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FLEVEL
MEMORANDUM REPORT ARBRL-MR-02891
MUZZLE-BLAST-INDUCED TRAJECTORY
PERTURBATION OF NONCONICAL ANDCONICAL BOATTAIL PROJECTILES
CL.Kevin S. Fansler
Edward M. Schmidt
C = January 1979
m VUS ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC
RESEARCH LABORATORY
ABERDEEN PROVING GROUND, MARYLANDiV
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rim• ja o/' et.vk name op, mcmfaetwur,' ,mo in th'a :vpo,.tdove
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S. TVOE OF REPORT & PERIOD COVERED
ZZLE- LAST-JNDUCED ,RAJECTORY .CRTURBATION OF ,NONCON CAL AND
CONICAL BOATTAIL PROJECTILE$-- / 1 2 .Final /
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I.Kevin S./ Fansler 90Edward M./SchmidtI
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NUMBERS(ATTN: DRDAR-BLL) -.---.--Aberdeen Proving Ground, MD 21005
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DDCIS. SUPPLEMENTARY NOTES
• ••MAR 22 107
1I. KEY WORDS (Continue an reveres side If necessary and
Idantily by block ntmb.r)
Spin-stabilized Projectie.4 BTransitional BallisticsFree-flight
BaliisticsProjectile Accuracy
/.-•> •zzle-blast loadings on F'pin-stabilized projectiles
are analyzed ar.d used tocompute rnsultent trajectory deviations.
Both conical and nonconical boattailrounds av:' treated.
Approximations are made that permit the expression for theforce on
the projectile to be integrated; the resulting equation is used
todevelop a universal momentum transfer function that can be
directly related toprojectile jump. Although nonconical boattail
configurations are more sensitiveto muzzle blast than conical
designs, the computed trajectory deviation ineither case is small
compared to the total measured dispersion of typical systemsq
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Ir
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS .. .. ...... ............... .
........
1. INTRODU'CTION .. ......... . .............. .............
7
II. ANALYSIS OF GASDYNAMIC LOADS. .. .. .. ....................
8
A. Region 1: Projectile Base Still Within the Bore . .. 8
B. Region 2: Projectile Base Out of the Gun Tube . . . 11i
III. RESULTS AND DISCUSSION .. ....... . ................
......12
IV. SUMM4ARY AND CONCLUSIONS....... ... . .. .. .. .. .. ..
...
REFERENCES .. ........... . .............. ................
19
AIPENDIX A. Expression for Transverse Momentum. .. ......21
LIST OF SYMBOLS. .. ......... . ................ ..........
23
IDISTRIBUTION LIST. .. ....... . ................ ........
25
.3
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I! i
LIST OF ILLUSTRATIONS
Figure Page
1. Flowfield sch'!matic ......... .................. ... 16
2. Experimental and extrapolated lift data for sphere-cones
................ ......................... ... 16
3. Momentum function for the conical S49 boattail ... .....
17
4. Momentum function for the square boattail ........... ...
17
5. Momentum function for the triangular boattail .... ......
18
6. Cross-section of nonconical boattail in muzzle plane 18
5
3 3OZI*3PA=I
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1. INTRODUCTION
On the modern battlefield, weapon systems must shoot
projectileswith a high degree of precision to lessen the chances of
enemy returnfire. Ideally, the weapon designer should be capable
ofquantifying the sources of dispersion in order to meet imposed
pre-cision standards. One source of perturbation to the desired
trajectoryiF the gasdynamic force experienced in the weapon muzzle
blast. Forfin-stabilized projectiles, dispersion caused by the
blast environmenthas been investigated 1,2; however, treatment of
spin-stabilizedprojectiles has been largely neglected. With the
introduction ofnovel boattail designs3 ,there is also interest in
comparing thesensitivity to muzzle blast of the nonconical boattail
configurationswith the sensitivity of the conical boattail
configurations.
The present paper eramines the influence of muzzle-blast
loadingsupon the trajectory of spin-stabilized rounds. With both
conical andnonconical boattail configurations, the analytical
approach is similar tothat used previously for fin-stabilized
projectiles: namely, the mostsignificant gasdynamic loadings are
assumed to occur within the quasi-steady, supersonic core of the
propellant gas jet. Pressure distribu-tions on the projectiles are
estimated utilizing steady-state flowtheory and experiments. This
procedure is not locally exact; however,it produces a reasonable
estimate of the overall muzzle-blast-inducedimpulse. Transverse
angular and linear momentums imparted by theblast are calculated
for triangular, square and conical boattailsand used to determine
the contribution to the dispersion of 155mm,M549 type
projectiles.
1. E. M. Schmidt, K. S. Fansler, and D. D. Shear,
"TrajectoryPerturbations of Fin-Stabilized Projectiles Due to
Muzzle Blast,"Journal of Spacecraft and Rockets, Vol. 14, No. 6,
June 1977,pp. 339-344.
2. K. S. Fansler, and E. M. Schmidt, "Trajectorj Perturbations
ofAsymnetric Fin-Stabilized Projectiles Caused by Muzzle
Blast,"Journal of Spacecraft and Rockets, IVl. 15, No. 1,
January-February 1978, pp. 62-64.
3. A. S. Platou, "An Improved Projectile Boattail," BRL MR
2395,US Army Ballistic Research Laboratories, Aberdeen Proving
Ground,Maryland, July 1974. AD 785520.
L7
FA=Vs u mJ.
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ng Il. ANALYSIS' OF GASDYNAMIC LOADS
Two distinct regions are considered. The first region for whicha
model is developed covers the flow following exit of the
rotating/obturating band fron the gun tube. When this band passes
the muzzle,the gas seal is released, but the boattail is still
within the tubeand the forward portion of thM projectile is
immersed in the expandingpropellant gases. The existence of
asymmetry in this fiow field dueto projectile angle of attack
results in transverse loads. For conicalboatcail rounds, these
loads alter the transverse momenium; however,due .o the c lindrical
'wheelbase' characteristic of non-conical boat-tail designs ,
momentum exchange occurs only when the non-planarsurfaces are not
contacting the bore. The second region of interestcommences as the
projectile base clears the muzzle and terminates asthe base passes
through the Mach disc of the propellant gas jet.
In both regions, the analysis seeks an upper-bound estimate
ofthe propellant gas loadings. This approach is supported by
previousupper-bound estimates 1 ' 2 showing that muzzle-blast-
induced dispersionof fin-stabilized projectiles is a negligible
component of the totplmeasured dispersion levels. Thus, it was not
worthwhile to seek moreaccurate, but lower magnitude,
estimates.
A. Region 1: Projectile base still within the bore
A schematic of the flow field is shown in Figure 1. For
con-venience, the flow behind the projectile is assumed to be sonic
(occursfor V a 700 m/s). At separation of the obturating band, the
pro-
pellant gases are released to expand into the atmosphere,
generatingthe characteristic muzzle-blast wave. Due to the presence
of theprojectile, this free expansion is constrained and if the
projectileis at an angle of attack, a1 , relative to the bore-line,
the propellant
gases will be deflected. Assuming that the projectile acts
somewhatlike a plug nozzle and that all of the flow is deflected
through theangle a1 at the sonic line, the resultant transverse
lift force cn the
projectile can be simply estimated from momentum
considerations:L1 = (p* + p*V. Au4 (I)
ul
where Au is the area of the muzzle that is unplugged (a function
of
boattail geometry), p* is the pressure at the muzzle, p* is the
gasdensity at the muzzle and V* is the gas velocity at the
muzzle.
Equation (1) is an estimate for a stationary projectile. It
maybe corrected to account for projectile velocity, Consider a
coordinatesystem moving with the projectile. The velocity of the
fluid in thissystem is given as Vr. If the fluid near the
projectile is deflected
r
8i
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to flow parallel to the projectile surface, the transformation
betweenthe stationary and moving coordinates is
Vr exp (ia) =V exp (ia) - Vp (2)
where a is the turning angle for the fluid in the gun-tube
coordinatesystem, V is the local velocity and V is the projectile
velocity.
* pWe obtain:
(I = ( V /V) a1 (3)
The largest value for a corresponds to the Mach number of the
fluidapproaching infinity. For Vp = V*, the limiting value of V is
3 V
p pand thus,
a = 2a /3 (4)
Thus to correct the lift force for projectile motion, we replace
a 1 by2a /3 in Equation (1). Using the isentropic flow relations,
we
obtain
2L= (y+l) p* AuaI (5)
The transverse momentum transferred to the projectile during
thetime between obturator separation and base emergence from the
muzzle isthe integral of the lift force over the time of passage.
Transformingthe variable of integration from time to distai,,e, t =
(X/D) (D/Vp),
pallows the momentum integral to be written:
2 (y+l) (DIV p)p*a, IAJ(X/D) (6)
where X is the distance traveled since unplugging started and D
is thebore dlameter of the gun tube.
Nonconical Boattails.
The value of P1 depends upon the boattail geometry because
of
the appearance of the vent area term, Au, under the integral
sign.
This expression is addressed in detail in Appendix A where it
isfound for nonconical boattail.- that
P1 = (16n/45) (y+l) (p*D3 /Vp)803 /2a 1 (X/D) 5/
2 (7)
where n is number of plane boattail surfaces and a is the
boattailangle. The total amount of momentum transferred to the
projectileby the venting flow is obtained by evaluating this
expression at
k the maximum travel of the round, i.e., when the projectile
base is at
I.
.. .........-...7,.. . .." .
-
the muzzle exit plane,. For nonconical boattails that terminate
in apolygon shape:
PIt = [5 /(90n ) (y+l)[pD 3/(0Vp)]a (8)
It is useful to define a nondimensiona) momentum transfer
function,P, similar to that developed in Reference 4,
T P/[(D/V )(Y+l)p*Aeqa 1] (9)
For nonconical boattail projectiles, the surfaces are treated
asminiature airfoils. In Reference 4, it was shown that the
equivalentlifting surface area for a multiple-finned missile is A
n= A/2 for
the lift vector in the angle-of-attack plane. Since the
nonconicaldesigns have only one surface to the "airfoil", the
following definitionis used:
k ei1 = PA/4. (10)
where A is the area of each plane surface.
This definition permits the momentum transfer functions to
bewritten (See Appendix A, Equations A9 and All)
TP = (128/15)(n/r) 3 (OX5D)5/2 (11)
and
P = (47 2 )/(15n 2 ) (12)it
Conical Boattails.
Using the same approach, the momentum transfer function for
aconical boattail design is
P1 = (n/3) (y+l) (p*D 3/Vp)O6a1 (X/D)2 (13)
2and in nondimensional form, with Aeq •D /4:
4. K. S. Fans Ler and E. M. Schmidt, "The Influence of
MuzzleGasdynamics Upon the Trajectory of Fin-Stabilized
Projectiles,"BRL R 1793, U. S. Army Ballistic Research Laboratory,
AberdeenProving Ground, MD, June 1975. AD B0O5379L.
I
-
i ...........
I (4 0/3(X/D)) (14)
B. ýeion 2: Projectile Base of the gun tube
Nonconical Boattails.
There are no data available describing the aerodynamics
ofnonconical boattails in reverse flow. Since the
three-dimensionalnature of the muzzle flow makes direct computation
difficult, the liftforce on the boattail will be estimated using
two-dimensional airfoiltheory. This approximation neglects any
effects of base bluntnessdue to truncation of the boattail;
however, the side force cn a bluffbody is generally smaller than
that on a slender body at an equal angleof attack. Therefore, the
current approximation should produce anupper bound on the
transverse force exerted on these nonconicalsurfaces by muzzle gas
loadings. The value of P2(X/D) can be obtained
[2directly from earlier results using the definition of A in
eqEquation (10). The center of force is taken as the area
centroid forthe plane surfaces. This center of force value would be
correctfor two-dimensional supersonic airfoil theory but may be
quite differentfrom the possible actual results. Nevertheless, for
this analysis. theapproximation should be sufficient since the
percent error betweendiffering possible moment-arm values would be
small.
Conical Boattails.
The transverse force on conical boattail projectiles is
estimatedusing experimental datas acquired on blunt, flared
sphere-conesat a variety of Mach numbers. These data were obtained
using a sphere-cone where the ratio of the diameter of the cone at
the sphere-cone junction to the base diameter was 0.8. The smallest
cone half-angle of this set of data was 100. For the purposes of
the presentanalysis, it is desired to compare the muzzle blast
sensitivity of a7 nonconical boattail with the sensitivity of a
similar half-anglecone. Thus, the data 5 are extrapolated to this
angle, Figure 2. Thecenter of force is taken to be halfway along
the boattail length.
5. A. D. Foster, "A Compilation of Longitvdinal
AerodynamicCharacteristics Includin9 Pressure Information for Sharp
andBlunt-nose Cones Having Flat and Modified Bases,"
SandiaCorporation Report No. SC-R-64-7311, January 196.5.
i'
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II1. RESULTS AND DISCUSSION
The M549 projectile is used as a basis for the computations
sinceflight data are available on this round with both the standard
7° boat-tail and a triangular boattail. The properties of the
projectiles aregiven in the tables below.
Table I: NS49 Parameters Independent of Boattail
M = 43.0 kgP
D = 0.155m
V = 670m/sP
p* = 1.52 x 107 Pa
y = 1.25
CM = 3.30a
CL = 2.95L
Ix = 0.13 kg-ra2
Table 11. MS49 Parameters Dependent on Boattail
Standard Triangular SquareConical NCBT NCBT
A (M2) 0.0189 0.028 0.016eq
@(deg) 7 7 7
AW(m) 0.295 0.293 0.340
0 m) 0.097 0.347 0.195
2I y(kg-m2) 1.93 1.71 1.71
Here A is the distance between the force acting on the planar
surfaceand the center of gravity;Z is the length of the
boattail.
The values of P through both the in-bore and exterior regionsof
interest are shown in Figures 3-5 for the three boattails
considered.
12
i. ilL•............................. ........................
~.....,, .... •,'L~a. .~
-
The coordinate X is the location of the projectile center of
pressurealong the bore axis, with X = 0 corresponding to the
muzzle. Sincethe momentum transfer function is cummulative, the
asymptotic value ofeach curve is equal to the total momentum P
transferred in the blast.
From Equation (14), it is apparent that the Pit for the conical
boat-
tail has an explicit dependence on Z/D and 0; however, for the
non-conical boattails considered (those taken back to the
intersection ofplanar surfaces), this is not the case. Pi and P are
independent of
it tall boattail and projectile parameters, i.e.r the curves are
'universally'applicable to any projectile with an n-sided
nonconical boattaillaunched with the velocity of the exit-gas speed
of sound.
The momentum functions are used to compute the transverse
angularand linear velocities imparted to the projectiles in the
blast region,using the approach of Reference 4. These velocities
are input to theaerodynamic jump relations 6 for conical boattails,
resulting in
E/a1 =lI+(CL A)/(CM D)] (y+l)p*A 2D/(MpVp)I t (i5)1 l(Leq '(p~p
t
Fcr nonconical boattails, the working equation, using Equations
(10)and (A12), is
3 3 2 -2O/al = [71 (y+l)/ 2 4 ] [p*D /(MpVp )] [ l-(CL A)/(CM
D)]I[Pt/n 0)] (16)a a
An expression for the first maximnum yaw may also be derived6 :
(17)
ICmaxI/a, = [2(y+l) A eqAD -2rpD CM /(21Jy)
For nonconical boattails, the working equation is, using
Zquations (A12)and (10),
I~ax/a =I 4 (1 2 2I¢maxI/a I f 7 3(y+l)/12] [AD p* Pt/(lyV n
0)]
• [(Ix't/1) - TpD SCM /(21 )1½ (18)2ay
Here o is the initial spin in radians per caliber.
6. C. H. Murphy, "Free Flight Motion of Symmetric Missiles,"U.S.
Army Ballistic Research Laboratories Report Number 1216,Aberdeen
heoving Ground, Maryland, July 19G3. AD 442757.
13
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fhho projectile parameters from Tab' ;s I and I I with the
above
relations are used to determine the va,;.e• in the following
table:
Table III. Launch Dynamic Characteristics
Standard Square TriangularConical NCBT NCBT
P 0.21 0.52 0.48t
k (deg/deg) 0.167 0.37 0.57Imax I /"lO/a! (mil/deg) 0.051 0.11
0.18
kýmaxl (degs for lowith a= 0.1°) 0.02 0.04 0.06
Comparing these results forc/aI with those obtained for
fin-stabilizedF 1
projectiles1 , we find that the values are similar.
Table III shows that the jump sensitivity of the
nonconicalboattail rounds is more than twice that of the standard
conical boattail.However, the last row gives the contribution to
the first maximum yawof this niuzzle-blast-induced jump. For a
reasonable distribution ofthe in-bore yaw level (lo,a 1 0.1°), the
resultant contribution to
the first maximum yaw is 0.060 in the worst case. This value is
judgednegligible compared with the frequentlg observed
first-maximum yawvalues for the M549 of approximately 5 . Since the
first-maximum-yawvalue is proportional to the aerodynamic jump, we
may infer that muzzleblast contributes only a small amount to the
total observed dispersion.
IV. SUMMARY AND CONCLUSIONS
A model is developed to determine the effect of muzzle
gasdynamicloadings upon the trajectory of spin-stabilized
projectiles for bothconical and nonconical boattail configurations.
The analysis consistsof two parts. First, the passage of the
boattail out of the gunfollowing separation of the obturator is
addressed. Second, the regionexterior to the muzzle but prior to
the projectile's passage throughthe Mach disc is modelled. The
conical boattail aerodynamics aredetermined from experimental data
acquired on sphere cones; two-dimensional airfoil theory is applied
to approximate the loadings onnonconical boattails.
Expressions for t'.. loadings :re integrated to determine
themomentum transferred to each of the M549 projectile
configurationslaunched at a muzzle velocity of 670 m/s from a 155mm
gun. The nor,-conical boattail configurations are found to be more
than twice as
14
do
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sensitive to blast as the conical design; however, in neither
case isthe contribution to dispersion due to muzzle gasdynamic
loadingssignificant when compared to the total dispersion of the
gun system.
r
15
I.!
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SONIC LINE
t '~BLAST WAVE
PATH LINES
MUZZLEEXIT PLANE
Figure 1. Flowfield schematic
3.0--EXPERIMENTS- REF. 8
- --- XTRPOLTEDCURVE
.0-
sUj
Uj
.0 102030405
MACH NUMBER
Figure 2. Experimental and extrapolated lift data for
sphere-cones
16
-
0. 25-
I: ~0.2 --
0.15-
I• 8.85f' 0. e
8-1 9 1 2 3 4 5 6
X IDe
Figure 3. Momentum function for the conical 549 boattail
0.6-
0.5
0.4 -
0.3
0.2 + IN-BORE PHASE- EXTERIOR PHASE
0 1 1 ,1. -
-1 0 1 2 3 4 5 6
,( Xo/D
f Figure 4. Momentum function for the square boattallf17 A
........................ •
-
1 0.5
0.4-
0.3
0.2 + IN-BORE PHASE
,- EXTERIOR PHASE
0[1_
[1 -2 -1 0 1 2 3 4 5I•: Xe /D
Figure 5. Momentum function for the triangular boattail
1,
I
I'
Figure 6. Cross-section of nonconical boattail in muzzle
plane
18
i•, r-.
S. ....... . .. . ,, ... .. .. .. , .. ,, . . = ,•. , ,i,• •.
=,u• L ;
-
-. . .
". . .. . .: .- .....
REFERENCES
1. E. M. Schmidt, K. S. Fansler, and D. D. Shear,
"TrajectoryPerturbations of Fin-Stabilized Projectiles Due to
Muzzle Blast,"Journal of Spacecraft and Rockets, Vol. 14, No. 6,
June 1977,pp. 339-344.
2. K. S. Fansler, and E. M. Sch:didt, "Trajectory Perturbations
ofAsymmetric Fin-Stabilized Projectiles Caused by Muzzle
Blast,"Journal of Spacecraft and Rockets, Vol. 15, No. 1,
January-February 1978, pp. 62-64.
3. A. S. Platou, "An Improved Projectile Boattail," BRL MR
2395,[ US Army Ballistic Research Laboratories, Aberdeen Proving
Ground,Maryland, July 1974. AD 785520.
4. K. S. Fansler and E. M. Schmidt, "The Influence of
MuzzleGasdynamics Upon the Trajectory of Fin-Stabilized
Projectiles,"BRL R 1793, U.S. Army Ballistic Research Laboratories,
AberdeenProving Ground, MD, June 1975. AD B005379L.
5. A. D. Foster, "A Compilation of Longitudinal
AerodynamicCharacteristics Including Pressure Information for Sharp
andBlunt-nose Cones Having Flat and Modified Bases,"
SandiaCorporation Report No. SC-R-64-1311, January 1965.
6. C. H. Murphy, "Free Flight Motion of Symmetric Missiles,"U.S.
Army Ballistic Research Laboratories Report Number 1216,Aberdeen
Proving Ground, Maryland, July 1963. AD 442757.
19
19
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APPENDIX A. EXPRESSION FOR TRANSVERSE MOMENTUM
In order to evaluate the integral in Equation (6), it
isdesirable to obtain closed-form expressions for A
U
A. Nonconical Boattail
Figure 6 shows the cross section of a boattail in the
muzzleplane. The bore radius is R and the area of the circular
segment canbe readily determined to be
SR2RA (a - sin2a/2) (AI)Ul
The expression in brackets may be expanded in series
yielding:
Au1 = D2 3a a 2 a- **)(2i~ Au5u '6- (a -T + I-0 a .(A2)
In keeping with the upper bound approach,the series is truncated
afterthe first term to give
A =D2a/36 (A3)u1
The distance BC, Figure 6, is
BC - R[l-cos(a)]
T (a2 _ a4/12 + .. )(A4)R2 4
However, BC is also related to the boattail angle and the
length, X,of the nonconical surfaces protruding beyond the muzzle
plane.
BC =_X tan8 (AS)
Assuming small e and taking the first term in the series
ofEquation (A4) gives
a2 = 40 X/D (A6)
which can be substituted into Equation (A3):
A (4D 2 /3) (X/D) 3 2 0 3/2(A7)
For an n-sided boattail,
A nA =(4nD2 /3)(X 3/2 3/2 (A8)u u1I
21
!.Ii..,,.
j -.-- -~ P -MM-PA." Aa&.*
-
This formulation for A may be substituted into liquation (6)uand
integrated to provide
P1 = (y+l)[16 nD3 p*al/45 V 1 O3/2 (X_/)S5 2 (A9)p
If the boattail terminates where the planar surfaces intersect,
then
am = it/n. (A10)
This permits the evaluation of the total momentum transferred
in-bore:
P = (n5/90 n4 )(y+l)(p*D3/AV ) a1 (All)
This type of boattail has a surface area that can be readily
computed.Since the surface is an ellipse formed by intersecting the
cylindricalbody with a plane at angle 0 to the axis,
A = A /0 (evaluated at the muzzle station)
= D2 (Tr/n) 3 /(60) (A12)
B. Conical Boattail
The unplugged area 4s
2 2Au = ir(R - r2) (A13)
where r is the diameter of the section of the boattail passing
themuzzle plane. Since R - r = X tan a :X 0,
Au =r(2R - X 6) Xe , (A14)
which may be substituted into Equation (6) to give
[l = (7/3) (y+l) (p*D 3/Vp) 0 (X/D) 2 1 (A1S)
22
L. ....
-
LIST OF SYMBOLS
a =angle defined in Figure 6
A = area of a plane surface of the nonconical boattail
A = -the equivalent area of an airfoil. in two-,eq dimensional
flow
A = the area at the muzzle that is unpluggedu
C DR = projectile drag coefficient in muzzle blast
CL slope of -che lift coefficient curve for forwarda flight
CM f slope of the static moment coefficient curve ina forward
flight
D~ th., diameter of the bore and projectile
I =axial moment of inertia
I =transverse w-iment of inertiaY
L = the lift force on the projectile
L 1 = lift force on projectile In the first part of
itstransit
£ = length of boattail
M p= projectile mass
n = the number of plane surfaces possessed by the boattail
p =pressure
P =the momentum given the projectile .
Pl momentum given the projectile during the first part ,:of the
projectile's transit :
S= the value of P nondimensionalized according toEquation (10)
:'
Pt the total amount of P imparted by the muzzle blast
t;
t
&23
-
LIST OF SYMBOLS (CONTINUEI))
t = time variable
V = local velocity of gases
V = velocity of projectilep
V = velocity of gases relative to a coordinate systemr located
on the projectile
X =position along the axis of the center of force
for the projectile relative to the muzzle
X = distance traveled since unplugging started
y = axis coordinate in the angle of attack plane that
isperpendicular to the x-axis
= maximum angle between projectile and gun-tube axis--angle of
attuck
= angle determined by a and fluid flow veloci~y relativeto the
ptojectile
= angle of attack upon emergency from muzzle
= angle cf attack upon emergency from muzzle blast
a, = derivativ, of a w•.th respect to distance in calibers
a/ = value of a' upoi emergency from muzzle bl.st0
= rr.tio of specific heats
A = distance from center of force to center of gravity forflight
through muzzle-blast region
= comrleY angle of attack
= roll angle of projectile
= boattail angle
e = aerodynamic jump
p local density of gases
24
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