’.
It is in order to reduce the pressure drag caused by boundary layer separation,
that objects are streamlined. Examples of the flow around high drag and low drag
objects are given in Fig. 4.3.
In this Figure, the upper sketch is for a fiat plate at right angles to its direction
of motion. In this case, the flow separates at the sharp corners producing a large
wake and hence a large drag. Even if the edges are significantly rounded, as for
the body in the following diagram, flow separation and hence high drag forces
still occur. It is necessary to use highly streamlined bodies before flow separation
is averted. A well streamlined shape is sketched in the final diagram. Most of the
drag on this final body comes from skin friction. For the same frontal area at the
same speed, this fully streamlined body could have a drag as little one hundredth
of the drag of the original flat plate.
However, a highly streamlined body is not normally a viable option for gunlaunched projectiles. Firstly it is not a good shape from an internal ballistic point
of view, being difficult to seal and to keep straight within the barrel. Secondly, as
will be explained later, it is markedly unstable directionally. Thirdly, it does not
provide enough volume for explosives or sub-munitions.
At low subsonic speeds, pressure drag for most bodies varies almost exactly
as the square of the speed, and, at these speeds, boundary layer separation
almost invariably leads to a large increase in drag. At supersonic speeds, however, this generalisation is invalid. There are situations at supersonic speeds
in which the presence of boundary layer separation significantly reduces the
drag below that which would be present in inviscid flow. This is due to the
appearance of another source of pressure drag at supersonic speeds – wave
drag. This is caused by shock waves in the airflow and these are discussed in
the next section.
Drag Due to Shock Waves
An inevitable accompaniment of movement at supersonic speeds is a system of
shock waves. The way in which these are produced is illustrated in Fig. 4.4.
Imagine a very slender object moving through otherwise stationary air at speed
V. At each instant, the object is pushing aside some of the air. Each of these ‘little
pushes’ is a small pressure wave which travels radially away from its point of
generation at the speed of sound, a. This pressure wave serves to warn the air
of the approach of the object so that the air can adjust itself to flow smoothly
around it.
Fig. 4.4a shows what happens when the object is moving at one half of the
ambient speed of sound, at a Mach number of 0.5. The circles represent the
positions that the pressure waves, generated at various times in the past, have
reached at the present time. For example, the circle labelled 1 is the present
position of the pressure wave which was generated when the object was at point
1. It can be seen that, although the waves travelling ahead of the object are closer
FIG. 4.2 Real and in-vacuo trajectories
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74 Military Ballistics External Ballistics – Part I 75
together than those travelling behind it, the fluid still receives warning of the
approach of the object.
Now let us look at the situation when the object is moving at twice the local
speed of sound ie at M  2. Instantaneously it looks like Fig. 4.4h. Now the
pressure waves which are attempting to travel ahead of the object, coalesce into a
straight line which intersects the line of flight of the object at the object itself.
This line, which is a very weak pressure pulse, is know as a Mach line, or in 3
dimensions, as a Mach cone.
It is clear that any air to the left of this line/cone can receive no warning of the
approach of the object. Since the speed of advance divided by the ambient speed of
sound has been defined as the Mach number, and the individual pressure waves
propagate outwards at the local speed of sound, it follows that the included angle
µ between the Mach line or the surface of the Mach cone and the line of flight can
be found as sin–1 FIG. 4.3 Flow past progressively more streamlined bodies (1/M). The angle µ is known as the Mach angle.
FIG. 4.4a Pressure waves from a source moving at half the speed of sound
FIG. 4.4b Pressure waves from a source moving at twice the speed of sound
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76 Military Ballistics External Ballistics – Part I 77
Some typical values for the Mach angle are:
M 1 1.5 2 3 10
 (deg) 90 42 30 19 6
which show how rapidly the angle decreases with increase of Mach number.
In the case of a realistic, less slender, object, the individual pressure waves are
stronger and travel correspondingly faster. The included angle of the common
tangent to the pressure pulses now produced is larger than the Mach angle and
the coalescence of the pulses produces a stronger pressure wave travelling at
rather more than the ambient speed of sound. This pressure wave is known as a
shock wave; it is physically very thin, typically less than 10–5m in air at sea level
and is the first indication which the air receives of the approach of an object
travelling at supersonic speeds. It can represent a pressure increase of several
atmospheres and the losses which occur as the fluid is compressed require a
constant supply of energy. This appears as an additional pressure drag force on
the body known as wave drag.
The first indication which the air receives of the approach of a supersonic
projectile is with the arrival of the bow shock. As the air passes through the shock
wave, it changes direction to pass around the nose of the body.
Since the velocity of the flow is reduced on passing through the shock wave and,
due to the increase in pressure, the temperature of the air and hence its speed of
sound is increased, there is a reduction in Mach number. The magnitude of the
reduction is dependent on the angle of the shock wave and the Mach number of
the oncoming flow. If the change in flow direction is too large, the downstream
Mach number becomes less than one and consequently subsonic. Under these
circumstances, the shock wave moves forwards, detaches from the nose, and
becomes, close to the body, a normal shock. This is illustrated in Fig. 4.5 which
shows the measured shock angles for 2-dimensional wedges at different flight
Mach numbers. In 2 dimensions, the shock angle for supersonic downstream flow
is quite well approximated by the sum of the wedge angle and the Mach angle.
For 3-dimensional noses, the turning angle is less and larger nose angles can be
tolerated without the shock detaching.
The main disadvantage of the detached shock is that it produces a large drag.
As the air moves further back over the projectile, it will encounter changes in
geometric shape. If the local slope of the body surface increases, another shock
wave will be formed. If reductions in surface slope are met, the air changes
direction through a broader region known as an expansion fan.
The larger the change in local slope of the surface, the stronger will be the
shock wave and the larger will be the drag produced by it. For bluff-nosed bodies,
the wave will be at right angles to the flow and this will generate large amounts
of drag. It is therefore important, if drag is to be kept small for supersonic
projectiles, that they have reasonably slender nose shapes. This has stability
implications which will be discussed later.
Flow patterns about two supersonic projectiles are given in Figs. 4.6 and 4.7.
Fig. 4.6 is the typical shape of a bullet or an artillery shell and it has an attached
bow shock. The low pressure wake and a tail shock can also be seen as can some
flow disturbances from the cannelure. Fig. 4.7 is a projectile with a blunt nose
and the detached, high drag, shock can be clearly seen. This projectile is a
training round for which high drag is an advantage in that it reduces the range
safety distance requirements.
The variation of three components of drag for a typical artillery shell shape
with flight Mach number are shown in Fig. 4.8. The rise in wave drag as Mach 1
is approached is noteworthy; as is also the tendency of the base drag to become
proportional to Mach number at higher Mach numbers. This also true of the total
drag. Skin friction is a relatively small proportion of the total but it becomes
more important for long thin projectiles.
At this stage it is worth commenting on flight in the transonic region. At
speeds close to the velocity of sound the projectile’s motion becomes difficult to
predict because the air resistance is changing extremely rapidly with speed.
Small changes in the projectile’s velocity, perhaps due to wind fluctuations, cause
marked changes in resistance. It is important that the projectile is flying steadily
when it enters the transonic region. Just outside the muzzle of the gun, it is
common for the flight of the projectile to be markedly unsteady. It is therefore
FIG. 4.5 Shock angle for wedges in supersonic flow
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78 Military Ballistics External Ballistics – Part I 79
undesirable to use a propellant charge giving a muzzle velocity which is in the
vicinity of the speed of sound. For this reason, muzzle velocities close to the speed
of sound are, if possible, avoided by the ammunition designer.
Yaw-dependent Drag
In general, a projectile will not fly with its body axis aligned with the direction
of flight. The angle between the body axis and the direction of flight is called the
yaw angle. It may arise from a variety of causes, as will be discussed later, but its
presence will usually generate a sideforce on the body. This sideforce will be
normal to the body axis and will therefore have a component in the drag
direction.
For small angles of yaw, the sideforce is proportional to the yaw angle; also for
small angles, the drag is equal to the sideforce times the yaw angle. Therefore,
the drag due to sideforce would be expected to vary with yaw angle squared and
this is found to be so experimentally.
At low speeds, the sideforce at constant yaw angle is proportional to speed
squared. Therefore this final drag component, like the other two discussed previously, is also approximately dependent on speed squared.
Drag Coefficient
The total drag of the projectile is the sum of the components described above.
In each case, the individual drag force is produced by a stress acting on an
area. These stresses may include normal stresses from pressures and tangential
shear stresses. Additionally, the forces generated depend on the density of the
fluid in which the motion is occurring. It is therefore logical to divide the drag
force on the projectile by the product of air density, forward speed squared and a
reference area. The denominator has the units of force and therefore the quantity
formed by this division has no units or dimensions. It is known as a drag
coefficient and written CD. In practice, for theoretical reasons, a 2 is introduced
FIG. 4.6 Shadowgraph of 7.62 mm bullet
FIG. 4.7 Shadow graph of high drag training round
FIG. 4.8 Components of drag
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80 Military Ballistics External Ballistics – Part I 81
into the numerator and, for projectiles, the reference area used is the frontal area
of the round.
In analytical form:
in which p is the fluid density, V is flight speed and d is projectile maximum
diameter or calibre.
The use of drag coefficient as a measure of the efficiency of a projectile is almost
universal. By measuring drag in terms of this coefficient, the first order effects of
air density, flight speed and projectile size are removed. Therefore in place of a
dependence on these quantities, the drag coefficient becomes essentially a function of Mach number only. This dramatically reduces the problem of data presentation since the drag characteristics of a given shape can now be presented by a
single curve of drag coefficient against Mach number.
This is illustrated in Fig. 4.9 in which the drag coefficient for the typical bullet
shape shown is plotted against Mach number. Note that there are no significant
effects of air density, flight speed or projectile size on this curve. Shown in Fig.
4.10 is the actual drag force in Newtons for a 7.62 mm round at sea level on a
standard day. Also given on this plot is the drag curve which would result if the
drag coefficient were truly constant ie independent of Mach number.
It should however be noted that if the properties of the fluid in which motion
occurs are significantly changed, this can also cause changes in the value of the
drag coefficient for a given shape.
For hand guns which are meant for shooting at close ranges, the ammunition is
not always designed to have low drag. If a bullet is designed to move extremely
fast while giving it a non-aerodynamic shape, it may produce more damage on
arrival than a more conventional bullet.
The Estimation of Drag Coefficient
The basic drag coefficient vs Mach number relationship can either be estimated
or it can be deduced from range measurements. Knowledge of this relationship
enables trajectories for the round to be predicted.
If an analytical approach to drag estimation is adopted, it is usual to consider
the drag of the projectile to be made up of individual contributions from the nose
or forebody, from the base, with boat-tail if present and from skin friction and
excrescence drag.
At low subsonic speeds, the drag of the nose is almost independent of shape.
Fig. 4.11 shows the variation for 4 different noses; the main difference is in the
Mach number at which the drag coefficient starts to rise, the more rounded the
nose, the sooner the drag rise occurs. This is due to the presence of small shock
waves in the flow, which are themselves a consequence of local supersonic
regions. These exist because the air is speeded up as it passes around the
projectile. Therefore there are regions in which the speed of the air is greater
than the speed of the object. Hence there will be regions in the flow for which the
local Mach number is greater than unity even when the flight Mach number for
the projectile is still less than 1. The bluffer the projectile, the greater the
increase in speed, therefore the lower the Mach number at which these supersonic regions occur.
At wholly supersonic speeds, nose shape is much more critical. A common
shape for supersonic noses is the ogive which is the shape formed by rotating the
arc of a circle about the axis of the shell. The construction of a simple tangent
ogive is illustrated in Fig. 4.12. The centre of the circular arc which forms the
nose is struck from an extended terminating diameter. The radius of the circle
AO is expressed in calibres and this kind of nose is known as a calibre radius
head (crh). In this case a 2crh since AO is 2 diameters in length.
Modern shells are often given an ogival nose of slightly different shape. It is
FIG. 4.9 Drag coefficient of 7.62 mm bullet
FIG. 4.10 Drag of 7.62 mm bulle
8 (Drag force) CD  V2d2
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82 Military Ballistics External Ballistics – Part I 83
produced by striking the arc from a line which bisects the angle AOC as in Fig.
4.13. This is a secant ogive head often referred to as a fractional calibre radius
head. The value of the fraction is the ratio of the radius of the circle which would
give a tangent ogive (AO) to the radius actually used (SR). In this case the shell
has a 2/4 crh since AO is 2 diameters in length and SR is twice AO.
Fig. 4.14 shows the variation of drag coefficient with Mach number for various
simple nose shapes at supersonic speeds. When it is recalled that drag force is
equal to drag coefficient times speed squared (and hence times Mach number
squared), it is clear that noses need to be fairly slender if very large drag is to
avoided at supersonic speeds.
Base drag is essentially due to the change in pressure on the base of the
projectile caused by boundary layer separation. It may be minimised by using a
tapered base – a technique known as boat-tailing. Fig. 4.15 shows the variation
in base drag for a tapered afterbody at low subsonic speeds for various
lengths of afterbody. The taper angle f is 9° which is about optimum, and this
brings the body to a point when the afterbody is 3.15 diameters long. It is
noteworthy that a boat-tail only 0.5 body diameters in length reduces the base
drag to 50% of what it would be for a completely bluff base.
Similar considerations apply at supersonic speeds. Fig. 4.16 shows the variation of base drag coefficient against Mach number for a plain base and a boattailed base. This time the taper angle is 6.5° which is about optimum at supersonic speeds.
Skin friction is normally estimated by ‘unrolling’ the surface of the projectile to
produce a flat plate and using semi-empirical skin friction formulae for flat plates
to give the coefficient. Excrescences are also usually tackled empirically, the
main contribution to drag being from the driving band.
Table 4.2 gives the drag breakdown for a typical artillery shell over a range of
Mach numbers, both for a shell with a boat-tail and for the same shell with a
plain base. It is noteworthy that the drag, expressed in coefficient form, is very
similar to that for the 7.62 mm round given in Fig. 4.9, in spite of the fact that
one is 20 times the linear size of the other. This illustrates the value of the use of
drag coefficients since the same shape at the same Mach number has the same
drag coefficient (though not the same drag), regardless of speed, size and air
density.
For the attack of armour, however, a very different shape, may be used. This is
essentially a long pointed rod travelling at very high speed, which, as will be
discussed later, must be fin stabilised. A typical drag breakdown for such a round
is given in Table 4.3.
As discussed above, an alternative approach to the estimation of drag is to
measure the speed decay of a round in flight in a ballistic range. Since the
deceleration of the round is produced by the drag force, it follows that:
in which V is the velocity drop over the time t.
FIG. 4.11 Nose drag at subsonic speeds
FIG. 4.12 Tangent ogive nose (calibre radius head)
FIG. 4.13 Secant ogive nose (fractional calibre radius head)
V d2
CDV —–   ———— . t
V 8 (mass)
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84 Military Ballistics External Ballistics – Part I 85
At high supersonic speeds, the product CD V is approximately constant and so
the rate of speed loss is a fixed proportion of the speed. For example, the 7.62 mm
round mentioned earlier, initially loses 7% of its speed everyone tenth of a
second.
Base Bleed
One way of reducing drag is to provide a gas generator in the base of the shell,
the gas from which is used to raise the base pressure and thereby reduce the base
drag. This technique is known as ‘base bleed’ and can increase the range of a
typical artillery shell by between some 10% to 30% though at the cost of some
reduction in payload. Fig. 4.17 illustrates the incorporation of a base bleed unit
into a typical artillery shell. Fig. 4.18 shows the marked effect of base bleed on
the range of a 120 mm experimental projectile.
A more sophisticated version of base bleed is ‘external burning’ which emits
gas through holes in the periphery of the base.
FIG. 4.14 Nose drag at supersonic speeds
FIG. 4.15 Boat-tail drag at subsonic speeds
FIG. 4.16 Boat-tail drag at supersonic speeds
TABLE 4.2
Drag breakdown for a typical shell shape
Mach number 0.5 0.8 1.0 1.2 2.0
Drag component:
Forebody 0.007 0.007 0.048 0.106 0.074
Skin friction 0.053 0.047 0.044 0.041 0.032
Driving band 0.005 0.005 0.010 0.015 0.011
Base with boat-tail 0.051 0.058 0.142 0.172 0.144
Base no boat-tail 0.128 0.176 0.214 0.270 0.212
Total with boat-tail: 0.116 0.117 0.244 0.334 0.261
Total no boat-tail: 0.193 0.235 0.316 0.432 0.329
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86 Military Ballistics External Ballistics – Part I 87
Ballistic Coefficient
The rate at which velocity decays is a measure of penetration; it is often
referred to by means of a quantity called the ballistic coefficient. The existence of
this quantity is a consequence of the fact that it is not normally possible to arrive
at an analytical solution to the problem of trajectory estimation. Before the
advent of high speed computers, the effort involved in trajectory computation was
prohibitive. Therefore it was assumed that the CD versus Mach number curve
(ie the Drag Law) was known and unique and the calculations were done (once!)
for this curve. Different designs were assumed to differ from this by a calibration
factor – the ‘shape and steadiness’ factor.
This is incorporated into the ballistic coefficient Co as:
mass
C0  calibre2  shape and steadiness factor
It follows that the deceleration of the projectile is given by:
The ballistic coefficient may therefore be viewed as a measure of the mass per
unit frontal area of a projectile times a drag efficiency factor and the deceleration
of the projectile is then inversely proportional to it. For example, a large ballistic
coefficient gives a small deceleration and hence good penetration and vice versa.
Because it may have been derived from a fit to range data which is different
from that which has been assumed by the drag law, its value may vary from
one trajectory to another even for the same projectile. It is still normally quoted
in lb/in2!
Aerodynamic moments
As already mentioned, a tube-launched projectile in ballistic flight will
normally experience yaw disturbances. These may be disturbances as it leaves
the tube due to tube motion or due to the effects of the propellant gases which
FIG. 4.17 A typical base bleed projectile
TABLE 4.3
Drag breakdown for typical APFSDS round
FIG. 4.18 The effect of base bleed on range
Mach number 3 4 5
B Wave drag 0.040 0.035 0.030
O Skin friction 0.070 0.050 0.045
D Base drag 0.080 0.040 0.030
Y Excrescence 0.070 0.050 0.045
Total body CD 0.260 0.175 0.150
F Wave drag 0.045 0.040 0.035
I Skin friction 0.070 0.065 0.060
N Base drag 0.045 0.025 0.015
Total fin CD 0.160 0.130 0.110
TOTAL CD 0.420 0.305 0.260
V2
CD
deceleration  8C0
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88 Military Ballistics External Ballistics – Part I 89
overtake it as it first leaves the tube, or due to cross-winds or gusts or simply due
to the effect of gravity in curving the trajectory downwards as shown in Fig. 4.2.
The presence of yaw generates a sideforce on the projectile and so there is a
general force R acting on the projectile. This force is usually split into 2 components. These may be either lift (L) and drag (D), or normal force (N) and axial
force (A). Lift is defined as the force at right angles to the direction of flight and
drag is the force opposing motion. Normal force is the force at right angles to the
body axis (often referred to as sideforce in these notes) and axial force is the force
along the body axis. These forces are shown in Fig. 4.19.
It is usually found to be convenient to use lift and drag (which are based on
flight direction) for performance calculations and normal and axial forces (which
are related to body axis positions) for stability analysis.
Theoretically, the local sideforce on a projectile is proportional both to the yaw
angle and to the rate of change of cross sectional area of the projectile.
Several consequences follow from this:
1. A slender body only generates a sideforce where its cross-sectional area is
changing.
2. A slender body which is pointed at the rear does not generate a net sideforce.
It does however, generate a nose up couple due to the upload on the nose and
the download on the rear.
3. Sideforce is proportional to yaw angle (for small angles).
For the slender projectile shape needed to minimise drag, especially at supersonic speeds, this produces a sideforce distribution similar to that sketched in
Fig. 4.20.
This sideforce exerts a total moment about the centre of gravity (cg) of the
projectile which is found by integrating the local sideforce times its distance from
the centre of gravity along the length of the projectile. This moment can be
represented by the total sideforce acting at a single point known as the centre of
pressure (cp). If nose up moments are reckoned to be positive, the relationship is:
Moment  Sideforce  the distance of the centre of
pressure forward of the centre of gravity.
Theoretically, the centre of pressure position aft of the nose apex can be
calculated from the formula:
Volume
Centre of pressure position  Projectile length –
Base area
Experimentally there is found to be a gradual rise in body normal force divided
by Mach number squared with Mach number, as shown in Fig. 4.21.
For a normal low drag shape, the centre of pressure is forward of the centre of
gravity and nose up yaw generates a corresponding nose up moment.
For the reasons advanced earlier in the case of drag, the aerodynamic moments
on a projectile are expressed in dimensionless form as a coefficient. The same
FIG. 4.19 Components of the force acting on a body in flight
FIG. 4.20 Body sideforce distribution due to yaw
FIG. 4.21 Body normal force divided by Mach number squared.
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90 Military Ballistics External Ballistics – Part I 91
normalising parameters are used, except that, because a moment is involved, an
extra length is required to make the ratio non-dimensional. The length used is
the projectile maximum diameter or calibre. The expression for the yawing
moment coefficient CM therefore becomes:
This coefficient, at least for small yaw angles, varies linearly with yaw angle
and since it is the rate at which the moment increases with increase in yaw angle
which is critical for stability, the derivative of this coefficient with respect to yaw
angle is the usual form in which this quantity appears in stability assessment ie
as CM/d which is written CM.
It is further deduced that CM measured relative to the projectile centre of
gravity is positive for a directionally unstable projectile, zero for a neutrally
stable projectile and negative for a directionally stable projectile. These terms are
defined in the next section. For all normal projectile body shapes, CM is positive.
Static Stability
The stability of a projectile is concerned with its motion following a disturbance
from an equilibrium condition. A distinction is made between the initial tendency of the projectile to return to its starting position after a disturbance and any
subsequent motion. Because the first of these can be analysed without any
reference to the resulting motion, it is termed static stability. Examination of the
motion itself is referred to dynamic stability analysis.
There are 3 classes of static stability which are possible. These are:
1. Stable – the projectile tends to return to its starting condition.
2. Neutrally stable – the projectile remains in its disturbed condition.
3. Unstable – the projectile tends to move further away from its starting
condition in the direction in which it was disturbed.
These 3 classes of stability can be illustrated by considering a cone on a
horizontal surface as shown in Fig. 4.22. In Fig. 4.22a, the cone is standing on
its base. If the apex is displaced sideways and then released, the cone will
return to its original, equilibrium position. It is therefore statically stable as
defined above. When placed on its side as in Fig. 4.22h, a sideways displacement simply produces a change in the rest position. The cone is then neutrally
stable. Finally, if the cone is balanced on its apex as shown in Fig. 4.22c, any
sideways displacement will cause it to tip further away from the initial condition
in the direction of the disturbance. The cone is now statically unstable.
This idea can now be applied to the case of a projectile in flight. If it is subjected
to a yaw disturbance then the sign of the resulting yawing moment generated
will also define the static stability. In the case of the low drag shape described
above for which the centre of pressure is forward of the centre of gravity, the
yawing moment generated by a yaw disturbance is in the same direction as the
yaw disturbance and hence will tend to make the projectile tumble. In other
words, the projectile is statically unstable. If the centre of pressure and centre of
gravity coincide, there will be no moment generated by a yaw disturbance and so
the projectile will be neutrally stable. Finally, if the centre of pressure is aft of
the centre of gravity, a yaw disturbance will generate a yawing moment tending
to remove the disturbance and so the projectile will then be statically stable.
There is clearly an important correspondence in the distance between the
centre of pressure and the centre of gravity and the static stability of the round.
This distance is called the static stability margin, or sometimes just the static
margin. It is positive for positive static stability, when the centre of pressure is
behind the centre of gravity, zero for neutral stability and negative for static
instability.
To minimise the longitudinal deceleration of the projectile in flight, it is
necessary to fly nose forwards. Fig. 4.23 shows the drag coefficient as function of
yaw angle for the 7.62 mm round mentioned earlier. If the round were to
tumble, the average drag would rise by a factor of about 10. Additionally, the
flight would become more variable and dispersion would increase. It is clear
therefore that some means must be found for stabilising the round to ensure that
the nose points essentially along the trajectory.
FIG. 4.22 (a) Cone on base – stable; (b) Cone on side – neutral; (c) Cone on apex – unstable
FIG. 4.23 Drag variation with yaw angle at M  2.5
8 (Yawing moment) CM  V2
d3
max
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92 Military Ballistics External Ballistics – Part I 93
The nature of the sideforce generated by yaw suggests one possibility. The local
sideforce is proportional to the rate of change of cross sectional area. Therefore, if
boat-tailing is used to reduce drag, it will make the instability worse. This is
because the static margin can be written:
Nose up moment about cg Static margin  – Total sideforce
The boat-tail, since its area is reducing, will generate a negative sideforce.
Hence, as shown in Fig. 4.24, the moment is increased while the total sideforce is
reduced, giving a more negative static margin and hence a further forward centre
of pressure and a greater instability. The remedy for this is to increase the body
cross section towards the rear. This will have exactly the reverse effect and so can
be used to stabilise the projectile.
The increase in area towards the base is known as a flare and so this is flare
stabilisation. It suffers from increased drag compared with a normal round but
may be used in special cases. For example at very high speeds when fins would
become too hot to be viable. The flared skirt is also seen in air gun pellets, which
otherwise do not have sufficient spin to be directionally stable.
Apart from the use of flares, which is not common, there are two standard
techniques of generating directional stability
1. By the use of fins.
2. By the use of spin.
These will be discussed separately below.
Fin stabilisation
As described earlier, it is only necessary to move the centre of pressure aft of
the centre of gravity to confer static directional stability on a projectile. This may
be done by providing a lifting surface aft of the centre of gravity of the round as
shown in Fig. 4.25.
In this Figure, the body lift is well ahead of the centre of gravity but is, as
explained below, rather small. The lift generated by the stabilising fins is behind
the centre of gravity and is relatively large. Taking moments about, for example,
the body nose apex, produces the position for the resultant of both lift forces. This
is the overall centre of pressure for the projectile with fins. If this is aft of the
centre of gravity, the projectile will be directionally statically stable.
Flat surfaces are far more efficient at generating lift than are bodies. This in
indicated in Fig. 4.26 in which the lift at a low subsonic speed from a fiat plate of
triangular form (a delta fin), a typical aircraft wing and a circular body with a
conical nose, all having the same plan area, are compared. The angle of incidence
used in this Figure is defined as the angle between the plane of the surface and
the upstream flow or flight direction. Note that both the fin and the wing give
much more lift per unit plan area at the same yaw angle than does the body.
The lift of a fin or wing varies with wing plan area and shape and with Mach
number. There is no simple general formula for the lift of wings and fins but Fig.
4.27 shows how the lift divided by Mach number squared varies for several delta
and rectangular planforms.
The progressive reduction in lifting effectiveness for Mach numbers above 1 is
FIG. 4.24 Effect of after-body on centre of pressure position
FIG. 4.25 Fin stabilisation
FIG. 4.26 Comparison of lift on wings and a body
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94 Military Ballistics External Ballistics – Part I 95
noteworthy compared with slender bodies, for which it increases slowly with
Mach number, as shown in Fig. 4.21.
Relatively small fins at the rear of projectiles therefore easily stabilise them, at
least up to moderate supersonic Mach numbers. Fin stabilised projectiles will
normally have a static margin of between 1 and 4 calibres, depending mainly on
the type of target to be attacked. In general hard targets require more stable
projectiles.
The main disadvantages of fin stabilisation are that room must be found for the
fins inside the barrel ie the body of the round must be sub-calibre (and be sabot
launched) or the fins must deploy after launch. The fins also increase the drag of
the projectile, typically by between 25% and 40%. However, the very low frontal
area does give the dart configuration good penetration.
Since the fins provide stability using the same mechanism as that which
produces instability in the body, ie from aerodynamic forces, the dart will remain
stable if it enters a dense medium, as long as it stays intact. This may be
contrasted with the behaviour of spin stabilised rounds which are discussed in
the next section.
Spin stabilisation
The basis of spin stabilisation is to be found in the properties of the gyroscope
and so these will be discussed first.
The Gyroscope
A gyroscope which is spinning very fast will have a large angular momentum
with respect to its spin axis. If it is assumed that the gyroscope has a moment of
inertia of A about its spin axis and it has an rotation rate of p, then its angular
momentum will be Ap. If an impulsive moment of mot is now applied at right
angle to the spin axis, then the resulting position of the net angular momentum
vector will have rotated by the small angle 88 from its original position as shown
in Fig. 4.28. Then   mt/Ap.
This shows that a gyroscope subjected to a moment acting at right angles to its
spin axis will actually rotate about an axis turned by 90 in the spin direction.
This motion, in a plane at right angles to the applied moment, is called precession. The rate of precession is determined by the ratio of the applied moment to
the angular momentum of the gyroscope.
The Use of Spin to Stabilise Projectiles
It has already been indicated that most projectile body shapes are aerodynamically directionally unstable. They can be made directionally stable by the use of
axial spin. The basic physical mechanism of this will now be examined.
Let us look at a spin stabilised projectile which is spinning clockwise as viewed
from the rear (as most modern spin stabilised projectiles do) and is aerodynamically statically unstable and suppose that it is subjected to a nose up yaw
disturbance in (say) the vertical plane. Because Cm is positive, the projectile will
experience a nose up moment attempting to increase the yaw angle. However, if
it is spinning fast enough, this moment, by the argument advanced above, will
cause the nose of the projectile to move to the right as shown in Fig. 4.29.
FIG. 4.27 Relative lifting efficiency of different wing planforms
FIG. 4.28 The gyroscope
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96 Military Ballistics External Ballistics – Part I 97
The projectile yaw angle is now the resultant of the original yaw, ie in the
vertical plane, and the consequent yaw (in the horizontal plane). The net yaw has
now therefore moved round in a clockwise direction when viewed from the rear of
the projectile. The argument can now be repeated and the consequence of the
original yaw disturbance is that the nose of the projectile describes a spiral of
reducing amplitude. This is shown in Fig. 4.30.
The main motion here is precession but it may be noticed that there is a small
higher frequency motion superimposed. This is normally present in gyroscopic
motion and is known as nutation. The amplitude and frequency of this motion, in
the case of a projectile in flight, are determined by the dynamic stability of the
projectile.
If the round has too little spin, it will tumble in flight. This reduces its range
and accuracy. A loud whirring noise is characteristic of an under-stabilised shell
in flight.
There are, in fact both upper and lower bounds to the amount of spin which can
be employed in the stabilisation of projectiles. Examination of the precession of a
gyroscope which is subjected to an external moment shows that, as the angular
momentum increases, the rate of precession reduces. This means that the response to a given disturbance is reduced and therefore it takes longer for the yaw
disturbance to be removed by the precessive motion. This means that too much
spin produces a round flying at larger yaw angles than are necessary and this
reduces range. Eventually, as the spin rate is made greater and greater, the
response of the round to a yaw disturbance is entirely inhibited and the round
flies as in the lower diagram of Fig. 4.31. A round in this condition is called
superstable.
The necessary spin rate for stability of small diameter rounds can be remarkably high. Table 4.4 gives the spin rates required for a variety of spin stabilised
FIG. 4.29 Shell gyroscopic stability
FIG. 4.30 Shell motion following a disturbance in yaw
FIG. 4.31 What is required for gyroscopic stability
TABLE 4.4
Spin Rates for Normal Stability for Different Calibre
Projectiles
Projectile Calibre (mm) Spin rate rev/min
Ll5 155 16500
MI 105 25600
M80 7.62 167000
CB10 5.56 334000
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98 Military Ballistics External Ballistics – Part I 99
projectiles at a constant speed corresponding to launch at sea level at a Mach
number of 2.5, ie 850 m/s.
It is also found experimentally that it becomes increasingly difficult to spin
stabilise projectiles as they become longer. An empirical length to diameter limit
of 7:1 is normally accepted as the maximum for spin stabilisation.
Finally, spin stabilised rounds entering a high density medium, such as body
tissue, will, if they remain undeformed, become directionally unstable and will
tumble.
Trajectory Prediction
Definition
The trajectory of a projectile is defined as the path taken by its centre of
gravity.
Gunnery Angles
The initial direction of the trajectory at the muzzle of a gun is specified in terms
of the following angles:
1. The angle of sight (S). This is the vertical acute angle measured from the
horizontal plane through the weapon to the line of sight
2. The angle of projection (P). This is the vertical acute angle measured from the
line of sight to the line of departure.
3. The angle of departure (D). This is the vertical acute angle measured from
the horizontal plane through the weapon to the line of departure.
Because the direction of the axes of the bore at the muzzle and the breech may
differ, the following angles are also used:
4. The tangent elevation (TE). This is the vertical angle measured from the line
of sight to the axis of the bore at the breech.
5. The quadrant elevation (QE). This is the elevation at which the gun must be
laid under the prevailing conditions in order to achieve the desired range.
The gun barrel tends to droop or bend slightly under its own weight. Droop is
specified in terms of the acute angle measured from the breech axis to the muzzle
axis. In addition, as the projectile moves along the barrel, the stresses produced
cause the barrel to ‘whip’ with the results that the initial direction of the
trajectory may deviate slightly from the axis of the bore at the muzzle before
firing.
This effect is called gun jump and is specified in terms of two components:
6. The jump J, which is the vertical acute angle measured from the muzzle axis
before firing to the line of departure
7. The throw-off, which is the acute angle between the projections of the muzzle
axis before firing and the line of departure in the horizontal plane.
These angles are indicated in Fig. 4.32.
The In-vacuo Model
Although air resistance has a very large effect on the trajectory of a projectile,
as discussed earlier, it is the gravitational attraction of the earth which causes
the characteristic curvature of the trajectory in the vertical plane. For this
reason, it is a useful preliminary to consider the form of the trajectory of a
projectile which is subjected only to the effects of gravity ie one with no drag,
though its predictions are, in general, highly inaccurate for all but the shortest
range trajectories and for the lowest drag projectiles.
The path of a projectile in a vacuum and in a uniform gravitational field for a
horizontal flat earth, is easily predicted and may be used to illustrate the effects
of aerodynamic forces by comparison with real trajectories, since, in a vacuum
these forces are absent.
The important in-vacuo results may be summarised as:
1. The trajectory is symmetrical about the vertical line through the apex of the
trajectory and is parabolic in shape.
2. The whole trajectory lies in the vertical plane containing the line of
departure.
3. The range depends on the muzzle velocity and the angle of departure
4. The range increases as the angle of departure increases, reaching a maximum at 45 and reducing to zero at 90 where the maximum height is reached
and the time of flight is a maximum.
5. The angle of arrival equals the angle of departure and the velocity at the
point of impact equals the muzzle velocity. The velocity is a minimum at the
apex.
6. The trajectory parameters do not depend on the nature of the projectile
because aerodynamic forces are absent.
FIG. 4.32 Gunnery angles
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100 Military Ballistics External Ballistics – Part I 101
The Point Mass Model
It has already been noted that, particularly for high speed projectiles, the air
resistance has a profound effect on the range performance of a gun. The simplest
useful trajectory prediction technique, known as the Point Mass Model, includes
both the zero yaw drag force and the projectile weight. Unfortunately, this model
is, in general, not analytically soluble and numerical integration must be used to
generate the predicted trajectory. The results differ from the in-vacuo model in
the following way:
1. The trajectory is not symmetrical about the vertex which is closer to the point
of impact than to the gun.
2. The vertex height is less than its in-vacuo value, though the differences are
usually quite small.
3. The angle of descent is greater than the angle of departure.
4. The impact velocity is less than the muzzle velocity because the projectile
does work to overcome its aerodynamic drag.
This is illustrated in Fig. 4.2. Table 4.5 shows how the presence of the atmosphere modifies the maximum range predicted by the in-vacuo model.
Comparative trajectories are illustrated for a range of QEs in Fig. 4.33.
When used as a fire control model, the Point Mass Model also incorporates
meteorological effects and drift caused by the earth’s rotation (the Coriolis effect).
The Point Mass Model has the advantages of requiring only a small amount of
data and of being easily solved by computer. In general, it is adequate for fin
stabilised projectiles and is used in the UK to obtain fire control data for these
projectiles. However, whilst this model can be used to represent the ranging
performance of spin stabilised shells by modifying the aerodynamic drag characteristics, this can only be achieved by introducing complex mathematical
procedures.
The main disadvantage of the Point Mass Model is that it makes no allowance
for equilibrium yaw and its effects. These will be described in the next section.
The Modified Point Mass Model
Equilibrium Yaw
In order to achieve maximum range and repeatable trajectories, it is necessary
for a projectile to fly with its nose following the curvature of the trajectory flight
path as shown in Fig. 4.31. This in turn, requires the projectile to rotate in the
vertical plane whilst in flight. If the projectile is fin stabilised, this will happen
automatically. For a spin stabilised projectile, rotation in the vertical plane will
be produced by a moment in the lateral plane, to satisfy the precessional behaviour of the round. For a round having clockwise rotation viewed from the rear,
this moment will be in the sense of moving the nose to the right of the trajectory
when viewed from above. For a properly stabilised spinning projectile, the
moment is generated by the round flying with its body axis inclined at an angle to
the local vertical plane of the trajectory. The angle between the body axis and the
vertical plane is known as equilibrium yaw. It is shown diagrammatically in Fig.
4.34. The progressive nose down rotation of the shell in flight is known as ‘slow
precession’ .
Equilibrium yaw does several things:
1. It increases the drag of the projectile in flight.
2. It produces a side force on the projectile, to the right for clockwise spin from
the rear.
3. It causes a component of flow across the projectile from left to right when
viewed down range.
The first of these gives a reduction in range which becomes a function of
quadrant elevation (QE) and muzzle velocity (MV).
The second gives a lateral acceleration to the projectile which causes it to drift
FIG. 4.33 Variation of trajectories with launch angle (QE)
TABLE 4.5
The Effect of Air Resistance on Range
Type of weapon Muzzle velocity Maximum range km
m/s
In-vacuo In air
300 mm Mortar 396 16 11
155 mm FH70 810 67 24
7.62mm SLR 840 72 4
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102 Military Ballistics External Ballistics – Part I 103
to the right when viewed down-range, the amount of drift depending on the
magnitude of the equilibrium yaw and the time of flight. For a long range shell
fired at high angle, it can amount to several hundred metres.
At very high angle of departure, the nose of the projectile may fail to follow the
trajectory at the vertex, with the unfortunate result that the projectile may fall
base first, at a reduced range and with a marked drift to the left.
The third effect is due to the combination of sidefiow and spin. The inviscid,
incompressible flow about a spinning circular cylinder is shown in Fig. 4.35. The
increase in velocity above the cylinder indicated by the closeness of the streamlines is reflected by a reduction in pressure. Similarly, there is a reduction in
velocity and hence an increased pressure beneath the cylinder. The net effect of
these pressure changes is to produce a lift on the cylinder which is known as
Magnus Lift. By the same argument there is a Magnus Lift developed on a
spinning shell which has a yaw angle. This lift causes firstly a small increase in
range and secondly, because the longitudinal line of action of the Magnus Lift
will not, in general, pass through the centre of gravity of the projectile, a moment
– the Magnus Moment.
The Magnus Moment tends to be very small but it can have a marked influence
on the flight of a projectile. This is because, like the stabilising gyroscopic effect,
the rotation caused is 90 out of phase with the disturbance. Consider the case in
which the Magnus Moment is nose up. If the yaw angle increased in flight due to
a disturbance, gyroscopic effects would tend to rotate the nose of the projectile
down. The subsequent precessive motion would lead to a progressive reduction in
the disturbance, as explained previously. However, the Magnus Moment will now
generate an incremental nose up moment which is opposed to the gyroscopic
motion and this leads to a reduction in projectile stability.
It is possible for the Magnus force to become large, for example when an antiaircraft gun is fired almost vertically. The yaw at the top of the trajectory then
becomes much greater and so the Magnus effect becomes more important and the
shell may, in fact, drift to the left.
The consequences of equilibrium yaw are therefore seen to be considerable. Its
inclusion in a trajectory model significantly enhances the predictive capabilities
of the model. A trajectory model including equilibrium yaw is known as a
Modified Point Mass Model. Again, numerical solutions using computers are
required. For conventional artillery shells, this model can provide extremely
accurate end point data and it will form the basis for the production of artillery
fire control data for the foreseeable future.
The Six Degrees of Freedom Model
A rigid body moving in free air has six degrees of freedom. The degrees of
freedom are the three components of translational motion of the projectile (up
and down, fore and aft and sideways) and the components of rotation about these
three directions. This is the ultimate trajectory model in that, at least in principle, it describes in detail the complex dynamics of a projectile. However, owing
to the indeterminacy of some of the initial conditions, and the aerodynamic data
which is required to be input, the model may not give significantly better end
point results than the Modified Point Mass Model. In addition, the model requires
considerably more computing power for its solution than does the Modified Point
Mass Model. Thus it is not suitable for routine fire control calculations.
Nevertheless, it does provide a powerful tool for the ammunition designer.
A comparison of the input data and approximate computing requirements of
the main trajectory models is shown in Table 4.6.
FIG. 4.34 Equilibrium yaw
FIG. 4.35 Flow about a spinning cylinder
TABLE 4.6
Input Requirements of Trajectory Models and Comparative
Computing Requirements
MODEL PM MPM 6DOF
Shell data 2 3 4
Aerodynamic data 1 5 8
Launch data 4 5 9
Core store 1 2 3
CPU time 1 2.5 250–2500
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104 Military Ballistics External Ballistics – Part I 105
Non-standard Trajectories
For fire control purposes, trajectories are computed for agreed standard conditions which may be considered to be the average of actual conditions. Small
changes from the standard conditions are known as variations and corrections for
these are made to tangent elevation, line or fuze setting. The variables with
which the field gunner is most concerned are range to graze, time of flight, line
and vertex height.
The standard conditions used in British field artillery are as follows:
1. Angle of sight  zero.
2. Muzzle velocity as laid down for the equipment and charge used.
3. Weight and shape of projectile as laid down.
4. Winds  zero.
5. Charge temperature 21°C.
6. Atmospheric data as ICAO.
7. Gravitational acceleration at sea level at 45 latitude  9.80665 m/s2
.
Corrections for non-standard conditions are applied as follows:
1. Range variations:
1.1. Range correction of the moment which allows for:
1.1.1. Head or tail winds (known as range winds).
1.1.2. Air temperature and density (note that if mean values are used
they are referred to as ballistic air temperature and ballistic air
density respectively).
1.1.3. Charge temperature.
1.2. Non-standard projectile correction which allows for a non-standard
weight or nature of the projectile, including fuze.
1.3. Rotation of the earth correction (see next section).
1.4. Effect of angle of sight.
1.5. Muzzle velocity correction.
2. Rigidity of the trajectory:
Small variations in the angle of sight are dealt with by assuming ‘Rigidity
of the Trajectory’. This assumes that, in order to hit a target which is above or
below the horizontal plane through the centre of the muzzle at the breech, the
trajectory which passes through the point on the horizontal plane which is
vertically above or below the actual target can be swung up or down to allow
for the height displacement; the concept of trajectory rigidity is only useful
for low-angle trajectories.
It is acceptable for anti-tank gunnery but errors arise in field gunnery.
These are:
2.1. The slant range is greater with a non-zero angle of sight.
2.2. The increased angle of projection gives a lower horizontal component of
velocity.
2.3. In the higher trajectory, the air density is reduced.
The first two effects lead to under-rallging as shown in Fig. 4.36. However
the third effect compensates for this to some extent and, for high velocity
guns at short ranges, this latter effect may even dominate causing overranging.
Variations for these reasons are known as ‘Non-Rigidity Variations’
3. Line corrections or corrections to the azimuthal direction of fire:
3.1. Line correction of the moment which counteracts the effect of cross
winds.
3.2. Drift correction due to equilibrium yaw.
3.3. Line correction due to rotation of the earth.
If a variation due to non-standard conditions is to the right of the line of
fire, as in the case of drift, then the correction required is taken as the same
angle to the left. Corrections to line are given in terms of mils:
6400 mils  360 degrees  6283 milli-radians
Hence a dispersion in line of 1 mil produces a lateral deviation of approximately R metres at a range of R kilometres.
4. Fuze setting corrections:
Time fuze settings must be related to the time of flight under standard
conditions. When conditions are not standard, the shell will not follow the
standard trajectory and will therefore have a different time of flight.
Effects of Rotation of the Earth
When considering long ranges and long times of flight, the effects of the
rotation of the earth must be taken into account. The rotational speed of the earth
at the equator is about 450 m/s, some 1600 km/h. At first sight, this appears to be
of no consequence since the gun, target and atmosphere are all travelling
together at more or less the same speed. However, the motion of the earth’s
surface is not in a straight line, so that, during the period that the projectile is in
flight, the target may be carried sideways or up or down by the earth to which it is
attached. These target motions appear as an error in the impact point of the
projectile; lateral deviations due to this effect are also called drift.
Fortunately, the rotation rate and dimensions of the earth are fixed, so that for
any known gun and target locations on the earth’s surface, it is possible to apply
corrections to the gun bearing and elevation to ensure that the projectile meets
the target. A rigorous treatment of this subject is beyond the scope of this book.
However, it is not difficult to get an adequate grasp of the principles involved.
There are three effects which can be distinguished: the first two primarily
affect the range of the projectile, the third mainly affects the bearing to the
target.
The first effect is most apparent when a projectile is fired vertically upwards at
the equator. During its period of flight the earth rotates beneath it so that the
projectile lands to the west of the launch point. This is known as ‘projectile lag’.
The second effect is indicated in Fig. 4.37. A gun at G is firing due east on the
equator at a target at T. Whilst the shell is in flight, the earth rotates beneath
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106 Military Ballistics External Ballistics – Part I 107
it so that when the shell arrives at the point P, which is where the target
originally was in relation to the gun position, the target has moved to T. Since
the shell is now above the earth’s surface, it will continue in flight and it will
therefore over-range, landing beyond T . It can be shown that the actual distance
by which the shell will over-range depends on the latitude of the gun, the
compass heading of, and range to, the target, the angle of descent at the target
and the time of flight.
These two effects are greatest at the equator and lessen towards the poles. They
have an opposing effect upon range; the first effect dominates at high gun
elevations and the second at low gun elevations. They are equal and opposite at a
gun elevation of about 60.
The third effect is shown in Fig. 4.38. In this case we are looking at a gun at G
firing due north from the equator at a target at T. This time whilst the shell is in
flight the gun moves to G1 and the target, which is at a more northern latitude
and hence at a smaller radius, moves a lesser distance to T1. The shell will
therefore land to the east of the target. The same argument can be applied to a
gun firing say west from the point labelled T in this Figure. This time the shell
will land north of the target. We can now generalise that, for guns fired in the
northern hemisphere, there will be an apparent drift to the right of the target by
an amount which will depend on the latitude of the gun, the range to the target
and the time of flight. For guns in the southern hemisphere, the drift will be to
the left.
As a rough guide to the magnitude of these effects, a 155 mm gun firing a high
explosive shell due east at a latitude of 52, on a nominal range of 24 km will
over-range by about 50 m and drift right by about 100 m. Over ranges of less than
5 km, the effects are generally much less than the shot to shot variations of the
point of impact, whilst for small arms the effects of the rotation of the earth are so
small that they are always ignored.
Cross-wind Effects
The effects of a cross-wind on the drift of a projectile are not quite as simple as
might be supposed. For example, it seems intuitively obvious that a projectile
launched into a cross-wind of 10m/s and flying for 50s will experience a down
wind drift of 500m. This is quite erroneous.
The actual situation is depicted in Fig. 4.39. Because the projectile has been
directionally stabilised, it turns into the relative wind. Its subsequent flight path
depends on how large is its drag. If, for example, its drag was zero, it would head
up-wind at exactly the same rate as the wind would carry it sideways because it is
turned exactly into the relative wind direction. It would therefore reach the
target as though there were no cross-wind.
Indeed, if the projectile were internally propelled like an artillery rocket or a
rocket assisted shell, such that it effectively had negative drag, it would pass
upwind of the target. A measure of the magnitude of the drag is the time of flight.
The in vacuo time of flight corresponds to zero drag, a shorter time of flight than
this corresponds to negative drag ie a net thrust, and a longer time of flight
corresponds to actual drag. Therefore it is the difference between the real time of
flight and the in vacuo time of flight that determines the drift due to a cross-wind.
Rockets
In this section we will look at the basic characteristics of a free flight rocket;
that is, a rocket which has all its guidance given to it by the launcher. Tactically,
we can distinguish two types of rocket, namely artillery rockets and rocket
FIG. 4.36 Rigidity of trajectory
FIG. 4.37 Effects of rotation of the earth (1)
FIG. 4.38 Effects of rotation of the earth (2)
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108 Military Ballistics External Ballistics – Part I 109
assisted projectiles (RAPs). Artillery rockets are launched at low speed from a
simple tube and contain a rocket motor which accelerates them up to speed.
Rocket assisted projectiles are fundamentally artillery shells which are launched
from normal gun barrels but which contain a small rocket motor to extend range.
The basic principles are however identical.
In each case, the motor consists of propellant, a combustion chamber and a
nozzle. The chamber is analogous to the chamber of a gun in which the propellant
is burned. The nozzle accelerates the products of combustion from the rear of the
motor. These gases emerge with a certain momentum, and according to Newton’s
Second Law of Motion, the rate of change of momentum of the gas leaving the
nozzle is equal to the force on the gas stream. This force, by Newton’s Third Law
of Motion produces an equal and opposite reaction on the nozzle. It is this latter
reaction, the thrust of the motor, which drives the rocket forward.
Rockets may have liquid or solid fuels. Generally speaking, liquid fuel rockets
have a higher energy content, while the solid fuel rockets are simpler to produce
and to handle. The relative merits of liquid and solid propellants are discussed in
Ref 3. In a vacuum, the maximum velocity of the rocket depends on the speed of
the emerging gases and on the ratio of the weight of the rocket before firing to the
weight when the propellant is all burnt. This ratio is called the ‘mass-ratio’. The
acceleration at any moment depends on the rate at which the fuel is being
consumed, the speed of emerging gases, and on the remaining weight of the
rocket. The maximum speed for the rocket will be reached when the fuel is
completely burnt (the ‘all burnt’ velocity).
When the rocket is fired in the atmosphere, these statements are only approximately true and external forces must be considered as well.
Artillery Rocket Accuracy
The principal disadvantage of free flight rockets is their relatively poor accuracy and consistency when compared with gun-launched ammunition. The main
sources of error are due to wind effects, thrust misalignment and in variations in
velocity at all burnt. Because of their low launch velocity, free-flight rockets are
significantly affected by cross-winds in the way discussed above. This may be
alleviated by using delayed opening fins. In this case the rocket is launched with
its fins closed: it is therefore directionally unstable and, in the presence of a crosswind, turns away from it. After a short period of time, the fins open and the
rocket, which is now directionally stable, turns into the wind. As the motor has
been burning during the whole of this manoeuvre the flight path looks like that
sketched in Fig. 4.40 and the gross effects of cross-wind have been removed.
Problems of thrust and centre of gravity misalignment can be reduced by
imparting a small amount of spin to the rocket at launch.
This ends the non-mathematical treatment of external ballistics. For those
readers who do not wish to go deeper, it is suggested that they miss out Part II of
this chapter and move on to Chapter 5. Anyone wishing to study the subject in
more depth should read Part II.
Self Test Questions
QUESTION 1 Calculate the Mach number of a projectile flying at 860 m/s at
sea level and at 5000 m.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 2 State how static pressure and velocity are qualitatively related
in a fluid flow by Bernouilli’s equation. Does this description
change at supersonic speeds?
Answer………………………………………………………………………………..
F ………………………………………………………………………………………….. IG. 4.39 Effects of cross-wind on drift
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110 Military Ballistics External Ballistics – Part I 111
QUESTION 3 What is meant by drag? Why are drag coefficients used for
projectiles?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 4 State the different kinds of drag experienced by a projectile in
flight.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 5 What is meant by boat-tailing and by base bleed? What are the
disadvantages of each?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 6 Explain what is meant by static stability as applied to a projectile in flight.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 7 Describe how fins can be used for projectile stabilisation and
define static margin.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 8 Why are muzzle velocities in the region of the velocity of sound
avoided by the designer of projectiles?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 9 Spin stabilised projectiles normally have a quantity known as
the gyroscopic stability coefficient within a limited range of
values. Give a physical reason for this and specify the likely upper and lower limits.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 10 Why is meteorological data required for artillery fire control?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 11 Why is it necessary to know your position on the earth’s surface
in order to accurately predict long range artillery fire?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 12 Give a brief account of the differences in using point mass,
modified point mass and six degree of freedom models for trajectory prediction.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
FIG. 4.40 Effect of delayed opening fins for artillery rocket
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112
External Ballistics – Part II 113
External Ballistics – Part II
Concept
In this section of external ballistics, an attempt is made to provide the reader
with a deeper and more theoretical study of some of the basic concepts presented
in Part I of this chapter. It is now necessary to refine our views by examining in
more detail the various aspects covered in a largely conceptual way earlier. A
study of this material is vital if a fuller understanding of the in-flight behaviour
of projectiles is required.
The lCAO Atmosphere
The variations of temperature, pressure and density within the atmosphere
can be determined from three equations. The first is the assumed variation of
temperature (T) in degrees Kelvin with pressure altitude (H) in metres:
0  H  11000
T  288.15  6.5  10–3 .
H
11000  H  20000
T  216.65
The second is the condition for vertical equilibrium:
in which p is the static pressure,  is the air density and g is gravitational
acceleration.
The third is the equation of state:
p  RT
in which R is the gas constant for air and is equal to 287.053 Nm/kg K.
If gravitational acceleration is assumed to be invariant with height, these
equations produce the following relationships between pressure, density and
height:
0  H  11000
p  (8.9620  0.20216  10–3•H)5.2559
r  (1.0488  23.659  10–6•H)4.2559
11000  H  20000
p  128240 exp( –0.15769  10–3•H)
r  2.0621 exp( –0.15769  10–3•H)
The density of air also varies with the amount of water vapour which it
contains, its humidity. Water vapour is less dense than air and therefore air
containing water vapour is less dense than dry air. The amount of water vapour
which air can hold varies substantially with the air temperature. Humidity is
usually quoted in terms of relative humidity which is expressed as a percentage
of water vapour present in the air compared with the maximum capacity of the
air at the same temperature. Zero percentage relative humidity corresponds to
completely dry air; a relative humidity of 100 percent indicates that the air
contains all the water vapour which it is capable of holding. It is then said to be
saturated.
The effect is however, normally very small. For example, air having a relative
humidity of 70%, a static pressure of 1010 mb and a temperature of 20°C will
have its density reduced by less than 1% when compared with dry air at the same
conditions of pressure and temperature. The data given in Table I of Part I of this
Chapter is for completely dry air.
In reality, the magnitude of gravitational acceleration falls with height above
the earth’s surface in accordance with Newton’s Law of Gravitational Attraction.
The assumption of a constant value for g therefore gives values for height which
are not exactly equal to the geometric height above sea-level. These heights are
known as geo-potential altitudes. The geometrical height at which the calculated
conditions of Table I apply can be found from the equation:
in which h is the geometric height above sea-level, H is the geo-potential altitude
given in Table I and r is the radius of the earth (6.356766  106
m).
The speed of sound (a) may be calculated from:
in which  is the ratio of the specific heats and may be taken to be 1.40 here.
If required, the dynamic viscosity is calculable from:
Fluid Motion
Basic Equations of Fluid Flow
The word fluid is used to describe both liquids and gases. It may be shown that
the equation relating static pressure and local velocity in a fluid is:
dp   rg dH
r
h   H
r  H
dp
 u du  0

1.458  10–6T3/2 kg
  T  110.4 ms
a   (gRT)
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114 Military Ballistics External Ballistics – Part II 115
For the case of an isentropic process, ie a process with no losses, which is
approximately valid for many fluid flow situations, pressure and density are
related by:
This leads to the relationship, for steady flow in which the effects of viscosity are
small,
At well subsonic speeds, the air density p, remains constant and Bernoulli’s
equation follows:
Dynamic Similarity
For a homogeneous fluid in motion, there are four different forces which can be
identified. These are pressure forces, viscous forces, elastic forces and inertial
forces. Pressure forces have been discussed above.
Experimentally, it may be shown that the viscous shear stress acting on a body
in motion is proportional to the coefficient of (dynamic) viscosity µ and to the
velocity gradient in the fluid. In dimensional terms, the velocity gradient is the
ratio of speed to length and this will therefore be proportional to V/L. Viscous
forces are therefore proportional to µ V/L times L2.
Elastic forces are related to the compressibility of the fluid and it has already
been shown that the behaviour of a fluid in terms of dimensionless quantities like
drag coefficient can be expressed as a function of the quantity Mach number. It
can be shown that Mach number represents the ratio of inertial to elastic forces
in a fluid.
Inertial forces are the forces which are involved in the acceleration of the fluid.
These are proportional to the mass of the fluid times its acceleration. In dimensional terms, the mass of the fluid is proportional to density times volume ie  L3
and acceleration (which is dV/dt ie VdV/dL) to V2/L. The ratio of inertial to elastic
forces in a fluid therefore can be written:
The ratio is known as Reynolds number and is written Re. The length is chosen to
be a characteristic length of the fiowfield or of the object within it.
The identification of these dimensionless parameters greatly simplifies the
collection, reduction and presentation of data. In the case of the drag of a given
shape for example, instead of the multi-variable equation:
Drag  f (p, V, size, µ, a)
in which a is the speed of sound in the fluid, we can write, much more concisely:
CD  f (Re, M)
the results of which, for a given shape, could be presented on a single sheet of
paper.
It follows that tests at the same Mach number and Reynolds number will
therefore yield the same drag coefficient (though not necessarily the same drag).
This equivalence is called dynamic similarity.
This enables small scale tests to provide data which can be applied full scale.
Where Mach number is less than about 0.5, the effects of compressibility are
negligible and only Reynolds number is significant. The application of dynamic
similarity is demonstrated here by comparing the results of two apparently
different tests reported by Schapiro in (1).
In this experiment, a small plastic ball with a diameter of 6 mm was dropped
into water and a large helium filled balloon with a diameter of 1 m was allowed to
rise in air. By measuring the steady rate of descent in one and ascent in the other,
the drag and hence the drag coefficient could be determined.
By weighting the balloon, the rate of ascent could be altered so that the
Reynolds numbers for the two tests could be made to be very similar. Therefore,
although these two tests appear very different, one in a liquid with a descending
small sphere and the other with an ascending large sphere in a gas, they are in
fact dynamically almost identical. The value of drag coefficient is therefore also
virtually identical and either test could be used to provide data for the other.
Dynamic similarity is the key to how we are able to predict, from measurements made in the wind tunnel on a model projectile such as a rocket, the forces
which the real rocket will actually experience.
Aerodynamic Drag
Skin Friction
In practice it is found that there are two different forms of boundary layer
which can exist – laminar and turbulent. The laminar boundary layer is characterised by flow steadiness and therefore a lack of mixing between adjacent layers
and a slow growth in velocity with distance from the surface. A turbulent
boundary layer is unsteady with much mixing between adjacent layers and the
mean velocity in it increases rapidly with distance from the surface.
Fig. 4.41 shows the local velocity within the boundary layer divided by the
velocity just outside the boundary layer plotted against the distance from the
surface divided by the boundary layer thickness. For the turbulent boundary
layer, in which the flow is unsteady, the mean local velocity has been used. In
practice, the laminar boundary layer is physically much thinner than an equivalent turbulent boundary layer and the skin friction is smaller than it would be if
the boundary layer were turbulent. The thickness of a laminar boundary layer
(lam) depends on the local Reynolds number Rex
in which the characteristic
p  constant

1
p  V2  constant
2
 p 1
 V2  constant
  1  2
L3
V2 L VL VL
ie or
L VL2
 
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116 Military Ballistics External Ballistics – Part II 117
length (x) is taken to be the downstream distance along the plate from the
leading edge. It is given by:
For a turbulent boundary layer, the growth rate is higher and is given approximately by:
The critical criterion which determines whether the boundary layer will be
laminar or turbulent, is, for smooth surfaces, the Reynolds number of the flow
based on streamwise length Rex
. For Reynolds numbers less than about 3  105, ie
for sub-critical Reynolds numbers, the boundary layer will be laminar. For
Reynolds numbers greater than about 5  105, ie for super-critical Reynolds
numbers, the boundary layer becomes turbulent. There is a transition region
between the two types of boundary layer. The growth of the boundary layer along
a flat plate is sketched in Fig. 4.42. Transition at a lower than usual Reynolds
number can be produced by roughening the surface of the plate.
For ballistic purposes we are mostly concerned with turbulent boundary layers
and two examples of the estimation of boundary layer thickness are given below.
For a 7.62 mm calibre, 28 mm long bullet, having a muzzle velocity of 840
m/s,
at sea level. A turbulent boundary layer is indicated with a boundary
layer thickness of 0.60 mm at the bullet base.
For a 9 m long rocket travelling at 600 m/s at an altitude of 5.5 km
A well established boundary layer is indicated with a thickness of 70 mm
at the rocket base.
A reasonable approximation to the skin friction drag of projectiles can be
obtained by ‘unrolling’ the surface of the projectile to give an equivalent flat
plate. The skin friction drag coefficient for turbulent boundary layers based on
the wetted area Sw can then be approximated by:
Pressure Drag
The condition of the boundary layer can have more effect than is evident from
skin friction alone however. A dramatic effect of the state of the boundary layer
and hence of the effect of Reynolds number can be seen in the special case of a
circular cylinder.
By application of Bernoulli’s equation, it might be supposed that in the wake,
where the velocity is low, the static pressure would be high. However, in order to
reach the wake from the free stream it is necessary to cross the separated
boundary layer in which viscous forces are important. This changes the constant
in the equation so that, in the wake, low velocities and low pressures exist
together. To a first order, the average pressure in the wake is the same as that at
FIG. 4.41 Boundary layer velocity profiles
FIG. 4.42 The boundary layer on a flat plate
lam
 5.2 Rex
0.5
x
840  0.028
Re   1.62  106
1.46  10–5
600  9
Re   2.4  106
2.31  10–5
2  skin friction 0.455
CF 
 
 V2
SW
(logRe)
2.58(1  0.2M2
)
0.467
turb
 0.37 Rex
0.2
x


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118 Military Ballistics External Ballistics – Part II 119
the separation point. This is normally quite close to the point of minimum
pressure on the body and so the wake pressure is usually quite low.
The mechanism of boundary layer separation may be grasped by considering
the flow over a circular cylinder. In the absence of viscosity, the flow pattern
would look like Fig. 4.43a. By application of the continuity equation and
Bernoulli’s equation, the pressure may be deduced to rise at the stagnation point
(indicated by S on the flow pattern) and then fall as the flow accelerates past the
cylinder until the position of maximum thickness is reached, at which point the
pressure starts to rise again. This is sketched in Fig. 4.43b.
In a viscous fluid, there would be a frictional force in the boundary layer acting
to slow it down. Since this is velocity dependent (it disappears as the velocity
tends to zero) it could not bring the flow to rest whilst the pressure is falling in
the direction of motion. However, beyond the position of maximum thickness, the
pressure starts to rise and the pressure forces and frictional forces are both acting
to slow the flow down. The result is sketched in Fig. 4.44.
As it is the fluid within the boundary layer immediately adjacent to the body
which is moving most slowly, it is this fluid which first comes to rest. At points
which are further downstream, the flow direction is actually reversed and examination of the resulting velocity profile indicates that there is a point of zero
velocity some distance above the solid boundary. The line joining these points of
zero velocity is sometimes called the separation streamline since it separates the
region of flow in which the fluid is continuing in the original flow direction, and
the region of flow in which the fluid direction has been reversed ie the wake.
(Strictly speaking this line is not a actually a streamline because flow crosses it.)
The origin of the separation streamline on the body is called the separation point;
it is the point at which the boundary layer separates from the surface.
The result of this is to drastically modify the flow pattern as shown in Figure
4.45a, which shows the flow past a circular cylinder in a real fluid, and the
resulting pressure distribution which is sketched in Fig. 4.45b.
The position of the separation point on a circular cylinder depends on which
type of boundary layer exists near to the potential separation point. The turbulent boundary layer, due to its larger velocity near to the surface and the mixing
which takes place within it, is more difficult for the fluid forces to slow down. It
therefore has a delayed separation point compared with the laminar layer and
therefore a smaller wake in which the pressure is higher giving a significantly
lower drag. The respective flow patterns are shown in Fig. 4.46.
The same type of behaviour is shown also by a sphere. The change in drag due
to the change in boundary layer separation point can be clearly seen in Figure
4.47 which is a plot of drag coefficient against Reynolds number based on
diameter for a sphere. If the flight Reynolds number is just below that for which
turbulent boundary layer separation would begin, surface roughness can be used
to promote early transition and hence a significant reduction in pressure drag.
This effect is most readily observed when the flight of a golf ball is considered.
The flow is modified as shown in Fig. 4.48.
Although the skin friction is increased for the roughened surface, the reduced
size of the wake and the increased pressure within it both reduce the pressure
drag of the sphere which predominates. The drag can drop to as little as one fifth
of the drag on a smooth sphere at the same speed. This is one of the two reasons
FIG. 4.43 (a) Inviscid flow about circular cylinder; (b) Pressure distribution
FIG. 4.44 Flow in the neighbourhood of separation
FIG. 4.45 (a) Real flow about circular cylinder; (b) Pressure distribution
13569_int.indd 118-119 11/15/11 11:20:49 AM
120 Military Ballistics External Ballistics – Part II 121
why golf balls are dimpled. (The second reason is that the Magnus effect is
enhanced for a body with a rough surface. The golf ball actually flies further in
air than it would in a vacuum for the same launch conditions. This is because the
Magnus Lift extends the range of the ball by more than the air resistance reduces
it.)
The general result that delaying boundary layer separation reduces drag is
only valid at subsonic speeds however. Since, at supersonic speeds, the pressure
that is reached is proportional to the angle the flow has turned through, delaying
separation at the rear of a body produces a lower wake under-pressure and hence
a higher drag. This is illustrated diagrammatically in Fig. 4.49.
In this case, early transition would therefore lead to a drag increase.
Shock Waves
A more detailed examination of the flow through shock waves is facilitated by
Fig. 4.50. This Figure is in the body-fixed reference frame and the approaching
air has a velocity of v1. This can be expressed in terms of the components parallel
to, ie vp and normal to, ie vn1, the oblique shock. The velocity component parallel
to the shock is unaffected by its presence and is therefore unchanged, whereas the
normal component experiences a pressure rise and slows down to give a reduced
normal component behind the shock of n2. Reconstitution of the total velocity v2
shows that the flow is now at a lower velocity and inclined at an angle  to the
initial flow direction. In 2-dimensions, this angle  is exactly the angle of the
surface slope which generates the shock wave and is what enables the supersonic
flow to turn and flow smoothly along the surface. Because the flow is now parallel
to the surface, the pressure on the surface is constant. It is also, for small turning
angles, proportional to the slope of the surface, both for positive and negative
surface slopes. It can be shown that the local surface pressure increment divided
by the dynamic pressure (a quantity known as the pressure coefficient Cp) is
related to the surface slope  by:
FIG. 4.46 Flow about a circular cylinder for laminar and turbulent separation
FIG. 4.47 Variation of drag coefficient with Reynolds number for a sphere
FIG. 4.48 The effect of surface roughness just below the critical Reynolds number
13569_int.indd 120-121 11/15/11 11:20:51 AM
122 Military Ballistics External Ballistics – Part II 123
In 3-dimensions, the flow turns through a smaller angle because it can expand
after it passes through the shock as shown in Fig. 4.51. In this case there is no
simple analytical relationship between surface slope and surface pressure.
The maximum turning angle in 2 and 3 dimensions for supersonic downstream
flow to be maintained is shown in Fig. 4.52.
It may be shown that, in air, the speed of propagation of a shock wave, (ie the
speed of advance of the wave normal to itself) divided by the speed of sound a, is
given by:
in which V0 is the propagation speed of the wave, p is the static pressure before
the wave, and p is the rise in pressure in passing through the wave. When V0/a
is large then p/p will be very large leading to large losses and hence the need to
do work on the air. This is shown by the large drag generated by bluff noses
illustrated in Fig. 4.14.
Aerodynamic Moments
As stated in Part I of this chapter, the local sideforce on a projectile is
theoretically proportional both to the yaw angle and to the rate of change of cross
sectional area of the projectile. The relationship which comes from Slender Body
Theory is:
in which N is the local sideforce, x is the distance from the body nose apex, V is
FIG. 4.49 Laminar and turbulent flow behind a truncated body at supersonic speeds
FIG. 4.50 The flow through a shock wave
FIG. 4.51 Supersonic flow past a cone
FIG. 4.52 Maximum turning angles for supersonic downstream flow
p 2
Cp 
 
 1/2V2 (M2  1)
V0 6 p   
1   a 7 p
dN dS  V2
sin  dx dx
13569_int.indd 122-123 11/15/11 11:20:52 AM
124 Military Ballistics External Ballistics – Part II 125
the flight speed, a is the yaw angle and S is the local cross sectional area of the
body.
Integrating this along the body and dividing by dynamic pressure times body
maximum cross-sectional area gives the overall normal force coefficient CN as:
where db
is the body base diameter.
The consequences which follow from these equations are:
1. A slender body only generates a sideforce where its cross-sectional area is
changing ie where dS/dx  0.
2. A slender body which is pointed at the rear and therefore for which db  0,
does not generate a net sideforce. It does however, generate a nose up couple
due to the upload on the nose and the download on the rear.
3. Sideforce is proportional to yaw angle (for small angles).
This sideforce exerts a total moment about the centre of gravity (cg) of the
projectile which is found by integrating the local sideforce times its distance from
the centre of gravity along the length of the projectile. This moment can be
represented by the total sideforce acting at a single point known as the centre of
pressure (cp). If nose up moments are reckoned to be positive, the position of the
centre of pressure is given by:
where Sb
is the body base area.
Now by integrating by parts,
where L is the body length. Hence, the centre of pressure position aft of the nose
apex is:
(in practice both this and the previous theoretical result for sideforce give
reasonable approximations to measured values at low Mach numbers but experimentally there is found to be a gradual rise in CN/d with Mach number, as
shown in Fig. 4.21).
Static Stability
Fin Stabilisation
The basis of the aerodynamic classification of fins and wings is in terms
of a combination of span and planform shape. It is called Aspect Ratio and is
defined as:
where span is the distance from tip to tip. The analysis of wing or fin lift is
normally conducted on the wing or fin which would be produced by joining two
opposing panels centrally to generate a single surface. This is known as the net
wing or fin. The net span is then the overall span of the wing or fin minus the
local body diameter. Low aspect ratio fins or wings (typically with A  2), are
called slender and do not suffer from the stall ie the loss of lift at incidence angles
above about 20, which is characteristic of high aspect ratio wings. This behaviour is also shown in Fig. 4.26.
The lifting effectiveness of a fin or wing may be expressed in terms of the rate of
change of lift coefficient with incidence angle ie CL/d. This is called the lift curve
slope. We have seen that, for slender bodies, it increases slowly with Mach
number, as shown in Fig. 4.21. For plane surfaces, the lift curve slope is strongly
dependent on planform shape and aspect ratio, as well as on Mach number. There
is no simple general formula for the lift of wings and fins but Fig. 4.27 effectively
shows how the lift curve slope varies with Mach number for several delta and
rectangular planforms of differing aspect ratio. The progressive reduction in
lifting effectiveness for Mach numbers above 1 is noteworthy. It may be shown
that lift and normal force coefficients are related by:
in which:
Interference Effects
There are appreciable interference effects on lift when a fin or wing is added to
a body. The first effect is that some of the lift on the fin is carried over onto the
body. This is because the fin upper surface pressures are, on average, lower than
those on the lower surface and the pressures at the root of the fin continue onto
the body. Therefore, there is an area on the body between the fin panels, over
which there is a pressure difference between the upper and lower surfaces
thereby generating an additional normal force on the body and hence lift.
CL  CNcos  CAsin
CD  CNsin  CAcos
CN  CLcos  CDsin
CA  CDcos  CLsin
Body volume Centre of pressure position  Projectile length  Base area
8N d2
b
CN     2 sin  V2
d2
max d2
max
V2 sin  0
Sb
xdS
xcp 
V2 sin  0
Sb
dS
0
Sb
xdS  LSb  0
L
Sdx and 0
L
Sdx  body volume 8.Lift 8.Drag CL  CD  V2
d2
max V2
d2
max
8.Normal force 8.Axial force
CN  CA  V2
d2
max V2
d2
max
Span2
Aspect ratio (A)  Plan area
13569_int.indd 124-125 11/15/11 11:20:52 AM
126 Military Ballistics External Ballistics – Part II 127
As a first approximation to the magnitude of this effect, the lift on the fin can be
averaged over the body as well. The magnitude of this additional lift is then equal
to d/b times the original fin lift, where d is the body diameter and b is the fin gross
span.
The second effect is that the fin incidence is locally increased by the flow
around the body, the body upwash. This is illustrated in Fig. 4.53. It may be
understood by reference to Fig. 4.45a. The component of flow normal to the body
is increased in velocity, just like the flow past a circular cylinder, the greatest
increase being adjacent to the body sides. When this flow component is added
vectorially to the axial flow component, the local incidence, and hence the lift, is
increased. It may be shown that, for a centrally mounted wing on a long body, the
lift enhancement is also by a factor d/b.
The net effect is therefore that the lift on the fin body combination is given by:
where the net fin is generated by removing the body and joining the root ends of
the individual fin panels to form a single flat surface, the net fin.
The interference effects on fin-body lift are shown diagrammatically in
Fig. 4.54.
When the fins are positioned at the rear of the body, as in APFSDS for example,
then there is less lift enhancement. As a first approximation, the net fin lift may
be assumed to be increased by the factor: 1  2(d/b).
When analysing the static stability of a fin stabilised projectile, it is not
necessary to look at different roll attitudes since the total lift of any number of
equally spaced fins greater than 2 is essentially the same regardless of roll angle.
Increasing the number of fins above 4 does not, however, increase the lift in direct
ratio to the number of fins added. For a net fin span to body diameter ratio of
unity, 6, 8 and 10 fins give approximately 25%, 43% and 55% more lift respectively than 4 fins. However drag is increased in direct ratio to the number of
additional fins.
Spin Stabilisation
The Gyroscopic Stability Coefficient
In the discussion in Part I it has been stated that the projectile must be
spinning ‘fast enough’ to confer stability. There is a correspondence between the
stability of a spinning top and the directional stability of a spinning projectile.
Consider a top having a moment of inertia about its spin axis of A and spin rate p,
at a small angle ex to the vertical, precessing at a constant rate  as shown in Fig.
4.55a. The moment of inertia about a transverse axis through the contact point is
B. The coordinate system in the vertical and horizontal directions is x,y and along
and at right angles to the spin axis is x,y.
From Fig. 4.55b the components of  in the x,y directions give the angular
momentum about the Ox and Oy axes respectively as:
and
Hence the angular momentum about Ox is, from Fig. 4.56:
FIG. 4.53 Body upwash effect on wing local incidence angle
FIG. 4.54 Wing-body interference effects
 B sin
A cos  Ap ie  Ap
d
Lift on fin with body  body lift  
1  
2 . net fin lift
b
13569_int.indd 126-127 11/15/11 11:20:53 AM
128 Military Ballistics External Ballistics – Part II 129
Now the moment in the vertical plane required to generate the precession
(which is the gravitational overturning moment) and which we will call here M,
is, from the equation of the gyroscope given in Part I, equal to the angular
momentum times the rate of precession in the horizontal plane ie:
M  Ap  B2

This equation can be solved for the rate of precession x, as:
(1)
This equation will have non-zero real roots corresponding to oscillatory motion
and hence to the top remaining upright, if:
This result can be applied directly to a spinning shell with the substitution of the
moment of inertia about a transverse axis through the centre of gravity for B and
with the gravitational overturning moment per unit tilt from the vertical M/a,
replaced by the aerodynamic moment per unit yaw angle M Hence, the spin rate
for neutral stability Po is given by:
in which M is the rate of change of yawing moment with yaw angle. This may be
viewed as the critical spin rate for the round for static stability in that it will
confer neutral static stability.
Any increase in the spin rate above this value will increase the stability of the
round and will increase its resistance to a yaw disturbance. This is expressed in
terms of a gyroscopic stability coefficient Sg defined as:
This will have a value of unity for a neutrally stable spin stabilised projectile
and a larger value for a stable projectile. It is usual to consider Sg  1.2 as the
lower bound for stability in practice.
If Sg is less than this value, the round is likely to tumble in flight leading to
shorter ranges and lower accuracy. However, it is possible to deliberately produce
rounds with Sg less than 1 in order to improve their ‘stopping’ capability.
In the design of artillery shells, the design value of Sg is normally taken to be
1.5, with 1.2 as a minimum. This gives a suitable balance between the two
extremes described earlier and allows the shell to be fired at quite high launch
angles to the horizontal.
For most conventional small arms ammunition, the design value of Sg is taken
to be around 2.
It may be noted that the equation for the frequency of oscillation in equation (1)
above can be written:
or (2)
in which
This shows that for values of Sg above 1, there are two distinct frequencies of
oscillation, one significantly higher than the other. The lower frequency motion
is usually, defined as the precessive frequency; the higher frequency motion
FIG. 4.55 (a) A spinning top; (b) Components of  in x, y axes.
FIG. 4.56 Components of angular momentum
Ap sin – B sin cos
which, for small  is approximately Ap – B.
Ap A2
p2
1 M        2B 4B2
B 
A2
p2
M

4B 
1
p0  (4BM) A
A2
p2
Sg  4BM
Ap 1    1  
1  
1/2  2B Sg
Ap   (1  ) 2B 1   
1  
1/2
Sg
13569_int.indd 128-129 11/15/11 11:20:53 AM
130 Military Ballistics External Ballistics – Part II 131
appears as a perturbation in yaw superimposed on the precessive motion and it is
known as nutation.
Still more can be learned from the expression for the gyroscopic static stability
coefficient by expressing it in a different way. Using the definition of CM used
earlier, the stability coefficient may be written:
Two further points emerge from this. Firstly that if stability of the round is likely
to be a problem, it will become evident that when the air density is high ie on a
cold day. It may also be noted that if the round enters a medium for which the
density is significantly higher than for air (such as human tissue), the round will
become very unstable and will start to tumble. This is in contrast to the behaviour of fin stabilised rounds. Secondly that, if the spin rate decays more slowly
than the forward speed, which it normally does, the round will become more
stable as it flies down range.
Finally, the spin rate of the round can be related to the twist in the barrel. In
the UK this is normally expressed in terms of the calibres of travel per revolution
n. Hence the spin rate can be written:
Substituting this in the equation above then gives:
at the muzzle. Now for a given barrel, n is fixed and therefore the stability will be
least when CM is largest.
A typical variation of CM with Mach number is shown in Fig. 4.57 for a
155mm HE round. It is clear that the maximum value of CM occurs close to Mach
1. Also shown on the Figure are the muzzle Mach numbers corresponding to
different charges. The deduction here is that directional instability, if it were to
be a problem for this shell, would become evident when firing at charge 3 or
charge 4.
There is a practical limit to the amount of twist which can be used in the barrel,
and as the length of a projectile is increased for a given calibre, the ratio of B to A
increases, as does the magnitude of the aerodynamic overturning moment, CM.
It follows therefore, that there is an upper limit to the length to diameter ratio for
a round which is to be spin-stabilised. This limit is normally reckoned to be
between 6 and 7. If a larger fineness ratio, ie a larger length to diameter ratio, is
required, then the round must be either fin or flare stabilised.
Aerodynamic Derivatives
For trajectory prediction, it is usually necessary to use a set of reference axes
fixed with respect to the earth. These are known as earth axes. In studying the
stability of projectiles it is more convenient to use a system of axes which are
fixed with respect to the body. These may have two forms – a set in which the axes
rotate with the body and a set in which they do not. The non-rolling axis set is
usually referred to as aeroballistic axes. The two axis systems are related by
spin rate. In both, the x-axis coincides with the body axis and the origin
of the coordinate system is located at the projectile centre of gravity as shown in
Fig. 4.58.
In the rolling axis system, angular movement in the xz-plane is known as pitch
() and in the xy-plane as yaw (β). Note that, in external ballistics, except in six
degree of freedom modelling, no distinction is normally made between pitch and
yaw and it is usual to find all angular excursions of the body axis referred to as
yaw. This is because, in external ballistics, axial symmetry is the norm and hence
pitch and yaw are indistinguishable. It is also possible, and sometimes convenient, to use a combination of total incidence σ and roll angle .
Rotations are defined with respect to the earth axes xo, yo, zo which are fixed in
space with xozo as the vertical plane and xoyo as the horizontal plane. Since
rotations are non-commutative, a convention is adopted to define angular displacement. This is that rotations are imposed in the order roll, pitch, yaw ie about
the xo, yo and zo axes respectively.
The forces and moments acting on the body are the axial force X, the side force
Y and the normal force Z, the rolling moment L, the pitching moment M and the
yawing moment N. Also shown in Fig. 4.58 are the moments of inertia A, B and C
and the rates of rotation p, q and r about the x, y and z axes respectively. Since it
is normal to analyse the flight of the projectile for small angles of incidence
only, the lateral velocities v and w are shown lower case to indicate that they
are small compared with the forward speed of the projectile U which is written
in upper case. It is general practice, however, again except for six degree
FIG. 4.57 Variation of yawing moment characteristics
2A2
p2
Sg  BpV2
d3
CM
2V
p  nd
8A2
Sg  Bd5
n2
CM
13569_int.indd 130-131 11/15/11 11:20:54 AM
132 Military Ballistics External Ballistics – Part II 133
of freedom modelling, to use V for forward speed when, strictly speaking, U, the
velocity along the x axis should be used.
For small angular perturbations between the body x-axis and the wind direction, the forces and moments thereby generated may be written as simple linear
functions of these perturbations. It is then possible to define the rate of change of
the aerodynamic quantities with respect to these perturbations ie to define
aerodynamic derivatives. The relationship between the aerodynamic forces and
moments and the most important perturbation variables are:
in which dmax is the projectile reference diameter and V is its flight speed. Note
that, because of the possible confusion with CL and CN ie the normal force and lift
coefficients, rolling moments and yawing moments (in the xy plane) are usually
written C1 and Cn.
The above definitions are not universal and differences such as factors of 2 and
minus signs can exist between the same coefficient as used by different authors.
Also forces and moments are sometimes quoted in the xy plane eg Cββ in place of
CN.
It is sometimes convenient to work in wind axes in which the relevant forces
are drag and lift. These are usually expressed as:
It may be noted that, for an axially symmetric body, the zero yaw values of side
force, yawing moment and lift CN0
, CM0
and CL0
are identically zero.
Dynamic stability
A good guide to the dynamic behaviour of a spinning projectile can be obtained
by analysing the equations of motion of the projectile with the simplifying
assumptions:
1. that the spin rate and speed remain constant
2. that the motion can be described in terms of small perturbations
3. that forces due to gravity can be neglected
The equations of motion for a spinning projectile with rotational symmetry can
then be written, in rolling body axis coordinates, as:
in which U is written in upper case whereas v and w are written un lower case to
show that they are small in size compared with U – this is the basis of the small
perturbation analysis.
Following Murphy in (2), it is convenient to introduce a complex yaw angle
FIG. 4.58 Convention system
Y  m( ˙v  rU  pw)
Z  m( ˙w  qU  pu)
M  B ˙q  (B  A) pr
N  C ˙r  (A  B) pq
V2
d2
max Axial force  Cx
8
V2
d2
max Normal force  (CN0  CN )
8
Vd3
max Magnus force  Cp p 16
V2
d3
max Yawing moment  (CM0  CM )
8
V2
d2
max Drag  (CD0  CD22
)
8
V2
d2
max Lift  (CD0  CL)
8
Vd4
max Spin damping moment  C1p
p
16
Vd4
max Damping moment  (CMq  CM
. ) q
16
Vd4
max Magnus moment  Cnp p 16
13569_int.indd 132-133 11/15/11 11:20:54 AM
134 Military Ballistics External Ballistics – Part II 135
ξ  β  i where β is given by v/U and   w/U. The aerodynamic force and
moment equations can then be expressed in the form:
By making use of the equations of motion and converting to a non-rolling body
fixed coordinate system, the following single equation for the complex yaw angle
is obtained:
in which:
The solution of this equation gives two time dependent motions with frequencies:
as shown for the top in equation (2), and damping:
in which the subscripts p and n refer to precession and nutation respectively.
For damped oscillations to occur, it is necessary for p and n  0
and since Q  0,   1  Sd ie Sd(2  Sd)  1/Sg
This criterion may be displayed graphically as shown in Fig. 4.59. If the projectile is statically stable ie CM and hence Sg are negative, either Sd(2  sd)  0
and the projectile is always dynamically stable, or Sd(2  Sd)  0 and there is a
maximum spin rate for dynamic stability.
For a statically unstable projectile, either Sd(2  Sd)  0 and there is a
minimum spin rate for dynamic stability, or Sd(2  Sd)  0 and the projectile is
always dynamically unstable.
FIG. 4.59 The stability diagram for small perturbations
FIG. 4.60 Motion in-vacuo
U2
d2
Y  iZ   CN 
8
U2
d3 pd Ud4
M  iN   CMp  iCM    (CMq  CM
.) (q  ir)
8 U 16
n2

.
 (2in  Q) 
.
   inQSD    0
Sg
d2
U md2
Q  CLa  Cq  8m 2B
Ap
n  2B
A2
p2
Sg  4BMa
2CL  (md2
/A)CMp
Sd  CL  (md2
/2B)Cq
Ap Ap p  (1  ) n  (1  ) 2B 2B
Q(  Sd  1) Q(  Sd  1)
p  n  2 2
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136 Military Ballistics External Ballistics – Part II 137
Trajectory prediction
The In-vacuo Model
Using the notation of Figure 4.60, the equations of motion in the horizontal
(x-direction) and vertical (z-direction) are, from Newton’s second law of motion:
where g is the acceleration due to gravity. Integrating these equations twice with
respect to time, with the initial conditions at time t  0:
gives:
Re-writing the first of these equations in the form:
(3)
and substituting in the second, gives:
(4)
which is the equation of a parabola.
Now the horizontal range of the projectile is the non-zero value of x when z is
zero. From equation (3), z  0 when x  0 or when:
ie
The maximum horizontal range is therefore obtained when   45°. The range is
then V2
0
/g
Substituting the expression for x back into equation (3), gives the time of flight
as:
The important in-vacuo results may be summarised as:
i. The range from the gun to the target is given by
This shows that the range increases with muzzle velocity squared; that
it varies with the launch angle reaching a maximum at 45 ie 800 mils,
after which it decreases reaching zero at   90 ie 1600 mils.
ii The time of flight Tf from the gun to the target is given by:
iii The whole trajectory lies in the vertical plane containing the line of
departure.
iv. The trajectory is parabolic in form and is symmetrical about the vertex
which is at a height (h) above the horizontal given by:
v. The angle of arrival equals the angle of departure.
vi. The velocity at impact equals the muzzle velocity because no energy is
dissipated; the loss in kinetic energy which occurs on the upward leg of
the trajectory is accompanied by an increase in gravitational potential
energy which is recovered on the downward leg.
vii The trajectory parameters do not depend on the nature of the projectile
because aerodynamic forces are absent.
viii. It follows from (i) that there are 2 values of o at which a given range less
than the maximum range can be achieved at a given muzzle velocity, one
with o above 45 and the other with o below 45; these are referred to as
high and low angle trajectories respectively. (Note that the time of flight
is always larger for the high angle trajectory as indicated by (ii).)
d2
x
m  mx  0
dt2
x
t  V0 cos0
1 x2
z  x tan0  g
2 V0
2 cos2
0
2V0
2
x  sin0 cos0
g
V0
2
x  sin (20)
g
d2
z
m  mz   mg dt2
1
x    mx.. dt dt  V0 cos 0t
m
1 1
z    mz.. dt dt  V0 sin 0t  gt2
m 2
dx dz
( x
. )  V0 cos 0 , ( z
. )  V0 sin 0 , x  z  0
dt dt
2V0
t  sin 0
g
2V0
Tf  sin 0
g
gTf
2
h  8
V0
2
Range  sin20
g
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138 Military Ballistics External Ballistics – Part II 139
The Point Mass Model
The presence of aerodynamic drag makes it necessary, in general, to use
numerical techniques to compute range, time of flight, velocity of arrival etc. The
equations of motion, in the absence of wind, now become:
where V2
 ẋ2
 ż2
and the initial conditions are the same as for the in-vacuo model.
The equations may be integrated by a step-by-step process such as RungeKutta or one of the more efficient solvers that are currently available. They are
first written as a set of first order differential equations by introducing the new
variables Vx ( ẋ) and Vz ( ż).
This can be illustrated by the special case for which an analytical solution
exists, namely a projectile for which the drag coefficient is proportional to the
reciprocal of the velocity and the air density is assumed invariant with height.
Writing CD  k/V makes the equations of motion
These are both first order differential equations of the variables separable type
and yield, with the appropriate boundary conditions:
These can now be integrated to give the following expressions for the trajectory
coordinates:
For a flat earth trajectory, the second of these equations can be solved for the nonzero value of t at which z is equal to zero. This is then substituted in the first
equation to give the range. The approximations used in this model ie CD ~ 1/M
and constant air density, are very nearly satisfied for APFSDS ie long-rod
penetrators and can be used to give quite a good approximation to their
performance.
Rocket Trajectories
So far we have been considering trajectories of projectiles such as cannon balls,
mortar bombs and shells which are unpowered in flight. Let us now consider the
trajectories of rocket propelled projectiles which are powered. The motion of a
rocket may normally be divided into three phases:
1. the launch phase
2. the boost phase, and
3. the ballistic phase.
In the launch phase, a launch tube is used to direct the initial motion of the
rocket. Launch dynamics is a complete study in itself and a good account of the
subject may be found in (3).
Once the rocket has accelerated to its maximum velocity at ‘all-burnt’, it enters
the ballistic phase. This part of the trajectory is similar to that for a shell and it
will not be discussed further.
The period of greatest concern for rocket accuracy is the boost phase between
launch and all-burnt, during which time the rocket is sensitive to initial disturbances, in particular, external winds and thrust misalignment. In the simplest
treatment of the rocket trajectory, which is reproduced here, it is assumed that
the launch is perfect and the rocket moves under the influence of gravity, axial
thrust and drag forces only. We can therefore assume that the rocket behaves as a
point mass.
Rocket Thrust
The action of a rocket motor depends on the conservation of linear momentum.
Consider the situation depicted in Fig. 4.61. The total momentum of the system
at times t and t  t remains the same.
FIG. 4.61 The conservation of linear momentum
V2
d2
CD x
.
mx..
 • 8 V
• d2
k
Vx   • Vx
8m
• d2
k
Vz   • Vz  g
8m
V2
d2
Cp z
.
mz..
  •  mg
8 V
d2
k
Vx  V0 cos0 exp   • t  8m
8mV0 cos0 d2
k
x  
1  exp   • t   d2
k 8m
8m 8mg d2
k 8mg
z   V0 sin0   
1  exp   • t    • t
d2
k d2
k 8m d2
k
8mg d2
k 8mg Vz  
V0 sin0  
exp   • t   d2
k 8m d2
k
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140 Military Ballistics External Ballistics – Part II 141
In the short interval of time t, a small amount of mass m is ejected with a
velocity V* relative to the rocket. Equating the momentum at time t and time
t  t gives:
(m  m)V  m(V  V)  m (V  V  V*)
Cancelling the common terms in m V and m V neglecting the second order term
mV, we obtain:
m V  m V*
Now m   dm/dtt where dm/dt is the rate of increase of mass and is negative
since m is decreasing with time. Dividing by t and letting t  0 we obtain the
differential equation of motion as:
V* is the effective motor thrust per unit mass flow and is known as the specific
impulse of the rocket. It may often be taken as a constant which depends only on
the propellant used. Some typical values are given in Table 4.7.
In the case of constant specific impulse, the equation of motion can be integrated in the form:
to give:
where Vo is the initial speed of the rocket and mo is the initial mass and V is now
the velocity at burn out.
In the above calculation the effects of drag and gravity have been neglected.
For artillery rockets for which the burn time is relatively short and hence the
launch longitudinal acceleration quite high and for which the launch angle is
quite low, this is a reasonable assumption.
Equations for Rocket Trajectories
If this assumption is not reasonable, the equations of motion along and at right
angles to the trajectory become, from Fig. 4.62:
In the first of these, D is the drag of the rocket with the motor burning. This can
be adequately represented by putting the base drag equal to zero in the free flight
drag estimate. Once again, a numerical integration scheme is required to obtain
the trajectory parameters.
The Modified Point Mass Model
Equilibrium Yaw
As explained in Part I, for a correctly spun shell the body axis follows the
curvature of the trajectory caused by gravity. Consider a projectile with a spin
rate p and a rotational moment of inertia A flying at an angle  to the horizontal
as shown in Fig. 4.63. The projectile will be subject to a rate of yaw d/dt
which, from the equation for the gyroscope, will be produced by a moment M such
that:
Now the moment is itself generated by the instantaneous equilibrium yaw angle
e ie M  M
.e. Hence:
FIG. 4.62 Rocket motion
TABLE 4.7
Typical values for specific impulse
Type of propellant Specific impulse
m/s
Solid Plastic composite 2500
Liquid HTP/hydrocarbon 2430
Oxygen/hydrogen 3560
Fluorine/hydrogen 3660
dV dm
m   V*
dt dt
v m dm
 dV   V* m0 V0 m
m0
V  V0  V* 
ln  m
dV
m   mV*  D  mg sin
dt
d
mV   mg cos
dt
d M  Ap dt
Ap d e  M dt
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142 Military Ballistics External Ballistics – Part II 143
Now the curvature of the trajectory is associated with a centripetal force Fc such
that:
But, from the triangle of forces, Fc  W cos and so 

 g cos/V. Substitution
into the equation for e, then produces:
showing that, all other things being equal, equilibrium yaw is a maximum at the
apex of the trajectory where  is zero. The equation can also be re-cast in terms of
the gyroscopic stability criterion, Sg, to yield:
The presence of equilibrium yaw causes a sideforce on the projectile which in turn
causes drift. The estimation of equilibrium yaw and its effects from the shell
angular momentum, moment characteristics and the rotation of the flight path in
the vertical plane is the key feature of the Modified Point Mass Trajectory Model.
This enables it to predict drift on a continuous basis as the computation proceeds.
It is found experimentally that the drift (D) increases with time of flight and,
for a given muzzle velocity, the magnitude is found to vary with the angle of
projection O, according to the law:
in which the constant C is found from firing trials.
Cross-wind Effects
A first approximation to the effects of cross-wind may be obtained from the
equation for in-vacuo time of flight and range given in Part I of this
chapter.
Since the in-vacuo time of flight is equal to:
then the magnitude of the drift (), due to a cross-wind of strength Vw becomes:
This is known as the ‘Rifleman’s Formula’.
Controlled flight
There is currently much interest in ways of altering the trajectory of projectiles
whilst they are in flight. It is desired to do this in a controllable way in response
to target information which may be obtained from the projectile itself for an
autonomous system, or from some other source. A simple way of making changes
to the range is by altering the drag of the projectile. This can be done relatively
easily, both for fin and spin stabilised rounds, by means of spoilers which alter
the drag of the round. Normally however, changes in flight direction are required
as well ie ‘heading’ changes. These require the generation of a sideforce on the
projectile.
For the case of fin-stabilised rounds, this is easily done, at least in principle.
The round is statically stable, by definition and is provided with lifting surfaces,
ie fins, towards the rear. Consider a round in ballistic flight ie in flight with no
side-force acting on it. Now we deflect the rear-mounted fins in one plane to
generate an angle between the plane of the surface and the incident flow. This
angle is called the angle of incidence. The consequence of this angular deflection
is the generation of a lift force on the fins behind the centre of gravity. This will in
turn produce an out of balance moment about the cg as shown in Fig. 4.64.
In response to this out of balance moment, the nose of the projectile will rotate
(upwards for the deflection shown). This generates a yaw angle between the body
axis and the direction of motion of the round which in turn produces a sideforce
and a moment about the cg. When the moment about the cg due to yaw is equal to
the moment due to fin deflection, the projectile will be in equilibrium again, as
shown in Fig. 4.65.
FIG. 4.63 Equilibrium of a spinning projectile
FIG. 4.64 Projectile subjected to control moment
Range
V0 cos0
Range   V 
Tf   V0 cos0
D 5 C tan0
W
Fc  
• V
g
Ap g cos e  • M V
4B Sg cos e  ApV
13569_int.indd 142-143 11/15/11 11:20:57 AM
144 Military Ballistics External Ballistics – Part II 145
To formulate this analytically, we introduce two new derivatives, the lift due to
fin deflection CL and the yawing moment due to fin deflection CM. These are
defined such that:
in which (CL)f is the fin lift curve slope based on the net fin plan area Sf and lf is
the distance of the fin centre of pressure aft of the projectile centre of gravity.
Note that no allowance is made for upwash effects in estimating the fin lift curve
slope because the derivative is with respect to the fin deflection, which is assumed
to take place at zero body angle.
The equilibrium angle of yaw t known as the trim angle, is determined per
unit fin deflection from the equation:
since the moments generated from each source are equal. The sideforce produced
in the equilibrium state Lt, is then:
from which the available lateral acceleration can be estimated. Note that if wings
are fitted to the projectile to increase the available side-force, the formulation is
identical but the wing contributions to force and moment are included in the
values of CL/d and CM/d used in the above equations.
This method of control is not available to spin-stabilised projectiles, though it is
being considered for terminal guidance for such rounds. In this case, the round is
de-spun and fins are deployed towards the end of the trajectory and manoeuvring
is limited to this phase of the flight.
The viable alternative for spin-stabilisation, which can also be used for finstabilised rounds, is impulse control. Here, an impulsive side force is applied to
the round by means of an explosive charge or small-thruster to produce a force of
magnitude F for a time t to give a change in heading angle :
as shown in Fig. 4.66.
Next phase
We have now looked at the ballistics involved in launching a projectile and in
its flight to the target. The next stage is concerned with its performance when it
reaches the target. In the case of hard target this is called terminal ballistics and
is the subject of the next chapter.
References
(1) Shapiro, A.H. (1961) Shape and Flow (the fluid dynamics of drag). Heinemann
Educational Books Ltd
(2) Murphy, C.H. (1963) Free Flight Motions of Symmetric Missiles
BRL Report 1216, US Army Ballistic Research Laboratory, Aberdeen Proving
Ground, Maryland
(3) Longdon, L.W. (Ed) (1983) Textbook of Ballistics and Gunnery Vol 1 HMSO
FIG. 4.65 Resultant equilibrium state
FIG. 4.66 Effect of impulse control
F t
  mV
V2
d2
Lift due to fin deflection  • CL • 
8
V2
d2
Yawing moment due to fin deflection  • CM • 
8
1  V2
Sf • (CL)f • 
2
1  V2
Sf1f • (CL)f • 
2
t CM
=  Cm
V2
d2
Lt  (CL•  CL•)
8
13569_int.indd 144-145 11/15/11 11:20:57 AM
146 Military Ballistics
147
5.
Terminal Ballistics – Part I
Self Test Questions
QUESTION 1 What principle enables wind tunnels to be used to predict the
aerodynamic forces which will act on a projectile in flight?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 2 For two geometrically similar shapes, what does equality of
Reynolds number imply?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 3 Under what conditions does roughening the surface reduce the
drag of a body in flight?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 4 Describe two interference effects which increase the lift of fins
when they are added to the sides of a body.
Answer………………………………………………………………………………..
………………………………………………………………………………………….
QUESTION 5 Describe the two components of motion which may be observed
after a spinning projectile has been subjected to a yaw
disturbance.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 6 If a 5.56 mm calibre bullet, with a muzzle velocity of 990 m/s
travels a distance of 30.48 cm in one complete revolution, estimate its spin rate.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 7 Assuming in-vacuo motion, what would be the maximum range
of a projectile fired with a muzzle velocity of 220 m/s.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 8 Describe two methods of flight control suitable for guided
projectiles.
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
Scope
Terminal ballistics may be defined as the study of the effects of projectiles on a
target. The conditions under which missiles impact against targets vary widely,
depending on strike velocity, strike angle, and the type of projectile and target. In
this chapter we shall be mainly concerned with the terminal ballistics of projectiles against armour.
Velocity Ranges
The range of initial velocities is, in many ways, the most fundamental consideration because velocity affects the variety of impact phenomena so much that
it can override almost any other consideration. The velocity range for a projectile
fired from a conventional gun is 500 to 1300 m/s, the nominal ordnance range.
The ultraordnance domain from 1300 to 3000 m/s is represented by warhead
fragments: and above this upper limit is the hypervelocity range, involving
shaped charges.
Angle of Attack
An important consideration in terminal ballistics is the angle at which the
projectile strikes the target. In the United Kingdom the ‘angle of attack’ is the
angle between the path or line of arrival of the projectile and the ‘normal’ angle
to the plate under attack: it is shown as angle  in Fig. 5.1. The normal angle is
defined as being 90° to the target plate. The angle of attack as defined here is not
common to all countries and some use the angle ex as the angle of attack. The
actual angle of attack will be dependent upon a number of factors such as the
stability of the projectile in flight and the type of projectile and target.
Characteristics of Projectiles
The penetrator shape is significant in determining the mode of target failure.
Pointed penetrators exhibit a piercing action in which target failure centres
about the projectile axis. Blunt shapes, on the other hand, exhibit a plugging
mode of perforation (see Fig. 5.3). The transition depends on penetrator shape,
and it is important to note that the effect of sharp and blunt shapes plays an
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148 Military Ballistics Terminal Ballistics – Part I 149
important role in establishing penetration resistance. The criterion for ‘sharp’
and ‘blunt’ shapes is often determined by the ratio of nose length/calibre. If this
ratio is greater than or equal to unity then it is termed sharp; if the ratio is less
than unity, then it is termed blunt.
To maximise penetration it is generally desirable for a penetrator to be long
and dense. There are basically two disadvantages that appear with increased
length: the first is an increased chance of bending mode failures, and the second is
an external ballistic instability for spin-stabilised projectiles. However, projectile
designs that use fin stabilisation avoid the latter problem.
Deformation of the projectile increases the diameter of the projectile and
therefore decreases the potential for penetration. Consequently, the penetrator
should ideally be difficult to deform. Steels, forms of tungsten carbide and
depleted uranium are all suitable for use as penetrators. The latter substances
have the advantage of high density, but brittleness defines definite limits on the
range of impact conditions for which these heavy materials can be used. Steel is
reasonably dense and can be given considerable toughness; its comparative
cheapness and availability are distinct advantages.
Characteristics of Targets
Roads, bridges, factories, tanks, ships and helicopters are all types of targets.
However, during an impact it is usually only an element of the target that is
attacked; for example an armour plated tank wall. Therefore, one convenient way
of classifying targets is to relate to their relative thicknesses. A ‘thin target’
assumes that for all practical purposes, stress and deformation gradients do not
exist. If the rear surface of a target exerts considerable influence on the deformation process during all or nearly all of the penetration then it is termed an
‘intermediate target’. A ‘thick target’ is defined as one in which the rear boundary
influences the penetration process only after substantial travel into the target
element, and lastly, a ‘semi-infinite’ target is one in which there is no influence of
a rear boundary on the penetration process. Another method of classification of
targets is their geometry. The target, if thin, may consist of single plates,
sandwich plates, or spaced plates; the shape may be flat, curved, or irregular.
An important part of penetration mechanics is an assessment of the material
characteristics of the target. Penetrability of substances is a quantitative
measure of particular interest to ballisticians. A comparison of performance of a
single projectile type against a spectrum of materials will establish some type of
order among them. Comparative penetrability is the basis for a crude classification of target materials as low resistant which mainly consist of soils, moderately resistant such as concrete and low-strength metal alloys, and highly
resistant, including the high-strength metals, alloys and ceramics. These classes
tend to correspond to different penetration phenomena, require different investigative approaches and correspond very roughly to the present classification of
targets by relative thickness and density.
The behaviour of a specified material is usually represented in terms of a model
that depends on its state and character. Most targets are characterised as solids,
but in some instances the target may act like a fluid. This happens when solids
are affected by the extreme pressures produced by hypervelocity impact. Solid
models include one or more of the domains of elastic, plastic, viscous, or hydrodynamic behaviour. Furthermore, an accurate specification of material response
of any solid is complicated by the extreme range of stresses to which an object is
subjected during penetration and by the diversity of deformation and failure
patterns that occur.
Due to the different types of projectile and target characteristics, it is easily
seen that forces of many different kinds can be induced in missiles when they
strike their targets. Projectiles will react to these induced forces in a variety of
ways. On impact, they may perforate the target, may penetrate the target, and
they may ricochet.
Attack of Armour
Penetration and Perforation
The defeat of a target often involves penetration or perforation of it by a
missile, so it is useful to distinguish between penetration and perforation.
Penetration may be defined as the entrance of a missile into a target without
completing its passage through it; perforation usually implies the complete
piercing of the target by the projectile.
Penetration
In general, the high-speed interaction of a long rod penetrator with a target can
be divided into three phases. The first phase is the impact phase where stress
FIG. 5.1 Angle of Attack
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150 Military Ballistics Terminal Ballistics – Part I 151
waves and stress levels are critical to initiate penetration. The second phase is
penetration where either or both the target and penetrator behave as though
they were fluid. This hydrodynamic failure is produced in those regions where the
pressures involved are well above the yield strength of the materials. The third
phase is the perforation phase which is discussed under its own heading later in
the chapter.
The interaction between a target and a penetrator is an interaction between
materials and can be explained in terms of the science of materials. The impact
phase is complex and often ideal situations are assumed. Idealised theories may
provide useful conclusions, and modifications may be subsequently made to
improve the model. The model may consider wave reflections in the penetrator
and stress waves in the target, for example. Wave motion within the penetrator and target can be accepted for low speeds, where there is no yield of
penetrator material to be considered. As the impact velocity increases,
the penetrator material begins to yield, the material flows plastically and
penetration starts. The impact phase may have important effects on the penetrator. Shortly after impact, the projectile suffers a radial tensile stress and the
striking end or nose of the rod mushrooms radially while pushing into the target.
At the same time, a plastic wave travels towards the free end of the projectile. In
some cases, when the impact stress is high enough, the rod or body of the
penetrator will fracture as the applied stress reaches the ultimate tensile stress
of the material. Such a stress level initiates axial cracks at the periphery of the
mushroomed region at the nose of the rod. The cracks propagate and shatter at
the stem of the rod or body. So, as the plastic wave travels down the rod it leaves
shattered material in its path.
In summary: at the moment of impact a stress wave is initiated in both the
target and the rod or body of the projectile and a very high level of stress is
applied on the impact area. The high stress level produced initially will produce a
crater and the projectile will penetrate the face of the target.
After the first phase of impact when the stress has been weakened and has
progressed away from the crater, an almost steady state is set up in which the
crater is being deepened at constant speed. During this phase the hydrodynamic
analogy proves useful because the pressures at the bottom of the crater are well
beyond the yield strength of the material in both the penetrator and the target
when the velocity is high. During penetration, the projectile may be partly
shattered; it will also have become plastic. Its nose will be advancing with a
specific velocity V and the crater may be imagined as advancing in an opposite
direction at a velocity U where V is greater than U (see Fig. 5.2). The crater
progresses by ductile failure and the target material flows aside. The pressure
created and the energy transferred by the impact melts the front of the projectile
and the bottom of the crater. The penetrator and target material behave as fluids
although not as perfect fluids. Because the back of the rod travels faster than the
front of the rod which in its turn travels at the speed of advance of the crater, it
follows that the projectile is constantly eroded. The material being eroded is
forced backwards, relative to the bottom of the crater, and flows between the
crater wall and the remaining portion of the penetrator. The projectile is thus
being consumed and turned into a hollow cylinder as illustrated. In summary,
provided the speed of impact is high enough, the projectile flows plastically
exerting a pressure on the bottom of the crater, thus deepening the crater. This
action causes the projectile to be inverted and turned into a hollow cylinder.
Perforation of Armour
The process of perforation is a complicated mechanism, which has not yet been
fully explained in the theoretical sense. In this section we will look at some of the
observed features encountered in these processes.
Plate failure is due to the interaction of a variety of mechanisms with one
predominating, depending on material properties, geometric characteristics and
impact velocity. The most frequent types, shown in Fig. 5.3, consist of fracture,
radial fracture, spalling, scabbing, plugging, front or rear petalling, or fragmentation and ductile hole enlargement.
Fracture
Failure involving fracture results in the perforation of thin or intermediate
targets. Fracture due to initial stress waves, which are stronger than the ultimate compressive strength of the target, could typically occur in weak, low
density materials. Radial fracture would be limited to brittle targets such as
ceramics.
Spall Failure (Scabbing)
Scabbing is a material failure due to the reflection of the initial compressive
wave from the far side of the plate and is a commonplace phenomenon under
FIG. 5.2 Impact of a long rod penetrator
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152 Military Ballistics Terminal Ballistics – Part I 153
explosive loading, a good example of this is the action of High Explosive Squash
Head (HESH).
Plugging
Plugging develops when a nearly cylindrical slug of approximately the same
diameter as the penetrator is set in motion by the projectile. Failure occurs due to
shearing produced around the moving slug. Plugs are most likely to be found in
very hard plates of moderate thickness. Its presence most frequently occurs when
blunt penetrators are used and it is sensitive to velocity and angle of attack.
Petalling
Petalling is most frequently observed in thin plates struck by ogival or conical
penetrators at relatively low impact velocities or by blunt projectiles near the
ballistic limit. As the material in the bulge on the back of the plate is further
deformed by the projectile, the elastic properties of the armour are eventually
exceeded and a star-shaped crack develops around the tip of the penetrator. The
sectors subsequently formed are then pushed back by the motion of the projectile,
forming petals.
Fragmentation
Fragmentation occurs when the target is composed of brittle material. The
fragments generated by a failed target themselves act as projectiles and must be
considered as penetrators when meeting any subsequent target.
Ductile Failure
The ductile type of failure is the kind most commonly observed in thick plates.
The perforation is accomplished by radial expansion of the plate material as the
projectile pushes through.
Predicting Failure
Numerous theoretical models have been developed in attempts to predict the
perforating ability of a projectile without conducting experimental test firings.
The behaviour of armour is so complex that none of these models is completely
satisfactory. In most of the theoretical developments, the volume of the hole
produced by the impacting projectile is assumed to be proportional to the kinetic
energy lost by the projectile when it perforates the plate.
Types of Projectiles
The first and still very common type of armour defeating projectile is some form
of solid shot which depends upon kinetic energy to penetrate armour. More
recently two forms of chemical or, more descriptively, explosive attack, have been
developed. The first is called High Explosive Anti-Tank (HEAT), which employs
a shaped or hollow charge principle to punch holes through armour. The second is
High Explosive Squash Head (HESH) which is a soft-nosed high explosive shell:
it pancakes on the armour before detonating and causes a scab to blow off the
inside.
Kinetic Energy
The kinetic energy form of attack is to use a solid projectile or ‘shot’ to impart
as much energy as possible on the target concentrated over as small an area as
possible. That is, it is desirable to maximise the relationship MV2/d2, or in other
FIG. 5.3 Perforation mechanisms
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154 Military Ballistics Terminal Ballistics – Part I 155
words the projectile should be a long thin rod. However, there is a conflict
between this requirement for shot shape, size and mass at the target and the shot
requirements in the gun and in flight. In the gun the shot should present the
largest possible cross-sectional area against which the propellant gases can act,
and it should be light. Therefore the shot should have a high value of d2/M and so
should be a short squat projectile made of a low density material. In flight, to
ensure the maximum kinetic energy is delivered at the target, it is important
that the shot loses as little velocity as possible on its way to the target. A heavy
dense projectile has a better carrying power than a lighter one, and hence will
have the potential for greater ranges. Furthermore, a thin projectile with a small
cross-sectional area will maintain its velocity better than one with a large crosssectional area. So a long thin dense shot is the requirement for the shot in flight.
Also, the longer and thinner the projectile provided that it is kept as heavy and
dense as possible, the better its penetrative performance. The depth of penetration for a projectile that survives impact without fracture is approximately
proportional to the impact velocity and the projectile length, and it increases with
the density of the penetrator. However, very long thin dense projectiles striking
armoured plate at high speeds are likely to ricochet and break-up due to material
failure. (1) gives a good practical account of the various forms of shot failure. The
differing requirements have been reconciled in the design of an Armour Piercing
Discarding Sabot (APDS) shot.
Until the adoption of APDS, various armour piercing projectiles were commonly used and they were generally modified in some way to overcome the
conflicting requirements outlined above. The current APDS consists of a small,
high density tungsten carbide core, sheathed in a sabot of light alloy which serves
to convey the sub-calibre shot through the gun bore and out of the muzzle.
Having performed this function, the sabot conveniently breaks away, reducing
air resistance and leaving the high velocity shot to expend its energy in penetration of the target where its smaller calibre results in less dissipation of energy.
Because of their considerable velocities, with muzzle velocities in the range of
1378 m/s, such projectiles have a high degree of accuracy and can usually be
employed at long ranges. To provide stability, the APDS round is spun and must
be fired from a rifled gun. For a kinetic energy round there are limitations in this
method, because the core cannot be made too long before it becomes unstable:
penetration of armour could be improved by making the core longer and thinner
and this can be achieved by fin stabilisation. The APFSDS can also acquire
higher muzzle velocities due to the use of a smooth bore gun.
An illustration of the major components of an Armour Piercing Fin Stabilised
Discarding Sabot (APFSDS) round-is shown in Fig. 5.5. However, as all the
energy the kinetic energy shot possesses at the target has to be developed and
imparted in the gun, there are a number of penalties to be paid for this. For
example, a large and heavy gun is needed to absorb the recoil energy, and there is
an increased effect of barrel wear compared with guns with lower velocity
projectiles.
Chemical Energy – Explosive Projectiles
Chemical energy in the form of explosive projectiles can be used to attack
armour in a number of ways; not all are equally effective. The targets attacked by
high explosive (HE) shells may be primarily damaged by fragments, by blast, or
by both. The usual type of HE fragmentation shell is illustrated below in Fig. 5.6.
The shell consists of a hollow streamlined projectile filled with high explosive and
provided with a fuze and exploder system so that it may be detonated at the right
time and place in relation to the target. As the detonation proceeds through the
explosive, the metal case begins to swell and to break up into a large number of
pieces of varying sizes which are thrown outward at high velocity. These fragments are effective against personnel and against such materials as guns,
soft-skinned vehicles, and aircraft, provided a vital part is hit.
The direction which the fragments take will be governed largely by the shape
of the projectile. For a spherical projectile when detonation starts at the centre,
the distribution of fragments should be uniform. For cylindrical projectiles most
F of the fragments would be concentrated in a fairly narrow beam of side spray. The IG. 5.4 Basic components of an APDS projectile
FIG. 5.5 Basic components of an APFSDS projectile
FIG. 5.6 Basic components of a HE fragmentation shell
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156 Military Ballistics Terminal Ballistics – Part I 157
maximum fragmentation effect is created when the greatest number of fragments of the maximum mass and velocity are produced. If the number of fragments is increased the chances of a hit on a target at a given distance from the
burst will increase. But, since the average mass of a fragment will be correspondingly reduced, the initial kinetic energy will be less and their velocity will fall
off more rapidly. In general a fragment from a modern shell will travel at about
1000 m/s and will weigh between 1.2g and 3.75g. The distribution in mass of
the fragments will depend upon many factors such as wall thickness, the ratio
of the weight of explosive to the weight of metal, and the type of metal. The
velocities of the fragments will be determined by the same sort of factors, but in
general a thin-cased shell will give rise to fragments having high velocities.
Fragmentation bombs and shells, mortar shells, grenades, mines, rockets and
other fragmentation missiles are all roughly similar in nature. They differ
primarily in the specific shapes used, the thickness of the shell wall, and in the
amount of explosive used.
Blast is the shock caused by the detonation of the high explosive filling. The
effects are most pronounced if the shell penetrates the surface of a target before
detonation. In the open environment, blast is effective for relatively small distances and fragmentation has more effect. The main practical uses of blast are in
the destruction of buildings and structural damage in general. Conversely the
use of straightforward high explosive shells with their blast and fragmentation
effects is not really effective against tanks. HESH and HEAT are, and we will
now discuss these in some detail.
Shaped Charges
The energy of an explosion can be concentrated by properly shaping the charge.
Early in World War II, it was discovered that when a hole in the explosive was
lined with a thin metal layer, the damage produced by the charge was greatly
increased. Hollowed lined charges are variously referred to as hollow charges,
shaped charges, lined charges and Munroe charges. Whatever the nomenclature,
they consist basically of a hollow liner of inert material; this is usually metal of
conical shape. The general effect of lining the charge is shown in Fig. 5.7.
Figure 5.7 (a) shows the effect produced by an ordinary cylindrical charge with
no cavity: Fig. 5.7 (b) shows the effect produced by a shaped but unlined charge,
and the third, Fig. 5.7 (c), the effect produced by a shaped and lined charge. This
effect is due to the fact that when a slab of high explosive material is detonated,
the detonating wave leaves the surface of the material at an angle. Therefore, if
the high explosive is formed with a cone-shaped hollow, the detonating impulses
emanating from the inner surfaces of the core will be focused and concentrated at
an external point on a line along the axis of the shell. The liner collapses under
the action of the explosive and gives rise to a stream of high velocity, high density
material that is capable of considerable penetration. The conical liner is usually
broken up by the explosion into three components; a plug, a jet and fragments.
Figure 5.8 is a diagram of the mode of deformation and break up of such a
conical liner.
High Explosive Anti-Tank (HEAT) Effect
In principle, HEAT works by using the shaped charge effect: that is by using
the energy available from the detonation of a charge of high explosive to collapse
and break up a metal liner into a metallic jet and plug.
When making a HEAT projectile the HE filling is formed with the conical
recess in the forward end, around a liner, which is normally made of metal. The
nose of the shell is hollow and strengthened so as to ensure a stand-off distance
when the filling detonates, which will bring the point of maximum concentration
of force against the surface of the target. The shell fuze acts immediately it is
decelerated and on detonation the tremendous release of energy creates a pressure wave.
The wave crushes the cone in upon itself. The cone is further deformed so that a
FIG. 5.7 The shaped charge principle
FIG. 5.8 Penetrative performance of a conical liner
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158 Military Ballistics Terminal Ballistics – Part I 159
high velocity jet of metal and gas is squeezed out along the axis of the cone by the
pressure wave. The stand-off distance is determined to ensure that the pressure
wave jet of gas and metal is fully formed at the point of impact with the armour
target. Penetration of the armour is dependent on many factors, the major ones
are stand-off distance, the type of explosive and cone angle, and the diameter and
thickness.
About 20% of the metal liner goes into the metallic jet which has a velocity
gradient from its tip of approximately 8000 to 9000 metres per second, to its tail
of about 1000 metres per second. The remaining 80% of the liner forms a plug
which follows the jet at a much lower velocity of the order of 300 metres per
second. The jet achieves its penetration solely by the intense concentration of
kinetic energy at its tip which exerts a pressure of about 200 tons per square inch
(3050 mega pascals) on the armoured plate. Under this intense strain the plate
gives radially and in doing so is permanently deformed. The penetration achieved
by a HEAT projectile is relatively good, and a small amount of high explosive
charge can penetrate a considerable thickness of armour plate.
Whilst penetration is important, it is not the main consideration: damage
behind the plate is the most important effect to achieve. HEAT causes damage
behind the plate in three ways; it can be with the jet itself, with the spall, and by
physiological and psychological effects against the crew by pressure, temperature
and flash. The jet will, having penetrated, disable anything in its path, but as
it is so narrow its chance of hitting anything inside the tank is comparatively
small. The main lethality effect comes from the spall. The wider the exit hole
on the inside of the tank, the more spall will be formed and the greater will be the
lethality. HEAT lethality is therefore often assessed on the basis of exit hole
diameter. Lethality, however, is obtained at the expense of penetration and vice
versa. The narrower the jet, the greater the penetration, but the less the
lethality. On the other hand, the wider the jet, the greater the lethality but the
less the penetration. The HEAT warhead designer must compromise between
these conflicting factors.
A number of factors influence the lethality amongst which are cone diameter,
material and thickness of liner, and stand-off distance. A good discussion of these
factors may be found in (1). Overall, HEAT is capable of achieving considerable
depths of penetration in armoured plate and is not affected by spaced armour.
There is however, one drawback arising from the use of the shaped charge shell
with a conventional gun. It is the creation of spin imparted from the rifling. The
general effect of this rotation is to diffuse the jet over a larger area, scooping out
wider, shallower depressions in the target surface, rather than the punching of
deep, narrow holes. As the rate of rotation is increased the penetration may be
reduced by as much as 50%. Consequently, the easiest solution is to fire a hollow
charge projectile from a smooth bore launcher or gun. Often this is impossible
because a rifled gun is required for other types of ammunition, so a slipping
driving band can be employed; this is the British solution for tank guns. In terms
of target effect, compared to other chemical modes of attack, HEAT is efficient
and economic in its use of high explosives, and therefore particularly applicable
to lightweight anti-tank weapon systems.
High Explosive Squash Head (HESH) Effect
Another method of attacking armour is HESH, or High Explosive Plastic (HEP)
as it is known to the Americans. Because it employs a considerable amount of
explosive, it has a considerable secondary use against bunkers, buildings and
personnel.
FIG. 5.9 Basic components of a HEAT projectile
FIG. 5.10 HEAT effect
FIG. 5.11 Basic components for a HESH projectile
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160 Military Ballistics Terminal Ballistics – Part I 161
In a HESH projectile HE is placed in contact with the armour, instead of being
separated by a ‘stand-off’ gap. It is base fuzed; it is thin-walled so that it can
squash against its target; and the nose contains a pad of inert composition which
tops the HE filling. When such a projectile strikes a hard surface, such as armour
or concrete, the thin shell walls collapse and the HE contact pancakes on to the
surface. The fuze then detonates the filling and, because it is now in direct contact
with the target, a high velocity compressive stress wave is propagated through it.
When this reaches the rear of the plate, the wave is reflected back through the
plate as a tension wave. The rebounding tension wave meets further waves
coming in the other direction. They combine, setting up a reinforced stress wave
which exceeds the tensile strength of the plate. As a result a large scab is torn off
the rear of the plate to act as a projectile inside the attacked vehicle.
There are a number of practical considerations, most of which are discussed in
(1). The presence of a pad of inert material in the nose of the projectile serves to
cushion the HE filling against the action of the impact. This avoids a premature
detonation, which would spoil the pancake effect and result in a relatively
harmless burst like that of a conventional HE shell. The most serious limitation
of HESH is that the effect is entirely defeated by spaced armour on the target.
Also, a HESH projectile striking at about 700 m/s will pancake on any plate
thicker than 6 to 8 mm; but its inherent kinetic energy will result in its
penetration of anything less. On the credit side, HESH is largely unaffected by
the angle of attack, and in fact by striking at an oblique angle, the effective area
of contact may be increased. The optimum penetration performance is actually
attained when the angle of attack is about 40°.
Despite its limitations HESH is particularly useful against many types of
target such as bridge supports and concrete buildings. It comprises a very
effective armour defeating projectile although it suffers from a relatively low
velocity and higher trajectory. Comparing instances when HEAT and HESH fail
to defeat, say, tank armour, HESH may still have a sufficient physiological effect
upon the crew to put the vehicle out of action.
To make the armoured vehicle designer’s task as difficult as possible, it is
important that kinetic energy, HEAT and HESH attack capabilities are kept.
This prevents the development of armour which can defeat particular types of
attack. For example, it would be comparatively easy to develop lightweight
spaced armour to defeat HESH, but it would be vulnerable to kinetic energy
attack. The converse is also true.
Mathematical Approach
Until this point in the chapter a descriptive approach has been followed. The
following section illustrates the various types of approaches that may be taken to
solve actual problems. Those readers not interested in the solution of terminal
ballistic problems should miss out the section.
FIG. 5.12 HESH effect
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162
Terminal Ballistics – Part II 163
Terminal Ballistics – Part II
Theory of Terminal – Ballistics
Approach
The theory of terminal ballistics is relatively new compared to the theory of
internal and external ballistics. The techniques of investigation for impact on
solid targets consist primarily of empirical relations (based on experiments),
analytical models, and computer modelling.
Empirical Relations
Empirical relations used in penetration mechanics are most useful when the
number of variables being correlated is small. The empirical relations commonly
used in penetration mechanics are based on experimental correlations of such
parameters as penetration depth P, crater volume C, the ballistic limit V50,* with
the dimensions of the projectile, mass m, the impact velocity V, and obliquity 
and the type of target and its thickness h. Examples are:
P/d  a0 mV2
/d3 Fundamental Armour Equation
P/d  a1 (mV2
/d)0.6993 Milne-de- Marre
P/d  2 mV2
/a2 d3 Morin
P/d  a3t ln (1  a4V2
) Dideon
E0  a5 d1.5 h1.4 Marre
C  a6 (1/2mV) Helie
In these equations d and E are the diameter and perforation energy of the
penetrator, t is the target density and ai are arbitrary constants. The above
relations avoid the problem of incorporating a large number of variables by
restricting the correlation to a single material; in this way the parameters of the
material description are reduced to a minimum or eliminated. When used with
strict adherence to this limitation, the empirical relation may be the best
available prediction. It shows that penetration mechanics is a complicated
phenomenon that depends on many parameters. However, empirical relations do
not provide any insight into the physical processes described, so we turn towards
other methods.
Analytical Models
Analytical models provide correlations similar to those from empirical studies,
but they also include relations between parameters of the system on the basis of
physical requirements. Simple representations may combine an empirical element with physical principles: for example, the resistive force can be assigned an
empirical formulation. The empirical force law is then used to determine a
correlation of some penetration parameter involving velocity, V, by solving an
equation of motion. The best known of these laws is the expression
Usually, in order to solve the penetration problem simply, the application of
the physical principles must be accompanied by restrictive assumptions.
Examples are ideal rigidity of the projectile, ideal plastic behaviour of the target
and specified modes of deformation in the target that lead to perforation.
Momentum or energy balances between striker and target have been successfully
applied. Other formulations have included: 1. target compression due to projectile
motion, the effect of friction and target inertia, 2. only target resistance and
inertia, and 3. the combination of target inertia, resistance to flow and friction.
Hypervelocities
The larger stresses occurring in hypervelocity impact permit neglect of rigidity
and compressibility of the striking bodies and the impact is viewed as fluid flow.
The description of material properties is then greatly simplified. Subsequently,
fluid mechanics can be used with correction terms for target and projectile
strength. Fluid models have been used to study the penetration of shaped
charges, and to study wave propagation in long rods simulating kinetic energy
penetrators.
Computer Modelling – Numerical Solutions
Because the theory of terminal ballistics is so complicated, most of the current
work uses numerical schemes. Briefly, we list some of the current computer codes
that are used to solve particular problems usually depending on the target size.
Goldsmith, (2), provides extensive references for further study. For thin plate
penetration and perforation, there are various computer codes covering this
rather large area of terminal ballistics. A computer solution for the deformation
of clamped circular plates under explosive loading, using the structural elastic/
plastic program DEPROSS, has been found to compare satisfactorily with
measured deflections and strains (3). STEEP (4) was employed to study the ejecta
characteristics behind thin plates of aluminium, copper and cadmium struck by
spheres at 7500 and 15000 m/s. The expansion of the debris cloud was traced and
* The ballistic limit V the mass axial momentum and kinetic energy of the ejecta were evaluated. 50 is defined as that velocity which penetrates for 50% of the time.
F  a7  a8V  a9V2
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164 Military Ballistics Terminal Ballistics – Part II 165
CRAM (5) computed perforation problems such as plugging and petal ling failure.
CRAM was also used for the study of penetration of equal weight, equal base
diameter steel cylinders, truncated cones, and ogival projectiles into steel and
aluminium targets at velocities up to 1000 mls. The monograph on High Velocity
Impact Phenomena (6) contains several papers on the subject as well as a
description of the CAMEO calculations for the behaviour of debris generated
upon hypervelocity perforation of thin targets and shields. Reference (7) contains
a section on penetration mechanics with several useful references to numerical
aspects of thin plate perforation. The impact of blunt and pointed tungsten
carbide and steel projectiles on steel armour and aluminium plates at velocities of
1000 to 1300 m/s was examined by the SHEP code. Penetration, plugging and
hole formation were explicitly treated (8). Applications to a numerical study of
hypervelocity impact, plugging failure in thin plates occurring during ballistic
impact and shaped charge jet formation are described in (9).
Thick Targets
Many of the numerical procedures already described either apply equally well
or may be readily converted to the case of thick targets, although here, the failure
processes are far more complicated.
Semi-Infinite Targets
Numerical methods have been applied to soil penetrations using the codes
WAVE-L and PISCES-DL2 (10). The results indicated that after the nondeforming projectile has been fully embedded in a given soil layer, the penetration process is steady. Disturbances of the soil do not extend far from the
projectile but there is generally separation of the soil from the projectile along the
nose. Deformations of the projectile are extremely small so that the projectile is
assumed rigid. Earth penetrators are intended to deliver a warhead whose
detonation either damages a subterranean structure or else causes extensive
cratering to a runway or road etc.
Summary
The ranges of impact velocity, thickness, angle of attack, and material characteristics are the basis for establishing some order among the approaches to the
subject of penetration mechanics. There is a great diversity among these approaches, which vary in sophistication from simple descriptions of test results to
detailed mechanical theories. The diversity of these approaches is due in part to
several aspects of the contributing phenomena. The range of impact speeds result
in mechanical phenomena whose descriptions are nonlinear and involve significant changes of material behaviour.
The simplest approach for a specific problem is to derive an empirical relationship, but this has the obvious disadvantage of non-applicability to other
situations. Analytical models are potentially the means for arriving at a simple
solution because these derive conclusions from fundamental concepts through
idealisations and simplifications and put the expressions in a tractable form.
Numerical solutions of sets of differential equations that represent the fundamental conservation laws and constitutive relations certainly promise to provide
the answers sought, but these have disadvantages of expense and complexity.
References
1 Goad, K. J. W. and Halsey, D. H. J. (1982). Ammunition; including Grenades and
Mines. Brassey’s Battlefield Weapons Systems and Technology, Vol. III.
2 Goldsmith, W. and Backman, M. E. (1978). The Mechanics of Penetration of
Projectiles into Targets, Int. J. Engng. Sci., Vol. 16. pp. 1–99. Pergamon Press.
3 Jones, N. (1971). Int. J. Solids Struct. 7, 1007.
4 Halda, E. J. and Riney, T. D. (1966). Doc. No. 66SD 409, General Electric Company,
Missile and Space Division, Philadelphia, Pennsylvania.
5 Sedgwick, R. T., (1968). Tech. Rpt. AFATL-TR-68–61. Air Force Armament
Laboratory, Eglin Air Force Base.
6 Kinslow, R. (Editor), High-Velocity Impact Phenomena, Academic Press, New York,
(1970) .
7 Army Materials and Mechanics Research Centre, Proc. Army Symp. Sol. Mech.
(1972). AMMRCMS Watertown, Massachusetts (1973).
8 Wagner, M. H. (1973). Proc. Army Symp. Solid Mechanics, (1972). pp. 196–208,
AMMRCMS 73–2, AD772 827, Watertown, Massachusetts.
9 Sedgwick, R. T. (1973). Proc. Army Symp. Solid Mechanics. pp. 209–219. AMMRC MS
73–2, AD 772–827.
10 Wilkins, M. L. (1963). Rpt. UCRL-7322, Lawrence Livermore Laboratory, Livermore,
California.
Self Test Questions
QUESTION 1 What are the main factors that affect impact phenomena?
Answer………………………………………………………………………………..
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QUESTION 2 What is the difference between penetration and perforation?
Answer………………………………………………………………………………..
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QUESTION 3 What three phases generally describe the interaction of a longrod penetrator with a target?
Answer………………………………………………………………………………..
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166 Military Ballistics
167
6.
Wound Ballistics
QUESTION 4 Mention some ways in which a target may be defeated by a
projectile without completing its passage through the armour.
Answer………………………………………………………………………………..
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QUESTION 5 What are the main features of an APDS round?
Answer………………………………………………………………………………..
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QUESTION 6 What are the advantages of an APFSDS round over an APDS
round?
Answer………………………………………………………………………………..
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QUESTION 7 Under what conditions would (a) blast be effective? (b) fragmentation be effective?
Answer (a)……………………………………………………………………………
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(b)……………………………………………………………………………………….
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QUESTION 8 How does a HEAT missile achieve its main effectiveness
against armour?
Answer………………………………………………………………………………..
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QUESTION 9 What type of anti-tank projectile would be relatively ineffective
against spaced armour?
Answer………………………………………………………………………………..
QUESTION 10 What are the main ways of investigating the subject of terminal
ballistics?
Answer………………………………………………………………………………..
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The Problem
Wound ballistics may be defined as the study of the motion of missiles within
the body and the wounding capacity of them. It is therefore basic to the understanding of the wounding effects of both bullets and fragments, which in the past
have caused over 90 per cent of combat casualties. A wound results from the
absorption of energy imparted by a missile when it strikes and penetrates tissue:
consequently the study of wound ballistics requires knowledge of the behaviour
of the bullet in flight and the effect it has on the tissues it penetrates. These
effects depend on the size, shape, composition and above all, the velocity and
stability of the bullet. In the body, the elasticity and density are the most
important factors which influence the retardation of the penetrating missile. The
soft tissues of the body, like water, are 800 to 900 times as dense as air and when
a bullet hits these tissues it nearly always becomes unstable; any angle of yaw
that is present will be greatly increased, with a subsequent increase in damage.
In fact trials have shown that a very slight yaw at impact may develop yaw
within the target of between 50 to 100 degrees. Generally, a bullet can cause
injury in one of three ways, depending on its velocity.
Causes of Injury
Laceration and Crushing
The main effect of low velocity, that is subsonic, missiles is to penetrate the
tissues and crush and force them apart. Only those tissues that have been
directly hit are damaged and the actual wound damage is comparable to that of a
knife wound. The laceration and crushing effect is not generally serious unless
vital organs or major blood vessels are directly injured.
Figure 6.1 demonstrates the typical path of a small subsonic projectile penetrating a gelatine block.* The bullet simply gives up its energy by creating a
small wound track formed by the displacement of the gelatine. The two other
types of injury occur with high velocity or supersonic projectiles.
* 20% gelatine and soap are often used in wound ballistic studies for simulating the human body.
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168 Military Ballistics Wound Ballistics 169
Stress Waves
When a high velocity bullet forces a wound track through solid tissue, it
compresses the medium in front of it and this region of compression moves away
as a stress wave of spherical form (see Fig. 6.2). The velocity of this stress wave is
approximately equal to that of the velocity of sound in water, that is, 1500 m/s.
The changes of pressure due to stress waves only last about a millionth of a
second, but they may reach peak values of up to 100 atm.†
Although the inertia of
the tissue does not allow the stress wave to bring about any transfer of tissue,
damage to the nerves, for example, may be caused at a considerable distance from
the permanent wound track.
Temporary Cavitation
The severe wounding effects caused by high velocity projectiles are mainly due
to temporary cavitation. When the bullet enters the body, momentum is transferred to the surrounding tissues. This momentum causes the tissues to move and
oscillate even after the passage of the projectile and thus a large cavity is created:
it is often approximately 30–40 times the diameter of the missile. This cavity, in
the space of a few milliseconds, goes through several pulsations, expanding and
retracting before reverting to a semi-permanent shape. These violent changes in
the tissues are sufficient to fracture bones, rupture organs and blood vessels and
damage nerves outside the immediate path of the projectile. The shape of this
temporary cavity is, except where yaw has taken effect, in the form of an
ellipsoid. Because the cavity has a subatmospheric pressure and is connected to
the outside by entry and exit holes, bacteria from the outside, together with
clothing and debris, are actively sucked into the depth of the wound.
Cavitation takes place mainly after the passage of the missile, and accounts for
the explosive nature of high velocity missile wounds. The greater the energy that
is imparted to the tissues, the greater is the size of the temporary cavity and the
more extensive the damage.
Ammunition Design and Wound Effectiveness
Logical Limitations
The design of projectiles has a great influence on the wounding effects. The
Hague Convention of 1899 banned bullets which have jackets with slits or an
opening at the point which would permit the jacket to strip upon impact with the
target. Such a design would allow the lead core to mushroom, causing very
serious entrance wounds. However, the Convention lacked a full appreciation of
such factors as yaw and velocity effects. This is because wound ballistics as a
science had not begun to develop.

1 atm is equivalent to 0.1 MPa
FIG. 6.1 Wound track of subsonic projectile
FIG. 6.2 Stress wave formed by high velocity projectile
FIG. 6.3 Temporary cavity formed by high velocity projectile
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170 Military Ballistics Wound Ballistics 171
Velocity and Stability
Pistol bullets have a relatively low amount of energy available to cause
damage. All rifle bullets have much more energy and if a rifle bullet is stable on
impact, it might go right through the target giving up only 10–20 per cent of its
energy. If it leaves unstable, it may give up 60–70 per cent of its energy, and
hence cause a more severe wound. In the extreme, if the bullet fragments on
impact, all the energy will be used up and a severe wound would result. The
external appearance of a bullet wound can be deceptive. If the bullet enters or
leaves the body end-on, then it will commonly have a small hole, irrespective of
the severe damage it may have caused during the passage through the body. If
the bullet enters, or more commonly, leaves the skin sideways on to some degree,
then the hole in the skin will be large and ragged, but the internal damage may
be no more severe.
Projectile Design
For the same amount of total energy expended, the design of the bullet can
make a profound difference in its effect on tissues. For example, a soft bullet will
flatten on impact, producing a much greater surface area and, therefore, greater
retardation. Such a design will produce a very early release of energy (see Fig. 6.5
(a)). If the bullet is an unstable jacketed bullet, the energy would be more rapidly
released as the bullet begins to yaw, Fig. 6.5 (b). If it is a stable jacketed bullet,
the energy is dissipated rather late and a longer wound track is observed, Fig. 6.5
(c). Obviously the length of the wound track is a measure of the stability of the
bullet in soft tissue.
Finally, two particular instances of the effect of design on wounding are worth
mentioning. The British .303 inch Ball Mark 7 created an unintentionally severe
wound. In this design the nose filling consisted of a metal plug. The nose tended
to break off and could cause break up of the bullet in a wound with consequent
serious wounding effects. The intention behind this design however had simply
been to improve the external ballistics of the bullet by placing the centre of
gravity as far back as possible. Other projectiles such as the 6.5 mm Japanese
rifle round had a thickened lead core at the base, which despite its small calibre
and lowish velocity, tumbled badly upon hitting tissue, and caused serious
wounds. Therefore, in recent years, bullet design has been concerned not only
with external ballistics and accuracy, but also with the wounding effects.
Body Armour
The Need
The idea of body armour has been with man since the earliest of times. In
modern life there is a requirement for lightweight armours capable of stopping
high velocity shot. This requirement covers both battlefield and internal security
situations. Early bullet-proof vests were made mainly from metal plates, but
they were generally clumsy. However, advances in metallurgy and synthetic
FIG. 6.4 Selected stills from a high speed cine film showing the temporary cavitation
FIG. 6.5 The respective wounding effects of a soft bullet, unstable jacketed bullet and
stable jacketed bullet
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172 Military Ballistics Wound Ballistics 173
materials have made it possible to produce protective coverings which are effective but at the same time light enough to be worn without discomfort.
The capacity of a high velocity projectile for causing bodily injury or damage,
on impact, is determined by its kinetic energy E. More accurately, for projectiles
of different shapes, it is convenient to define damage capacity as E/A where A is
the contact area on impact. The armour used must resist penetration of the
projectile, and also dissipate most of the kinetic energy of the shot. For high
velocity bullets this is extremely difficult to achieve within the severe constraints
of weight and bulk. Also, because the body is in close contact with the armour,
and the vital organs are susceptible to relatively small amounts of kinetic
energy, the problem is further complicated. Differing solutions to the problem
have been produced.
Metallic Armour
Metallic armour acts by totally rejecting the bullet and preventing any penetration. It usually breaks up a projectile and distributes the energy over a large
surface area of the plate. Metallic armours are the best material for stopping high
velocity rifle bullets, and it is the only material successful in stopping steel-cored
bullets or armour-piercing projectiles. Modern techniques can produce a light
low-alloy steel armour which is used in light jackets for total ballistic protection
and is also put into doors and seats of vehicles and helicopters. Also, metallic
armours can usually survive more than one bullet impact in the same area. The
ability to withstand such close and repeated shots is something that other
armours do not have and is a strong factor in choosing metal for certain applications. On the other hand, all metallic armours suffer from the disadvantage of
spalling on projectile impact: this is a result of shock wave reflection from the
back surface. Another disadvantage of metallic armour, illustrated in Fig. 6.7, is
that if the plate is in direct contact with the body a shock wave can be transmitted to the underlying tissues and damage from cavitation can occur.
Reinforced Plastic Armour
Reinforced plastic armour protects by partially rejecting the bullet and
partially absorbing it. In the process each bullet strike partly destroys the area it
hits. Plastic is therefore not as good as metal at defending against more than one
hit over a small area and the area of armour which has been struck must be
replaced if the overall protection of the plate is to be maintained. Moulding the
plastic provides easy shaping for a particular application. In acceptable weights
it offers protection only against low-velocity bullets although it is particularly
good against blast and fragmentation attack from grenades or shells.
Ceramic Armour
Ceramics are characterised by their hardness and low density. Armourpiercing bullet design favours high density, high hardness core materials, in
particular tungsten carbide. For the effective shatter of tungsten carbide bullets,
the ceramic layer should exceed the hardness of the bullet, so something like
boron carbide is typically used for a ceramic armour. Ballistic attack is defeated
by the material starring, cracking and generally degenerating rapidly. Any one
area cannot accept more than one hit, and to limit the effects of this it is generally
used in the form of small tiles, each one of which will absorb one hit. It needs to be
backed with some extra form of protection such as textile armours, but does offer
FIG. 6.6 The permanent record of cavitation effects as shown in standard soap blocks very high levels of protection from bullets of all normal calibres. The major
Cavity produced by a 6 mm steel sphere
travelling from left to right with an impact
velocity of 1100 m/s.
Cavity produced by a 7.62 mm NATO ball
round travelling from left to right with an
impact velocity of 850 m/s.
FIG. 6.7 Possible cavitation effects with metallic armour
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174 Military Ballistics Wound Ballistics 175
disadvantage of ceramic armours is that they are expensive. It is also necessary
to replace any damaged tiles which makes this type of armour suitable only for
specialised uses.
Textile Armour
Textile armours are a series of layers of heavy-weave cloth such as nylon or
Kevlar sewn together. They will stop any round nosed bullet, for example about
16 layers or plys will stop a 9 mm bullet travelling at 330 m/s (see Fig. 6.8). The
bullet is caught in the heavy weave as it hits, and the nose is rapidly deformed
into a rounded mushroom. This mushroomed bullet is then unable to force its way
through the mesh of fibres and stops. The material, despite the number of layers,
is flexible and can be easily tailored into a waistcoat. There is a possibility of
injury to the wearer, due to the cavitation that occurs upon impact, but the extent
of any injury is generally considered to be relatively small.
Transparent Armour
Transparent armours are generally made from one of three materials; glass,
polycarbonate or acrylic plastic. With all of these some degree of lamination is
necessary to break up the impact of the projectile’s stress waves, and prevent
sudden shattering. Glass armour is the only transparent armour which can be
made to withstand rifle bullets, but there are obvious serious drawbacks to its
use: the weight, and the need for lamination behind the glass to prevent damaging splinters flying off with each impact are the most important. Plastic armour,
such as polycarbonate sheets, is virtually unbreakable and makes a useful riot
shield. In thick laminations it offers good protection against low-velocity bullets,
but not against rifle calibres. It also has the advantage that it generally does not
crack.
Composite Armour
The concept of composite armour is to dissipate a large amount of kinetic
energy without transmitting it to the target situated behind the armour. This
may be achieved in three ways: firstly, it slows the projectile to reduce the kinetic
energy; secondly, it induces the projectile to break-up and decreases the kinetic
energy per unit area, and thirdly it absorbs residual kinetic energy.
In practice a triple composite armour consisting of an outer layer of metal, an
inner layer of ceramic and a backing layer of plastic material provides an
effective shield. The metal layer slows down the projectile while the ceramic
layer induces shatter of the projectile and finally the plastic layer, since it is
capable of elastic deformation, absorbs the kinetic energy. In addition, the plastic
provides a protective backing and by a suitable choice may minimise stress
reflections and hence spalling. Such a composite armour offers significant advantages over metallic armour. At 60 kgm2 it will protect against a 7.62 mm shot, as
compared with over 100 kgm2 required for steel armour. Composite armour also
offers a better performance than metals for varying angles of attack. However,
although composite armour uses the advantages of the individual armours from
which it is constructed, it also suffers from various disadvantages: it has a limited
re-use capability because the progressive breakdown of the ceramic layer is not
easily repairable; in addition it is expensive to manufacture.
Composite armours are now replacing the other types of armour in many
situations. This is due to the comparative lightness and effectiveness of the
ceramic layer. Although the processing of ceramics is expensive, new technology
will probably reduce the cost. In the end, however, the particular type of armour
used ultimately depends upon the likely form of attack.
Self Test Questions
QUESTION 1 What physical quantity causes a wound to result?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 2 What factors influence the retardation of a penetrating missile?
Answer………………………………………………………………………………..
QUESTION 3 What causes temporary cavitation?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
FIG. 6.8 The non-penetrating impact of a 9 mm bullet travelling at 330 m/s against
16 plys of Kevlar body armour
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176 Military Ballistics
177
7.
Ballistics Instrumentation
QUESTION 4 Is it necessarily true that projectiles with the same kinetic
energy create the same wounding effects?
Answer………………………………………………………………………………..
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QUESTION 5 What are the main requirements of body armour?
Answer………………………………………………………………………………..
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QUESTION 6 What are the advantages and disadvantages of metallic armour
over other types of armour?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
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QUESTION 7 What are the advantages and disadvantages of ceramic armour?
Answer………………………………………………………………………………..
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QUESTION 8 How does textile armour prevent penetration?
Answer………………………………………………………………………………..
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Introduction
All ballistic phenomena occur at high speeds, and a great number of complex
interactions occur; to study these phenomena it is necessary to use specialised
measuring equipment. In this chapter we shall look at various high speed
photographic techniques applicable to all branches of ballistics, followed by
external and in-bore instrumentation. We are usually concerned with microsecond photography, projectile velocity, displacement, yaw and spin, as well as
pressure measurements up to several thousand mega pascals.*
High Speed Photography in Ballistics
The Problem
High speed photography is used directly in all branches of ballistics. As
ballistic photography is such a wide-ranging subject we shall only present those
techniques which are well established and capable of easy use†
. In general, all
events in ballistics take place quickly, and in order to obtain clear pictures
exposures are usually very short. As object movement becomes faster, exposure
must decrease and illumination must increase or the image must be artificially
enhanced electronically.
Photographic System
A photographic system consists primarily of a light source, optics, a shutter
and a means of recording the photographic image. The particular type of photographic system chosen depends upon the duration of the event. An indication of
typical time durations for various illuminating sources is as shown in Fig. 7.1.
For example, the displacement of a 7.62 mm projectile travelling at 840 m/s is
approximately 1 mm in 1.2  10–6 seconds. To ‘freeze’ such a motion requires a
light source which has a duration considerably less than this, and so we use a
spark. If we were to use a microflash system which has a typical time duration
flash of 10–5 seconds, the bullet would be blurred. If a single event is to be
recorded then a continuous light source with a rapid shutter can be used in
* 1 mega pascal is equivalent to about ten atmospheres of pressure.

Fuller (1) gives a more detailed account of high speed photography in ballistics and many further
references.
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178 Military Ballistics Ballistics Instrumentation 179
conjunction with stationary film; if however a time history is required it is usual
to use either high speed motion film or to have fixed film with opto-mechanical
scanning of the image across the film or synchronised with a moving film. For a
limited number of images a sequence of sparks or flashes can be recorded in
superposition on a stationary film.
Microflash Photography
With the microflash system an intense flash of a few microseconds duration is
produced when a high voltage is discharged through a gas-filled lamp normally
containing argon or xenon, and this flash can be used to provide both the
illumination and the short exposure necessary for a sharply defined image of a
rapidly moving object, as illustrated in Fig. 7.2.
In general this technique is only suitable for subsonic projectiles. This is
evidenced by the blur on Fig. 7.3. It is worthwhile comparing this photograph
with Fig. 7.12 using Ballistic Synchro. Microflash photography has applications
in ballistics work for the study of projectiles in flight to observe an irregularity in
behaviour or damage that might be sustained during the firing process. A series
of photographs of a particular event can be obtained by using multiple flashes.
The same system may also be used to measure the spin of projectiles. To achieve
this lines are painted longitudinally on the shell body which is photographed in
flight in the light of two microflash lamps spaced a short distance apart down the
range. The time interval between the two flashes is recorded and the rotation of
the shell during this interval can be measured from the photograph: from the
measurements the rate of spin can be calculated. Triggering the flash at the
required moment may be performed with the breaking of a light beam, an
electrical circuit, or by sound detection.
Spark Photography – Shadowgraphs
Sparks are certainly the oldest form of illumination for ballistic photography
and their popularity and use continue. The great advantage of this method lies in
its short exposure time. The spark, which is usually less than 0.5 microseconds in
duration, is produced by discharging a high voltage through a gas such as argon
on the opposite side of the projectile from the camera, and the light emitted from
the spark then ‘shadows’ the object. When obtaining a shadowgraph of a supersonic projectile the spark is often triggered by a detector sensing the nose shock
wave from the projectile. The shock wave is seen as a shadow because the large
pressure disturbances generated by the shock wave create density gradients in
the shock wave which refract the light away from the direct path, as evidenced by
the appearance of darkened lines.
By triggering several sparks after known time delays successive images may
be recorded on the film.
Spark photography is used for studies of armour protection and penetration,
yaw measurement, air resistance, aerodynamic performance of a particular
shape of projectile, and surrounding airflow around an event when it is subject to
a pressure front.
Schlieren
The Schlieren technique is another method for detecting the density change in
the field of view. All of the light emitted from a small light source is focused on to
a small obstruction: only the light refracted by the pressure disturbance in the
field of view is focused on the photographic plate.
FIG. 7.1 Time scale for various illuminating sources
FIG. 7.2 A microflash photograph of a 105 mm shell in flight
FIG. 7.3 A microflash photograph of 120 mm APDS in flight. The blurring is caused by
long duration of flash
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180 Military Ballistics Ballistics Instrumentation 181
High Speed Cine Photography
Slow Speed Cine
By filming at a high framing rate and projecting at the normal 16 or 24 pictures
per second (p.p.s) the time scale is extended and a recorded event can be viewed in
slow motion. Conventional cine cameras employ a mechanical system of film
transport where claws engage in perforations along the edge of the film and pull
it past the aperture frame by frame. The movement is intermittent because the
film is held stationary in front of the aperture during exposure. The result is a
sharp image.
Figure 7.9 shows a rocket launch taken at a frame rate of 40 p.p.s. with an
exposure time of 1/1000th sec. The filming speed of the conventional type of
camera is limited ultimately by the mechanical strength of the film base; above
1000 p.p.s. the film will tear. Hence, to achieve higher framing speeds a different
FIG. 7.4 Experimental arrangement for spark photograph
FIG. 7.5 Diagrammatic representation of the shadowgraph
FIG. 7.6 Multiple spark photograph
FIG. 7.7 Schlieren photograph of sphere
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182 Military Ballistics Ballistics Instrumentation 183
system of film transport is required. This is provided by the continuous running
type of camera.
Medium Speed Cine
A medium speed camera gives up to about 10,000 full p.p.s. with 16 mm film
and over 20, 000 p.p.s. with quarter frame. It is extremely versatile and probably
the most widely used in ballistics work. Here the film is run through the camera
continuously at high speed and is not held stationary during exposure. To
prevent blurring of the image due to film movement, an ‘optical compensator’ is
incorporated between the lens and the film, illustrated in Fig. 7.10. This takes
the form of a rotating glass block which is usually geared to the film drive. As the
block rotates, light rays from the lens are refracted to an extent which depends
upon the angular position of the block and the effect is to move the image in the
same direction that the film is travelling. Definition is not quite as good as that
produced by the conventional camera. Continuous running cameras of this type
can operate satisfactorily at filming speeds up to 104 p.p.s. If a standard 30 m film
spool is accelerated to this rate, the film will reach a speed of 273 km/h and be
fully exposed in less than three quarters of a second. Correct triggering of the
event is therefore critical.
The framing rate is controlled by the motors driving the sprocketed drive wheel
and take-up spool. Useful recording time is governed by the time used in accelerating the camera to operating speed. It is common practice to record a time base
on the film edge to allow accurate film speed determination during analysis. This
is achieved by a light emitting diode which simply flashes a small light spot on
the edge of the running film at a known rate, for example at every 1 /1000th sec.
FIG. 7.8 The Schlieren system
FIG. 7.9 Rocket launching
FIG. 7.10 Compensation for film movement by rotating glass block
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184 Military Ballistics Ballistics Instrumentation 185
Lighting
The very short exposure time associated with fast filming speeds makes high
intensity lighting necessary. For most purposes suitable lighting can be provided
by photoflood lamps. Typically, these are rated at 750 or 1000 watts each and
have built-in reflectors so that the light can be focused onto the subject. A more
sophisticated form of lighting is known as ‘synchronous flash’ or ‘strobe illumination’ which is produced by a stroboscopic unit. Briefly, this provides a high
intensity repetitive flash of short duration by means of a gas discharge lamp. It
can be synchronised to the camera so that it flashes at the same rate as the
filming speed; that is, a separate flash is produced for each exposure. The
available flash energy provides a more intense illumination than from photoflood
lamps.
Ballistic Synchro
Ballistic synchro is a powerful technique using the medium speed camera
without its compensating block in a streak mode. This technique produces high
quality front-lit photographs showing projectile condition, yaw and spin rate, as
well as the operation of sabots and obturators. The method used to compensate for
projectile motion is shown in Fig. 7.11.
High Speed Cameras
Beyond a certain acceleration rate, film fails mechanically and higher framing
rates must be achieved by alternative means. There are a number of types of high
speed camera. Many allow the film to remain stationary and sweep the image
across it by reflection from a revolving mirror or prism. They are constructed in
two forms, either as streak or framing cameras. Streak photography is obtained
when the camera is used without any film motion compensation at all It is
particularly applicable to explosive studies: the record is taken as a continuous
exposure as the film passes the lens. Figures 7.13 and 7.14 show the two types of
camera arrangements. They are capable of a rate of several million p.p.s. A large
amount of light is required to illuminate the subject, and problems may occur due
to the limited number of frames per film length. The problem of event and mirror
position synchronisation may also occur.
Image Converter Cameras
Image converter cameras have two main advantages: they can electronically
enhance the brightness of an observed event and as a result can give very short
exposure pictures on ordinary film. Framing speeds of a typical Image Converter
type camera can range from 10, 000 to 2 x 107 p.p.s. with exposure times down to
10ns.* Figure 7.15 shows the internal layout of such a camera.
FIG. 7.11 Ballistic synchro technique
FIG. 7.12 A ballistic synchro record of 120 mm APDS
FIG. 7.13 The rotating mirror streak camera
* 1 ns  1  10–9 secs.
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The event is imaged onto a photo-cathode and the stimulated electron beam is
focused onto a phosphor screen. In passing through the image intensifier the
original brightness of the image becomes enhanced to the point where a conventional film camera can photograph it from the phosphor screen. By electronic
control the electron beam carrying the image can be shuttered and scanned
across the phosphor to enable multiple consecutive images to be separately
recorded. Figures 7.16 and 7.17 show respectively the types of image converter
photographs taken in normal mode and streak mode.
Flash Radiography
In some ballistic applications the only way to obtain suitable pictures is by
using X-rays, which are produced by electron bombardment of a metal anode. The
wavelengths normally used for flash X-ray photography range from 10–4 to 10–1
nano-metre with durations of 1 microsecond or less.
Initially only single exposures could be obtained, but developments have led to
repeated discharges in a single tube giving up to 150–200 p.p.s. Currently,
framing rates of 106 and higher have become possible. Up to five pictures may be
recorded on the same film frame, but this leads to loss of definition; for higher
framing rates a continuous film is more common. A recent development has been
the use of image intensifiers where the X-ray image is changed into an optical
image which can then be photographed by an ordinary cine camera.
Radiographs of internal functions in guns (see Fig. 7.18) require high energies
in order to achieve penetration. Typical measurements may be the observation of
FIG. 7.14 The rotating mirror framing camera
FIG. 7.15 Image converter camera schematic
FIG. 7.16 Lead pellet impacting on hard target. Framing rate  1  105 p.p.s.
FIG. 7.17 Streak photograph of 7.62 mm bullet travelling at approximately 840 m/s
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188 Military Ballistics Ballistics Instrumentation 189
projectile movement in the bore with time or the behaviour of parts of the
internal mechanisms under firing stresses. The region just beyond the muzzle
where the projectile is moving through the expanding propellant gas is a particularly difficult region for photographic observations due to the large amount of
flash; so it is a suitable subject for the radiograph technique, as is the mechanism
of sabot detachment and integrity of the projectile itself. By photographing
against known datum lines the projectile yaw can also be seen.
During the external ballistics phase of the flight conventional photography is
usually more effective. However, X-rays can be used to study the condition and
operation of internal mechanisms such as fuzes. In terminal ballistics many
target materials of great interest are metal or other materials which are opaque
to normal photographic methods. Flash radiography thus offers a unique method
of observing the interactive process between projectile and target during penetration. Another advantage of radiography is the ability to photograph through
the luminosity often associated with impact. Radiography is also useful in wound
ballistics, where conventional photography is of course limited.
External Ballistics Measurements
In this section some of the more common forms of measuring equipment
suitable for the study of external ballistics are described. As previously mentioned, we are usually concerned with the displacement of projectiles, their
velocity, retardation, yaw and spin rate.
Photodetector Counter Chronometer (PCC)
PCC equipment is an opto-electronic system for measuring the velocity of a
projectile by timing its trajectory very accurately over a known distance. It
consists of two light intensity detector screens and a velocity computing counter
with interconnecting cables. Different types of detector screens are available,
depending upon the size of the projectile and the angle of elevation at which they
are fired. The detector screen comprises a lens, slit and photodetector coupled to
an amplifier: it is arranged so that light from the sky immediately above the
detector screen is collected by the lens and focused on to the slit through which it
passes to the photodetector below. The passage of the projectile interrupts some of
the light reaching the photodetector causing a sudden reduction in photodetector
current which triggers the counter.
FIG. 7.18 X-ray shadowgraph of 0.45 automatic during firing
FIG. 7.19 PCC equipment
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190 Military Ballistics Ballistics Instrumentation 191
To measure velocity the projectile is fired over two detector screens placed a
known distance apart. The pulse from the first detector screen starts the counter
chronometer, and the pulse from the second one stops it. As the distance between
the screens is known, the velocity can be measured by the velocity computing
counter. On large ranges shells may be fired over many photoelectric detectors
spaced at known intervals. The time taken by the shell to reach each of the
detector screens in turn is measured by electronic counters and a computer can
convert these results to provide a print-out of retardation at corresponding
velocities down range. Since retardation is markedly dependent on any yaw that
occurs during flight, photography may be used to record the attitude of the shell
simultaneously.
Principle of Radio-Doppler
If a radio transmitter emitting continuous waves is directed towards the base of
a shell moving directly away from it, the base of the shell will reflect some of the
radiation. The return signal can then be picked up by a receiver located near the
transmitter. If this receiver is also fed directly from the transmitter it will
simultaneously receive two sets of waves which will interfere, one set direct and
one set via the shell. For certain positions of the shell along the trajectory the two
sets of waves will be in phase and will reinforce each other so that the receiver
will record a maximum signal; for intermediate positions of the shell the two
waves will arrive out of phase and a minimum signal will result. Consequently
the receiver will show a series of maxima and minima as the shell recedes from
the emitter. The total length of the waveform is the sum of the transmitted and
reflected waves, that is, a  b. If the shell moves a distance of one wavelength ,
the total length of the two waves becomes a  b    (a  /2)  (b  /2).
Therefore, the time between each maximum is the time taken for the shell to
travel half a wavelength (/2). Using the relation speed  distance/time, we have
where V is the velocity of the shell. Thus the number of maxima per second,
which is the so-called Doppler frequency fD will be given by:
Since the wavelength  is known, a measurement of fD gives a measurement of
the velocity of the projectile V over a large part of the trajectory. Most commercial radio doppler units also provide facilities to produce a retardation record
from the range of velocities measured. In general the radio-doppler system
possesses advantages over a PCC system because it can be used in adverse
weather conditions such as fog or thick mist which would defeat an optical
method; neither does it require any apparatus to be set up forward of the gun.
Typically it has the disadvantages that a sufficiently large signal is not reflected
from the base of a small-calibre projectile to allow its use in that case and it is
necessary to have a fairly good idea of the expected velocity. However, neither of
these disadvantages apply in the case of the calibration of guns in the field
because it is usual to already have a knowledge of the approximate size and
velocity of the projectile.
Yaw Sonde
The yaw sonde is the only satisfactory means at present available for
measuring yaw and spin behaviour of shells in free flight throughout most of the
trajectory and for all angles of fire. Although different versions are used in the
United Kingdom and North America, the principles are basically the same. The
equipment consists essentially of a miniature radio transmitter which is frequency modulated by a solar cell when the cell is periodically illuminated by the
sun’s rays passing through a pin-hole. The solar cell is ‘V’ shaped and mounted
parallel to the shell’s axis (see Fig. 7.21). It is located at right angles to a pin-hole
in the casing of the shell. The equipment is fitted in a package having the same
shape and characteristics as the nose cone of the shell which it replaces and must
be capable of withstanding the typical 30,000G force that is experienced on firing.
The shell is fired in a plane at right angles to the sun’s rays. In flight, the image
of the sun, formed by the pin-hole, sweeps across the ‘V’ shape solar cell, crossing
first one arm of the ‘V’ and then the other, during each revolution of the shell. The
position at which the sun’s image sweeps across the ‘V’ cell arms depends on the
angle between the axis of the shell and the sector to the sun. Figure 7.22
illustrates the principle of the ‘V’ cell.
Transmission of the information from the sonde starts approximately 0.4
seconds after firing and is picked up by a ground receiver. The output consists of a
train of pulse pairs, one pair having been transmitted for each revolution
FIG. 7.20 Principle of radio-doppler

time between maxima  ––
2V
2V
fD  ––

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192 Military Ballistics Ballistics Instrumentation 193
throughout the shell’s trajectory. The time between successive pulse pairs defines
the spin rate, the time of each pair defines the angle of yaw.
The information can be stored on magnetic tape and used to calculate yaw, spin
rate, precession and stability. A typical example is shown in Fig. 7.24.
Shot Position Indicator
The shot position indicator is an electronic target which automatically records
the X and Y co-ordinates of a supersonic projectile as it passes through the sensor
plane.
The system can be used for measuring the position of a single shot, or the
positions of the individual shots in a burst of automatic fire; it will work over a
range of calibres from 4 mm to greater than 30 mm. The co-ordinates of the
projectile are found by measuring the point of impact of the wave front on two
orthogonally mounted sensor rods. If the projectile passes normally through the
target plane, the shock wave front expands as a circle centred on the position of
the projectile.
The two points of impact of the shock wave front on the two axes are then
related to the X and Y co-ordinates of the projectile (see Fig. 7.26). The point is
FIG. 7.21 Principle of yaw sonde
FIG. 7.22 Principle of the ‘V’ cell
FIG. 7.23 Interpretation of pulse record
FIG. 7.24 Results from yaw sonde
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194 Military Ballistics Ballistics Instrumentation 195
found by measuring the difference in arrival time between longitudinal vibrations at each end of the rod. The shock wave at the impact point sets up a system
of longitudinal and transverse waves in the rod material. The longitudinal waves
travel faster than the transverse waves and so reach the end of the rod first. Two
transducers mounted on the ends of the rod convert the longitudinal waves to two
electrical signals; by measuring the difference in arrival times of those signals
the point of impact 1 may be found from 1  (t1  t2)V/2 where V is the velocity of
the longitudinal waves in the rod material. When the shock wave strikes the
middle of the rod, the two arrival times are equal and will be zero. This implies
that the origin of the co-ordinate system is at the centre of the rod and that a
positive or negative sign must be used to distinguish readings from either side of
the centre point. The sign convention follows the normal Cartesian co-ordinate
system, illustrated in Fig. 7.27.
In-bore Pressure Instrumentation
The Problem
In-bore instrumentation consists of the determination of the loading and response of projectiles and projectile components during in-bore travel. In-bore
travel is defined as beginning at the time of projectile ramming (shot-start) and
continues through the interior ballistic cycle and muzzle exit until the projectile
is no longer influenced by the propellant gases. There are two prime objectives:
the first is to quantify the specific loading and response as actually experienced
by the projectile and gun; the second is to accurately simulate these loading
conditions accurately in mechanical devices other than guns for both cheaper
testing and for establishing models. The principal measurements required are
FIG. 7.25 Shot position indicator
FIG. 7.27 Cartesian co-ordinate system
FIG. 7.26 Position of shock wave front
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196 Military Ballistics Ballistics Instrumentation 197
pressure and projectile travel; the latter leads to projectile velocity and acceleration. If the measuring equipment is carried in the projectile then it must
withstand large forces due to large accelerations, both longitudinally and transversely, because of vibration in the bore during travel. Additionally, the information must be extracted from the projectile, and this requires either flexible
wire connections or radio telemetry. It is clear that gun-mounted rather than
projectile mounted instrumentation provides an easier environment. Some of the
currently available measuring techniques used in internal ballistic instrumentation are described in the following paragraphs.
Gas Pressure Measurement
Two basic types of pressure gauge are used. There are gauges which only
provide a record of the maximum gas pressure reached during the firing process,
for example, the Crusher gauge, and then there are gauges such as the
Piezoelectric gauge, which provide a continuous record of pressure/time
variation.
Two types of crusher gauge are in common use, one fitted flush with the inside
of the gun barrel wall and the other which is located in the propellant for large
guns.
An assessment of the maximum pressure reached in the gun barrel and
chamber during the firing process is obtained by subjecting to this pressure a
cylinder or sphere of copper and measuring the extent to which it is permanently
compressed or crushed. A typical example would be a copper ball of 4.76 mm
diameter. The crusher gauge response represents only about 80% of peak pressure but it can be calibrated against the piezoelectric gauge to read full peak
pressure. Although not an extremely accurate gauge it is adequate for making
comparative measurements.
The piezoelectric gauge is again usually placed flush with the interior wall of
the barrel. The gauge uses the phenomenon that certain crystals, when subjected
to pressure, develop an electric charge proportional to the applied pressure. This
charge is generated at an unusually high impedance, in the range of 109–1014
ohms: the cause is the high impedance connecting cables which are necessary to
prevent loss of signal charge. The charge is fed into a charge amplifier to convert
to a lower impedance and also to produce a voltage proportional to the input
charge. The signal can then be recorded. Recordings are commonly made into
digital storage oscilloscopes or on to magnetic tape and subsequently processed.
In-bore Projectile Movement
The Problem
Methods are required to provide information on the variation of axial displacement of a projectile in a gun during firing.
Microwave Interferometer
In a microwave interferometer, a microwave source of known frequency produces a signal which is fed to an antenna. The microwave energy is directed by a
cheap replaceable reflector so that it propagates down the barrel of the gun. Some
of the incident signal is reflected from the moving shell and picked up by the
antenna. When the instantaneous received and transmitted signals are compared
there is a cyclic change in phase difference which can be calibrated in terms of the
projectile motion. Various commercial microwave bands are used according to the
size of gun barrel and to the delineation of motion that is required: ‘X’ Band
(10,000 MHz) and ‘Q’ Band (35,000 MHz) are the most common. They give
interference maxima every 15 mm and 4.5 mm respectively when used in their
fundamental modes. The number of maxima is therefore a measure of the total
distance traversed.
The only continuous method for measuring in-bore velocity is to use the
interferometer, where the frequency of the output signal from the interferometer
is proportional to shell velocity. This frequency shift, known as the Doppler Shift,
can be used to evaluate shell velocity.
Laser Interferometry
Because of its shorter wavelength the laser provides superior discrimination to
that obtained from a microwave system. However, because the frequency shift
associated with even low velocity movement becomes impracticably high it is
necessary to provide automatic electronic processing to deal with the very large
numbers of interference maxima. The use of a CO2 laser in preference to the more
commonly available helium-neon laser increases the distance between interference maxima from 0.315 µm to 5.3 µm: this enables projectile displacement to
be made with greater accuracy than with the ‘Q’ Band microwave source where
the distance between maxima is 4.5 mm.
FIG. 7.28 Crusher gauge
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198 Military Ballistics Ballistics Instrumentation 199
Direct Optical Method
A light beam reflected from the front of a shell can be directed as shown in
Fig. 7.30, to a light sensitive detector and so provide a direct optical method of
measurement. When the shell is in its start position the light beam strikes the
centre of a plane mirror placed on the shell nose; the reflected beam from the shell
then passes through an optical processing system. As the shell moves towards the
muzzle, the reflected beam path changes and an image moves across the face of
the detector device. If the detector consists of a graduated coated piece of glass,
and focusing arrangement, then the light intensity picked up by a photo-detector
indicates the shell displacement.
Bore-Wire Resistance Method
In the bore-wire resistance method a wire collecting cup is connected to the
nose of the shell. A multi strand bore-wire of constant resistance per unit length
FIG. 7.29 Basic interferometer system
FIG. 7.30 Direct optical method
FIG. 7.31 Bore-wire resistance method
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200 Military Ballistics Ballistics Instrumentation 201
is connected tautly between the cup and a solid mount. A constant current supply
is connected to the bore-wire, with a return path through the shell body and
driving band to the gun barrel; the shell acts as a low-resistance moveable
terminal. As it moves, the bore-wire is gathered in the cup and the decrease in
bore-wire length produces a change in resistance which can be detected as a
voltage change at any point between the cup and the current supply.
Barrel Contacts
Insulating probes can be inserted through holes drilled in the barrel wall to
make barrel contacts. When the shell makes contact with them an electrical
circuit is completed and provides signals for recording the time of arrival. It is
very important that the probes are placed at the correct depth to ensure each
probe operates at the same relative shell position.
Barrel Strain Gauges
In this technique, strain gauges are mounted circumferentially at a number of
positions on the exterior of the barrel. They then produce the required information by acting as ‘time of arrival’ gauges.
Linear Displacement Transducers
Linear displacement transducers use a coil assembly wound on a hollow
cylinder through which a rod acting as a transformer core moves. As the conducting core is moved through the coil the impedance rises and gives the required
measurements.
Projectile Acceleration
Accelerometers
All gauges used for in-bore work are deflection-type accelerometers which
basically consist of a seismic mass and some, preferably linear, force-resisting
process. The level of acceleration is detected by the ‘deflection’ of the restoring
system as it responds to the inertial force on the seismic mass. In Piezoelectric
accelerometers the seismic mass is supported by a piezoelectric material. When
the crystal is stressed by the mass under acceleration an electric charge is
generated which can then be processed using charge-amplifier techniques.
Capacitive Accelerometers
Capacitive accelerometers consist of deflecting a diaphragm replacing the
distinct ‘mass’ and ‘spring’ sections of other gauges: as a result acceleration may
be measured by observing the change in capacitance between the insulated metal
diaphragm and the gauge housing.
Projectile Rotation
The measurement of rotation in the bore can be carried out using optical or
microwave methods. In both cases a beam is transmitted from an external source
down the barrel where it is returned by special reflectors placed on the nose of the
shell.
Thermal Measurements
Thermal measurements related to internal ballistics phenomena are difficult
to make and it is not certain whether existing techniques provide sufficient data
and accuracy for them to be used with confidence.
Reference
1 Fuller, P. W. W. (1979). High speed photography in ballistics, pp. 112–123 ICIASF ‘79
RECORD.
Self Test Questions
QUESTION 1 In ballistics, what is the essential requirement of any photographic system?
Answer………………………………………………………………………………..
QUESTION 2 How is it possible to measure the spin rate of projectiles using
the microflash system?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
FIG. 7.32 Linear displacement transducers
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202 Military Ballistics
Answers to Self Test Questions
Chapter 1
QUESTION 1 Leonardo da Vinci.
QUESTION 2 Galileo Galilei.
QUESTION 3 a. Rigid body dynamics.
b. Fluid dynamics.
QUESTION 4 Ballistic pendulum.
QUESTION 5 Drag as a function of the properties of air was recognised.
QUESTION 6 To allow the use of streamlined spun-stabilised projectiles with
a consequent increase in range and velocity.
QUESTION 7 Bourne fired the powder in a small metal cylinder and the
extent to which the lid rose gave an indication of the ‘strength’
of the powder.
QUESTION 8 The law relating to pressure and density at constant volume for
gases.
QUESTION 9 The lack of sophisticated pressure measuring equipment and
high speed photography.
QUESTION 10 a. Forensic ballistics.
b. Wound ballistics.
Chapter 2 – Part I
QUESTION 1 Propellants do not require atmospheric oxygen for combustion.
QUESTION 2 To achieve location, obturation (sealing) and spin of the
projectile.
QUESTION 3 Composition dependent: burning rate constant, pressure index,
force constant and co-volume. Shape dependent: ballistic size
and form function.
QUESTION 4 Ignition, shot-start, engraving, peak pressure, all-burnt,
muzzle exit, maximum velocity.
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 3 For what studies is spark photography used for?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 4 What are typical framing rates for
(a) slow speed cine?
(b) medium speed cine?
(c) high speed cine?
Answer (a) …………………………………………………………………………..
Answer (b) …………………………………………………………………………..
Answer (c) …………………………………………………………………………..
QUESTION 5 What photographic system would be particularly useful for
studying the interactive process between projectile and target
during penetration?
Answer ……………………………………………………………………………….
QUESTION 6 What information can yaw sonde provide?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 7 What two basic types of gauges are available to measure gas
pressure inside a gun barrel?
Answer………………………………………………………………………………..
…………………………………………………………………………………………..
QUESTION 8 What system is available for measuring a continuous record of
in-bore velocity?
Answer………………………………………………………………………………..
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204 Military Ballistics Answers to Self Test Questions 205
QUESTION 5 Sustained pressure, late all-burnt, strong muzzle flash and
blast, and high but inconsistent muzzle velocity.
QUESTION 6 Very slow burning propellants in relatively large chambers.
QUESTION 7 The surface area increases during burning.
QUESTION 8 Advantage: reduces barrel distortion. Disadvantage: reduced
cooling promotes erosion during repetitive firing.
QUESTION 9 Kinetic energy of projectile  32% of total energy, and
rotational energy  0.15%; so rotational energy ÷ kinetic
energy 
QUESTION 10 The propellant gases are cooled by expansion and contact with
the bore and chamber walls.
Chapter 2 – Part II
QUESTION 1 Burning rate is equal to twice the rate of regression.
QUESTION 2 The total propellant mass and granule shape are retained, but
the ballistic size is halved so as to produce the same propellant
vivacity.
QUESTION 3 Mass of gas liberated  Cz.
QUESTION 4 a. Remains stationary.
b. Decelerates.
QUESTION 5 The ability to model flame spread during ignition, and pressure
oscillations throughout the firing sequence.
QUESTION 6 The Noble-Abel equation only describes the state of a stationary gas; it does not take the velocity and acceleration of the gas
into account.
Chapter 3
QUESTION 1 Blast shock wave and bottle shock consisting of a barrel shock
and Mach disc. These appear in both the precursor and main
blast fields, though the main blast field is much more intense.
QUESTION 2 Slight acceleration and yawing of the projectile.
QUESTION 3 The speed of sound.
QUESTION 4 Preflash, primary flash, muzzle glow and intermediate flash.
QUESTION 5 Flash blast.
QUESTION 6 172 dB.
QUESTION 7 The recoil momentum is equal to the total forward momentum,
which includes the momentum of the propellant gases.
Chapter 4 – Part I
QUESTION 1 Mach number is defined as flight speed over the ambient speed
of sound. At sea level on a standard day, the speed of sound is
340.29 m/s. Therefore Mach number is 860/340  2.53. At 5000
m on a standard day, the speed of sound, from Table 4.1 is
320.53 m/s and the Mach number is 2.68.
QUESTION 2 From Bernouilli’s equation, the sum of static pressure and
dynamic pressure, which contains V2 is constant. Hence, if the
velocity rises the pressure falls and if the velocity reduces, the
pressure rises. Bernouilli’s equation strictly only applies at low
subsonic speeds but the qualitative relationship above is still
valid at supersonic speeds.
QUESTION 3 Drag is the force which directly opposes motion on a body
moving through a fluid. It depends on the shape, size and speed
of the object, and the density, viscosity and compressibility of
the fluid. For a given shape, the drag coefficient depends mainly
on Mach number and this is therefore a convenient way of
collecting and presenting data.
QUESTION 4 a. There are three different kinds of drag experienced by a
projectile in flight, viz:
Skin friction; this is a direct result of the viscosity of the air.
Pressure drag; this may have two components, one indirectly
due to viscosity. This causes the average static pressure
acting on the base of the projectile to be lower than that
acting on the nose. The other component, which appears a
speeds close to and above the speed of sound, is due to the
presence of shock waves.
Yaw-dependent drag; if the projectile yaws in slight, it generates a force normal to the body axis. This has a component in
the drag direction.
QUESTION 5 Boat-tailing ios the reduction in base area by means of a
tapered afterbody. Its purpose is to reduce drag. Its main disadvantage is that it reduces the static stability of the projectile.
Base bleed is also a way of reducing drag by introducing gas
into the wake and hence increasing base pressure. its main
disadvantage is that it is difficult to achieve repeatable burning
characteristics, hence trajectories tend to vary from occasion to
occasion for nominally identical launch conditions.
100
0.15%  —  0.5%
32
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206 Military Ballistics Answers to Self Test Questions 207
QUESTION 6 Static stability is the tendency of a projectile to return to its
equilibrium state following a yaw disturbance. Projectiles may
be statically unstable, neutrally stable or statically stable
depending on the sign of the static margin (see next question).
QUESTION 7 Static margin the distance of the centre of pressure aft of the
centre of gravity of a projectile. If the centre of pressure is
forward of the centre of gravity, a nose-up yaw disturbance will
generate a nose-up moment which will accentuate the disturbance. Adding fins at the rear of the projectile will move the
centre of pressure backwards because of the lift which they
produce at incidence. If the centre of pressure is moved aft of the
centre of gravity, static stability will result.
QUESTION 8 The transonic zone is the region of greatest instability.
QUESTION 9 The gyroscopic stability coefficient Sg is a measure, among
other things, of the spin rate of the projectile. Too little spin and
hence a small value of Sg will allow the round to tumble in
flight giving reduced accuracy and range. Too much spin and
hence a high value of Sg will also reduce range and could lead to
fusing difficulties for high explosive shells.
QUESTION 10 Trajectories are critically dependent on the forces acting on
the projectile in flight. The forces are, in turn dependent on air
density, viscosity, speed of sound and hence on temperature.
Additionally, the natural wind affects the trajectory. This is
all meteorological data and is required if accurate fire control
prediction is to be achieved.
QUESTION 11 Firing East or Wesy with long times of flight can give significantly different ranges than those predicted by a flat earth
trajectory model due to a combination of the earth’s rotation
and itsa surface curvature. For the same reason, rounds fired
in the Northern hemisphere will land to the right of the target
and in the Southern hemisphere, to the left.
QUESTION 12 The Point Mass trajectory model contains only projectile
weight and drag. It generates no information on projectile yaw
angle and sideforces are therefore absent. Drift effects have to
be introduced artificially. Six Degree of Freedom models
attempt to predict all of the projectile motions in detail and
require much information on projectile aerodynamics and
initial conditions. They are also relatively slow in operation.
The Modified Point Mass model is a good compromise between
the two since it models equilibrium yaw and drift but with
little more information, time, or computer memory requirements than the simple Point Mass model.
Chapter 4 – Part II
QUESTION 1 That aerodynamic forces and moments expressed in coefficient
form are unique functions of Mach number and Reynolds number. Therefore tests on geometricallt similar bodies at the same
values of M and R, will yield the same coefficients.
QUESTION 2 Dynamic similarity is inertial and viscous forces ie these forces
will have the same ratio in both cases. Hence boundary layer
behaviour for example, will be identical for both shapes ..
QUESTION 3 For subsonic bodies with boundary layer separation producing a
large low pressure wake and at Reynolds numbers just below
the critical value ie when the boundary layer separation is
laminar but where a small increase in flow speed would cause
transition to turbulence before separation could occur.
QUESTION 4 The upwash effect; this increases the local inccidence and hence
lift of the fin close to the body side.
The carryover effect; this increases the left of the body due to
the pressure difference on the fins being ’carried over’ onto the
body.
QUESTION 5 Precession and mutation as shown in Fig. 4.30.
QUESTION 6 The bullet will take 0.3048/990 seconds for one revolution and
therefore its spin rate is 3238 rev/s.
QUESTION 7 Maximum range in-vacuo is given by Vo
2
/2g ie for a muzzle
velocity of 220 m/s, maximum range is 4936 m.
QUESTION 8 For fin stabilised projectiles only, moving fins. For both fin and
spin stabilised projectiles, impulse control.
Chapter 5 – Part I and Part II
QUESTION 1 a. Angle of attack.
b. Strike velocity.
c. Target configuration and composition.
d. Projectile configuration and composition.
QUESTION 2 Penetration may be defined as the entrance of a missile into a
target without completing its passage; perforation is the complete piercing of the target by the projectile.
QUESTION 3 a. Impact phase.
b. Penetration.
c. Perforation.
QUESTION 4 a. Fracture.
b. Spalling.
c. Scabbing.
d. Petalling.
e. Fragmentation.
f. Plugging.
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208 Military Ballistics Answers to Self Test Questions 209
QUESTION 5 a. High density core.
b. Discarding sabot.
c. Long thin rod for optimum penetration.
d. Low drag profile.
e. Spin stabilisation.
f. High muzzle velocity.
QUESTION 6 a. Higher muzzle velocity.
b. Greater penetration.
QUESTION 7 a. In destruction of buildings and structural damage in
general.
b. Open environment.
QUESTION 8 Through spall and by physiological and pyschological effects
against the tank crew by pressure, temperature and flash.
QUESTION 9 HESH.
QUESTION 10 a. Empirical.
b. Analytical.
c. Experimental.
d. Numerical.
Chapter 6
QUESTION 1 Absorption of kinetic energy imparted by missile.
QUESTION 2 Elasticity and density of tissue.
QUESTION 3 The transfer of projectile momentum to surrounding tissues.
QUESTION 4 No. For the same amount of total energy expended, the design
of the bullet can make profound differences in its effect on
tissues.
QUESTION 5 a. Resist penetration of the projectile.
b. Dissipate most of the kinetic energy of the shot.
c. Satisfy the restraints of bulk and weight.
QUESTION 6 Advantages:
a. Effective against high velocity bullets.
b. Able to withstand close and repeated shots.
Disadvantages:
a. Spalling.
b. Possible effects of cavitation.
c. Heavy (although light low-alloy steels are available at
extra costs).
QUESTION 7 Advantages:
a. Extremely hard.
b. Lightweight.
Disadvantages:
a. Anyone ceramic tile cannot accept more than one hit.
b. Expensive.
QUESTION 8 The heavy weave of the material deforms the nose of the projectile, so preventing the mushroomed bullet from forcing its way
through the mesh of fibres.
Chapter 7
QUESTION 1 To ‘freeze’ the motion.
QUESTION 2 Lines are painted longitudinally on the projectile body which is
photographed in flight in the light of two microflash lamps
spaced a short distance apart down range. The time interval
between the two flashes is recorded and the rotation of the shell
during this interval can be measured from the photograph.
QUESTION 3 a. Armour protection and penetration.
b. Yaw measurement.
c. Air resistance.
d. Aerodynamic performance of a particular shape of projectile.
e. Pressure disturbances in surrounding airflow.
QUESTION 4 a. 24 p.p.s.
b. 10,000 p.p.s.
c. Several million p.p.s.
QUESTION 5 Flash radiography (X-ray shadowgraph).
QUESTION 6 A measure of yaw and spin rate, precision and stability, for
shells in free flight for most of the trajectory.
QUESTION 7 a. Crusher gauge.
b. Piezoelectric gauge.
QUESTION 9 Interferometer.
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210 211
Bibliography Index
Abrasion 29
Accelerometer 200–1
Aerodynamic
forces 71–87, 115–21
moments 87–94, 123–4
Aerodynamics 71–94
Air resistance 100, 154
All-burnt 26–7
Ammunition
design and wound effectiveness 168–70
Angle of attack 147, 160
Armour attack 83, 149–53, 159
Armour piercing discarding sabot
(APDS) shot 154
Armour piercing fin stabilised discarding
sabot (APFSDS) shot 12, 126, 154–5
Atmosphere 67–9, 112
Atmospheric conditions 68
Axial moment of inertia 131
Ballista 2
Ballistic
coefficient 86–7
size 18, 39
synchro 184
tables 85–6
Ballistics
externa11–3, 67–146
history 1–7
instrumentation 177–202
intermediate 51–66
internal 4–5, 9–49
terminal 5–6, 147–66, 162–6
wound 6, 167–76
Barrel
contacts 200
cooling 31
distortion 32–3
lagging 33
life 28–34
materials 29–30
shock 52, 54
temperature 30
Base
bleed 84–7
drag 82–3
Bernouilli’s equation 117
Blast
intermediate ballistic 53–4, 60–4
precursor 54–5
suppression 61–3
terminal ballistic 156
Boat-tailing 83, 85
Body
armour 170–5
Bore erosion 29–31
Bore-wire resistance method
198–200
Bottle shock 52–3
Boundary layer 115–19, 122
separation 73, 117–21
Breech 4, 33
Bullets 78, 80–1, 167, 170
Burning rate 15–16, 37–8
Calibre 9
radius head 81
Cameras 177–86
Cartridge 10, 12
Cavitation 168, 172–3
Ceramic body armour 173–4
Chemical energy explosive attack 155–6,
159
Cine film 180–3
Closed-vessel 17–18, 39–41
Combustion chamber 108
Composite body armour 175
Computer modelling
‘Anti-personnel Weapons’, Stockholm International Peace Research Institute, Taylor and
Francis Ltd., London, 1978.
‘Ballistic Range Technology’, Advisory Group for Aerospace Research and Development,
NASA, Maryland.
Goldsmith W., and Backman M. E., ‘The Mechanics of Penetration of Projectiles into
Targets’, Int. J. Engng. Sci. Vol. 66 pp. 1–99, Pergamon Press, 1978.
‘High Velocity Impact Phenomena’, edited by Kinslow R., Academic Press, New York,
1970.
‘International Ballistics’, HMSO, London, 1951.
Krier and Summerfield, ‘Interior Ballistics of Guns’, American Institute of Aeronautics
and Astronautics, New York, 1979.
‘Oerlikon Pocket Book’, Zurich-Oerlikon, 1952.
Schmidt E. M., ‘Muzzle Devices – A State of the Art Survey’, USA Ballistic Research
Laboratories, Aberdeen Proving Ground, Maryland, 1973.
‘Textbook of Ballistics and Gunnery’, edited by Langdon L. W., HMSO, 1983.
13569_int.indd 210-211 11/15/11 11:21:15 AM
212 Index Index 213
internal ballistics 45–8
point mass trajectory 100, 138–9
rocket trajectory 99–103, 136–8
terminal ballistics 163–4
Cone angle for HEAT 158
Corrosion 28–9
Co-volume 17–18, 41
Cross-wind 107, 142
effects 107, 142
Crusher gauge 5, 196
Damping
moment 133
Density
air 113
Decibels 63
Direct optical method 198–9
Doppler frequency 191
Drag 3, 12, 71, 133
aerodynamic 71–87, 115–21
coefficient 79–84, 117
rocket 3, 12, 71, 133
types 72–3
Drift 105–7
Driving band 11–12
Ductile failure 150, 153
Dynamic
similarity 114–15
stability 90, 133–5
Efficiency, propulsive 24–5, 27
Elastic force 114
Empirical force law 163
Energy distribution
internal ballistics 24–5
muzzle exit 25
recoil 25–6
Energy liberated by propellant 41–2
Engraving of driving band 22
Equilibrium yaw 100–3, 131, 142
Erosion 29–31
Exit pressure 54
Explosive projectiles 153–61
External ballistics 1–3, 67–146
history 1–3
measurements 189–94
Fin stabilisation 83, 92–4, 124–7, 143
Firing sequence 21–8, 30
Flame
spread 21
temperature 17–18
Flash 55–60
blast 60–4
hider 56
suppression 56–62
Flash radiography 187–8
Flechette 12
Fluid motion 70, 113–21
Force constant 17–18, 30, 40–1
Form function 19–20, 30, 38–9
Fractional calibre radius head 81
Fracture 151
Fragmentation 153, 155–6
Fragments 155–6
Friction
drag 72
Frictional force 25
Fuels, rocket 109
Grains 13–16
Granules 13–16, 18–20
Gravity 113
Gun 9–11
Gunpowder 13
Gyroscopes 95–6
Gyroscopic stability 127–30
Hague Convention 1899 168
Hearing damage 60, 62–4
High explosive projectiles
anti-tank (HEAT) 153, 156–61
fragmentation (HE) 153, 155–6
plastic (HEP) 159–61
squash head (HESH) 152–3, 156,
159–61
High speed
cameras 184–5
photography 177–88
History of ballistics 1–7
Hypersonic speed 70
Hypervelocity impact 163
Ignition 10, 15–16, 21
inhibitors 19
temperature 15
Image converter camera 185–6
Impact 105, 172, 194
In-bore
pressure instrumentation 194–7
projectile motion 197
projectile motion instrumentation
197–200
Inertial force 114
Inhibitors 97
Injury 60, 167–8
Instrumentation 177–202
Intercontinental ballistic missile 67
Interferometry 197
Intermediate ballistics 51–66
Internal ballistics 4–5, 9–49
history 4–5
in-vacuo model 99, 136–7
Jet
metallic, HEAT effect 158
reaction gun 33
Kinematic viscosity 68
Kinetic energy 158
(KE) attack 158, 172
projectile 172
Laminar flow 115, 122
Laser interferometry 197
Lauch, rocket 181–2
Law
burning rate 37–8
empirical force 163
Newton’s Second Law of Motion 108
Newton’s Third Law of Motion
108–9
Piobert’s 4, 13–15, 37
Leading 29
Lift force 88, 92–3, 126–7
Lighting 184
Linear displacement transducers 200
Lined charge 156
Long-rod penetrator 147–8, 150
Lubrication of projectiles 29
Mach
disc 52, 54–5
number 70, 74, 76–7, 80–1, 86, 115,
125
Magnus
effect 103, 120
moment 102–3
Mass-ratio 109
Metallic body armour 172–3
Microflash photography 178–9
Microwave interferometry 197
Moderators 62
Modified point mass model 100–3, 141–3
Moments of inertia 131
Motion
in-bore 197
in vacuo 107
Multi-tube propellant 18, 20
Munroe charge 156
Muzzle
brake 62, 64–5
energy 25
flash 27
gas flow 51–4
Muzzle velocity 3, 22–3, 32
Newton, Sir Isaac 2–3
Second Law of Motion 108
Third Law of Motion 1O8–9
Law of Gravitational Attraction 113
Noble-Abel equation 40
Noise 60–1
Nozzle 108–9
Nutation 96–130
Overpressure 60–1
Penetrability 149
Penetration 148–51, 158
Penetrators 147–8, 150
Perforation 149, 151
Perturbation 130, 132, 135
Petalling 153
Photodetector counter chronometer
(PCC) 189–90
Photography 177–88
Piezoelectric gauges 196–7
Piobert’s law 4, 13–15, 37
Plug 152
Point mass model 100, 138–9
Plugging 152
Precession 95–7
Preflash 55
Pressure 114
drag 72–3
gas 4, 22, 25, 196–7
gauges 4, 61
in-bore 194
index 16–17
peak 22, 25–7
shotstart 22
Pressure waves 51, 74–5
external ballistics 74–5
intermediate ballistics 51
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214 Index Index 215
Primer 10, 20–1
Probertised rifling 31–2
Progressive granules 19, 27–8
Projectile 11–13, 24, 147–8
acceleration 200–1
design 170
drag 3, 12, 71, 133
drift 105–7
lubrication 29
rotation 201
shape 155
stability 145
types 11–12, 69–70, 153–9
Propellant 5, 13–20, 38, 140
additives 30, 33
composition types 5, 13–14
grains 13–16, 18–20
granules 13–16, 18–20
shape 156–7
vivacity 38
Quasi-static shock wave 52
Radio doppler 190–1
Radiography 187–8
Range 71, 87, 104–5, 107, 147
Recoil 11, 25–6, 64
Recoilless gun 33–5
Reinforced plastic body armour 173
Reynolds number 114–16, 119
Rifling
probertised 31–2
Rockets 3, 5, 107–9, 139–41
Launch 181–2
Rotating mirror cameras 186
Rotation of the earth 105–7
Sabot 11, 64, 154
Scabbing 151–2
Schlieren system 179, 181–2
Shadowgraph 179, 188
Shaped charges 156–7
Shock, bottle 52–3
Shock wave 179
external ballistic 73–8, 121–3
intermediate ballistic 52–4
reflection 172
Shot 11–12
position indicator 193–4
Shot-start 22, 195
Silencer 61–2
Skin friction 72, 83, 115–17, 119
drag coefficient 72
Slivers 20, 27
Smokeless powder 13
Sound
intensity 61
speed of 51–2
waves 52
Space curves 22–3
Spalling 151–2, 158, 172
Spark photography 179–81
Specific impulse 140
Speed
hypersonic 70
subsonic 69–70
supersonic 69–70, 81, 122
transonic 69–70, 77
Speed of sound 51–2
Spin 159, 178
damping moment 133
rate 97, 128, 130
stabilisation 3, 95–8, 127–30
Stabilisation
fin 83, 92–4, 124–7, 143
spin 95–8, 145
Stability 170
coefficient 130
gyroscopic 95, 127–30
Streak camera 184–5, 187
Stress waves 150, 160, 168
Suppressants
blast 61–3
burning rate 19
flash 56–60
Suppressors
blast 61–3
flash 56–62
Targets 105–6, 148–50, 159, 164
Temperature
air 68
barrel 30
flame 17–18
gas 42–52
ignition 15
propellant 30
Terminal ballistics 5–6, 147–66, 162–6
Textile body armour 174
Thrust 139–40
Time curves 22–3
Trajectory
in-vacuo 99, 136–7
models 99–103, 136–8
non-standard 104–5
prediction 98–107, 136–42
rocket 139–41
Transonic zone 77
Transparent body armour 17 4–5
Turbulence 52, 117
Turbulent flow 122
Van der Waals’ equation 39–40
Velocity 3, 52, 118–19, 147, 170
ranges 147
Viscosity 68, 72
Viscous
boundary layer 117
force 114, 117
Vivacity 38
Wake 73, 117–19
Wear, barrel 28–33
Web size 18
Wind effects 107, 142
Wound ballistics 6, 167–76
X-ray photography 187–8
Yaw 87–97, 130–1
disturbance 87–8, 91, 95–7
equilibrium 100–3, 131, 142
sonde 191–3
Yawing moment 91, 131
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