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STECHNICAL REPORT ARBRL-TR-0?432

AERODYNAMIC CHARACTERISTICS OF THE

30MM XM788EI AND XM789 PROJECTILES

Robert L. McCoy

DTICOctober 1982 ELECTSNOVO 319.82

US ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLAND

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lechnical Report AI.BRL-TR- 02432 ./4.Ab'2/ .? -__4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

Aerodynamic Characteristics of the 30mm FinalXM788E1 and XM789 Projectiles 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(a) 3. CONTRACT OR GRANT NUMBER(s)

R. L. McCoy

9. PERPORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUMBERS

U.S. Army Ballistic Research LaboratoryATTN: DRDAR-BLLAberdeen Proving Ground, Maryland 21005 RDT&E 1L62618AH80

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US Army Armament Research & Development Command October 1982US Army Ballistic Research Laboratory (DRDAR-BL) 13. NUMBEROF.PAGES

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on rwverse side if neceesary and Identify by block number)

30mm XM788E1 (TP) Projectile Aerodynamic Drag30mm XM789 (HEDP) Projectile Gyroscopic StabilityAerodynamic Characteristics Yaw Limit-Epicycle

20. ABSTRACT (Cou•true o re•erse &hb It nr*ct*r7a nd mdideonlif by block number) (bja/ajb)

Spark range tests of the 30mm, XM788EI (TP) and XM789 (HEDP) projectiles wereconducted to determine the aerodynamic characteristics at supersonic, transonic,and subsonic speeds. Both projectiles demonstrated adequate gyroscopic sta-bility at all speeds tested. The data show the existence of a slow arm limit-cycle yaw at high subsonic speeds, and a limit-epicycle yaw at lower subsonicspeeds. The effect of the limit-epicycle on total drag is discussed.

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TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS ............. ..................... 5

LIST OF TABLES ................. ........................ 7

I. INTRODUCTION ............ ......................... 9

II. TEST PROCEDURE AND MATERIAL ........... .................. 9

III. RESULTS .................................... 10

A. Drag Coefficient .................. ................. 11

B. Pitching Moment Coefficient .... .............. .. 11

C. Gyroscope Stability ........ .................... ... 12

D. Lift Force Cuefficient. ....... ................... 13

E. Magnus Moment Coefficient and Pitch DampingMoment Coefficient . ................... 13

F. Flight Dynamic Predictions ................ 15IV. CONCLUSIONS......................................... 17

V. RECOMMENDATIONS ........ .................. . . . . . 17

REFERENCES . . . .. .. .. .. . . ... . . . . . . . . . . 49

LIST OF SYMBOLS.......... . . .......... . . . . . 51

DISTRIBUTION ..................... .......................... 5s

Aocession For

DTIC TAB

justification-_ -

Dirstribtion/

Availability Codes

Avail and/or

3

* V. _ . .- . -. - ...

LIST OF ILLUSTRATIONS

Figure Page

I. Photograph of 30mm XM789 and XM788E1 Projectiles ................. 23

2. Photograph of the Cross Section of "Potted" XM759 Fuze ........... 24

3. Sketch of 30mm, XM78BE1 TP Projectile ............................ 2S

4. Sketch of 30mm, XM789 HEDP Projectile ............................ 26

5. Shadowgraph of TP Projectile at Mach 2.30 ........................ 27

6. Shadowgraph of HEDP Projectile at Mach 2,30 ...................... 28


8. Shadowgraph of HEDP Projectile at Mach 1.85 ...................... 30




12. Shadowgraph of HEDP Projectilet a Mach 1.00 ............. o....... 34

13. Shadowgraph of TP ProjecLiie at Mach 0.74 ........................ 35


15. Zero-Yaw Drag Force Coefficient Versus MachNumber, XM788E1 ...................................... . 37

16. Zero-Yaw Drag Force Coefficient Versus MachNumber, XM789 .................................................. 38

17. Yaw-Drag Force Cc fficient Versus Mach Number .................... 39

18. Zero-Yaw Pitching Moment Coefficient Versus MachNumber, XM788EI 1....#.......................... .. .. 40

19. Zero-Yaw Pitching Moment Coefficient Versus MachNumber, XM789 . . . . . t e. . . . . . . o. . . .. . 4

20. Lift Force Coefficient Versus Mach Number ........................ 42

21. Zero-Yaw Magnus Moment Coefficient Versus Mach Number ............ 43

22. Zero-Yaw Pitch Damping Moment Coefficient VersusMach Number .................................................... 44

I I I I I I II I I5

.- 7 .

LIST OF ILLUSTRATIONS (Continued)

Figure Page

23. Cubic Magnus Moment Coefficient Versus Mach Number ............... 4S

24. Cubic Pitch Damping Moment Coefficient Versus Mach 4bNumber.......................................................

25. Fast and Slow Arm Amplitudes Versus Mach NumberXM789 ......................... ......................... 47

26. Spin Damping Moment Coefficient Versus Mach NumberXM788 ......................................................... 48

6

LIST OF TABLES

Table Lage

I. Average Physical Characteristics of the 30mm,XM788E1 and XM789 Projectiles .................................. 18

II. Aerodynamic Coefficients of the 30mm, XM788E1Projectile ........ .. ...... ...... .......... ...... ................ 19

III. Aerodynamic Coefficients of the 30mm, XM789Projectile ...................... ........ . ....... ........ .. 0.04. 20

IV. Flight Motion Parameters of the 30mm, Xfl788E1Projectile ........................ ........... .. . . ............. 21

V. Flight Motion Parameters of the 3L0nm, XM789Projectile ................... .. ................................ 22

I.

I. INTRODUCTION

The first series of spark photography range tests for the 30mm, M230Chain Gun ammunition was conducted at the Ball stic Research Laboratory (BRL)in November, 1978. At that time, only the XM788 Target Practice (TP)ammunition was available for testing, and the results of a sixteen-roundfiring program were reported in Reference 1. The early XM789 High-ExplosiveDual-Purpose (11EDP) ammunition had experienced both in-bore mechanicalfailures and down-range airburst problems, and the possibility of a fuzingproblem aggravated by a limit-cycle yaw at transonic and high subsonic speedswas raised as a program concern. Since no spark tinge tests were conducted onthe early HEDP ammunition, neither the limit-cycle yaw nor the effects ofmoving fuze parts on the limit-cycle could be predicted. The BRL haddiscussed this potential problem with the office of the Project Manager for30mm Ammunition as early as 1979, and recommended that both standard and

q "locked-up" fuzes should be included in the XM789 aeroballistic range tests.

In the meantime, the XM788 TP projectile had undergone design changes.The later design of the XM789 HEDP projectile body incorporated both rotatingband and base section modifications, and the original TP projectile no longerlooked like the re-designed HEDP. In addition, some XM788 projectiles hadfailed in-bore, and fragments of the broken shell bodies were observed tostrike the blast suppressor. The proposed "fix" for the TP projectileconsisted of changing both the shell design and the steel, and the new shell,designated the XM788EI TP, was available for the second series of spark rangetests.

The 30mm Mann barrel required for the tests was received early in 1981,as were the test authority and funds. The required ammunition components werereceived in April 1981, and firings were conducted in the BRL AerodynamicsRange between 24 April 1981 and 15 May 1981. The test plan called for thefiring of eighteen TP rounds and eighteen HEDP rounds, distributed over a Machnumber range from 2.3 down to 0.75. Four additional round- of HEDP with"lock-up" fuzes were also fired at the lowest test velocity, to provideinformation on possible flight dynamic effects due to moving fuze parts.

II. TEST PROCEDURE AND MATERIAL

The 30mm projectiles, XM789 HEDP, with fuze, PD, XM759, and the XM788E1TP are shown in the photograph of Figure 1. The "locked-up" fuze models wereproduced by first setting the fuze in the armed position, then "potting," orinjecting epoxy resin into the fuze to solidify and lock all ioving parts intoa rigid structure. The "potting" added approximately 1.5 grams weight to theXM759 fuze. Figure 2 is a photograph of the cross-section of a potted fuze,and shows the cylindrical rotor aligned with the firing pin.

I. R. L. McCoy, "Aerodynacmic Characteris tics of the 30mnr XMZ8S Projectile,BaZZistic Research laboratory Memorandwn Report ARBRL-MR-03019, 112 ,; 1250.AD A086096.

9

Physical measurements were taken on a sample of five TP and five HEDProunds, and on three HEDP projectiles with potted fuzes. Figure 3 is a sketchof the XM788EI prcjectile, and Figure 4 is a similar sketch for the XM789,with XM759 fuze. Average physical characteristics of the projectiles arelisted in Table I.

The test rounds were fired from a 30mm Mann Barrel, Serial No. 57,designed to be an interior ballistic duplicate of the M230 Chain Gun barrel.Twist of rifling was constant, with helix angle of 60 30' (27.6 calibers/turn). Various propellants and charge weights were selected to achieve therequired test Mach numbers. No yaw-inducer was used on any rounds in thetest.

A total of twenty rounds of XM788E1 TP were successfully launched, aswere nineteen XM789 HEDP rounds, and four HEDP projectiles with pottedfuzes. The aerodynamic data from the forty-three successful flights arepresented in this report.

III. RESULTS

The range data were fitted to solutions of the linearized equations ofmotion and these results used to infer linearized aerodynamic coefficients,using the methods of Reference 2. The actual projectile force-moment systemoften is not strictly linear, and given sufficient data, the actual non-linearbehavior can be determined from the free-flight range results. 3 For the 30nmXM788E1 and XM789 projectiles, sufficient data were obtained to permit deter-minction of some non-linear aerodynamic coefficients. A more detailedanalysis of non-linear effects is presented in the various subtopics of thissection which discuss individual aerodynamic coefficients.

A useful by-product of tests conducted in the BRL aerodynamics ranges isthe high quality shadowgraph information obtained. Figures 5 through 14 wereselected for locally small yaw (angle of attack less than 1/2 degree) andcommon Mach numbers, to &:;..: c"!"ýrison of the flowfields around the TP andHEDP projectiles.

The round-by-round aerodynamic data for the XM788EI TP projectile areIisted iii Table i, and the aerodynamic data for the XM789 HEDP are given inTable III. Free-flight motion parameters for the XM788E1 and XM789 are listedin Tables IV and V, respectively.

2. C. 21. ?.MIrph, "Data Reduction for the Free Flight Spark Ran qes," BaliisticResearch Laboratorzes Report No. 900, February 1954. AD 35833.

3. C. H. MIurphy, "The Measurement of Non-Linear Forces and Moments by Meansof Free Flight Tests," Ballistic Research Laboratories Report No. 974,FebPUCruc, 1956. AD 93521.

10

- . .-- - - - - -

A. Drag Coefficient

The drag coefficient, C., is determined by fitting the time-distance

measurements from the range flight. CD is distinctly non-linear with yaw

level, and the value determined from an individual flight reflects both thezero-yaw drag coefficient, C0 o, and the induced drag due to the average yaw

level of the flight. The drag coefficient variation is expressed as an evenpower series in yaw amplitude:

CD = C + C 62 + .....62 ()

where C0D is the zero-yaw drag coefficient, CD is the quadratic Ydw-drag0 62

coefficient, and 62 is the total angle of attack squared.

Analysis of the drag coefficient data for the two 30ram projectiles showedthat the zero-yaw drag coefficients were, for practical purposes, identical.However, the yaw-drag coefficient for the TP projectile was found to besignificantly higher than that for the HEDP design, at both supersonic andsubsonic speeds.

The yaw-drag coefficients obtained from the analysis were used to correctthe range values of C to zero-yaw conditions. Figures 15 and 16 show the

variation of CD with Mach number for the XM788E1 TP and the XM789 HEUP shell0

respectively. Figure 17 shows the behavior of the yaw-drag coefficients for

the two projectiles. Although the CD0 2 curves are significantly different,

the resulting total in-flight drag values, for the expected average yawlevels, differ by less than 2% at supersonic speeds, and less than 4% atsubsonic speeds. rhe TP round will experience slightly higher total drag atall speeds than will the HENP round.

B. Pitching Moment Coelffici•C ent

The range values of the pitching moment coefficient, CM , were fitteda

using the appropriate squared-yaw parameters from Reference 3. CM was foundM

to vary significantly with yaw level for the 30ram projectiles. The pitchingmoment is assumed to be cubic in yaw level, and the coefficient variation isgiven by:

C =C + C 62CM M 2 (2)S % (2 )

CIOI0,--

II1i

where CM is the zero-yaw pitching moment coefficient, and C2 is the cubic

MIO

coefficient.

For both the TP and HEDP projectiles, the value of C2 at supersonic

speeds was found to -14; the corresponding value at subsonic and transonicspeeds was -8. These values were used to correct the range values of CM to

Ci

zero-yaw conditions. Figures 18 and 19 show the variation of CM with Mach

number for the TP and HEDP shell, respectively. The TP round shows approxi-mately 3% higher pitching moment than the HEDP round at subsonic and transonicspeeds; at high supersonic speeds the curves for the two shells are practi-cally identical. Note that the CM for the potted fuze rounds is about 3%

a0

lower than that of standard HEDP shell at MV = 0.15. This effect reflects the

forward shift in center of gravity location caused by adding 1.5 grains ofpotting resin into the fuze.

C. Gyroscope Stability

The 30mrm, XM788E1 TP and XM789 HEDP projectiles have ample launch gyro-scope stability when fired at standard velocity (805 metres/second) from the60 30' twist rate (27.6 calibers/turn) of the M230 barrel. The average launchgyroscopic stability factor (Sg) of both projectiles at standard atmosphericconditions is 3.0. SincP neither projectile will ever be fired at reducedvelocities, and the ratio of axial spin to forward velocity increases continu-ously for flat trajectories, the s ightly lower values of S observed in9Tables IV and V for the lower Mach numbers will never occur in field firings.

The 30ram projectiles art oesigned to be launched in forward fire from anaircraft. The addition of the aircraft's velocity vector to the gun muzzlevelocity has the effect of reducing the projectile's launch gyroscopic stabil-ity factor by the ratio:

/ WiV.. .

VA/C V Muzzle

where VA/C is the forward velocity of the aircraft.

The following table shows the effect of aircraft speed on launch Sg, forboth TP and HEDP ammunition in foward fire.

12

VA/C Sg

(Knots) kaunch)

0 3.00150 2.50300 2.11450 1.80550 1.64

At an aircraft speed of 550 knots, the M230 Chain Gun ammunition still

has more than adequate gyroscopic stability.

D. Lift Force Coefficient

The lift force coefficient, CL , was also analyzed by the method of

Referonce 3. If the lift force is assumed to be cubic in yaw level, the

coefficient variation is given by:

CL = L + a2 62

where CL is the zero-yaw lift force coefficient, and a2 is the cubic

coefficient.

No significant value of the cubic lift coefficient could be found, foreither projectile tested, at any speed. Further analysis 5howed that nosignificant difference existed in the range values of CL , between the TP and

athe HEDP shell. Figure 20 shows the variation of CL with Mach number forthe two projectiles. a

The lift force coefficient is not as well determined from spark rangetests as is the pitching moment coefficient. This tact is reflected in thelarger rcund-to-roind data scatter observed in Figure 20, compared with thepitching moment data plotted in Figures 18 and 19.

E. _Manus Moment Coefficient and Pitch Damping Moment Coefficient

The Magnus moment coefficient, C and the Pitch Damping Mome2nt

CoefFicient, (CMq + CM%),. are discussed together, since if cither coefficient

is non-linear with yaw level, both coefficients exhibic non-linear coupling inthe data reduction process 3. Due to mutual reaction, the analysis of

13

loo

CM and (CM + CM.) must be performed simultaneously, although the aero-Pc q a

dynamic moments are not, in themselves, directly physically related.

If the dependence of both the Magnus moment and the Pitch Damping momentare cubic in yaw level, the non-linear variation of the two moment coeffi-cients is of the general form:

C M 2+ C 62C pa Pa0 (4)

(CM + CM.)=(CM + CM;)o + d2 62 (5)

where CM and\ (CMq + C are the zero-yaw values of Magnus and Pitch

Damping moment coefficients, respectively, and C2 and d2 are the associated

cubic coefficients.

In Reference 3 it is shown that the non-linear coupling introducedthrough the data reduction yields the following expressions for range values[R-subscript] of CMpa and (CMq + CM%):

C C + 62 + d 6 2P R Po eTH (6)

[CM C = (CM+ + C+ 2 22 + d 2 2L q CMJ R Mq CM) 6eHT 2 eHH (7)

where the above effective squared yaws are defined as:

2 = KF 2 + KS2 + (4ý KF2 - PsKs2)/4 - (8)

eHT 2 (1 Y/Ix) +~ 4 4)(KS2 K F') /(4-(10)

14I

eHH (1 )

The remaining symbols are defined in the List of Symbols in this report.

Preliminary analysis of the XM788E1 and XM789 data showed no significantnon-linearity in either CM or CM + CM., at supersonic speeds. However, at

pa q Etransonic and subsonic speeds, the two coefficients showed strong non-linearcoupling, and it was necessary to perform the analysis on both coefficientssimultaneously.

The data rounds were first separated into Mach number groups, by roundtype, and individual values of the two cubic coefficients were obtained.Next, a statistical analysis was performed, which yielded the useful resultthat there was no significant difference between the TP and HEDP rounds, ineither linear or cubic coefficients. The further result that the potted fuzErounds were statistically the same population as standard HEDP rounds com-pleted the preliminary analysis, and all round types were then combined forfinal analysis.

The combined data rounds were again separated into Mach number groups,and final values of the cubic coefficients were obtained at five average Machnumbers through the transonic-subsonic range. The values of C2 and d2 fellnearly along straight lines, and a weighted linear ledst squares fit of thecubic coefficients was used to correct the range values C and C + C to

M CM Cmepa q a

zero-yaw conditions. Figures 21 and 22 show the variation of tM and

C M + CM % , respectively, with Mach number. Figures 23 and 24 show the

variation of the cubic Magnus moment coefficient, C 2, and the cubic pitch-

damping moment coefficient, d2 , with Mach number. Figures 21 and 22 include

irndividual round coefficients, corrected to zero-yaw values, and visually con-firm the fact that no significant differences in Magnus or pitch-dampingmoments exist between the TP and HEDP rounds. The same two figures also implythat no significant flight dynamic difference should be observed between stan-dard and "locked-up" XM759 fuzes, with the HEDP shell.

F. Flight Dynamic Predictions

The damping rates, xF and XS' of the fast and slow yaw modes indicate the

dynamic stability of a projectile. Negative X's indicate damping; a positive

X means that its associated modal arm will grow with increasing time.

15

For a projectile whose Magnus or pitch-damping moments are non-linearwith yaw level, the damping rates also show a non-linear dependence on yaw.In Reference 4, Murphy successfully predicted the effect of a cubic Magnusmoment on the damping rates by means of an amplitude-plane analysis. However,the amplitude-plane technique becomes cumbersome for a projectile with bothMagnus and pitch-dariping moment non-linearities. The effects of changi,,y epi-cyclic frequencies along the trajectory, in addition to variations in thevalues of aerodynamic coefficients with Mach number further increase thedifficulty in application of the amplitude-plane method to long-range flightdynamic problems.

For purposes of this report, the BRL six-degree-of-freedom (6-DOF) tra-jectory program5 was used to predict the long range flight dynamic performanceof XM788E1 and XM789 ammunition. The 6-DOF program results for the prototypeXM788 projectile had previously been compared with numerical integration ofthe damping rate equations', and the agreement between the two methods wasvery close. Since the 6-DOF trajectory program includes all the effects ofchanging axial spin and aerodynamic coefficient variations with Mach number,it represents the best approximation to real-range flight dynamic prediction.

The 6-DOF trajectory program was run for an XM789 HEDP projectilelaunched at a muzzle velocity of 805 metres/second, and with an initial yawrate of 20 radians/second, which produces a first maximum yaw of about 2-1/2degrees. Analysis of the trajectory program output shows the fast arm dampedto an insignificant level at 1,000 metres range, where the flight Mach numberis approximately one; the slow arm at the same range has damped to around 0.1degree in amplitude. The above situation persists until approximately 1,300metres range (M = 0.85), where the slow arm amplitude begins to grow. At

1,500 metres range (M = 0.77), the slow arm has grown to a local maximum

value of about 2.1 degrees, and at this point the fast arm, which had remainedvery small since the low supersonic region, begins to grow. The fast armamplitude attains a maximum value of approximately 1.6 degrees at about 1,750metres range (M = 0.69). Thus, the dynamic performance of the XM789 shellat subsonic speeas is described as a "limit-epicycle," rather than the slowarm limit-cycle motion previously observed1 for the XM788 projectile. Figure25 shows the predicted variation of the fast and slow arm amplitudes with Machnumber for the XM789 shell. A similar trajectory was run for the XM788E1,with substantially identical results.

The dashed curves shown in Figure 25 must be considered as estimatedtrends, because they are based on extrapolated aerodynamic data. No firingswere conducted in the BRL Aerodynamics Range below Mach number 0.70, due tohigh gravity-induced trajectory curvature at low velocities. For the 6-DOF

4. C. H. Murphy, "Free Flight Motion of Symnetric Missiles," BallisticResearch Laborato.ries Report No. 1216, July 1963. AD 442757.

65. R. F. Lieske and R. L. McCoy, "Equations of Motion of a Rigid Projectile,"Ballistic Research Laboratories Report No. 1244, March 1964. AD 441598.

16

trajectory simulations, all aerodynamic coefficients were assumed constantbelow Mach number 0.7, and the spin damping moment coefficient, C£ , from

pReference 1 was used for simulations of the current 30mw projectiles. Sincethe presently available aerodynamic data has "run out" at a real range ofapproximately 1,750 metres, and the intended use of the 30mm gun systemenvisions firing to ranges of three kilometers and beyond, it is ;iighly rec-ommended that a long-range instrumented flight dynamic test similar to thdt

recommended in Reference 1, be conducted for the XM788E1 and XM789 snell, toverify the estimated trends of this report.

The spin damping moment coefficient, C , was not measured for theXp

XM788E1 and XM789 projectiles. The spin damping data previously obtained forthe XM788 TP shell are considered sufficiently accurate to describe the spinperformance of the current projectiles, and the C p curve presented in

Reference 1 is reproduced in this report as Figure 26.

IV. CONCLUSIONS

The 30mm, XM788E1 TP and XM789 HEDP projectiles, when launched at amuzzle velocity of 805 metres!second from the M230 barrel, are gyroscopicallystable in either forward or side fire, from aircraft at speeds up to 550knots.

The spark range data predict that a limit-epicycle yaw exists, for bothshell at subsonic speeds. The predicted slow arm amplitude is approximatelytwo degrees, and the fast arm amplitude is around one and one-half degrees.The limit-epicycle yaw will produce drag increases of between four and eightpercent over the zero-yaw values at subsonic speeds.

V. RECOMMENDATIONS

It is recommended that a long-range flight dynamic experiment be con-ducted to verify the predicted subsonic limit-epicycle behavior of the XM788EIand XM789 projectiles. The flight dynamic tests should be conducted using theM230 Chain Gun, and the flight dynamic information could best be obtainedusing the 30mim yawsonde fuze currently under development at the BRL.

We have recently been advised that the M230 Chain Gun barrel has beenchanged from the constant-twist rifling used in the present tests to a gain-twist form of rifling. It is recommended that a limited re-test of XM788E1and XM789 ammunition be conducted from the new barrel, at standard muzzlevelocity, to insure validity of the present aerodynamic data for rounds with adifferent engraving pattern on the rotating bands.

17

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,"t'- !f•' t tf~f., | __ I1] I 1 I iI 1I . ..J[I ~, ,I 1444 +4 + t'i iti.i--jl4 I itl T 4 4i m" ''

't f -• ' t , 4 -.

+ •+" + +: t + E ;" - t -+ 4

"T I T4 I I t I f t t -- ,JýHfý +

j ++ + +4 ! + ............+f ! 1 I , +

't t 1i~ t

4' 1 1 It f

T, tt 4 it M 4~~- . -t 4+', + 1 l J ,

t I'_ 4L " + " "'. .I •t t 4 t i t 1c 4~ vt .f 4 1 + + . .

i • -It _L JL tL _L J J J• .L L • 1 ... ' 1

-~~ - - - - -r - , -- IT

, T4 I4

T t 4 1 tI _-f.~ 4 -

4.'

Hr1

24 4 S - 44 4t+*-f4 _41i 44

4-t--- t-- -+ 4 4V 4

-44

't 4-AA ±4t 44 - 4j

it 4 44 +~ (Ai ti- ~ -$

fI -44.

44 4- 4 4

--4 + 4f4-

4j -t i-T--tf - 4-

4 4 ------ jI T A

.... ---- . jt .

.V ~ ... ... ......

IcL' ti 4-

4** 4.

+ y 4 1414 i {I~ -i'~4 4ý 4:fH

14t TT-,Tj1---7-1- -

----------

TII

_11

Sco

Ern

CLj

£t -

J-~l A.0Ci

48

44- -- ý+ -A 111I

REFERENCES

1. R. L. McCoy, "Aerodynamic Characteristics of the 30rnm XM788 Projectile,"Ballistic Research Laboratory Memorandum Report ARBRL-MR-03019, May1980. AD A086096.

2. C. H. Murphy, "Data Reduction for the Free Flight Spark Ranges,"Ballistic Research Laboratories Report No. 900, February 1954.AD 35833.

3. C. H. Murphy, "The Measurement of Non-Linear Forces and Moments by Meansof Free Flight Tests," Ballistic Research Laboratories Report No. 974,February 1956. AD 93521.

4. C. H. Murphy, "Free Flight Motion of Symmetric Missiles," BallisticResearch Laboratories Report No. 1216, July 1963. AD 442757.

5. R. F. Lieske and R. L. McCoy, "Equations of Motion of a Rigid Projec-tile," Ballistic Research Laboratories Report No. 1244, March 1964.AD 441598.

49

LIST OF SYMBOLS

a2 = cubic lift force coefficient

C2 = cubic static moment coefficient

C2 = cubic Magnus moment coefficient

D r af F-tor4gCD (1/2) p V2 S

CD = zero yaw drag coefficient0

CD62 = quadratic yaw drag coefficient

CL = Lift FaCQ Positive coefficient: Forcea (112)pV 2 S 6 in plane of total angle of

attack, at' ._L to trajec-

tory in direction of at.

(at directed from

trajectory to missile

axis.) 6 = sin at.

CN Normal Forge Positive coefficient: Forcea1/2) p Vz S 8 in plane of total angle of

attack, at, _Lto missile

axis in direction of a.

CN = CL + CD

r Positive coefficient: MomentH (112) p VZ S 6 increases angle of attack

at.

CM = Magnus Moment Positive coefficient: Momentp rotates nose _.. to plane of

V/ V at in direction of spin.

CN Manus Force Negative coefficient: Forcep-cc acts in direction of 900p(/2 ip, V2 S (P

1/2 p VS • rotation of the positivelift force against spin.

t• 51

LIST OF SYMBOLS (Continued)

For most exterior ballistic uses, where a : q, 6 .-r, the definition of thedamping moment sum is equivalent to:

C C Damping Moment Positive coefficient: Moment* ~~M M _ _ _ _ _ _ _ _ _

q (1/2) V2 S d qVd) increases angular velocity.

C Roll Damping Moment Negative coefficient: Momentp (1/2) p V2 S d decreases rotational veloc-

(LV ity.

CPN center of pressure of the normal force, positive frombase to nose

, = angle of attack, side slip

* (• + 02) 1/2 -= sin- 6,total angle of attack

A fast mode damping rate

negative A indicates damping

XS slow mode damping rate

p = air density

= fast mode frequency

= slow mode frequency

c.m. = center of mass

d = body diameter of projectile, reference length

d2= cubic pitch damping moment coefficient

Ix= axial moment of inertia

Sr. 52

'o4... ,•


I - transverse moment of inertiay

KF = magnitude of the fast yaw mode

KS5 magnitude of the slow yaw mode

X= length of projectile

m = mass of projectile

M= Mach number

p = roll rate

q, r = transverse angular velocities

qt = (q 2 + r2)1/2

R = subscript denotes range value

s = dimensionless arc length along the trajectory

sd 2= ---- , reference area

Sd = dynamic stability factor

Sg = gyroscopic stability factor

V = velocity of projectile

VMuzzle launch velocity of projectile

VA/C = aircraft velocity

53


'C Effective Squared Yaw Parameters

K F + K- 2

2 K = 2 • * F- K K26eF KF2 + KKS2

S 6es KF

(K\ 2 + , (2 - K F2s)

TeTH - (OF -

2 (ý Ks2Z KF2)eH , -0

4S

pm

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cVi - Defense Technical Information Center · failures and down-range airburst problems, ... The 30mm projectiles, XM789 HEDP, with fuze, PD, ... common Mach numbers,