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AO-A096 196 ARMY ARMAMENT RESEARCH A ND DEVELOPMENT COMMAND WATERETC F/9 19/4COMPUTATION SCHEMES FOR SENSITIVITY COEFFICIENT OF EXTERIOR BAL--ETC(U)JAN 81 C N SHEN, J J WU
UNCLASSIFIED ARLCB-TR-81002 SBIE-A DE NO 0 Il NL
SEMI
I flfflfllIfllIflfl/f//III//I/IIIIIfll~fflfEE/
lll 8 32
1111112 LA 1.6
MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS- 1963-A
A ~ e) LEVEL~TECHNICAL REPORT ARLCB.TR-81002
COMPUTATION SCHEIES FOR SENSITIVITY COEFFICIENT OFEXTERIOR BALLISTICS WITH VELOCITY SQUARE DAMPING
Julian J. Wu
'I
January 1981U US ARMY ARMAMENT RESEARCH AND DEVELOPMEN COMMAND" LARGE CALIBER WEAPON SYSTEMS LABORATORY
i) BENetT WEAPONS LABORATORY
WATERVLIETO N. Y. 12189
MM No. oo7000204 D T ICDA Project No. 1564018136GRN T 11
PRON No. IA0215641AIA E,, B
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READ INMTUCTINSREPORT DOCUMENTATION PAGE BEFORE COMPLEM PORNI. REPORT NUMBER 2. GOVT ACCESSION NO. RECIPIENT S CATALOG NUMBER
ARLCB-TR-81002 f1YV '7W94. TITLE (and SAbIle) S. TYPE OF REPORT A PERIOD COVERED
COMPUTATION SCHEMES FOR SENSITIVITY COEFFICIENTOF EXTERIOR BALLISTICS WITH VELOCITY SQUARE-DAMPING S. PERFORMING Oo. REPORT NUMBER
7. AUTNOR(*) . CONTRACT OR GRANT NUMBER(e)C. N. Shen and Julian J. Wu
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK' Wp LAREA A WORK UNIT NUMBERSeSanet Weapons Laboratory AMCMS No. 36KA7000204Watervliet Arsenal, Watervliet, NY 12189 DA Project No. 1564018136GRN,DRDAR-LCB-TL PRON No. 1A0215641A1A
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US Army Armament Research & Development Command January 1981Large Caliber Weapon Systems Laboratory IS. NUMBEROF PAGESDover, NJ 07801 3614. MONITORING AGENCY NAME A ADDRIESS(I'f dldfftnt ha Con0oltndi Office) IS. SECURITY CLASS. (of thle report)
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1S. SUPPLEMENTARY NOTESPresented at the 1980 Army Numerical Analysis and Computer Conference,NASA Ames Research Center, Moffett Field, California, 20-21 February 1980.
19. KEY WORDS (Cent/uee en eva" side If eoeeernv and ldontl& by block nuabeu)
Ballistics with Drag Computational SchemeSensitivity Coefficients Iteration ProcedureTarget HittingInitial Velocity and Elevation Angle Correction
21 ARSRACr (CMam mw. 1Nmeew eMd lIp bf lok NoThe principal equation of exterior ballistics vith velocity square dampingterm has been integrated analytically in obtaining the solution for tangentialvelocity in terms of the elevation angle and other parameters. Using thevariational method, four equations are obtained. The first one is derivedfrom consideration of terrain slope and the second one is determined by hittingthe target. The third, and fourth equations are variatoof the renE, E(O'DO&N RIvE16E£)
JAN 1 Emrosse , MVSImosore UNCLASSIFIEDSECURITY C:LASFICAlIrm OF ThI s 8 tln (e. de ned
SIeCURITY CLASSIFICATION OF THIS PAGE(IlUm DO* MW
20. Abstract (Cont'd)
elevation drag functions, respectively.
The computation involves integrals which can be evaluated analytically if thedrag coefficient is relatively small. in simplifying thq computational procedurgwe can assign the launch and impact slopes and then compute the drag functionsand the terrain slope. However, this procedure is in reverse order becausephysically the terrain slope is known a priori the rounds are fired. If theterrain slopes and launch slopes are given first, an iteration procedure incomputation is required to solve for the impact slopes. The sensitivitycoefficients and the range ratios are then computed and plotted for variousterrain slopes and launch slopes as the drag coefficients are varied.
SCURITY CLASSIPICATION OF THIS PASCM,, DMM Mee
,., \ r , ,V,
TABLE OF CONTENTS
INTRODUCTION 1
PREVIOUS WORK 2
RESULT OF PAST RESEARCH 3
DYNAMICAL EQUATIONS FOR TRAJECTORIES 4
SOLUTION FOR NONDIMENSIONAL RANGE 5
VARIATION OF THE NONDIHENSIONAL RANGE AND THE SLOPE FUNCTION 6
THE SOLUTION FOR ELEVATION 7
TERRAIN SLOPE FROM LAUNCH POINT TO TARGET POINT 8
VARIATION OF THE RANGE AND ELEVATION DRAG FUNCTIONS 9
SOLUTION FOR SENSITIVITY COEFFICIENT 11
ITERATION OF SOLUTIONS 13
EVALUATION OF INTEGRALS 14
DETERMINATION OF THE EXTREMAL VALUE OF THE DAMPING COEFFICIENT 16
COMPUTATION PROCEDURE 18
Iteration Procedure 18
Sensitivity and Range Computation 18
NUMERICAL SOLUTIONS 19
CONCLUSION 19
REFERENCES 24
APPENDIX A A-i
APPENDIX B B-I
±
"I
TABLESpage
I. RANGE AND SENSITIVITY COEFFICIENTS FOR FLAT TERRAIN 22
II. RANGE SENSITIVITY COEFFICIENTS FOR 15* TERRAIN 23
LIST OF ILLUSTRATIONS
1. Forces, Slopes, and Initial and Final Parameters for a A-4Trajectory.
2. Impact Angle vs. Launch Angle for Different Target Slopes and B-2Damping Coefficients.
3. Nondimensional Range vs* Launch Angle for Horizontal Target and B-3Different Damping Coefficients.
4. Nondimensional Range vs. Launch Angle for Upgrade Targets and B-4Different Damping Coefficients.
5. Sensitivity Coefficients vs. Launch Angle for Horizontal Target B-5and Different Damping Coefficients.
6. Sensitivity Coefficient vs. Launch Angle for Upgrade Targets B-6and Different Damping Coefficients.
Accession For
NTIS GRA&IDTIC TAB E5Ur -Toi'.-ed 51
By-Di5trib)ution/Availability Codes
D iAvail and/or
DhjSpecial
INTRODUCTION
The research on sensitivity coefficients on ballistics has been discussed
in two previous works. The first paper1 matches the sensitivity coefficient
of exterior ballistics to that of interior ballistics, while the second paper2
presents the sensitivity coefficient of exterior ballistics with velocity
square damping. This report continues the work of the latter 2 and extends the
analytic study in order to give numerical solutions of the problem.
The design of a gun involves numerous parameters. These parameters
should be in such a combination that the best first round accuracy is given.
While the shell leaves the gun it has perturbations for the muzzle elevation
angle and the muzzle velocity. The ratio of the two is the sensitivity coef-
ficient of the interior and the exterior ballistics. It is desired to compen-
sate for errors due to uncertain changes of muzzle velocity by the automatic
response of the muzzle elevation angle within the gun system. With a correct
design this compensation can be made by matching the exterior ballistics to
the interior ballistics through the analysis of gun dynamics. This process is
called passive control since there is no external measurement involved nor
instrumentation needed for control. This general problem can be formulated by
first investigating the sensitivity coefficients for exterior ballistics with
velocity square damping.
IShen, C. N., "On the Sensitivity Coefficient of Exterior Ballistics and ItsPotential Matching to Interior Ballistics Sensitivity," Proceedings of theSecond US Army Symposium on Gun Dynamics at the Institute on Nan and Science,Rensselaerville, NY, September 1978, sponsored by USA ARRADCOH.
2Shen, C. N., "Sensitivity Coefficient of Exterior Bal. istics With VelocitySquare Damping," Transactions of the Twenty-Fifth Conference of ArmyMathematicians, ARO Report 80-1, January 1980, pp. 267-282.
IE
PREVIOUS WORK
In the second paper the principal equation of exterior ballistics has a
drag term which, in this case, is proportional to the square of the velocity
in the tangential direction of the projectile. The sensitivity coefficient is
expressed as the ratio of the initial elevation angle deviation to the initial
precentage velocity deviation. The work is to find analytically the sensitiv-
ity coefficient of the exterior ballistics with velocity square damping which,
comes from the nonlinear air resistance for a projectile. This principal
equation is integrated analytically In obtaining the solution for tangential
velocity in terms of the elevation angle, together with all the necessary ini-
tial conditions. The horizontal range and the vertical range are also
expressed as integrals of certain function of the elevation angles. In order
to obtain the sensitivity coefficient it is necessary to find the perturba-
tions of the horizontal and vertical ranges. This procedure Is similar to
that of evaluating differentiation under the integral sign. The perturbation
of the ranges is the sum of the perturbations due to the initial velocity, the
initial elevation angle and the impact elevation angle. By setting to zeroes
the range perturbations we can group the coefficients of the perturbations
into two separate equations. The ratio of the perturbations for initial
elevation angle to that for initial velocity is the sensitivity coefficient
for exterior ballistics that we are seeking.
2
RESULT OF PAST RESEARCH
The following results are concluded as:
1. The principal equation of exterior ballistics is derived with the
trajectory slope as the independent variable.
2. The closed form solution for the horizontal component of trajectory.
velocity is determined for the case of exterior ballistics with velocity
square damping.
3. The nondimensional range is obtained in terms of an end slope
function and a range drag function.
4. Variations of the nondimensional range are expressed as variations of
launch velocity.
5. Variations of the range drag function are in terms of the variations -
of the range drag integral.
6. The range drag integral has parameters in the integrand as well as
the upper and lower limits. The variations of the integral are found.
7. The partial derivatives of the range drag integrand are evaluated.
8. The variational equation for the range is in terms of elements
involving three integrals as coefficients of three variational parameters.
9. The variational parameters are that of launch velocity, the launch
elevation angle, and the impact elevation angle.
10. The average of the end slopes is equal to the terrain slope times the
range drag function minus the elevation drag function.
11. Variations of the nondimensional elevation are expressed as
variations of the end slopes and the variations of the drag function.
3
12. The variational equations for the elevation are determined similar to
that for the range.
13. Eliminating the variations of impact slope, 6qj. from the set of two
variational equations gives the ratio of the coefficients of 6vo/v o and 6qo/
(qo-qi).
14. The sensitivity 660/(6vo/vo) may be obtained by dividing this ratio
6qo/(6vo/v o ) by the quantity (1+qo 2 ).
However, numerical calculation of this problem was not carried in that paper. 2
DYNAMICAL EQUATIONS FOR TRAJECTORIES
After several transformations as given in the paper 2 and summarized in
Appendix A, the dynamical equations are simplified as
C
du = - u3 (l+q2)1/2dq (1)q
u2dx - -- dq (2)
g
I U2dy -- qdq (3)
In the above equation the independent variable q is the projectile slope which
is related to 0 as
q - tan (4)
and the dependent variable u is related to v and 0 as
u-v cos e (5)
2 Shen, C. N., "Sensitivity Coefficient of Exterior Ballistics With VelocitySquare Damping," Transactions of the Twenty-Fifth Conference of ArmyMathematicians, ARO Report 80-1, January 1980, pp. 267-282.
4
where g - the acceleration due to gravity
c - the drag coefficient of the projectile
*v - the velocity of the projectile
e - the path inclination (elevation angle)
x - the horizontal distance of the projectile
y - the altitude or vertical distance of the projectile
The solution for horizontal component of velocity u can be obtained by
integrating Eq. (1) to give
L1-uo2 p(q) - pO(qO)]} (6)
where
p(q) - q(l+q2)l/ 2 + tn[q + (1+q2)1/ 21 (7)
po(qo) - qo(l+qo2 )1/2 + tn[qo + (1+qo 2 ) 1 / 21 (8)
U 02 = Vo
2 sec-20 0 w V02 (1+qo2) - 1 (9)
and
qo tan 00 (9a)
Finally, Eq. (6) takes the form
22V2 2 H(q ,qoVo 2 ,c/g)U2 . ....- {1+ 2 (10)1+qo2 I - H(q,qovo, c/g
wherec Vo 2
H(qqovo2 ,c/g) --- -[ P(q) - Po(qo)J (1I)g 1+qo 2
SOLUTION FOR NONDIHENSIONAL RANGE
In determining the range x for the trajectory the closed form solution of
u2 in Eq. (10) can be substituted into Eq. (2) to obtain the solution in
integral form as
5
44- 3 ' ' ; . -
Vo2 _qi R(q,qovo2 tc/g)
x- " - - --------) [(qj-qo) + f --- dq (12)
where xj - range at impact point
xo - range at initial point
and ql - projectile slope at impact point.
To non-dimensionalize the range, Eq. (12) is divided by the factor v02/g
as
X(xj,xo,vo)/A(qi,qo) - Gx(q ,qo9vo2 ,c/g) (13)
where the nondimensional range is
X(xixo,vo) - (xi-xo)g/vo2 (14)
the slope function is
qo-qiA(qo,qi) - !; °2 (15)
1+0
and the range drag function due to air resistance is
1 qi H(q'qo'vo23 c/g)
Gx(qj,qov 2 ,c/g) f ---------- dq (16)qo-qi qo 1 - H
VARIATION OF THE NONDIKENSIONAL RANGE AND THE SLOPE FUNCTION
In order to obtain a first round hit of the target one of the conditions
is that the variation of the range should be zero, i.e., from Eq. (14)
6(xi-xo) - 0 (17)
We take the perturbation for the nondimensional range from Eq. (14) as
8X 6(xl-x o ) 26vo-------- ----
x Xl-x o vo
6
6x 26vo-- -0-.. (18)X vo
The variation of the slope function A in Eq. (15) becomes
6A 6qi 6qo 2qo 6qo- ---- + (19)
A qo-ql qo-qj 14<o
Next, taking the variation of Eq. (13) and using the expressions given in Eq.
(18) and (19) we have6X 6A 6GS- = --- (20)x A Gxor
6Gx 26vo 6qt 6q0 2qo6qo
.. . . . -- - - --- - ....- (21)Gx vo qo-qi qo-qi 1+qo
This gives the variation of the range 6Gx in terms of the variations of the
initial velocity 6vo, and the variations of the initial slope 6q0 and that of
the impact slope 6 qi.
THE SOLUTION FOR ELEVATION
The differential equations for elevation was given in Eq. (3) and the.
solution for u is In Eq. (10).
Substituting Eq. (10) into Eq. (3) gives
Vo2 H(q,qov o 2 ,c/g)dy - - -------- - --------------- lqdq (22)
g(l+qo2 ) I - H
Integrating the above one obtains
V02 qi2 q2 qi qHYi.yo ------ --- + --- dq] (23)
-o 8(1o 2) 2 qo 1-H
Rearranging yields the relationship between the range Y, the end slope
function A, and the elevation drag function Gy.
12Y(yi,yovo)/A(ql,qo) = - (qo+ql) + Gy(qlqovo2 ,C/g) (24)
2
where the nondimensional elevation is
* g(Y-yo)
V02
A is given in Eq. (15), and the elevation drag function Is
I q qH(q,qovo 2,c/g)dqGy(qiqovo2 -c/g) f-------- ------ - (26)
qo-qi qo 1-H
TERRAIN SLOPE FROM LAUNCH POINT TO TARGET POINT
If Eq. (25) is divided by Eq. (14) with the 4id of Eqs. (16) and (26),
one obtains
Yi-yo (1/2)(qo+qt) + Gy...A M = ----- (27)xi-xo -
where m Is the terrain slope from launch point to target point, a constant
parameter. Therefore,
(1/2)(qo+qi) + Gy a mGx (28)
We use Eq. (27) to find the variational equation for the elevation.
-Taking the variation of Eq. (28) for any given m, we have
(1/2)(6qo+6qi) + 6Gy - a 6GX r 0 (29)
It to noted that 6Gy and 68G are related in Eq. (29), with also the variations
6qo and 6 qi.
K _ _ _ _ _ _ _ _8
VARIATION OF THE RANGE AND ELEVATION DRAG FUNCTIONS
Work was performed on evaluation of the drag function in the previous
paper2 which is summarized in Appendix B. This gives the respective varia-
tions of the range and elevation drag functions as
26v o 6q 6qo6Gx - av --- + ai ----- - ao ----- (30)
Vo qo-qi qo-qi
26vo 6 qi 6qo
6Gy a bv ---- + bi ------ bo ----- (31)Vo qo-qi qo-qi
where1
av - I--]-----1121 (32)qo-qi
Hi
ai - -- ------ (33)1-Hi qo-qi
2qo c V02 dpo
ao 112 - ----- Iii + (-) ----- --- 102] (34)80w~ 1q~,qo+qi g 1+q0 2 dqo
1bv - .----- J121 (35)qo-qj
qjHj Ibi ---------- ll (36)
I-Hi qo-qi
and2qo 1 c V0
2 dpo
J12 + J11 + (-) 0- --
, 0 qo-q g l+q 2 dq0 o
2Shen, C. N., "Sensitivity Coefficient of Exterior Ballistics With VelocitySquare Damping," Transactions of the Twenty-Fifth Conference of ArmyMathematicians, ARO Report 80-1, January 1980, pp. 267-282.
9
In turn, the integral 111, 112, and 102. and other terms are given as
follows._ R(q,qoVo 2 ,cIg)
1-l(qo,qi) f . --- dq (38)qo I-H
qjH
112(q0,qi) -fq 4jj q (39).qo
H
102(q°,qi) f I --- dq (40)
Hi - (q-ql) (41)
anddpo
. 2(1+qo2)1/2 (42)dqo
Moreover, the integral J11 , J12. and J0 2, and other terms are given as
follows. .g(
qj- f qI(qlqovo2-C/)/(1-H) dq (43)qo~i
J12 = qR/(l'lt)2 dq (44)qo
02 I q/(l-H)2 dq (45)qo
and1
SJ1 (46)qo-q
10
SOLUTION FOR SENSITIVITY COEFFICIENT
It is noted that both Eq. (21) and Eq. (30) give the variation 6Gx In
terms of variations 6vo, 6qiq and 6qo and the equations are independent of
each other. Rewriting these equations as:
26v° dqj 6qo 2qo6qoIt 6Gx =Gx .. .. ... +...y (47)6G~~~-GL ---- ------ ------- 4 1(7
Vo qo-qi qo-q i 0
26v° 6qi 6q°6GK iav +ai ----- - o .- (48)
vo qo-qi qo-qi
Subtracting Eq. (48) from Eq. (47) and then dividing by Gx gives
av 26v° ai 6 qi a° 6q° 2q°6q°------------------- -----------i[i-- ----- --[ -G x vo Gx qo-qi G+ qo-qi + -l+qo 2 (49)
It is also noted that Eq. (31) expresses the variation 6Gy in terms of
variations 6vo, 6qi , and 6qo, and Eq. (29) related 6Gy and 6G, with the same
set of variations. These are rewritten as
26vo 6qi 6qo=G V-- + bi --- + bo --- (50)
4 vo qo-qi qo-qi
6qi - - 6qo +- 2m6G x - 26Gy (51)
By substituting Eqs. (48) and (50) into (51) we have
26vo 6q1 6qo6q -6qo + 2m[a v ---- + ai ----- o - - - - -
Vo qo-qi qo-qi
26v o 6 qi 6qo
-2[bv . + bi ----- - bo (52)vo qo-qi qo-qi
11
or
(-2mav+2bv) 26v o -2maj+2b i 6qi -2mao+2bo 6qo0 .---------------- + (- + ---------- . (1 +- --------- ----- (53)
qo-qi Vo qo-qi qo-qi q0-qi qo-qj
Equations (49) and (53) are in terms of the variations SVo, 6q1 , and 6qo.
Sensitivity coefficients are ratio of 6qo and 6vo, thus 6qi can be eliminated
by combining these two equations:
-2mav+2bv ai av -2mai+2bi 26vo0 --------- )(1---) + ( + -- )( .. -------
qo-qi GX GX qo-q1 Vo
-2mao+2bo ai+ ( + -------- )(1 +
qo-qi
ao 2qo(qo-qi) -2mai+2bi 6qo(1 +--------------)( +-.........)1 (54)
Gx 1 02 qo-qi qo-q(
If we define the ratio S* as
6qo/(qo-qi)S* -- (55)6Vo/Vo
Then from Eq. (54) this ratio is
-2mav+2bv ai av -2mai+2bi-------- +-) +0 + --M ,----------
qo-qi _, qo-qiS* -(-2) --------------------------------------------- (56)
-2mao+2b o ai ao 2qo(qo-qt) -2mai+2bi0- ---------- )+ (1 +------- -1--------
qo-qi 1+qGX "o, -) qo-q "
The sensitivity coefficient is defined as
8qo/(+qo2)S 0-- (57)avo/vo
12
which give the relationship
S - ---- -s* (58); o
ITERATION OF SOLUTIONS
The terrain slope m is usually provided before the computation. Equation
(27) shows that this terrain slope is in terms of the range drag function, Gx
in Eq. (16) and the elevation drag function Gy in Eq. (26). In turn, these
drag functions are expressed in terms of the projectile starting slope qo and
its impact slope qi. Computationally the solution of a for given qo and qI Is
a straight forward substitution procedure using Eqs. (27), (16), and (26).
However, since the terrain slope m is known In advance physically and the
impact slope qi is not, we compute instead the impact slope qi for a given
starting slope qo in Eq. (27). Rewriting this equation we have
2 q1 qH(qqovo2 ,c/g)dqq -qo+ ------ f ---
qo-qi qo 1 - H
2m qi H(qqovo2 ,c/g)dq+2m--------f (59)
qo-qi qo 1 - H
One can guess an initial value for qt on the right side of the above equation
and perform the integration. The value for qI on the left side can be used as
the initial value for qI on the right side for the second trial. This itera-
tion procedure continues until the value of qI converges to a numerical solu-
tion as its limit.
13
Equation (59) may be expressed as
2 2.qi - qo -.-----. J1 + 2m -1------1 (60)
qo-qi qo-qi
where III and J11 are given in Eqs. (38) and (43). These integrals will be
evaluated in the next section.
EVALUATION OF INTEGRALS
If the drag coefficient cvo2/g is small compared to (1+qo 2) the denomina-
tor terms (1-H) and (1-H)2 in Eqs. (38) through (40) and (43) through (45) may
be neglected. One obtains from the above equations the following
qiIII a 112 9 f Hdq (61)
qo
102 a qi - qo (62)
Jll g J12 q dq (63)
qo
I
J02 - (qi2-qo2) (64)2
Equation (11) for H may be rewritten as
IH- K ---- [p(q) - Po(qo) (65)
where
K cvo2/g (66)
and p(q) is given in Eq. (77). Then the integrals III and 112 are
111 i 112m KM (67)
14
where
MH - (Pqinq1 Pqmqo + (q0-ql)p0 (q0 )J (68)
and
P f p(q)dq
I (1+q2)3/2 +- qtnfq + (1+q2)l/2] (1+q2)1/2 (93
(9
Similarly the integrals il and J12 are
ill as J1 1(1 (70)
where
N---- Qqmql -q + - (qo q1 )p0 (q0 )l (71)
and
Q f qp(q)dq
1 31 1-- q(l+q2 )31 2
-- q(l+q2)11 2 + (-q2 +-t~ 1q)/1 (2
48 2 8
In the last section the Iteration equation (59) for the solution for qj
now becomes
qi -- qo + 2(mGx-GY) (73)
where
- 1 (qo-qi)-1 K (74)
and
- - (qo-qi)-l K (75)
We are now able to compute qi If the quantities m and qo are given.
15
The coefficients used in the variation of range and elevation drag
functions in Eqs. (32) through (37) give
a- - (qo-qi)-l 10 (76)
ai - Ki-(p(qi) - po(q0)J - (qo-qi)-1 M (77)
2qo dpo
- ( 0"iY]M--- (qo-qi)I 78
- - (qo-qi) 1' 10 (79)
bi - KI-qj~p(qI) -po(q 0 )] (qo-qi)-1 NJ (80)
and2q0 dpo
bo-K[- - (qo-qi) 1 ]IN - (qo -qi2) ;; (81)1+o2 d 0
With qo and qj known, we are able to compute the coefficients of three a's and-
three b's as given in the above equations. It is nov ready to substitute all
the coefficients into Eq. (56) to compute S*. By Eq. (58), In turn, one
obtains S, the sensitivity we are seeking.
DETERMINATION OF THE EXTREMAL VALUE OF THE DAMPING COEFFICIENT
The assumption that the denominator terms (1-li) may be dropped In
evaluating the integrals depends on
H <<1 (82)
From Eq. (65) this is equivalent to
11(q) -Kj- 1 - [p(q) -po(q 0)j} << 1 (83)
16
where
p(q) - {q(l+q2)l/ 2 + In[q + (l+q2)l1/2] (84)
It is noted that
H(qo) -0 at q -qo (85)
The extremal value of Rl(q) can be evaluated at q - qi
1
IHI 4 IHlextremal - IH(qj)I - K [p(qj) - Po(qo)]i << 1 (86)
The above is conservative in that we use the largest value of JHl for
determining the range spread of K. For example if qo - 1 and qi - -1, then
1+-- o2 -P(qo) " (1+Xn 2.414)/2 - 0.940 (87)
1
- - - - - -(qi) " (-l+tn 0.414)/2 - -0.940 (88)
then
KI-0.940 - 0.9401 << 1 (89)
or
K << 0.532 (90)
and if qo r, and qi -r3, then
1po(qo) - [(1.732)(2) + In 3.7321/4 1.195 (91)
1
1 p1o P(qi) - [(-1.732)(2) + n 0.2681/4 - -1.195 (92)
then
K1-1.195 - 1.1951 << 1 (93)
17
K << 0.418 (94)
The maximum of K we chose in this paper Ismax(cVo2/g) = max K - 0.20 (95)
which satisfies the above requirements for the approximation used in
evaluating the integrals.
COHPUTATION PROCEDURE
The computation procedure consists of two parts, the iteration procedure
for the impact slope qj and the sensitivity and range computation after
obtaining qi.
(1) Iteration Procedure - By giving the launch slope qo one can compute
such quantities as po(qo) from Eq. (7), Pqq 0 from Eq. (69) and Qqq 0 from Eq..
(72). The iteration procedure starts by assuming an initial impact slope qj.
Then we can obtain Pq=ql and Qqq from Eqs. (69) and (72), respectively. In
addition, we can compute H and N from Eqs. (68) and (71), respectively. By
assuming the value of K (i.e., cvo2/g) in Eq. (66) the Integrals Ill and Jll
are calculated from Eqs. (67) and (70), respectively. With these integrals
one can find a new value for the impact slope ql by using Eq. (60). This
iteration procedure continues until the value of qi converges to a numerical
solution as its limit.
(2) Sensitivity and Range Computation - With the iterated solution of the
impact slope qj known, one can proceed to find the sensitivity coefficients
and the range. We start to find the range drag function Gx and the elevation
drag function Gy from Eqs. (74) and (75), respectively. Next the coefficients
av, ai, So, by, bi, and bo can be evaluated by Eqs. (76) through (81).
18
Substituting these functions and coefficients into Eq. (56) we can get S*,
thus the sensitivity coefficient S can be obtained readily from Eq. (58).
The range X can also be computed by combining Eqs. (13), (15), and (74).
NUMERICAL SOLUTIONS
We are using the nondimensional damping coefficient K - cvo 2/g - 0.2 for
illustration purpose. The terrain slopes are assumed to be m - 0 and m -
0.2679 which is corresponding to a terrain angle of 0 and 15 respectively.
The launch angle 60 are varied as follows:
for m - 0, 7.5* 4 Bo 4 82.5* at intervals of 7.5o
for tan-1 m - 15, 22.5 • 80o 75* at intervals of 7.5*
The results are shown in Tables I and II, and plotted in Figures 2 through 6.
Figure 2 gives the iterated impact angles for various launch angles. The
effect of K to the range is fairly large as shown in Figures 3 and 4. How-
ever, the effect of the nondimensional drag coefficient K to the sensitivity
coefficient is not very large except near the critical sensitivity which mag-
nitude is unbounded as shown in Figures 5 and 6.
CONCLUSION
This report is the third of a sequence. The findings are summarized as
follows.
19
From the first paper I we have:
(1) The error increments in hitting a target are derived and set to zero
for a bull's eye landing.
(2) The sensitivity coefficient is defined as the ratio of increments of
the initial elevation angle to the increments of the natural logarithm of the-
velocity.
(3) Without air resistance or damping the maximum range is found to be
located at an optimal initial elevation angle, which is half of the sum of 900
and the terrain angle.
(4) Without damping the sensitivity coefficients become infinite at
maximum range and zero at minimum range of zero.
(5) The sensitivity coefficient is positive for high trajectories (above
45* initial elevation angle) and negative for low trajectories (below 45*
initial elevation angle).
(6) Due to uncertainty of muzzle velocity deviation a gun may be designed
by matching the interior ballistics to the'exterior ballistics. This is to
give automatically and instantly a proportional deviation of its initial
elevation angle when the shell leaves the muzzle during firing for the first
2round.
IShen, C. N., "On the Sensitivity Coefficient of Exterior Ballistics and ItsPotential Matching to Interior Ballistics Sensitivity," Proceedings of theSecond US Army Symposium on Gun Dynamics at the Institute on Han and Science,Rensselaerville, NY, September 1978, sponsored by USA ARRADCOH.
20
The second paper 2 investigated the problem when the principal equation of
exterior ballistics has a drag term which is proportional to the square of
velocity. The paper is to find analytically the sensitivity coefficient. A
sequence of nonlinear transformation was used before the sensitivity problem
can be studied. The variation of various parameters is investigated,
including the variation under an integral sign.
The present report continues the work of the second and converts it into
a different analytical form that is suitable for numerical computation. The
solution for sensitivity coefficients is condensed into a single formula with
the terms readily computable from a set of parameters. An iteration procedure
is used in the computation. Some integrals are approximated in order to
obtain their analytical solution forms.
The effect of velocity square damping to the range can be large,
especially at the maximum range. However, the effect of velocity square
damping to the sensitivity coefficient is usually small, except near the
4 maximum range.
2Shen, C. N., "Sensitivity Coefficient of Exterior Ballistics With VelocitySquare Damping," Transactions of the Twenty-Fifth Conference of ArmyMathematicians, ARO Report 80-1, January 1980, pp. 267-282.
21
TABLE I. RANGE AND SENSITIVITY COEFFICIENTS FOR FLAT TERRAIN
2For m 0 cvoo qo ei(degree) qj x S
7.5" 0.13165 0 - 7.5000 -0.131652 0.258819 -0.2679490.20 - 7.7808 -0.136641 0.249773 -0.268020
15.00 0.26794 0 -15.0000 -0.267949 0.500000 -0.5773500.20 -16.1583 -0.289738 0.465538 -0.578836
22.50 0.41421 0 -22.5000 -0.414213 0.707106 -1.0000000.20 -25.0595 -0.467574 0.636169 -1.011472
30.00 0.57735 0 -30.0000 -0.577350 0.866025 -1.7320500.20 -34.2094 -0.679841 0.755507 -1.804402
37.50 0.76732 0 -37.5000 -0.767326 0.965925 -3.7320500.20 -43.1708 -0.938106 0.821550 -4.422821
45.00 1.00000 0 -45.0000 -1.000000 1.000000 00.20 -51.4995 -1.257153 0.835103 14.3160
52.50 1.30322 0 -52.5000 -1.303225 0.965925 3.7320500.20 -58.9394 -1.660305 0.798363 2.645294
60.0 ° 1.73205 0 -60.0000 -1.732105 0.866025 1.7320500.20 -65.5374 -2.198108 0.714006 1.342668
67.50 1.73205 0 -67.5000 -2.414213 0.707106 1.0000000.10 -69.0956 -2.618147 0.647751 0.9150740.20 -71.6039 -3.006806 0.585294 0.796916
75.0" 3.73205 0 -75.0000 -3.732050 0.500000 0.5773500.10 -76.0085 -4.013342 0.459463 0.5298020.20 -77.5372 -4.524636 0.417197 0.465626
82.5" 7.59575 0 -82.5000 -7.595754 0.258819 0.2679490.10 -82.9614 -8.099269 0.238697 0.2458990.20 -83.6382 -8.969194 0.217891 0.217213
22
TABLE 11. RANGE AND SENSITIVITY COEFFICIENTS FOR 15* TERI,7N
For a - +0.2679 cv02
80 qo - O(desree) ql X S
22.50 0.41421 0 6.932362 0.121586 0.249772 -0.2785960.10 6.783131 0.118944 0.245353 -0.2787370.20 6.619309 0.116045 0.240893 -0.278891
30.00 0.57735 0 - 2.379286 -0.041550 0.464175 -0.6341120.10 - 3.002055 -0.052443 0.448368 -0.6361740.20 - 3.751089 -0,065562 0.432262 -0.638617
37.50 0.76732 0 -13.03583 -0.231526 0.628688 -1.214643
0.10 -14.38786 -0.256530 0.598311 -1.2301680.20 -16.14129 -0.289416 0.567092 -1.250663
45.00 1.00000 0 -24.90070 -0.464200 0.732100 -2.7327360.10 -26.98236 -0.509137 0.688258 -2.8779360.20 -29.80982 -0.572933 0.642910 -3.107193
52.5' 1.30322 0 -37.50354 -0.767425 0.7673640.10 -39.96784 -0.838143 0.714885 32.5356210.20 -43.37056 -0.944680 0.660410 13.127275
60.0* 1.73205 0 -50.10622 -1.196250 0.732075 2.7317080.10 -52.42571 -1.299732 0.678076 2.4058320.20 -55.62687 -1.461933 0.622002 2.031230
67.50 2.41421 0 -61.97076 -1.878413 0,628640 1.2143580.10 -63.73921 -2.026837 0.580688 1.1055130.20 -66.17359 -2.264476 0.531007 0.968816
75.00 3.73205 0 -72.62684 -3.196250 0.464108 0.6339620.10 -73.71086 -3.422142 0.428564 0.5814360.20 -75.20812 -3.787022 0.391874 0.514330
82.5° 7.59575 0 -81.93802 -7.059954 0.249690 0.2784890.10 -82.41359 -7.508239, 0.230773 0.2556520.20 -83.41359 -8.238227 0.211307 0.226622
23
REFERENCES
I. Shen, C. N., "On the Sensitivity Coefficient of Exterior Ballistics and
Its Potential Matching to Interior Ballistics Sensitivity," Proceedings of
the Second US Army Symposium on Gun Dynamics at the Institute on Man and
Science, Rensselaerville, NY, September 1978, sponsored by USA ARRADCO.
2. Shen, C. N., "Sensitivity Coefficient of Exterior Ballistics With Velocity
Square Damping," Transactions of the Twenty-Fifth Conference of Army
Mathematicians, ARO Report 80-1, January 1980, pp. 267-282.
24.
I
APPENDIX A
DYNAMICAL E .UATIONS FOR TRAJECTORIES AND VARIABLE TRANSFORMATIONS
For a constant mass travelling in a vertical plane with no life and
applied thrust, but having drag and velocity vectors contained in the plane of
symmetry as shown in Figure 1, the dynamical equations of motion are:1
dx--- v tos 0 (l),dy
dx--- v sin O - 0 (A2)dt
dOm(g cos a +v -- ) = 0 (A3)
dt
d2x D cos(a4)
It is noticed that deviations due to anomalies in the azimuth direction are
not considered here.
By differentiating Eq. (Al) with respect to t one obtains
d2x d- (v coo 0) (AS)
Substituting into Eq. (MA) we have
d D coso-- (v cos)-------- (A6)dt m
IShen, C. N., "On the Sensitivity Coefficient of Exterior Ballistics and Its
Potential Matching to Interior Ballistics Sensitivity," Proceedings of theSecond US Army Symposium on Gun Dynamics at the Institute on Man and Science,Rensselaerville, NY, Septmber 1978, sponsored by USA ARRADCON.
A-1
NEW
Solving for de/dt in Eq. (A3) one obtains
d -g cos 0-- - (A7)dt v
Equation (A7) indicates that the differential equations can be transformed
from the time domain in t to the angle domain in 8. Equations (A6), (Al), and
(A2) are divided by Eq. (A7) in achieving this transformation as
d(v coo 0) Dvde mg
dx v2
- - - -- (A9)' dO g
dy- - -- tan 8 (A10)
dO g
Equation (A8) is called the prinicpal equation of exterior ballistics.2 It
can be integrated if the drag D is a known function of velocity v.
The head wind drag is a velocity square damping term given as
D - mcv2 (All)
where
c - cw(- d2 )(p/2) (A12)
4
cw - the dimensionless resistant coefficient
d - the diameter of projectile
and p - the air density.
2Shen, C. N., "Sensitivity Coefficient of Exterior Ballistics With Velocity
Square Damping," Transactions of the Twenty-Fifth Conference of ArmyMathematicians, Alto Report 80-1, January 1980, pp. 267-282.
A-2
Thus the principal equation of exterior ballistics (Eq. (A8)) becomes
d cv3-; (v com- (A13)
d g
A further transformation of the dependent variable in necessary by letting
u -v cos 6 (All4).
where u Is the horizontal component of the projectile velocity. Then the
dynamical Eqs. (A13), (A0), and (A10) become
du--- U3 sec3 6 (AMS
dO d- sec2 0 (Al6)*
a - -- BGC2 0tan 8 (A17)
To simplify further the form of the dynamical equations another transformation
of the independent variable is made by letting
dO sec2 0 l+q2 (Ali)
A-3
Pezme er tw aTr~etraecor
A-
APPENDIX B
VARIATION OF THE DRAG FUNCTIONS
As given by Eq. (54) in the previous paper2 the variation of the range
drag function 68X 1112 26Vo
qo-qi vo
Hqsq Ill ] 6 q j+ ---
l-Hq-q i qo-qi qo-qi
2qoi 12 I11 € V02 dpoI0 2 6qo
+ -2 ---- - ... - --1+qo qo-ql g 1+qo2 dqo qo-qi
It Is noted that the difference of the end slope is not zero, i.e., qo-qi # 0. "
Therefore, the problem does not become singular. We have expressed the
variation 6G. In terms of the variational parameters.
As given by Eq. (77) In the previous paper2 the variation of the
elevation drag function 8Gy Is
J12 26v oqo-qj vo
ql~q-qj 311 6qI
+ [ - . ...I'Rq,,qi qo-qi qo-ql
2qoJ 12 Ji1 c Vo2 dPoJ0 2 6qo
14% qo-qi g 14.qo dpo qo-qi
2Shen, C. N., "Sensitivity Coefficient of Exterior Ballistics With VelocitySquare Damping," Transactions of the Twenty-fifth Conference of ArmyMathematicians, ARO Report 80-1, January 1980, pp. 267-282.
I-1
I~ .. .. T
I I T
I: T II
I I T'
I . . . I.. . I
vI
, + i
: t'N
•j I
Fiur 2. ImatAgev.Luc nl o ifrn
Tage Slpe an apn ofiins
B-2,,
I I I I I T I I I I II . . . I I . I I I I I I I I I I I . I I I I . I . . I . I I I I I I I I I
5 1 1 1 1 1 T I 1 1 1 1
T I I I I I I I I I T
1 1 T I I I I I I I 1 1 1 1 1 a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Ej-r
I Ff 1 1 1 T-T
I T-FFI-T-Ei 1-1 rT
T-r-VIT-1-1 I ) I-T
I I I T I I I Li I :aF i -- T
T j I I I
I T
I A I I I- I I I
FT
I I I I I III A 1 1 1 1 1 1
. .........
I I .. I I IT I 1 T
1 , 11 1 ! : I t l l l i l II F T 1 1 1 7 1 1 1 1 1 1 1
i t - 1 11 1 [ - : I f i i l i ; j i l-
T I I I T-T
iiiiiij
ITIif
G- -1
1 1 j-
IF I f I
t I-F I I T-1 I
.711
w1l;1 ...a I I .I I I A4
Figure 3. Nondimensional Range vs. Launch Angle forHorizontal Target and Different DampingCoefficients.
11-3
[(
I
I II
I I1 i I T
I F i I: F j II I I T I I . .I i I Ti I I
U T an IffertI ing o e i
l I I i 1 T1
l i t J . .. .
A• I
;T1
SI I•F
,I I I ,
,~t I i~
Figure 4. Nondimnsional Range vs. Launch Angle forUpgrade Targets and Different Daming Coefficients. :
3 -4
tT
I T I I
I T
1 I I I I L
T ii I :
I i
Fiue .Sestiiy offcens Lanc Angl.fo
I MA r,:a.
Hoiota agen Difeen Daming
i ,
Coefficients.
B9; 3-5
Uprd Tagt an DifrI Dapn Coefficients.I I
I A
rt r I- 1.7-
Figure 6. Sensitivity Coefficient vs. Launch Angle forUpgrade Targets and Different Damping Coefficients.
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