the ballistics processor of a multiple processor airborne

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

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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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Approved for public release; distribution unlimited.

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4. TITLE (and Subtitle)

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)

9. PERFORMIMG ORGANIZATION NAME AND ADDRESS

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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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

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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

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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

PRINT

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.

PRINT

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


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


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

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


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


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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101

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


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

>0 JAN765 NOV 7*24 JAN79

2 3 8J «

2 U 9 U 1

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