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MEMORANDU -I - 'P BR -R-3768 DETERMINATION OF AERODYNAMIC DRAG AND EXTERIOR BALLISTIC TRAJECTORY SIMULATION OR THE 155MM. DPICM, M864 BASE-BURN PROJECTILE ROBERT F, ESKE E+~TIC... UL .42 , , APPROVED Mi PUBLI, RUE ISIR S' N UJNUI. T' " Available COPY/ :'Best v , U.S. ARMY LABORATORY COMMAND BALLISTIC RESEARCH LABORATORY ABERDEEN PROVING GROUND, MARYLAND 6o 0 028
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155 Mm Drag and Exterior Ballistic

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Page 1: 155 Mm Drag and Exterior Ballistic

MEMORANDU -I -'P BR -R-3768

DETERMINATION OF AERODYNAMIC DRAG

AND EXTERIOR BALLISTIC TRAJECTORY SIMULATIONOR THE 155MM. DPICM, M864 BASE-BURN PROJECTILE

ROBERT F, ESKE

E+~TIC...UL

.42

, , APPROVED Mi PUBLI, RUE ISIR S' N UJNUI. T'

" Available COPY/:'Best v

, U.S. ARMY LABORATORY COMMAND

BALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

6o 0 028

Page 2: 155 Mm Drag and Exterior Ballistic

UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE

Form ApprovedREPORT DOCUMENTATION PAGE OM8No. 0704-OI8

P. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS

UNCLASSIFIED2a. SECURITY CLASSIFICATION AUTHORITYr 3. DISTRIBUTION /AVAILABILITY OF REPORT

2b. DApproved for public release; distribution2b. DECLASSIICATION DOWNGRADING SCHEDULE is unlimited.

4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S)

BRL-MR-3768

6. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATIONU.S. Army (If applicable)

Ballistic Research Laboratory I SLCBR-LF

6c. ADDRESS (City, State, and ZIP Code) 7b. ADDRESS (City, State, and ZIP Code)

Aberdeen Proving Ground, MD 21005-5066

Ba. NAME OF FUNDING/SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION (it applicable)

US Army Research.Developmentn- Enginprin2 CenterI

8c. ADDRESS (City, State, and ZIP Code) 10. SOURCE OF FUNDING NUMBERSPROGRAM PROJECT TASK WORK UNITELEMENT NO. NO. NO. ACCESSION NO.

Picatinny Arsenal, NJ 07806-5000 F18Y30 77MlAJ

11. TITLE (Include Security Classification)

Determination of Aerodynamic Drag and Exterior Ballistic Trajectory Simulation for The155mm, DPICM, M864 Base-Burn Projectile

12. PERSONAL AUTHOR(S)

Robert F. Lieske13a. TYPE OF REPORT 13b. TIME COVE:ED 14 DATE OF REPORT (Year,1Mont5,OaY) 1,5. PAGE COUNT

Memorandum Report _ FROM TO___ 1989 January 13716. SUPPLEMENTARY NOTATION

This report supersedes IMR-891, dated June 1987.:7. COSAT1 CODES 18. SUBJECT TERMS (Continue on reverse if necessary ana identify by block number)

FIELD GROUP SUB-GROUP Base-Burn Projectile Dopper Radar Data

19 01 Aerodynamic Drag Flight Performance

01 01 Trajectory Modeling19. ABSTRACT (Continue on reverse if necessary and identify by block number) -

HAWK Doppler radar data collected at Yuma Proving Ground,4Ar-L for the 155mm, DPICM,

M864 base-burn projectile have been reduced for the purpose of determining the aerodynamic

drag. The estimated- base drag reduction during base-burn motor functioning showed a verygood correlation with time of flight and to a lesser degree with local atmospheric air

pressure. No correlation was evident with flight Mach number provided an effective basedrag coefficient is assumed which is just a function of Mach number. This result suggestsa simple addition to the Modified Point Mass Trajectory Model for the exterior ballisticsimulation of the M864 base-burn projectile. K, ,e c A rJee 41 C

•~.2# ' C C /.-',-

20. DISTRIBUTION /AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATIONr-UNCLASSIFIED/UNLIMITED 10 SAME AS RPT. - DTIC USERS UNCLASSIFIED

22a. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE (include Area Code) 22c. OFFICE SYMBOLROBERT F. LIESKE 301-278-3577 $SCBR-LF-T

aU o rm 1473, JUN 86 Previous editions are obsolete. SECURITY CLASSIFICATION OF THIS PAGE

UNCLASSIFIED

Page 3: 155 Mm Drag and Exterior Ballistic

Acknowledgement

The author would like to express his appreciation to Mr. Joseph A. Hurfffor preparing the computer programs required to calculate the aerodynamic dragresults and to Mrs. Margaret S. Wilson and Messrs. Joseph M. Wall and JosephW. Kochenderfer for assembling and reducing the 155mm, DPICM, M864 base-burn projectile range data. The author also wishes to acknowledge the numeroushelpful comments and suggestions received from Dr. James E. Danberg.

Aooession For

TIS GRA&IDTIC TABUnannounced I]Justification ,

IB.Distribution/

Availability Codes

IAvail and/orDist Special

AO

Page 4: 155 Mm Drag and Exterior Ballistic

Table of ContentsPage

List of Figures .. .. .. ... ... ... ... ... .... ... ... ..... vii

I. Introduction. .. .. .. .. ... ... ... ... ... .... ... ... .... 1

11. Reduction of HAWK Doppler Radar Velocimeter Flight Data .. .. .. .... 1

111. Determination of Aerodynamic Drag Coefficient .. .. .. .. ... ... .... 2

IV. Determination of Base Drag Reduction Factor. .. .. .. .. ... ... .... 3

V. Results. .. .. .. .. ... ... ... ... ... ... ... .... ... .... 3

VI. Analysis of Aerodynamic Drag Results. .. .. .. .. .... ... ... .... 4

VII. Exterior Ballistic Trajectory Simulation Model. .. .. .. .... ... .... 5

VIII. Conclusions .. .. .. .. ... .... ... ... ... ... ... ... ..... 7

References .. .. .. .. ... ... ... ... ... ... ... .... ... .. 27

List of Symbols .. .. .. .. ... ... ... ... .... ... ... ... .. 29

Distribution List. .. .. .. ... .... ... ... ... ... ... ... .. 31

Page 5: 155 Mm Drag and Exterior Ballistic

List of FiguresFigure Page

1 Physical characteristics of the 155mm, DPICM, M864 base-burn projectile. 8

2 Aerodynamic drag coefficient versus Mach number for round number 4315,with inert base-burn motor, fired with propelling charge M4A2, 7W, at aquadrant elevation of 748 mils ................................ 9

3 Aerodynamic drag coefficient versus Mach number for round number 4376,with inert base-burn motor, fired with propelling charge M119A2, 7R, at aquadrant elevation of 748 mils ................................ 10

4 Aerodynamic drag coefficient versus Mach number for round number 4341,with inert base-burn motor, fired with propelling charge M203E2, 8R, at aquadrant elevation of 748 mils ................................ 11

5 Base drag reduction factor versus time of flight for round number 5089 firedwith propelling charge M4A2, 7W, at a quadrant elevation of 500 mils. . . 12

6 Base drag reduction factor versus time of flight for round number 1034 firedwith propelling charge M4A2, 7W, at a quadrant elevation of 750 mils.. . . 13

Base drag reduction factor versus time of flight for round number 1013 firedwith propelling charge M4A2, 7W, at a quadrant elevation of 1150 mils. . . 14

8 Base drag reduction factor versus time of flight for round number 1044 firedwith propelling charge M119A2, 7R, at a quadrant elevation of 500 mils.. . 15

9 Base drag reduction factor versus time of flight, for round number 1050 firedwith propelling charge M119A2, 7R, at a quadrant elevation of 750 mils.. 16

10 Base drag reduction factor versus time of flight for round number 4202 firedwith propelling charge M119A2, 7R, at a quadrant elevation of 1150 mils. 17

11 Base drag reduction factor versus time of flight for round number 4216 firedwith propelling charge M203E2, 8R, at a quadrant elevation of 499 mils. . 18

12 Base drag reduction factor versus time of flight for round number 4329 firedwith -propeling charge M203E2, 8R, at a quadrant elevation of 748 mils.. 19

13 Base drag reduction factor versus time of flight for round number 4219 firedwith propelling charge M203E2, 8R, at a quadrant elevation of 1147 mils. 20

14 Aerodynamic drag force coefficients for the 155mm, DPICM, M864 base-

burn projectile .......................................... 21

15 Factor f, versus Mach number for the 155mm, DPICM, M864 projectile. 22

16 Factor 12 versus time of flight for the 155mm, DPICM, M864 projectile. 23

vii

Page 6: 155 Mm Drag and Exterior Ballistic

List of Figures (Continued)Figure Page

17 Residuals of the stepwise multiple regression analysis versus time of flight. . 24

18 Residuals of the stepwise multiple regression analysis versus Macli number. 25

19 Residuals of the stepwise multiple regression analysis versus local air pressure. 26

Vii

Page 7: 155 Mm Drag and Exterior Ballistic

I. Introduction

The Modified Point Mass Trajectory Model 1.2 is the primary method of trajectorysimulation used in the preparation of Firing Tables. This model requires three typesof input data: projectile mass properties, aerodynamic coefficients and the performanceparameters determined from range testing. This report discusses an initial attempt todetermine, for trajectory modeling, the aerodynamic drag of the 155mm, DPICM, M864base-burn projectile (Figure 1) from the HAWK Doppler radar data.3 .4 A detailed treatiseof base-burn projectile ballistic modeling technology is reported by Niles-Erik Gunners,Kurt Andersson and Rune Hellgren in Reference 5, Chapter 16, "Base-Bleed Systems forGun Projectiles".

II. Reduction of HAWK Doppler Radar Velocimeter FlightData

The frame of reference for all vectors is aground-fixed, orthonormal, right-handedCartesian coordinate system with unit vectors ( 1, 2 and 3 ). The 1 axis is the intersectionof the vertical plane of fire and the horizontal plane and pointing in the direction of fire.The 2 axis is parallel to the gravity vector, Y, and opposite in direction. The 3 axiscompletes the right-handed coordinate system.

The slant range rate of change as measured by HAWK Doppler radar is recorded ona digital tape. The first step is to smooth the data and determine the time derivative.Least squares fits (second-degree polynomials) to the data are determined for 0.56 secondintervals (fifteen point smoothing) along the trajectory. The slant range rate of change (i)and time derivative of the slant range rate of change (i) are obtained from the quadraticfit at the midpoint of the fifteen point interval.

An estimated trajectory is generated separately using the projectile mass properties,launch data, atmospheric conditions, estimated aerodynamic coefficients and estimatedbase drag reduction factor during base-burn motor burning as a function of time. Thetrajectory is adjusted, using factors on base drag reduction during motor functioning andlift, to match the observed impact data. A trajectory velocity (, ) is calculated using theHAWK radar smoothed slant range rate of change (i) and the estimated trajectory slantrange rate of change (it ) and velocity (Urt) as follows:

,= ( / ,)t4 (1)

where:

it = ut cos(rt, Ut) =

Page 8: 155 Mm Drag and Exterior Ballistic

Note: Subscript t refers to quantities determined from the estimated trajectory and thosewith subscript r are obtained using both the HAWK radar data and the estimatedtrajectory.

A trajectory acceleration (u"4) was calculated using the time derivative of the HAWK radarslant range rate of change (i) and the estimated trajectory, time derivative of the slantrange rate of change (i't) and acceleration (ut), using the following two formulations:

"4 = ( i/,'t),- + [( t F-_ )/ ] (2)

and

('. / U) (3)

where:

7,; = {,rt -' 1 r4 ) + (f,.; ")] - (f,.t t),it}/rt

and

rt =

The mean of the results were similar for both of the u-, representations, however, thevariation (spread) of the results were significantly improved using equation 3 and it wasused for determining the results presented.

III. Determination of Aerodynamic Drag Coefficient

The mass of the projectile, atmospheric conditions, estimated trajectory data and theDoppler slant range rate of change and time derivative of the slant range rate of changeprovide the necessary inputs to determine the aerodynamic drag. The following inversesolution of the point-mass equations of motion is then used to compute the aerodynamicdrag (C,). -

CD, = -1(Tr - U-1 - 8- A) 8m/ (rpd ' ) (4)

2

Page 9: 155 Mm Drag and Exterior Ballistic

IV. Determination of Base Drag Reduction Factor

The base-burn motor is intended to increase the range of the projectile by reducingthe base drag. An initial estimate of the aerodynamic zero yaw drag force coefficient (CDo)and the base drag component (CDD) was determined for the M864 base-burn projectile,with an inert base-burn motor, using the semi-empirical drag estimation model known asMcDrag.6 A base drag reduction factor (IBD) was then defined to quantify the amount ofbase drag reduction as follows:

fBD = (Cro - CD,) /C,, (5)

A base drag reduction factor of unity indicates that all the estimated base drag is elimi-nated, whereas a value of zero means none of the estimated base drag is eliminated. A fBDgreater than one indicates that thrusting is present, since more than just the base draghas been eliminated.

V. Results

The aerodynamic drag results, obtained from the procedure described above, arepresented for a sampling of the M864 projectiles, including three M864 projectiles withinert base-burn motors. These projectiles were fired at Yuma Proving Ground, AZ duringMay 1987. The base-burn motors were designed to burn out at approximately 30 seconds.The aerodynamic drag coefficients for the projectiles with inert motors are presented asa function of Mach number and for the projectiles with live motors base drag reductionfactors are presented as a function of time of flight.

The sampling includes projectiles with inert and live base-burn motors fired withpropelling charges: M4A2, charge 7W; M119A2, charge 7R; and M203E2, charge 8R. Thesampling contains an inert base-burn motor fired at a quadrant elevation of 748 mils andlive base-burn motors fired at three quadrant elevations (approximately 500, 750 and 1150mils) with each of the propelling charges.

Figures 2 through 4 present the aerodynamic drag coefficients versus Mach numberfor projectiles -with inert base-burn motors fired with each of the propelling charges ata quadrant elevation of 748 mils. Figures 5 through 13 present the base drag reductionfactors versus time of flight for projectiles with live base-burn motors fired with each ofthe propelling charges at quadrant elevations of approximately 500, 750 and 1150 mils.

Figures 2, 3 and 4 show excellent agreement of the aerodynamic drag coefficientsfor the inert base-burn motor projectiles fired with the three propelling charges. Fig-ures 5 through 13 show a good correlation of the base drag reduction factor with timeof flight for the live base-burn motor projectiles. The base drag is consistently reducedby approximately 50 percent and the results show an increase in this percentage for thehigher quadrant elevations indicating a possible correlation with local atmospheric pres-

3

Page 10: 155 Mm Drag and Exterior Ballistic

sure. There is some irregularity in the base drag reduction factor for the transonic velocities(Mach numbers: .95 to 1.05). This is identified where evident on some of the figures. Theirregularity is probably due to the error in the transonic aerodynamic inputs and/or theMach number determined from the HAWK radar data. The base drag reduction factorsalso indicate a delay in base drag reduction at the beginning of approximately one secondand a base-burn motor burnout time of approximately 33 seconds.

The large variation in the base drag reduction factor for the high quadrant elevationrounds is probably due to a combination of the following: the magnitude of the totaldrag coefficient, the data reduction methodology including the estimated trajectory, theestimated aerodynamic coefficients and the HAWK Doppler radar capability.

VI. Analysis of Aerodynamic Drag Results

An aerodynamic drag coefficient CD0 was determined for the M864 projectile basedon the inert base-burn motor projectile results. In addition, an estimated inert, base dragcomponent CD, was determined for the M864 projectile with an inert base-burn motorby subtracting the McDrag estimates of head, skin friction, band and boat tail drag fromthe determined CD for the inert M864 projectile. Figure 14 presents the determined CD"and CD, functions of Mach number.

The base drag reduction factors for times of flight between three and 33 seconds wereanalyzed using a stepwise multiple regression technique.7 The base drag reduction factorswere fit as a function of time of flight and the difference in a reference (standard) airpressure (P,) , 1013.25 mb, and the local air pressure (P) divided by P, to account forthe apparent variation with quadrant elevation. A review of the residuals as a functionof Mach number suggested an additional function of Mach number, f,, which forms amodified base drag coefficient, (CDB. fi), for the M864 base-burn projectile. Figure 14presents the modified base drag coefficient, (CDD, fl), while Figure 15 presents the f,factor as a function of Mach number. The value of (CD.. fl) is 25 percent higher for Machnumbers less than 0.9 and 11 percent lower at Mach number 2.0 than the theoreticallydetermined CD, . A theoretical discussion of drag reduction for base-burn projectiles ispresented in Relerence 5, Chapter 16.

The stepwise multiple regression fitting function including fi is as follows:

[(CO - CD)/(C 3 ,D fi)] = f2 + h (P, - P)/P, (6)

Where:

" Factor f2 is a function of time of flight.

" Factor f4 is a constant.

Note: A factor f3 multiplying f2 is added in the final formulation.

4.

Page 11: 155 Mm Drag and Exterior Ballistic

The factor, f2, as a function of time of flight, determined by the stepwise multipleregression process for the nine M864 base-burn test projectiles (Figures 5 through 13), ispresented in Figure 16 and the constant, f4, determined simultaneously, was 0.30. Theresiduals of the stepwise multiple regression fitting process are presented in Figures 17, 18and 19 versus time of flight, Mach number and local air pressure respectively. The rootmean square error of residuals is 0.12 and the residuals show that the fitting process hasremoved the symmetric biases due to time of flight, Mach number and local air pressure.

Summarizing:

" An aerodynamic drag coefficient (CD0 ) was determined for the M864 projectilebased on the inert base-burn motor projectile results.

" A base drag component (CDv,) was theoretically determined for the M864 pro-jectile with an inert base-burn motor based on CD0 minus the fore body dragestimated by McDrag.

" A factor, f,, multiplied by the inert projectile base drag component (CD.) isused to represent an effective inert base drag coefficient for the M864 base-burnprojectile as a function of Mach number.

" A factor, f2. is used to represent the effect of time on drag reduction during theburning phase of the M864 base-burn projectile.

" A change in drag reduction with local air pressure was assumed to account forthe observed variation of the M864 base-burn projectile base drag with quadrantelevation. This apparent effect with local air pressure was further assumed totake the form f4 ( P, - P ) / P, , where the factor, f4 , is a constant.

VII. Exterior Ballistic Trajectory Simulation Model

The reduction in base drag during base-burn motor functioning is shown to be afunction of time of flight and to a lesser degree a function of local air pressure. A constant,f3, multiplied by f2 is added to create a simulation methodology containing flexibility formatching experimental radar and impact or air burst data. The constants, f3 and f4,can then be used effectively as parameters for matching the measured impact or air burstrange firing data.

The combined base drag reduction term, - ( CD, )[ f2 f3 + f ( P, - P ) / P, ], isthen added to the projectile's inert total drag force coefficient term, CD. + CD., (Qa,) 2 ,of the Modified Point Mass Trajectory Model for Rocket Assisted Projectiles.

The total drag computed by the Modified Point Mass Trajectory Model for RocketAssisted Projectiles is then as follows:

Page 12: 155 Mm Drag and Exterior Ballistic

{CDo -(CDf)f 2 f3 + (P - P)/P,] + CD (Q )} V( 7 )8m

Where:

* The factor, fo, can be used as a function of quadrant elevation, instead of theballistic coefficient, for matching experimental impact or air burst range firingdata for projectiles without base-burn motors. A value of 1.0 was used for fobecause this parameter was not used for matching the experimental firing data ofthe M864 baoe-burn projectile.

Note: The ballistic coefficient, m, / ( fo d 2 ), as a function of quadrant elevationis currently in use as the parameter for matching experimental impact orair burst range firing data. The use of fo, previously identified as i, hasthe advantage of being a nondimensional factor.

* The term, ( CD,, f, ) ( f2 f3 ), is used to represent the drag reduction as a functionof Mach number and time of flight and for matching experimental impact or airburst range firing data of individual or groups of projectiles.

e The factor, f, , as illustrated in Figure 15, is used to represent the change indrag reduction as a function of Mach number. f, was determined from theanalysis of the HAWK radar data.

e The factor, f2, as illustrated in Figure 16, is used to represent the change indrag reduction as a function of time of flight. f2 was also determined fromthe analysis of the HAWK radar data.

* The factor, f3 can be used as a parameter for matching experimental impactor air burst range firing data.

* The term, ( CD,, fi )[ f ( P, - P ) / P, ],is used to represent the drag reductiondue to an apparent effect of the local air pressure.

* The factor, f4, is a constant during base-burn motor functioning and is zerothereafter. f 4 was determined to be 0.30 based on the analysis of the HAWKradar data.

* The factor, Q, is a constant used to compensate in part for the approximationsin the Modified Point Mass Trajectory Model. A value 1.2, normally used forartillery projectiles, was used for this parameter.

In the above model the nondimensional factor, fo, replaces the ballistic coefficientas the parameter used for matching experimental impact or air burst range firing data ofprojectiles without base-burn. In that role, fo would normally be a function of quadrantelevation. The factors, f, through f4, should be effective parameters for representing theHAWK radar results and matching the experimental impact or air burst range firing dataof the 155mm, M864 base-burn projectile.

6

Page 13: 155 Mm Drag and Exterior Ballistic

VIII. Conclusions

The aerodynamic drag coefficients determined for the M864 projectiles with inert base-burn motors show excellent agreement for the flight Mach numbers of projectiles fired withpropelling charges: M4A2, charge 7W; M119A2, charge 7R; and M203E2, charge 8R.

The reduction in base drag during base-burn motor functioning correlated very wellwith time of flight and to a lesser degree with local air pressure. These results support thesimplistic addition of a base drag reduction term to the acceleration due to drag equationof the Modified Point Mass Trajectory Model for Rocket-Assisted Projectiles for simulatingthe ballistic trajectory of the 155mm, M864 base-burn projectile.

7

Page 14: 155 Mm Drag and Exterior Ballistic

155mm, DPICM, M864 Base-Burn Projectile

Sketch

Dimensions

Length of Projectile calibers 5.79

Nose Length calibers 3.42

Cylinder Length calibers 1.86

Boattail Length calibers .50

Boattail Angle degrees 3.00

Mass Properties

Mass kgs 46.95(lbs) 103.5

Mass of Fuel kgs 1.21(lbs) 2.67

Center of Gravity cm from nose 58.8(in from nose) 23.16

Moments of Inertia

Axial kg-m 2 .158(lb-ft 2) 3.75

Transverse kg-m 2 1.657(lb-ft) 39.32

Figure 1. Physical characteristics of the 155mm, DPICM, M864 base-burn projectile.

8

Page 15: 155 Mm Drag and Exterior Ballistic

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References

1. Lieske, R.F. and Reiter, M.L., "Equations of Motion for a Modified Point Mass Trajec-tory," TS Army Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland,BRL Report No. 1314, March 1966. (AD 485869)

2. NATO STANAG 4355 (Draft Edition 1), The Modified Point Mass Trajectory Model,February 1988.

3. Lieske, R.F. and Kochenderfer, J.W., "Determination of Performance Parametersfor Fin-Stabilized Free-Flight Missiles," US Army Ballistic Research Laboratory,Aberdeen Proving Ground, Maryland, BRL Report No. 1349, December 1966.(AD 809790)

4. Lieske, R.F. and MacKenzie, A.M., "Determination of Aerodynamic Drag from RadarData." US Army Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland,BRL Memorandum Report No. 2210, August 1972. (AD 750564)

5. Nils-Erik Gunners, Kurt Andersson and Rune Heligren, National Defense Research In-stitute (FOA) Tumba. Sweden, "Base-Bleed Systems for Gun Projectiles," Chapter 16,Volume 109. Dated 1988. Progress in Astronautics and Aeronautics, Gun PropulsionTechnology. Published by the American Institute of Aeronautics and Astronautics,Inc.. 370 L'Enfant Promenade, SW. Washington, DC 20024.

6. McCoy. R.L.. " 'McDrag' - A Computer Program for Estimating the Drag Coefficientsof Projectiles.- US Army Ballistic Research Laboratory, Aberdeen Proving Ground,Maryland. BRL Technical Report ARBRL-TR-02293, February 1981. (AD A098110)

7. Breaux. H.J.. Campbell, L.W. and Torrey, J.C. " Stepwise Multiple Regression Sta-tistical Theory and Computer Program Description," US Army Ballistic ResearchLaboratory. Aberdeen Proving Ground, Maryland, BRL Report No. 1330, July 1966.(AD 639955)

27

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28

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List of Symbols

Symbol Definition

CD0 zero yaw drag force coefficient

CDy, inert base-burn motor, base drag component of zero yaw dragforce coefficient

CD, radar determined drag force coefficient

d reference diameter of projectile

fA factor as a function of quadrant elevation, previously identifiedas i, used for matching experimental range firing data

factor used to represent the drag reduction as a function ofMach number

f2 factor used to represent the drag reduction as a function oftime of flight

f3 factor used as a parameter for matching experimental rangefiring data

14 factor used to represent the drag reduction due to an apparenteffect of local air pressure

acceleration due to gravity

rn fuzed projectile mass at time t

ni, reference fuzed projectile mass

P air pressure at trajectory position

P, reference (standard) air pressure (1013.25 mb)

CD, (Qa,)2 yaw of repose drag term in the "Modified Point Mass TrajectoryModel"

rt trajectory estimated, slant range magnitude

- trajectory estimated, slant range

HAWK radar determined, rate of change of slant range with time

rtime derivative of the HAWK radar determined, slantrange rate of change

it trajectory estimated, rate of change of slant range with time

rt trajectory estimated, time derivative of the slant range rate ofchange

29

Page 36: 155 Mm Drag and Exterior Ballistic

List of Symbols (Continued)

Symbol Definition

ttrajectory estimated, velocity of the projectile with-respect-to theground-fixed axes

ut trajectory estimated, acceleration of the projectile with-respect-tothe ground-fixed axes

velocity of the projectile with-respect-to the ground-fixed axis,determined from HAWK radar data and estimated trajectory

U. acceleration of the projectile with-respect-to the ground-fixed axes,

determined from HAWK radar data and estimated trajectory

v speed of projectile with-respect-to air

F velocity of the projectile with-respect-to air

IF velocity of the air with-respect-to the ground (wind velocity)

A acceleration due to Coriolis effect

p density (specific mass) of air

30

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Page 38: 155 Mm Drag and Exterior Ballistic

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Page 39: 155 Mm Drag and Exterior Ballistic

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