THE BALLISTICS PROCESSOR OF A
MULTIPLE PROCESSOR AIRBORNE
TACTICAL SYSTEM.
Harry Andrew Jupin
OBrarsi'
Naval Postgraduate School
ffiontete^ California 9394Q
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A MULT I
THE EA
PLE PROCE
LLISTICS PROCEOF
SSOR AIRBORNE
SSOR
TACT] CAL SYSTEM
by
Ha rry Andrew J"up in
June 197 5
Th e s i s Advisor
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The Ballistics Processor ofA Multiple Processor Airborne TacticalSystem
5. TYPE OF REPORT i PERIOD COVERED *
Master's Thesis;June 19756. PERFORMING ORG. REPORT NUMEES
7. AUTHORf*;
Harry Andrew Jupin
6- CONTRACT OR GRAHT HUM3>r.T<(e)
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Naval Postgraduate SchoolMonterey, California 93940
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June 19 7 5
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16. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on revoree elde It necetaary and Identity by block number)
Ballistics processor of a multiple processor airborne tacticalsystem
Three INTEL- 8080 microcomputersGeneral second order Runge-Kutta method of integration of the
equations of motion of unguided air- t o- surface we apons20. ABSTRACT (Continue on reveree elde It neceeemry end Identity by block number)
This thesis developed the ballistics processor of a multipleprocessor airborne tactical system. The multiple processorsystem consisted of three INTEL- 8080 microcomputers: theexecutive processor, the navigational processor and theballistics processor. The ballistics processor utilized a
general second order Runge-Kutta method of integration of theequations of motion of unguided air-to-surface weapons. The
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^D I j°n"3 1473 EDITION OF 1 NOV 6S IS OBSOLETE
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UNCLASSIFIEDfliCUHITY CL ASSI FIC * TION OF THIS P » G Ef«'»i nn Del. En(er«
19. Key Words
Ballistics Trajectory Algorithm
20. Abstract
ballistics processor computed sufficiently accurate and timelysolutions to enable the executive processor to extrapolate anaccurate release point for the weapon. The algorithm permittedcomplete flexibility in release conditions and allowed a
complete arsenal of air- to- surface weapons currently carried onthe A7-E aircraft. The cost of the ballistics processor usingcurrent "off the shelf" components is $1635 and the entiretactical system was estimated to cost $3891.
JDD 1473Form
, 1 Jan 73S/N 0102-014-6601 JimJlSSIEIEIL
StCUKITY CLASSIFICATION OF THIS PAGEf*7»»n D*(* Enffd)
The Ballistics Processorof
A Multiple Processor Airborne Tactical System
by
Harry Andrew JupinLieutenant, United States NavyB.S., Muskingum College, 1968
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOLJune 1975
Naval Po
Jrtor-
ABSTRACT
This thesis developed the ballistics processor of a
multiple processor airborne tactical system. The multiple
processor system consisted of three INTEL-8080 microcomputers:
the executive processor, the navigational processor and the
ballistics processor. The ballistics processor utilized a
general second order Runge-Kutta method of integration of
the equations of motion of unguided air-to-surface weapons.
The ballistics processor computed sufficiently accurate and
timely solutions to enable the executive processor to extra-
polate an accurate release point for the weapon. The
algorithm permitted complete flexibility in release conditions
and allowed a complete arsenal of air-to-surface weapons
currently carried on the A7-E aircraft. The cost of the
ballistics processor using current "off the shelf" components
is $1635 and the entire tactical system was estimated to cost
$3891.
TABLE OF CONTENTS
I. INTRODUCTION 10
II. BACKGROUND - 15
A. THE NAVIGATIONAL COMPUTER 16
B. THE BALLISTICS COMPUTER 18
C. THE EXECUTIVE COMPUTER 18
D. THE INTERCOMPUTER COMMUNICATIONS NET 19
III. BALLISTICS PROBLEM 22
A. BASIC ASSUMPTIONS 22
B. DERIVATION OF THE EQUATIONS 2 3
C. RUNGE-KUTTA APPLICATION -- 24
D. CLASSES OF WEAPONS 2 5
1. Streamlined Bombs and Bullets 25
2. Drogued Bombs 26
3. Cluster Bombs 26
4. Unguided Rockets 26
E. COMPRESSION OF DRAG DATA 27
F. CURVE FITTING THE DRAG COEFFICIENTS 29
IV. FORTRAN SIMULATION 38
A. BOMB DROP SIMULATION 38
B. OVERALL SIMULATION 41
V. PL/M IMPLEMENTATION 45
VI. CONCLUSIONS 4 7
APPENDIX A PROGRAM FLOW CHARTS AND PROCESS GRAPHS -- 49
APPENDIX B PROGRAM DEFINITION OF VARIABLES 69
FORTRAN AND SIMULATED PL/M OUTPUT 7 2
FORTRAN PROGRAM 74
PL/M PROGRAM 84
LIST OF REFERENCES 102
INITIAL DISTRIBUTION LIST 103
LIST OF TABLES
Table
I. REFERENCE DRAG CURVE COEFFICIENTS AND CUTS 33
II. WEAPON CONSTANTS IN DECODE 34
III. SAMPLE ERRORS IN IMPACT RANGE CALCULATIONSFOR THE GENERAL -PURPOSE BOMB MK8 2/SNAKEYE/UNRETARDED 39
IV. SAMPLE ERRORS IN IMPACT RANGE CALCULATIONSFOR THE SPECIAL-PURPOSE BOMB MK40/MOD 0/CONICAL TAIL 39
V. SAMPLE ERRORS IN IMPACT RANGE CALCULATIONSFOR THE PRACTISE- PURPOSE BOMB MK76/WITHLUG 40
VI. FORTRAN TEST CASES (TIME INTERVAL, MK83) 4 2
VII. PL/M TEST VERIFICATION (TIME INTERVAL,MK83) 46
LIST OF FIGURES
Figure
1. Attack Aircraft's Tactical System'sInterconnection -- 11
2. The Intercomputer Communication Net 20
3. Curve Fitting Cd 30
4. Thrust Versus Time for a Typical Rocket 36
ACKNOWLEDGEMENT
The author wishes to express his thanks to Associate
Professor Kodres for his overall assistance and guidance
throughout the project, and his keen insight into micro-
computer programming. Special thanks go cut to my family,
Jeanne and Tim, who have constantly served as an inspiration
and stabilizing influence throughout the preparation of this
thesis
.
I. INTRODUCTION
Military airborne tactical systems are used to aid the
pilot and bombardier/navigator in their specific functions.
For an attack aircraft, the tactical system is used in the
following manner:
1. to navigate to a given target
2. to issue steering commands enroute and/or to guide
the autopilot
3. to display to the crew the current target location
in relation to the aircraft
4. to solve ballistics equations which enable the crew
to release a weapon at the appropriate instant.
The presently operational systems employ a small
general purpose computer, such as the IBM 4 PI system. In
this thesis, the ballistics processor of a multiple processor
organization of a typical tactical system for an attack air-
craft is presented. The system's architecture made use of
three INTEL- 8080 microcomputers, each of which was dedicated
full time to its assigned function. The purpose of the
multiple processor organization is to exploit the presently
emerging inexpensive microprocessor technology to solve
military tactical problems. The system's organization is
shown in Figure 1. Because microcomputers presently have
no hardware mult iply-and-divide functions, the programmed
software mult iply- and-divide functions caused the microcomputers
10
INERT I AL
NAVIGATION
SYSTEM
DOPPLER RADAR_— i—
—
NAVIGATION
DISPLAY
_L_t"?7
AIR DATA
COMPUTER
NAVIGATION COMPUTER\*,
V
DISPLAY
KEYBOARD H>-
ARMAMENT
SELECT
—</
—>-EXECUTIVE COMPUTER
WEAPONS
RELEASE
iHLCMmJAUTOPILOT
FIGURE 1. ATTACK AIRCRAFT'
S
TACTICAL SYSTETS INTERCONNECTION
11
to run on a typical mix of operations approximately six times
slower than the presently operational computer systems. By
reorganization of the computations and the use of table
look-up techniques, the described system was capable of meet-
ing the real-time requirements of an airborne weapon delivery
system.
The ballistics computer uses the best estimate of the
position, dive angle, altitude, and velocity of the aircraft
in its computation. It is also provided with the weapon
identification (for the particular drag characteristics)
and the wind velocities. From this information and the
target's altitude (above Mean Sea Level), the computer
determines the down range travel and time of fall of the
unguided weapon (i.e., the impact coordinates in the target
plane)
.
The accuracy with which a ballistic weapon can be
delivered against a target depends greatly upon the accuracy
of the sensor- supplied release parameters as well as the
accuracy with which the ballistics equations are solved to
predict the weapon's down range travel.
Most aircraft unguided air-to-ground weapons can be
described as ballistic projectiles. The only forces acting
on them after release from the aircraft are gravity and
aerodynamic drag. Bullets, streamlined bombs, drogued
(retarded) bombs, cluster munitions and unguided rockets
(after motor burnout) are all ballistic projectiles. Guided
12
weapons and weapons developing lift are not ballistic pro-
jectiles nor are they considered in this development,
This thesis first describes the over-all tactical system
in order to give the reader a perspective of how the ballistics
processor functions within the entire weapon system. Included
in this background material is a discussion of the naviga-
tional computer, the ballistics computer, the executive com-
puter and the intercomputer communications network.
The basic calculations involved in solving the ballistics
problem is described next. The assumptions used in the pro-
blem formulation are stated; the derivation of the particular
differential equations is presented; and the application of
the Runge-Kutta integration method is shown to be suitable
for current operational airborne weapon systems. The four
general classes of weapons and how the algorithm handles each
integration is defined. The unique method of compressing the
drag data for all twenty-eight weapons is shown to be a simple
curve fitting of the drag coefficients with respect to seme
standard.
The simulation was initially programmed using FORTRAN
for ease of testing and debugging. The important phases in
programming were in the DERIV and RUNGE subroutines where a
close study of the optimum calculation methods was conducted.
The program was tested for accuracy of results by comparison y
with the published range tables (NAVAIR 01-lC-lT-l).
Upon completion of the FORTRAN implementation, the re-
sultant program was translated into PL/M, a high level
13
language compatible with the selected microcomputer (INTELLEC
80) . The selected test cases were verified correct in the
PL/M version of the program.
14
I I , BACKGROUND
In order to give the reader a complete understanding of
how the ballistics processor functions, this section is
devoted to describing the overall tactical system in which
the ballistics processor is an important part. The first
section describes the function of the navigational computer,
the sensors which provide data to the computer and the
algorithm used to compute present position.
The ballistics computer will be completely defined in
the remaining sections of this thesis and will only be
briefly reviewed here. Using the best estimate of position
and velocity at a given instant, together with information
of the weapon's drag characteristics, the ballistics
computer solves a system of four differential equations and
calculates the impact coordinates of the weapon in the target
plane.
The executive computer controls the display and generates
commands to the autopilot and weapon release mechanism. This
computer is interrupted asynchronously as fresh data are
developed in the navigational and the ballistics computers.
The executive computer extrapolates from the data the appro-
priate release instant of the weapon.
The intercomputer communication net is described next.
The star-like communication net provided a simple and rapid
means of information transfer.
15
A. THE NAVIGATIONAL COMPUTER
This subsystem is present in all tactical systems. In
operational systems, a subprogram (module) is periodically
executed to update the present position by the change in
position in the last time increment.
In this computer, the sensor instruments provide digital
data to the computer at specified time intervals , controlled
by a clock. There are four sets of sensor instruments which
provide the data:
1. The Inertial Navigation System (INS)
2. The Doppler Radar (DR)
3. Air Data (AD)
4. Radar Set (RS)
.
The INS system provides attitude (roll and elevation) and
velocity increments in the x, y, z directions. The DR is a
velocity sensor that uses the doppler principle for the
continuous measurement of aircraft ground track speed and
drift angle. The AD computes outputs of modified corrected
static pressure, pressure altitude and mach number from
static and pitot tube pressure inputs. The RS provides
target azimuth, elevation, and range signals to the naviga-
tional computer. A magnetic compass measures the aircraft
heading in relation to magnetic, north. In order to navigate
accurately in this mode the wind speed and direction are
determined by using the "sensed" velocities of the aircraft.
Altimeter measurements are also sampled in this system to
update the present position altitude.
16
There are four basic modes in which this system operates.
The crew has a choice of using both the INS (velocities) and
DR outputs to minimize the Schuler oscillations. This is the
most reliable and accurate mode. If the crew suspects that
either the INS or the DR system is malfunctioning, they may
use only the data from one of the sources. The least
reliable and inaccurate mode is the last resort, when both
the DR and the INS are malfunctioning. The AD mode uses the
externally estimated wind vector and true airspeed of the air-
craft to determine the position in latitude and longitude.
In all operating modes, the same basic technique of smooth
ing input data is used. The least squares curve fitting
technique is used to fit a quadratic polynomial to the most
recent n (n $ 16) data values. Use of the Legendre
orthogonal polynomial functions is made in order to make the
numerical calculations as accurate and rapid as possible.
Smoothing the short time increment data values will generate
less total error in the results.
Using the smoothed data values, the appropriate calcu-
lations are made to estimate the distance increments in the
past time interval. As soon as the new value is generated,
the display of the navigational computer is refreshed, the
executive computer is interrupted and this most recent
estimate is transferred to the executive computer. If the
navigational computer is functional, the crew has information
on the present position, independent of the executive
computer
.
37
The least squares curve fitting technique enables the
system to diagnose sensor malfunction quite easily. If the
sensor generates fluctuations in the input data above a
certain threshold, then the sensor can be diagnosed as mal-
functioning. Similarly, if two sensors disagree above an
allowable threshold, then an automatic data analysis will
suggest which sensor is failing. If the crew concurs in
this diagnosis, the operating mode of the navigational
system can be changed to exclude the faulty sensor.
B. THE BALLISTICS COMPUTER
The ballistics computer is provided with the present
estimates of the position, dive angle, altitude, and
velocity of the aircraft. It is also provided with the type
of weapon and wind conditions. From this information and
the target's altitude, the computer determines the down-
range travel of the weapon, if released at this instant.
The assumption of no guidance after release from the air-
craft is still applicable. However, the weapon may have a
rocket assist, or a retarded mode of fall.
C. THE EXECUTIVE COMPUTER
The executive computer controls the main display and
generates steering commands to the autopilot and firing
pulses to the weapons release mechanisms. One of its
critical functions is to extrapolate from the data generated
by the ballistics computer the weapon release point.
18
A real time clock is used to record the time at which
the asynchronous data were generated by the ballistics
computer. The position of the impact point in the target
plane can be described by a quadratic function of time.
Therefore, the real time release point (adjusted for time
delays) can be extrapolated very accurately from these data
without additional extensive calculations.
D. THE INTERCOMPUTER COMMUNICATIONS NET
In order to keep the design of the intercomputer
communications net simple and inexpensive, the starlike
network was chosen. The executive computer is the center of
the star and one input port is multiplexed to the two output
ports of the navigational and ballistics computer as shown
in Figure 2.
The intercommunication takes place in the following
manner: An initializing phase of the program interrogates
the keyboard to obtain data about the initial position of
the aircraft. The latitude and longitude are entered in
degrees and minutes to the nearest 1/10 minute; the
altitude is initialized at the height above mean sea level.
These data are made available to the navigational computer.
Also the type of weapon and the position of the target (if
known in advance) is made available to the ballistics
computer. Similarly, the mode of navigation is determined
(if possible prior to flight) and relayed to the navigational
computer.
3 9
SEMSORS
iIN PORT 2
fER BALLISTICS COMPUTERNAVIGATION COMPU"
OUT PORT 1 IN PORT 1 IN PORT 1 OUT PORT 1
t4
^
tj
IN PORT 1
:CUTIV
OUT PORT 1
EXfr. COMPUTER
IN PORT 7 OUT PORT \ OUT PORT 2
4 <7 t,
/ KEYBOARDRELEASE
MECHANISMDISPLAY
FIGURE 2, THE INTERCOMPUTER
COMMUNICATION NET
20
The execution phase of the computer program makes use of
the real time clock in the executive computer which is
asynchronously inrerruping the navigational computer in
order that sensor information be transferred via input port 2
to the navigational computer. When the navigational computer
has completed its calculations, it interrupts the executive
computer and transfers the calculated current position
sequentially via input port 1 to the main computer. The
amount of information sent is 9 bytes, and the transfer is
accomplished in about 250 micro-seconds.
Similarly, when the ballistics computer has completed
its calculations, it interrupts the executive computer and
transfers its information sequentially via input port 1 to
the executive computer. Only 6 bytes of information are
sent in approximately 170 micro- seconds . Between interrup-
tions, the executive computer updates the display using
output port 2. Calculations are continuously made to extra-
polate the release instant of the weapon, and the release
mechanism is activated when appropriate via output port 3.
2!
III. BALL I STICS PROBL EM
The equations of motion governing the trajectory of a
ballistic projectile are a simple-appearing set of second-
order differential equations whenever the drag is neglected.
However, when a reasonably accurate model of the aerodynamic
drag caused by the projectile's motion through the air mass
is included in these equations they are rendered nonlinear,
and no general solution in closed form has been found that
is satisfactory for solution by an airborne fire-control
computer.
A. BASIC ASSUMPTIONS
The mathematical model was chosen to represent reality
as accurately as possible. The armed forces publish range
tables for various weapons and therefore the same model was
chosen as the one used to construct these tables (NAVAIR 01-
1C-1T-1)
.
The equations of motion were developed assuming the
projectile is a point mass acted on only by the force of
gravity and the retardation forces due to air resistance.
The trajectory was restricted to a plane by ignoring the
crosstrail effects of the wind. These crosstrail effects
and the component of wind which affects drag are considered
in the executive computer.
22
The assumptions are summarized below:
1. The Earth is flat and nonrotating.
2. The gravitational attraction is constant.
3. The projectile is a point mass.
4. The projectile is not powered (except rockets) and
has a constant mass.
B. DERIVATION OF THE EQUATIONS
The weapon's fall is assumed to be governed by a drag
force which is proportional to the square of the velocity.
Newton's equation of motion becomes
m dVt/dt = - g m - ||vt||2
Cw Vt/ ||vt||
where the only two forces acting on the weapon are gravity
and aerodynamic drag.
Vt - total velocity
m - mass of the projectile
g - force due to the Earth's gravitation
Cw - drag coefficient which depends on the
characteristics of the weapon and may vary
with time (in case the weapon has two drag
curves), and it varies slightly with airspeed.
In order to solve this differential equation, initial
conditions, that is the position and velocity, must be
specified. The most convenient coordinate system is height
"y" above sea level and "x" downrange travel, which is zero
at the time of release.
23
In the x and y component form, Newton's equation becomes
dVx/dt = - ||vt|| /m * Vx * Cw
dVy/dt = - g - ||vt|| /m * Vy * Cw
dx/dt = Vx
Dy/dt = Vy
x(0) = downrange travel
y(0) = present altitude
Vs(0) = present horizontal velocity
Vy(0) = present vertical velocity
C. RUNGE-KUTTA APPLICATION
The choice of the "best" Runge-Kutta (R-K) formula for
a particular problem was not entirely obvious. The lower
order formulae are simple, but require more integration
steps than the more accurate higher order formulae. On the
other hand, the complexity of the higher order formulae
may require so much computer time that this becomes the
overriding consideration. Generally, it was found to be
safer to use the lower order formulae when discontinuities
in the coefficients of the differential equations can be
expected. Such discontinuities frequently occur in bomb
trajectories (e.g., drogue deployment or thrust termination)
The second-order R-K formula also afforded the greatest
flexibility in the distribution and size of the integration
steps. These considerations, along with their simplicity,
led to the choice of the second-order R-K method as the most
appropriate for the ballistics algorithm.
24
Let UI = Vx Fl = -1/m * |Vt|| * Vx * Cw
U2 = y F2 = Vy
U3 = Vy F4 = -1/m *||Vt
||* Vy - Cw - g
The differential equations may be expressed as
dUi/dt = Fi(Ul,U2,U3,U4) i = 1,2,3,4
Therefore, the Runge-Kutta second order method used in the
program becomes
mli = At * Fi(Ul,U2,U3,U4) i = 1,2,3,4
m2i = At * Fi(Ul + cAtmll,U2 + cAtml2,U3 + cAtml3,
U4 + cAtml4) c = 0.7, i = 1,2,3,4
U'i = Ui + l/2c * ((2c - 1) * mli + m2i) i = 1,2,3,4
The procedure computes new values for the variables
until the remaining time of fall, DTV, (as computed for a
vacuum release) , is less than the value of the next integra-
tion time increment. The procedure then uses the value of
DTV for the final integration step size, and in the calcula-
tion of the down range travel of the weapon.
D. CLASSES OF WEAPONS
1 • Streamlined Bombs and Bullets
The process described above represents streamlined
bombs quite well. After being fired, a bullet is really just
a small, streamlined bomb and was handled as such. The muzzle
velocity (3300 feet per second) of a particular gun-bullet
combination was added to the aircraft velocity.
25
2
.
Drogued B ombs
These bombs are more complex in their trajectory
modeling. Their common characteristic is that they are all
released in a relatively low-drag configuration. At some
time after release, they deploy vanes or a parachute which
greatly increases the total drag. To compensate for this
change of drag characteristics, the initial integration step
size was chosen to coincide with the deployment of the high
drag devices. Thereafter, there was no difficulty in calcu-
lating the trajectory of the drogued weapons.
3. Cluster Bombs
These weapons are released as large, low-drag
containers. At a predetermined point in their trajectory,
which is time dependent, they use some method to dispense
smaller weapons. These smaller weapons are of higher drag
than the container. The ballistics computer will predict
the impact point of the center of the pattern rather than
individual positions of the bomblets. The method of applying
the algorithm is to calculate the trajectory of the container
to the point where it dispenses the small weapons; switch
to a different drag curve which describes the motion of the
pattern center; and integrate down to the target altitude.
As in the retarded bombs, the initial integration step was
chosen to coincide with the act of dispensing the small
weapons
.
^ Unguided Ro ckets
Rockets present several unique complexities. They
change weight during flight; they have thrust as well as
26
drag; and they slew around after firing, because their
launches may not line up with the aircraft direction of
flight. The algorithm treats unguided rockets like a
cluster bomb with the first stage having an average thrust
as well as an average thrust time. This provides sufficient
accuracy and saves computer storage that would be taken
storing the thrust profile of each rocket.
E. COMPRESSION OF DRAG DATA
The behavior of a particular weapon's drag coefficient,
Cd, as a function of Mach number must be known to calculate
impact ranges for that weapon. Normally, this information
is given in tabular form with some weapons having more than
100 points in the table. Most aircraft have the capability
to carry many different types of weapons. It is desirable
to store all the Cd information of an aircraft's weapon
repertoire in the airborne computer to avoid loading the Cd
data for each different combination of weapons. This
requires storing drag coefficient data in a more efficient
form than a table.
The following factors influence the choice of a scheme
to approximate the drag coefficients for a given set of
weapons
:
1. Computer storage
2. Computer computation time
3. Delivery envelope of each weapon
4. Allowable downrange and time-of-fall errors.
27
It would be ideal to use one general, fairly powerful
expression to fit all the drag tables for the weapon
repertoire of a given aircraft. This is quite difficult to
accomplish because weapons like guns and rockets require
Cd values for Mach numbers much higher than free-fall bombs.
As a consequence of this generally acceptable tabling pro-
cedure, extra storage is spent on weapons that do not require
the extra capability.
A significant saving can be realized when it is
recognized that several free-fall weapons have drag
coefficient curves that differ only by a multiplicative
factor. The Mk 80 low-drag series are one such group of
weapons with the Mk 84 drag curve conventially taken as the
reference. Similarly, the Mk 106, CBU bomblet, and Sadeye
bomblet are another group with the Garve drag curve as the
reference. The specific region of interest of many more
drag curves can be approximated by multiplying these two
reference drag curves by a factor and then translating them
along the Cd and M axes until they match the original drag
curves
.
To further aid in the explanation of approximating
weapon drag curves by this method, let Cd(M) and M be the
drag coefficient and Mach number of a weapon drag curve.
Let Cd (M) be the drag coefficient of a reference drag curve
and let A, B and DM be the multiplicative factor, drag
coefficient translation, and Mach number translation, respec-
tively, that when applied to the reference drag curve will
28
approximate the particular weapon drag curve of interest.
That is:
Cd(M) * A * Cd(M + DM) + B
A hypothetical weapon drag curve, Cd, and reference drag
curve, Cd, are shown in Figure 3. For simplicity of discus-
sion, the Mach number intervals (M2 -Ml) and (M4 - M3) are
equal. Suppose the Cd curve is multiplied by an A so that
the Cd and the A * Cd curves are nearly identical. For the
Cd and A * Cd curves to be a congruent, the A * C d curve must
be translated along the Mach axis an amount equal to DM, and
along the Drag Coefficient axis an amount equal to B.
For many weapons, the above stated equation will not give
an accurate fit over a wide interval, of Mach numbers; instead,
A, B and DM should be chosen so that the Cd fit is best for
the Mach number interval that corresponds to the delivery en-
velope for a particular aircraft.
Considering the weapons currently implemented on a typical
attack aircraft, (A7-E), both the Mk 84 and the Garve reference
drag curves are necessary to handle the free-fall weapons.
Instead of extending these curves to handle guns and rockets
that have a much larger Mach number range, a rocket-gun
reference drag curve is used.
F. CURVE FITTING THE DRAG COEFFICIENTS
In the previous section, the reference curves needed and
how they were used to approximate weapon drag curves for use
29
UJ
LU
<_>
(JD
A * Ld
fll M3 112 NAG I
FIGURE 3. Curve Fitting Cd
30
in the algorithm was presented. In this section the reference
drag curves versus Mach number is represented. The form of
the function used will minimize computer storage, logic, and
computation time for the allowable error in downrange and
time of fall for the different weapons. It was decided that
each of the three reference curves be divided into three
regions with Cd expressed as a second-order polynomial in
Mach number for each region.
Cd(M) = aO + al * M + a2 * M2
The coefficients, aO, al, and a2 depend on the region the
Mach number M is in. The drag coefficient for a particular
weapon is expressed as:
Cd(M) = A * (aO + al * (M + DM) + a2 * (M + DM)2
) + B
As a further step to save computer storage and reduce the
computation time for the algorithm, another compression of
drag data was undertaken. The computation C * (PI/8 * Cd)
,
CKDG, is made in the Runge-Kutta portion of the algorithm
2where C is the bomb factor, d /W, where d is the weapon
diameter in feet and W is the mass in slugs. One constant
per weapon in computer storage and two multiplications per
integration step is saved by expressing CKDG as
CKDG = CF * CKDG + DKG
where CF = A * (d/d)2
* W/W
31
CKDG = d /W * PI/8 * Cd
DKG = d2/W * PI/8 * B
In the above equations, d and W are the assumed diameter
and mass for the reference curve Cd. The final form of CKDG
in the algorithm is
CKDG(M) = CF * (bO + bl * (M + DM) + b2 * (M + DM)2
) + DKG
Table 1 gives a list of bO, bl, and b2 for each region of the
three reference curves and the values of CF, DM, and DKG, are
tabulated in Table 2.
Thrust appears in the algorithm in the total drag
function
HH = TH/V - RHO * CKDG * V
where TH = thrust/mass
RHO - air density
V = weapon velocity
and CKDG was previously defined in the above equations.
Figure 4 shows thrust versus time for a typical rocket
approximated by a constant, thrust = total impulse/Tl,
where Tl is decreased from actual motor burn time in order
to make thrust equal to the average thrust of region 2. The
mass of the rocket, W, obviously decreases during rocket
motor burn time. However, it is adequate for W to be
32
TABLE 1. Reference Drag Curve Coefficients and Cuts.
Coe fficientRetcrence Curve
Mk 3 A G arve Rocket- Gun
Re g ion 1 :
bOblb2
Re g ion 2 :
bOblb2
Re g ion 3 :
bOblb2
Cut
First cut, CT1
Second cut , CT2
5729E-030.00.0
4.67840E-020. 1097116 .65480E-02
0. 116380150.2 17643899
.
76706E-02
0.834
0.977
3.5350393. 3477822.872624
11.2616527.4162521 . 73083
23. 791544 4.2607714.49960
0.622
0. 885
0. 1041150.2303470. 167644
0. 1940370.4014780. 164612
7.33246E-022 . 03275E-022.44682E-03
1 .032
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36
approximated by a constant, by varying W until dov/nrange
error is minimized. The second stage of the rocket, for time
after weapon release greater than Tl, is free fall and TH
will be zero.
37
IV. FORTRAN SIMULATION
A. BOMB DROP SIMULATION
The initial ballistic, trajectory algorithm for digital
airborne fire control computers used in the simulation of
Traj , Deriv, and the Runge routines, was developed by the
Boeing Company, Seattle, Washington. The research was
completed under a Naval Weapons Center contract in September
1970.
As previously mentioned, the Second-Order Runge-Kutta
integration scheme was chosen because of its flexibility
and simplicity. The results, as evidenced in Tables 3, 4
and 5, compared favorably to the published data (NAVAIR
01-lC-lT-l), with mean errors of 0.101 for the down range
travel. The low-drag bombs showed relatively little
sensitivity to the step size, so long as the maximum
integration step, (DMAX) , was less than or equal to 5
seconds and the normal Initial Integration Step, (DTI),
was less than or equal to 3 seconds. However, the dual
stage weapons (IDN0>17) did show a large sensitivity to
the step size used during the integration scheme. The
accuracy desired, required the DTI value to be consistently
less than or equal to 2 seconds with the average value of
1.05 seconds, for the final eleven weapons.
The ultimate verification of the step size for the
Runge-Kutta (Deriv also) method was conducted by varying
38
TABLE 3. Sample Errors in Impact Range Calculations for theGeneral-Purpose Bomb Mk 82 /Snakeye /Unre tarded .
Release Conditions ImDac t Ranee, ft Impac t Rang e
Velocity, - altitude,Angle
,Ballistic Algor i thm error
,
kno t s ft deg Tables ft
400 2500 8039 80395000 -10 9158 9159 + 1
12000 -30 10536 10537 + 1
12000 -45 7530 7531 + 1
450 2500 8998 899 5 -3
5000 -10 10025 10023 -2
12000 -30 11340 11341 + 1
12000 -45 7998 7999 + 1
500 2500 9946 9940 -6
5000 -10 10843 10839 -4
12000 -30 12067 1206712000 -45 8406 8407 + 1
TABLE 4. Sample Errors in Impact Range Calculations for theSpecial-Purpose Bomb Mk 40/Mcd 0/Conical Tail.
Release Conditioiis Imnact Ranee, ft Impact RangeVelocity
,Altitude
,Angle
, Ballistic A.1 gorit hm error ,
knots ft de 2; Tables ft
400 2500 8305 8301 -4
5000 -10 9434 9432 -2
12000 -30 10847 10846 _ i
12000 -45 7691 7690 - 1
450 2500 9329 9324 -5
5000 -10 10342 10339 -3
12000 -30 11670 11669 -1
12000 -45 8160 8159 -1
500 2500 10348 10342 -6
5000 -10 11200 11197 -3
12000 -30 12401 12402 + 1
12000 -45 8559 8559
39
TABLE 5. Sample Errors in Impact Range Calculations forPractise-Purpose Bomb Mk 76/ with Lug.
the
Release Conditions Impact Ranee, ft Imp act RangeVelocity ,
Vlt i tude , Angle , Ballistic Algo r i thm errorkno t s ft deg Tables ft
400 2500 7633 7617 -165000 -10 8719 8710 -9
12000 -30 9991 9992 + 1
12000 -45 7232 7235 + 3
450 2500 8463 8459 -45000 -10 9493 9487 -6
12000 -30 10712 10720 + 8
12000 -45 7669 7674 + 5
500 2500 9268 9255 -135000 -10 10195 10192 -312000 -30 11337 11340 + 3
12000 -45 8038 8039 + 1
40
the time interval from one to five seconds. The TRAJ
integration scheme was simultaneously tested using several
release conditions to demonstrate the model's flexibility in
simulating bomb loft, level flight release and dive bombing
conditions. The emphasis, at this point, was on the number
of functional calls to RUNGE for the projectile to reach the
target altitude. The results, by altering At, showed changes
ranging from 0.00°6 to 0.44% for the time of fall, and 0.00%
to 0.46% for the downrange travel, with a substantial decrease
in the number of calls to RUNGE also evidenced (Table 6)
.
B. OVERALL SIMULATION
The PRINT routine was a means of simulating the various
cockpit displays in the tactical attack aircraft. The output
consisted of the release altitude, dive angle, and airspeed,
as well as the resulting downrange travel and the time of
fall of the weapon.
The INPUT routine simulated the collecting and passing
of the sensor data to the various processors. The sensors
included in the attack aircraft are the Inertial Navigation
System, Doppler Radar, Air Data and the Radar Set. Specifi-
cally, the velocity, dive angle, altitude and the identifi-
cation number of the weapon (usually provided via the armament
panel) are required for the execution of the program.
The SETDAT routine completed the interface between the
executive computer and the ballistics computer. Its purpose
was that of assigning the constants their specific nonvarying
41
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values and reinitializing the variables to zero at the start
of each new weapon's integration. The routine also performed
various calculations including the determination of the
velocity in the x and y directions, the conversion of raw
velocity to feet per second and the calculation of the release
angle in radians.
The weapon dependent constants required for the algorithm
were stored in DECODE. Also, DECODE contains the necessary
logic to load the correct constants for the particular
weapon the crew has selected. After the first pass, there
is no need to go through DECODE before entering the algorithm
for a trajectory calculation. The actual scheme used for
DECODE in reality will depend on the particular aircraft and
its fire control requirements and how the algorithm is inter-
faced in the airborne computer.
44
V. PL/M IMPLEMENTATION
To implement the model on the INTEL 8080 microcomputer,
the floating point arithmetic package was used to maintain a
four hexadecimal digit accuracy during calculations. The
floating point variable therefore occupies three eight-bit
words: two words for the variable mantissa and one word for
the mantissa sign bit and the variable exponent. The man-
tissa of the variable is viewed as a normalized binary number
with the left most digit always one.
In order to manipulate these variables, an arithmetic
subroutine library was supplied by Professor Kodres. At
present, there exist routines that add, subtract, multiply,
divide and take the square roots of these floating point
variables
.
The actual PL/M version of the ballistics program was
translated directly from the working FORTRAN version. All
the associated flow charts, process graphs, program listings,
and the simulated output can be found in the appendix.
The PL/M routines have been compiled and tested. The
results compare very favorably with the FORTRAN results.
The PL/M verification of the test parameters is included in
Table 7.
45
TABLE 7. PL/M Test Verification (time interval, MK 83)
RELEASECONDITIONS
BALLISTICTABLES
—DT = 3.0
COMPUTAT3 ON TIME
Trial 1 Trial 2
10°350 kts
1000 ft11.716701,
11.69 -.171(4)
6694. 5-. 0931.2(sec)
1.1(sec)
10°350 kts
3000 ft17.299826.
17.28 -.053(6)
9 8 2 . 5 -. 5 3
1.8 1.7
10°650 kts500 ft
13.8914192.
13.82 -.49%(5)
14140. -.3631. 3 1.2
10°650 kts
3000 ft20.85
2 8 5 9.
20.81 -.193(7)
2 8 2 . - . 1 8 3
2. 2 2.1
0°
300 kts3500 ft
14.857 3 8 2.
14.83 -.133
(5)7 3 79. -.033
1.6 1.4
0°
300 kts15000 ft
30.981516 4.
30.95 -.09 3
(11)1 5 1 5 6 .
-. 4 3
3.2 3.0
0°
450 kts5000 ft
17.821312 7.
17.81 -.053(6)
1 5 1 2 2 .- . 3 3
1. 8 1.7
-20°400 kts
5500 ft12.817 9 6 3.
12.80 -.0 73
(5)7963. 0.003
1.2 1.1
-20°550 kts
7000 ft13.50
113 8 7.
13.47 -.223(S)
11388. +. 0131. 2 i. 2
-40°450 kts
5000 ft8.18
46 89.
8.18 0.003(5)
4 6S9. 0.0030.9 1.0
-60°400 kts
70 ft9.61
5194.
9.60 -.103(4)
319 5. +.0230.3 0. 8
4 6
VI. CONCLUSIONS
The objective of this thesis was to demonstrate a micro-
computer based digital system which will perform the same
functions as the attack portion of a presently implemented
airborne tactical system. The main purpose of the project
was to verify the feasibility of implementing the most com-
plex arithmetic task of the airborne tactical computer by
use of an INTEL 8080 microcomputer at an estimated cost of
$3891 for the entire tactical system.
Because of the rapidly decreasing hardware costs of the
microcomputer and significant increases in processing speeds
which the microcomputer industry is presently realizing,
these new, innovative systems, which are in all ways equivalent
to the presently available avionics systems, can be developed
at a hardware cost which is nearly 10 times smaller. The
software development costs are also likely to be smaller
because the system can be subdivided into smaller, independ-
ently operating subsystems, each of which will perform a well
defined task.
The lack of hardware multiply-and-divide operations in
the microcomputer is compensated by careful process analysis
to reduce the number of multiply-and-divide operations. This
allows three ,: slow" (in the computer sense) computers to
solve the same problem that one "fast" computer is presently
solving, at a great hardware cost reduction. The necessity
for careful analysis of the problem may even disappear soon,
47
because the microcomputer manufacturers are likely to
incorporate the hardware multiply-and-divide operations.
This thesis demonstrates the feasibility of using the micro
computer as a systems' building block for airborne tactical
systems
.
APPENDIX A PROGRAM FLOW CHARTS AND PROCESS GRAPHS
Symbols Used In Flow Charts
( )Start/Stop
Output
Card Input
Processing Symbol(Arithmetic Operation)
Conditional Symbol
Subroutine Call
Return Statement
O Program Entry Point (label)
49
Symbols Used in Process Graphsinput output
1. Bite Operations (8 bits):
ADD
MULTIPLY
SUBTRACT
DIVIDE
ASSIGNMENT
2. Floating Point Operations:
ADD
MULT
DIV
SQRT
ADD (Subtract) Routine
MULTIPLY Routine
DIVIDE Routine
SQUARE ROOT Routine
3. Subroutine Calls
)ERIV First Derivative Routine
50
MAIN PROGRAM
c START
IJ
SET= -1
ISETDAT
5SWITCH= 1
5OINPUT
c STOP J^(T)
IIDN0=1
KIDNOSTEP 1 <y
I DNO=NUMI D ( I I DNO
)
SET = 0.0
51
MAIN PROGRAM (PART 2)
SETDAT
5DECODE
1IANG=1
STEP 1KDEG
<h
SLTHETA=DEG(IANG) *RAD
IY=1STEP 1
kiy<h
2SET =1.0
5
<]—(4
SETDAT
52
MAIN PROGRAM (PART 3)
SWITCH= -1.
TRAJ
£>(5
53
INPUT SUBROUTINE
C ENTRY }
vVELOCITY
2ALTITUDE
vDIVE ANGLE
vWEAPON ID
NUMBER
54
PRINT SUBROUTINE
C ENTRY
I3
GLOBAL
VARIABLES
OUTPUT
DATA
2.
HEADING
INFORMATION
55
SETDAT SUBROUTINE
c ENTRY
I)
GLOBAL
VARIABLES
7G= 32 .174
RAD= .0174533**A= 0.7
AA= 5/AYT=VYK= --5.0
FRACT== 0.5
RETURN
CFORMl= 0.0CFORM2= 0.0DM1= 0.0DM2= 0.0DKG1= 0.0DKG2= 0.0VMUZ= 0.0VE= 0.0SL= 0.0DMAX= 5.0
***ITYPE= -1
FN= 0.0IBOTH= 1
**T= 0.0
5
X±J
1U= VKTS(IV) *1.6878DEL= ATAN(VE,U)V=SQRT (U*U+VE*VE)VXA= (V+VMUZ) *COS(THETA-DEL)VYZ= (V+VMUZ) *SIN(THETA- DSL)
iRETURN
RETURN
* SET VALUES FOR PL/M ARE 1, 2, 3
** T IS TM AND A IS A IN PL/M.
*** ITiTE HAS A VALUE 3 FOR -1 IN PL/M.
56
DECODE SUBROUTINE( ENTRY )
GLOBAL
VARIABLES
c4IDNO )
—&E
G>
G>
REF= 4
DKG1= 2.55E-03DTI= 3.0
£> IREF= 4
DKG1= 6.29E-03DTI= 3.
G>—t>PREF= 4
DKG1= 4.01E-03DTI= 3.
-£>jlREF= 2
DMAX= 3
.
CF0RM1=. 3.92E-03DKG1= 2.75E-03DTI= 2.
-IX IREF= 2
DMAX= 3.
CF0RM1= 3. 91 E- 03
DKG1= 6.36E-03DTI= 3.
31
31
31
£>nIREF= 4
DMAX= 2
DKG1= 0. 021266DTI= 1.
IREF= 1
CFORMl- 2.5704
DTI= 3 .
o©
-o 3]
57
DECODE SUBROUTINE (PART 2)
G^H
12
14
IREF= 4
DMAX = 3.
DKG1- 9.77E-03DTI- 2.
O IREF= 1
CFORMl- 2.064DTI- 3.
GG^ I>REF = 1
CFORMl = 1. 34 31
DTI= 1.
tx IREF- 1
CF0RM1= 1.21DTI- 3.
© ^ IREF= 1
CFORM1 =1.0DTI- 3.
€> IREF- 1
CFORMl^ 3.12
DKG1= -1.22E-03DTI- 3.
t>( 31
31
31
31
31
15 O IREF- 1
DMAX- 3.
CFORMl- 3.4972DTI- 2.
£> IREF- ]
CFORMl- 1.605DTI- 3-
17 ~t> IREF- 4
DMAX- 3.
DKG1- 7.3 3F-0 3
DTI- 1
.
31
58
DECODE SUBROUTINE (PART 3)
18
19
20
21
O IREF= 1
ITYPE= 1
IBOTH= 2
DKG1 = 7.33E-03CFORM2= 1.69E-02DM2= 0.38DKG2= 0.17166DS= 0.6617SL= -0.000269DTI= 2.
t>
1>
t>
€>0
IREF= 1
ITYPE= 1
I BOTH= 2
CFORMl= 2.0754CFORM2= 0.2217DS= 4.267DTI= 1.
5
IREF= 1
I TYPE- 1
IBOTH= 2
CFORMl= 2.2973CM1= 0.32DKG1 = 8.17E-03CFORM2= 1.11E-02DM2=0.41DKG2=0. 16885DS= 4.06DTI= 2.0
31
IREF= 1
I TYPE = 1
r
I
IBOTH= 2
CFORMl= 2 2404
CFORM2= 1178DS= 4.0DTI= 1.62
31
59
DECODE SUBROUTINE (PART 4)
220-
24
25
26
27
28
o IREF= 1
ITYPE= 1
IBOTH= 2
DKG1= 9.77E-03CF0RM2= 2. 3 IE- 02DM2= 0.38DKG2= 0.23287DS= 0.679SL= -0.000303DTI = 1.622
Ol
CH IREF= 3
ITYPE= 2
CFORM1- 0.82CFORM2= 1 .
FN= 1746.
DS= 1.4225DTI= 1.
•O IREF= 4
ITYPE=DS= 0.98DKG2= 1.48DTI= 0.31
C> IREF- 4
ITYPE=DS= 0.89DKG2= 2.
DTI= 0.22
£> IREF= 4
ITYPE=DS= 0.89DKG2= 2.70
-OlIREF= 2
ITYPE= 2
CFORM1- 0.1514CFORM2= 0.1514DS = 0.5DTI= 0.8
31
IREF= 3 r
1
DMAX= 1.5CF0RM1= 2.9964DKG1 = -0.014992VMUZ= 3300.DTI= 0.5
o©
31
31
31
31
31
60
DECODE SUBROUTINE (PART 5) ©—-aQ2JO 1 ^Tref V i
&
33
34
O CC(1, 1,1) =
CC(1, 2,1):
CC ( 1 , 3 , 1 )
=
CC ( 2 , 1 , 1 ) =
CC(2,2,1)=CC(2,3,1)=CC(3,1,1)=CC ( 3 , 2 , 1 )
=
CC ( 3 , 3 , 1 )
=
CT(1,1)=0.CT(2,1) =Q.
:1.6E-03:0.0
0.04.7E-02-0.10976.7E-02-0.11640.2176-9.7E-02834977
i> CC ( 1 , 1 , 1 ) = .1041CC (1,2,1 )=-.2303CC(1, 3,1)= .1676CC(2,1,1)=-CC(2,2,1)=CC(2,3,1)=-
-.1940
.4015-.1646
CC ( 3 , 1 , 1 )
=
7.3E-02CC(3,2,1)=CC(3,3,1)=CT(1,I)= 1
CT(2,1)= 1
-2.0E-022.44E-0303230
-oT retURN
O©
CCU, 1,IB0TH) = 3.5350CC(1,2,1B0TH)= -3.3478CC(1,3,IB0TH)= 2.8726CC(2,1,IB0TH)= 11.2617CC(2,2,IBOTH)-- -27.4162CC(2,3,IBOTH)= 21.7308CC(3,1,IB0TH)= -23.7915CC(3,2,IBOTH)= 44.2607CC(3,3,IBOTH)= -14.4996CT(1,IB0TH)= 622CT(2,IB0TH)= 885 f
Io©
61
TRAJ SUBROUTINE
iD= DS+SL*U
£
C ENTRY
i3
CF= CFORMlDM= DM1DKG= DKG1MSTG= 1
X= 0.0
T= 0.0VX= VXAVY= VYATH = FNYA= Y
Y= ALT(IY)
5RUNGS
5DTV= 1/G*(VY+SQRT(VY**2+2.*G*V)
)
D= DTI
(F)
MSTG= 2 I
TEH '
MSTG=6 TABLE=27
o>* ITYPE= -1 IN FORTRAN
3 IN PLM
-O
_iD= DMAX
MSTG= 1 M>©PLMMSTG=TABLE 1=
62
TRAJ SUBROUTINE (FART 2)
©-
(7>o
DKG= DKG2
DM= DM2CF= CFORM2TH= 0.0
o
o DKG= DKG2
DM= DM2
CF= CF0RM2TH = 0.0
D= DTV
I PLM TABLEl=MSTG= 1 ! MSTG=
1RUNGE
DTV=1/G* (VY+SQRT(VY**2+2.*G*Y)
)
T= T+DTVX= X+DTV*VXSWITCH= 0.0
IPRINT
RETURN
63
RUNGE SUBROUTINE
C ENTRY
I3
GLOBAL
VARIANCES
IAD= A*DYO= Y
VXO= VXVYO= VY
2RHO=2.4E-04-Y (6.8E-08-Y(6.7E-13)
)
5DERIV
5Y=YO+AD*VYAP1= AP2
ANl = AN2
VX» VXO+AD*ANl
VY= VYO+AD*APl
RHO=2.4E-4-Y(6.8E-8-Y(6.7E-13)
)
DERIV
*A IS A IN PLM
** T IS TM IN PLM
XT= T+DX= X+D*(VXO+AA*(VX-VXO)
)
Y= YO+D* (VYO+AA* (VY-VYO)
}
VX= VXO+D*(ANl+AA*(AN2-ANl)
)
VY= VY0+D*(AP1+AA*(AP2-AP1)
)
64
O HXX 2 srH
<? <
g ? 1
1 . *—
3
>o
X>
DO >co
CQ3
r w 1r^ CO
5
V, \
CM
Eh
saa<
aa<
1i r r t
S VEhJ SUB
7E-08
!
S1 s
I l_co
.-1
1 2<
rV
J
5h
<Eh
D
II ii
i
w
^ <QQ<
>Mswa
(
iH
, / J
A—
&
' L D
Y1
i
toO
1
Ws^\Z r-
Eh
5s
CQ, D
1 CO
coo
i
w WV //\ ^
CO
niH
1
wr^
^J ( r\
D< C
>
•v S
oX>
O5
* r
sl-
1II
=F~^ —1
!
65
9 © € ©
<5
inwy§
wo
B
66
DERIV SUBROUTINE
c ENTRY )
GLOBAL
VARIANCES
£IR£G= 1
5.V= SQRT(VX*VX+VY*VY)
CM= V* (8.9E-04+3 . 2E-09*Y)+DM
^JlIREG= 2
^xr^CKDG= DKG+CF* (CC ( I REG , 1 , MSTG) + (CC(IREG, 2 , MSTG) +CC ( I REG , 3,MSTG) *CM) *CM)
VHH= TH/V-RHO*CKDG*VAN2= HH*VXAP2= HH*VY-G
SL
RETURN
67
68
APPENDIX B: PROGRAM DEFINITION OF VARIABLES
SYMBOLPL/M if
FORTRAN varied DEFINITION
A
AA
ALT
AN1,AN2
AP1,AP2
CC
CF
CFORM1
CFORM2
CKDG
CM
CT
DEG
DEL
DKG
DKG1
DKG2
DM
DMAX
DM1
DM2
DS
Al Runge Kutta parameter
0.5/Runge Kutta parameter
Weapon release altitude
Runge Kutta variables (x direction)
Runge Kutta variables (y direction)
Matrix of drag curve coefficients
CKDG stretch factor
CKDG stretch factor for the first stage
CKDG stretch factor for the second stage
Bomb coefficient times drag coefficient
Mach number
Matrix of Mach curve coefficients
Weapon release angle
ArcTan (ejection velocity/weapon velocity)
Shift in CKDG
Shift in CKDG for the first stage
Shift in CKDG for the second stage
Shift in Mach number
Largest integration step size
Shift in first stage Mach number
Shift in second stage Mach number
Integration step size
69
SYMBOLPL/M if
FORTRAN varied DEFINITION
DTI
FN
FRACT
G
HH
IANG
I BOTH
IDNO
IREF
IREG
IV
IY
MSTG
RAD
RHO
SET
SL
T
TH
THETA
U
V
vx
TM
Initial integration step size
Equal to the Thrust for the first stage
Parameter for Runte Kutta integration(value between zero and one)
Acceleration due to gravity
Total drag function
Number of release angles integrated
Key determining if two drag curvesare required for the weapon
Identification number of each weapon
Reference for Mach curve coefficients
Mach region index
Number of velocities integrated
Number of altitudes integrated
Weapon stage index
Conversion factor for radians
Air density
Key to initializing variables
Slope coefficient for retarded snakeyefins deployment time
Elapsed time of flight from weapon release
Thrust/Mass of rocket
Adjusted weapon release angle
Aircraft speed
Velocity of weapon
Velocity component of V in the x
direction
70
SYMBOLPL/M if
FORTRAN varied DEFINITION
VXA VX at weapon release
VXO VX at start of present integration step
VY Velocity component of V in the ydirection
VYA VY at weapon release
VYO VY at start of present integration step
VZ Muzzle velocity of guns
X Weapon ground range from release
Y Weapon altitude above sea level
YO Y at start of present integration step
YT Target altitude above sea level
71
FORTRAN AND PL/M RESULTS
IDNO VKTS ALT DEGFOR^PRAN PL/M
TOF RANGE TOF RANGE
1 450. 3000. -10. 10.26 7515. 10.27 7518.
2 450. 3000. -10. 10.41 7393. 10.41 7395.
3 450. 3000. -10. 10.32 7467. 10.32 7470.
4 450. 3000. -10. 10.73 7189. 10.74 7191.
5 450. 3000. -10. 10.91 7085. 10.8 7 7078.
6 450. 3000. -10. 11. 13 695 5. 11. 14 6957.
7 450. 3000. -10. 10.32 7466. 10.33 7469.
8 450. 3000. -10. 10.59 7292. 10.6 7294.
9 450. 3000. -10. 10.29 7492. 10.29 7495.
10 450. 5000. -10. 10.25 7521. 10. 26 7524.
11 450. 3000. -10. 10.25 7529. 10.26 7532.
12 450. 3000. -10. 10.24 7536. 10.24 7 5 39.
13 450. 3000. -10. 10.22 7547. 10.23 7550.
14 450. 3000. -10. 10.31 7478. 10.31 7480.
15 450. 5000. -10. 10.40 7424. 10.41 742 7.
16 450. 3000. -10. 10. 26 7 516. 10.27 7 519.
17 450. 3000. -10. 10.50 7368. 10.51 7371.
18 450. 3000. -10. 16.55 3852. 16.57 3847.
19 450. 300 0. -10. 20.84 4620. 20.87 4619.
20 450. 3000. -10. 14.92 5771. 14.94 5768.
21 450. 3000. -10. 16.83 5484. 16.87 5 4 81.
22 450. 3000. -10. 17.96 3299. 17.98 3296.
23 4 5 0. 5000.j
-10. 10.67 10 9 4 0. 10.69 10935.
FORTRAN AND PL/M RESULTS (continued)
IDNC VKTS ALT DEC
FORTRAN PL/M
TOF RANGE TOF RANGE
24 450. 3000. -10. 7.82 12814. 7.85 12815.
25 4 50. 3000. -10. 31.70 1657. 31.70 1656.
26 450. 3000. -10. 36.29 1384. 36.30 1382.
27 450. 3000. -10. 41.58 1213. 41.58 1210.
28 450. 3000.... .. _ . . ..
-10. 20.56 2444. 20.57 2442.
73
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LIST OF REFERENCES
1. Naval Weapon Center Technical Publication 5416,A Ballistic Trajectory Algorithm for Digital AirborneFire Control Systems , Duke, Arthur, and others
,
September 1972.
2. Hildebrand, F. B., Introduction to Numerical Analysis,
p. 485-495, McGraw Hill, 1956.
3. Naval Weapons Laboratory Technical Report 2741, A-6Program Revision , Kuty, L. R. , January 1974.
4. Lambert, J. D. , Computational Methods in OrdinaryDifferential Equations
, p. 114-156, Wiley, 1973.
5. Lapidus, L., Numerical Solutions of Ordinary DifferentialEquations , p . 39-137, Academic Press, 1971
.
6. Naval Weapons Laboratory Tactical Manual BallisticTables, NAVAIR 01-lC-lT-l, October 1973.
7. Pease, J. A., No Drop Bomb Simulation Using Micro-Computers , M.S. Thesis, Naval Postgraduate School,Monterey, California, 1974.
8. Syracuse University Research Corporation TechnicalReport 73-129, A6A-A6E/Tram , Rodems, J. D., and others,March 19 73.
102
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 2
Cameron StationAlexandria, Virginia 22314
2. Library, Code 0212 2
Naval Postgraduate SchoolMonterey, California 93940
3. Chairman, Department of Aeronautics 1Naval Postgraduate SchoolMonterey, California 93940
4. Assoc. Professor U. R. Kodres, Code 72Kr 1
Department of Computer ScienceNaval Postgraduate SchoolMonterey, California 93940
5. Lt. Harry A. Jupin, USN 1
920 Miller AvenueClairton, Pennsylvania 15025
6. Burt Chase, Code KBC 1
Naval Surface Weapon CenterDahlgren, Virginia 22448
7. T. H. Zehner • 1
Grumman Aerospace CorporationTram BuildingPlant 07, Department 471Calverton, New York 11933
8. Commander Naval Air Test Center 1
System Analysis GroupWeapon System GroupPatuxent River, Maryland 20670(Attn. F. Phillips)
103
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