Nasa Contractor Report Nasa Cr-19324 Sol1 D

NASA CONTRACTOR REPORT

NASA CR-19324

(HAS A-CR- 150324) SOLID-PROPELLA NT PCCKET N77-28213 UCTOB IITEBIAL EALLIS'IIC FEfiFCSdBNCE VARIATION ACALYSIS, PHASE 2 Final Report (Auburn Univ,) 127 p HC A07/= A01 CSCL 21H Unclas G3/2J 39225

SOL1 D-PROPEUANT ROCKET MOTOR INTERNAL BALL1 STlC PERFORMANCE VARIATION ANALYSES (PHASE TWO)

By Richard H. Sforzini and Winfred A. Foster, Jr. Aerospace Engineering Department Auburn University Auburn, Alabama

Final Report

September 1976

Prepared for

NASA - GEORGE C. MARSHALL SPACE FLIGHT CENTER Marshall Space Flight Center, Alabama 35812 TECHNICAL REPORT STANDARD TITLE PAGE 1. ocmnar wn. 12. GWERNISNT ACCESSION NO. 13. RECIPIENT*S CATALOG YO. 1

Variation Analysis (Phase Two)

7. AUTnOR(S) 0. PERFORMlNG ORGAN1 ZATIOW REPCR r U Richard H. Sforzini and Winfred A. Foster, Jr. 9. PERFORMING ORGANIZATION HAM€ WO AOORESS 10. WOAK UNIT NO. Auburn University School of Engineering 1 1. COWfRICT OR GRANT NO. Auburn, Alabama

12. SPONS~AINGA~E~V MAW ANO libGEssp Cantrac ,or Report National Aeronautics and Space Adainistration Washington, D. C. 20546 14. SPONSORING AGENCY COOE

1s. SUPPLEMENT-Y MOTES I 16. MSTRmThe report presents the results of continued research aimed at improving the Iassessment of off-nainal internal bal?istics performance including thrust imbalance *between two large solid-rocket motors (SR)Is) firing in parallel. Previous analyses by th authors using the Monte Carlo technique (NASA Contractor Reports NASA CR-120700 and CR- 144264) have been extended to pedt detailed investigation of thrust imbalance and its first time derivative throughout 1.w burning times of SRn pairs. Canparison of results with flight performance data from Titan IIIC SRMs shows good agreement. Statistical correlations of the thrust imbalance at various times with corresponding nominal trace slopes suggest several alternative methods of predicting thrust imbalance to that presented in earlier reports. A separate investigation of the eff~ctof circular-perforated grain deformation on internal ballistics is discussed, and a modified design analysis computer program which permits such an evaluation is presented. Canparisons with SRM firings indicate that grain deformation may account for a portion cf the so-called scale factor on burning rate between large motors and strand burners or small ballistic test motors. Thennoelastic effects on burning rate are also investigated. Burning surface temperature is calculated by coupling the solid phase energy equation containing a strain rate term with a model of the gas phase conibustion zone using the Zeldovich-Novozhilov technique. Comparisons of solutions with and without the strain rate term indi-cate a small but possibly significant effect of the thermoelastic coupling. An approach is suggested for using the Monte Car10 program to assess the degree of predictability of SRM ,performance at various phases in a development program; however, it has been determined that the type of data needed for the assessment is not generally available. A number of changes to the computer programs listed in CR-144264 that have been made to improve their accuracy and flexibility are presented.

(6. DISTRIBUTION STATEMENT Unclassif ied-Unlimi ted i "' XORDs A. A. McCool I Director, Structures 6 Propulsion Lab.

I I 19. SECURITY CLASSIF. (d thlm =pot) 20. SECURITY CLAS SIF. (of thlm pa*) 21. NO. OF PAGES 22. PRICE Unclassified Unclassified I NTIS l. J ~PC- Form 3292 (*Y 1969) The authors express appreciation to personnel at the George C. Marshall Space Flight Center for their many useful suggestions which materially aided this investigation and in particular to Hessrs. B. W. Shackelford, Jr. (NASA Project Coordinator) and E. P. Jacobs.

The suggestion of Dr. L. 8. Caveny of Princeton Univer- sity to use the Zeldovich-Novozhilov technique for modeling transient burning rate phenomena proved quite useful in the thermelastic analysis and is also gratefully acknowledged.

The participation of the following from the Aerospace Engineering Department at Auburn University is likewise appreciated: Mr. D. F. Smith, graduate student, who assisted in modification of the design analysis computer prcgram for the effects or' grain deformation and in the evaluation of sample cases, and who developed the closed form approximation of grain deforma+.ion which was used in the Frogram; Ms. L. S. Garrick, graduate research assistant, vho performed the statistical computations ~ndprepared a number of figures, Hr. H. Aultman, Jr., applications EDP programmer, whcl assisted in obtaining and plotting computer results; and Mrs. M. N. McGee who typed the bulk of the manuscript.

Appendices ...... 63 A . A Computer Program for Analysfs of SRH Pair Imbalance Data During Specified Time Intervals ...... 64 B . A Computer Program for Analysis of SRH Pair Imbalance Data at Specified Times 9uring Operation ...... 70 C . The SRH Design Analysis Computer Program with Option Permitting Evaluation of Grain Deformation Effects ..... 75 LIST OF FIGURES

Fig. 11-1 CalComp plot of thrust imbalance versus time from 0 to 100 secs. for 30 pairs of theoretical Titan IIICS~...... 6 Fig. 11-2 Tolerance limits for thrust imbalance from 0 to 100 secs. for 30 pairs of theoretical Titan IIIC SRMs (Probability 0.90 for 99.7% of the distribution). ... 8

Fig. 11-3 Tolerance limits for first time derivative of thrust imbalance from 0 to 100 secs. for 30 pairs of theoretical Titan IIIC SRHs (Probability 0.90 for 99.7% of the distributfon) ...... 9 Fig. 11-4 Comparison of Monte Carlo tolerance limits for 30 pairs of Titan 1114 SRMs with flight test analysis for 21 pairs (K = 3.68 and 3.86, respectively; probability of 0.90 for 99.7X of the distribution ...... 11

Fig. 11-5 Comparison of contractor's thrust iinbalance during web action time analysis (~ef.7) with toleranze limits obtained by the Monte Carlo program for 30 space shuttle type SRM pairs (0.90 probability for 99.7% of distribution)...... 12

Fig. 11-6 Thrast imbalance versus time from 0 to 105 secs. for 30 theoretical pairs of Space Shuttle type SRMs obtained from the Monte Carlo program ...... 13

Fig. 11-7 Correlation coefficients between thrust imbalance and thrust imbalance rate at various times during operation for 30 pairs of Titan IIIC SWs ...... 15 Fig. 11-8 Scatter diagram: mean of absolute values of thrust imbalance for 30 theoretical Titan IIIC SKM pairs versus the absolute value of mcsn time. rate of change of thrust for the 60 motors ...... 17

Fig. 11-9 Scatter diagram: standard deviation of thrust imbalance for 30 theoretical Titan IIIC SRM pairs ..---.,rsus the absolute value of the mean time rate of change of thrust for the 60 motors...... 18 Fig. 111-1 Sector of cross-scction of solid propellant with and witl~outpressurization and combustion...... 21 LIST OF FIGURES (Continued)

Fig. 111-2 Grain geometry notation for analysis of solid- propellant strains. The insulation and liner may be considered a part of the propellant ...... 21 Fig.. 111-3a Comparison of approxirate solution for tangential bore strain (Ref. 4) with finite element solution method of Ref. 9 ...... 25 Fig. 111-3b Continued comparison of approximate solution for tangential bore strain (Ref. 4) with finite element solutionmethod ofzef. 9 ...... 26 Fig. 111-4 Comparison of theoretical and experimental results for a Titan IIIC/D SRM with no scale factor on strand burning rate ...... 29 Fig. 111-5 Coatparison of theoretical and experfmental results for a Castor TX 354-5 SRM with no scale factor on small ballistic test motor burning rate...... 30 Fig. IV-1 Typical finjte eleuent for axisymmetric problem. ... 37 Fig. IV-2 Schematic of finite element model (no scale) ..... 40 Fig. IV-3 Comparison of results with increased pressurization rate to baseline example ...... 45 Fig. IV-4 Comparison of results with increased elastic modulus to baseline solution ...... 47 LIST OF TABLES

Table 11-1 Sample printout for time interval imbalance analysis computer program ...... 5 Table 11-2 Sample printout for time slice imbalance analysis computer program ...... 10 Table 11-3 Results of correlation and regression analysis of Monte Carlo thrust imbalance and tkrust imbalance rate for Titan IIIC SRMs (AF = b Ar' + c) ...... 16 Table IV-1 Baseline example data ...... 42 Table IV-2 Results for baseline example ...... 43 Table IV-3 Results for modified pressurization rate example. . . 44 Table IV-4 Results for modified elastic ~nodulusexample. . . . . 46

Table A-1 Computer program for analysis of SRM pair imbalance data during specified time intervals. . . . 65

Table B-1 Computer program for analysis of SRM pair imbalance data at specified times during operation ...... 71

Table C-1 Example data sheets for design analysis program withgraindeformation ...... 77

Table C-2 Sample computer printout for design analysis program with grain deformation ...... 79 Table C-3 SRM design and performance analysis ...... 80 NOMENCLATURE

English Symbol Definition Units Used- a Propellant burning rate coefficient. in~sec-~si~ c Specific heat. in-lbf /lbmeF

C Coefficient of variation; i.e., the ratio of the v - standard deviation to the mean.

Strain tensor. - =ij E Modulus of elasticity. lbf /in2 F Thrust . lbf K Statistical confidence coefficient. -

T Total impulse. lbf-sec -t R Length of propellant grain. in. m Mass or ratio of inside to outside radius. slugs or - m Mass of propellant gases generated per unit time. slugs/sec G n Burning rate exponent or number of observations - of a statistically distributed variable.

P Pressure. lbf /in2 r Burning rate. in/sec r Correlation coefficient - C . R Radius. in. s Standard deviation of a sample of a statistically units vary distributed variable. s Square root of the second moment of a sample units vary 0 distribution about an assumed zero mean.

S Burning perimeter. in. t Time. sec. NOMENCLATURE (Continued)

English Symbo 1. Definition Units Used

T Temperature in thermoelastic analysis. OR

Grain temperature and standard grain temperature, OF T~r*Tref respectively. x Radial coordinate in themelastic analysis. in. - X Value of general statistically distributed units vary variable.

BY' Distance propellant has burned from initial in. lateral surface of circular perforated grain and from other initial surfaces, respectively.

Greek Symbol a Linear coefficient of thermal expansion /OF

8 Volumetric coefficient of tlicrmal expansion /OF

6 The Kronecker delta (1 when i=j, 0 when i#j) - il A Change or difference in quantity. units vary

E. Strain. - rl Compressibility. in2/lb f h Thermal conductivity. in-lbf/in-sec OF v Poisscm's ratio. -

P Density. slugs/in3 u The standard deviation of a statistically units vary distributed variable; i .c., the square root of the second momcnt about its man value. u Temperature sensitivity of burning rate at P constant pressure. /OR

T Thickness. in. NOMENCLATURE (Cont hued)

Subscripts DeIinition av Average or mean value.

C Motor case or compressed state of propellant just outside the heat-af f ected zone. e Grain exterinr surface position.

1 Grain exterior surface position or initial condition. max Maximum value.

Unpressurized end unheated state of propellant or in the thermoelastic analysis the ambient transient condition.

Propellant.

Radial coordinate.

Surface condition.

Steady state

Transient

Propellant web.

Distance burno,d.

Axial coordinate.

Tangential. coordinate. I. INTRODUCTION AND S-Y

This report presents the results of research performed at Auburn University during the period October 28, 1975, to September 30, 1976, under Modificati~nsNos. 17 and 18 to, the Cooperative Agreement, dated February 11, 1969, between NASA Marshall Space Flight Center (MSFC) and Auburn University. The prjncipal objective of the research was to further develop techniques for theoretical assessment of solid rocket motor (SIZM) internal ballistic performance to include statistical investigation of thrust imbalance of pairs of SRMs firing in parallel as on the booster stage of the Space Shuttle.

The theoretical thrust imbalance of motor pairs has been previously investigated statistically by application of che Monte Carlo technique (Refs. 1-3). The results of this investigatic;: include a computer program which selects sets of significant variables on a probability basis and calculates the performance cnaracteristics for a large number of motor pairs using a mathematical model of the internal ballistics. Com- parison of such a statistical analysis for TITAN IIIC motor pairs with test results shows the theory underprcdicts the standard deviation in maximum thrust imbalance by 20% with variability in burning times matched within 2% (Ref. 1).

The referenced research disclosed a number of ways in which both nominal and off-nominal performance prtdictions for single and ?airs of SRMs could be improved. The present report deals with the results of research to produce these improvements.

Recent efforts by MSFC to describe tkrust imbalance characteristics to be anticipated in a concise way that would he meaningful for systems performance studies have indicated that the Molite Carl program would be useful for this purpose. Tbo supplementary computer programs are included in this report to permit the critical imbala:~ceparameters to be determined on a statistical basis throughout the operating times af the SRMs. The principal critical imhnlance paranietcrs arc the thrust imbalance and its first time derivative. Data for these and other parameters of interest; e.g., impulse inhalance; arc gcncratcd by the PIonte Carlo program and stored on ~agnetictnpc. One program is uscd to determine the maximum values of the parameters of intercst during specified time periods. Thc second program computes statistical tolerance limits as a function of time based on the Monte Carlo sample. CalComp plots of the parameters of interest versus time are obtainable for both individual SU1 pairs over- laid on etch other and for thc tolerance limits. Thcsc are illustrated in the report. Comparison of program rcsr~ltswith flight test results for Titan IIIC are given and sllow gnod agreement.

The correlation coefficients between thrust imbalance and its time derivative wcrc dctcrmjncd at various times. Tl~eresults show considerable variation in thc goodness of the colrclation at various operating times. A linear regression analysis shows little generali&.~tionis possible as to the relation between these variables at the various operating times. On the other hand, there appears to be excellent correlation between thrust imbalance and the nominal slope of the thrust time trace which suggests several alternative approaches to that used in Refs. 1 and 2 for determining thrust imbalance.

Another portion of this research treats certain effects of grain deformation on the internal ballistics of SRMs. The report explains how the deformation can affect the burning rate of the propellant and account for a significant portion of the "scale factor'' between large and small motor burning rate. The important factor is the tangential strain of the burning surface of the propellant grain. A method developed by Smith (Ref. 4) has been used to estimate the strrin at various times. The method was coupled with the design analysis computer program (Ref. 2) to analyze the effects on internal ballistics for circular-perforated grains. Comparisons are presented of the internal bailistic results with and without grain deformation and with actual test data for two different SRMs. The modified design analysis computer program is listed in the report along with a sample solution. Variations in the propellant and motor case properties (especially variation in the propellant modulus) could produce variations in the grain deformation causing each SRM of a pair to perform differently. However, further verification of the deformation analysis is in order before the results are incorporated into the Monte Carlo program.

A third general aspect of the present investigation is that of changes ic grain temperatures produced by high strain rates. Previous research (Ref. 2) indicated that thermoelastic coupling could have a significant effect upon the grain temperature distribution near the burning surface during highly :ransient chamber pressure conditioi~s. This evaluation was based upon the assuwtion of a specified fixed surface temperature. How- ever, during transient conditions the surface temperature is not fixed but depends upon both the combustion chamber pressure and the thermoelastic cov~plingitself which will alter the heat transfer from the combustion zone. Also, the previous analysis considered the effect of surface regression on he energy balance only in an approximate and in- tuitive way. In the present work, the problem was solved more rigorously. The surface regressiorl term is included in tlie energy equation along with the strain rate (thcrrioclast;~) term. 'Ilic surface telnpcrature is calculated by couplinr thc enerlry equ'ltion \

Solution of the lifferentinl equation for the mergy balance of the solid propellant grain subject to tlic appropriate boundary conditions yields the temperature distribution and tlic s~~rfaccregression rate. Tile solutions are ohL nined r~sinrthe numrsri cn? tccl~niclucof Fostcr (Refs. 2 and 6) applied to a circular-pcrforntc.cl casts-l%ondcd grain of fiilite length subjected to an increasing cl~nr~~bcrpl-cssurcb. Comparisons are made of the results with and without thennoelastic coupling. It is concluded from the comparison that the effect of themelastic coupling is very slight. However, it may not be insignificant. Additional research is indicated, particularly with respect to possible vlacoalastic effects.

Another facet of internal ballistic performance variation considered is the ability to assess the quality of ballistic predictions. It is important to know the confidence with which the performance of an SRM as a function of time may be predicted at any phase cf a motor development program. In general, predictability will improve as more motors of a single type are manufactured and tested. As shown in this report, the Monte Carlo program provides an approach to assessing the degree of predictability both for single SRMs and pairs at various phases in the develqpment program. However, further investigation has shown that it may prove impractical to obtain the type of desis~and manufacturing data necessary to establish predictability by this method.

The final portion of the report identifies additions and other changes that have been nade to the two computer programs listed in Ref. 2: the Monte Carlo program and the design analysis program. Each change is identified by page number and line, as listed in Ref. 2. The changes are for the most part refinements in analysis or program logic. A few minor errors in the programs have been found and corrected. The most extensive addition is the improvement of the use of tabular values for burning surface area during tailoff in the Monte Carlo program. This had been accomplished earlier for the design analysis program. XI. TItl MONTE CARLO EVALUATION OF PERIWRMANCE PARAMETERS

The ability to predict the thrust imbalance between a pair of SRE(s firing in parallel, such as on the Space Shuttle or Titan IIIC, as a function of tinre is essential to the total vehicle system desigo. To do this with a reasonable degL-ee of accuracy without large scale testing programs has obvious economic advantage. As has been discussed in Refs. 1-3, the Honte Carlo technique provides a reasonably accurate and economical means of predicting the performance variations of pairs of SRMs. In the present work, the utility of the Honte Carlo program has been extended to permit further evaluation of performance parameters. The Honte Carlo technique is used to generate, as in Refs. 1-3, the performance of a theoretical population of motor pairs. Provision has now been made, however, for the data generated by the Monte Carlo program to be in the form of a magnetic tape containing the thrust imbalance versus time data for each SRn pair. The changes to the Monte Carlo program necessary to prodace the tape are listed in Section VI of this report.

The tape generated by the Monte Carlo program may be analyzed with respect to certain critical performince characteristics by the use of two new computer programs described in this section of the report. Such an analysis is extended for a sample case to a separate investigation of statistical correlations for a number of sets of peifo~mnceparameters. As is discussed, one particular correlation may have important general practical application.

The Imbalance Analysis Computer Programs

Because of the various possible applications for the statistical analysis of the thrust imbalance versus time data and to maintain a certain amount of generality, the analysis was done in two parts.

The objective of the first part of the analysis w,-c, given any time interval during which the motors are firing, to determine the mean value and the standard deviation about this mean of the ahsolutc value of the maximum thrust imbalance and of tllc tine within the prescrihcd interval when this thrust imbalance occurs. This is calculated using a data search routine which is compatible with tl~rdata tape eencrntcd kj the Monte Carlo program. The saw calculations were also made for tllc time rate of change of thrust imbalance within the prescribed time interval.

The computer program for the analysis of imbalance data during specified time intervals is listed in Appendix A. A sample of the computer output is shown in Table 11-1. In addition, computer generated plots may be made by superimposing curves rcprcscnting the thrust imbalances and thrust imbalance rates for each motor pair of the populaticn. This feature is quite useful in deterndning the limits and general behavior of the imbalances for a particular population of motor pairs. ,In cxamplc of this graphical output is given in Figure TT-1. Thc latter example is bascd upon a tape of the

Figure 11-1. CalComp plot of thrust iml~nlanceversus time from 0 to 100 sccs. for 30 pairs of theoretical Titan IIIC SR!k. .performance of 30 pairs of Titan IIIC SR& prepared using the input values described in Table 2 of Ref. 1.

Any number of intervals may be specified for analysis. Also, similar data is obtained as output for the absolute value of the maximum inpulse imbalance and the times at vhich they occur. The absolute impulse inbalance is defined by the relation, t f

where F1 and F2 are the thrusts of the two motors of a pair. maximum absolste impulse imbalance clearly occurs at the last tabulated time ~oint within the interval.

The objective of the second part of the analysis was, given any time during which the motors are firing, to determine the statistical limits of the thrust imbalance and the first time derivative of the imbalance. In this analysis these limits are taken to be fl(2so, where so is the second moment of the distribution about an assumed zero mean and 5 is the two- sided tolerance factor determined by the size of the sample being considered and the specified probability that a specified percentage of the distribution will be included within the limits. A separate data search routine is used to make these calculations and the results are presented in the form of computer generated plots. These plots determine a statistical imbalance envelope for the population. Sample plots are shown in Figures 11-2 and 11-3 based again on the 30 Titan IIIC pairs. Table 11-2 illustrates the computer printout for this data. The computer program used for the time slice analysis is listed in Appendix B.

The second program also generates data similar to that described above on impulse imbalance and absolute impulse imbalance. In the case of the absolute impulse imbalance the upper limit is taken as x + Kls where K1 is the one-sided tolerance limit, s is tile sample standard deviation, and x is the true mean because the +- K2so limits would have little significance in this case.

In order to establish the validity of this analysis, a comparison was made between a sample of 30 Titan IIIC SRM pairs obtained from the Monte Carlo program and data furnished by PlSFC on 21 Titan IIIC flight tests. Figure 11-4 shows that good ajirecment was obtained. Another comparison was made between 30 Space Shuttle type SRf pairs from thc Monte Carlo program and the contractor's preliminary predicted inbalance envelope (Ref. 7) during web action time. This comparison also showcd good agreement with regard to the shape of the envelope as can be secn in Figure 11-5. In- dividual SN1 pair results are sllown in Figtire 11-6 for the period 0 to 105 seconds for the 'Ionte Carlo cvnluation. The contractor's prediction was based on the estimated 3 sipsprezd bctwecn nntched motors considered Figure 11-2. Tolerance limits for thrust imbalance from 0 to 100 secs. for 30 p~irsof theoretical Titan IIIC SRMs (Probability 0.90 for 99.72 of the distribution). b w

0 0 a0 rn

(.000 e - *=

h CY I 0 4-er X - $? cn \ rn m c3 w w= F -= a%.a

Wa -0 =& 9=m- A I 4: ah =a -0 . L?- 3 a=, )-a- m w- I

0 a. e, QD, * Figure 11-3. Tolerance lit,,itr; for first time derivative of thrust i~~t~nlance fro^ .'. to 100 sccs. for 30 pairs of thcorcticnl TiLan lIIC SNfs (Probability 0.90 for 93.72 of t!,~c:jstribution). Table 11-2. Sample printout for time slice imbalance analysis computer program.

ruts TI*€ srtc~.AS r.nEn rr o.so srcs TWE + oa - KZ~SIWA LWIT AOOUT A ZERO MEAN FOR THE rnRust lnsrirnce 1s r,srrrc 04 rw fnt + OR - r2*s1C*~ LlnlT AWTA ZERO REAN FOR fM ~HRUS~IM~ALANCE RATE IS b.b9WE 03 LU~UC

rnts TIHE SLICE MAS rraEr rr 15.35 SECS THc + UZ - KZ*SI&nP LIqIT Ah3UT A ZERO REAN FOR In€ THMUST tMUALAkCE IS C.Sb52E 04 Lw THE + OR - KZ~SIC~ALIYIT Aanul A ZERO YEAY FOR THE THRUST IN~ALINCE RAIE IS 5.0566E 02 LWISK

tnts r13~SLICE *AS TAIC~N AT 15.~9SECS fni aa - KZ*SICM ~I*IT raour A ~EROMEAN FOR THE THRUSC I~~ALANCE IS 4.4337~ 04 L~F TWE + oa - RL*SICNA LIWI ALICUI A ZERO nEAu FOR rn~rnrcusr IM~ALINCE RATE IS b.03se~ u LWIS~C

In15 TIME SLIC~ d4t TAKEN ~1 25.33 SECS rnf Oa - r2*SlcYr r1mlT rftju1 A LiiO HEAN FOR In€ lHauSr IMaACAnCk IS3.&+7cE 04 L~F IMF OM - KZ*SIWA LI-11 UI;LIT A CE*U WAN FOR THL 1nRvsr I~L)ALASC~RATE IS +.eorac 02 LUIICC

THIS Il3E %ICf .AS lACC9 A1 3J.JJ StCS TnC Oi - a2-sltnA LI*IT L~PJI4 Ltao MANFOR IN InRuSr IMBALANCE IS 3.135%€ 04 LbF Tnt t 09 - uZ+SlGu ~I*llAnCdr A ZEi(tl HEAN FOR IHt THkUSI III8ALAhtCE RAIL IS 3.40296 03 LIFISEt

lnlS 11*E SLICE hrS TPCi-4 AT 35.00 SCCS TWF t oa - nz*slcnr ~t-11ra;uT A IL?.O MEA~foh In€ 1wCIuSf IM~ALANCE IS 3.1bZIE 02 LBF IHE OR - K2*51tYA LIMlr l~uitlA LEi4IJ MEAN fUR 1ME IHRUSI lM8ALANCE RATE IS Z.94lbE 02 LI)/sEC

TnlS TIME SLICE m4S TAKE% A1 45.U SECS InE t ca - rz*,lb:?n LIYI~ h;r(rur A IE~O HEAN ~I)R In€ rnnusl lNJ4LIhCE IS 2.uZobE 05 LPF TNE 0Z - UZ*$Ib'4P l.1~~1AOOUI A IEkO PltAh; )OR THE ThMJS1 IN84LANCL RATE IS J.5042L 02 LOF/SLc

ThIS 1lnt SLICE WAS 1hKEN AT 53.55 SECS ThE 01 - KZ*SIGXA ~1.411LaCUT A ZcdG REPN FOR THE THRUST IHdALPkCE IS 2.bBU9E 04 LIP 1Ht JA - UZ*SI(;XA Lllll ALOul A Lid0 MtAh FOR THE THRUST IML(ALANCE RATE IS L.SJ0oE 02 Lw/SCC I

-- -Flight Test Analysis Nonte Carlo Program

I I !

TIME (seconds)

Figure 11-4. Comparison of Monte Carlo tolerance limits tor 30 pairs of Titan 111-C SRMs with flight test analysis for 21 pairs (K -3.68 and 3.86 respectively; probability 0.90 for 99.7% of the distribution). Figure 11-5. Conipari son of contractor ':; ~hrustinhalance during web action tirrc nnalysi :; (Rcf . 7) with tolerance limits obtained by thc flontr Carlo program for 30 space shuttle type SRM pairs (O.?? j)robc~1,i1it.y for ?9.7% of distribution). Figure 11-6. 'I'hrttst iml~nlnnceversus tinlc from 0 to 105 secs. for 30 theort.ticn1 ~):iirr; of Spnce Sllr~ttletype SRtts obtained from tllc Montc Cnrlo program. possible in throat erosion rate, delivered specific impulse, burn rate, and propellant weight. The combination of the variations produced a 2.0% burn time differential which is high with respect to the 0.85% differential predicted by the Monte Carlo program. We attribute the difference in the two results primarily to the much higher burn rite difference (Cv=0.42X) used by the contractor with respect to that used in the Monte Carlo program (Cv=0.06X) .

Correlation of Thrust Imbalance with Thrust Imbalance Rate

For possible use in systems performance analysis, the correlation between thrust imbalance and its first time derivative was investigated at various operating times. This was done for both algebraic and absolute values of the variables using the data from the Monte Carlo program for the 30 pairs of theoretical Titan IIIC SRMs. At each of the 20 time points considered the correlation coefficient rc between the variables was determined. The results are given in Figure 11-7. Also, a linear regression line was establi;? ed for each time point. The slopes and intercepts of the linear regres:.!an lines are listed in Table 11-3.

Examination of she results shows that there is a better correlation of the algebraic values than the absolute values in the region from 10 to 80 seconds, but the correlation coefficient for the algebraic values changes sign shortly before web time (107 seconds). Both the algebraic and absolute values show poor correlation during the 15 seconds preceding and excellent correlation for the 15 seconds following web time. The varying characteristics of the regression lines at the various time points prevent further generalization of the rrsults.

It is possible that better correlation might he obtained by examining the relationship between thrust imbalance at one time and the thrust imbalance rate at a slightly earlier time. Such correlations have not been attempted because the potential applications of such works have not developed as originally anticipated. However, the analysis has suggested correlations between other variables which may have important consequences as discussed below.

Correlation of Thrust Tmbalnncc.-- with Nominal Thrust Rate

Investigation of the correlation between thrust imbalance and nominal thrust-time trace character~sticshave revealed some interesting results. The mean of the absolute values of thrust imbalance at a given time was found to have a correlation coefficient of 0.976 with the obsolute value of the nominal slope cf the thrust-time trace at the same time. This is based on analysis of the previously discussed Titan IIIC Monte Carlo data. A similar correlation of the standard deviation of the absolute values of thrust imbalance with the nominal slopc of the thrust-time trace gives a sample correlation -0efficient of 0.954. Figures 11-8 and 9 are scatter diagrams for the variables. \!hetl the thrust imbalance was correlated with both thrust slopc and thrust, the correlation was not significantly improved. r I 1 1 I .c + + - Absolute Values *+ 0 -Algebraic Values JF * + + .I + + $ + 41 +

I0 + I - +

0 tl f I I

0 4 b 0 0

0 0 6 G L

TIME (scconds)

Figure 11- 7. Correlatjon coefficients between thrust imhalance and tllrust imbalance rate at various times during operatSon for 30 pairs of Titan ITTC SWs. Table 11-3. Results of correlation and regression analysis of Monte Carlo thrust imbalance and thrust imbdance rate for Titan IIIC SRMs ... (AF b AT + c). i Algebraic Values Absolute Values , b Time (sec) b c =I= c r c I 0.5 -0.6900 -179 -0.0774 1.1291 8114 0.1590

10.6 -74.5114 -2895 -0.8109 50.6067 3871 0.49C5

20.0 -59.8190 -1872 -0.9242 56.6595 528 0.7763 -1- 30.0 -7.9085 -3215 -0.8043 3.3994 4329 d. 4C01

40.0 -70.3361 -24 1 -0.9054 6.0127 1434 0.7739

50.0 -90.7287 -820 -0.8476 67.9270- 1960 0.63 26 60.0 -147.0134 -463 -0.5535 101.9460 4138 0.3736

70.0 -101.8451 171 -0.8339 88.4369 1603 0.6678

80.0 -54.3596 -412- -0.8092 44.2926 1659 0.6154 90.0 -46.6494 -lob9 -0.5239 46.2998 2862 0.5319

100.0 -53.3827 -992 -0.5424 47.8307 3028 0.4708 -- 10' ' -4.5388 -232 -0.211.0 3.6304 0.2421 - -- 1( -0.0436 1187 -0.0907 t-i-0.0171 1::: -0.0526 ------i03.0 0.5521 3362 0.7057 0.5111 4803 0.6596 r 104 .O 0.7497 13709 0.7070 0.3391 22439 0.3124 ..-- 105.0 1.2287 3778 0.4265 0.8371 42578 0.4449 - b 106.0 -27.8496 2534 -0.9731 26.9581 2929 0.9324 *- - 110.0 -21.2082 --1279 -0.9858 21.1880 27 3 0.9651 120.0 -7.4078 -14 1 -0.9863 7.4302 -i23 0.9670 -- . 125.0 -0.5615 3520 -0.3174 0.2900 4899 0.1642 1 -.

I + I J I

t - 0.5372 IfaVI + 5056 a l~~l Correlation coefficient = 0.9540

I I 8 c + + 0 10 2 0 30 40 5 0 60 70 Cc ABSOLUTE VALUE OF TIME RATE OF CHANGE OF THRUST (lbf x lom3) V) Figure 11-9. Scatter diagram: standard deviation of thrust imbalance for 30 theoretical Titan IIIC SRM pairs versus the absolute value of tho mean time rate of change of thruat for the 60 motors. The nominal slopes of the thrust-time trace were determined by colputi3g the mean of the slopes for the 60 SRHs at each ti= point.

It should be noted that the 20 time points on which the latter correlations were based vere not randomly selected; they were selected to cover all portions of the trace for the previous analysis of the relationship between thrust imbalance and imbalance rate. Although the time points vere not selected with a knowledge of the values of the present variables of interest, the manner of selection introduces a question on the random nature of the variables (thrust imbalance and thrust slope), so that further investigation of the correlation should be accomplished before attempting additional statistical analysis. However, it appears that there is a strong relationship between the variables in question.

The relationship between the thrust imbalance and the slope of the thrust-time trace has at least tM important potential applications. The first can be accomplished only if the regression analysis proves to be general, that is, it must be shown that it is reasonable to expect that the same regression line fits all SRHs or at least members of a family of SRMs. If it does, the thrust imbalance predictions could be based directly on the slope of the trace of a new SR?! without Monte Carlo evaluation. Also, although it has been shown (for Titan IIIC's, at least) that the Monte Carlo program gives reasonable representation of the SRM pair idalance, it would bc a logical step to test the correlation directly against actual performance d~ta. This suggests a second application - that of using the regression analysis to establish imbalance scaling relationships betvecn different SWs. The regression analysis for the Monte Carlo program wo~ld be compared to that for the actual moror to establish the scaling relationship. The result would then be applied to a regression analysis of a Monte Carlo evaluation for a new SRT, (e.g.. the Space Shuttle) to establish the imbalance limits for the new SRN pairs. This could be done whether or not the same regression line Pics the two types of SRMs. Investigation of these applications is beyond the scope of the current research. 111. EFFECT OF GRAIN DEFORMATION OE INTERNAL BALLISTICS

The circular-perforated (c.p.) portion of the grain of an SRM of the Space Shuttle type may deform as much as 1.2 inch in the radial direction under pressurization from the chamber gases. At a 30-inch bore radius, this results in as much as a 4% increase in the initial burning perimeter of the grain bore. This can affect the chamber pressure and the apparent burning rate of the propellant. The effects will decrease as burning of the propellant web progresses because the displacement of the burning surface relative to its unpressurized position will decrease. The ability to quantitatively assess these particular effects of grain deformation on internal ballistics should improve predictability of performance of individual motors, especially for the first motor of a new design where the "scale factor" on burning rate is uncertain.

Analysis of Grain Deformation Effects

The underlying hypothesis on which the ballistic effects of the grain deformation is based is that at fixed pressure the regression rats r of the pressurized propellant surface just beneath the heat-affected C zone is independent of the state of strain. To show that this is plausible, first consider the burning perimeter of a sector of solid propellant. The burning perimeter consists of a thin zone (liquid and/or solid) in which the physical properties are degraded to a state where the propellant develops negligible shear stresses. Therefore the regression rate of this zone (r ) should be independent of the precise state of strain calculated just benefth the degraded zone which will be somewhat thinner than the solid phase heat- affected zone. The concept needs experimental verification, but appears consistent with the known features of the solid-gas transition region,

In the present analysis, it is assumed the thermoelastic coupling discussed in Section IV on this report is absent. Thexmoelastic coupling may possibly alter the depth of and the temperature distribution within the heat-affected zone and thereby make burning rate sensitive to strain. However, as discussed in Section IV, it is expected to be significant only during highly transient conditions; i.e., during ignition and very rapid depressurization.

If we now consider a control surface (See Fig. 111-1) separating the regian of degraded propellant from the remaining propellant, a quasi-steady state mass balance across the heat-affected zone, which for the purpose of this analysis is considered of zero thickness, yields

where pm is the density of the degraded propellant and p is the density of the unheated propellant just outside the heat-affected z%e. It is apparent that r and p are pressure dependent but should not depend on the state of strainmundernzath the degraded surface. Also, when as in the principal case of interest here, Poisson's ratio v is close to 0.5, it is intuitively clear and may easily be shown that the density of the unheated propellant can be expressed as a function of pressure (and modulus) only since Pressurized

Heat-Affected Zone (Control surface) Unpressurized

Propellant '-\

Figure. 111-1. Sector of cross-section of solid propellant with and without pressurization and combustion.

Figure 111-2. Grain genrnctry notation for analysis of solid- propelln.lt strains. Tile insulation and liner may be consLdered a part of tile propellant. vhere c is strain, P is pressure, E is modulus of elasticity and 8, r and r refer to the tangential, radial and axial directions, respectively. Thus Eq. 111-1 should govern the burning rate of the propellant regardless of the state of strain and r is a function of pressure only (for fixed Initial grain temperatureCand erosive burning characteristics) . Next, the ef feet of the deformation on internal ballistics is examined, We consider only a circular-perforated grain for which tze is independent of the tangential coordinate at one axial position. The mass generated per ualt tlme per unit length at the control surface separating the heat-affected tone of the propellant from the unheated propellant is given by

where S is the burning perimeter, a is the linear ci.:.fficient of thermal expansion and the subscript o refers to the unpress: rized and unheated (reference grain temperature) state of the propellant. The approximation of Eq. 111-2 has been used in Eq. 111-3 to determine the effect of pressurization on the unpressurized density, and that density has been further modified for the small effect of thermal expansion between the standard temperature T at which density is measured and the actual grain tempe- Ref rature T ~r' A second modification of customary ballistic analysis is the way the tlme At required to burn an increment Ay nonnal to the surface is calculated. The usual relationship is

where r is the burning rate deduced iron strand burners or small ballistic test motors modified with a "scale factor" reflecting the estimate of the change in burning rate between the strands or test motors and the SRn under consideration. The comparable equation to Eq. 111-3 used with Eq. 111-4 is

For the present we disregard any scale factors and devise an appropriate relationship for At in terms of by and r applicable to the deformed grain problem. First, note that the r shoufd be the rate at which the unpressurized and unheated propellant ?egresses because the Ay is to be used for further computations of surface area as an increment of undeformed prapellant. Next, it is assumed that the changes in length (R) of the grain produced by pressure and thermal loading are negligible. As shown in Ref. 4, longitud- inal (axial) grain strain due to pressurization is on the average over the length of the grain approximately a half order of mgnitude less than the corresponding tangential strains. The axial strains are negative while the tangential strains are positive. Although a quantitative assessment has not been made, when heating of the surface of the propellant is considered, the axial strain at the point of interest; i.e., just beneath the heat-affected zone, will become less negative because of the restraint of the unheated propellant to the expansion of the heat-affected propellant. "lis will decrease the extent of any change in length of the propellant relative to the increase in perimeter over that due to pressurization alone. Referring to Fig. 111-1, we obtain from a mass balance between deformed and undefonred propellant

P S 1r P S L(l+~~)r~ (I11-6) pooo- PC0

The sme result may also be obtained by direct combination of Eqs. 111-3 and 111-5. With this ro, Eq. 111-4 then applies to the deformed grain situation.

Data to give r are obtained frm ballistic test motors or strand burners. As previo&ly noted, no scale factor is considered. Indeed, it is suggested that the grain deformation may account for a significant portion of the scale factor. The strands are under essentially a state of hydrcstatic compression. The same is very nearly true for the typical ballistic test motor because of the rigidity of the relatively thick cases and the relatively thin webs. It should be recalled that rc should be independent of the state of strain. However, the burning rate of small test motors (or strands) is deduced by div!ding the undistorted web thickness (or strand length) by time. Therefore, to determine r the regression rate c of the propellant surface just beneath the heat-affected zone of the SRM, the test motor (or strand) rate must be reduced by the multiplying factor [1-P(1-2v)/E]. The rc, of course, is subject to the usual laws governing its relationship with initial grain temperature, pressure and velocity (erosive burning).

Simplified Determination of Grain Deformation

For a first analysis of the effect of grain deformation on internal ballistics, a simplified model of the deformation was developed. We are concerned for the present only with the modification of the internal burning surface of a circular-perforated grain for which it has been shown that the important parameter is the tangential bore strain E The E will change 9' 9 as burning progresses and also will vary in general with axial position. Smith (Ref. 4) working under the direction of this project has shown that as an approximation the E may be considered independent of axial position. 6 Smith extended the method used by Vandenkcrckhove (Ref. b) for calculation of grain stressesin case-bonded propellant with rigid motor cases to the calculation af propellant strains in non-rigid motor cases. His results for tangential strains at the bore are given by

where P is the radial stress at the propellant-case interface as given by e Here the geometric notation is given by Fig. 111-2, Pi is internal pressure, v Is Poisson's ratio and E is modulus of elasticity. The subscripts p and c refer to the propellant and motor case, respectively. Details of the analysis are given in Ref. 4.

Snith compares results for tangentlal, radial and axial strain with a aore exact solution obhined using a computer program devised by Brisbane (Ref. 9). The two solutions for the tangential strain are illustrated in Figs. 111-3a and b for a 140-inch diameter motor segment 300 inches long for various web thicknesses. The results are given in non-dimensional form and are applicable to -tors of various sizes vith the same ratios rw/Re, r,/% and Illband material properties. Study of Figs. 111-3a and b shows that the simplified solution, which assumes the same strain at all axial positions, in general, overestimates the strain with respect to the more exact solution. The worse case is at the largest web thickness to outside radius ratio (TAUfRO). Here, the strains calculated by the two methods are within 30% over two-thirds of the grain length and the comparison improves rapidly with decreasing web thickness (increasing distance burned). It should be noted that the strains calculated by both Refs. 4 and 9 are thme due to pressure loadings only. Expansion of the propellant in the heat-affected zone will tend to produce additional tension of the solid surface just outside the heat-affected zone (the surface of interest). This will compensate somewhat for the apparently excessive strains calculated by Eq. 111-8.

We feel that the approximate analysis is adequate for a first approach to analysis of the described effect of deformtion on internal ballfstics. It will permit rapid calculation of the bore strains fcr various configura- tions, material properties and distances burned. Additional investigation performed by the authors under this project using the program of Brisbane shows that tangential bore strain, the quantity of primary importance in the analysis, is changed by less than 5% by solid-phase thermal gradients so that them1 effects may be neglected in this first analysis. The reason the thermal gradient changes the tangential bore strain so little is that the gradients are confined to only a very thin zone at titc surface. The bulk of the strain is due to pressurization of the entire propellant web.

Modification of the Desigrn Analysis Computer Frog=

The design analysis computer program of Ref. 2 has been modified to permit computation of the internal ballistics with the deformed grain. The program is listed in Appendix C along with instructions on preparation of new input data cards and a sample problem solution.

Bore deformation of c.p. grains or grain segments is evaluated by the simplified approach discussed earlier. Only deformation of the internal burning surface of c.p. grains or segments is considered. The calculation of the deformations of the ends of c.p. grains and star grains is a formid- able task. Any solution would appear to be inconsistent with the simplified analysis of deformtion and internal ballistics being presented and well beyond the scope of the present research. The design analysis treats the ends of c.p. grains and star grains with an unmodified burning ratio. This is accomplished by computing a separate burned increment Ay' applicable to c.p. grain ends and the entire star grain. The Ay' (computer symbol YETA) E: from Ref. 4 0 E e (from Ref. 9 r,/R, = 11140 Ep = 550 psi Ec = 30 x lo6 psi = 0.499 v~ vc = 0.300

Figure 111-3a. Comparison of apprcximate solution for tangential horc strain (Ref. 4) with finite element solution met1102 of Ref. 9. ce from Ref. 4 from F.ef. 9 r,/R = 11140 Ep = 550 psi E, = 30 x 10 psi up = 0.699 v, = 0.300

Figure 111-3b. Continued conipnrison of approximate solution for tanpcntinl bore strain (Ecf'. 4) with finite element oolution metliod of Ref. 9, is related to the Ay applicable to the c.p. grain lateral surface by the equation

The Ay' thus calculated gives the regression of the star grain surfaces and ends of the c.p. grain in the same time increment that the c.p. lateral surface regresses a distance Ay. The modification of the bore surface of the c.p. grain due to deformation is accounted for in the mass generated equation by the use of the program variable XETH which is the fraction of the total burning bore surface associated with the c.p. grain. The factor (1 + XETH*ETHETA), where ETHETA = €8, multipli:~ the undeformed total burning bore surface area when the deformation option is elected by setting ISO-1. The program will calculate the internal ballistics without grain deformation when ISO=O, The program changes that have been made in the design analysis of Ref. 2 to permit the alternative computations +ire identified by the symbol , in the left-hand margin of the listing. Also, the modifications of the desigv analysis program discussed in Section VI of this report have been incorporated into the program listed in Appendix C.

In modifying the program, it was found convenient for the purposes of minimizing the number of modifications and for preserving the ability of the program to make calculations with no deformation to change the usage of SUMAB In the program. Before the modification (or presently when ISO=O) SUMAB represents the surface area at a given distance y burned normal to the entire surface. With the modification (when ISO=l) because of the difference between y and y' the SUMAB represents the area at a distance y burned over a part of the surface and y' over another part. This creates an error in the calculation of the weight of propellant burned (WP2) which is obtained by integration of SUMAB with respect to ;-alone. Therefore the calculation of WP2 should be disregarded when ISO=l. To calculate hP, the arithmetic average of WP2 and the weight of propellant burned (WP1) calculated from the mass discharged rate, the program now sets WP2=WP1 when ISO=l. Comparisons of weights calculated by the two methods can still be made by comparing WP1 calculated with ISO=1 with WP2 calculated with ISO=O. The latter is stili a valid calculation for total weight of propellant burned even with grain deformation. Also comparisons have been made for the two sample cases discussed next and the UP1 is not significantly different for ISO=O and ISO=l.

Comparison of Theoretical and Experimental Results

To test the validity of the hypothesis regarding the effect of grain deformation on internal ballistics, the modified program was used to predict the performance of two entirely difrcrent rocket motors and the results compared with actual performance data. In czch case the burning rate used to predict the performance was that obtained from ballistic test motors or strand ourners; no scale factor was applied.

The first comparison made was for a Titan 1IIC/D configuration. The basic input parameters used for the pl'ediction are those given by the mean values in Table 2 of Ref. 1 with the following exceptions: The propellant density had been changed to 0.06325 15m/in3 because additional data on this has become available. The strand burning rate coefficient obtained from a number of batches of propellant loaded into the same SPA (No. D5) for which the comparison is to be made was used for the prediction. The burning rate coefficient used (0.06466 with no scale factor) was deduced from Ref. 10. Five burning slot faces were assumed in the program calculations. A table of tabular values was prepared so that the burning surface areas calculated by the unmodified program more nearly match the known geometric surface versus distance burned characteristics of the Titan IIIC/D configuration over web action time. Aft-end closure propellant surface representation was included in the tabular values,

Propellant and motor case properties used in the analysis of the deformed propellant grain are given on Fig. 111-4. The figure is a comparison of the program results for aft-end stagnation chamber pressure versus time with and without deformation and with the results of analysis of actual test data given for Titan IIIC/D motor number D5, in Ref. 10. This particular motor was selected because its performance is more or less typical of the Titan XXIC/D and because of availability of strand burning results on the same propellant loaded into the large motor,

A similar comparison of theoretical and experimental results was made for the Castor TX354-5 SRM. This is a 31-in. dia. motor with an entirely c.p. slotted gra4.n which is described in detail in Ref. 11. The SRM is also described briefly in Ref. 12 where input parameters applicable to an early version of the design analysis program are identified. A complete listing of input values for the modified design analysis is given in Table C-1 of the present report where the Castor motor is used as a sample problem to illustrate use of the program. The input values differ from those used in Ref. 12 because of the program changes that have been made and because we have improved our representation of the grain geometry, particularly with respect to use of tabular values. Again, t'_:. burning rate coefficient used has no scale factor between the Castor motor and the small ballistic motor data from which the coefficient was deduced. Vacuum thrust results are compared (See Fig. 111-5) for the Castor motor. The vacuum thrust results for the actual motor were based on analysis of static test results and are tabulated in Ref. 11 as typical performance values. The strand burning rate data is also typical but are not otherwise identifiable with the specific large motor.

Analysis of the two comparisons shows that the theoretical predictions are substantially improved by consideratior, of the deformations. Additional comparisons are needed to confirm the efficacy of the new model. Modifics- tion of cther parameters can also make improvements in prediction possi5le. A prime example is the use of the bur-iing rate scale factor. However, the scale factor is most often a~pliedwithout specific theoretical justification and usually is only accurate after data on actual SREI firings has been obtained. On the other hand, the deformation effect has a basis in theory. If it can be further validated, it will not only by itself improve prediction capabilities but also clear the way for more accurate assessment of other specific ballistic phenol~~enasuch as erosive burning and nozzle throat erosion. i'or example, consider the discrepancy between the prediction with grain deformation and the test results for the Titan IIIC/D. It is obvious that the areas under the two traces are not the same. This is suggestive of a difference between the nozzle throat erosion rate used in the theoretical prediction (time independent, pressure and size dependent) and the actual erosion rate. The application of a scale factor to the theoretical burning rate coefficients

would not improve the area discrepancies and would mask the true throat erosion and/or buildup phenomena.

Variations in the propellant properties used in the deformed grain program can affect the results. Likewise variations betveen SRHs of a pair firing in parallel can contribute to thrust imbalance. As shown in Ref. 4, the effect of propellant ~nodulusis especially important. Bovever, additional ccafirmation of the deformed grain model, preferably including direct expera.mental verification, should be obtained before the effact is hcorporated into the Monte Carlo program.

Equations 111-3 and 111-7 which give the basic burning rate theory of the rode1 may also be coupled to more rigorous descriptions of the grain deformation and/or more rigorous models of the other aspects of internal ballistics than those used in the modified design analysis program. IV. THEmoELASTIC ANALYSIS

The problem to be discussed in this st~tionis the effect on the solid propellant burning rate produced when heat is generated by mechanical deformations ~fthesolid propellant. The basic theory and the numerical solution procedure used in the thermoelastic analysis as vell as the results which have been obtained are presented.

The basic formulations and concepts associated with the coupling betveen thermal and mechanical loadings can be traced to the vork of Duhamel (Ref. 13). Other early investigators include Lord Kelvin (Ref. 14) and Joule (Ref. 15). It was Duhamel vho developed the relationships used to express the distribution of strain in a body subjected to differential heating.

Basic Theory

Consider a sprall element of an isotropic elastic solid subjected to an arbitrary stress which is redfrom its surroundings and subjected to a temperature change, AT. The additional strain in the elenent is given by the tensor BATbi /3, where 8 is the coefficient of volumetric expansion and dij is the Kronil cker delta. From this it follows that if the strain tensor is originally given by eij, then the new strain tensor would be eij - BATdij/3 and this would be the strain used in the constitutive relation. The effect of a nonuniform temperature distribution becomes equiva- lent to that of a body force per unit volume, given by - ($/s)V(AT), rl being the compressibility and V the gradient operator. Thus, once AT is known from the solution of thc heat conduction problem, the system of equations for the disp:ace~ent field is fully defined. As can be seen, this approach postulates the effect of the temperature gradients on the state of strsin, hut assumes that the heat transfer is totally unaffected by the state of strain. This result is upheld rigorously when the medium is in mechanical and thernal equilibrium, but its application to time dependent problems is not sati~factory. However, Duhamcl and Neumnn do suggest that a term proportional to Jfkk/at, the rate of change of dilatation, be included in the heat conduction analysis. Tile present work investigates this tern's effect on a tl~ermoelasticanalysis, and in turn, on the propellant burning rate.

Much effort has been expended in recent years to obtain solutions for problems involving themmechanical coupling. This has been due, at least in part, to the necessity for structural analysis of bodies subjected to a nonisotheml, transient tenperature distribution. One of the better examples of a body subjected to this type of loading is the solid-propellant rocket motor. The general analytical formulation of the problem is well known and can be found in 7;arious publications, of which Ref. 16 is typical. The prcblem to date has hccn obtaininfi solutions for problems whosc geometry and loading conditions, both thermal and mechanical, are sufficiently general to he of practical interest. Setftral approaches appear to be suitable for analyzing problems in- wolving thennomechanical coupling. The first possibility is that of ob- taiuing analytical solutions of the governing equations. There have been several such solutions in the literature in recent years. However, the approach to now has been severely limited as to choice of geometry, loading and boundary conditions. The second possibility utilizes the finite elemnt method and entails the approximation of the full set of the governing, coupled, nonlinear partial differential equations by a set of nonlinear algebraic equations which are then solved nllmerically. This finite elemznt method is considerably more flexible than the first approach vith regard to the variety of geomtrfes, loading and boundary conditions which can be treated and h3s been applied to several classes of problem. Bouever, this nonlinear finite element method also appears to be soeevhat limited due to the colblexity of solving large numbers of simultaneous nonlinear algebraic equations. Also added complexity is introduced if it is desired to intresti- gate the effect of rwnlinear aaterial properties (except for materials which obey relatively shple matbtical laus), arbitrary loadings, and boundary conditions.

A third possibility is to approximate the transient nonlinear solution by a series of linear solutions. In this approach the variable, time, is included in what basically can be described as a time stepping procedure. At each time step, the initial them1 and mechaaical states are established based ou the conditions which are specified either by a direct input or by the conditions which existed at the end of the preceding time step. Ihe solution then proceeds by obtaining the temperature distribution in the body for the given thermal loading. The thertal load may include applied temperature distributions on the external boundaries and/or at points within the body, heat flux through the boundaries, and convective heat sources (or sinks) distributed throughout the body. The internal heat sources are particularly important since they are used to account for the coupling phenomena. Ihe solution then proceeds by using the coraputed temperature distribution as zn input to the stress analysis portion of the coqmtational procedure. Determination of the deformations, strains and stresses due to the cosbined loading is then made. @rice this is done for a particular time interval, the coupling term, which is a function of the marerial prcperties (which may be functions of teuperature), the local temperature, and the local strain rate, can be evaluated. With this information the computation then proceeds to the next time step where the coupling term is included as an internal heat source in the heat transfer analysis.

This third approach, the one used in this research, appears to eliminate most of the problems associated with arbitrary time dependent loadings, temperature dependent material properties, and also some of the numerical coxcputational problems associated with the nonlinear finitc element method described above. This reduces the numerical computation difficulty because the problem is now formulated as a set of linear algebraic equations. The main objection to this approach is obviously the linear modeling of a nonlinear problem. However, if the mathematical modeling of the geometry, the loading, the material properties, and the time step is done carefully, this method is sufficiently reliable for the solution of a considerably larger group of problems than was heretofore practical.

The work presented here is restricted to axisynmretric bodjes. This does not ilply a restriction of the method, but is made because of the applicability to the problem at hand.

Problems in thenmelasticity require not only the solution of the equations of elasticity, but in addition those equations necessary to describe the thed state of the body being analyzed. In nmny problems the them1 state can be sufficiently defined by determining the distribution of temperature throughout the bdy and also any changes in this distribution vith respect to time. Quite often as in the present vork it is convenient to obtain the temperature distribution from the conservation of energy equation. The discussion vhich follous will be primarily concerned with the energy equation since it is the equation vhich will be used to determine the influence of the thewloelastic coupling on the propellant burning rate. Tbe elasticity equations are not presented because they are used in a more or less classical manner.

As in Ref. 2, the conservation of energy equation for a control volume wving with a velocity r(t) with respect to an observer situated on the boundary (~0)of the control volurae and modified to account for voluuetric heat release or absorption within the solid phase is

where x is measured radially outward from the surface, ;represents the sum of three translational strain ccmponents at the point under consideration, and the entire last term represents the heat dissipated by elastic deformations.

The tvo terms which are >f particular interest in the present analysis are the first and last terms on the right-hand side of Eq. IV-1. As mentioned above, the last tnrm represents the contrib~tionof heat from the elastic deformation of the body and can be calculated directly if the instantaneous local teqerature is known as well as the time rate of change of the translational strain components. Once the initial temperature and strains are prescribed. the temperature at any point within the control volume at any time is obtained from the solution of Eq. (IV-1). The strain rates can then be calculated at any instant of time once the mecknical load is prescribed as a function of time. Hence, with regard to its calculation, the thermoelastic coupling term offers very little difficulty.

On the other hand the first tern on the right-hand side of Eq. IV-1 requires that the instantaneous value of burning rate be calculated. One might initially propose that the burnfng rate r(t) could be easily obtained from the pressure ?(t) which is required for the therunelastic term by ex- tending the "standard" burning rate law for steady-state burning to the transient problem as

However, as pointed out in Ref. 5, the instantaneous burning rate r(t) may differ greatly from that obtained from a steady-state analysis due to a highly complex heat feedback aechanism between the various regions in and near the combustion zone. Therefore, it is necessary to construct a model to account for this thermal feedback mechanism in order to properly evaluate r(t).

(lne such model for determining the transient burning rate developed by Zeldovich and Novozhilov, referred to as the 2-N model, is described in Ref. 5. The present work uses this model to evaluate r(t) which is then incorporated into the conservation of energy equation. As in Ref. 5, the 2-N method uses measured steady-state burning rates as a function of pressure and ambient temperature. This data is then used to construct the heat feedback function from the gas to the solid in the proper instantaneous form to be used to study transient burning problems. The assumption which is made to allow use of steady state data is that the rate processes in the gas phase and at the propellant surface can be considered quasi-steady in the sense that their characteristic times are short. compared to the pressure transient. The validity of this assumption is demonstrated in Ref. 5. What develops from this is that the steady state heat flux and the transient heat flux have the same functional form, which is written for the transient problem as

vhere Ts is the instantaneous solid surface temperature and To is a fictitious temperature which exists infinitely far from the solid surface. Now, assuming the pyrolysis law to be of the Arrhenius type

and the empirical expression for r in terms of p and To to be given by where up is the temperature sensitivity coefficient at constant pressure, which for this work .'.s expressed as a linear function of pressure by

be can solve simultaneously Eqs. IV-1, 4, and 5 to obtain the unknowns r, Ts and To.

Numerical Solution Procedure

The finite element method as used in this study is based on the work 'of Becker and Parr (Ref. 17). The basic program used for the finite element analysis is that of Brisbane (Ref. 9) and is primarily an extension of that given in Ref. 17. The method used is also quite similar to that given by Wilson and Nickel1 (Ref. 18).

The use of the finite element method involves the division of the region of interest into a number of subregions which normally have a relatively simple geometric shape. Usually a polygon is used for two-dirsensional problems and quite often triangles are used. In this work the problem is axisymmetric and a circular ring element with a triangular cross-section is used. A typical element is shown in Fig. IV-1. The subregions, or elements, are connected at discrete points called nodal points. These nodal points are usually at the corners of the polygon as will be the case in the present work.

For the heat conduction problem a simple relationship for temperature and its first derivative as a function of position within the element is assumed. One of the more common assumptions and the one made here is that both temperature and its first derivative are linear functions of position within the element. Hence, for the axisynnnetric problem the temperature and its gradient can be written as

and

It should be noted that T(x,z) and aT(x,z)/at are considered to be evaluated at a particular time; e.g., t = t*, and hence, no explicit time dependence is expressed in the above equations.

Once this approximation has been made, it is substituted into the basic field equations which describe the heat transfer process. The result is a set of linear algebraic equations, the unknowns of which are the temperatures which exist at the nodal points. Figure IV-1. Typical finite clement for axisymtric problem. Basically the same procedure is used for the elasticity problem except that the displacements are approximated instead of the temperature. Another linear approximation is used for the displacements. This approximation is then substituted into the field equations for the elasticity problem and a set of algebraic equations is obtained for the nodal point displacements. These nodal displacements are used to evaluate the constants in che linear displacenent assumption which is then appropriately differentiated to obtain the strains at a particular time.

It is now appropriate to examine in more detail the computational procedure which is used and the manner in which the coupling between the thermal and mechanical loading as well as the burning rate term are introduced into the numerfcal computational process. The solution technique consists of solving separately the energy equation and the elasticity equations at each tine ctep. Outputs from each are then used as inputs to the other, thereby forming a closed loop computational scheme. The calculation of the temperature dependent term which appears in the stress-strain relations is common to both the uncoupled and coupled sol~tions. It is calculated from the apparent element tenperature which is obtained from a geometrical weighting of the four nodal point temperatures which are computed in the heat conduction part of the analysis.

The computation of the thennoelastic coupling term which appears in the energy equation, Eq. IV-1, requires some knowledge of the deformation history of the body in order to compute the strain rate. For the work presented here it is only necessary that the state of strain at time, t - At, and at tine, t, be known, where At is the time incremeat between steps. This requires that the strain components for each element be retained at the end of each time step. The old values are then replaced by the new state of strain which is then used in the next time step. The initial strain has been taken to be zero, but no significant problems arise if a non-zero initial strain conditioa is used. For an isotropic body, the therroelastic coupling tenn in Eq. IV-1 involves only the dilatational coqonents of the strain tensor. The axisymetric solution requires that three strain components be kn;wr, for each element. However, the coupling tenn in Eq. IV-1 contains tlic r;um of the three dilatational components so that only the trace of the strain tensdr need be retained for each element. ?his results in a considerable savinp of computer storage. The strain rate for each elerent can then be corputed by taking the difference between the trace of the element strain tensor at time, t - At, and at time., t , and dividing by the increment At. The resulting linear approximation is then transferred to the heat conduction analysis. The remaining variables in the coupling tern involve the temperature and material properties of the element. ?he temperature is knohm from the calculations described above and the material properties are either prescribed constants or are given in tabular form as a function of temperature. If the material properties are given as constants, a direct substitution is made and the coupling term is evaluated directly. If the material properties are given in tabular form, a linear interpolation routine is used to obtain values not given in the table. The term containing the burning rate requires that the 2-N model be used to evaluate the instantaneous burning rate r(t) based on the eldsting surface temperature at the end of each time step. The burning rate term is then added to the heat balance for each element. The heat flux at thpl surface is computed using Eq. IV-3 and this becomes the heat flux boundary condition for the next time step,

The numerical example presented here is a simulated ignition of a large SRM. The motor is modeled as a straight circular perforated grain wfth ar, initial bore diameter of 60 inches and a web thickness of 40 inches. The grain is encased in a 0.5-inch thick steel case. The motor is subjected to a time dependent increase in pressure on the bore and end surfaces, but burning is allowed in a dfrection normal to the bore surface only. The variables in the analysis are assumed to be distributed symmetrically about the motor centerline.

Finite element model

It is well known that solid propellants are extremely good insulators and that the heat-affected zone of a burning propellant is of the order of about 10 mils. The finite element model is constructed to analyze this region with reasonable accuracy. A schematic of the finite element model is shown in Fig. IS'-2. Finite elements with a radial thickness of 1 mil are used for the first 20 mils of the propellant web (Steady state temperature profiles show that this will mre than adequately cover the heat- affected zone under most normal operating conditions). The remaining web is modeled using finite elements which have a radial thickness of approximately 10 inches. The steel case is modeled by two elements 0.25 inches thick. The vertical dimension of each element is 10 inches. The coarse- ness of the grid in the vertical direction is justified because the variables under consideration (i.e., temperature, displacements, strains, etc.) do not have gradients in the vertical direction. To further reftne the numerical calculations, each rectangular cross section element described above is subdivided into four triangular elements of the type illustrated in Figure IV-1.

As the propellant regresses, a new finite element grid is constructed at each time step. The first 20 mils of the propellant is always modeled by elements of the size described above. The remaining elements, except those associated with the steel case, become decreasingly small as the web thickness diminishes. This is done to assure that the same degree of qccuracy is maintained with regard to the propellant In or near the heat- affected zone.

Initial conditions

The initial conditions are prescribed as: 1) no initial strains exist; 2) an initial uniform pressure exists over the entire propellant surface; 3) the initial temperature distribution corresponds to that which would Figure IV-2. Schematic of finite element model (no scale). exist if the propellant were burning under steady state conditions at the existing pressure. The equation for this distribution is

where T is taken to be the initial propellant bulk temperature; 4) the initialoheat flux at the burning surface is obtained using steady state values in Eq. IV-9.

Boundary conditions

The boundary conditions are: 1) The propellant and case remain bonded together, but the propellant-case assemblage is unconstrained; 2) the temperature of the bore surface at any instant of time is uniformly distributed: 3) only the bore surface la subjected to the heat flux calculated from Equation IV-3 (i-e., the r?-.!ds of the propellant are assumed to be insulated); 4) the pressure at any instant of time is uniformly distributed over both the bore surface and the ends of the propellant. The pressure gncreases exponentially according to the relation

P = Pmax - (pmaX-pi) exp [-fit / (pmex-pi) ]

Numerical Results and Discussion The numerical results presented here show how the propellant burning rate is affected by thermoelastic coupling and also how these effects are modified when certain parameters are varied. The results do not represent a detailed parametric study but are presented to show the general influence of certain parameters.

For comparison purposes, a "baseline" case was established as a reference for making comparisons. Table IV-1 shows the basic data used for the baseline analysis. The results obtained for the baseline example are shown in Table N-2 for 50 milliseconds. The time period chosen is sufficiently long to allow the dynamic effect on burning rate to decrease to within a few percent of the quasi-steady state rate and, as can be seen in Table IV-2, the influence of thermoclastic coupling on the dynamic rate has practically vanishcd. The notation for burning rate in Table IV-2 and in later tables, rSS, rt)wlo and rtIw represent the quasi-steady state rate, transient rate without thermoelastic coupling and transient rate with thermoelastic coupling, respectively. This table indicates that the influence of t!iermoelastic coupling is present but that its si2nificance with regard to burning rate decreases very rapidly with time. Table IV-1. Baseline example data.

Burning rate coefficient (a) Burning rate exponent (n) .35 Pyrolysis law (Eq . 1-4) constant (As) 882.28 in/sec Activation energy (Es) 28800 BTU/lb-mole Thermal condtict ilrity (A) 5.1 x 10'~BTU/sec-in-OR Density (PI .063657 lbm/in3 Specific heat a constant volume (c) .24 BTU/lbrn-OR Modulus of elasticity (E) 550 psi

Linear coefficient of thermal expansion (a) 5.27 x ~O'~/OR Poisson's ratio (v) .499 Initial pressurization rate di) 14000 psilsec Initial ambient pressure (Pi) 14.7 psi

Maximun chamber pressure (Pmax ) lCC0 psi

The first parameter to be varied is the pressurization rate. This is done by charging the initial pressurization rate used in Eq. IV-10 from 14,000 to 28,000 psifsec. TL-, results obtained for this example are shown in Table IV-3.

The effect on the significance of the ,nermoelastic coupling produced by changing the pressurization rate can be seen in Fig, IV-3. This figure shows the logarithm (base iO) of the absolute value of the percent difference between the transient rate without thermoelastic coupling and the transient rate with thermoelastic coupling for both the increased pressurization rate and baseline examples as a function of time. As can be seen, increasing the pressurization rate has a sip-ificant effect on the thermoelastic phenomena. This result is consistent since the thermoelastic term in Eq. IV-1 has a strain rate factor which is proportional to the pressurization rate.

The second parameter altered was tl~celastic modulus of the propellant. Not only docs the elastic modulus affect the strain and hence the strain rate, but it is also a multiplying factor in the tt,ermoelnstic term of Eq. IV-1. The elastic modulus was changed from 550 to 2,000 psi. This change is somewhat arbitrary, but it is known that a propellant does exhibit a considerably highcr effective elastic modulus under rapid loading conditions. Thc results obtai.ncd for this example are shown in Table IV-4. Note that rss and rt)wlo arc the same as in Table IV-2, since these values are not affected by t lc el astic modulus.

The effect of this change on the tl~crmoclasticcoupling is shown in Fig. IV-4. wi.i inpresented as the logaritlim (basc Ig) of the absolute value of tlic pcrccnt diif:rrr~ce between the Table IV-2. Results for baseline example

Time '6s 't)v/o 't)w (met) (inlsec) (in/sec) (in/sec) 1 ,1223 .I122 .I122 2 .I403 .I430 .I398 3 .I546 .I820 .I795 4 .I666 ,2153 .2140 5 ,1770 ,2386 .2 379 6 .I864 .2563 .2559 7 .I948 .2703 .270b 8 .2024 .2813 .2812 9 .2095 .2900 .2900 10 ,2160 .2969 ,2969 11 .2222 .3C2 3 .302 3 12 .2279 ,3065 .3066 13 .2333 ,3098 .3100 14 .2384 .3125 .312 7 15 .24 33 .3146 .3156 20 .264 3 .3214 .3224 25 ,2815 .3259 .3268 30 .2959 .3304 .3311 35 .3082 .3350 .3357 4 0 .3189 ..1399 .3404 45 .3283 .3449 .3453 50 .3367 .3449 .3503 TabJ? IV-3. Results for modified pressurization rate example. T r r Time s8 t)w/O t )W (nmec) (in/sec) (inlsec) (in/sec) 0.01 ' I I I I I C 10 2 0 30 4 0 SO Time (msec)

Figure IV-3. Comparison of results with increased pressurization rate to bascline example. Table IV-4. Results for modified elastic modulus example

Time rss rt )"/o 't)" (rsec) (ialsec) (b/sec) (inlsec) (Time (rrsec)

Figure IV-4. Cowparison of results with increased elastic modulus to baseline solution. trausient rates vith and vithout coupling for both the baseline and dfied elastic mdulus examples. The effect of a larger modulus is very sirilar to that of an increase in pressurization rate, For this example, the effect is significantly larger than for the increased pressurization rate. P-is should not be taken to rean that the influence of the elastic mdulus is =re significaut than that of the pressurization rate, A arch -re detailed study wuld be required to draw any conclusion vith regard to the relative Lolpartance of these two parameters.

Indeed, many other paraaeters vhich affect both the burning rate and the thenmelastic coupling should be subjects of future parapetric studies. hngthese vould be the constants in the burning rate lav and the appli- catiorr of the burning rate law over a vide range of temperatures. The coefficient of thermal expansion is also subject to the local terperature and state of strain. The experimental data required regarding the variation in the coefficient of thermal expansion under the conditions which exist during the firing of a solid rocket motor has not been found. lbo other parameters vhich deserve consideration are the thermal conductivity and Poisson's ratio. The conductivity influences the depth of the heat- affected zone and hence the heat cransfer feedback mrechanism. Poisson's ratio could significantly affect the state of strain if it were much less than 0.5.

Wany of the parameters discussed above vould require a considerable experimental effort to determine the behavior of these variables under the conditions experienced during the firing of a rocket -tor. Hwever, considerable data is available for the elastic modulus with regard to thermal and aechanical loading conditions. Hence, future efforts should first be directed toward incorporating a mre rigorous model of the modulus including the influence of both the thermal state and loading rate.

In sumnrary, a model has been constructed to analyze the influence of thermoelastic coupling on propellant burning rate during transient operating conditions. The analysis utilizes a model of the combustion process which is derived from steady state burning rate data and which is more readily applied than some other models considered such as presented in Ref. 19 which require a more detailed model of the flame structure. Eximples have been presented xhich show that the thermoel~~ticcoupling docs produce a small, but significant, effect during the initial portion of an ignition process. The results show that increasing the rate of pressurization and/or increasing the elastic modulus increases the thennoelastic effect. Several variables are nentioned as possible candidates for a rigorous parametric study and/or experimental research; in particular, the propellant modulus. V. PERFORPWCE PREDICTABILITY INVESTIGATION

In designing an SRn it is important to establish the limits within which the performance of the SRH may be expected to be at all points in its operating time. These limits will be more narrow when the quality of the prediction is higher. Poor predictability yields broad limits which in turn lead to conservatism and inefficient, uneconomical designs. The problem addressed here is how to assess the degree of predictability so that limits can be set vith confidence.

'Ihe quality of a ballistic prediction is degraded by two general failings: 1) model inadequacies and 2) uncertainties of basic materials and dimeasional characteristics used to evaluate the model. Examples of -el inadequacy include inability to model exactly ignition transients and even equilibrium burning rate as a function of all pertinent variables. Likewise, for non-homogeneity of chemical species and, to a lesser extent, grain temperature. Even the dimensional specifications are often not portrayed precisely; e-g., the ovality of a nominally c.p. grain and the eccentricity of the grain bore with respect to the motor centerline. Al- though the modeling is constantly being improved, it must be admitted that all ballistic models leave much to be desired.

For examples of the second general shortcoming of ballistic analysis which degrades predictability, uncertainties of basic material and dixen- sional characteristics, we have only to consider such parameters as burning rate, throat erosion, propellant density and again, mandrel ovality and misalignment. These and many other parameters are subject to processing and manufacturing variations. Nov, although the statistical variation of these variables about some mean value nay generally be deduced from previous motors, for any particular design the mean value itself is a matter of some uncertainty until a numher of motors have been built and tested. The uncertainty in general decreases with each additional firing. This explains why the predictability limits become narrower with additional motors tested. The problcn here is to identify quantitatively the degree of predictability for all points in time in the development program. Per- haps the predictability of the first motor should be stressed; however, because it influences tihe basic design which may be impractical to change later.

Application of arbitrary safety margins to allow for deviation from predicted performance could, of course, lead 'lo either overly conservative or unsafe designs either of which is undesirable. The Monte Carlo program, on the other hand, provides an approach for establishing limits on performance on a logical probability basis.

Reference 2 shows how the Monte Carlo program can be adapted to permit statistical variation in mean values from motor to motor as well as statistical variation about the mean value. To evaluate the effect of the uncertainties on the limits of performance, it is only necessary to ascribe to each variable a variability (e.g., a standard deviation for a normal distributed population) of its mean value corresponding to that deduced from experience on other programs. This variability will identify the uncertainty of the corresponding variable. Each variable may also be assigned distributions corresponding to more or less well established variations about the shifting (uncertain) mean values. After evaluation of a large nder of motors selected on a probability basis in the manner of the Monte Carlo program, the performance limits versus time are established. 'Ihe uncertainties can be adjusted to obtain predictability limits at various tinres in the development program as more information of the mean values is obtained. Also, both pair and individual motor population performance variations may be determined.

It is customary in ballistic predictions to adjust the constants in the burning rate relation used to obtain improved predictability as data from firings is obtained. The improvement is often dramatic. The improvement may actually be due to adjustment for inadequacies of the burning rate coefficient. This suggests that model inadequacies may at least partially be treated as uncertainties in the approach outlined here. Care must be taken, however, in deducing values for uncertainty from previous programs to segregate those that are due to actual variation in input parameters from those that are due to model inadequacies.

Unfortunately, it now appears impractical to obtain the data necessary to establish uncertainties of any type as our queries of industry and government sources and survey of the literature reveals. It is most difficult to recover documentation on what values of a specific variable were used in an original design calculation versus those thzt were later established to be more valid in various stages of the development and production programs. This is the type of information required for the suggested predictability assessment.

So our efforts to find a better way to establish predictability limits have proved somewhat abortive. Nevertheless, we have recorded the basic approach in the hope that it gives imp~tusto acquisition and documentation of more detailed statistical data on design and manufacturing variables. VI. CHANGES TO PREVIOUS COMPUTER PROGRAMS

In this section, changes to the two computer programs listed in the Appendices to Ref. 2 are given. Each change is listed separately and identified with a page and line number counted from the top of the page. Brief explanations of the reason for each set of changes are also given. The changes are for the most part refinements in analysis or program logic. A fev minor errors in the design analysis program have been found and are corrected herein. The most extensive change is the improvement of the use of tabular values of burning surface area during tailoff in the Monte Carlo program. This wes accomplished earlier for the design analysis program.

I The specific modifications are given in the form of new or revised cards. The cards should be punchi.6 beginning with column 7 except that state- rent numbers are punched to end in column 5 or unless otherwise noted. The card numbers given are for reference to the MSFC and Auburn University program iistings only. Numbers for cards that have been deleted are marked with an asterisk. Modifications Nos. 1 through 15 apply to the Monte Carlo program and Nos. 16 through 22 to the design analysis program.

Modification No. 1 - Card No. 00087050

Purpose: To eliminate statistical analysis for single mtors.

Add card between lines 6 and 7 p. 89:

IF(NRUNS .EQ. 1) STOP Modification No. 2 - Card Nos. 00169800 6 00170000

P;lrpose: To permit use of fractional number of slots.

Change line 24 p. 106 to:

Change line 26, p. 106 beginning in card column 6 to:

Modification NG, 3 - Card Nos. 00077400* 6 0007750W

Purpose: To eliminate instability in mass balance iteration during tailof f near transit ion pressure (PTRAN) . Delete lines 6 and 7 p. 87 -Modification No. 4 - Card Nos. 00069900 & 00070000 Purpose: To elidnate occasional instability in mass balance iteration caused by too small a distance burned increment at web time

Change line 27 p. 85 to: IP(Y .GE .ANS .AND KOUNT ,EQ .O) DELY=ANS-YLW .05*DELTAY Change line 28 p. 85 to: lbdification No. 5 - Card Nos. 00004400, 00174400, 00005100,00045800*, 00045900*, 00046000*, 00084750

Purpose: To improve accuracy of calculation of action time; i.e. time at which specified low tnrust level is reached during tailof f.

Add to line 28 p. 107 and line 44 p. 71:

Add to line 3 p. 72

Delete lines 26, 27 and 28 p.80.

Insert between lines 31 and 32 F. 88:

Modification No. 6 - Card Nos. 00061950, 00061951, 00061952, 00062000 Purpose: To eliminate unnecessary calculations when there is no erosive burning.

Add 3 cards between lines 43 ,-ad 44 p. 83:

Change statement number on line 44 p. 83 from 3 to 131: Modification No. 7 - Card Nos. 00002600, 00106800, 00044651, 00075000, 00075300, 00075301, 00075900, 00075901, 00115850, 00115851, 00115900, 00114150, 00114151, 00114200, 00163150, 00164000, 00075600

Purpose: To improve logic of the use of tabular values of burning surface area during tailoff.

Add to lines 26 p. 71 and 18 p. 93:

,ABTT

Add between lines 14 and 15 p. 80:

Change line 30 p. 86 to:

Change line 33 p. 86 (continuation card required) to:

Change line 39 p.86 (continuation card required) to:

Insert 2 lines between lines 12 & 13 p. 95:

Prefix line 13 p. 95 with statement number: 91

Insert 2 lines between lines 43 and 44 p. 94:

Change line 44 p. 94 to:

Insert between lines 5 and 6 p. 105: Add to lines 14 and 18 p. 105: -ABTT

Change line 36 p. 86 to:

Modification No. 8 - Card Nos. 00081450, 00081550, 00104950, 6 00104951. Purpose: To provide printout to indicate DPOUT and XOUT limits have been exceeded if appropriate.

Insert between lines 46 and 47 p. 87: IF(ABS (DPcDY) .GE .DPOUT) WRITE (6,1890) Insert between lines 47 and 48 p. 87:

IF(Y.GE.XOUT) W~ITE(6,1891)

Insert 2 lines between lines 41 and 42 p. 92:

1890 FORMAT (* DPOUT LIMIT EXCEEDED')

1891 FORMAT (' XOUT LIMIT EXCEEDED*)

Modification No. 9 - Card Nos. 00185151, 00185152, 00191651, 00191652, 00196550, 00196551

Purpose: To output imbalance data to tape (see Section 11).

Insert 2 lines between lines 39 and 40 p. 109 and 8 and 9 p. 111:

Insert 2 lines between lines 9 and 10 p. 112

99998 FORMAT(4E16.9)

99999 FORMAT(I10) Modification No. 10- - CardNos,00007402, 00014002, 00015202, 00015951, 00030700, 00030800

Purpose: To allow card input of radial grain temperature gradient. Insert between lines 26 and 27 p . 72:

Insert between lines 44 and 45 p. 73 beginning in card column 1:

C * 2 FOR CARD INPUT OF TEMPERATURE GWDIENTS *

Insert between lines 8 and 9 p. 74:

Insert between lines 15 and 16 p. 74:

Change lines 19 and 20 p. 77 to:

READ (IREAD, 3700) TUBULKO, TBULKE

2700 READ (IREAD, 3700) (YTAB (ITAB) ,T;ABA(ITAB) ,TTABB(ITAB) , TTABC (ITAB) ,

Modification No. 11 - Card Nos. 00031300, 00031500, 00032800, 00036350, 00039500, 00077100, 00098900, 00099700, 00099701

Purpose: To permit user to specify the extinguishment chamber pressure (PEXT) and the grain reference temperature (TREF) . See also Modification No. 13 for other necessary changes for TREF.

Add to lines 25, 27 and 40 p. 77 and line 11 p. 79:

,PEXT ,TREF

Insert between lines 33 and 34 p. 78:

C * PEXT IS THE PRESSURE AT WIICII THE PROPELLANT EXTIEGL ISIIES *

Change line 3 p. 87 to:

Add to line 29 p. 91 inside closing parenthesis: Add to line 36 p. 91 inside closing parenthesis (additional continuation

card required) :

Modification No. 12 - Card Nos. 00077000, 00077010, 00077020, 00079100 Purpose: To improve logic of mass balance iteration during tailoff

Change line 2 p. 87 to:

Insert 2 lines between lines 2 and 3 p. 87:

IF (PEAR. GE .PTRAN) DPCDY = PBAR*DADY / (1 -N2 ) /ABAVE

PONOZ = PONOJ+DPCDY*DELY

-. -pe line 23 p. 87 to:

4. CONTINUE Modification No. 13 - Card Nos. 00036960, 00048300, 00046100, 00055000, 00058200, 00060500, 00077600, 00048500, 00048700, 00048800, 00048900, 00049000, 00049600, 00049700, 00055200, 00055300, 00055400, 00055500, 00055900, 00056000, 00058400, 00058500, 00058600, 00058700, 00059100, 00059200, 00060700, 00060800, 00060900, 000610GC, 00061400, 00061500, 00077800, 00077900, 00078000, 00078100, 00078500, 00078600

Purpose: To allow user to specify a grain referc.1ce temperature (TREF) See also Modification No. 11.

Insert between lines 33 and 34 p. 78:

C * TREF IS THE DESIGN TEMPERATURE OF THE GRAIN *

Change 00.0 in the following places to TREF:

Line 3, p. 81; line 29 p. 80: line 22 p. 82; line 6 p. 83; line 29 p. 83; line P p. 87; line 5 p. 81; lines 7, 8, 9, 10, 16 and 17 p. 81; lines 24, 25, 26, 27, 31 and 32 p, 82; lines 8, 4, 10, 11, 15 and 16 p. 83; lines 31, 32, 33, 34, 38, and 39 p. 33; lines 10, 11, 12, 13, 17 and 18 p. 87. Modification No. 14 - Card Nos. 00179500, 00179520, 00179540, 00196553

Purpose: Tocreate a tape containing the thrust versus time data for each SRM. The data is written on unit 9 and the appropriate job control cards must be provided for this unit. This modification was accomplished to obtain nominal trace characteristics for statistical correlation of thrust imbalance with nominal trace characteristics (See Section 11).

Change line 21 p. 108 to:

Insert 2 lines between lines 31 and 32 p. 108:

Insert between lines 9 and 10 p. 112 after Modification 7 insertion:

Modification No. 15 - Card Nos. 00012200, 00012500, 00012600, 00012700, 00013900

Purpose: To improve clarity of option descriptions.

Insert line 26 p. 73 between OVALITY and ANALYSIS:

Change description line 29 p. 73 to:

FOR PLOTS, TNIULAR OUTPUT AN3 STATISTICATd ANALYSIS

Change description line 30 p. 73 to:

FOR TABUIAR OUTPUT AND STATISTICAL ANALYSIS

Change description line 31 p. 73 to:

FOR PLOTS AND STATISTICAL ANALYSIS

Change description line 43 p. 73 to:

FOR TAPE INPUT OF TEMPERATURE GRADIENTS

See also Modification 13 (between lines 44 and 45 p. 73) Modification No. 16 - Card Nos. 00030200*, 00030300*, 00020451, 00020452, 00020651, 00020652, 00020653, OC020700

Purpose: To correct error in calculation of burning rate coefficients Q1and 42 which occurs with input of grain temperature gradients.

Remove Q1 and 42 equations, lines 15 and 16 p. 158, and insert same between lines 13 and 14 p. 156.

Insert 3 lines between lines 15 and 16 p. 156:

IF(ITEMP) 10005, 10006, 10005

loo05 Q~=A~*EXP(PIPK* (1. -NI) * (TGX-TREF)

Q?=A2*EXP (PIPK* (1. -N2) *TGR-TREF) ) Change line 16 p. 156 to:

Modification No. 17 - C-rd Nos. 00032151, 00032152, 00032153, 00033154, O(lQ32200

Purpose: To eliminate iteration for burning rate when there is no erosive burning.

Insert 4 lines between lines 34 and 35 p. 158:

132 IF(PN0Z .LE .PTRAN) RNOZ = Ql*PNOZ**Nl

IF(PNOZ.GT.PTRAN) RNOZ = 42 ?NOZ**N2

GO TO 4

Change statement number on line 35 p. 158 from 3 to 131

Modification No, 18 - Card Nos. 00037300 h 00037400

Purpose: To eliminate occasional instability in mass balance iteration at web time which occurs when y (DELY) increment just prior to begi~ningof tailoff is toc small.

Add to lines 38 and 39 p. 159: kdification No. 19 - Card No. 00045100* Purpose: To eliminate instability in mass balance iteration during tailoff when the chamber pressure crosses the transition pressure PTRAN.

Delete line 20 p. 161. Wification No. 20 - Card Nos. 00045000, 00045010, 00045020 Purpose: To correct error in selection of N2 and N1 vhen PTRAN is crossed during tailoff

Change line 19 p. 161 to:

Insert 2 lines between lines 19 and 20 p. 161:

Hodification No. 21 - Card Nos. 00124800, 00124900, 00125000

Purpose: To allow the plotting of additional data poinzs so that the user may use smaller y increments.

Change 200 to 999 throughout lines 6, 7, and 8 p. 178.

Modffication No. 22 - Card Nos. 00019810, and OC020600

Purpose: To correct an error in the place of calculation of the characteristic velocity. This error would occur when TRG#TREF and CSTARTfO.0. causing a small accumulative error in CSTAR over the operating time.

Insert between lines 14 and 15 p. 156:

Change line 15 p. 156 to: VII. CONCLUDING REMARKS

The continued research in internal ballistics performance variation has produced considerable improvement in the flexibilit) of the Monte Carlo and design analysis computer programs. In addition to a large number of Pinor improvenrents in both programs, two new computer program codes now pennit more detailed examination of the thrust imbalance characteristics defined by the Monte Carlo program. Sample evaluations for the Titan IIIC and comparlsons with flight test data show good agreement throughout the operating thof the mtors and thus provide additional confirmation of the hteCarlo nrethod. Statistical correlation studies of the sample r~sultssuggest several new and possibly important approaches to predicting thrust imbalance.

Design analysis program modifications now permit evaluation of the internal ballistic effect of general deformation of circular-perforated grains. Although additional validation of the wdel is needed, application of the method to two differenr: SRMs indicates that the deformation may account for a substantial portion of the so-called "scale factor" on burning rates.

The research in the thermoelastic coupling of propellant strain rate with grain temperature and burning rate has culainated in an apparently successful method for theoretical evaluation of the phenomenon. Assessment of the influence of thermoelastic coupling indicates a small but definite effect which warrants further investigation. REFERENCES

1. Sforzini, R. H., and Foster, W. A., Jr., "Honte Carlo Investigation of Thrust Imbalance of Solid Rocket Hotor Pairs," Journal of Spacecraft and Rockets, Vol. 13, No. 14, April 1976, pp. 198-202.

2. Sforzini, R. H., and Foster, W. A., Jr., "Solid-Propellant Rocket Hotor Ballistic Perfomnee Variation Analyses," Find Report, NASA Contractor Report CR-144264, Engineering Experireent Station, Auburn University, September 1975.

3. Sforzini, R. H., Foster, W. A., Jr., and Johnson, J. S., Jr., "A knte Carlo Investigation of Thrust Imbalance of Solid Rocket Hotor Pairs," Final Report, NASA Contractor Report CR-120700, Auburn University, Navember 1974.

4. Smith, D. F., "Solid Propellant General Deformations and Their Effect on Fator Performance," Master of Science Thesis, Auburn University, Scheduled for Submission in October 1976.

5. Sunmerfield, H,, Caverty, L. tl., et &, "Theory of Dynamic Extinguish- mt of Solid Propellants with Special Reference to Nonsteady Heat Feedback Law," AIAA Journal, Vol. 8, No. 3, March 1971, pp. 251-258.

6. Foster, W. A., Jr., "A Method for the Transient Nonlinear Coupled niermoelastic Analysis of Axisymmetric Bodies Using Finite Elements," Ph.D. Dissertation, Auburn University, Auburn, Alabarna, 1974.

7. Baker, J. S., "Equilibriun Thrust Imbalance Analysis for TC-227A-75," Memorandum transmitted to K. W. Jones, MSFC, by L. G. Bailey, Wasatch Division, Thiokol Chemical Corporation, Letter No. 7010fLB-75-547, 13 November 1975.

8. Barrkre, M., Jaumotte, A., Fraejis, De Vaubeke, B., and Vandenkerckhove, J., RockeC Propulsion, Elsevier, Amsterdam, 1960, pp. 277-281.

9. Srisbane, J. J., "Heat Conduction and Stress Analysis of Solid-Propellant Rocket Motor Nozzles," Technical Report S-198, U.S. Amy Missile Conunand, Redstone Arsenal, Alabama, 1969.

10. "Propellant Burning Rate Investigation," Final Report, UTC 4310-73-51, to Space and Missile System Organization, USAF Systems Command, United Technology Center, Sunnyvale, Calif., Oct. 31, 1973, p. 3-27 and p. 4-5. "Technical Description of the TX 356-5 Rocket Motor," Huntsville Division, Thiokol Chemical Corporation, Control No. U-67-1005A, Uay 9, 1967.

Sforzini, R. H., "Design and Performance Analysis of Solid-Propellant Rocket Motors Using a Simplified Co~puterProgram, Final Report, NASA Contractor Report CR-129025, October 1972. -1, J . K. C ,, "Second Henmire sur les PhSnodnes Thermodcaniques ," J. de 1'Ecole Polytechnique, Val. 15, 1837.

Kelvin, Lord (Sir William Thompson), Mathematical and Physical Papers, Vol. 3, Cambridge University, Press Warehouse, London, 1890, pp. 1-260.

Joule, J. P., The Scientific Papers of James Prescott --* Joule Vol. 1, The Physical Society of London, 1887, pp. 413-473.

Hermann, G., "On Variational Principles in Thermoelasticity amd Heat Conduction," Quart-Appl. Math., Vol. 21, No. 2, 1963.

Becker, E. B., and Parr, C, H., "A9plicab'sn of the Finite Element kthod to Heat Conduction in Solids," 1 -mica1 Report S-117, U.S. Army Hissile Command, Redstone Arsenal, Alabama, 1967.

Wilson, E. L., and Nickell, R. E., "Application of the Finite Element Wethod to Heat Conduction Analysis," Nuclear Engineering and Design, Vol. 4, No. 3, 1966.

19. Krier, H., --et al, "Nonsteady Burning Phenomena of Solid Propellants: Theory and Experiments,"AIAA Journal, Vol. 6, No. 2, February 1968. pp. 278-285. A. Analysis of SRH Pair Imbalance Data during Specified Time Intervals

B. Analysis of SRM Pair Imbalance Data at Specified Times during Operation

C. Sf@! Design Analysis with Gption Permitting Evaluation of Grain Deformation Effects APPENDIX A

A COWUTER PROGRAM FOR ANALYSIS OF SRH PAIR IMBALANCE DATA DURING SPECIFIED TINE INTERVALS

The computer program listed in this appendix is used to analyze a data tape containing motor pair data which is generated by the Honte Carlo per- formnce analysis program described in Refs. 2 and 3. This program analyzes the data for a prescribed number of time intervals. The variables investigated are the absolute values of thrust imbalance and its first time derivative, the impulse imbalance and the absolute impulse imbalance. For each variable the program calculates the maximum value for each pair during the time interval, the maximum value occurring in all time intervals for all pairs, the mean value for each time interval and the standard deviation for eac5 time interval. The program also determines the mean value and standard deviation of the time at which the lnaximum value of the variable occurs within the time interval.

Input Data

The discussion below gives the general purpose, order and FORTRAN coding inf ormstion for the input data.

Card 1 Number of time intervals to be analyzed (110)

Col. 1-10 Number of time intervals (NSETS)

Card 2 Time interval data (110, 2F10.4)(one card for each time interval)

Col. 1-10 Number of motor pairs to be analyzed (NAMAX)

11-20 Beginning rime of interval (ATHIN)

21-30 Ending time of interval (ATMAXI

Program Listing

The program listing is presented in Table A-1.

Output

The output of the program is discussed in Section 11 of the main report where a sample output illustration is also given. Table A-1. Comguter program for analysis of SRH pair imbalance data during specified time intervals.

DlYENSION FOIFF(LOdO)rTPLUT( IOUb) iOFOUT( 1003) OIdENSION ~Ar4AX153)rATYIh(5O)~ATHAX(SG) CALL GSIZEt 500orllorl121) READISr101) NSElS 09 7300 IPLT=lr3 RErJIhL) 8 00 6330 L=l rNSETS

IF( IPLToEQo2oAAlDoL l€40 1) CALL PLOT( 240 r0.r-3) IF( IPLloEQ.ZoANCoL~GTo11 CALL F'LOT(4orOor-3) IF(IPLToEOo3) CALL PtUT(9orOer-3) XMAX=3,0 'iMAX=J.i) XOYAX=3.3 IOHAX=i).O REWINO 8 IF(LoGTo1oANDo IPLToE0.1) CALL PLOT(26o~0o,-3) IF(IPLT~EQe1) RtAD(Sr131) hAMAX(CIsATMIN(L)sATMAX(LJ N:4AX=NAHAX( L) TYIN=AIAIN(L) TMAX=ATMAX(L J dRITf(6,133) L~hMASvT#!N~TMAX 03 1330 N=LsNdAX DFHAX=3.0 13FWAX=3,3 RE AiJ(Br 131J NX r

GO TO YO 89 WRITElb,Lil72) N,DFMAX 90 WRITE(6qlObl TOFHAX CALL SIGdAR(OFMPX* SlrS2, XMAXS~XHAXM*N,NMAX) CALL SIG3AA(TDFMAX,S3,54,TMXXS#T4XXi.l#IY,luRAX) IF(KOUNT.NEoO1 NX=KOdhT IF( 1CoNf 091 NX=IC IF(N-11 141 14~16 14 CALL SCALE (FDIFFIB.OINXII~ IF~FDlEF(NX+lI.LT*Oo3oAhOo 80 O*FOIFF (NX+21+fUIfF (NX+l J oLEoOo9) 2F19STV=F 3IFF(NX+l)*2o IF(FDIFF1NX+l) .LT.o.~)~~~uo~~.o*FDI~F(~.u(+~~+~o~F~~NX+~).GT.O.O) ZFIRSTV=-ABS(d.*FDIFF(NX+21) IF(FDIFF(NX+lJ .Gk,0,31 FIkSTV=-~FDIFF(NXtl)t8~i)*FDIFF(NX+2))+LI OELTAV=-Fl kSTV/4. O 16 CONT INUE FOtFFINX+11=F IHSTV FO IFF(NX+Z) =dELTCV IPLOT(VXtl)=Tr4IN TPLOT(NXt2)=(TMbX-lMIN)I5.0 IF(NeEQ. ~.AND~L.EJ.L~ANu~~PLT.~O.~)CkLL PLUT(GOOIL.S~-3) IF(N.NE.1) G3 TO 999 GO TO (1,2,31rIPLT i CALL AX1 S( 0.3,0o0? 'TrlRClST IMBALAIvCE (LUF 1',22,8o0,93.3, ZFIASTVgIJECTAVb GO ro 4 2 CALL A%IS(003#0*0, ' IYPULSt IMoALANCE ( L8F-SECS) '926,8.0,9B.OI 2F 14STVrUtLTAV) GO 10 4 3 CALL AXIS(d.O*O.O, 'A3SOLUTE IMPULSt LMBALANGE ILdF-StCS)', 237.8. C193.0, ZFI~~STVIOELTAV) 4 CuLL AXIS~LI.~~~.O~'TIME(SLCS 1' r-llr5-O~OoOtTPLOl(NX+l) 2vTPLIJT(NX+Ll I 945 CALL PLOT (0.140,31 C4LL LIX<( lPLLITrFOItF,~~~,l,c)~lJ 1331 C3 JTIJdF. LlJ TO (2*69?),1?L I 5 &SITE (tr lOuJ XYAX dKITt(url3G) TXXX kR ITE (6,109) kidITE(b,lds) XMAX!I W2ITE(6r111) HRiTE(6~10'+1 XMAXS d@ITEI6rllO) TMXXM W4ITEla,ll2) TMXXS GJ T3 8 6 WRI TE (6,1341 ) XMAA Table A-1 (Cont 'd)

kRITE(6,1O6I TMXX WH IT€ (61139) WRITE16,1041) XMAXM bIRITEtb,LllJ WRITE(b,lO41) XYPXS WRICE(6~LLO) TMXXM WR ITE(6,lLLJ TMXXS GO TO 8 7 WR11Ei6,1342) XMAX ARlTt(6,106) TMXX bi~lTE(6,10Y) dR ITE(6,lOGZJ XSlAXM WRfTt(6~111I WRITE(6,1041) XMAXS hRITE16rL13) INXX9 WRITEL6p11L) TYXXS 6 IF(IPLT.~E.A) G(1 TO 6000 REWIIJO 8 00 LOO0 N=lpNMAX DF DHAX=O.O TDFOMX=d*O KEAD(3, 1011 NX KOUNT=O K= 0 1C=O 03 5333 J=l,NA CALL IhPUI (1,F IPLT IF( 1,Ll.TMIN) K=J IF(T.LT.1i-\I&) GO TO 5000 I= J-K SF(T.GI .T?4AX.AND.KOU~\T~[:b:~O) KOUNT=I-1 IF(KUUrdT 13v13,5333 13 TPLOT I I)=T FD IFF ( II=F IF (T.LT.TMAX.ANUeJ.GCo~4X) IC=I 5330 CON T IYUk 16 IV,LIIJNI .:JE. J) NX=KC)UKT lF(IC.iJt.3j IrrX=IC I)F~~)OT(1~=FOIkF(l)/lPLCJT( 11 1'1F_)fiAX=Ar,S(L,t33f( 1) J 10FUMX=IPLUT(lI 00 3d33 J=ZpNX UFOUT (J)=(i-dIFF(J1-FDIFF(J-1 1)/(7PLUT(J)-TPLbT(J-1)1 AF U=AdS( UFIIUT (J11 IF ( AFl).Gr .DFili.lAX) ToFi)MX=T?LCr (J IF ( CFD.CT.Df-DMAX I ijFUclAX=AdS (OFOUT (JJ ) IF(UF DMAX .GT .XdMAX J IDWAX=Ti)kL)MX 3000 IF(L)~UM~X.GT.XI)'~AX)XU,qAX=DFUMAX Table A-1 (Cont'd)

URITE(6r108) N,OFDMAX WAITE(6r\13J TOFDHX CALL SIGdAR(DFO4AX~S5,S6~XOHAXSrXOHAXH~N~NH4X) CALL SICBARITDFDMXrS7rSB~TOWAXS~TOHAKH~tY~NM4X) IF IN-1 15, 159 17 15 CALL SCALE(UFOOT,a.OrNX,l IF(UFDUT (NX*lJ ~LT.O~O~AND~8~O*OFOOT(NX+2)+DFlOT(NX+l )oLEoOeOl 2FIRSTV=DF00T(NX+1)*2. IF(OFDOr(NX+l J ~LT~3~OaAND~8~3*0FOOT~NX+Z~+~FO~T~NX+~1oGT.OoO) 2F IRSTV=-ABS (8. *DFDOT (NXt2 1 ) IF(OFOOT(NX+lJoGE~3,3) FIkSTV=-~OFO0T~NX+l)+B~3*OkOOT~NX+2))*2~ DELTAV=-F IRSTV/4.O I7 COVT1 NU€ OF3UT(NX+li=FIKSTV DF~~T(NX+~I=O~LTPV TPLOT(NX+L)=TNIN TPLOT (YX+2)=(TM&X-TMIN)I20.0 IF(NmEU.L.ANDeIPLTeEU.1) CALL PLOT(9.0rO.Ot-3) IF(N.Eu.~) CALL AXIS(0.0~0.0~'THKUS~ IM3ALANCt KATE (LBS/SEC)'r31, 28mO~90o0,FI4STV~OELTAV) IF (NeEUolJCALL AXIS(O.~,~~OI'TIYE (SECS) 'r-l1,20.~O~O,TYLOTtNX+l) 2,TPLOT (NX+2) 1 CALL PLOT(3.0,4.0r3) CALL LINE(TPLOT~UFUOTIXX,~,O~L) 2000 COYTiNUt dR ITE(6r 103) XDMAX HRITE(brll3) TOMAX WRIIE(6~10Yl ~iiITt(b~lJ5)XDYAXM O~GZNALPAGE 1 ~RlTE(brll1) OF POOB QUO dKITt(6~105)XDYAXS WRiTE(b, 114) TOMAXM MRIT€(6,115J TOMAXS 63 JO COiqT IWE 7000 C3hr INUE CALL PLCT (0.3,0.09999 1 103 kOHMkT(4k16.9) 191 FO3MAT ( I AS, LF10.4) 132 FJi?MAT(3(SX,f 16.9) 1 A33 FJdp!&T( lH11 YX,'Tt115 I3 1 IMt INTtKVAL !VdMdkH8 IG1//, lOX,*TtiEKE ARE 2 ' ,I+,' SETS [i); OI1TA FUft THIS 11Mk INTEt

105 FORHAT ( lOX, ' THE A8 SOLUTE VALUE OF THE MAXIMUM THRUST IMdALANCE RAT iE DURING THIS TIM6 INTERVAL IS 'rlPEll.4,' LBF/StC') 106 FORHATI1OX~'THIS IMBALAIVCE OCCURS AT ' ,F 7.2~' SECS' p/) 107 FORMAT (LOX, ' THE ABSOLUTE VLLUE OF THE MAXIMUM THRUST IMdALANCE FOR 2 HOTOR PAIR NUHBEH ''16t' 1S ',1PE11.41' LBF'I 107 1 F ORMAT (19X1' THE AdSOLUTE VALUE OF THE HAXIHUM IMPULSE 14BALANCE FO 2R YOTOR PAIR NUMBER '914,' IS 'rlPE11eQr' LBF-SECS') 1072 F3RMAT( 10Ag 'THE HAXIHUM ABSOLUTE IMPULSE IM8 2AL4NCE FOR MOTOR PAIR NUMBER 'e 14,' IS * ,lPEllo4,' LBF-SECS') 108 FORHATI 1OX'' THE ABSOLUTE VALUE OF THE MAXIMUM THRUST IHdALAkCE HAT 2E FOR MOTOR PAIR (YlJYBkH 'r14t' IS ',lPElleQ~' LaF1SEC'J 109 FORMAT(13Xt' THE MEAh VALUt OF') 110 FOR%AT(/,~OXI'T~JE HEAN VALUE CF THE TIME FOR THIS IMBALANCE IS '9 2F 7.21' SECS't/I 111 FURMATIt~1JX~'THESTAN3ARD DEVIATION OF') 112 FORHAT I lOX,'THE STANOAK0 OEV IATION OF THE TlME FOR THIS IMBALANCE 215 ',FTo2r' SECS't/) 113 FO~NATI1OX,'TlIIS IMdALANCt KATE OCCURS AT '1f 7.2~' SECS',/J 114 FOKHAT(/tldX,'THk HE4N VALUE OF THE TIME FCR THIS It48ALANCE KATE I 2s 'rF7.2,' SECS't/J 115 FOHe4JT(ldX,'THE STANI3ARD DEVIATION OF THE TIHE FOR rHIS IH8ALANCE 2HATE IS *,67.29' SfCS't/) STOP END APPENDIX B

A COMPUTER PROGRAM FOR ANALYSIS OF SRM PAIR IMnALANCE DATA AT SPECIFIED TIMES DURING OPERATION

The computer program listed in this appendix is used to analyze a data tape containing motor pair data which is generated by the Monte Carlo performance analysis program described in Refs. 2 and 3. This program analyzes the data at a prescribed number of discrete time slices and calculates the statistical tolerance limits about a zero mean value for the thrust imbalance, thrust imbalance rate and impulse imbalance. For the absolute impulse imbalance the statistical tolerance limits are computed about the true mean value.

Input Data

The discussion below gives the general purpose, order and FORTRAN coding information for the input data.

Card 1 (3F10 .O)

Col. 1-10 One sided K-factor (SIGK1)

11-20 Wo sided K-factor (SIGK2)

21-30 Initial time point to be plotted (TMIN)

Card 2 (2110)

Col. 1-10 Number of time slices to be taken (N)

11-20 Number of niotor pairs to be analyzed (NMAX)

Card 3 Time Slice Card (FlO.0) (one card per time slice)

Col. 1-10 Time at which analysis is to bs mde (TSL) -Program Listing The program listing is presented in Table B-1.

The output of the program is discussed in Section I1 of the main report where a sample output illustration is also given. Table B-1. Computer program for analysis of SRM pair imbalance data at specified times during operation. DIMENSIOi'J F(1000)tFO( 1000) tT( 10001~TSL(SO(5~tFOIS~SOO~~F~S(500~ D164ENS10N F1(50~)~F31(~01tF1M(503),FDIM~503)~FISM~500) DIHENSION 51(500)tSZ(5UO~ tS3(530) tSQ(S30) CALL GSIZE(530otllot1121) READ(St103) SlGKIt SIGKZtTMIN READ(5t 101) NpNMAX WRITE(btlO5) NMAXtSIGKltSIGKZtN 101 FORMAT(21 LO) DO 4033 KsLtN 4000 REAO(St1031 TSL(K) 100 FORMAT(8FLOoO) DO 99999 fPLT~lt3 RE'rJIND t3 00 3000 L=1 tNYAX CALL If.PUT( TIF tNX, IPLT) 102 FORHAT (4E1603) FD(l)=F(l)/T(l) DO 1000 J=2,NX 1003 Fil(J)=(F(J)-F(J-1) )/(T(JJ-T(J-1)) DO 3000 K=l,N CALL INTRPl(F,TtNXtTSL(K) tF I(K) CALL INTRPL(FDtTtNXtTSL(K) tFOI(K)) 104 FOKMAT(10X,3E16o9) CALL SIGdA3IFILK)rSA(K)~SL(K),FIS(K),FIM(K)tLtNMAXvIPLT4 3000 CALL SIGdAH(FO1 (K)tS3(K) ,S4(KI,fDISLK) ,FOIM(K) tLtNM~X,IPLT) 00 1 1=1,N FD 1st I )= SIGKZ*FDIS( I) IF(IPLToE4.3) GU r0 11 FIS(I)= SIGKZ*FIS(I) GO 70 1 11 FiSM( I)=FIM( 1)-SIGKl*FIS( 1) IF (FISM( I).LEeO.O) FISH( I )=Om0 FIS( I)=FIPl( I)+S[GKl*FlS(I) 1 CONTINUE DO 4 I=1,11 MRITC(brld6) rSC( 1) IF ( IPLT .I\(E. 1) GO TO 33 ~RITti(brld3) kISI I IvFDISI I) G3 TO 4 33 Ii(IPL1 .Nt.2) GO Td 40 WiIITt(6~113) FIS(1 I r,L) TO 4 40 MK:TG(6t1321 FIN( I) YQ iit (6,123) F IS( I) kRlTE(69133) F ISM( I) 4 CONTINUE IF( IPLT-1) 839 83970 83 CALL PLO~(~.O~L.~,-N Table 8.~1(Cort'd)

GO 10 90 70 CALL PLOT(9.3,0-0,-3) 90 CONTINUE CALL SCALE(FISt8*OtN,l) IF(FIS(N+l)oLT.O.O.ANDo8oO*F IS(N+L~+FIS(N+l) oLEoOm0) 2FIRSTV=FIS(N+l)*2. IF (FIS(N+l) .LT.d~0oAfJ9~8~3*FISIN*Z)+FIS(N+tJeGToOoOJ 2FIRSTV=-AJS(a.*FIS(N*2)) IF(F IS(N+LI oGt.3.0) FIkSTV=-(FIS~N+l)+8oO*FIS(N+2))*2o DELTAVz-F IRSTV/G.O FIS(Nt1) =FIRSTV FIS(N+2)=OFLTAV TSL(N+l) =TMIN TSL(N+2)=(TSL(N)-TSL(;v+l) )/5.3 IF(IPLToNE.1) GO T0 10 CALL AXIS(9o3,O.O~'THKUST IMBALAACE (LBF)',22,8.0,90.0t 2fIRSTV,DLLTAV) GO TO 15 10 IF( IPLT.NE.21 GO TU LO CALL AXIS(\).O,doO,' IYC'ULSE IMBALANCE (LdF-LtCS)' 928, JmO,93eOt 2FIKSTV,9tLTAV) GO TO 15 20 CALL AXIS(0.0,O.O 9 'A850LUTE IMPIILSE IMBALANCE [LBF-SECS)'. 37t 8.09 290oOgF IRSTVtOt LTAJ) 15 CALL AX1 S(deOt4oOq'l IME (SECS)'~-l1,5od~OoOtTSL(N+l) 2,TSLIN+Z) CALL PLC)T(0.,4.,3) CALL LINt(TSL,FIS,Ntl,d~l) 00 2 I=l,N f 15( I I=-FIS( 11 2 IF;IPLT.EQ.3) klS(I)=fISM(I) CALL PLOT13.,4.93) CALL CINE(TSL,FIS, 391 I IF( IPLT.IJE.~) GO I'. 35 9 CALL PLUT19.3,deLir . CALL SCALC(FDIS,~~. r tl) fF (FUIS( J+1 ).LTedm\ .ANI)~d.O~FOlS(rJ+?)+FI)IS(~J+1)~LE.G.O) ZkIi?STV=FDlS(N+l)*L. IF (FUIS(fJ+l).L T.(:.~.AI\~U.~.~~~OIS(N+~)+~OIS(N+L).GT.U.J) 2Fl;STV=-AdS(3.*FDIS(i4+2) 1 IF(FLJIS(IJ+~).Gt.O.O) FIhSTV=-(FI)IS(N+l)+8.O*EL)IS(N+2) )*OIZ. CcLTAV=-FIRSTV/4.0 FO IS(N+l)=FIKSIV FU ISI N+2 1 =DtLT AV CALL AXIS( 3.3,d009' THHtJST IMBALA'JCC HA?€ (LHF/SECJ' ,319 28.3~93mO~FIKSTV~OtLTAV) CALL AX1 S(UoO*'t.d,'TIME (SECSI',-lL,3. 93mOtTSL(N+l) Z,TSL(N+ZJ) Table B-1 (Cont'd)

CALL PLOT(OoOr4-0r3) CALL LINE(TSL9FDIS rNrlrOlI) 00 3 l=lrN 3 FDISll)=-FDISI I) C4LL PLOT(000~4a0,3) CALL LIAE(TSLrfD1S~~rArOlII 99999 IF( IQLT.E3*3J CALL PLOT(OoOrOaOr999) WE TURN 103 FORYATt 10x1 Z'lHE + OR - KZ*SIGYA LIMIT ABOUT A f ERO MEAN FOR THE THRUST IMBALA 2NCE IS 'rlPE11.4,' LdF'r/r13XrrTHt + OR - KZ*SIGHA LIMIT AaOUT A 2 2EaO MEAN FOR THE THRUST IMBALANCE RAlE IS 'rLYE1104r' L8F/SECar//) 103 FOiWAT(1Hlr9Xp'THERE AHE ' r14ra SETS OF MOTOR PAIR DATAgr/rAOX,'TH 2E Oh€ SIDED K FACTOK IKl) FOR THIS SAMPLE SIZE IS 'rF7.3r/r 313X~'THE Tdil SIOED K FACTOk (KZ) FOA 1HIS SAMPLE SIZE IS 'rF?a3r/ ZrLOXr'Tkit hESULTS Ai(E CALCULATED AT 'r14r' TIME SLICES'r///) 106 FI)AHAT(~~XI@T!~ISTIHE SLICE HAS TAKEN AT '9F7.2~' SkCS'J 113 FaRMAT(13Xv 28THE + OR - KZ*SIGMA LI4IT ABOUT 4 LERO MEAN FOR THE IHPULSE INBAL 2ttNCE IS 'r13Ella4~' LBF- SFCSrr//) 123 FOitqAT (10x1 2 er~t+ K~*SIGMAcIntT ADOUT THE MEA~FOR THE ABSOLUTE I~PULSE 2 IHBALANCE IS '~lPEllr41' LBF-SEC') 135 FORYAT; LOXI 2 'TdE - <1*SlGH4 LlHI1 AOOUT THE HEAN FUR THE AaSOLUTE IMPULSE 2 I%ALANCE I$ 'iLPEL1.4rr LdE-SECar//) 132 FLJKHAT(~~XI'T~IEHEAN VALLk OF THE ABSOLUTE IMPJLSE IMRALANCE 15 '9 ZIPEILaQ~'LdF-SECS' ST CIP EN 3 SU3K0U1 INE SIGSAk(XrXIrXIi~SIGXrOX~1COUidT~NrIPLT) XN=FLdAT (N) IF(ICCUNTeCT*l J GO TO 1 XIL=O *o XI=do0 I XIZ=XIZ+X*+2 X1=XI+I. HX =X I/X;4 XIs=x:**z IFI IPLT-3) 213~2 2 ARC=( XI2fXNl ;iI ;a 4 3 E2G=( XI2/XN)-( XIS/XN**LI Ii(AKG-LE ,3-01 AkG=O-0 C SIGX=S3RT(ASG) PE TURN Fa0 Table B-1 (Cont'd) SUBROUTINE 1NTRP1(Y,ltNrTlrOY) DIMENSION Y(N) 9TlN) Nl=N-1 DY =o*o 00 1 I=l,Nl IF(ll~GE.1 (t)eANOeTTeLT,T( 1+1)) DY=((Yt1+11-YI I))/IT(I+L)-141) JJ 2*( 11-14 L 1 )+Yt t IF (OYmYEmO-0. RETURN 1 CONTI NU€ RE TURN END SU&GOUTlNE INPUT~TrDXtNX,IPLT) DIYENSION 1~1000)tOXI13GO) REAO(8r 100) NX GO T0 (lrZt3)rIPCT 1 READ(8tLOO) (11 I)tOX( I) rDirDZtl=LINX) GO TO 4 2 REAU(8r233) (T(l)rDlrOX( l)tDZ,I=l~NX) GO TO 4 3 READ(Br2OO) ( T(I)r01,02rOK( I )tI=ttNX) 4 RE TURN 100 FORHAT(1 la) 203 FLl8HAl (4E 16.9) €NO THE SRH DESIGN ANALYSIS PROGRATI WITH OPTIOX PERMITTING EVALUATION OF GRAIN DEFORWTION EFFECTS

This appendix contains the instructions for the preparation and arrangement of the data cards. In addition, a complete listing of the program statements is given. The program was written for use on an IBX 370/155 computer and requires approximately lOOK storage locations on that machine. The program also is designed to be used with a CALCOHP 663 drum plotter. me plotter requires one external storage device (magnetic tape or disk). However, anly minor program modifications are required to eliminate the plotting capability of the program.

Input Data

lbe discussion below gives the general purpose, order and FORTRAN coding information for the input data. All of the Lta, except for the changes described herein, are identical to the input data described in Appendix B of Ref. 2.

Card 4A Replaces Card 4A of Ref. 2. Ignition, inert weight and grain deformation options (4X,I1,9X,I1,9~,11)

Col. 1-4 IGO =

0 For no ignition calculatio~s

1 For ignition calculations

0 For no inert weight calculations

1 For inert weight calculations

0 For no grain deformation effects 2s C1 For grain deformation effects Card 9 Kcplaces Card 9 of Ref. 2. Uniform temperature and gzain detor- mation constants (3 cards).

Card 9A Vniform temperature card (input only if ITEW=l) (5X,F10.0)

Col. 15 TCR =

6-15 Value of TCR -75- Card 9B Grain deforaration constants (input only if IS0 1) (7X,E12.4,

10XsE12 .4,9X,F8.5,9X,P8.5)

Col. 1-7 PWD

9 Value of PMOD

30-41 Value of CX)D

51-58 Value of PMJ

68-75 Value of CMI

Card 9C Grain deformatior. constants (input only if ISCb1) (10XsET2.4,

10X,F8.5)

CO~. 1-10 ALPTS

11-22 Value of ALPTS

33-40 Value of TAW:

Table C-1 represents an example set of data. Table C-2 is a sample computer printout obtained with this input data.

Table C-3 presents the complete program listing. The program has bee? designed to prsduce graphical representations of the computatior-al results on the CALCOMP plotter. To delete the plotter compilation requirements, dummy subroutines my he substituted for the following subroutines: CSIZE, PLDT, LINE, AXIS, and SYMBOL.

ORIGINAL PAGE IS OF HNR QIYA !,I7 \- 0-m' PAGE 18 OPwoaQUW

d 0 C, C)N 0 I 0 00 0 I I Q YI (Yw w C - -. 4 C e4 n 0 *I Qm J . dC)Ont*01(\...... 0 o-oamruor. N mmmmmwm a OvYCVcY t IV) u-au * w ow-U a a0 e. Y) - 4

(Y.m d. TABLE C-3

C ********************************************************************* C SRH DESIGN AN0 PERFORMANCE ANALYSIS * C * PREPARE0 AT AUBURN UNIVERSITY * C UNDER NOD. NOe 14 To COOPERATIVE AGREEMENT WITH C C * NASA MARSHALL SPACE FLIGHT CENTER C * C * 8Y C * Re Ha SFORZINI AND We A. FOSTER* JRe * C AEROSPACE ENGINEERING OEPBRTMENT * c .* SEPT EHBER 1975 a .C * .c * bC * HOOIFIEO FOR GRAIN OEFORHATtON INTERNAL BALLISTlC EFFECTS )r bC * a bC * BY * bC * Re He SFORZINI AND OAVIO Fe SMITH a bC AEROSPACE ENGINEERlNG OEPARTHENT bC * SEPTEMBER 19 76 c C $**+***SSSf*$8**+*+*****4******************<****4********************* C C INTEGER GRAIN REAL MGEN*MDIS~MNOZ~M&~*JROCK*N~L~ME~~ME~ISP~ITOTvHU~HASSeISPVAC REAL NlrNZvNSEGvKl*K2vKEHvKENvNSvLCCeLTAY REAL M2,MDBARv ISPZe ITVACvKAvKBvLAMdOAv ITV COHMON/CONSTl/ZWvAtvATvTHETA,ALFAN COMMON/CONST2/CAPGAM~ME~0UTE~ZETAF~Ta~H0~GA~E~C6AME~T~P€~ZAP€ COMMON/CONST3/SvNSvG~AIN~NTA8YvNCAKO COMMON/C~NSl4/OfLOIvOOvZOvOl COMMON/VARIAL/Y~T~DELY~OELTAT~PONOZ~PHEAD~RNOZ~RtiEAOvS~MAdvPHMAX COHMON/VARIAZ/ABP~RT~ASSLCIT~A13NOZ~APHEAOvAPNOZ~~A~Y~A~P2~A~N~vAdS2 COMHON/VAKIA3/1TOT~ITVAC~JKOCK,ISP~ISPVAC~M~~SvMh~Z~SGvbUM~T CO~4MON/VARIA4/RNTrdHT~SUM2~K1~K2~H3~RtfAVE~RNAVEvkdAR~YB~KUUNTvTL COYMON/VARIAS/A8~AIN~ABTO~SUMOYvVCIvAdTTvPT~AN COMMO~~/VARIA~/WP~~CFe~PvRA0ERvtf'Sv VCvFLAST vTLASTvDJvPClNTOT,kPl COMMON/VARIA7/TIMXvFVv ITVvNX COMMCIN/VARlAB/YOI b CSMMON/VA~IAY/E~HETA~RHO~~RATP~PMCO~~CMCO~P~~U~CMU~ALPTS~RATG~ b COM?lLJN/VARIZO/Yt TA 9XETHv I50 COMMON/IGNL/KAvK8~dFS~RHO,LvP1JiIGvTf 1vT I2~LS1GvQlvNlvU2vN2 CUMMON/IGN~/~LPHA~~€~A~PB~G~~~~G~C~LT~GPA~TOF~L~P CDMMON/PLOTT/NUHFLlf 161 vIPQvNUUPv IPTv IGP DiMENSI3N YTAb(301 vTTAB(30) DATA PIvG/30 19159~32- 1'25/ CALL GSCZE (416.rll-O,1100) CALL PLOT(6.25r2-~-3) iOP=O REA0[5rSaOI NRUNS Table C-3 (~ont'd)

C *******************************t****************************** C * READ IN THE NUMBER GF C;NFIGURATIONS TO BE TESTED 4 C ****************************************4**************************** NIABY=O NC ARO=O 00 901 t=ltNRUNS NEXTR=NTABY-NCARO IF INEXTR 11901t A901 t 1902 1902 REAO(5~1903) (OLtD2~03tD4tD5tDbtIEX=leNEXTR~ 1901 WRITE(bt602) i REAU(5,lllll) NTABtNTABY RE AD( 5,499) SUMOY I .".!?SG 2MiY e? .CELTAT rRN2LrS::EAC~SiiHA~rPiii.iA~~ 5~142~ IT ~~T~RHT~RNT,R~~R~OK~~RHAVE~RNAVE,HBAR~ITVAC~SUPMT~PONTUT C *S*S*l**St*$****8*****************8*4******4***********4********4**** C * SET INITIAL VALUES OF SELECTED VAKIAdLES EQUAL TO ZERO * C * ***NOTE*** THESE VALUES MUST BE ZEROED AT THE BEGINNING OF * C * EACH CONFIGURATION RUN * c **********+************************************************************ b READ (St4911 IGOtIWOtISO READ(St+931 1YOt(NUMPLT(JJ)tJJ=1tlb~tlTEYP C ...... - C* READ IN THE USER'S OPTIONS * C ** * C * VALUES FOR (GO ARE * C 0 FGR hO IGNITILN TQ4NSIENT CALCULATIGNS C * I FOR IGNIT ION TRANsltNT CALCULAT IONS * C * VALUES FOR IN0 ARE * C * 0 FCj8 NO INERT *EIGHT CALCULATIONS * C * 1 FOR INERT *EIGHT CALCULAI ACNS * bC * VALUES FOR IS0 ARE * bC * 0 FOR NO GRAIN DEFORr4ATION EFFECTS * bC * 1 FOR GRAIN DEFORMArlGN BALLISTIC EFFECTS qr bC ***YOTE*** if GRAIN-ZlALL STAA GKAIR), ISO MUST BE 0 * C * VALUES FOR iPO AAt * C * 0 FOR NO PLOTS * C * 1 FO3 PLOTS OF EUUIt IDhIUM BllRhlNG ONLY *** C * 2 FUR PLUJS (IF IC;l\iIlA'J~~Ih/bhSIthT LlNLY * C * 3 FOd PLCTS Of- BIITP IGNITIUh TKANSIEhT 4NU * c * EC;UILIUCIU'-l HUIJ:'IkG * C * VALUES FOk NUHPLT(JJ1 hk, Ih3T I\ELUI htC tCF. IP(I=u) * C * 0 IF SPECIFIC PLOT 15 NUT DLSI:ILU * i. 1 IF ,PECIFIL PLOT IS YESIRED 4 C * ORDER OF SPECIFICATION UF hUMPLTLJJ) IS * t * 1 FHEAD VC TIi4E * C 2 PONOZ VS TIME * C * 3 PtltAC Ah0 rGF'lZ VS TIME 01 C 4 RHEAD VS TIP * C * 5 HNLZ VS J lME * Table C-3 (~ont'd)

C * 6 RHEA0 AND RNGZ VS TIME 1000 CON1 lRUE C 7 SUMAB VS TIME C * 8 SG VS TIME * C * 9 SUMAB AND SG VS TIME a C * 10 F VS TLAE t C * 11 FVAC VS TIHE * Cc* A2 F AN0 FVAG VS TINE Cce 13 VC VS TIME * C .* 14 SUMAO VS YB L C * 15 SG VS YB t C * 16 SUMAB AND SG VS YB C * VALUES FOR ITEMP ARE * C * 0 FOR TEMPERAtURE GRADIENT * C * 1 FOR UNIFORM TENPEfiATURE C NTAB IS THE NUMBER OF Y STATIONS FOR NHICH TABULAR c C * TEMPERATURES ARE SPECiF IED C * NTABY IS THE kUH8ER OF Y STATICNS FOR WHICH TABULAR AREAS c C * ARE SPEClF lED * C ...... WRITE ibr492) IGOt IWOeISO WRITE(br494) lPOt~NUMPLT(JJ)tJJ=lt16ItIlEMP HKITE(6r 11112) NTAB,NTAdY READ~St501)RN2Nl,RHO,AlvNlr ALPHA,RETA~i3UpCSTAR C *******4******4***8***4***+4S*44***44It****k*****44*******44****k***** C * READ IN BASIC PROPELLANT CHARACTER1 ST ICS * C * * c*+*rc MOOIF [CATION MADE 5/3/76 c * RNZNl IS THE RATIO OF THE NOMINAL VALUES Of THk OUkNING RATE * C * EXPONENTS ABdVE AND BELOW TtiE TKbhSI 7 ION PRESSURE * C * RHO IS THE DENSITY OF ThE PROPELLANT IN LBM/IN**3 t C * Al IS THE BURNING RATE COtFFICIENT BELGW TYE TRANSITIUN * C * PRESSURE * C * N1 IS THE aURNING RATE EXPONENT BELCW THE TRANSITION PRESSURE * C * ALPHA AN0 BETA ARE THE CONSTANTS IN THE EROSIVt BURNING 4 C * RELATION OF HIIOILLARC AN0 LENOIH * C * MU IS THE VISCOSITY OF THE PROPEL1 Ah1 GASFS * C * CSTAR IS THE CHAKACTEA IS1IC tXH!iUST VtLOCI TY IN F T/SFG * C ******4***4****~****OI***aiC*~*SIt84***~0**4*~~+4****4******~****~****** HRITE :b,603) HHO~Al,Nl~kLPHA,B€TA,~'U~C5TARwRh2hl AHO=RH0/32.174 :.€AD( 5, 532) L ,TAU,DE,DTI ,THETA,ALFAh.LTAP,XT,ZO,CSTART ,PTRAN C ******+******~******~~#rO4*rC**44*******94*4*~*******4*4*4****4**4****4 - * READ I& BASIC MOTOR OIMENSIONS rg c * * C * L IS THE TOTAL LENGTH OF THE GRAIN IN IhCHES * C * TAU IS THE AVERAGE dEb THICKNESS OF THE CONTROLLING GRAIN * r* LENGTH IN INCHES * Table C-3 (Cont'd)

C DE IS THE DIAMETER OF THE NOZZLE EXIT I,J INCHtS * C * 011 IS THE INITIAL DIAMETER OF THE NUZZLE fHKOAT IN INCHES 8 C * THETA IS THE CANT ANGLE OF THE N3ZZLE WITH RESPECT TO THE C * MOTOR AX1 S IN DEGREES * C * ALFAN IS THE EXIT HALF ANGLE OF THE NCZZLE IN OEGREES L C * LTAP IS THE LENGTH OF THE GRAIN AT THE NOZZLE EN0 HAVING t C * AODIfIONAL TAPER NOT REPRESENiED BY ZG IN lNCHES * C * XT tS THE DIFFERENCE IN WEB THICKkESS ASSOCIATED hIlH CTAP a C * LO IS THE INITIAL OIFFERENCE BEiWfEN WE& THICKNESStS AT THE * C * HEAD AND AFT ZNDS OF THE CONTRGLLING GRAIN LENGTH * C * CSTART IS THE fEHP5AATURE SENSITIVITY OF CSTAR c C * AT CONSTANT PRESSURE * C * PTRAN IS THE PRESSURE AdOVE WHlCH ThE BURNING HATE EXPONENT * C * CHANGES * C ************44*44***********4*****O************4******4*4**4********* NZ=NL*RNZNL 42=Al*PTRAN** I Nl-N2) WR ITE(6,604) L ,TAU,UEtDTI ,THETA,ALFAN,LTAP,XT,ZO,CSTART tPTRANwh2 THETA=THETA/57,2YS!d ALFAN=ALFAN/57.29578 READ(S,503) OELTAY ~XOUTIDPOUT~ZETAF,TB,H8tGAM,EHREF ,PR€f 9 1DTREF,PIPK,TREF,GAME~PkXT IF{ITEMPmNE~OIGO TO 10300 READISp700) (YTA61 ITAt3),TTABI ITARJ~ITAt3=l,NTAB) WRITE(6p701) tYIAB~ITAB)~TTAU(ITAB)piTAU=l,NTA~) GO TO LOO04 10000 REA0(5riiI0011 TGk C *$**+****t4*******4*4*********4****~4**f********4*4******~***~~*4**** C * REAO IN BASIC PERFOAMANCE CONSTANTS 4 c * * C * OFLTAY IS THE DESIKEU BURN IPdCfiEMtN' OUKING TAILOFF Ih INCHES * C * XOUT IS THt UlSTANCE BURNED IN INCHES AT WHICH THE PKUPELLANT * C * BREAKS UP * C * OPOUT IS 'HE DEPRESSUfiIZATiON RATE IN LU/Ih**3 AT kHlCH THE * C * PKOPELI-ANT IS EXTIkGUI ShfG * C 4 ZETAF IS THE THRUST LCISS CUE; FkClfNT * c * IS THE ESTIMAT~.~DUKN 1j;rt IN S~CU~DS I@ C * HB IS THE tSTI1YArf:O tlU8hOUT ALTITUL)E I~JFttT * C * A2 IS THE i3UKRING DATE COtf-6lCIEhT AdCVE THt THANSITJUN * C * PRESSURE * C 4 CAM IS THE RAT10 OF SPtCiFIG HtAlS FOR Ttit PHOPELLPNT GASES * '1 * EKREF IS THE WLFERENCC JdRUAT ~HUSIILNHATE IN iN/S€L * C * TGR IS THE TEMPtRATUKE Of THE GRAIN Ih CtGREtS F * C 4 PREF IS TIiE KEfEFtfNCt NUZZLt STAGNATICN PHESSUHt IN Lt)/lN**i * i * OrREF IS THt HEFtKEYGt TtiKtlAT DIAMtTkd IN INtHtS * c * PlPK IS IWE ItMPtKLTURt StNSiTILITY CCEFf lC!ENI OF PRFSSURE * c * AT CdNSTANT K PEN DtGREL f- 4 c 4 TREF IS WE OESICN TEMPERATURE uk TPE GRAIN IN OEGKE~SF 4 Table C-3 (Cont'd)

C GAME IS THE EFFECTIVE GAMMA A1 THE NOZZLE EXIT PLANE * C PEXT IS THE PRESSURE AT WHICH THE PROPELLANT EXTIhGUISHES IN * C LBIIN**Z I - c tt~*~t*~++++w~*****t***t********e***t*****~****~***c***~r*t~****c***t. 10004 WRITE(~~~~~~OELTAV~XOUT~DPOUT~ZETAF~~~~HB~GAM~ERR€F~PREF~DTR&F ~,PIPK~A~,TREFIGAHE~PEXT IF11TEMP.NE.O) WRITEZ6~10002) TGR NCAR D= 0 NO UM= 0 IP 1x0 WN la. 85 z= to s=o.o NS=O.O KOUNT=O ABMAIN=O.O

A0 lO=O lO ABTTsO- TLAST=L DE LY=OtL TAY b ElHETA=OO b PMU=O. 5 ? AC PTS=Oo ? YETA=Oo ? DYETA=Oo b PMOD- 13030o ? YE=OoO b XE TH=Oo TOP=GAM+ 10 €30 T=GAM- 1 ZAP=IUP/(Z.*BOT) CAPGAfi=SQRT (GAM )*120 /TOP )**ZAP VOPE=GAblE+ 1 l BOTE=GAME-1- ZAPE=TOPE/( 20*BOTE CGAYE=SdRT(GAME)*l2~/TOPE )**ZAP€ AE=P I*UE*DE/Qm CSTARN=CSTAR 1 IF(XToLEmO.0) Ti=OoO :F(ITEMPoNEoO) GO TO 10d03 CALL INTRPI~TTA~~YTA$~NTAB~Y~TGHIO) .;***a MOOIFICATION MAOE 1/7/76 QL=AA*EXP(PIPK*( lo-Nl)*( TGK-TREW QZ=AZ*EXP(PIPK*( 10-N2)*I TGR-TREF) 1 WRITEl6qlOll YqTGR 10003 CSTAR=CSTARN*[A.+CSTART*(TGR-TREFI) C**t* MODIFICATION MADE 1/7/76 IF( ITEMP110005q10006q 10305 Table C-3 (Cont'd)

10005 QL=Al*EXP(PIPK*I 1.-Nlj*( TGR-TRkF) QL=AZ*EXP1PIYK*[ 10-NpJf [TGH-TREF j i 10006 IFIXToLEoOoO) GO TO 40 TC=(Y-TAU+XT+Z/~O)*LTAP/XT IF (TLoLEoOoO) TL=OoO IF 4TLoGEoLTAP) TLsLTAP 40 IF (Ti QlrGlt*Z 41 OT=DTI GO TO 43 .42 RAOER=ERREF*((PoNOZ/PREF)**r)o8)*((DTREF/DT)**O-2~ OT=DT+(2.0*RADER*DELTAT) 43 AT=PI*DT*DT/4* EPS=AE/AT IF (IGOoEOeOeORoYoGT,0.0) GO TO 900 REA0(5,97) KA,KB,UFS,CSIG-PMIGrTI l,TIZvRRIGtOELTIG,PBIG C $$$*$***LS*$***********8f.******************4************************* C * READ IN VALUES kE9UIRED FOR IGNITIOh CALCULATIONS * C * **+NOTE*** NOT KEUUIgED IF 160-O c C * * C * KA ANG KB DEFINE THE CHARACTERISTIC VFLOCIlY IN FTISEC * C * CSTR = KA + KR * PfiESSURE s C * UFS IS THE FLAME-SPAEAOING SPEED IN Ih/SfC 4 C * CSlG IS THE CHARACCERISTIC VELOCITY OF THF IGNITER IN FT/SEC * C * PMIG IS THE MAXIMUM IGNITER PRESSURE IN LOS/IN**L * C * TI1 IS THE TIME OF MAAIMUM IGNITER PKESSUKE IF* StCONDS * C * TI2 IS ThE TIMEI IN StCUhDSl FOR 'Ht IGNITER PKESbURE TO I c * UROP 10 10 PER CEIVT Of- MAXIMUM VALUt(PM1G) * C * RRIG IS THt AVERAGE REGRESSION RATL- UF TIiE FIRST HALF OF THt * C * IGNITER PRESSURE TIME TRACE 1N LBS/IN**2/SEC * C * OELTIG 15 THE IIME INCKEMFNT FUH IGNITIGN TKAN:IErUT * C * CALCULATIONS IN SECONDS * C * P0IG IS THE BLdWUUT PKESSURt UF THE MAIN MOTOR BLOrdOUl PLUG * C * IN LBS/IN**2 * C *4******************444**tr*********4**4*********44*******4********4** HRITE(bt8421 KA~KB,UFS~CSIG~PMIG~~TI~~~RIG~I:~LT~G~PBIG 900 IF(IHOokOe3.0KoYoGT~Oo~)GO TO 832 KEAOIbv603) ~TEMPISIVMAP~SIGMAS~X~~X~~SYCN~PI~L)CC~PSIC~~~LC~LC lC,NStZ,tlCNISYNNUM, PSIS,PSIA,Kl vK2vD51IhS,LLL ~~MS,KEH~K~~,U!-II~~~~TAU ZL, WA C ~*~$**4~4$**~**~**~*4L:C****O~**~*rt.c4ic$*444*~b<*v444*~*4*4**4444****** C * READ Ih BAS 1C PKUPtKTIES KtUUIKED 6CR hEIGHT CALCULAT IC!vS le * ***NOTE*** NOT REQUIKEO IF IWU=d * C * * C * DTEMP IS THF MAX EXPECTED INCHtASt IN TEYPERATUKE ABr VE L C * CONUIIIUNS 3RDtR Wt!ICti MAIN TRACk WAS CALCULATEO IN t C * DEGKCkS f AHREILlit IT * C * SIGMAP IS THE VAKIATIUbi '5 PHYAX rc C * SIGMAS IS THt VI\RIATI(IP~ :h CASE MAIERIAC YIELD STRENGTH * Table C-3 (Cont'd)

C * X1 IS THE NUMBER OF STANDARO OEVIAT IONS IN PHMAX TO Bt USfD * C * AS A BASIS FOR DESIGN C * X2 IS THE NUHOER qF STANDARD DEVIATIONS I& SY TU BE USED AS * G A BASIS FO: )€SIGN * C * SYCNOM IS THE NOMINAL YIELO FTRENSlH UF THE CASE RATERIAL * C * IN LBS/INCH a C * DCC IS THE ESTIMATED MEAN UIAHETER CF THE LASE IN INCHES * C * PSIC AS THE SAFETY FACTOR ON THE CASE THICKNESS * C * @€LC IS TI'€ SPECIFIC WEIGHT OF THE CASE MATERIAL IN LBS/IN**~ * C .* LCC IS THE LENGTH OF THE CYLINORICAC PORTION OF TYE CASE * C * INCLUDING FORWARD AND AFT SEGMENTS IN INCVES * C * NSEG CS THE NUMdER OF CASE SEGMENTS * C * HCN IS THE AXIAL LENGT!i OF THE NOLZLE CLOSURE IN INCHES * C * SYNNCY IS THE NOMINAL YlELO STRENGTH OF THE NOZZLE MATERIAL * C * IN LBS/INCH * C t,. 6 PSIS IS THE SAFETY FACTOR ON THE NOZZLE STRUCTURAL MATERIAL' * u* PSlA IS THE SAFETY FACTOR ON THE NUZZLE ABLATIVE MATERIAL * C * K1 AND K2 ARE EMPIRICAL CONSTANTS IN THE NOZZLE WTe EQUATION * *+ FSIINS IS THE SAFETY FACTOR ON NOLLLE INSULATION * 4- a DELINS IS THE SPECIFIC WEIGHT PF TYE INSULATION 1N LBS/IN**3 * ,301 CONTiilUE C * Ktti IS THE EROSION RATE OF INSULATICN TAKEN CONSTANT * C * EVEHYHHERE EXCEPT AT YHE NOZZLE CLCSURE Itr II./SEC C * KEN IS THE ERUS10il RATE OF lNSULA1 ICN A1 THE NOLZLE CLOSURE * C * IN IN/SEC * C * OLINE!? IS THE SPECIFIC WEIGHT GF THE LINkH IN LBS/Id**3 * C * TAUL IS THE THICKNESS OF IHE LINER IN 1NC!iES * C * HA IS ANY ADDITIONAL WCIGHT NUT CONSICkhFD ELSEWHERE IN LBS * C C*************************~-~1Y*************\-***~***444********~****** WRITE (6,610) DTEMP, SIGMAPtSIGMASt XlqX2t SYCNClt4tOCC~PSICtDELC~L 1CC ~NSEG:!iCN~SYNNOMrPSISrPSIA,Kl,KZrPSI INSqUEL INS~K~H~K~II~LLINEdt Tfi 2UI- *HA 872 1;ONTI NU€ b rFIIS0,EQ.O eORa YmGTm 0.0) GD TO 80 b Hi 40( 5,5351 PMOOqCMODt PMUqCMUt ALPT SqTAUC ,C ,C *******~e****4**44**0*4*t*~*rgr**ItO~4;****4***4**4*~*4**4*W*****e*4***& bC Q READ IN THE CONSTANTS FUR GKAIN iJtFUKPATION * bC ***NOTE*** NUT HEGUIKED ;F ISC=O * bC * PMOO IS THE ELASTIL M03ULUS CF Ttit PRGPELLArJT IN PSIA [AT TH2 * bC * TEMPERAT~HE TLRI * 3 + Cr4OD IS THE ELASTIC MUUULUS OF 7t:F C4SC (id TtiE bULK TtMPERATURt * ..A OF THE GRAIN) IN LBS/IN**2 PC 4 PMU IS POISSOhxS RATIO FOR THE PKOPI~LLPNT (o) THE BULK TLMPCKATGRE* br $ OF THE 3RAI:i) * CNU IS POISSOPJrS AATIO FOR THE CASE iai I'if BULK TFMPtRbrURC Of * bt * THE GRAIN) * bC * ALPTS IS THE LINEAR COEiFICIFNr OF THtRMbL EXPANSION OF THE * bC * PROPELLANT IN IN/IN * Table C-3 (Cont'd) bC TAUC IS THE CASE THICKNESS IN INCHES * ,C ,C ********************************************************************* b WRITE(6*799) PI4001 CWOO*PHU*CHU*ALPTSpTALC 80 CONTINUE CALL AREAS IF(YaLEm3.0) VC=VCI fF(ABS(ZW)mGToOoO) GO TO 20 'F(SUHABoLE.0-OJ GO TO 31 X= (ABPORT+ABSLOT)/SUHAB 90 ';hOZ=AT*X/APNOZ*(2 .*( La+BOT/2o*UNl*!4Nl)/ TOP) **ZAP if (ABS(MNOi-MNLImLEm0.002) GO TO 2 IZ.4 L-MNOZ GO TO 90 2 VNOZ=GAH*CSTAR*~NOZ*SQRT(~(~~/TOP~**~TOP/~OT~I/~L~+~OT~~~*H~~Z*HNO 12) J PRAT=1le +BOT/2~*HNOZ*HNOZ) ** 4-GAH/BOT) JROCK=AT/APNOZ SUHYA=OELY*(AbP2+ABNZ+ABS2) IF (Y-EQoOeO) SUMYA=OoO VC=VC +SUHY A w If IiSO-EQ.0) GC' TO 9 b RATR=ZI*t/OO + OI/W b RATRZ=RATR**? b RATP=(~~*RATK?*(L~~PMU**~)/(~~~RATR~~~/(QM~O*DO*~~~~PHU*CMU~/~C~OD b l*TAUC*Z,)-PMU*(I,+PMU)+( io-PNU**Z)*L A-+RATkZ)/( lm-RATd2)) b 9 CONTINUE IF~Y*GTeOIO) GO TO I1 c**** HOOIFICATION MADE 1/7/76 PON~Z=(Q~*RHO*CSTAR*SUMAB/AT I**( I*/( ~~-N~~)*(~~+(CAPGAN*JROCK)**~/ 12- )**(N~I[ i.-rvl) IF~PONOZ~GT~PTRAN)PONOZ=(Q~*RHU*CSTAR*SUPA~/AT)**~~~/~~~-NZ)~*~~~+ l(CAPGAH*JROCKJ**Z/Z- I**(lJZ/( 1a-NZ)J MO IS=AT*PONOZ/CSTAR P2=PONOZ PONOZ Z=PONOL PNOZ=PRAT*PONOZ P4=Z.*HOIS*VNOZ/IAPHEAD+APNOZ)+PNOZ. lFIGRAlN,Ed.3) PG=MDIS * VNOL/APNOZ + PNCZ 5 PNOZ=PHAT*PUNOZ PHEAD=2.*MDIS*VNOZ/(APHEAD+APNGL)+PNOZ IFIGKA INeEO13) PHEAO=#DI S * VI\~Z/AP~UL+ PNOL IF(PHEADmCEePTRAN)RHEAD=QL*PHEAO**Ni if (PHEA0-CraPTtiAk) RHEAD=CiZ*PhEAO**NZ ZI T=MOIS*X/APNOZ HN l=RHEAO PHEADZ=PHEAD c*++* MOO IF ICAf ION MADE 1/9/76 3 IF(ALPHA) l?lrl32q131 P * C C 0 0 W W Q, C 4 P 4. C h, c c ry r IU- (~r w~-~~-~.~z~c~o-o~-~~~o~c,-x,;~BoP~~~cID;~-~~-rnx-rnx-o-=- osn3xnsza~~mmnomnz-oqcnxz<'oozznooIxn--E; 02 QZ PvrV B < < 3- 4 xr xr ** 3 - x rn 9 cv 0 Dl B1 *o c. C C - 9x G x f g 0 * r -C 4. rub kt) 0 a LI l r* -+ -* 1 C3 l a* -4- *w *. z z 0 IN *z *Z a -4 D Q 0 XI *O *O Y n c1 a 7 DC (rc er~ U X, 0 .*r a* a* 0 e CF lu 4)- ** ** + + + h\ Wt OX D D - I- m h, a c w kt- 4 r 3 V, V, e ee PW B* r r 1 *I *r *r C: C ru XX ;red PO 4 4 l >D xx xx - - 9 *I( o* 0s + + P XT, \* \* r. D 'x ruw hlu wlu h, r C *- LI- m- + * - Q+ 44 44 DW \ ZP \* \4+ C 0 P cx -* -* tZ x -C re re 0 0 C -* *a *a fu lu 0 t *\ I) + - -II - e- 9 9 - wc rzr fuP L Z 9 e9 * ** lu r 4b ID m* m* * + W mx %. X* * s- D 0 Qtu Qh) 0: Cr *G -* -* z 7 P * wm wm I Table C-3 (~ont'd)

GO TO 5 6 RE=Z~*~DIS*X*L/((APNOL+APHEAC)*HU) IF(IG0oNE-OoANDoYoLEoO-OJ CALL IGNITN IF(YoLEmO.OJ WRITE(6r AOL) RE PONJ=PONOZ CALL OUTPUT 10 IF(YoLE, ,OS*TAU) GO TO 16 StNK1=VC/(CAPGAM*CSTAR1**2*RBAR*OPCDY/l2- HASS=-Ol*MOIS AN S4=Y+10* O*DELTAY IF(KOUNTaCT-0) GO TO 16 IF(ABSISINKl)oLEoNASS.ANOoANS4.LE.ANS-XTJ GO TO 18 GO TO 16 18 OELY=lOo*DELTAY GO TO 55 16 DE LY=OEL 1AY 55 YLED=Y Y=Y+DELY ANSzTAU-ABS(Z/ZO) c***4 HOUIFICAlIdN MAOE 1/9/76 IF (YoGE- ANS-ANDoKOUIJToEQoO) OELYzANS-YLEC+OoOS*DELTAY IFIYaGE.ANSoAKDoKOUNToEQo0) Y=ANS+OoOS*OELTAY DELTAf=2,*OELY/(WHAVE+RNAVE) b DELTAT=UELTAT* ( ( 1,-PONOZ*( 10-20*PMU) /PMOOI *Il.+ALPTS* ( TGR-taEF 1) I* b 1*3/f lo+tTHETA) b DYETA=DELY*( (1o+ALPTS*(TGR-TREF J I*( 1,-PGhC:Z+I Io-Z0*P!lU)/PMtJ0) )**3/ b 1(1 ,+€THE Tb l b YE TA=YEfA+DYETA b IF ISOoEQoiJI YElA=Y SUM2= SUHAB RN2-HN02 RHZ=RHEAO AVE2=AVE 1 GO TU 1 11 MDIS=AT*PONtlZ/CSTAR GO TO 5 12 DPCDY=(PHEADL+POPJOZZ )/[KNAVk+RHAVt) *l)ROY +lI'HEAUL+PUNOLZ)/( I AUPZ+AB lNZ+AOSZI*2e)*DAUY IFIAHS~DPCDY),Gt,UPOUToOR.Y,GE.XOUl) GC TC 25 SINK~=VC/(CAPGAM*C~~~K~**~*RGAH~~)PCOY/~L~+~P~IEAU~+PC;NULL)/Z~*~RNAV iE+&HAVE)/L- *IAUPZ+ABNZ+ABSL) /( LLo*(C5TAk*CAP(;AM)**2) STUFF=NGEN-SINK1 MU IS-STUFF PONOL-MDI S*CSTAR/A 1 IF(YoGEeOoY*(ANS-XT) )PL)luI:Z=PI:PJJ+OPCUY*CELY IFLSTUf F oGE -GMINoANO.SlUFFoLEoGMAX) GO TC 14 GO TO 5 14 PL=PONOZ Table C-3 (Cont'd)

PONJ=PONOZ PONOZL=(Pl+PZJ /2- PZ=PONOZ P3=QHEAD PHEADZ=( P3*P4) /ZI P4 =PH E AD NO IS=AT*PONOZ/CSTAR OELTAT=ZI*DELY/IRHAVE+RNAVE) b OELTAT=DELTAT*(~1--P~NOZ*(1--2-*PMUJ~PHOO)*(l~+ALPTS*~TGR-TREFJJJ* b 1*3/( Lo+ETHETA) Z=L+UELTAT*( RNAVE-RHAVE) T=T+DELTAT IF(YoLT.ANSJ CALL OUTPUT lF(YoLT,ANSJ GO TO 10 ZW =z SUHBA= SUMAI P l=PONOZ RHZ=RH€AD RNZ=RNOL RAVE=AVEl ABAAIN-SUMAB ABTO=OoO WRITE(br51) 2G ANSZ=TAU+ABS( ZW/20 J KOUNT=KOUNT+ 1 1FLKOUNToEOol)CALL OUTPUT IF(KOUNToE9ol)GO TO 10 DE LYW =DE L TAY DY 2-DELY W IF(ZH1 32~32.33 32 IFtYoLToANS2oAND,AaSIZW)-GToOYZ) GO TO 211 SUMAB=ABMAIN+ABlT GO TO 31 211 SU:4DY=SUMDY+DELYW SUMAB=(l.+SUM~Y/Zd-DELYW/(Z~*ZWl)*ABTO-tSUMDY/Z~-D€LY~/(Z~*ZM~J*AB lMAIN+ABTT GO TO 31 33 lF(Y.LToANS2*AND-ZWoGToOY2) GO TO 21 SU&;AB=ABTOtABT T GO TO 31 21 SUMDY=SUMDY+DELYH SUMA~=(~.-SUMDY/~W+DELYW/(~-*ZW)~*A~MAIN+(SUMOY/ZW-DELY~/(Z.*ZMJ)* 1ABTOtABTT 31 IF(SUMABoLE.O-0) PONUL=POkOL/2* IF(SUMAOaLE.Oo0) GG TO 25 MDIS=AT*PONOL/CSTAR ABAVE=(SUMAB+SUMf%A 1 /20 SUMYA=DELY*ABAVE Table C-3 (Cont'd)

VC=VC+SUMYA DADY=ISUHAB-SUMBA) /OELY PBAR=1Pl+PONOZ 1/20 SUHBA-SUHAti 22 IF(Pi3AR,LE.PTRAN)DPCOY=P€iAR*CAUY/(lo-NA)/ABAVE IF (PtlAR,CT,PTRANl DPCDY=PBAR*OAOY/( i 0-bJ2) /ABAV€ PONOZ=PONJ+DPCDY*O€LY IF(PONOLoLEo3-0) PUNOZ=O-0 IF(PONOLoLEoPEXTl GO f0 25 c*+** MODIF lCATION MADE 5/3/76 IF(PONOZoLEoPTRAN) Q2=01 IF(PONOZmLEo PTHAN) NZ=Nl RNOZ=QZ*PONUZ **N2 c+r+* MOOIF ICATlON MADE 1/9/76 RH €Ad= RNOZ RBAR= IRHEAD+RAVE )/2- b IF( ISOeEOoO) GO TO 405 b GU 10 401 b 605 CONTINUE b MCEN=RHOS*(KNUZ+RHEAO)/2 o*SUMA6 b IF(ISO.EU-1) MGE~v~=MGEN*(~~+ETHETA*XETH) b IF ~IS0oEO.O) MGENL=MGEN GMAX= 100002tMO IS GM IN=0.399d*t40 IS SINKl=VC/lCAPGAM*CSTAHI**Z*RBAR*DPCDY/ lZo+PBAR*ABAVE/( 12o*(CAPGAM* 1CS TAR 1 **2 1 *i\dAR STUFF=MGENl-SI hK1 MDIS=STUFF IF lSTUFFoGEoGMIN,ANUoSTUFF.LEoGMAX) GO TC 23 PBAR=lPL+PUNOZ J/2, GO TO 22 23 RHAVE =tdH2+RHEAC)/2- KNAVE=(KNZ+RNUL I/Z, RHZ=RHEAU RNZ=RNOZ PHEAD=PUNOZ KAVE-YtiFAD P 1 =P~IWZ PO NJ- PONOL MDIS=AT*PONOL/CSTAi4 IF/ABS(DPCDY),Gt.DPilUT) GO TO 25 IF(YoGEoXOUT) GO 10 25 DtLTAT=20*3tLY/(KHAVE+RNAVEl DELTAT=UELTAT*I ( 1.-I'ONIJL*~ 1.--2o*PMU)/PMOI;)*I l*+ALPTb*(TGR-TRCF 1) I* 1*3/1 1otLT)j.k JA) Z= L+DELTAT* (l

GO TO 10 25 RHEAD=OoO RNOZ=RHE AD PHEAD=PONOZ HD IS=AT*PONOZ/CSTAR WRITE(4t318) DELTAT=2.*DELY/(RHAVE+RNAVE) DELTAT=DE~TAT*~~~~-PONOZ*~~~-~~*PHUI/P~OD~*~~~+ALPTS*~TGR-TREF~~~* 1*3/( l.+ETHETA) 1= T+DELTAT CALL OUTPUT TIHE=T DELTAT=,S rf N=TIHE+S. PHT=PHEAD SG=Oo 0 29 T=T+DELTAT PHEAO=PHr/EXP( CAPGAM**2*AT*CSTARj JC*{ T-T IME) *I20 1 PONOZ=PHEAO MD IS=YUhUZ*AT/CST AR Y=Y+. S*RHEAD CALL OUTPUT IF(ToLT~TIM~AND~PHEADoGE~O~O4)GOTO 29 WP l=G*SUHMT WP2=RHO* I VC-VC I)*G WP=L kPL+WP2 1 /2, IF( ISO.EQ.11 nP=WPl IS P= ITOT / dP ISPVAC=I TVAC/WP FAV= ITOT/T FVACAV=I TVAC/T PONAV=PONTOi/T LAMBDA=( VC-VC I)/VC WRlTE(b1102) ~PllWP2rWP~PHMAXlISP~ISPVAC~IIOi~ITVAC~FAV~FVACAV~PO~ ~AV~VCIIVC,LAMBDA IF(!kO.E3.0) GG TO 903 P?4EOP=PHMAX*( l.+XL*512:1AY )*EXP(PIPK?UTEMP) SYC=SYCNOY*(l.-XZ*SIGXAS) TAUCC=PSIC*PXtOP*OCC/(2**SYC 1 wCC=PI*TAUC~*CCC*UELC*LCC*~ 1.+LNSEG-1, )*(4O.*TAUCC/LLC)) TAUCD=TAUCCi2. WC H-2- 5*P i /Z. *OCC*OCC*TAUCO*DECC WCN=6.5*PI/2**OCC*HCN*TkUCO*DELC WC=WCC+wCH+WCN EPSIL=AE/PI/DI 1/OT 1% HN=Kl*DTI*DTt/(1.+.S*SlN~ALFAN~)*~~CPSIL-S~~T~EPSlL~~*PMEOP*DTl*PS 1IS/SYNNOM+KZ*T*PSI A) WINS=T*PSIINS*DELINS*OCG*P~*IKEH*~DCC**~O+~S+NS)*TAU/~~+O~~~/ Table C-3 (Coqt'd)

lPSIINS*(CCC-TAU*(S*NSI))+KEN*-80*HCN) WL=TAUL*DLINEH*PI*DCC*(DCC/Z-+LCC+HCN) WI=WC+WN+WINS+WL+WA )?H=w1 +WP ZE TAH=dP/WM RATIO= ITOT/WM WRITE(b,b05) WRITE(6,601) PMEUP,TAUCC,WC~~N,WI~S,WL,WI,WM~ZETAM,RATIO 903 CONTINUE ND UM= 1 IF ( 1POeNE.O-AND- IPOoNEoZ) CALL OUTPUT 901 CONTINUE IF(13P-NE.dJ CALL PLOT (0-0,0-0,999) STOP 500 FORMAT(X,F~~LI~X*F~~L~~X~F~*3,7X7FU*5,7X,Fd-5t/qlOX, 1 F7.2~4k7FboZtrtXrFt.2r8XtFlO. 1rtXpt t3.2) 6d4 F~RMAT(//~L~X,LI'~~~A~ICr.1UTOH L)ltiEluSlUI1S~/rl3X~3tiL= ~F8.L~/,13Xv~H1 1AU= ,Fb-2r/rlJX,4tiDL= ' 1 I't11-4t/,13Xr5ti~)TI= rLPt11~4~/~13X~7HT~ETA=r 1PE11-4,/r13Xt7t1ALP JHAN=, 1PE11.4r/r13X,bt4LTAP= r lPtllo4,/r 13X,4t1XT= ,lPtll.4,/r 13X14HZ 40= r lP€11.4r/~l3X1tlHCSTARl= ,lPt11~4~/,13X1ItlPTtiAlu= ~lPElI~G~/rlJX 5*4kINL= 1DELl.':) 10001 FORMAT ( SXrF 10-U) 700 FOkMAT~2FlOo4) 701 FU&MAT(ZOXI'Y= '*~PE~~.~,~~XI'TGH=', 1PE11.4) Table C-3 (Cont'd)

503 FORMAT(7X,F6m3,5X,F7-2,7X,F7-2,7X,F So4,3X,f6a2,3X,F8aO,/, 15XtF7m4,8X,F8-5,SX,F8.2,7X,F7e3,5X,F7e5p/,5X,F 7-395XrFta4, 25X,F6-1) 10002 FORMATI13X,'TGR= 'tlPE11.4) 606 FORMAT(//rlSX,27HBAS(C PERFORMANCE CONSTANTS,/,13X,BHDELfAVn eF6.3 It/ ,13X,bHXOUT= rF8=2,/rl3X,7HdPOUT= rF80 l,/r ldX,7HZETAF= ,F7obt/,l 23X,4HTB= rF6m 1,/,13X,4HHB= rFB*O,/, ljX,ShGAM= ,F7r4,/,13X,7HEHHEF= 3 ,FB-S,/rl3X,6H?KEF= ,F8-2,/,13X,7HOTREF= ,F7-3 4,/rljX,6HPIPK= rf8-5,/, 13X,4HAZ= ,F 8m5,/,13X,6HTREF= ,F7.3t/r13X16 5HGAMEz rF7-4,/,l3X,bHPEXT= pF6-1) 97 FORHAT(3X,F7m 1,SX1F6-4,bX,F8-1,7X,F7o1,7X,F7mL,6X,F5-3,/,4X,F5-2, L ?X,F7.1,9X,F5.3, 7X,F7m3) 842 FORMAT(ZOX, 18HIGNITION CUNSTANTS,/,13X,4HKA= ,F7,1,/t13X14HK8= r A F?m4t/r l3XvSHUFS= ,kdm l,/rl3X,6HCSiG= ,F70 ll/r13X,bHPMIG= r 2 F7-lr/r13X~SHTIl= rF6=3,/,13X,5HT12= ,F5-2,/,13X16HRR1G= 9 3 FB-Lr/,l3X,8HDELTIG= rFb-3,/rl3X,bHPBIG= ,f7r3r//) 600 FORMATI 21XeF6-2, lOX~F6~3,13X,F6a3,6X,F5-2, /, 5X.f 502,10X,F 10 1-2,7XtF7e2,9X1FSo2 ,8XeF6-3,/ ,GX~F8~LmtIX,F4mO~7X,F?m2,10X,F10r2,&X9 2F5m2t/t7X1FS-2,6X,F7o4,6XtF7m4, lOX,F5-2t lOXrF7-4,/,6X,F7-4,7X,F7o4 3rlOX,F7o4,8X,F?r4r6X,FY-Z) 610 FURMAT( 20x1 19HINERT WEIGHT INPUTS,/,13X, 17HDTEMP= ~~PE~~-~v/,~~X,&HSIGMAP=,lPE11-4,/, 13X18HSIGMAS= ,1PEllr 24,/,13X,GHXl= rlPEll,G,/rl3X,4HXZ= ,lPEllm4~/r13X,8HSYCl\10~~=vlPE11 3.4r/rl3X,SHUCC= ,lPtllo4,/,13X,6HPSIC= r lPE11.4,/r 13X,GHDELC= 9 lPk 41lo4t/rl3X,SHLCC= rlPtll04r/rl3X,6t-1NSEG= rlPfllobr/r13Xc5HHCN= VIP 5Ello4r/r 13X,BHSYNNUMf lPEllr4~/,13X,6PPSIS= 1PEllm4r/~13X~6HPSIA 6= rlPEl1-4,/,13X,4HKl= rlPtllm4~/,13X,4HKL= ,lPE11-4,/pl3X,BHPSIIN 7S= rlPElI-4,/rl3X,BHUELINS= tlPE11-4,/ rl3X,SHKEH= 9 1PE11-Gl/,13Xp5 BYKEN= rlP€1104r/,l3Xt8HDLINER= ,lPE11~4,/r13Xv6HTAUL= ,lPE11-+,/,1 93~;4HwA= ,1DE1l04) 101 FORMAT(///,33Xr~5H****$:***4**.zI*49:4**44*4*/,, EQUI 1LiB2IUM aUKNING ***r/,33X,29H***********O*4**44*4t***4*~**,//,3~X~ 225HINITIAL REYNOLOS NUMBER= llPk11.4) 51 FORMAT(~~X~~~H**********@**~*****Z~**~*~/~~XH**AOFF BEGINS 1****,/r37Xv23H4*******0:****8**4*4*4**9//] 318 FORMAT(37Xr23H*********************f*~;~3X23 HALF SEC0NI.I T 1RACE9/r37X,23H*******X**t**@*8*it*****v//) 102 FOKMAT(13X,>HdPl= ,lPt11-4r/r13X,5HkP2t rlPE11-4r/r13X14HdP' mLPE1 11o4r/,13X,7HPHMAX= tlPfll~4r/r13X~~tiISP=~lPEll~4~/,13X~BHISPVAG= 2,lPirllo4r/rl~Xr0ti1T~IT= r lPf ll.'>~/,ljXr ltiITVAC= ,LPEllm4,/,13X,5HF 3AV= T~PE~~O~~/,~~X~~H~VAGAV=rIl'E11o4r/r13X,tltiPONAV= rlPE1104~/,1 ~3>:15HVCI= r lPElAoGr/r lJX,SHVCF= 9 lPEllo4,/,13X,8tiLAML)DA= r lPEJ 1-41 605 FOKMAT(///r42X25r4MOTOH WE IGt+T CALCULAT IOhSl 601 FORMAT(13X,23HMAX EXPECTLD PRtSSUR€= rlPk11.4r/,l3X,28HCYLINORICAL 1 CASE IHICKNESS- 9 lYE11.4r/r 13Xv9HCASE h1= r lPE11.4,/, 13X,llHNOLLL 2E dT= rlPE11-4r/,l3X, lSHINSUCAT1CN kT= 1 lPEll-&r/,13X, 1OHLINER hT= 3 rlPE11-4,/t13XllbH10TAL INEKT kT= ~1PE11e4~/~13X~ZOHTUTALMOTOR W 4EIGHlt ,LPEll-4,/* 13X,?HZETAM= ,lPt11.4, /t 13x921HKATIU OF ITOT TO 5WM= ,lPE11.4) €NU Table C-3 (~ont'd)

SUBROUTI NE AREAS c *********************************4****4**************4**4************ C SUBROUTINE AREAS CALCULATES BURNING AREAS AND POkT AREAS FOk * C CIRCULAR PERFORATED (C,P, GRAINS AND STAR GRAIhS OR FOR A C * COHBINATION OF C.P. AN0 STAR GRAINS t C ...... INTEGER STAR,GRAIN,ORDER,COP REAL MGEN~MDIS~MNOZ,MNl,JROCK~NtLtHE1,ME~ISP,ITOT~MU~HASS~~SPVAC REAL LGC1,LGNI ,NS,hN,NP,LGSl ,NT,LTPwLGC,LS,LF REAL MZ,MUaAR, ISP2t lTVAC~Ll,LZ,LFW,LFhSQD COHMON/CONSTl/Zh,AE,ATtIHETA,ALFAN COHMON/CONST3/S,NS ,GRAIN ,NTAdY vNCAR0 COHHON/CONST4/0tLDI~DO~ZOtOI ~OMMON/VARIAl/Y~T~DELY~DELTAT~PUNOZ~PHEAO~RNOZ~RHEA~~SU~A0~PHMAX COMMON/VARIA2/ABPORl~AdSLOT,AbNOZ~AYHEAD~APNOZ~~ADY~AbP2~AbN2~A~S2 COMMON/VARIA3/1 TOT* ITVAC,JROCKt iSP, ISPVAC,MOISpMNOL, SGpSUh;MT COMMON/VARIA4/RNT~RHT,SUF.12~Rl~R2~R3,RHdVE~RNAVEtROAK~YB~KOUNT~TL COMMON/VARIAS/AOMAIN,ABTO~SUMDY,VCI1ABTT~YTfiAN COMMON/VAR IAd/YDI b COMMON/VAR123/YETAtXETHIlSO DATA PI/3eA6159/ ABPC=O,O ABNC=O.O ABSC=OoO ABPS=O 00 ABNS=3,0 ABSS=O,O DAdT=O,O SG=30 0 VC IT=O,O ANUM=PI/4. PI02=PI/2. RNT=RNT+RNOL*DELTAT RHT=RHT+kt4EAl)*OkLT AT IF(Y-LE,O,O) AGS=O,O K= 0 IF(ABS(ZW).GToOoOJ K=l YB=Y IF(KoEQ. 1) Y=Y B-SUMDY/2. 2 IF (K.EQ.2) Y=YB+ABS(Li~1/2.-SUMOY/2~ L IFIIS0,EQ.O) YErA=Y Ic(Y,LE.O.O) KEAD(5,SOO) INPUT,GRAIN,STAR~NT,CROEH,COP C ***$t44**4~4~**4*4**IC****8~994*4:G*ItO.*f4444#4*4**~~~4********4**~4**~ C * RtAu THE TYPE OF INPUT FUR THE PHIIGRAY Ah0 THE BASIC GKAIN 4 C + CONF IGURAT ION AND ARRANGEMENT * C * VALUES FOR INPUT AkE * C * 1 FOR ONLY TABULAR INPUT C * 2 FOR ONLY kOUATION INPUTS (EQUATIONS ARE dUiLT 4 Table C-3 (Cont 'dl

C * INTO THE SUBROUTINEI * 'C * 3 FOR A COMBINATION OF 1 ANC 2 * C * VALUES FOR GRAIN ARE * C * 1 FOR SrRAICHT CoPo GRAIN 4 C * 2 FOR STRAIGHT SVAR GRAIN * -C * 3 FOR COMBINATION OF COP. AND STAR GRAINS t C * VALUES FOR STAR ARE lWAGON WHEEL IS CCPiSlDERED A TYPE OF C * STAR GRAIN IN THIS PROGRAM) * C * 0 FOR STRAIGHT COP. GRAIN C .* 1 FOR STANDARD STAR 4 C * 2 FOR TRUNCATED STAR 4 C * 3 FOR MAGON WHEEL * C * VALUES FOR NT ARE * -C * 0 IF THERE ARE NO TERMINATICN PORTS * C * X WHERE X IS THE NUMBER OF TERdINATION PORTS * C * VALUES OF ORDER ESTABLISH HOW A COMBIhATION COP* AND STAR * C *. GRAIN IS ARRANGED 1 C * 1 IF DESIGN IS STAR AT HEAO END AND COP. AT NOZZLE * C * 2 IF DESIGN IS COP. tl HEAD END AND CoPo AT NOLZLE * C .* 3 IF OESiGN IS COP. AT HEAD END AND STAR AT NOLZLE * C * 4 IF DESIGN IS STAR AT HEAD END AND STAK AT NOLZLE C * ***NOTE*** IF SRAIN=l( VALUE CF ORDER MUST BE 2 I 1000 CON1INUE C * ***NOTE*** IF GRAIN-2, VALUE CF OROER MUST BE 4 * C * VALUES FOR COP ARE 1APPLICPt3LE TO COP* GRAINS UhLYJ 4 C. * 0 IF BOTH ENUS AWE CONICAL CR FLAT * C * 1 IF HEAD END IS CCINICAL OR FLAT AND AFT END IS * C * HEMISPHERICAL rc C * 2 IF BOTl4 ENDS ARE HEMISPEERICAL * C * 3 IF HEAO EN0 IS HEMISPHERICAL AND AFT END IS 4 C * CONICAL OR FLAT * c 44*******4*4*4****484*4***8********4*************************4*44**** IF (YoLE.O.0) HRITE(6rG07) IF(Y.LE.O.0) WRITE(6#600) INPUTgSRAINgLTAR~NTgORDER,COP IF(INPUT,EQm2) GO TO 12 IFlYoLEoOoO) GO TO 6 If(KoEQ.2) GO TO 91 IFIKoEQ. L)Y=YB IF (YTmLEoY) GO TO 8 9 DENUM=YT-YTZ SLOPEl=(Ai3PK-ABPK2 )/DEKdM SLOPE2=( ABSK-AOSK2 /DENOM SLOPE3=(ABNK-ABIJKZ ) /DtNOM SLdPEI=IAPHK-APHKZ )/DENOM SLOPE5=( APNK-APNK2 ) /DENOM Bl=ABPK-SLOPEl*YT BZ=AdSK-SLOPEZ*YT 83zABNK-SCOPE3*YT Table C-3 (Cont'd)

BlsAPHK-SLOPE4*YT BS=AYNK-SLOPE5*YT ABPT=SLOPE1*Y+i31 ABST=SLOPEZ*Y +62 ABNT=SLOPE3*Y+B3 AP HT- SLOPE4*Y +84 APNT=SLOPE5*Y+BS YB=Y IF (KoFQo 1) Y=YB-SUMDYIZo 91 IF4 INPUToEQ.3) GO TO 3 GO TO 52 6 READ1 5~5071 YT,ABPK,ABSK~ABNKIAPHK~APNK~VCIT NCARD=NC ARO+ 1 C **S*4********4*4**4***********4*4$************44*~***********4******* C * READ IN TABULAR VALUES FUR Y=O.O ihOT RtOUIRED IF INPUT=2) * C * * C * ABPK IS THE BURNING AREA IN THE PORT IN I&**2 * C * AUSK IS THE BURNING AREA IN THE SLOTS IN IN**2 t C * ABNK IS THE OU&NING AKEA IN THE NilZZLF END 1N 1N**2 ~r c * APHK IS THE PUKT AKtA AT TtlE HtAO EhD IN IN**2 4 C * APNK IS THE PORT AREA AT ThE NOZZLE EhD IN IN**2 * c 4 VCIT IS THE IhITIAL VOLUME UF CHAMBE& CkSfi ASSOCIATED WITH * C * TABULAR INPUT IN IN**3 P: C **t*S*$$*4*****44+44:************4**4***44******************4*+******* WRITE16p610) WRITE(6,583) AOPK,ABSK,ADNKvAPHK,APNK WtlITE(6r584)VCIT ABPT=AtlPK Ad Sl=ABSK ARNT=ABIVK APHT=APHK APNT=APNK YT 2=Y 1 IF(INPUT.EQ.31 GO TO 3 VC I=VCIT GO TO 52 8 YT?=YI ABPKZ=A~SPK ABNK?=Ai\hK A13SK2=AdSK APHt(Z=APHK '$:'YK2=APNK KEAiJL5,SOS) YT,AOPK~ABSK~ABNKTAPHK~APNK NC Af?O=NCARL)+ 1 L *+*9**S#841*49~3*$****9i~4***$0****444*6:*~+~$*4*****4**4**44*~*4*4*~~* C * READ IN TAHULAK VALUkS FOk Y=Y (hU1 hEOUIKEU FOR INPUT=Z) * C * (NOTE THAT TABULAR VALUE CAeOS FOR Y GT 0 CO NO1 IMMtOlATELY * C * FULL3W THOSE f-OK Y kQ 0 IN THE UAlA DtCKI 4 Table C-3 (Cont'd)

C **********************************4******4*************************** YRITE16t611) YT WRITE16r583j ABPK~ABSKIABNK~APHK~APNK GO TO 9 L2 ABPT=O.O AdNT=3*0 AdST=O.O 3 IFLGKAIN.NE.2) GO TO 4 ABPC=O.O 3ABNC-0 -0 A8 SC=0.0 GO TO 7 4 IF IY.LE.O.0) READ(SrSQ1) D010ItOECOIrS,THETAGtLGC I,LGNI pTHETCNtTHE L TC H C *************4******************4**~**************************4****** C * READ IN BASIC GEOMETRY FOR C.P. GRAIN LNOT REQUIRED FOR * C * STRAIGHT STAR GRAIN) * C * DO IS THE AVERAGE OUTSIOE INITIAL GRAIN UIAMETER IN INCHES r C * Dl IS THE AVERAGE INITIAL INTERNAL GRAIN CIAMETER 1N INCHES 9 C * OELDI IS THE OIFFEHtNGE bETWEEN THE INITIAL INTERNAL GRAIN $ C * DIAMETER AT THE NOLZLf tND OF LGCI AN0 01 Ih IhCHES C * S IS THE NUMBER OF FLAT BURNING SLOT SICES (NOT INCLUDING * C * THE NOZZLE END) * C * THETAC IS THE ANGLE THE NOZZLE EN0 CF THE GRAIN MAKES JITH * C * THE MUlOK AXIS IN DEGREES * C 8 LGCl IS THE IN1 TIAL TOTAL LENGTH OF THE CIRCULAR PERFORATION * C * IN INCHES * C * LGNI IS THt INITIAL SLANT LENGTH OF THt BUfiNIkG CONICAL * C 4 GRAIN AT THE NOZZLE END IN INCHES * C 4 THETCN IS THE CONTRACTION ANGLt OF THE BDNDED GRAIN IN OEG- * C * THETCH IS THE CONTKACTIUM ANGLE AT THE HEAU tND IN OEGKtES 4 C $848~~~*t****~I**4+44~~~~4:~~*~4~~O~~~tf4~~44*~~4~~*4~4*~~*~*~*4~~~~~* IFlY.LE.U.0) WR1TE(b,6011 00~0I~UELOI~S,THETAG~LGCI~LGNI~IH€TCN~TH lETCH IF IY.LE*O~O)THETAG=THETAG/57o29578 IF (Y .LEodmO) THETCN=THETCN/578 IF 1Y~LEoO.O)THETCh=TttETCH/57.L457~ OUSQO=30*UO DISOD=dI*DI BNUPl=ANUM*DOSQD TLL-TL ff(3RDER-GE.3 TLL=O*O YDI=Z.*Y+DI Y3ISQD=YDI*YDI ABSC=S*ANUM* ( OOSQU-YDISOD) IF(ABSC.LEsO.0) ABSC=UeO IF(YDI.GT.DOi GO TO 100 IFITHETAC.GTo0.08727) GC! TO 101 Table C-3 (Cont'd)

IF(COP.EQ.0) GO TO 700 IF ICOP.EQ.11 GG TO 701 IFiCOP.EQ.2) GO TO 702 CHCKL=OOSQD-YDISQD IF(CHCK1.LT.O.O) CHCKl=O.O LGCzLGCI-I SPHT (DOSQD-DISQDI-SQRTt CHCK 11 J/Z.-Y*COTANd THETCN) GO TO 710 702 CHCKl=D0SQD-YOISQO IFICHCK1.LToO.O) CHCKL=O.O IF(CHCK1.LT.O.O) CHCKl=O.O CGC=LGCI-(SQRT (DOSOD-OISQO1-SQRTICHCKl)) GO TO 710 701 CdCK2=00SQD-(YDI+OELOI)*t2 IF (CHCK2oLT.O.O) CHCKZ=O.O CGC=LGCI-4 SQRT(DOSU0-lDI+L)ELD1)**2)-SQRT (CHCK2) 112. 1-Y*COTANI THETCH) GO TO 713 700 LGC=LGCI-Y*ICOTAN(THETCN)+COTANITHETCH)) b 710 ABPC=PI*YDI*(CGC-TLL-S*YETA) A!3IJC=O.O GO TO 732 101 CONTINUE IF(COPoEQ.O.ORoCOP.EQ*1J GO TO 720 CHCKl=OOSQU-YOISQO IF (CHCK1 .LT.O.Ol CHCKl=O.O A~PC=PI*YUI*(LGCI-(S~U~(OOSOO-DISQU1-SQRTlCHCK!~112.-TLL b 2-( S+TAN( THETAG/Z. I 1*YETA) GO TO 730 b 720 AaPC=Pl*YOI*~LGCI-Y*COTANITHETCH)-TLL-IStTAN(THETAG~2.))*YETA) 730 IF (C0P.f 4.l.OHoCOP.EP.Z) GO 10 731 A~NC=P~*(LGNI-Y*COTAN~TH~~AG+~HETCN)-Y*TAN(TH€TAG~~~))*~OI+ 1 DELDI+YtLGNl*SINLTHETAG)+Y*SlNLTHETCN~/SIN(tHETAG+TtiETCN) 1 GO TO 732 731 IF(Y.LE.O.0) GO TO 7311 GO TO 7312 7311 R7=~~DI+DEL011/2.+LGNI*SIN(~HETAG~~~~COSITHETAG~-SIN(THETAG)* 1 jdKT((DU/L.)+*Z-( IdI+UE.LLIl~/Z.+LCFJI~SIPi(ltlET~G) )**i!) 7312 IF (R7+YmLT. lt30/20 )*COS(l t4trAG) GO TO 11 11 1 A~~~C=P~*~LC,~I~+(A./S~N(THETAG))~~((LlU/2. 1-LGNi*SINITHETA1;1 I-( OI+OELOI)/2.)-Y*COTAN(Tl~ElAG)-Y* TAN( THtTAG/2-) )*IAOI+DELDI 2/2.tY+D0/2.) GO TO 22222 Lilll KPK=SQHl(((D0/2.)**2)-R7**2~-SQRT1I~OO/2.1**2)-IR7~Y~**2) AdNC=PI*(LGNI-HPK-Y*TAN[ THE TAG/^.) I*( IUl+DELDI )/2.+SQRT( (DO/ 1 2m~**2-IR7+YJ**2)*SINITHETAG1+Y+IH7+Y)*CUS~THE~AG)1 222 22 CONT INU€ 732 IF (AUPC-LE.O.01 ABPC=O.O IF (AUNC.LE.O.0) ABNC=O.O Table C-3 (Cont'd)

GO TO 5 100 ABNC=O,O ABPC=O,O 5 DH=D1-LO APHT=ANUfl*(DH+2.*RHT)**Z IF(APHT ,GE ,ANUH 1 APHT=BNUH IF(K,LTo2) APHTl=APHT APNT=ANUH*(OI+DELDIt2.*RNT)**Z IFIAPNT,GE,BNUM) APNT=BNUfl IF(GRAINoNEe1J GO TO 7 ABPS=OoO ABSS=O,O ABNS=OoO GO TO 50 7 IF(Y-LEoO-OI READ( St5021 NStLGSI .NPtRC,FILLtNN C ...... c REAO IN BASIC GEOMETRY FOR STAR GRAIN 1NGT REQUIkED FOR 4 C STR4IGHT C-Po GRAIN) 4 C NS IS THE NUMBER OF FLAT BURNING SLOT SIDES (NOT INCLUDING C * THE NOZLLE END) * C LGSI IS THE INITIAL TOTAL LLNCTH OF THE STAR SHAPED * c * PERFORATED GRAIN IN INCHES c NP IS THE NUMBER OF STAR POINTS C C * RC IS THE AVERAGE STAR GRAIN OIJTSIOE RACIUS IN INCHES * C * 6ILL IS THE FILLET RADIUS IN INCHES 4 C * NN IS THE NUMBER OF STAR NOZZLE END HURNING SURFACES 4 C ********4**u***************v***************************************** IF(Y*LE~OoOl WRITE(bt602) NS~LCSI~NP~RCIFILL~~N PIDNP-PI /NP RC SdO=RC*RC IF ( ISOoEQol) YE=Y IF4 ISOoEQo11 Y=YETA FY=FILL+Y FY SQD=F Y*eY IF(STAK.EQ.1J GO TO 20 IF(STARoEQoZ) GO TO 201 IFIY-GT-900) GG TO 179 AD(5p42Al TAUWW~LI~LLIAL~~A~~ALPHA~~HH C ****C+4*t**49~**04*+4P@*43*4*f***t*I**P**4*******4****************4** C * REAO IN GEOMEiKY FUK MAGON uHEEC (NCT REQUIRED FOR STANDARD * C * OR TRUNCATED STAR GRAINS) + TAUUW IS THE THICKNESS OF TH~PROPELLANT WEB IN INCHES * C * L1 AND L2 ARE THE LENGTHS OF THE TkO PARAlLEL SIDES OF THE * ; * TWO SET' OF STAR POINTS IN INCHES C * ALPHA1 AND ALI'HAL Adk THk ANGLES 6EThEEN TPE SLANT SIDES OF * C THE STAR POINTS CORRESPONDING iC L1 ANC LZr KESPECTIVELYt* C * AND THE CERTER LINES OF THE PCINTS IN CEGREES f * HW IS HALF THE HIOTH OF THE STAR POINTS IN INCHES * Table C-3 (Cont'd) ...... WRfTE(6t4221 lAUkWtLlrL2,ACPHAltAC4HA2,Hk AL PHAl =ALPHA1/57-29578 ALPHAZ=ALPHAZ/ 57,29578 AL PZ=ALPHA2 XL 2=L 2 LFWRC-TAUWW-F ILL LF WSQD=LF W*LF W THETFH=ARSIN( HW+FILLIILFWJ SLF~=LFW*SIN(THETFkl 179 XKK=O SC=O,O €NUH=(RCSQO-LFwSQD-FYSQD)/~~~*LFW*FY) AL PHA2=ALP2 L2=XL2 190 YT AN=Y*TAN( ALPHA2/2,) COSALP=COS ( ALPHA2 SI NALP=SIN(ALPHAZ) IF(YTAN,GT,LZI 80 TO 182 IF (FY,Gl.SLFH) GO 10 181 SGW=NP*(L~-L,*YTAN+I SLFW-FILL)/SINALP-Y*COTAN(ALPHAZ)+FY* 1 (PID2+THETFW)+(LFW+FY)*(PIONP-THETFW)) GO TO 183 181 fF (Y,Gl.TAUWW) GO TO 184 SCW=NP*(FY*LPIDKP+ARSIN(SLFW/FY11+iZIOluP-~HET~Wl*LFH) GO TO 183 184 SCW=NP*F Y*( THElFW+ARSIN( SLFWIFVI-ARCOSlEAUM) 1 GO T@ 183 182 YPO=-SLFW IF(ALPHA2,GEmPIOZ) GO TO 222 Q=-F ILL+L2*TAN 1 ALPHA2)-Y/CCSALP XP I=(-Q*T AN( ALPHAZ 1-SQRT (-Q*iJ+F YSQD/CCiSALP*COSALP 1 )*COSALP*cOSALP YPI=XPI*TAhr(A;PHAZ1+Q XPO=(YPO-Q)*COTAN(ALPHAL) GO TO 223 222 XPI=Y-L2 YP I=-SORT(FYSQ0-XP I*XPI 1 XPO=XP I 223 FYLS=SQRT(SLFd*SLFW+XPI*XPI) XP102=(XPi-XP01~(XP1-XP0) YPIO2=(YPI-YPLl)*( YPI-YPO) tF(FY,CT.FYLS) GO TO 186 IFIY-GE-TAUaW) GO TO 185 SGH=NP*(SURT (xPI02+YPI02 )+FY*IPID2+THETF h-ARSIN(XPI/FY l+#LFdtfVl* 1 (PIDNP-THETFW)) GO TO 183 185 SGJ=NP*(SQRT(XP~~~+YP~U~)+~Y*(PIDZ-ARS!N(XP~/~Y)-ARCOS~ENUM))) GO TO 183 Table C-3 (Cont'd)

186 IF(YoGToTAUWW) GO TO 187 SGN=NP*IF Y*(PIDNP+ARStN( SLF~(/FY) !+(PIDNP-THETfbdj*LFW) GO TO 183 187 SCY=NP*FV*LTHETFWtARS!Nt SLFW/FY 4-ARCOS (EhUMl I 183 IF(SG4oLE.a-0) SGUt3.0 IF(Y,G~.G*OI GO TO 188 AGS2=05*( P~*RCSQD-NP*LFW*SLFC~*(COS~THETF~)-SIN(THE~~W~*COTAN~ALPHA & ~)-~~*(L~+FIcL*TANIALPHA~/~.))/LFM)-(P~-~o*NP*F 2 ILL*(LZ+SLFW/SINAC~*LFw*LP1ONP-THETFk'~+(PIDhP+PiDZ-l~/SINALP~* 3 FILL/2-)) AGS=AGS+AGS2 188 CONTINUE SG=SG+SGU IF(KKKoEd.1) GO 10 24 L2=L1 AL PHAZ-ALPHA1 KKK=1 GO TO 190 201 IF(YoLEoUo0) READ(SrS031 RPvTAUS C *t**C***+**********+*$****S***+****************************************** C * READ IN GEOMETRY FOR TRUNCATED STAR (NCT REQUIRED FOR C * STANDARD STAR OR WAGGN v~HEEL) t C RP IS THE INITIAL RADIUS CF THE TRUNCATION IN INCHES 4 c * TAUS IS THE THICKNESS OF THE PHOPELLA~T WEB AT THE BorTon * c * OF THE SLOTS IN INCHES * c *****4*************0*9:4****4~*0*4**8~48***+***8***4******44********** IF(Y-LE-0.01 dRITE(6.603) RPtTAUS THETAS-PIDNP RPY=RP+Y LS-RC-TAUS-FILL-RP RPL=RP+LS lHETSl=lHETAS-ARSIN(FY/RPY) IF (THETSloCE.Oo0) GO TO 110 IF(Y-LE-TAUS) GO TO 103 THETAC=ARSINl(RGSQD-RYL*RPL-FYSOD)/{Z~*FY*RPLIl IF ITHETAC-GE-3-01 GO TU 104 IF(Y*LT-HG-RP) GCI TO 105 SG =O. 0 Gil 16 14 103 S'J=2~*NP*(dPY*THETSl+LS-~RYY*COS~THETA5-THETSlI-kPI+P~02*FY) GO TO 14 LC4 SG=2.*NP*(RPY*THETSl+LS-~RPY*COS(TH€TAS-THElSlJ-RF)+FY*THETAC) GO TO 14 105 SG=Z-*NP*( i3PY*THETSl+SORT (RCSQO-F YSkO)-SbkT( RPY*RPY-FYSdD) 1 14 IF(YoCEo0.3) AGS=PI*IRC.SQD-HP*KPJ-NP*~PI*FILC~kILL/2o+Z~*LS*F~LL) GO TO 31 110 THEIAF=THETAS THETAP=2,*THETAS Table C-3 (Cont'd)

TAUUS-TAUS GO TO 111 20 IF(YoGTa0oOI GO TO 1791 READ(Sr5041 THETA^ v THETAPeTbUWS C ...... C * READ IN GEOMETRY FOR STANOAR0 STAR 1k0i REQUIRED FOR * C * TRUYCATEO STAR OR HAGON WHEEL) * C * THETAF IS THE ANGLE LOCATION OF THE FILLET CENTER IN UEGREES * C * THETAP IS THE ANGLE OF THE STAR POIhT Ih DEGREES C .* TAUWS IS IHE HE8 1HICKNESt OF THE GRAIN IN INCHES C *****C************************c*************************************** WRITE16,604) THETAFvTHETAPvTAUWS THETAF=THETAF/ 57-29578 TdETAP=THETAP/57-29578 THETAS-PI /NP THETS1=lo00 111 LF=RC-TAUHS-F ILL 1791 CYUM=(Y+FILC)/LF DNUM=SINI TH€TAFI/SIhi(THETAP/20 1 ENUM=(RCSQD-LF*LF-FYSQOI/(20*LF*FY) FNUM=SIN(THETAF)/CCS( rHErAP/Z. 1 IF(CNUMoLEoFNUM1 GO TO 106 IF(Y~LE.TAUWSICO TO 107 SG=ZI*hP*FY*(THETAF+ARSlNISIhITHETAF)/CNUHJ-ARCUSfENUM)1 GO TO 23 106 IFI Y-LEO TAadS) SG=Z.*KP*LF*I D~UM+CNUM*~PICL+THETAS-THETAP/~O 1-CUTAN( THETAP!Z.) +THETAS-THETAF J IFIYoLE.TAUnS) GO TC 23 SG=Z~*NP*(FY*~ARSIN(ENUM~+THETAF-THEiAP/Z.ltLF*ONUh-FY*COTAh(TH€TA 1P/2,)) GO TO 23 107 SG=2-*NP*LF*(CNUM* (THETAStAHSIN(SIN(THETAF J/CNUM) J+lHETAS-1HETAFJ 23 IF(THETS1oLE.OoO) GO TO 14 IF(YmLEIOmO) ASS=PI*RC**2-NP*LF*LF*ISII\i(THETAFl*(CUS(Tff€TAf~- lSIN(THETAF )*COIk~.cITHETAP/Z.) )+THETAS-THETAF+2-*F ILL/LF *ISIN( THtTAF 21/SIN( THETAP/Lml+THETAS-TFETAF +FILL/(LI*Lf I*lPIO2+THETAS-THE 3TAP/Z.-COTAN ( TtiETAP/2.11 1 ) 1 24 CDNTINUE 31 IF (SG-LE mO-UI SG=GoO IF(KmE3oO.ORoK.EQmZ) SGN=SG IFIKmtE.1) SCti=SG IF(Y-LE-Oo0) SGZ=SG IF{K,EQm2) GO TO 37 RAVEDT=R 1+( SG+ SGZ)/Lm*HbAR*0ELTAT RYOT=RZ+1 SG+SGZ)/Z.*UNAVE*OI:LTAT RHOT=R3+(SG+SG2)/2o*KHEVE*UELTAT R1 =RPVEDT RZ=RNDT Table C-3 (Contld)

R3=RHDT SG2= SG CO TO 38 37 IFIKOUNToNEml) GO TO 39 SG3= SG R4=R1 RS=RZ Rb=R3 39 RAVEDT=R4*LSC+SG3)/2m*Rt)AR*DELTAT RNDT=RS+( SG+SG3 )/2 -*RNAVE*DE LTAT RHDT=R6+( SG+SG3 )/2-*RHAVE*OELTAI R4=RAVEDT RS=RNDT R6=RHDT SG 3sSG 3a ABSS= ( AGS-RAVEDT) *NS IF(ABSS-LE-0-0-OR- SG-LEoOmO) ABSS=OoO ABNS=( AGS-RNDT 1 *NN IF(ABNSmLE-0-OoOR-SGoLE-0-0) ABNS=O*O IF(ORDER-LE.2) ABPS=(LGSI-Y* (NS+hNJ )*SG IF(OKDERoLE~2) GO TO 36 ABPS=(LGSI-TL-Y*(NS+NN))*SG 36 PIRCRC-P I*RCSQD APHS'P IRC3C-AGS+RHDT IF(APHS.GE.PIRCRC,UR,SC.LE,O,O) APHS-PIRCRC APNS=PIRCRC-AGS+RNDT IF (K-LT,ZJ APHSl=APHS IF(APNS-GE-P IRCRCI APNS=PIRCRC IF (1SO.EQo11 Y-YE 50 IF (NToEQoOoO) GU TO 371 *********************************************************************IF(YoLE.0-GI READ( 5,506) LTPIOTPVTHETTPITAUEF F * READ IN GEOYETRY ASSOCIATED hITH TERPlhATIOh PORTS (NOT 4 * REQUIRED IF NT=Ol * * LTP IS THE INITIAL LENGTH OF THE TERMINATION PASSAGES * * IN INCHES 4 * DTP IS THE INITIAL DIAKETER OF THE TERMINATION PASSAGE * * IN INCHES * * THETTP IS THE ACUTE AWL€ BETdEEN TBE AXIS OF THE PASSAGt * * AND THE MOlOR AXIS IN IIEGREES * * TAUEFF IS THE ESTIMATED EFFECTIVE WED THICKNESS AT THE E TERMINATION PORT IN INCHES t 4******************************8****4************4*4********************* IF(Y-LE-0-0) WRlTE(br606l LTF,OTP,THETTP,rAUEFF THETTP=THElTP/57*29578 DAOT=NT*3~1Q159~~(DTP+Z~*Y)*[LTP-Y/SIN~THETTPll-~DTP+2o*Y~**2/4~+ l[Y+DTP/Z* )*(DTP/L- )*( l*-l-/SIN( THETTP) 1 J IF (YmGEoTAUEFF) DABT=O.O Table C-3 (~ont'd)

371 IF(Y-GToOoO) GO 10 52 IF(NToNEoOo0) GO TO 45 LTP=O.O DT P=OoO 45 IFIGRAINoNE.2) GO TO 49 CGCI=OoO LGNI=OoO 01 SQO=O,O DOSQD=Qo *RCSQD 49 IFtGRAINeEOo 1) LGSI=O.O VC I=~,~*(ANuM*~IS~~*(LGCI+LGNIt(ANUM*OOSQD-AGSi* I LGSI+NT*LTP*ANUH*DTP*DTP)+VCIT 52 BaP=o,o BBS=OoO BBN=OoO ABPORT=ABPT+ABPC+ABPS+DABT+BBP IF (K,LT,Z) X€TH=A6PC/(ABPORT+l~E-20) At3 SLOT=A8ST+AOSC+ABSS+HBS ABNOZ=ABNT +AdNC+ARhS+ijdN ABTT=ARPT+ABST+ABNT IF(KoGEo2I GO TO 55555 SUHAB=ABPORT+ABSLOT+ABNOL 55555 CONTINUE IF(KoEQeO) GO TO 99 IF(KoEQ.1) ADMAIN=ABPORT+AOSLOT+ABNOZ-d8TT K=K+ 1 lF(K.GTeZl GO TO 69 GO TO 2 69 ABTO=AdPORT+ABSLOTtAUNOZ-AWT 99 CONTINUE IF(Y.GT.O.01 GO TO 70 AdPl=ABPCIRT ARNl=ABNOZ ABSl=ABSLOT 70 ABPZ=IARPL+ABPORT~/2. ABN2=( ABNl+AdNL)L )/2- A3S2=(A&S1+ABSCLlTJ/2- IFIINPUT-EQ.1) GO TO 76 GO TO ~11,7L,73,74),OROEA 71 APHEAD=APHSl APIUOZ=APNT SG=SGH GO TO 75 72 APHEAD=APHT 1 APNOZ=APNT SG=O. 0 IF(GRA1NoEQo31 SG=(SGH+SGNI/Z. CD TO 75 Table C-3 (Cont'd)

73 APHEAO=APHTl APNOZ=APNS SG=SGN GO TO 75 74 APHEAD=APHS1 APNOZ=APNS SC=SGN CO TO 75 76 APHEAD-APHT . APNOZ=APNl 75 Y=YB 01 F F=SUHAB-SUM2 DADY=DIFF/DELY ABPI=ABPORT ABNl=ABNOZ ABS1-ABSLOT IF(ZWeCEeOe0) GO TO 77 ABMl-ABHAIN ABMAl N=ABTO A8 TO=ABMl 77 RETURN 500 FORMATL9X~I2,9X,I2,8XpI2,6X,F4eO,9X, IZ97X, 12) 607 FORMATl//,20Xvl9HGRAtN CONF IGUWATION) 600 FORMAIL13X~7HINPUT= ,12r/,13X,7HGRAIN= rIZ,/rl3X,6HSTAR= ei2,/,13X lrQHNT= rF4ed,/t13X,7HO8DER= rIZr/rl3XeSHCOP= r12r//) 507 FORMATL 6X,Fbe2~lOX~E11~4~1~X~E11~4~8X~E11~4~/~22X~Elle4~ 19X,E11.4,8X,E11.4) 610 FORMAT (/13X,40HTABULAK VALUES FOR YT ECbAL ZERO READ IN8 583 FORMAT(~~XISHABPK=~lPElle4,5X,5HAt)SK=, 1PElle4rSX~5HAbNK=riPElle4~ 1 SX~SHAPHK=~~PE~L~~I~X~~HAP~K=~~PE~~~~,//) 584 FORMAT(13X,5HVClT=,lPE11~4~//1 505 FORMAT(6X~F7o3~9X~~l1-4~1UX~E11~4~8X1E11e4~/~22X~Ell~4e9X~€ll~4~ 611 FORMAT L/13X,23HTABULAR VALUES FOR YT= rF7e3,SH READ IN1 501 FORYAT(5X,F8e2,6X,F7e3,9X~F7.3,5X~F6.&9X~F8e5,/,7X~F8-2,7X,F7o2,9 lX,FBe5,9X,F$eS) 601 FORMAT(ZOX,19HCoPe GRAIN GEOMETRY,/,13X,4HDO= ,F8e2,/,13X,4HDI= rF 17o3r/*13Xt7HUELOI= 9F7e3r/r13X13HS= ,FoeZ,/rl3X,BHTHETAG= rF9-5,/, 213X,6HLGCI= rF8*2r/r13X,6HLGNI= ,f7.29/9 13X,BHTHETCN= ,F9o5,/,lSX, 38HTHETCHz ,F9*5,//) 502 FUHMAT(5X~F6e2~7X,FL).2,5X~F4oO~5X,Fd.Jr9X~F7m3~5X~f4eO~ 602 FORMAT(lSX,l9HBASIC STAR GECMETRY,/*l3X,4HNS= rF6e2,/,13X,6HLGSI= IrF8-2,/, l3Xt4HNP= ,F!ieO,/, 13X14HKc= ,FBe3,/,ljX,bHF ILL= rF7r3pir13 2XvGHNN= ,F4.0,//) 42? FO~MATL3~bX,F5e2)~2(10X~F?e51,6X~F5e2) 412 FORMAT fLOX,2OHWAGON WHEEL GECMETRY ,/ ~~X~~HTAUWH=,F6-2,/113X, P 4HL1- rF6e2r/r13X,4HL2= ,Fbe2,/v13X,8HALPHAl= rF9eSr/rl3X, 2 dHALPHA2= ,F9*5,/,13X,4HHW= ,F6eZ,//) 503 FORMAT[SX,F 7m3,7X,F7e3J Table C-3 (~ont'd)

603 FORMATIZOXI~~HTRUNCATEDSTAR GEOHETRY,/,l3X,4HRP~ rF703r/r13Xe6HTA lUSf ,Ftr3,//) SO4 FORMAT19XeF 8oSe9X,F 8o4,8X,F7o3) 604 FORMAT(ZOXe22HSTANDAHO STAR GEONETRY~/~A~XI~HTHETAF=rF90Sr/.13X18 LHTHETAP= ,F9obr/r13Xs7HTAU~S= ,F7.3,// 1 SO6 FORNATI7XvF7mZe?X,F6~Z,lOX,F B-SelOX,F7*3) 606 FORMATI~OXIZ~HTERMINAT~ONPORT GEOMElRY~/r13XvSHtTP= ,F7*2e/113X,S LHOTP= 1F6.2e/e13XleHTHETTP= rF8rS,/~l3XrGHTAU€Ff= rF7r3,//) END Table C-3 (Cont'd)

SUBROUTI NE OUTPUT C ...... C SUBROUTINE OUTPUT CALCULATES BASIC PERFORMANCE PARAMETERS C AND PRINTS THEM OUT AS A FUNCT IGlv Of DISTANCE BURNED * C * (WEIGHT CALCULATIONS ARE PERFORMED IN THE MAIN PROGRAM) C * i IS THE TIME IN SECS * C Y IS THE DISTANCE BURNED IN INCHES C * RNOZ IS TdE NOZZLE END BURNING RATE Ih iNCHES/SEC C * RHEA0 IS THE HEAD END BURNING RATE IN INCHES/StC a C * PONOZ IS THE STAGNATION PRESSURE AT THE NOZZLE END IN PSIA * C PHEAD IS THE PRESSURE AT THE HEAO END OF THE GRAIN IN PSIA * C PTAR IS THE PORT TO THROAT AREA RATIO It C * MNOZ IS THE MACH NUMBER AT THE NOZZLE END OF THE GRAIN C * SUMAB IS THE TOTAL BURNIRG AREA OF PROPELLANT IN 1N**2 * C * SG IS THE BURNING PERIMETER IN INCHES OF THE STAR SEGMENT * C (IF ANY) * C * PATM IS THE ATMOSPHERIC PRESSURE AT ALTITUDE IN PSIA * C * CFVAC iS THE THEORETICAL VACUUM THHbST COEFFICIENT * C * FVAC IS THE VACUUM THRUST IN CtiS C * F IS THE THRUST IN CBS AT AMBIENT PRESSURE C * ISP IS THE OELIVEREO SPECIFIC IMPULSE IN SEC AT AMBIENT * C * PRESSURE * C * CF IS THE THEORETICAL THRUST CCEFFICIENT AT AMBIENT PRESSURE * C * VC IS THE VOLUME OF CHAMBER GASES IN IN**3 * C * MDOT IS THE WEIGHT FLOHRATE IN C8/SEC * C * CFVD IS THE DELIVERED VACUUM THRUST CCEFFICIENT * C * ITOT IS THE ACCUMULATEO IMPULSE IN L&SEC OVER THE * C TRAJECTORY * C * ITVAC IS THE ACCUMULATED VACUUM IMPULSE IN Lb-SEC . C ISPVAC IS THE DELIVERED VACUUM SPECIFIC IMPULSt IN SEC 1000 CONT INUE C * UP IS THE EXPENDED PROPELLANT WEIGHT IN LB * C * RADER IS TYE NOZZLE THR04T EROSION RATE IN IN/SEC * C * EPS IS THE NOZZLE EXPAhSICN RATIO * C * ALT IS THE ALTITUDE IN FT C * DT IS THE NUZZLE THROAT DIAMETER IN Ih C * APHEAD IS THE tlEAO END PORT AREA, IN IF\i**2 * C * APNOi IS THE NOZZLE END PChT AREA Ih IN4q2 * C * COF IS THE CHARACTEK IS1IC THRUST COEFFICIENT * c * CFD IS THE DELIVEREO THiiUST COkFF ICIEhT AT AMBIENT PRESSURE * -r. * ETHETA IS THE TANGENTIAL STRAIN OF THE C.P. GRAIN AT THE BORE * * RHO5 IS THE OENSITY OF THE PROPELLANT IN SLUGS/IN**3 AT THE * bC * CHAMBER PRESSURE AND TEMPERATURE TGR * bC * RATP IS THE RATIO OF EXTERNAL TO INTERNAL PRESSURES ON THE * bi * C.P. GRAIN * bc * RATRZ IS THE S3UARE OF THE RATIO OF BCRE RAOIUS TO OUlSIOE bC * RADIUS OF THE CeP. GRAIN * bC * XETH IS THE FWACTIOh OF THE TOTAL BURAING BORE SURFACE * Table C-3 (~ont'd)

bC * ASSOCIATED UITH THE COP, GRAIN bC * YETA IS THE DISTANCE BURNEO IN INCHES FOR THE ENTIRE STAR * bc * GRAIN AN0 THE COP- GRAIN €NOS iHEN THE GRAIN OEFORMATION * bC * MODIFICATION IS IN USE 1 C ...... REAL MGENtMDIStMhOZtMNlt JhOCKvNvCvHElt HE* ISPt ITOTtflUvMASSvISPVAC REAL M2tMD3AR~ISP2tITVACtMOUT~ISPV COMM3N/COYSTl/ZW~A€vATvTHETAtALFAN CONMON/CONST~/CAPGAM~C~E~BOTE~ZETAF~TB~H~~GAME~CGAM€~TOPE~ZAPE CO?~MOY/VARIAL/Y~T~OELY~OELTATtPONOZvPHEAD~RNOZ~RHEA~vSUMAB~PHMAX COMMON/VARIA~/ABPORT~ABSLOT~ABNOZ~A?HEAD~~PNOZ~OACY~ABP~~A~N~~ABS~ COMMON/VARIA~/ITOT~ITVAC~JROCK~ISP~ISPVAC~MDIS~HNOZ~SG~SUMMT COMMON/VARIA~/A~MAIN~ABJO~SUMOY~VCI~A~~TT~PJRAN COMMON/VAKIA~/WPZ~CF~wP~RAOER~EPS~VC~FLAST~TLAST~OT~PUNTOT~WP~ COMNON/V.~KIA~/TIHE tFVv ISPVvNX b COHMON/VARIA3/ETHETA~fiHO5~RATP~PHOOtCMOO~PMU~CMUtALPTS~RATR2

, b COMMOY/VARI20/YETAvXETHe ISfl COMMON/IGNl/KAvK~tUFS~RHOtL~PMIG~TIlrTI2~CSIGtQ~tNltQZ~NZ COMM9N/PLOTT/NUMPLT( 161 v IPUv hDUMqNP9 IOP OIMENSION TPLOT(~~~~~PNPLOTI~~O)~PHYLOT(~~~)~~FPLOT~~~OI~FVP~OT~~OO 1) ,RNPLOT(230) ~KHPLUT[~OG~9YBPLOT (200) tAt3PLOT (200 J tSGPLUT(200) vVCPL 201 (230) DATA G/32* 1725/ IF(NDUM.EQ.1) GO TO 2 ME l=7.0 NP=NP+l Y B=Y VC X= VC IF(YOLE.0.O) M2=MDIS MPQAR=(MZ+MDIS)/2- SUMMT=SUMMT+MDBAK*DELTAT WP 1=G*SU4MT WPZ=RHO* (VC-VZ I)*G WP=( dPl+WPZ)/Zo b IF (ISOaEdeI) WP=WPl PTAR=l./JROCK 17 ME=SQRT(Z-/DOTE*( TOPE/Ze*i AE*MEl/AT 1 **(Io/ZAPE)-lo J i IFIARSIYE-MEl).Lt.O,002) GO TO 9 ME l=HE GO TO 17 9 CONTINUE PRES=( 1.+BOT€I2.*ME*MEJ**[-GAME/BOTE) AiT=HB*(i/TB)**(7o/3.) PATM=14.696/EXP(O04J10JE-04*ALTJ IF (MDISeLE.O.OeOHePONC~I~LEoOo0)GO TO 45 COF=CGAPlE*SQRT( L.*GAMt/8OTE*( 1.-PKES** [tjCT€/GAMEi J 1 CF=COF+AE/AT*( PRkS-PATM/PONOZ) CF VAC.=:CF+AE/AT*PATM/PUhc)Z Table C-3 (cL..t'd)

CFO=(COF*(Lo+COS(ALFAN))/2.+EPS*PRES)*LfTA+EPS*PAlH/PONOZ CF VD=CFD+EPS*PATNIPONOZ F=COS t THETA) *PONOZ*AT*CFO IF (FoCE.OoO) Fro-0 IF(Y-LE-000) FZsF FBAR=(F+f2)/2, F VAC=COS( THETA )*PONOZ*AT*CFVO IFIYoLE~OoO)FVZ=FVAC FVBAR=IF V2+F VAC)/2 MOOl=MOf S*G ISP=F/HDOT ISPVAC=FVAC/MOOT IT01=1TOT+FBAR*OELTAT Ir JAC=I TVAC+FVBAR*OELTAT IF(Y.LE.0-O)PONZ=PONOZ QONBAR=(PONZ+PONOZl/Z- PONTOT=PDNTOT+PONBAR*OELTAT OO NZ= PONOZ MZ=MOI S F2=F FVZ=FVAC IF (PHEAU.GT,PHMAX) PHMAX=PHEAD GO TO 47 45 CF VAC=O. 0 FVAG=O.O F=O,O 47 WRITE(be1) T~YB~RNOZvRHEAO~PCNOZ~PHEADsPTAR~MNCZ~SUMAB~SG~PATM~CFV LAC~FVAC~F,ISP~CFIVCX,MDOT~CFVOsITOT,ITVACpIS?VAC,WP~RAr)ER~EPSpALT 29DT9 APH€AD,APNOZICOF ,CFD IF( ISO.EQeOJ GO TO 200 HRITE(6v5) ETHETA~RHOS,HATPIRATR~~XETH~YETA 200 CONT INU€ IF 4 IPOoEQoO) RETURN TPLOT(NP)=T PNPLOT (NPI=PONOZ PHPLOT(NP)=PHEAD FPLOT(NP1-F FVPLOT (NP)-FVAC RNPLOT(NPI=RNUZ RHPLOT (NP)=RHEAD YBPLOT(NP)=YB ABPLOT (NP)=SUMAB SGPLOT INP)=SG VCPLOT(NP)=VC RE TURN 2 hlP=NP*Z IOP=1 DO 1004 1=1916 Table C-3 (Cont'd)

IF(NUHPLT(I).EQ.lI GO TO 1003 GO TO 1004 1003 GO TO ~10~20~30~40~50t55t60~~75~80~90~95~97~100~1L0~~15~~1 10 CALL PLOTIT(TPLOT,'TIME (SECS)'tllrPHPLCT,'PHEAO /PSIAl8rlZ, 1 PNPLOT~'PONOZ'~5rNP,l~'OU~HY'~5) GO TO 1004 20 CACL PLO'~IT(TPLOTI'TIME ISECS)'rllrPNPLOT,'PONOZ (PSIA)'~lZ,PHPCOT lr@PHEAD fPSIA)',12tNPtl,'OUnMV',S) GO TO 1004 30 CALL PLOTIT(TPLOT,~TIHE (SECS)',ll,PHPLOTr'PHEADe,5,PNPLOT lV8PONOZ'~S,NPV~~'PR€SSURE (PSIA)' ,151 GO TO 1004 40 CALL PLOTLT(TPLOTI'TIIJE (SECS)',ll,RHPLOTq'RHEAD (IN PER SEC)'rlBp ~PHPLOTI'PHEAO (PSIA)',~ZINP, L,'OUMMYat5) GO TO 1004 50 CALL PLOTIT(TPLOT,'TIME (SECS)'tll,RNPLOT,'RNOZ (IN PER SEC)',17, 1PNPLOT,'PONOZ fPSIA)',lZ,NP~lr'OUHMY'~5J GO TO 1004 55 CALL PLOTIT(TPLOT,'TIME ISECS)'~ll~RHPLOT~'RHEAD',5,RNPLOT~ 1 'RNOZ',~~NPI~,'BURNING RATE (IN PER SECJ',ZS) GO TO 1004 60 CALL PLOTITLTPLOTI 'TIME 4SECS)*,Ll~ABPLCT,'TOTAL BURNING AREA (SQ lIN)'~Z6,PNPLOT,'PONOZ't5~NP~ l,@DUMHY'~S) GO TO 1004 70 CALL PLOTIT(TPL0T~'TIME (SECS)'rllpSGPLOTp'STAR PERIMETER (IN1'119 l,PNPLOI,'PONOZ' rS,NP,l,'DUMMY',S) GO TO 1004 75 CACL PLOTIT(TPLOT,'TIME (SECS)',lltABPLCT,'TOTAL BURNING AREA (SQ lIN)'t26rSGPLOT,'STAR PERIMETER (INI'~19~hP,Z~'OUHMY',5) GO TO 1004 80 CALL PLOTIT(TPLO1,'TIME fSECS)',lltFPLOTt'THRUST (LBS)'V~~~PNPLOT* ~'PONOZ'~SINDI~,@DUMMY'~~) GO TO 1004 90 CALL PLOTITlTPLOT,'TIME (SECS)',lL,FVPLOT,'VACUUH THRUST (LBSI'rl9 1,PNPLOTp'PONOZ' ,SrNP,lp'OUMMY'fiI GO 10 1004 95 CALL PL0'fITiTPLOT~'TIME (SECS)@,11~FPLOT,'THRUSTr~6,FVPLUT, 1 @\,'ACUUY THRUST'r13rl\lP~3,@THHUST (L8SIa,lZ) GO TO 1004 97 CALL PLOTIT(TPLUTt'T1ME lSECSI',LltVCPLOl,'CHAMBER VOLUME (IN**3)' 1 ,~~~PNPLOTI'PONOZ',~~NP*~,'OUMMY@~~) GU TO 1004 100 CALL PLOT IT(Y BPLOT ,'BURNED DISTANCE ( IN) ' ,20,ABPLOT, 'TOTAL BURNING 1 AREA (SO INI'~26~PNPLOT~'PUhOZ@~5,NP,lp@DUMHY'tS~ GO TO 1004 110 CALL PLOTIT(YBPL0T ,'BURNED DISTANCE 4 IN)',Z~,SGPLOT,'STAR PERlHETE 1R IIN)'tl9,PNPLOTr'PONOZ',5~NP,l,'DUMMY' 951 GO TO 1004 Table C-3 (Cont'd)

115 CALL PLOTIT(YBPL0T r'8URNED DISTANCE ( IN) ' ,201ABPLOTe 'TOTAL BURNING 1 AREA (SQ IN)' ,26rSGPLOT,'STAR PERIMETER ( IN)'e199NP,2,'3UMMY',5) 1004 CONT I NUE RE IURN A FORMAT(13Xr6HTIHE= tF7-2r12Xr3HYt ,F6.2,/r13X,bHRNOZ= ~lPElla4r9H I RHEAOt rlPEll-4,9H PONOZo rlPE11.4,9H PdEAO= ~1PElL.4e/r13X,6HP 2TARt rlPEllo4rQH HNOZP r lPEll.4r9H SUMABs ,lPEll.Ct9H SGt r 3lPEllo4~/rl3Xr6HPAT~~rlPE1104r9H CFVACs rlPEllo4e9H FVAC= rlPE 4llr4r9H F= r lPEl1.4a/,13X16H ISP= IPEl l.419H CF= rlPE11. 549 9H VCs r lPElle4~9H MCOT= rlPE11.4r/r13XebHCFVD= rlPE11.419 6H ITOTs rlPE11.4r9H lTVAC= sLPE11.4e9H ISPVACt ,iPEll.41/r13X,6 7HWP= r1PElloGr9!1 RADER= r IPElle4,9H €PSI slPEll.4r9H AL Tt 8 r 1.PEll.4r/~l3Xr6tiDT= rlPEli.4r9H APHEAO= r1PE:1.4r9H A#NOZ= 91 QPE11.4r9H COF= r lPEll.4r/r13Xr6H CF0= r lPEll.4,/) 5 FOAMAT( 13XrBHETHETA= tlPEllo4e9H RHOS= r lPE11-4,9H RATP- 9 1PE1 11o4r9H RATRZz r lPEll.Gt9H XETH= lPE11o4r 7H YETAS r lPElle4r//) END Table C-3 (Cont'd)

*********************************************************************SUBROUTINE IGNITN * SUBROUTINE IGNITN CALCULATES THE PRESSURE RISE DURING f * THE IGNITION PERIOD * ASIG IS THE IGNITER ThKOAT AREA IN IN**2 * UIGTOT IS THE TOlAL WEIGHT OF THE ICIVITER Pr.OPELLAYT IN LBS * MfGAV IS THE IGNITER AVERAGE MASS FLOk RATE OVER THE FIRST * HALF OF THE IGNITER BURNING TICE IN LaS/SEC PCIG IS THE IGNITER PRESSURE IN LBS/IN**L * *******************4*********************4*************************** REAL K(~),L~KAIKB~JROCK~JZ~MIG~MIGAVpMSRN~ME~MOIS~MNO~~MNOZI~M~~ REAL NLtN2vMIGAVE COMMON/CONSTl/ZNtAEvATvTHETA,ALFAN COMMON/CONST2/CAPGAMvME,BOTE~ZETAFvTt3vH8~GAME~CGAHE~TOPE~ZAPE COMMON/VARIA~/~~TIG~OELY~OELTAT~PCNOZ~PHEAD~RNOZ~RH~AD~SUMA~TPHMAX COMMON/VARIA2/A8PO~TvABStOT~A8NOZvAPHEAOvAPNOZvOAOYvA8P2~A6N2~AESZ COMMUN/VARIA3/ITOT~ITVAC~JROCK~ISP~ISPVAC~MOISvMNOZvSG~SUMMT COMMON/VARI A5/AdMAIN* ABTOtSUMOY vVC1 *ABTT vPTRAN COMMON/IGNl/KAvK~vUFSvRHOvLvP~IGvTIlvT12 tCSIG~ULtNlvQZtN2 COMMON/IGN~/ALPHA~BETA~PBIG~RRIGT~ELTIG~X~TOPIZAP COMMON/PLOTT/NUMPLT(16)tIPOvhOUYvIPTtICP DIMENSION 819) DATA Al~A2~A3,A4/m17G76v~1.55L481vla2O5536~eL7l~b5/ OATA B(l)v6(2)*d(3lvt3(4) v0(5)/0mvm4va455737~1eve296978I OATA 0(6)tY(7Jv8~8)~8~9)/aL5876t~21t)1t-3-Ci5L)96Sv3e832864l ***********************************4*********************************4* * THE A'S ANU H'S ARE CONSTANTS FOR THE HUNGE-KUTTA INTEGRATIUIV * *****4**4*****************4***************~********4************4**** DATA G/32e1725/ XXX=.OS*PONOZ 1PLUG=O PONOZ I- "'NO2 RhEADI 'AD RNCLI=s ,L PHEADI=#H€AO DELTT=DEC TAT 01 SM-MD IS DELTAT-OELT IG SUYAH I=SUMAU MNCLI =MNOZ MNOL=daiI RH FAD=O, 0 R%OZ=OeO MD IS=O*O AS i=O.O TIGI=OaO PC 1=14.696 TIG=O ,0 Table C-3 (~ont'd)

PCYEW=lCo696 SUHAB=Oo 0

PCIe1Co696 *a PHEAOtl6o696 PONOZ=14o696 SLOPE=SUf4ABI /L GZ44PGAM*CAPGAM J2= JROCK* JROCK GJ=GZ*JZ/Z. HIGAV=.Z*AT/G AS1CI4o*MlGAV*CSIG/~4~*PHIC-RRfG*4TIZ~T11)) Y1GTOT~G*HIGAV*~Sa*(ffZ-TIL)I6~) MIGAVE=HIGAV*G YRfTE(6r999) ASIG~Y1GTOTrHIGAVE WRlTE(6.10J 18 NNWO WRITE (6s30) PC tG CALL OUTPUT 9 CONTINUE 00 8 Nalr4 IF(NoEQo11 PC+PCI IF(NaEQ.2) PC=PCI+B(Z)*K( 1) IF(NoEQo3) PC=PCI+B(S)*K( L)+B1b)*K(Z) IF(NoEGo4) PC=PC1+8(7)*K( 1)+0(8)*K(2)+K13) TlG=TIGI +Bi N)*DELT 1C SUMAB=AB I+SCOPE*UF S*B(N) WELT tG

IFI SUMAa lGT SUHAB 1 ) SUMAB= SUMAS 1 PHEAD=PC lF(HDfSoNEoOo0) PHEAO=PC*(lo+GJ) 1F(PHEAUoLEoPTRAN)RHEAO=Qi*PHEAO**Nl IF(PHEADoGT oPTRAN)RHEAD=&Z*PHEAD**NZ IFITIGoLE,TIl) PCIC=PMIG*TCG/TIl 1F (l!GIGT,TIl,ANO~PCIG~GT~PHEAO)PCIG=PHIG-HRIG*ITIG-fIl) IF(PC1GoLEaPHEAD~ PCIG-PHEAD HIC=OoO IF(PC IG,GT,PHEAO,AND~TIG~LE~TI2/2~)WIG=PCIC*ASIG/CSIG CETR=KA+Kt3*PC MOIS=PC*AT/CSTR IF(PCoLE-PBIGmANDelPLUGoEQoO) GO TO 7 IPLUG=l FNdZ=r4NOL I PNOZ=PC* :--GJ ) ZiT=HDI S*X/APNOZ RN L=NHEAO AZ=ALPHA*ZlT**o8 XL=UF S*T IG IF(XLsGT-C) XL=L 4 EX=XL**. 2*EXF ( BETA*RNl*P HO/Z 111 Table C-3 (Cont'd)

IF (PNOZ~LE~PTRAN)RNOZ=RN~-(RN~-Q~*PNOZ**N~-AZ/EX 1 / 4 lo+AZ*8E TA*kHo/ Z(ZIT*EX)) IF I PNOZ~GT~PTRAN)RNOZ=RN~-~RN~-Q~*PNOZ**N~-AZ~EX~/t l.+AZ*BE TA*RHOl ZIZIf*EX) ) IF(ASS(RN1-RNOZ)oLEoOoOO2) GO TO 5 RN l=RNOZ GO TO 4 7 MOIS=OoO MN OZ=Oo 0 PNOZ=PC RNOZ=RHE AD 5 CONTl NU€ MSRWRHO*SUHAB*(RNOZ+RHEAO)/~~ DENOM=[VCI/(~~~*CSTR*CSTR*G~))*(~,-~~~*K~*PC)/CSTR) OPDl= t HIG+HSRH-MO IS) /O€NOM IF(DPDi~tloO~Oo~ND~PCollo20o0)DPOT=OoO Kt N)=DELT IC*DPDT 8 CONT lNUE PCNEW=PCI+Al*K( 1)+A2*K(Z)+A3*K(b) PHEAO=PCNE W IF(HDISeGToO.0) PHEAO=PCNEW*(lo+CJ) PONOZ=PCNEU XXY=AaS ( PONOZ-PONOZ I ) IF~PCNEW~LE~~~OG~*PCI~AND~$UNAB~E~~SUMABI~ANO~XXY~LEXXXGO TO 13 A0 I=SdMAB TlGI=T IG PC I=PCNE W NY N=NN&+ 1 IF(NNVoGEoSJ GO TO 18 GO 10 9 13 CONTtNUE CALL OUTPUT WRITE(br30) PClG DELTAT=DELTT HDIS=DISM SUMAB=SUMAB I PONOZ=PONdZ I RHtAD=RHEADI P.NOL=KNUL I PHEAD=PtiEADI MIYUZ=MNUZI IF(IPOoNEo2oANO~IPO~NEo3JGO TO 53 NOUM= 1 CALL OUfPUT NO UM= 0 53 CONT INUE IPT=O RE TURN Table C-3 (Cont'd)

999 FORYATl///r20Xv25HICNITER SI ZE CALCULATICIiS, /, ~~XISHAS~G=.FI~~I~I 1 13X~IH~IGTOT=rF7~2~/~13X~bHMICAV=~F8~3,///) 10 FORMATI~~XIZ~H**************.~****~******/ IGNITION 1TRANSIENT ****,fr33X,28H*****************+*r*+**++**) 30 FORM~T(~~XI~HPCIG=1PElA-4) END

SUBROUTINE INTKP~(YITIN.TTvOY.ICHK) DIMENSION Y(N),T(N) Nl=N-1 OY=O,O IF (ICHKJ 212.3 2 00 1 I=LrNl IF(TT,GE,T(I),AND,TT~LT~T(I*1)1 OY=~(Y~I+1)-Y~II~/lT~I+l~-T(I))) 2*I TT-T( I))+YII IF IDY ,NE,O-OI RETURN 1 CONTINUE 3 DO 4 I=l.NL IF{TT,LE,T([ ),AND.Tl.GT.TL I+l)) DY=((YI 141)-Y( I))/(T~I+l)-Ti1)) 2*( TT-111) )+Y( I) If (OY.NE,OoOJ RETURN 4 CONTINUE RE TURN EN 0 Table C-3 (Cont 'd)

*********************************************************************SUBROUTINE PLOTIT(X~XHDR~KX~Y~YHDR~NY~T~~HDR~NT,NP,NPLOT~OUMMY~ND~ I SUBROUTINE PLOTIT PLOTS TkC DEPENDE&f VARIABLESv Y AND Tt C * VERSUS AN INDEPENDENT VARIABLE* X C XHORt YHDR. AND THOR ARE THE HEADINGS FOR THE X. Ye AND T C * AXES, RESPECTIVELY 8 C * KX. NY. AND NT ARE THE NUMBER OF CHARACTERS IN THE XI Yt AND C T AXES HEADINGS, RESPECTIVELY (PAX OF 32 IN EACH) C NP 1S THE NUHaER OF POINTS TO BE PLCTTEO PLUS 2 4 C VALUES FOR NPLOT ARE $: C * 1 FOR Y ONLY PLOTTEO VERSUS X * C 2 FOR Y AND T PLGTTED VERSUS X ON SANE AXES C WITH INDIVIOUAL SCALES C * 3 FOR Y AND T PLOTTED VERSUS x ON SAME AXES C * WITH SAME SCALES * c ounnv is THE HEADING FOR THE ooueLE AXIS (NPLOT=~) c C * NO IS THE NUMBER OF CHARACTERS IN DUMPY C *************************************-****************************** DIMENSION XHOR(8)tYHDR(B)~THDRA8)~OUMMY(8J~X4NP)~Y(NPJ~T(NP) NX=-KX NH=NP-1 NN=NP-2 IF(NPLOToEQo1) GO TO 9 CALL SCALE(Tt4o rNNv1) 9 CALL SCALE(XvBmtNNt1) CALL SSALE(Y.4- rNNw1) IF(NPLUloNEo3) CALL AXIS(O~~OorYHDRrNYtGotl8OotY(NMjtY(NP)) IF(NPL0ToEQ-3) CALL AXIS(O-~OerDU~MY~NDI4,,ldOo~YtNM)~Y(NPj) CALL AXIS(O-IO~~XHDA~NX~~~,90-,X(hM) tX4;JP)) IF(NPLOToEQo1) GO TO 12 DO 11 I=ltNN 11 T( II=-T4 I) 12 00 13 I=ltNN 13 Y( I)=-Y(I) CALL LINE(Y.XtNN.ltOpl) CALL PLOT(OoiOor3) IF(NPLOToEQo1) GO TO 24 IF(NPLOJOEOo2) CALL PLOT (O-i--5r2) IF(hPLOToEQ.2) CALL AXIb(O,i-oSiTHDRiNTt4otltiO-~T(NMJ~T(NP)) CALL LINE~TvXiKNiltOv2) DO 25 I=lrNh 2s T( II=-T( I) 24 DO 26 IzlrNN 26 Y( I)=-Y( I) IF(NPL0T-EQ-1) GO TO 32 CALL SYMBOL(-4o35t*52t ?ltltOotO) CALL SYMdOL(-bm2vo52relv2iOovO) CALL SYMBOL(-4e3~o65~~l~YHOR~93-~FcY) CALL SYMUOL(-4-15io65iol~ThOR~90-iNTI 32 CALL PLOT(Bo5iOot-3) RE TURN EN 0

QU S GOVERNMENT PRINTING OFFICE 1977-7400491354 REGION NO. 4