International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Drag Brakes Variation for Range Correction of
Artillery Projectile
R. B. Sarsar 1 and S. D. Naik2
Applied Mathematics and Reliability Group, Armament Research and Development Establishment,
Pashan, Pune 411021, Maharashtra, INDIA.
Abstract
Improvement in accuracy of artillery projectile is of prime importance for gun designers. Generally it is associated with range
extension. Various methods and modifications in the design are applied to correct the range and deflection of artillery projectile
like impulse thruster, pulsejet, drag devices, reciprocating canards. Out of these methods one is drag brake deployment which is
analyzed in this paper. A trajectory corrector module consisting of disc is placed at the aft of the fuze. Once deployed, it increases
the frontal area and hence the drag which reduces the overshoot range. Analysis has been carried out for increased frontal area,
initiation and duration of deployment and optimum range correction with varying diameter of drag brakes using simulation. The
model used is six degrees of freedom equations of motion.
Keywords: Trajectory correction systems, Drag brakes, range correction, Fuze, Equations of motion, Artillery projectile,
Trajectory, Accuracy.
Nomenclature
ϕ
θ
ψ
α
β
δ
γ
Ax
Ay
Az
C-xyz
C-tnb
m
O-XYZ
u
v
w
Roll angles in degree
Pitch angle in degree
Yaw angle in degree
Angle of attack in degree
Side slip angle in degree
Flight path angle in degree
Flight yaw angle in degree
Rotation matrix about x axis
Rotation matrix about y axis
Rotation matrix about z axis
Body coordinate system
Velocity coordinates system
Mass of the artillery projectile
Inertial coordinate system
Velocity component in x direction
Velocity component in y direction
Velocity component in z direction
1. INTRODUCTION
The role of field artillery is to provide fire support to other arms by backing up attacks, providing defensive fire,
neutralizing an enemy's gun emplacement. Long and medium-range artillery projectiles are used for indirect fire. These
are area weapon and largely suffer from dispersion at impact point if fired against a specific target. Reduction in
collateral damage and delivering highly accurate fires, dictates the need for much improved round accuracy. Variations in
temperature, wind, the barrel wear of the weapon, and the charge are some of the factors affecting during flight of a
projectile.
Trajectory correction is defined for a given trajectory to reduce the dispersion at point of impact. It is achieved by
applying corrections to the path of the body in flight using different means like drag brakes. Trajectory correction is
applied in two directions: along range and perpendicular to range (drift). Such systems are known as two dimensional
projectile trajectory corrector systems which include two types of aerodynamic surfaces to be deployed during the flight.
The first type of surface provides one-dimensional range correction, which is a drag device that acts as an airbrake. The
second type of surface provides the correction in cross range, which affects spin and the normal force of the projectile inflight. These types of surfaces are known as spin device which are also stored within the projectile at launch. The
initiation and duration of deployment of the drag devices and spin devices determine the percentage reduction in range
and drift.
Improving artillery projectile accuracy is very important for designers and lot of work has been carried out in last decade.
A simple analytical approach has been studied by C.Grignon et al [1] which talks about deflection correction capabilities
Volume 2, Issue 9, September 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
obtained with spin control, direct lift and pitch control. C.Grignon et al determines that change in direct lift and pitch
characteristics seems to offer significant deflection correction, where as spin control is not efficient enough for primary
deflection control even for the minimum spin admitted by the gyroscopic stability criterion. Stockenstrom [2] presented a
paper on range dispersion reduction where author has observed that for the ERFB-shaped projectiles, the drag
augmentation device needs to impart a drag multiplier factor upto two in the Mach number range 0.9 to 1.6. The design
of a canard-controlled mortar projectile using a bank-to-turn concept is presented by Rogers and Costello [3]. Rogers and
Costello used PID control for set of two reciprocating fixed - angle maneuver canards to perform trajectory corrections
and PD control for set of two reciprocating fixed-angle roll canards for regulating projectile spin rate to zero and setting
the airframe in the desired roll orientation.
Corriveau et al [4] used impulse thruster to correct the trajectory of a conventional artillery projectile. Results indicated
that locating the impulse thruster at or aft of the center of gravity provides greater drift correction for a given impulse
magnitude. Corriveau et al assessed that impulse thrusters with a burning intensity between 15 and 20 Ns would provide
adequate course correction for a 105 mm artillery projectile. The work has been continued by Way P et al and gave a
novel method for correcting the trajectory of spin- and fin-stabilized projectiles using pairs of impulse thrusters located
away from the center of mass [5]. Gupta et al [6] studied a trajectory correction flight control system which consists of a
lateral pulsejet ring mounted on the rocket body. Each pulsejet on the ring imparts a single, short-duration, large force to
the rocket in the plane normal to the rocket axis of symmetry. A range correction module which is fitted onto a spin
stabilized ballistic projectile for correcting range error is designed by Brandon et al [7]. In this work Brandon et al used
sixteen semi-circular plates to create a blunt cross sectional area in front of projectile to increase the drag and effectively
slow down the projectile. Harkins et al has given a drag-brake deployment method and apparatus for range error
correction of spinning gun launched artillery projectiles in [8]. Bar et al has discussed about drag device with different
design [9]. In their work, the drag device is placed approximately at half the height of the fuze with flap-shaped sectors.
The drag device in launch position forms a radial ring which is closed or has gaps.
The work carried out in this paper discusses the effect of change in drag brakes surface area on range correction. The
analysis has been carried out for various frontal areas of drag brakes and studied the effect of it on range using six degrees
of freedom equations of motion. An optimum design of drag brakes can be obtained using this analysis for required
correction in the range.
2. DRAG BRAKE CONCEPT
The concept of drag brake with a single deployment scheme has been discussed by Hollis [10]. In addition, Hollis has
proposed the requirement of power to deploy the drag brake and suitable DC motor to actuate it. The intent of this paper
is to replace standard fuze of conventional artillery projectile with trajectory corrector module, without any changes
within the ogive shape of the artillery projectile. Drag brakes are placed at the aft part of fuze as shown in Figure 1.
Adaptability of trajectory corrector module is maintained at design level.
Figure 1: Artillery Projectile with Drag Brakes
The aim of this work is to estimate the effect of change in diameter of fuze on range. When the drag brakes are deployed
during the flight, it increases the frontal area of the fuze which indeed increases the drag acting on the projectile and
reduces the range. The frontal area of fuze changes with variation in drag brakes diameters. The drag brakes are divided
into four parts as shown in Figure 2.
(a)
(b)
Figure: 2 (a) Front view with Drag Brakes deployed (b) Front view without drag brakes
The analysis is carried out for the varying drag brakes diameters. The additional parameters considered are point of
deployment of drag brakes and deployment duration. Further analysis is carried out for one time deployment of drag
brakes during the complete flight and multiple time deployment of drag brakes. Multiple deployments of drag brakes are
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
applied for get the optimum range correction. The simulation is carried out using six degrees of freedom model for
projectile motion.
3. MATHEMATICAL MODEL
A general mathematical model is developed using Newton’s second law of motion to study the dynamical motion of an
artillery projectile. Motion of the projectile is affected due to the forces and moments acting on it. The forces and
moments on the projectile are acting in different frames, for example, drag force and lift force are represented in velocity
frame and gravity force acts in inertial frame. To generate scalar equations of motion, transformation matrix between
each frame is required to be defined. Using transformation matrices, forces and moments can be resolved in particular
directions. A solution of equations of motion gives the trajectory of the projectile.
3.1. Coordinate System: The vector equations obtained from Newton’s second law are resolved in three directions with
the help of three reference frames as shown in Figure 3.
1. Inertial Frame (O – XYZ)
2. Body Frame (C – xyz)
3. Velocity Frame (C – tnb)
Figure 3: Coordinate System
The relations between these frames have been developed using transformation matrices obtained by applying Eulerian
rotations sequentially. Transformation matrices are developed with the assumption that anticlockwise sense is positive.
Transformation matrix from Inertial frame to Body frame is denoted by AIB. It consists of three rotations and hence is
obtained as product of three rotation matrices [11].
AIB  Ax   Ay   Az  
cos  cos


AIB   sin  sin cos   cos  sin 
cos  sin cos   sin  sin 
cos sin 
cos  cos   sin  sin sin 
cos  sin sin   sin  cos 
 sin 
sin  cos   1

cos  cos 
where ϕ, θ & ψ are roll, pitch and yaw rotations respectively. ABI is obtained by taking transpose of AIB.
Transformation matrix from Inertial frame to Velocity frame is denoted by AIV and defined as
AIV  Ay  Az  
cos  cos 
AIV     sin 
 cos  sin 
cos  sin 
cos 
sin  sin 
 sin  
0   2 
cos  
δ and γ are defined as angle of elevation and drift respectively. AVI is transpose of AIV.
Transformation matrix from Body frame to Velocity frame is denoted by ABV and defined as
ABV  Ay  Az  
cos  cos 
ABV     sin 
 cos  sin 
Volume 2, Issue 9, September 2013
cos  sin 
cos 
sin  sin 
 sin  
0   3
cos  
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
α and β are angle of attack and sideslip angle respectively.
3.2. Equations of Motion: The dynamical motion of the projectile is studied with the help of Newton's second law of
motion. The equations of motion are converted in body frame with the help of transformation matrices. The forces taken
into consideration are gravity force, drag force, lift force, Magnus force and damping force. The moments considered are
Spin damping moment, Spin driving moment, Overturning moment, Magnus moment and Damping moments. The six
degrees of freedom model which consists of three scalar equations for linear motion (force equations) and three for
angular motion (moment equations) is developed and scalar equations of motion are given below [11] - [12]:
1
 gravityx  (drag x  DBx )  lift x  coriolisx  magnusx  damp x 
m
1
v 
 gravity y  (drag y  D By )  lift y  coriolis y  magnusy  damp y
m
1
w   gravityz  (drag z  D Bz )  lift z  coriolisz  magnusz  damp z 
m
1
p 
SD  SDr 
I xx
u 


 4 
 5
 6 
 7 
q 
1
OM y  MM y  DM y
I yy

 8
r 
1
OM z  MM z  DM z 
I zz
 9

In these equations (u, v, w) denotes velocity of the projectile in x, y and z directions respectively. m is the mass of the
projectile. Ixx, Iyy and Izz are moment of inertia in x, y and z directions respectively. All the forces are resolved
componentwise to get the scalar equations for transverse motion. SD denotes the spin damping moment, SDr denotes
spin driving moment. OM, MM and DM are overturning moment, Magnus moment and damping moment respectively.
Scalar equations for angular motion are obtained by resolving these moments in x, y and z directions. Definitions of all
the forces and moments could be referred in literature [13].
The drag induced by the drag brakes is denoted by the vector DB and it is defined as follows.
DB  0
when drake brake is not deployed
1
DB   S1VCd V
2
when drake brake is deployed
where ρ is air density, V is velocity of projectile, S1 is surface area of drag brakes ΔCd is additional drag coefficient.
A simulation study has been carried out and analysis has been done to study the effect of drag brakes on projectile range.
The equations of motion are integrated using forth order Runge – Kutta method.
4. SIMULATION AND ANALYSIS
4.1. Simulation Data: The simulation has been carried out for 155 mm artillery projectile. The initial conditions used for
simulation are m = 44.80 Kg, V = 878 m/s, θ = 30 deg, x = y = z = 0. Mass of the projectile includes weight of drag
brakes also. It is assumed that the weight of each drag brake is 0.012 Kg. The deployment of drag brakes increase
diameter of fuze and the analysis is carried out for the variation of fuze diameter from 60mm to 100mm with the
assumption that 50mm being the diameter of fuze without drag brakes. As drag brakes are hosted in the fuze, the
deployment of drag brakes with diameter more than 100mm is not possible because of space constrains. The
deployment of drag brakes with the diameter of fuze of 60mm increases the frontal area by 1.5 times whereas for
100mm diameter of fuze the frontal area increases by 4 times. It is assumed that projectile will take 4 - 5 seconds for
stabilization itself after the muzzle exits and 2 seconds to deploy the drag brakes.
4.2. Simulation Results: The analysis is carried out for deploying the drag brakes at various stages of trajectory and once
the drag brakes are deployed it will remain open throughout the trajectory. Deploying drag brakes in early stage of
trajectory results in maximum range correction. As the diameter of drag brakes increases, it improves the correction
in the range. Hence maximum range correction is observed for maximum diameter of drag brakes deployed at the
earliest. With fuze diameter of 60mm deployed at 10 seconds during the flight corrects the range by 219m whereas
the same drag brakes gives the correction of only 3m if deployed at 50 seconds. Deployment of drag brakes of 100mm
fuze diameter gives maximum correction of 1347m (Figure 4).
Volume 2, Issue 9, September 2013
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
1400
Fuze dia 60mm
Fuze dia 70mm
Fuze dia 80mm
Fuze dia 90mm
Fuze dia 100mm
C o rrectio n in R a n ge ( m )
1200
1000
800
600
400
200
0
10
15
20
25
30
35
Drag Brake opening Time ( s )
40
45
50
Figure 4: Effect of deployment time on range
The simulation analysis has been carried out for deployment duration with deployment start at 10 second of flight time. It
is observed (Figure 5) that long duration deployment of drag brakes results into large correction in range. Deployment of
drag brakes of fuze diameter 60mm at time of 10 seconds gives correction of 125m whereas deploying same fuze diameter
for completer time of flight gives range correction of 219m. Similarly for 100mm fuze diameter, deployed for 10 seconds
starting from flight time of 10 seconds gives 820m range correction whereas for complete flight time deployment1347m.
1400
1200
60mm
70mm
80mm
90mm
100mm
Co rrectio n in R a n g e ( m )
1000
800
600
400
200
0
10
15
20
25
30
35
Deployment duration ( s )
40
45
50
55
Figure 5: Drag brakes deployment duration
Ctrl start
time (s)
10
Table 1: Drag Brakes deployed for 10 seconds
Correction
Correction
Correction
Correction
60mm Fuze
70mm Fuze
80mm Fuze
90mm Fuze
dia (m)
dia (m)
dia (m)
dia (m)
125
271
437
619
Correction
100mm Fuze
dia (m)
820
20
56
120
194
277
367
30
23
53
85
124
163
40
10
22
37
54
73
50
2
8
11
18
23
It is observed that deployment of drag brakes early for small duration gives maximum range correction whereas deploying
for small duration in later stage of trajectory gives less correction in range. Table 1 shows deploying drag brakes with
60mm fuze diameter at 10 second of time for 10 seconds duration gives 125m correction in range but deploying at 50
second of time for 10 seconds duration gives 2m correction only.
5. CASE STUDY FOR MULTIPLE DEPLOYMENTS
One time deployment results into the range correction and increase the diameter is dictated by the required correction in
the range. In case if the range correction required does not lead to the exact increase in the diameter of drag brakes then
multiple deployment may be the solution for it. In this case depending on the correction required, the variation in the
diameter can be calculated and applied at each step. An algorithm has been developed for the same in terms of velocity
and applied for a particular case. The selection of fuze diameter is based on following algorithm.
a. Calculate the ideal trajectory for exact range.
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b. Select the time of deployment of drag brakes.
c. Calculate the difference between the actual velocity of the projectile and ideal velocity.
d. Select the drag brakes such that the difference of velocities becomes zero or closer to zero.
e. Deploy the drag brakes for one second.
f. Repeat the step b to d for remaining time of flight.
5.1. Case Study: The case study is carried out for the data given in section 4.1. It is observed that the uncontrolled
projectile achieved 25180 m range with total time of flight is 63.8 seconds and vertex is achieved at 28.75 seconds.
The remaining velocity at time of impact is 346 m/s when fired at a target on 25000m. It means that a correction of
180m is needed to hit the target. The analysis is carried out for multiple deployments of drag brakes during the
flight get the maximum of 180m correction in range. At the time of drag brakes deployment, calculate the impact
point using current trajectory data and find the overshoot distance. Depending upon overshoot distance the suitable
drag brakes are opened to correct the range. It is observed that deploying the drag brakes just after the vertex height
gives 100% correction in range. Figure 7 shows the requirement of fuze diameter for drag brakes once it is deployed
at flight time of 31 seconds. It is observed that initially only 60mm fuze diameter drag brakes is required and in later
stage of trajectory requirement of drag brakes is varies between 80mm to 100mm to achieve the optimum range
correction.
0.1
0.095
F u ze dia ( m )
0.09
0.085
0.08
0.075
0.07
0.065
0.06
0.055
30
35
40
45
Time ( s )
50
55
60
65
Figure 7: Fuze diameter during flight
6000
w/o drag brakes
with drag brakes
5000
A ltitu de ( m )
4000
3000
2000
1000
0
0
0.25
0.5
0.75
1
1.25
1.5
Range ( m )
1.75
2
2.25
2.5
2.75
4
x 10
Figure 8: Trajectory
Figure 8 shows the effect of drag brakes on the trajectories of uncontrolled and controlled artillery projectile. The
differences in velocities due to drag brakes are shown in Figure 9, it is observed that deployment of drag brakes
throughout the flight reduces the remaining velocity by 11m/s which give the required correction in range.
Further analysis is carried out for given projectile data and study suggests that a DC motor of the power 300W is required
to extract and retract the drag brakes during the flight [10].
Figure 9: Effect of drag brakes on velocity
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 9, September 2013
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6. CONCLUSION
Range correction using drag brakes is studied in this work. The effect of varying drag brakes frontal area on projectile
range has been analyzed. One time deployment of drag brakes at early stage of flight gives maximum range correction.
To increase the accuracy artillery projectile multiple deployment scheme can be used. The result shows that deploying
drag brakes early for small duration also gives sufficient correction in range. The deployment time can be predefined
depending on required correction of range. Optimum correction in range is possible with the help of multiple
deployments algorithm.
References
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Ballistics, Tarragona, Spain, pp 747 – 754, 2007.
[2] Stockenstrom A, “A Simplified Approach to Range Dispersion Reduction”, 20th Int. Symposium on Ballistics, pp
180 - 186, 2002.
[3] Rogers J, Costello M, “Design of a Roll-Stabilized Mortar Projectile with Reciprocating Canards”, Journal of
Guidance, Control, and Dynamics, Vol. 33, No. 4, July – August 2010, P. No. 1026 - 1034.
[4] Corriveau D, Berner C, Fleck V, "Trajectory Correction Using Impulse Thrusters For Conventional Artillery
Projectiles" 23rd International Symposium on Ballistics, Tarragona, Spain, pp. 639 – 646¸2007.
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[6] Gupta S K, Saxena S, Singhal A, Ghosh A K, " Trajectory Correction Flight Control System using Pulsejet on an
Artillery Rocket", Defence Science Journal, Vol. 58, No. 1, pp 15-33, 2008.
[7] Brandon F J, Hollis M S L, “Drag Control Module for Range Correction Of A Spin Stabilized Projectile”, US Patent
no. 5,826,821, 1998.
[8] Harkins T E, Bradford S D, “Drag-Brake Deployment Method and Apparatus for Range Error Correction of
Spinning, Gun Launched Artillery Projectiles”, US Patent no. US 6,345,785 B1, 2002.
[9] Bar K, Bohl J, Kautzsch K, “Spin-Stabilized Projectile With A Braking Device” US Patent no. US 6,511,016 B2,
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[10] Hollis M S L, "A Report on Preliminary Design of a Range Correction Module for an Artillery Shell", Army
Research Laboratory, Report number ARL-MR-298, 1996.
[11] R L McCoy, “Modern Exterior Ballistics”, Schiffer Publication, 1999.
[12] Sarsar R B, Mukhedkar R J, Naik S D, “Study on Extending Range of Artillery Rocket Using Control Surfaces”,
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[13] Anonymous, Text Book of “Ballistics and Gunnery”, Her Majestic Stationary Office, Volume I, pp 476 – 483, 198.
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