UNCLASSIFIED AD 4 24008 - apps.dtic.milaspect of the analysis. The general features of TEA-Z and...
75
UNCLASSIFIED AD 4 24008 DEFENSE DOCUMENTATION CENTER FOR SCIENTIFIC AND TECHNICAL INFORMATION CAMERON STATION. ALEXANDRIA, VIRGINIA UNCLASSIFIED
Transcript of UNCLASSIFIED AD 4 24008 - apps.dtic.milaspect of the analysis. The general features of TEA-Z and...
UNCLASSIFIED
AD 4 24008
DEFENSE DOCUMENTATION CENTERFOR
SCIENTIFIC AND TECHNICAL INFORMATION
CAMERON STATION. ALEXANDRIA, VIRGINIA
UNCLASSIFIED
rTECHNICAL MEMORANDUM TM-1263
TEA-2
TRAJECTORY ERROR ANALYSISCOMPUTER PROGRAM
FOR
TWO-DEGREE OF FREEDOM TRAJECTORIES
BY
HAROLD J. KOPP
JOHN N. NIELSEN
OCTOBER 1963
PICATINNY ARSENALDOVER, NEW JERSEY
Technical Memorandum TM-1263
TEA--
Trajectory Error Analysis Computer Program
for Two-Degree of Freedom Trajectories
by
Harold J. KoppJohn N. Nielsen
October 1963
Reviewed by;3 EP TORChief, Computing and Analysis Section
Approved by: A, __
W. R. BENSONChief, Engineering Sciences Laboratory
FELTMAN RESEARCH LABORATORIESPICATINNY ARSENAL, DOVER, N.J.
TABLE OF CONTENTS
Page No.
ABSTRACT 2
INTRODUCTION 3
PROCEDURE 4
A. Mathematical Basis 4B. Computer Technique 13C. Computer Usage 23
1. Rocket Trajectory Input 24Rocket Trajectory Sample Input Case 28
Z. Ballistic Trajectory Input 34
Ballistic Trajectory Sample Input Case 35
RESULTS AND DISCUSSION 40
APPENDICES
I. TEA-2 FORTRAN Listing 45
I. Sample Problem Output 65
DISTRIBUTION LIST 70
L1
ABSTRACT
Estimates of the cumulative error associated with thethird-order Runge-Kutta solution of the two-degree of freedomequations of motion are presented. These estimates constitute asimple, but not rigorous, approach to automatic increment selection.Details of the computer program TEA-Z which utilize these equationsare described. Sample calculations are included. TEA-Z is betweenZ. 0 and 1. 7 times faster than the previous trajectory program inaddition to providing greater consistency and reliability.
2
INTRODUCTION
Numerical solutions of two-degree of freedom trajectoryproblems are approximations to the exact solutions. Intelligentand efficient employment of high-speed computers for solving thesetrajectory problems requires that the errors resulting from numeri-cal methods be controlled. This report presents the equations anddescribes the computer techniques used in TEA-Z, a computer pro-gram which calculates integration step length automatically. Theintegration step length is computed such that the estimated error inthe range and/or percentage error in altitude at impact comparedto maximum altitude lies within prescribed limits. This program,therefore, provides ammunition designers with the ability to cal-culate trajectories such that the cumulative numerical error at theimpact point is tolerable and of pre-specified magnitude.
The TEA-Z trajectory program requires the usual input ofdrag, thrust, and mass tables as well as initial conditions for velocity,angle, and the coordinate position. In addition, one must furnishallowable percentage errors for the range and/or maximum altitude.Details concerning input and output are included in Section C of theProcedure.
Equations for error estimation and automatic step selectionare presented in Section A. Program limitations and computer floware described in Section B. Appendix I contains a FORTRAN listingof the program. Lastly, sample calculations are presented inAppendix II.
3
PROCEDURE
A previous report, Technical Memorandum 1262, establisheda rigorous mathematical foundation for this work. Results fromthat report are utilized in conjunction with additional approximationsto provide a set of equations for TEA-2. Section A contains thisaspect of the analysis. The general features of TEA-Z and detailedinstructions for its application are presented in Sections B and C,respectively.
A. Mathematical Basis
Trajectory equations of motion for a point-mass, two-degreeof freedom system are:
X = V cos 0 X (t=0) = 0
Y V sin 0 Y (t=0) = Yo*_ T ~zv (/)
- - g sin 0 - JKDd V/M V (t=0) = Vo
= -g cos O/V 0 (t=O) = 0
where T(t), m(t), ffiy), KD(V, j ), g(y), and d are given. They
symbolize thrust, mass, air density, the drag coefficient, gravi-tational acceleration, and particle diameter, respectively. Thequantities V and 0 are the magnitude and orientation of the particlevelocity, respectively. Particle position coordinates are specifiedby x and y. Figure 1 illustrates these quantities.
4
YOv
yo
Figure 1The Coordinate System
An approximate solution to the set of equations (1) is obtainedby using a third-order Runge-Kutta method. The approximate solutionis given by:
Ue U, + ( Cu Cs (,2 C&0 0 (2)
where
K20 = ' ' ki , ,0. ti ) 4A v,)
C, h)
5
Coefficients corresponding to the Gill modification are used.
They are:
C1 = . 62653829
C2 = -. 55111241
C4 = -. 48268182
C 5 =. 85614329
Application of the Runge-Kutta solution method to equations (1)introduces truncation error which is dependent on the degree of ap-proximation and rounding error which follows from the limitationsof the digital computer. Errors at the i-th time step propagateforward in time and introduce "propagation error. "
The aforementioned errors were treated at length in TM-126Z.It was shown that a bound for the cumulative error can be obtainedfrom consideration of propagation, truncation, and rounding errors.Results from TM-1262 are combined with assumptions and approxi-mations in this section. The set of equations so derived provideestimates of the cumulative error (not rigorous error bounds), and isemployed in the TEA-2 program. The assumptions, limitations, andequations used in TEA-Z follow.
Assumptions:
1. The speed of the point-mass does not change by more than5% per time increment.
2. The orientation of the velocity vector does not change bymore than 2 degrees per time increment.
3. The difference between the values generated by a singlestep of length (h) and two steps of length is only due to truncationand rounding errors.
4. Quantities undergoing arithmetic operations and those instorage possess a relative error less than 10 - 8.
5. Sine and cosine routines used on the Picatinny ArsenalIBM 709 have a relative error less than 10-8.
6
Derivations contained in TM-lZ6Z established rigorouserror bounds. This report is not concerned with rigorous errorbounds, but with error estimates. Consequently, bounding valuesfor the variables used in TM-1262 will be replaced by local values.Moreover, estimates for truncation error based upon the differencein the variables for a single step of length (h) and two steps oflength (h/2) will be employed. Appropriately modified equationsfrom TM-1Z6Z follow. Estimates of cumulative errors at the i-thpoint which result from errors at the n-th point are:
IKI (6
ei erx Cit,>([ C 'L62"
C4i = 0()3 ,' ,h4-(..-,) c(o 4
C = 0G., ,i - , . , K- 0(4 -,.'
'C4 - 0(37 KaO(2-0 rA I(/"1FI
eC, rn) 5--ji
7
kc'cg, ,,j <'/ 7
and = 0(440 (Vr - '4 e7
0 0"d OCq 033 OCOeR(2)
= -,, Ow 7 "r52 OO7
a l s o , c S- ( / )
OC 3 4
, = .- o .- )
The propagation error coefficients are given by:
1C= Cs e , s = (/)
% = (.o - .0,14 c A .0O03 ) (c 2)
1.20o, ; 0.28 -.ooohz (,9.2)
= (0, .08)(J.0-.02C 0- .o04C 2 , .10 09.5)
= (0 .07 04 (09.4)0(5- .o/- /.Oc - .04c2 -. 0002 c (9.5),C 4= 60C /0;) (A.02 A .016 C) ,
,C4, = (C 0,.o) (1.,40 -.oa 02C- .0005 C2) 9 .7), .4 k20 -A .2J 0. 005 c 8
where c ,,204 2 V 2 (20)
8
Next one invokes the assumption that the difference betweenthe values resulting from a single step of length (h) and two steps oflength (h/2) is only due to truncation and rounding errors. Subtractionof the rounding error yields:
max< o R Y, 0> (21J.1)
Ay nMeX <Y(h) y2) -o,,hY 0 > 2 )
S= mayXK1()"e(,.:?)I- <1e, .4,o (21.1)
where the rounding error coefficients may be approximated by:
0( 1 . 9 xo -
-7
YR 1.X7O(VR = 12.9 +.2.0 /2.c25 xJ
-7.
0(&qe = 2.6 -Y ,O
It follows from equations (15), (23), (92), (93),. (94), and(95) of FRL-TM-1262 that
AI Z (23)
Z4 (24)
Thus,
O v (Ilr 3 I 41V +A) 41,i (25)7 Yh "
CKO- r (0(43 Al 20- a4 I" YAO HV ) -01411 (26.)
7 Vh'9
Also,
7 11h -4(27)AI~no
Oyr 64__ (271)7 yh 4
Time increments are selected such that the estimatedcumulative errors based upon local (that is, n-th step) propagation,truncation, and rounding errors satisfy specified allowable errorcriteria for the x and/or y variables at the impact point. That is,
and/or N - ey - (Z)
where the subscript f denotes the impact point and the quantities O,and ey are the allowable errors in the x and y variables at impact,respectively. Equations (7) and (8) require the number of stepsbetween the n-th and f-th points. Moreover, estimates of the range and/ormaximum altitude will be required since the allowable error is specifiedas a certain percentage of these quantities.
Equations (28) and/or (29) are solved by the method of doublefalse position with a convergence criterion of h -
where ii .O) Z> 0 ' and Ii (h2) 0, i =, 2. The starting value of theiteration h. .oO/Y followed by 3.5ho =h 2 if fi(h0 0 or by• Iha = h, if 1(Ao) - 0 . The maximum step size is . 035. V/g which
is consistent with assumption 2. Time increments less than. 000035V/gare not permitted.
The following formulas are employed to obtain the required
estimates:
(1-n) = /h (30)
where /) [J' Y/;, 2 .ym] 4,4, (3J)
or A , Stich ,, y(t,) ($1.,)
10
where tm is the starting point for the terminal ballistic phase of thetrajectory. The time tm will be specified in the thrust table.
The terminal ballistic phase (Phase IV) impact range andmaximum altitude are estimated from a linear drag model. They aregiven by:
x 6
SV'- 2 A4u)
Va= uP- .. 3 )
U, S n n -a)AwL,.., = Li. -(l3,3, .. / ,)
u.,= uL (z33, z
where
Yf 0o dfnes /
A ui - (0, +.9995)1 0 > U/,[,o,. -/) W-4 ")m1.9995)/ 1[20
Yn f11
The following approximation is made for the modeltrajectory:
e = [. 00234 lb/ft3] X 10"m
wherem=0 0 ft t_ y Z 58,000 ftm=l 58, 000 ft4 y z 100,000 ftm=2 100, 0010 ft . y e 160, 000 ftm=3 160,0 00 ft- y e 230,000 ftm=4 230, 000 ft y y ,e 300, 000 ftm=5 30, 000 ft y z 400, 000 ftm=6 400, 000 ft ! y Z 600, 000 ftm=7 600, 000 ft _y
Estimates of the dependent variables for an intermediateballistic phase (Phase II) from tj to tg follow:
= -( 9 5;17e~ A pk341 _)
0 - - (4)9= o.5. i 1,9
X 4' 0.6y, Cos&,(4-.)(6Y9, - . 0. 8 , 5o ,9 - 07(a), = y,* ~o.8y, ;,,o,:4-i,.) (37)
The velocity change for a thrust phase from th to tk has beencalculated from linear drag approximation. Estimates for thrust phasesare:
2 = (30)
12 (40o)
where
. 76 7' 1 - 1
Y2 (43)
- 6 C 0 (4 )/2 (44)
B. Computer Technique
A number of refinements have been incorporated in TEA-Z inorder to generate reasonable error estimates. These refinementsinclude:
1. Conservative range estimates.
2. Insuring that the truncation error coefficients do notdecrease by a factor of more than . 8 from the last time they werecomputed unless a phase change occurs.
3. Averaging the estimated error at the (n+N)-th step suchthat it is one-half the sum of estimates based upon local values at then-th and (n+N)-th steps provided that N steps have been taken since thelast step selection and that a phase change does not occur at n+N.
4. The acceptable percentage error is doubled if thecriteria cannot be satisfied.
13
The TEA-2 trajectory program may be utilized to computetwo-dimensional, point mass trajectories for one and two stagerockets and/or ballistic projectiles and consists of the followingphases:
Phase I Acceleration of Booster and Main StagePhase II Coasting of Main StagePhase III Acceleration of Main StagePhase IV Free-flight of Main Stage
Any of the above phases may be excluded in the computations;for ballistic trajectory computations, only Phase IV is required.
The following are program conditions:
1. Control systems are not considered2. A flat earth model is assumed3. Winds are not considered4. ARDC Standard Atmosphere, 19595. Constant thrust or thrust variable with time6. Thrust modification for atmospheric pressure changes7. Constant linear weight change or weight variable with
time for rocket boosted phases8. kD or CD drag coefficients may be used
9. Form factor to alter drag coefficient
Thrust values for the acceleration phases (I & III) are obtainedin the computations by linear interpolation of a thrust (lb) versustime (seconds) table when table values are presented; however, aconstant thrust force may be presented instead of tables.
A constant weight change (burning rate, ib/sec) or a table ofweight (lb) versus time (seconds) may be introduced into the computa-tions for rocket boosted missiles. Using the table, weight is obtainedin the program by linear interpolation. When tables are not presented,the weight is decreased during thrust phases by a factor of integrationincrement (seconds) multiplied by the burning rate for each step ofintegration.
14
Tables of drag coefficients (CD or kD) versus mach numbersmay be presented for each acceleration phase to be computed and forthe free-flight phase. If only Phase IV (i. e., ballistic trajectory)is to be computed, then only the drag coefficient table for free-flightis required.
A constant drag coefficient may be presented for the entiretrajectory. (See description for Card #1.)
A generalized and a detailed flow chart are presented in orderto depict the program logic. The corresponding FORTRAN listingcan be found in Appendix I. Reference should be made to the flowcharts prior to any modification due to the complexity of the branchingoperations in the body of the program. This program requires 7024words of core storage.
15
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C. Computer Usage
The procedures for preparing input data to trajectory programTEA-Z are discussed in this section, and examples of typical casesare presented and described.
Input data is punched on cards which are loaded with the programdeck onto tape. Data sheets which may be used for presenting requireddata for key-punching are illustrated in the descriptions of sampletrajectory cases to be computed. The computer program reads inputdata from Tape AZ which is the standard for IBM FORTRAN MonitorSystems.
Descriptions of input requirements will consist of two parts:(1) Rocket trajectories and, (Z) Ballistic trajectories.
23
1. Rocket Trajectory Input
Card #1 contains nine one-digit and three two-digit controlparameters from columns 1 through 15. Columns 16 to 20 are fora decimal input number. These parameters are to be presentedacross card #1 as follows:
Card #1 .... NEXT(l), NEXT(Z), NEXT(3), NEXT(4), KDCON,NTHRST, KD, M, NXY, JN, NR, NS, GAMMA
Format (911, 312, F5. 0)
where:
NEXT(1) .... equal to 0 for not computing Phase Iequal to 1 for computing Phase I
NEXT() .... equal to 0 for not computing Phase IIequal to 1 for computing Phase II
NEXT(3) .. equal to 0 for not computing Phase IIIequal to 1 for computing Phase III
NEXT(4) .... equal to 0 for not computing Phase IVequal to 1 for computing Phase IV
KDCON .... equal to 1 for variable drag coefficients(use tables)
equal to 2 for constant drag coefficisit
NTHRST .... equal to 1 for using thrust tablesequal to 2 for not using thrust tables
KD ........ equal to 1 for using kD coefficientsequal to 2 for using CD coefficients
M ......... equal to I for using weight tableequal to 2 for not using weight table
24
NXY ...... equal to 1 for step selection based on rangeerror criterion
equal to 2 for step selection based onmaximum altitude
equal to 3 for step selection based on totalrange and maximum altitude
JN ........ number of trajectories to be computed for thesame conditions of error limit, thrust anddrag coefficients (a two digit integer)
NJR ...... trajectory identification number for firsttrajectory to be computed ( a two digit integer)
NS ....... number of integration steps to be computedfor error estimations (a two digit integer- ifleft blank on card, set equal to ten in program)
GAMMA ... percent error limit for trajectory computations(if left blank on input card, GAMMA is set equalto 5. 00 in program)
Cards #2 to #7 contain the drag coefficient table for Phase I. Cards#2 to #4 consist of the table mach numbers, and cards #5 to #7 containthe corresponding drag coefficients.
Cards #8 to #13 contain drag coefficient tables for free-flight phases(Phases II and IV). Cards #8 to #i0 are for the table mach numbers, andcards #11 to #13 are for the corresponding drag coefficients.
Cards #14 to #19 contain the drag coefficient for Phase III. Cards#14 to #16 are for the table mach numbers, and cards #17 to #19 are forthe corresponding drag coefficients.
Either kD or CD coefficients may be used in the tables; however,values that are to be used must be specified by the control parameterKD in card #1. All of the coefficient tables may be eliminated as inputwhen in card #1 KDCON = 2.
25
Cards #20 to #25 contain the thrust table for Phase I. Thistable is not required when a constant thrust force is to be used; andfor this case, the control parameter NTHRST in card #1 must equal2, and a thrust value must be presented in card #46. Cards #20 to#22 are for the table time values, and cards #23 to #25 are for thecorresponding table thrust values.
Cards #26 to #31 contain the weight table for Phase I andmust be presented only if in card #1 M=l. Cards #26 to #28 are fortable time values, and cards #29 to #31 are the corresponding tablevalues of weight.
Cards #32 to #37 contain the thrust table for Phase III. Thesecards are not required when a constant thrust value is to be used; andfor this case, NTHRST must equal 2 in card #1 and a constant thrustvalue must be presented in card #46. Cards #32 to #34 are for tabletime values, and cards #35 to #37 are for the corresponding tablethrust values.
Cards #38 to #43 contain the weight table for Phase III and mustbe presented when constant burning rates are not used (in card #1 M=1).Cards #38 to #40 are for table time values, and cards #41 to #43 are thecorresponding table values of weight.
Data field widths for all input valves are nine columns, andeight values may be introduced on one card. Tables are required onlyfor phases that are to be computed. Each table that is required by theprogram will consist of six input cards; therefore, if data is not availablefor completing the maximum table requirements, blank cards must beused to supply the proper number of cards.
The last three cards required for input (cards #44, 45, and 46)will be listed along with input symbols, and these cards are requiredfor each rocket trajectory to be computed. Using the same errorcriteria, thrust, weight, and drag coefficient tables, "IJN" trajectoriesmay be computed by varying initial conditions, configuration, etc., incards #44, 45 and 46.
26
Card #44 ..... THDEGO, VO, WGT, DIA, XO, YO, TO, YF
where:
THDEGO... initial quadrant elevation, degreesVO ........ initial velocity, feet per secondWGT ...... total weight, poundsDIA ....... missile diameter, inchesXO ........ initial range, feetYO ...... initial altitude, feet
TO ........ time at start of trajectory, secondsYF ........ terminal altitude, feet
Card #45 ..... THRST, DOTM, AREA, BWGT, THRZ, DOTMZ,AREAZ, PRESO
WHERE:
THRST ..... booster thrust, poundsDOTM ..... booster burning rate, lb per secondAREA ..... nozzle throat area of booster, sq. in.BWGT ..... booster weight (w. o. propellant), pounds
THRZ ...... main stage thrust, poundsDOTMZ .... main stage burning rate, lb per secondAREAZ .... nozzle throat area of main stage, sq. in.PRESO .... static test atmospheric pressure
AREA, AREAZ and PRESO are to be presented for input onlywhen the thrust is to be modified in the computations for atmosphericpressure changes. For this correction not to be performed, AREAand AREA2 must be made equal to zero or left blank on the inputcard.
27
Card #46 ........ TIM(l), TIM(2), TIM(3), DWRT(I), DWRT(2),
DWRT(4), CKD, FACT
where:
TIM() ....... end of Phase I, secondsTIM(Z) .. ..... end of Phase II, seconds
TIM(3) ....... end of Phase III, secondsDWRT(l) ..... output spacing for Phases I and III, secondsDWRT(2) ..... output spacing for Phase II, seconds
DWRT(4) ..... output spacing for Phase IV, secondsCKD ......... constant drag coefficientFACT ........ form factor (if left blank on input card,
form factor is set equal to one)
ALL DATA PRESENTED ON CARDS #2 TO #46 MUST HAVE
DECIMAL POINTS AND HAVE FIELD WIDTHS OF NINE COLUMNS.
Rocket Trajectory Sample Input Case
The rocket sample case is for a one stage rocket which wouldrequire computing Phases I and IV of the program.
The following conditions represent the trajectory input requirements:
1. CD drag coefficient tables2. constant burning rate (5 lbs/sec)3. constant thrust (100 lbs)4. step selection based on range error criteria5. 10 integration steps per estimate
6. quadrant elevator, = 45. 07. initial velocity = 1000.0 fps
8. weight = 100. 0 lbs9. diameter = 11.0 inches
10. burning time = 4.0 seconds11. output spacing every At12. form factor = 1.013. 5.0% error limit
Z8
For the above conditions, the following data must be presentedin card #1:
NEXT(l) = 1, compute Phase INEXT(2) = 0, do not compute Phase IINEXT(3) = 0, do not compute Phase IIINEXT(4) = 1, compute Phase IVKDCON = 1, variable drag coefficients presentedNTHRST = 2, constant thrustKD = Z, using CD drag coefficients
M = 2, not presenting weight tables(constant burning rate)
NXY = 1, step selection based on range errorcriteria
JN = 01, compute one trajectory
NJR = 01, trajectory numbered 1NS (blank) , ten integration steps per estimateGAMMA (blank) , 5. 0% acceptable error
An illustration of the presentation of data on card #1 may be seenon page 33.
The drag coefficient tables for Phase I and Phase IV as presentedfor key-punching are shown on the table data sheet on page 32. Thesetables are punched onto cards #2 to #13, and the punched cards areillustrated on page 33.
Cards #14 to #19 are not required for input as Phase III is notto be computed.
Cards #20 to #43 are not required for input as thrust and weighttables are not to be presented.
The following data must be punched on card #44:
THDEGO (45. 0), quadrant elevation, degreesVO (1000. 0), initial velocity, fpsWGT (100. 0), missile weight, lbsDIA (11. 0), missile diameter, inches
Values for XO, YO, TO, and YF do not have to be punched onthe card for this case as they equal zero.
29
I
The following data must be punched on card #45:
THRST (100. 0), constant thrust force, lbsDOTM (5. 0), burning rate, lbs/sec
Values for AREA, BWGT, THRZ, DOTMZ, AREA2, and PRESOdo not have to be punched on the card as this data is not required forthe given rocket.
In card #46, only the following need be punched:
TIM(l) (4.0), end of Phase I, seconds
TIM(2) and TIM(3) are not required as Phases II and III are notbeing computed.
DWRT(1), DWRT(Z), and DWRT(4) are left blank as output is tobe printed for every integration increment, at.
CKD is left blank as table values of drag coefficients are presented.
FACT is left blank as a form factor equal to 1. 0 is required;however, a value of 1. 0 may be punched for FACT.
Data presented for key-punching onto cards #44, #45 and #46may be seen on the input sheet on page 3 1. An illustration of this dataon cards may be seen on page 33.
30
kL
INPUT FOR TRAJECTORY PROGRAM TEA-2
For Ballistic and Rocket Trajectory
QUADRANT ELEVATION, degrees 45.INITIAL VELOCITY, ft/sec /0 ,TOTAL WEIGHT, lbsDIAMETER, inchesINITIAL RANGE, ftINITIAL ALTITUDE, ftINITIAL TIME, secTERMINAL ALTITUDE, ft
For Rocket Trajector,
BOOSTER THRUST, lbs /04.BOOSTER BURNING RATE, lb/sec ..BOOSTER NOZZLE AREA, sq. inchesBOOSTER WEIGHT (EMPTY), lbsMAIN STAGE THRUST, lbsMAIN STAGE BURNING RATE, lbs/sec .MAIN STAGE NOZZLE AREA, sq. inches __M4
STATIC TEST ATMOS. PRESSURE, lb sec "ft 4
For Rocket Trajectory
TIME AT END OF PHASE I, sec 4.oTIME AT END OF PHASE II, secTIME AT END OF PHASE III, secOUTPUT SPACING FOR PHASES I & III, secOUTPUT SPACING FOR PHASE II. secOUTPUT SPACING FOR PHASE IV, secCONSTANT DRAG COEFFICIENTFORM FACTOR
For Ballistic Trajectory
OUTPUT SPACING, secFORM FACTORCONSTANT DRAG COEFFICIENT
31
Table Data Sheet for TEA- - Trajectory Program
Name Date .
Argument 0.0 0.6 0.9 A0. .1 /.2 ASd7 c1. /.4 .5 . 6 /.. 20 .
5.0 _
F .nrction .3 ./3 .14 .J75 .255 .277 .26 Z .275
CO ..26 2-1 ..2, _. .,'6 ./6/ •L .pase . .132
III- -.- -
Argument 0.0 o 0 .q 0 - -
,-A no. 1.4 1.5 .8 .0 .. 5 3.o 1-8
Function .5 .13 .14 .175 .255 .R77 .-?" ..?75
L(Ah- - -- -W
Argument
Function.
Argument I
Function -,
~32
ROCKET TRAJECTORY INPUT
14__U__I_____11 *71FiiZi5i~iTT~ 4 15365 b35 S.7 $93160161S2 635616 61351611- 76 7179 F
I'~ 4T i F 85 - Tii ' 3222272 3033 3243 5375 55 434 79 m
1.4.01.1.1'.5.1.1
1 34 56T71 111 1823141516117 IN291-2222272280'23 31 4444444614 455152 5A4 M53 1 156016162 6316406 7U 6138 727- W..13 .13 .14 ITS5 .U . 1 . i
714 1]i :j-j 1 1 4 -,' 2 37 9 303 333313435 -38 340141 4 43144 4 -141 49352 54- 5 16 161 62 63-4 6 1 7
.1510
9!1815 to ' 2633 3 304443444 41 4850151 52 AR 55.5385038536j 465 6838a386 131
I I 9 1.0 13. I 1-~~3
8 1 7 7 1 181 9 1 I ? I I I -
6-34 1 . 5 374555657 15- 221121 6127 2 .- 532 3al3435 743736-a4T424344456-4"15 52 5453 57585 I234511 813 -
*41 :1 1-1D88;65-0 .83825 01 163 123 A-5 U11
I 13, .. 14
- I I V I 2 1 111 1 2 1 1 0I i I I
1-2-3rr -7 6 5 1-5]5:'?1 111 1Ik 5-1 1 Z222314 4'5' 2. p 332 334 35 - 31 3% 414Q431444545 5 55 w45 1 862 643 605 m138 7
:22-322 58-8 348556*58?7 222232431178 J3435 37 36 A 45442 44 6 aa 58 523 6 51156 363 d644 6163 8 L7
-3145 Oi 11011 121314 15 1617581 "6122 261 2 3532 435 37 13 8 4 1 649 8545 474 0 1Wj 5 j571M A 162436645 U 8
so ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . ..~l'"* 7 S 5 27 5 '5 7 ' *7 08 588 71745 0 4.. .. 5 r4 S . so 33 4 m so
1 5- 9 # 8 9 12 -521 14 13 A I 60 9220 2, 12 23 26 n 29 2? 26f 34 I 31 2 55 34 39 M 30 2 9 4044 42 43000
4444444444444444444444444444144444444444444444444444444444444444444444444 444
55s555555i55~55555555s555555566155 1 1 555555555
6 666 66 6 6j 66 6 6 6 616 6 6 S6 66 6 6 j66 616 6 6, 6 6 6 6616 66 6,6661661 6 SGi6 66 l6 66 6 6ii66 5;j.6S66666 6fOT TWO RT~F FOUR FlV E[ six I SEE I EGT
99 9~ 99 9919 99 99 99 9 999 9 99 99 9 9 9 9 999 99 99999999 9j 9 19.88 1 3 1 i I 888I125384855555111212 2124521;; 78 .!X5 32313435 36814742414445 144149;15 3 ,555313163423646586838385t7 1172831451 3558
S200 'F.6:[9 A4328
33
T
2. Ballistic Trajectory Input
Card #1 contains the same values as in card #1 described forrocket trajectories on page 24.
Card #2 to #7 contain the drag coefficient table. Card #Z to #4are for the table mach numbers, and cards #5 to #7 are for the cor-responding drag coefficients. This table is not required whenKDCON = 2 in card #1. For this case, the constant drag coefficientmust be presented in card #9.
Either kD or CD drag coefficients may be used, and valuesemployed must be specified by the control parameter KD in card #1.
Card #8 ...... THDEGO, VO, WGT, DIA, XO, YO, TO, YF
where:
THDEGO ...... initial quadrant elevation, degreesVO ........... initial velocity, feet per secondWGT ......... total weight, poundsDIA .......... missile diameter, inchesXO ........... initial range, feetYO ........... initial altitude, feetTO ........... time at start of trajectory, secondsYF ........... terminal altitude, feet
Card #9 ..... DWRT(4), FACT, CKD
where:
DWRT(4) ..... output spacing, secondsFACT ........ form factorCKD ......... constant drag coefficient
ALL DATA PRESENTED ON CARDS #2 TO #9 MUST HAVE
DECIMAL POINTS AND HAVE FIELD WIDTHS OF NINE COLUMNS.
34
V
Ballistic Trajectory Sample Input Case
The following conditions are for the sample ballistic trajectory:
I. CD drag coefficient table2. initial velocity = 1000. fps3. quadrant elevation = 45.004. weight = 100.0 lbs5. diameter = 11. 0 inches6. output spacing every & t7. form factor = 1. 08. ten integration steps per estimate9. 0. 5% error limit
For the above conditions, the following data must be presentedin card #1:
NEXT(l) = 0, do not compute Phase INEXT(2) = 0, do not compute Phase IINEXT(3) = 0, do not compute Phase IIINEXT(4) = 1, compute Phase IVKDCON = 1, variable drag coefficients presentedNTHRST = 2, no thrustKD = 2, using CD drag coefficientsM = 2, not presenting weight tablesNXY = 2, step selection based on altitude error
criteriaJN = 01, compute one trajectoryNJR = 01, trajectory numbered 1NS = 10, integration steps per estimateGAMMA = 0. 5, percent acceptable error
An illustration of the presentation of data on card #1 may beseen on page 37.
The drag coefficient table for the ballistic trajectory as presentedfor key-punching is shown on the table data sheet on page 38 . Thistable is punched onto cards #2 to #7, and the punched cards are illustratedon page 39.
35
The following data must be punched on card #8:
THDEGO (45. 0), quadrant elevation, degreesVO (1000. 0), initial velocity, fpsWGT (100.0), missile weight, lbsDIA (11. 0), missile diameter, inches
Values of XO, YO, TO, and YF do not have to be punched onthe card for this case as they equal zero.
Card #9 may be left blank. No value is required for CKD as atable of drag coefficients is to be presented, No value is requiredfor DWRT(4) as output is required every integration step, At. Novalue is required for FACT as the form factor must equal one.
Data presented for key-punching onto cards #8 and #9 may beseen on the input sheet on page 37 . An illustration of this data oncards may be seen on page 38
36
INPUT FOR TRAJECTORY PROGRAM TEA-Z
For Ballistic and Rocket Trajectory
QUADRANT ELEVATION, degrees -INITIAL VELOCITY, ft/ secTOTAL WEIGHT, lbsDIAMETER, inches /1,INITIAL RANGE, ftINITIAL ALTITUDE, ftINITIAL TIME, secTERMINAL ALTITUDE, ft
For Rocket Trajectory
BOOSTER THRUST, lbsBOOSTER BURNING RATE, lb/secBOOSTER NOZZLE AREA, sq. inchesBOOSTER WEIGHT (EMPTY), lbsMAIN STAGE THRUST, lbsMAIN STAGE BURNING RATE, lbs/secMAIN STAGE NOZZLE AREA, sq. inchesSTATIC TEST ATMOS. PRESSURE, lb sec 2ft 4
For Rocket Trajectory
TIME AT END OF PHASE I, secTIME AT END OF PHASE II, secTIME AT END OF PHASE III, secOUTPUT SPACING FOR PHASES I & III, secOUTPUT SPACING FOR PHASE I, secOUTPUT SPACING FOR PHASE IV, secCONSTANT DRAG COEFFICIENTFORM FACTOR
For Ballistic Trajectory
OUTPUT SPACING, sec .. ....FORM FACTORCONSTANT DRAG COEFFICIENT
37
Table Data Sheet for TEA-2 Trajectory Program
Name Date
Argument . .. 1.8 .. / /.Z. /3
5.0 7.0 90_ o _ ?. 0 50.
Function *j3 ./3 .14 .5 .. ?-45 .277 .. 2c4 .275
,/ . .Z.Z.-- 2 ./,o .. '0
Argument
Function
Argument
Function
Argument
Function.
Argument -
Function
38
BALLISTIC TRAJECTORY INPUT
ool, ~ZZP,Gi,, oP.S c o81 3 J01 1 12 0114 14 F, 191200 22 22364 53 221 29 3 2 jjj~3 J
4 ~ 34~4 4 d!36 L2 U_)5 Q6 4 w12 68107 11 9 IS
1.4 5. 617I.
11.4 1 1 1----.--3 .4
.0 0P. It.o IS .A .... 00 1A --- •4 '1.( 1 a - - I 1* 0 I I_
I 31J I jis . A1 M . 19 II .Z441,H 1 FS .,54S! ,, ,.-As .,~l H10o i 27 :m-is 1i , I ____. _.-___,
I fI LS 2 f) t l l I I I
n - - -!-!?L1~9 L p
.. - To s. I_ I) 1* 1s 1 1t 77) , *1 *
___22__31_4_2_____________________4_____1,_+11F =.. 6 pa 1
I 3 3 5 1 8 ) I 42 711I -71 ? , 5i 3 a ?7 % 11 * t Da 1 3 t 9 a 39 -g Wn , 4 4,, 84 4It 4
111111 I111 IIf 1 If111111 1111 .6 1 11111111 Soi 1. 1111 "tlllill2 J-
3 3 33 33 33 33 33 33 33 33 3 333 33 33 33 33 33 3 133 33 33 3333 3 3 333 33 3 33 33 33 33 33 33 3 333 3 33 3 ; 33 3333
55555555555555555 555555 5 55555555 5 '555555555155 ~ 555555 ~55 555555I666I6 6 6~~ 661 . I 1 6 616666 6 6 1G16 6666 8Gj 6 6
14 YW0 1 140 OU R- iEsl S16 '16 EIGHT" 1 17 ' 7" 7 11 11/l"' "7 1 1117777711 l7 71 7 7 7 1 T" i I I IW77 1 7171 7 1i7 7 7 7 777711 7
88 888 88 a81888a88888888888888888 88888888888a 888a8888 18
9 9 9 99 19 9 f 9 9 9 9 9 9 J 99 9j999 19 9 9 91 999 9 9 99 9 999 99 9 99 9 9 99 9 9 9 9 99 99 919, 7 )194 5 1 I Iile1 121,1 14 1, 16I1 11 .: 7, ? 4'.' t 3 SB !) 3?7 3'W JA~3 3,2 3 I),1 41 42 4344 45 874148 47, 'J 52 51 ,4 S £ '.iS£ um U b)U 65 3,, ... o: 11137 I Ia ? 16 1' ?1 F9.110
39
RESULTS AND DISCUSSION
The TEA-Z digital computer program for the IBM 709 hasbeen tested on a substantial number of trajectories. This programhas always selected reasonable time steps. Time steps for TEA-Zare dependent upon the accuracy specified by the user as well asthe problem characteristics. That is, problem initial conditions,body drag characteristics, and vehicle thrust characteristics areimportant parameters and exert a significant influence on the timeincrements used by TEA-Z.
Analysis of output for typical ballistic flights has shown thatthe estimated range error is somewhat more than 10 times too con-servative. That is, an estimated range error of 1% by TEA-Z wouldcorrespond to an actual range error less than 0. 1%o.
Four typical trajectories were computed in order to establishtiming estimates. The program limitation to a maximum angularchange per time step of two degrees places an upper limit on compu-tational speed. This limitation explains the small difference incomputation time between the 5% and the 0. 576 ballistic trajectoriespresented in Table 1.
The first ballistic trajectory corresponds to the sample input,and sample output presented in Section C and Appendix II, re-spectively. A similar trajectory and the-time steps required for lessthan 0. 5%6 range error are presented in Figure 2. Actual range erroris probably much less than 0. 5%o in this case. Figure 2 illustratesthat maximum time steps were taken from 10.. 483 seconds to 33. 837seconds.
Figure 3 illustrates a typical rocket trajectory which issimilar to the ballistic case. Initial step size is much smaller inthis case. The average step size is 250 less than for the ballisticcase.
40
Table 1 lists two typical ballistic trajectories and two typicalrocket trajectories. Both 5% and 0. 5% range accuracies were re-quested for these trajectories. These problems were run both onTEA-Z and on the "standard" two-degree of freedom program whichutilizes a fixed time increment which is an input quantity. Thisinput time increment is based on the user's experience. The valuesreported in the table are typical of those presently used. Under theassumption that the four cases presented with flight history con-stitute the "average" problems, the use of TEA-2 constitutes a timesavings of 2. 1 and 1. 7 for the 5% and 0. 5% accuracy specifications,respectively. This will result in a cost reduction of 35, 900 dollarsper year or 28, 500 dollars per year for the respective requiredaccuracies. Cost savings are based on current usage of 19 hoursper month at a cost of 300 dollars per hour. The 5% requested ac-curacy should be the most widely used case.
Computation time is approximately 2. 5 to 3. 5 times fasterthan the "standard" program for the usual ballistic and rockettrajectories. However, TEA-Z speed is restricted by output timewhich results in the computation times reflected in Table 1. Compu-tation times including terminal output and error table output only arealso included in Table 1. Clearly, detailed flight history should beprinted out only when absolutely necessary.
Very high drag bodies such as parachutes cannot be treated bythis program.
41
0 00
-44
0 N 0 N
&4 ~L 'f , N 0 -fS- - - N.
o0
6-4
- -- 4
H 01
[-4)
04 04
4 coNN cc ~ 1
tc - N Nr 0
"-4-4
00
NDN
%0-N
(d~ 0 0
1 42
(AX C-0 1 X 980'i)1;v
co
- 0
-4E
-, U-4.
/ 0
/ ~z1
I- 4
UU
w 44
43.
(A X0 f Xo 990 *I)/IV
C%
LA
-Ve
-En
-VZ/0
/u
N OiI 0
Z U<,
-
a) .4444
APPENDIX I
FORTRAN LISTING
THE TEA-2 SOURCE PROGRAM IS WRITTEN IN FORTRAN LANGUAGE.IT REQUIRES SINE, COSINE, SQUARE ROOT, AND EXPONENTIAL LIBRARYSUBROUTINES* ALSO REQUIRED ARE THE ATMOSPHERE (ATMOS) ANDTABLE SEARCH (TABLE) SUB-PROGRAMS WHICH ARE LISTED AFTER THEMAIN PROGRAM. A FORTRAN LISTING OF THE MAIN PROGRAM FOLLOWS.
CIEA-2 TWO-DIMENSIONAL9 TWO-STAGE ROCKET OR BALLISTIC TRAJECTORYC (WITH 1959 ARDC STANDARD ATMOSPHERE)C COMPUTING AND ANALYSIS SECTIONCC TRAJECTORY ERROR ANALYSIS *. TWO DEGREES OF FREEDOMC ANALYSIS ** LT. H.J. KOPPPH.D * EXTENSION 72264C PROGRAMMING ** MR. J.N. NIELSEN * EXTENSION 73230CC THIS PROGRAM SELECTS THE INTEGRATION STEP LENGTH SUCH THAT THEC ERROR IN THE TOTAL RANGE IS LESS THAN A PRESCRIBED AMOUNT AND/OIRC THE ERROR IN ALTITUDE IS LESS THAN A PRESCRIBED AMOUNT.C
DIMENSION TABLI(24,2),TABL2(24,2)tTABL3(24,2)TRSTI(292),I TRST2(24,2)tTMAS(2492),TMAS2(2492)tNEXTi5)tTIMI3),2 DWRT(4)tXD'OT(3)tY'D'OT(3)tVDOT(3)tTHOOT(3)DIMENSION TN(200)tXN(200)tYN(2CO)tTHN(200)tVN(200)tEX(Z00)I EY(200),ET(200)tEV(200)COMMON RHOPRSBGRAVCMACHKDCONNATMOS
CC FORMAT STATEMENTSC
100 FORMAT (91193129F5.0)101 FORMAT (8F9.0)104 FORMAT (7FlO.0)105 FORMAT (8F5.0)106 FORMAT (IHI,43Xt3lH COMPUTING AND ANALYSIS SECTION/27Xt66H TWO DIM
IENSIONAL9 TWO-STAGE ROCKET OR BALLISTIC TRAJECTORY PROGRAM/41Xt37H2 *WITH 1959 AROC STANDARD ATMOSPHERE*//)
108 FORMAT (8HI TIME,7X,6H RANGE,6Xt9H ALTITUDEt4X,9H VELOCITY,4X,I 6H THETA5X95H MACHt5X53H KD,7X,7 H THRUST,6X,7H WEIGHTSXP2 8H DENSITY)
109 FORMAT (8H TIMETX,6H RAN'GEt6X,9H ALTITUDEt4X99H VELOCITY,4Xt1 6H THETAt5Xt5H MACH,5X,3H KD,7Xt7H THRUST96Xt7H WEIGHTtSX#2 8H DENSITY)
110 FORMAT (14XtBH X-DERIV5X8H Y-CERIVSX,8H V-DERIVt3Xt9H TH-DERIV,I 38XSH DRAGt6X,9H PRESSURE//)
Il FORMAT (F9.3tZF14.3,Fl2.3tFI1.3,FIC.4,F9.4,FL4.3,F11.3tE5So)112 FORMAT (9X,2FL4.3,FI2.3,FLI.3,33XtFl1.3,E15.5)113 FORMAT (46X920H END OF TRAJECTORY,13)115 FORMAT (lH1946X,26H COASTING OF PAIN STAGE)114 FORMAT (3BX,44H ACCELERATION OF BOOSTER AND MAIN STAGE)116 FORMAT (IHL,44X,30H ACCELERATION OF MAIN STAGE)117 FORMAT (IHI,45X,29H FREE-FLIGHT OF MAIN STAGE)118 FORMAT (lHI,36X,59H ACCELERATION OF BOOSTER AND MAIN STAGE 0
IFF LAUNCHER)121 FORMAT (4TXt26H COASTING OF MAIN STAGE)122 FORMAT (45X,3OH ACCELERATION OF MAIN STAGE)123.FORMAT (46X,29H FREE-FLIGHT OF MAIN STAGE)
45
126 FORMAT (IH1.56X96H TEA-2/34Xt52H TRAJECTORY ERROR ANALYSIS ** TWOIDEGREES OF FREEDOM//)
127 FORMAT (5X,71H ANALYSIS ** DR. H.J. KOPP * EXTENSION 72264 * REPIORT TO BE PUBLISHiED/5X,50H PROGRAMMING ** MR. J.N. NIELSEN * EXTEN2SI[N 73210//)
128 FORMAT (8X,103H THIS PROGRAM SELECTS THE INTEGRATION STEP LENGTH SIUCH THAT THE ERROR IN THE TOTAL RANGE IS LESS THAN A/5X9111H PRESC2RIBED AMOUNT AND/OR THE ERROR IN ALTITUDE IS LESS THAN A PRESCRIBE3D AMOUNT. THE ERROR CRITERION USED FOR/5X923H THIS TRAJECTORY WAS4 **)
129 FORMAT (8X,85H THE ABOVE CONDITION WAS S4TISFIED, AND THE COMPUTED1 ERROR IN TOTAL RANGE IS EQUAL T0tE15.8,9H PERCENT.//)
130 FORMAT (8X,93H THE ABOVE CONDITION C0ULD NOT BE SATISFIED, AS THEICOMPUTFD ERROR IN TOTAL RANGE IS EQUAL TO/6XtE15.8,9H PERCENT.//)
131 FORMAT (28X,39H THE ERROR IN TOTAL RANGE IS LESS THAN ,F6.2,I 9H PERCENT.//)
132 FORMAT (28X#45H THE ERRBR IN TERMINAL ALTITUDE IS LESS THAN ,F6.291 29H PERCENT OF MAXIMUM ALTITUDE.//)
133 FORMAT (50X,22H FINAL ERROR ESTIMATES/57Xt6H *****)134 FORMAT (5Xv7H RANGE ,E15.8914H RANGE .ERRR ,E15.8911H ALTITUDE ,
I E15.8,17H ALTITUDE ERROR ,E15.8/5Xt7H THETA vE15.8,2 14H THETA ERROR tE15.8tI1H VELOCITY 9E15.8,3 17H VELOCITY ERROR ,E15.8//)
135 FORMAT (51X,19H INITIAL CONDITIONS/57X,6H *****)136 FORMAT (8X,7H RANGE EI5.893X9lCH ALTITUDE tE15.893X,7H THETA ,
1 E15.8,3XtIOH VELOCITY oE15.8//)137 FORMAT (47X,25H TABLE OF ERROR ESTIMATES/56X96H *****)138 FORMAT (1HI,40X,37H TABLE OF ERROR ESTIMATES (CONTINUEC)/56Xt
I 6H *0*o.)139 FORMAT (11X,5H TIME,17Xt6H RANGE,16X,9H ALTITUDE,15X,6H THETAt16X#
1 9H VELOCITY/30Xt12H RANGE ERRBR,IOX,15H ALTITUDE ERRFRt9X,2 12H THETA ERRUR,IOX,15H VELOCITY ERROR//)
140 FORMAT (6XtEl5.8,4(8XElS.8)/21X,4(8XEI5.8)//)141 FORMAT (5X,24H IMPACT BETWEEN RANGE OF,E15.8,9H FEET ANCtEIS.8,
I 5H FEET//5X936H IMPACT CENTER PREDICTED AT RANGE 0FtEIS.892 21H FEET AND ALTITUDE 0F,E15.8,5H FEET)
142 FORMAT (8X,90H THE ABOVE CONDITION COULD NOT BE SATISFIED, AS THE1CUMPUTED ERROR IN ALTITUDE IS EQUAL TO/6X,ELS.8,29H PERCENT OF PAX21MUM ALTITUDE.//)
143 FORMAT (8X#91H THE ABOVE CONDITIONS COULD NOT BE SATISFIED. THE COIMPUTED ERROR IN TOTAL RANGE IS EQUAL TO/3XvE15.8t56H PERCENT, AND2THE COMPUTED ERROR IN ALTITUDE IS EQUAL TOE15.8,29H PERCENT OF MA3XIMUM ALTITUDE.//)
144 FORMAT (8X,82H THE ABOVE CONDITION WAS SATISFIED, AND THE COMPLTEC1 ERROR IN ALTITUDE IS EQUAL TO,E15.8,11H PERCENT OF/5X,17H MAXIPUM2 ALTITUDE//)
145 FORMAT (8X,83H THE ABOVE CONDITIONS WERE SATISFIED. THE COMPUTED EIRROR IN TOTAL RANGE IS EQUAL T0,FI5.8,9H PERCENT,/4Xt48H AND THE2CIMPUTED ERROR IN ALTITUDE IS EQUAL TO,E15.8,29h PERCENT OF PAXIMU3M ALTITUDE.//)1 READ INPUT TAPE 2,100, NEXT(I),NEXT(2),NEXT(3),NEXT(4)tKDCNtI NTHRSTKCMNXYJfINJR,NSGAMMA
CGO TO (70,T4)tKDCON
CC REAC STATEMENTSC
70 IF (NEXT(l)) 727297171 READ INPUT TAPE 2t101, TABLI72 READ INPUT TAPE 2,101, TABL2
46
IF (NEXT(3)) 14,74,7373 READ INPUT TAPE 2,101, TABL374 GO TO (75,2),NTHRST75 IF (NEXT(1)) 77t77,404
404 READ INPUT TAPE 2,101, TRSTIGO TO (76,7hM
76 READ INPUT TAPE 2,101, TMAS77 IF (NEXT(3)) 2,2,7878 READ INPUT TAPE 2,101, TRST2
GO TO (79,2),M79 READ INPUT TAPE 2,101, TMAS22 READ INPUT TAPE 291019 THDEGOvVO9WGTtDIA,XOYOvT9,YF
IF (NEXT(l)) 405,405,406405 IF (NEXT(3)) 407,407,406406 READ INPUT TAPE 2,101, THRSTDOTMA REABWGTTrR2,DSTM2,AREA2,PRESE
READ INPUT TAPE 2,101, TIMtDWRT(l),DWRT(2)tDWRT(4)tCKOFACTGO TO 408
407 READ INPUT TAPE 2,101, DWRT(4),FACTrCKO405 WRITE OUTPUT TAPE 3,106
CC CONVERSION OF INITIAL VALLES TO CORRECT UNITSC
T = TOx = XOV = YO
V = VOTHET = THDEGO/57.29577THETO -THETRWGT = WGTRMASS = WGT/32.17405RMASSO RMASSDOTM =DOTM/32.17405DOTM2 = DOTM2/32.17405BMASS = BWGT/32.17405CMACH = 0.0DIA = DIA/12.0DIASQ =DIAwDIAAREA =AREA/144.
PRESO = 144.0*PRESODWRT(3) = DURTMlIF (NS) 1001,100191002
1001 NS = 101002 IF (GAMMA) 1003,1003,10041003 GAMMA = 5.01004 IF (FACT) 1005,10,05,10061005 FACT = 1.01006 GO TO (5092,5090),NTHRST5090 TRST1(191) = TO
TRSTI(1,2) = THRSTTRST2(ltl) = TIMM2TRST2(1,2) = rHR2OELTHR -0.1.TIM(l)DELTRS - 0.1.(TIM(3)-TIM(21)DO 5091 1 2,24TRSTI(fi,) =TRSTI(I-1,1) + DELTHRTRST1(lt2) =THRST
TRST2(191) aTRST2(1-I11 4 DELTRSTRST2(192) THR2
5091 CONTINUEC
47
C INITIAL VALUES FOR ERROR ANALYSIS RBUTINEC5092 AXR = 1.9E-7
AYR = 1.9E-7ATR = 2.6E-7AVTL - 0.0
ATTL a 0.0AXTL a 0.0AYTL = 0.0EXLN = 0.0EYLN = 0.0EVLN = 0.0ETLN = 0.0EXNN = 0.0EYNN = 0.0EVNN = 0.0ETNN = 0.0EVN = 0.0ETN = 0.0EXN = 0.0EYN = 0.0YOU = 0.0OtFY = 0.0DTBNE = 0.0TIME = TIM(M)NEG = 0NPSS = 0NA = 0NB = 0NTERM = 1NEST = 1NN = 2CNN = NNNO = I
NSTEP = 1YMAX = YIF (THET) 59919599105992
5991 JMP - 2GO TO 5993
5992 JMP I 15993 JUMP 1
GAM2 = .01GAMMAKPHASE - I
OELT = V/32174.050TH DELTEPS = 1.0KU = IKUK =ASSIGN 7007 TO KAPASSIGN 42 TO NWRTGO TO (6001,6002t6001tNXY
6001 NNXY a 1GO TO 6003
6002 NNXY - 26003 ASSIGN 602 TO NERR
CGB TO (80t8l),KD
80 CF - FACTGO TO 82
81 CF = 0.39269908.FACT
48
82 J I1NNN - 1NATMOS I
88 N = 1LINE * 2ASSIGN 64 TO NCNT
CC INTEGRATION CONSTANTSC
Cl = 0.62653829 * DELTC2 = -0.55111241 * DELTC3 = 1.4072559 * DELTC4 = -0.48268182 * DELTEPSLON = 0.001 * DELTWRIT = T-EPSLON
CIF (NEXT(N))409409,410
409 NNN x 2
GO TO 50410 WRITE OUTPUT TAPE 3,11461 WRITE OUTPUT TAPE 3t109
WRITE OUTPUT TAPE 3,110C
3 O 45 K = 1,3GO TI NCNTt(62t64)
62 WRITE OUTPUT TAPE 3,108WRITE OUTPUT TAPE 3,110ASSIGN 64 TO NCNTCONTINUE
C64 CALL ATMOS(Y,V)
GO TI (I0,29),KDCON10 GO TI (17,25,2t,25),N
CC TABLE 18K-UPS FOR DRAG COEFFICIENTCC DURING FIRST BURNING STAGEC
17 CALL TABLE(TABL1,CMACHCK~t24)GO TI 29
CC DURING SECOND BURNING STAGEC
21 CALL TABLE(TABL3,CNACHCKD24)GO TO 29
CC DURING FREE-FLIGHTC
25 CALL TABLE(TABL29CNACHCKO,24)CONTINUE
29 GO TO (95i96)tNTHRST95 GO TO (200,969300996)9N
CC THRUST TABLE FOR FIRST BURNING STAGEC
200 CALL TABLE(TRSTIoTtTHRST924)GO TO (18,96996)tK
18 GO TO (19,96)tM19 CALL TABLE(TMASTRMASSt24)
RMASS a RMASS/32.17'05
49
GO TO 96
C, THRUST TABLE FOR SECOND BURNING STAGEC
300 CALL TABLE(TRST2vTtTHRSr,24)GO TO (22,96996)tK
22 GO TO (23,96)tM23 CALL rABLE(TNAS2tTtRMASS924)
RMASS - RI4ASS/32.I.7405CONTINUE
C,.C. DERIVATIVES
C.96 CCD = CF*CKD
DRAG = CCD*RHOVVDIASQDOM = DRAG/RIMASSCCOS = COSF(THET)CSIN -SIIJF(THET)XDOT(K) = V*CCOSYDOT(K) = V*CSINGO TO (501050295019502),N
501 PRESS =PRSB*ARGU
THRUST =THRST + AREA*(PRESO-PRESS)VfOT(K) =(IVHRUST/RMASS) - GRAV*CSIN -DOM
RWGT = 32. 1?405*RMASSGO TO 503
502 VDOT(K) =-GRAV*CSIN - DOM503 THDIT(K) =-GRAV*CC2S/V
C34 GO TO (35,43P44h9K35 GO TO NWRT,136,39,41,42)36 IF(DWRT(N)) 37937,3837 ASSIGN 41 TO NWRT
GO TO 4138 ASSIGN 39 TO NkRT
CONTINUE39 IF(WRTT-T) 40,40,4240 WRIT =T+OWRT(rl)-EPSLON41 THDEG =57.29577*THET
THflOT =57.2957THDOT(l)
WRITE OUTPUT TAPE 3,111, TXYV,THOEGtCMACHCCDTHRUSrRWGTRHIWRITE OUTPUT TAPE 39112, XCBT(l),YCOT(1),VCZT(1),THDDTDRAGPPESSLINE = LINE + 1IF (28-LINE) 63,63, 42
63 ASSIGN 62 TI NCNTLINE = ICONTINUE
CC. MODIFIEC THIRD ORDER RUNGE-KUTTA INTEGRATIONC
42 X = X + CL.XtD0T(K)Y = Y+CL*YDOT(K)V w V+CI.VOT(K)THET = THET+CL*THDIT(K)GO TI 45
C43 X = X +C2*XDIT(K)
Y YeC2*YDOT(K)V =V+C2*VDOT(K)THEFT a THET+C2*THDIT(K)
50
GO TO 45C
44 X = X + C3*KITK-1) + C4*XCOT(K)Y= Y+ C3*YDOT(K-fl+C4.YOT(K)V = V+ C3.VOMT(K-1).C4*VDMT(K)THET =THET+C3*THDOT(K-1)+C4.THCOT(K)T = T+DELTRMASS = RMASS-t0MTM.CELT
45 CONTINUEcC H.KOPP ERROR ANALYSIS ROUTINE (S1M'PLIFIEDIC
NSTEP - NSTEP + IGO TO (599,599,9,92)KU
599 IF (NN - NSTEP) 6019601992601 GO TO (13l1,92bvNTERM
131.1 IF (Y-YMU) 92,1312913121312 GO TO NERR,(602,603,605)
C602 ASSIGN 603 TO NERR
NSTEP - NN - 1.VV VTHTH =THETXX = Xyy = YRMT = R?4ASSTT = TMOTT a DELTGO TO 3
603 ASSIGN 605 TO NERRASSIGN 42 TM NWRTNSTEP = NN - 2DH = DELTVH = VTHH = THETXH = XYH =
CC, COMPUTATION OF PROPAGATION ERROR COEFFICIENTSC
C = (2.O*RHO*CCD*DIASQ.V.V)/(AMASS*GRAV)CSQ = C*CPC = COSF(THET)PS = SINF(THET)PS - ABSF(PS)A13 = PC(I1.03 - .014.C + .C003*CSQ)A14 = 1.2*PS + .28 - .OO0l.CA23 = (PS4.08).(1.0-.02*C+.0004*CSQ)A44 = 1.2*PS +.2L + .0003.CPCP7 =PC + .07A24 a A44*PCP7A33 = .019 *(-1.1 + C*(.04 - .OOC2*C)lA34 a (PC4.1)*(1.02+.018*C)A43 = PCP7*(1.4-.02*C+.0G03*CSQ)
CDELTA = A33*A44 - A34*A43AVR = (THRUST/RWGT)*2.9E-7 + f.'3.3E-7 + 2.OE-7AFT - A33-A44ARGM =AFT*AFT + 4.0*A34*A43EPS a 0.5.(A33 + A44 + SORTF(ARG4))
51
GKK a IEPS-A33)/A34EPS = GRAV*EPS/VV = VVTHET = THTHX = xxV = YYT =TTRMASS =RMT
DELT =0.5*DELTC
604 Cl = 0.62653829*DfLTC2 = -0.55111241*VELTC3 = 1.4072559*DELTC4'= -0.48208L82*OELTGO TO 3
605 EHI = EPS*D:HASSIGN 602 TO NERRCLAM = 1.0 + EHEH02 a 0.5*EH*EH
CC COMPUTATION OF TRUNCATION ERROR COEFFICIENTSC
A = ABSF(VH-V) - DH.UTHRUST/RMASS)*I1.OE-O + GRAV*7.OE-8 4DIM*I 19.0E-8)IF (A) 50019500295002
5001 A = 0.05002 B = ABSF(THH-TiqET) - (GRAV*.0tI.OE-8)/V
IF (B) 5003t5004950045003 B = 0.05004 DH3 =DH**3
CC =1.75*V*DH3AVT (A*ABSF(A33) + VH*A3'..B)/CCIF (AVT-AVTL) 4002t4001#4001
'4001 AVTL - 0.8*AVTGO TO 40,03
4002 AVT =AVTL
AVTL =0.8.AVT
'003 CONTINUEATY = (A43*A +- VH*A'.'..)/CCIF (ATT-ATTL) 4005t400414004.
40,04 ATTL - 0.8*ATrGO TO 4006
'.005 ATT ATTLATTL z0.8*ATT
4006 CONTINUEPHB = A'4.AVT-A3'..ATTPHOB = A44*AVR - A3'..ATIRPSB - -A'.3*AVT + A33*ATTPSBB - -A43*AVR + A33*ATREVOV = EVN/VHVH4 =VH*DH*DH3TH3 = 2.0'DH3QQ = 1.1428571/VH4ONV? z DI4.V*7.OE-8DX2 =ABSF(XH-X) - OHV7IF (0X2) 5005,500695006
5005 DX2 a 0.05006 AXT - QQ*DX2
IF IAXT-AXTL) 40089400940'074.007 AXTL a 0*8*AXT
1 52
GO TO 40094.008 AXT =AXTI
AXTL O.8*AXT4009 CONTINUE
0Y2 = A8SF(YH-Y) - DHV7IF (DYZ) 5007,5008,50:08
5007 0Y2 = 0.05008 AYT = QQ*DY2
IF (AYT-AVIL) 4011'4010940104010 AYTL = XVI8*AYT
GO TO 60054011 AYT = AYTL
AYTL - 0.8*AYTCC TRAJECTORY ESTIMATES FOR THRUST PHASESC6005 CYK = Y
QYH = VQXK = XQXH = XQVK =VQVH = VQTHK = WETQMH = RMASSDMOIT = 00114GO TO (600696026,600896028),NEST
6006 DO 6366 If = 1930IF (TRSTltII,1) - T) 6366t6166,6266
61-66 QTK = TRST1(I112)QTH = TRSTt(II+lt2)QMH = TMASUI1+1,2)/32.17405CONT TRST1(11e1,1) - TRS71(II,1)GO TI 6466
6266 QTK =TRSTI(II-192) + ((T-TRST1(1I-1,1))/(TRSTI(II,1)-TRSTtUII-1,1M)) *(TRST1(1I I2)-TRSTL( 11-1,2))QTH =TRSTI(II92)QMH =TMAS(II92)/32.17.05CONT =TRST1III,1) - TGO TO 6466
6366 CONTINUE6466 KIER I
GO TO ('r 101,6011 ) 9M6008 DV' 6388 11 = 1930
IF (TRST2(II,1) - T) 6388,61889628e6188 QTK = TRST2(!I,2)
0TH - TRST2(II+1,2)QMH = TMAS2(1141#2)/32.L74C5CONT = TRST2(11e191) - TRST2(11,1)GO TO 6488
6288 OTK = TRST2(II-192) + ((1-TRST2(I1-1,1))/(TRST2(1I131-TRST2(II-1,11)))* (TRST2(I112) - TRST2(II-192))0TH = TRST2(II,2)QMH = TMAS2(1I,2)/32.17405CONT = TRST2(II,1) - TGO TO 6488
6388 CONTINUE6488 KIER =2
DOnflT = 01112GO TI (6010,6011)tM
6010-OM = (QMH-RMASSI/CONT
53
GO TO 60126011 QM = -DM:DOT
QMH = RMASS - DMDOOCONr6012 QT = (QTH - THRST)/CO'NT
ALPHA = GRAV'SINF(THET)BETA = RHO.DIASQ*QVK*CKD*CFAQT08 = ALPHA + QT/BETA
QEX = BETA/CMUIN = QM + BETAQVH = QTH/BErA - CQMH*AQTV,8/UN) 4((JVK *AGTge.RMASS/UK THRST/
1 BETA)*((RMASS/QIAH)'*QEX)PT3V = 0.4*(QVH+QVK)*COINTCSKAY = COSF(QTHIK)QXK = CXK + PT3V*CSKAYQYK = QYK + PT3V*SINFIUTHK)QTHK = QTHlK - 2.0*GRAV*CSKAY.CONT/(QVK+CVP)QVK = QVHII = 11 + 1
6013 00 6021 1 = 11,30CALL ATMOSCQYKtQVK)GO TO (6014,6oI6hvKLER
6014 QTK = QTHQTH = TRSTI1iI,2)CONT = TRSTI(I, 1) - TRSTI(I-1,1)TMOUT = TRST1(191)GO TO (6614,6615htKCCON
6614 CALL TA8LE(TABLICMACHCKC,24)e615 GO TO 16015t60t9),M6015 QMlK = CMH
CMH = TMASU,92)/32.17405GO TO 6018
C6016 QTK = QTH
QTH = TRST2(19Z)CONT = TRST2(l,l) -TRST2{ 1-191)TMOUT = TRST2(I,1)GO TO (6616,6617),KOCON
6616 CALL TABLE(TABL39CMACHCKC924)6617 GO TO (60l7,60t9btM6017 QM:K = QMH
QMH4 = TMAS2(li,2)/32.174056018 QM = (OMH-QMlK)/CONT
GO TO 60206019 CM = -OMDOT
CMK = QMHQMH = CMK - 0,MEOT*CONT
6020 QT = (QTH-QrK)/CONTSNIKAY = SINF(QTH!K)CSKAY = COSF(QTHK)ALPHA = vGRAV*SNKAYBETA = RHO*DIASQ*QVK*CKC*CFAQTZB = ALPHA + CT/BETAUIN Q M + BETACEX =BETA/CM
CVH =(QTki/BETA) - (QMH*AQ;TOB/UN) 4(QVK +(AQTOB*QPK/LUN) -(QTK/
I BETA)l*((QMK/CMH)**QEX3PT3V = O.40(VH+QVK)*CONTQXK = QXK +PT3V*CSKAYQYK = QYK +PT3V*SNKAYCTHK a QTHK - 2*OuGRAV*CSKAY*CONT/(VK+VHt)
54
QVK = QVHIF cTIME-TMOUT-CONr.1.OE-4) 60229602296021
6021 CONTINUE6022 GO~ TO (6024,6028),KLER6024 IF (NEXT(2))6028,6028,6025
CC TRAJECTORY ESTIVATE FOR INTERMECIATE BALLISTIC PHASEC6025 QMK =QMH - BMASS
CONT =TIM(2) - TIMM IGO TO 6027
6026 QMK =RMASS
CONT =TIM(2) - T6027 PT6V 0.6*QVK.CONT
CSKAV = COSF(QTHIK)SNKAY = SINF(QTf-IK)QXK = QXK + PT6V*CSKAYQYK = QYK + PT6V*SNKAYQTHK QTHK - (GRAV.CSKAY/CVK)*CONTGO TO (66271,6,628)tKCCON
6627 CALL TABLE(TABL2tCMACHCKC,24)6628 VVK = QVK + (-GRAV*CSKAY - RHZ*CF.CKD.QVK*eVK.0IASQ/QMK).CONT
KLER = 211 = 2DMlOT = DOTM2GTH = TRST2(II-1,2)GO TO 6013
CC TRAJECTORY ESTIMATE FOR TERPINAL BALLISTIC PHASEC6028 U = SINF(QTHK)
QDOTY =QVKWU
TMTN =(01DOTY +SQRTF(QDZTY*QDOTY *2.0*GRAV*CYK))/GRAVGO TO (6129,6128,6129),NXY
6128 GO TO (6129,5014),JPf129 CSK = COSF(QTHiK)
Q00OTX =0.6*QVK*CSK
PLUS 1.0/(QVK*CSK)GO TO (6227,6228),KDCON
6227 CALL ATMOS(QYKQVK)CALL TABLE(TALIL2,CMACHCKC,24)
6228 BET = 0.0023769ODIASQ*QVK*CKD*CF/32.L74C5SNOCS = U/CSKDU = 0.05*(U+0.9995)DUQ'G = U/32.17405
C6229 IF (QYK-5.8E4) 6029i6030,6C316029 BOG = BET
GO TO 6043(030 BOG = 0.l.BET
GO TO 60436031 IF (QYK-1.0E5) 6030,6032p6C336032 BOG = 0.01*BET
GO TO 60436033 IF (QYK-1.6E5) 6032,6034,60356034 BOG = 0.001.BET
GO TO 60436035 IF (QYK-2.3E5) 603496036,6C376036 BOG - BET*I.OE-4
GO TO 6043
55
1037 IF (QYK-3.0E5) 6036,6038,6C396038 BOG = BETeI.OE-6
GO TO 6043f039 IF (QYK-4.0FF5) 603896040,6C416040 BOG = BFT*l.OE-8
GO TO 6043C041 IF (QYK-6.0E5) 6040,60,42,6042f042 BOG = BET*1.OE-13
Cf6043 AA = BOt;.SNOCS + PLUS
OM;U = (1.0+U)*(1.0-U)ROOT = SQRTF(OMJ)AU =(ROOT*AA - BOG*,U)**2
CQDOTY = U/(2MU*AU)CYH = QYK +0.8*O1DOTY*DUOGQDOTX = 1.0/(AU*ROOT)QXH = QXK + QD 0TX*flUOGGO TO (644496046),JM'P
6444 G0 TO (60,44,6046),JU4P6044 IF (QYH - QYK) 604596045960466045 YMAX =QYK
JUmP = 26046 IF (QYH-YF) 60A9,6048960476047 QYK = QYI4
QXK w QXHU = U - DlU
IF (1.0+U) 60489604896229C6048 XMAX = QX-
GO TO 50146049 XMAX = QXK + (CXH-QXK)*(QYK-YF)/(OYK-QYH)
CC ESTIMATES OF ERRORS OVER N STEPSC5014 JUMP = I
IF (NN-NSTEP) 800298002,80018001 CNN = NSTEP8002 CL'AMN = CLAM**CNN
GL (CLAMN-1.0)/(CLAM-1.0)VHH VH*DHPHH- ABSF(PHBDH3 + PHOB)PSH =ABSF(PSB*DH3 +PSBB)IF (PHH-PSH) 40139401494014
4013 CONE = PSH/ABSF(CELTA)GO TO 4015
4014 CONE = PHH/ABSF(OELTA)4015 ETOK = ETN/ABSF(GKK)
IF (EVOV-E1BK) 40169401794C174016 CTWO = ETOK
GO TO 40184017 CTWO = EVOV4018 CTHR = A13 + GKK*A14
CFOR - A23 + GKK*A24C12 = CONE + CTWOEVIN = V*(-CONE + CLAMN.C12)ETLN = GKK*EVLN/VEXIN a EXN + VHH.(CNN*(AXTo0H3 + AXR) - CBI4E*CTHR + CTHR*C12*GLIEYLN a EYN + VHH*(CNN*(AYT*0H3 + AYR) - CNNE*CF@R + CF:UR*C12.GL)
56
IF (NN-NSTEP) 8003,8003980048003 EVN a 0,5*(FVLN + EVNN)
ETN = 0.5*(ETLN * ETNN)EXN - 0.5*EXLN + EXNN)EYN = 0.5*(EYLN + EYNN)GO TO 8005
8004 EVN = EVLNFTN = ETLNEXN = EXLNEYN a EYLN
8005 GO TO (71119722297333)tKPHASECC FRROR ANALYSIS INITIALIZATION FOR START IF PHASEC7222 KPHASE = 3
NN = 4NSTEP = 2CNN 2.0AVTL = 0.0ATTL 0.0AXTL = 0.0AYTL = 0.0
EVNN = 0.0
ETNN = 0.0
EXNN = 0.0EYNN = 0.0
EXAVE = EXNEYAVE = EYNEVAVE = EVNETAVE = ETNTOO = TXOO = X
Y00 = YVO0 = VTHOC = THETRMOO = RMASSDTH = V/32174.05ASSIGN 42 TO NWRT
GO TO 637C7333 KPHASE = 1
T = Too
X = X0'Y = Y0eV = VO'
THET = THO0RMASS = RMIO
EXN = EXAVEFYN = EYAVEFVN = EVAVEETN = ETAVE
ASSIGN 36 TO NWRTGO TO 609
C7111 IF (NN-2) 700197001970027001 T = TO
X = XO
Y = YO
V = VI
57
VzTHET =THETO
RMASS =RMASSO
EXN = 0.0EYN = 0.0EVN = 0.0ETN = 0.0
1002 CNN = NSNN =NSNSTEP 1ASSIGN 3,6 TO NWRTIF (SENSE SWITCIH6) 300493006
3004 CALL PDUMP(ZMA13vI)3006 CONTINUE
CC ESTABLISHING STARTING VALUES FOR DOUBLE FALSE POSITIONC
609 KLK 1DT= 0.035*V'H/GRAV
610 CLAM = 1.0 + EPS*DTGNN = TMTN/DTCLAMN = CLAM**GNNGL =(CLAMN-1.0)/(CLAM-1.0)DT3 =DT**3
PHI ABSF(PH3*DT3 + PIH86)PST =ABSFIPSB*DT3 + PS8B)IF (PHT-PST) 4020,402l,4C21
4.020 CONE = PST/AUSF(DELTA)GO TO 4022
4.021 CONE = PHT/ABSF(CELTA)4022 ETOK = ETNIABSF(GKK)
EVOV = EVN/VHIF (EVOV-ETOKJ 4023,402494C24
4023 CTWO = ETOKGO TO 4025
4024 CTWZ = EVOV4025 C12 = CONE + CTWO
VHH = VH*CTGO 10 (4026,4027),NNXY
4026 EXNN = EXN + VhH*(GNN*(AXT*CT3 + AXR) - CkNE*CTFR + CTHR*C12*GL)XIDT = EXNN - GAM2*XMAXGO TO 4028
4027 EYNN = EYN + VHH.(GNN*(AYT.CT3 + AYR) - CONE*CESA + CFOR*C12*GL)XIDT = EYNN - GAM2*YMAX
C4028 GO TO (6lI96l5,615v6l59617),KLK611 IF (XIDT) 612,613,613612 0TH = DT
GO TO 630613 Xifl2 =XIVT
DT2 DT614 DT =0.1.01
KIK =KIK + 1GO TO 610
615 IF (X!DT) 616,614,614616 XID1 XIDT
DT1 DTGO T0 618
617 GAM2 * 2.O*GAM2GO TO 609
C
58
CC DOUBLE FALSE POSITION ROUTINE FOR FIDELT)
618 PTO = (r0Tl.XID? - tT2cIofl)/(XIC2 -X101)
GNN = TMTN/DTOCLAM = 1.0 + EPS*DTO
CLAR'N = CLA M**GNNDT3 = tJTO**3
VHiH = VH*DTOGL. = (CLAMN-1.0)/CLAM-i.0)PHT = ABSF(PH,,3*0T3 + PH63B)PST = ABSF(PS8*0T3 + PSB8)IF (PHT-PST) 4030t403194031
4.030 CONE = PST/ABSF(CELTA)GO TO 4032
4031 CONE = PHT/ABSF(DELTA)4032 ETOK = ETN/ABSF(GKK)
IF (EVOV-ETO:K) 4033,4034,40344033 CrWO = ETOK
GO To 40354034 CTWO = EVOV4.035 C12 = CONW + CTWO
GO TO (5020,5')21)PNNXY5020 EXN'N = EXN + VtiHH(GNN*(AXT*CT3 + AXR) - C3NE.CT141R + CTIR*C12*GL)
XIflO = EXNIN - GAM2.XPAXGO TO 5022
5321 EVNN = EYN + VHH*(GNN*(AYT*CT3 + AYVR) - CIONE*CFOR + CFgR*CL2*GIJ,xinoI = EVN'N - GAM2*YMAX
5022 DIFH = (DT1-CT2)/DT2IF (0.05-ABSF(ICIFH)) 661,660,66C
660 GO rI (5025,5026),NNXY5025 GO TO (5029,50?9,5027),NXY5026 GO TO (502995U29,5028),NXY5027 KIK 1
DT = fTONNYY 2GO TI 610
5028 NINXY = 15029 DTH = DO
NEG = 0NPFS =0GO TO 630
661 IF (XIO) 619,620,620619 NEG =NEG + 1
NPOS =0
GO TO (621,626)#NEG620 NPOS =NPOS 4 1
NEG =0
GO TO (621,62'flNPOS621 IF (XIDO) 622,623,623622 DTI DO
GO TO 618623 0T2 =OTO
X102 XIOGP TO 618
626 IF (XIO1) 627,628,628627 X102 a X102/2-0
59
NEG - 0
GO TO 6LB
628 X!01 X10112.0NEG = 0NPOS z0GO TO 618
629 IF (XIDI) 628t627,621C
630 GO TO (632,632,632t631)iN631 NN =NS
CNN =NSKPI'ASE 1 INSTEP =1
CPO TO 636632 CNN = NS
IF (CNN*CTH+T - TIMMN) 63696369633633 CNN = (TIM(N)-T)/DTH
NiN = C NNCCN = NiNIF (CNN-CCN) 634,634,635
634 CNN = CCNGO TOJ 6355
635 CNN = CCN + 1.0f355 NSTEP = 1
NN a CNN6356 NEST a NEST + 1.
IF (NEXT(NEST) 6356,6356#635?(35? KPHASE = 2
0TH = (TIM(N)-T)/CNN636 CLAMN = CLAM**CNN
GL = (CLAM'N-1.0)/(CLAM-1.0)EVNN -VH*(-CONE + CLAMN*C12)ETNlN = GKK*EVNN/VHEXN!N = EXN + VHH*(CNN*(AXT*CT3 + AXR) - CONE*CTHR + CTI4ReCL2*GL)EYNN = EYN +VHH*(CNN*(AXT.CT3 * AYR) - CONE.CFIR + CFSR*CI2*GL)
GO TO (7006,7003,7006,637)tKU7003 YOU = YF + EYN
OLEY = (EYNN - EYN)/CNNERRA z -DLEY
7006 EV(NO) = EVNET(NO) =ETN*57.29577EX(NO) aEXNEY(NO) =EYNTN(NO) = TT + DHXN(NO) =XHYN(NO) = YHVN(NO) =VHTHN(NO) z THH * 57.29577NO = NO + 1.
637 DELT = 0THCl =0.62653829*DELTC2 = -0.551L1241*DELTC3 = 1*4072559*DELTC= -0.48268182OELT
GOl TO (92,9249)KPHASEC
49 THOEG = 57.2957THETTHOCIT - 57.295f.THCIT(3)
60
RWGT - 32.17405 * RMASSWRITE OUTPUT TAPE 3,111t TXYtVtTHDEGCMACHCCOTHRUSTRWGTRHIWRITE OUTPUT TAPE 3tll2t XCST(I)YDT()tVCBT(ItTHDTtDRAGPRESS
50 N = N+lIF(NEXT(N)) 50,50951
51 GO TO (3,55,56,5997020trN
55 THRUST = 0.0DOTM = 0.0AREA = 0.0RMASS = PMASS-BMASSRMOO = RMASSRW'GT = 32.17405 * RMASSGO TO (801,802),NNN
801 WRITE OUTPUT TAPE 3,115GO TO 999
802 WRITE OUTPUT TAPE 3,121NNN = 1GO TO 9919
.c56 THRST = THR2
DOTM DOTM2AREA = AREA2/144.0KASE = 2
GO TO (803,04)tNNN903 WRITE OUTPUT TAPE 3,116
GO TO 999804 WRITE OUTPUT TAPE 3,122
NNN = 1GO To 999
C59 THRUST = 0.0
DOTM = 0.0AREA = 0.0
GO TO (805,806),NNN805 WRITE OUTPUT TAPE 3,117
GO TO 999.806 WRITE OUTPUT TAPE 3,123
NN = IC
999 EPSLON = 0.001*DELTNEST = NTIME = TIM[N)
ASSIGN 42 TO NWRTASSIGN 64 TO NCNTLINE = 2WRITE OUTPUT TAPE 3,109WRITE OUTPUT TAPE 3,110GO TO 3
CC TEST FOR END OF TRAJECTORYC
92 GO TO KAPd700797010t7013)7007 IF (YnOT(I)) 7008331308 KU - 2
ASSIGN 7010 TO KAPYMAX YJMP = 2
7010 IF (Y-YOU) 7012#7011,70117011 YOU = YOU + CLFY
L
YLST =VGO TO 3
7012 ASSIGN 42 TO NWRTNTERM - 2YONE - YYTWO a YLSTDTTWO = -DELTDELT3 = -DTTWO*(YONE-YU)/(YTWOa-YONE)DELT =DELT3ASSIGN 7013 TO KAPGO TI 604
7313 IF(Y-YOU) 7014t7015970157014 YONE = Y
NB = 0NA =NA + 1IF (NA-2) 6081,608096080
6080 YTWO = YOU +0.5*(YTWO-YO'U)6081 DONE aDELT3
DELT3 = ((YVWI-YOU)*DT8NE - IVIE-YIU)4()TTWO)/(YTWO-YUNE)DELT = DELT3 - DONEGO TI 7016
7015 YTWO = YNA =0NB = NB + IIF (NB-2)6083,6082,6082
6082 YINE = YOU + 0.5.(YlNf-YIUj)6083 DTTWO = DELT3
DELT3 = ((YTW3-V2ti).OT0NE - (YINE-Y0,U)*DTTW)/YTW--Y0tE)DELT = EL13 - OTTWO
7016 IF (YTWO-YINE-.01*EYN) 70L7,7017,60410k? GO TI (70l897018,70l9,7020),KU7018 XUNO = X
KU = 3NN =NSCAIN = NSTEPNSTEP = 1TT = TXH = XTKH = THETYH = YOUVOU = YFDLEY a 0.0VH = VVLST YDONE =0.0
CH = 0. 0ASSIGN 1010 TO KAPGO TO 6005
1019 EV(NO) = EVNET(NO) = ETN*57*29577EX(NO) = EXNEVINO) = EYNTN(NI) = TXI4(NO) a XYN(NO) = YV?4(NO) = VTHN(NI) a THET 57.295771KU - 4NN =NSCNNh a NSTEP
62
NSTEP a 1T = T
XH = XTHH = THETYH = YOUVH = VYLST = YD = 0.0YOU = YF-EYN
OLEY = ERRADTONE 0.0ASSIGN 7010 TO KAPTHDEG = THET*57.29577WRITE OUTPUT TAPE 3,111, TXY,VtTHDEGtCMACHCCDTHRUSTRWGTRHU
401 WRITE OUTPUT TAPE 3,113, NJRWRIT = T + 1000.ASSIGN 42 TO NWRTGO TO 6005
C
7020 LINE = 27XDOS = XWRITE OUTPUT TAPE 3,126WRITE OUTPUT TAPE 3,l27WRITE OUTPUT TAPE 3,128GO TO (5051,5053,5052),NXY
5051 WRITE OUTPUT TAPE 3,131, GAMMAPERX = 100.*EX(NO)/XN(NO)GO TO 5054
5052 WRITE OUTPUT TAPE 3,131, GAMMA
5053 WRITE OUTPUT TAPE 3,132, GAPMAPERY = 100.*EY(NO)/YMAX
5054 IF (GAMMA-100.*GAM2) 5055,5C56,50565055 GO TO (5551,5552,5553),NXY5551 WRITE OUTPUT TAPE 3,130, PERX
GO TO 50575552 WRITE OUTPUT TAPE 3,142, PERY
GO TO 50575553 WRITE OUTPUT TAPE 3,143, PERXtPERY
GO TO 50575056 GO TO (5554,5555,5556),NXY5554 WRITE OUTPUT TAPE 3tl29, PERX
GO TO 50575555 WRITE OUTPUT TAPE 3,144, PERY
GO TO 50575556 WRITE OUTPUT TAPE 3,145, PERXPERY5057 WRITE OUTPUT TAPE 3,133
WRITE OUTPUT TAPE 3,134, XN(NO)tEX(NO),YN(NO),EYINO),THN(NB),I ET(N0),VN(NO)tEV(NO)WRITE OUTPUT TAPE 3,135WRITE OUTPUT TAPE 3,136, XZ,YOtTHDEG0,VBWRITE OUTPUT TAPE 3,137WRITE OUTPUT TAPE 3,139
NOO = NO - I10 5059 KE =29NO0WRITE OUTPUT TAPE 3,140, I"N(KE),XN(KE),YNIKE),THN(KE)tVN:(KE)t
1 EX(KE)*EY(KE)vET(KE)#EV(KE)LINE = LINE + 3IF (56-LINE) 5058,5O5895059
5058 WRITE OUTPUT TAPE 3,138
WRITE OUTPUT TAPE 3,139
63
i
LINE a 55059 COJNTINUE
GO TO (5060t5060,5060,5060,5061),N5060 WRITE OUTPUT TAPE 39141, XLNIXOSStXN(N@)tYF51061 NJR =NJR + I
AN J N-1IF CJN) 1,1,2
C
64
APPENDIX II
This appendix contains output for the sample calculationdepicted by Figure 2. The input for this computation is illustratedin Section C. The first portion of the output consists of a table ofrange, altitude, speed and angle as a function of time. Also in-cluded are the time derivatives of these quantities, the mach number,the drag coefficient, the thrust, vehicle weight and drag, the airdensity, and the air pressure when required.
The second section of the output is devoted to the error analysisroutine. This begins with a description of the error specification andprovides the estimated error for comparison. A table of range error(feet), altitude error (feet), speed error (feet/second), and angularerror (degrees) as a function of time follow. This page is terminatedby bounds for the impact range which are based upon a considerationof the numerical error envelope surroundin g the computed trajectory.
65
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DISTRIBUTION LIST
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Feltman Research LaboratoriesMr. L. H. Eriksen, Director ) 1Mr. R. Frye, Deputy Director) (in turn)
Engineering Sciences LaboratoryMr. W. R. Benson, Chief 2
Computing and Analysis SectionMr. J. Spector, Chief 3- 5
Mathematical Analysis UnitDr. L. F. Nichols, Chief 6- 7Dr. H. J. Kopp 8- 17Mr. B. Barnett 18
Digital Applications UnitMr. I. Klugler, Chief 19 - 28Mr. J. Nielsen Z9- 33Mrs. E. Stochel 34
Digital Systems UnitMr. D. L. Grobstein, Chief 35
Special Assistant to Chief, CASDr. S. L. Gerhard 36
Aeroballistics SectionMr. A. A. Loeb, Chief 37- 38
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Engineering Research SectionMr. G. Demitrack, Chief 45
Engineering Analysis UnitDr. F.K. Soechting, Chief 46
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Pyrotechnics LaboratoryMr. S. Sage, Chief 47 - 48
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Long Range Atomic Warheads LaboratoryMr. H. W. Painter, Chief 56
Tactical Atomic Warheads LaboratoryMr. G. I. Jackman, Chief 57
Atomic Ammunition Development LaboratoryMr. J. G. Drake, Chief 58
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Ammunition Development DivisionMr. R. Vogel, Chief 6Z
Warheads and Special Projects LaboratoryMr. F. E. Saxe, Chief 63
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Artillery Ammunition LaboratoryMr. E. H. Buchanan, Chief 65 - 66
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ATTN: AMSMU-SS-SC 78AMSMU-SS-EC 79AMSMU-RE-R 80
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72
