The Effects of Outage on Performance Statistics for Bipropellant Rockets JULY 1967

Transcription

1 AIR FORCE REPORT NO. SAMSO-TR AEROSPACE REPORT NO. TR-01Se( )-l rh ' lo' <Oi A < The Effects f Outage n Perfrmance Statistics fr Biprpellant Rckets JULY 1967 Prepared by JOSEPH W. DUROUX Vehicle Systems Department Systems Design Subdivisin Applied Mechanics Divisin AEROSPACE CORPORATION Piepared fr SPACE AND MISSILE SYSTEMS ORGANIZATION AIR FORCE SYSTEMS COMMAND LOS ANGELES AIR FORCE STATION Ls Angeles, Califrnia --v I ^..r THIS DOCUMENT HAS BEEN APPROVED FOR PUBLIC RELEASE AND SALE; ITS DISTRIBUTION IS UNLIMITED CLEARINGHOUSE lrfeder-v! Scien?-?>r Tchn c-i' l~lfmatin Spting :ld Va 22151

2 Air Frce Reprt N. SAMSO-TR Aerspace Reprt N. TR-0158(3ll6-50)-l THE EFFECTS OF OUTAGE ON PERFORMANCE STATISTICS FOR BIPROPELLANT ROCKETS Prepared by Jseph W. Durux Vehicle Systems Department Systems Design Subdivisin Applied Mechanics Divisin AEROSPACE CORPORATION El Segund, Califrnia July 1967 Prepared fr SPACE AND MISSILE SYSTEMS ORGANIZATION AIR FORCE SYSTEMS COMMAND LOS ANGELES AIR FORCE STATION Ls Angeles, Califrnia This dcument has been apprved fr public release and sale; its distributin is unlimited p**..*«** '. v«

3 FOREWORD This reprt is published by the Aerspace Crpratin, 1 Segund, Califrnia under Air Frce Cntract N. F C The reprt was prepared by Jseph W. Durux f the Systems Design Subdivisin, Applied Mechanics Divisin, in rder t dcument research carried ut frm March 1966 t December 1966 under the abve cntract. This reprt was submitted n 11 September 1967 t Lt. Cl. Allen W. Thmpsn, SMVTD, fr review and apprval. The authr wishes t acknwledge the assistance f Philip H. Yung, Cmputing and Data Prcessing Center, Aerspace Crpratin, wh prepared the numerical analysis and cmputatin prgram given in the Appendix. Apprved D. Willens, Directr Systems Design Subdivisin Applied Mechanics Divisin Rbart J. Whalen, Grup Directr Space Launching Systems Develpment Directrate Manned Systems Divisin Publicatin f this reprt des nt cnstitute Air Frce apprval f the reprt's findings r cnclusins. It is published nly fr the exchange and stimulatin f ideas. VtyJy^-' 1 " Allen W. Thmpsn, Lt. Cl., USAF Asst. Deputy Directr fr Engr and Test Dept. fr 624/632A

4 ABSTRACT The distributin f perfrmance errrs in biprpellant rckets is determined as a functin f engine and prpellant lading parameters and cntributing errrs. The distributin is fund t be highly skewed, s that the prbability f exceeding three sigma can be significantly greater than it wuld be if the distributin were nrmal. Thus, three sigma as a minimum perfrmance limit is nt justified. A detailed analysis is required in each case t determine the relatinship between perfrmance margin and prbability f missin success. Als, a realistic methd f ptimizing the prpellant lading bias is develped. The methds previusly used have resulted in fuel biases that are usually much lwer than the ptimum. Cmputer prgrams have been develped t perfrm the analyses described in this reprt, bth fr the single-stage and the multistage case. i

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6 CONTENTS FOREWORD li ABSTRACT iii SYMBOLS vlll I. INTRODUCTION 1 II. THE OUTAGE FUNCTION 3 III. THE OUTAGE DISTRIBUTION 9 IV. THE COMBINED DISTRIBUTION FOR ONE STAGE 15 V. THE MULTISTAGE CASE 19 VI. THE CHOICE OF BIAS 25 VII. EXAMPLES OF RESULTS 35 VIII. CONCLUSIONS AND RECOMMENDATIONS 47 APPENDIX 49 REFERENCES 59

7 . FIGURES 1. Outage Functin 6 2. Transfrmatin f Density Functin Outage Distributin Distributin f S Prbability (P a ) Versus Deviatin Frm Mean Errr; Titan III. Stage Prbability (P-) Versus Deviatin Frm Mean Errr: Titan IIIC, Vehicle Prbability (P a ) Versus Bias; Titan IIIC. Stage Optimum Bias Versus Perfrmance Decrement; Titan IIIC. Stage Prbability (P a ) Versus First Stage Bias: Titan IIIC. Stages 1 and Minimum P a Versus Secnd-Stage Bias; Titan IIIC. Stages 1 and Prbability (P 0 ) Versus Deviatin Frm Mean Errr; Titan IIIC. Vehicle 11 46

8 L TABLES I. Cntrl Card, Data Set 1 51 II. Values f IJ (J = 0 8), Data Set 1 52 III. Frmat f Card, Data Set i 52 IV. Vectr Data, Data Set 2 53 V. Frmat f Card, fiata Set 2 54 VI. Data Deck Setup (One Case) 55 VII. Prductin Deck Setup 55 VIII. Output, Output Set 2 57 IX. Output Frmat 58

9 SYMBOLS Rman Uppercaae Lettera A the deviatin f S frm "5 in multiples f <r s C the deviatin f shutdwn perfrmance frm mean perfrmance K the errr scaling factr fr each stage N the number f biprpellant stages S X - 1 W the sum f nn-utage perfrmance errrs in ne stage X the ttal perfrmance errr caused by ne stage W Z the fractinal utage equal t m L Rman Uppercase Letters With Superscripts F*() the discrete prbability that () = 5 Rman Uppercase Letters With Subscripts F (Z) the discrete prbability that the fractinal utage has the value Z P the cumulative prbability that X > 3? + a<r x r that T > T +»",-!^ R, the xidizer-t-fuel burning rati b w R the laded rati f usable xidizer t usable fuel equal t -_* L f

10 L R the nminal value f R. n b T. the ttal perfrmance errr caused by the first i stages W. the weight f usable fuel W T the ttal weight f usable prpellant equal t W- + W W the weight f usable xidizer W the utage weight Z the maximum allwable utage Rman Lwercase Letters With Superscripts f ( ) a prbability density functin truncated by cmmand shutdwn Rman Lwercase Letters With Subscripts f,( ) the cntributin t f ( ) frm fuel utage f ( ) the cntributin t f ( ) frm xidizer utage fr - ( ) the prbability density functin fr C ] unless therwise defined Greek Lwercase Letters a the deviatin f X frm If r f T frm T in multiples r f tr r a,. x t ß the prpellant lading bias and the mean value f S 6 the limiting magnitude f S fr which zer utage ccurs R L \ equal t -5 R b u a dummy variable fr integratin the ttal errr crrespnding t the cmmand shutdwn perfrmance level equal t T" + Ccr

11 mmmm Greek Lwercase Letters With Subscripts ß the ptimum prpellant lading bias <rr -i the variance f [ ] Superscripts and Subscripts ( ), refers t the situatin in which fuel utage ccurs unless therwise defined ( ). (r numerical subscript) refers t the i r numbered stage ( ) refers t the situatin in which xidizer utage ccurs unless therwise defined

12 SECTION I INTRODUCTION This study primarily aims t determine the effects f utage n the distributin f errrs in the perfrmance f biprpellant-staged rckets, and t develp a methd t ptimize prpellant lading bias. In cmputing perfrmance errrs fr rckets, three basic assumptns have been standard: a. All cntributing errrs are nrmally distributed. b. All cntributing errrs are independent f each ther. c. Except in ne special case, all f the transfer functins which relate the cntributing errrs t the perfrmance parameters are linear ver the ranges f the errrs. The special case is assciated with the utage effect, characteristic f a liquid biprpellant system. While investigatin int the validity f all the preceding assumptins is desirable, this study deals nly with the special case in Assumptin c abve. Therefre all the ther assumptins are cnsidered valid in this reprt. The special case cmbines three f the cntributing errrs t frm a highly skewed distributin. This distributin ften cntributes mre t the ttal perfrmance errr f a stage than all f the ther cntributing errrs cmbined. Thus the distributin f all f the cmbined errrs can be cnsiderably skewed. Until nw, this cmbined distributin has nt been investigated adequately. The specific bjective f this study is t determine certain distributin characteristics f cmbined perfrmance errrs. The imprtant characteristics determine the prbability f exceeding a certain maximum allwable adverse errr r the adverse errr which will nt be exceeded with a particular allwable prbability. Als fr this purpse, the variances given fr the cntributing errrs are assumed t be ppulatin variances estimated with sufficient cnfidence s that the prbabilities cmputed frm them are cnservative. -1- ftstt,,,,.- ;:.^,..,:..ifc.r... - 'T.jm-r. :vi*;v-,.-;....u-v..^-

13 *fr SECTION 11 THE OUTAGE FUNCTION Outage is defined nly in reference t a biprpellant. It is the usable weight f ne prpellant cmpnent (fuel r xidizer) left in the tank when the usable prtin f the ther cmpnent is cmpletely cn- sumed. Outage degrades the stage perfrmance by increasing the burnut weight and decreasing the cnsumed prpellant. Errrs in the amunts f fuel and xidizer laded, and in the predicted xidizer/fuel burning rati cause utage. The basic utage functin has been described in previus reprts (e.g., Refs. 1, 2, 3, and 5). explanatin. Hwever, t derive the functin here shuld aid in All quantities will be defined as the derivatin prgresses. Definitins are als given in the symbls sectin in the reprt frnt matter. The laded rati R. (r mixture rati laded) is the rati f the weight f xidizer W t weight f fuel W, laded n the stage and usable. The term "usable" excludes prpellant trapped in the tanks r lines r therwise wasted, even when n utage exists. A useful intermediate parameter in this analysis is \, whi<;h is the rati f R, t R,, where R, is the xidizer/ J_i b fuel burning rati averaged ver the burning time. R, is ften called the mixture rati burned. R. W X = R^ R = R-W- < 1 ) b K b f It is further useful t define a parameter S s that S = \ - 1. Then W b f f (2) W = R^ W r + R. W JS b f b f -3-

14 The actual amunt f utage must be specified differently in tw different regins. When the laded rati equals the burning rati, n utage exists. At this pint, X = 1 and S = 0. When the laded rati is less than the burning rati, an excess f fuel (a deficiency f xidizer) is laded and a fuel utage ccurs. Frm Eq. (1) bserve that X < 1 and S is negative fr this fuel utage situatin. Similarly, when S is psitive (X > I) an xidizer utage ccurs. Separate expressins are required fr fuel utage and xidizer utage. Fuel utage is equal t the usable fuel minus the fuel cnsumed by cmbining with the usable xidizer. Similarly, xidizer utage is equal t the usable xidizer minus the xidizer which cmbines with the usable fuel. Hence, if ne uses Eq. (2) W W -. = W, - =-i zf f R, = - W f S when S s 0 (3a) W = W - R. W, = R. \V.S when S ^ 0 z b f b 1 (3b) where W, is the fuel utage and W is the xidizer utaee. zf B z 6 The utage may be mre cnveniently expressed as a fractin f Wj., the ttal usable prpellant. Z, and Z are the fractinal utages fr the tw cases W L = W f + W = W f (i + R b X) = W f (l + R L ) (4) Z f W = "W zf TT-R- when 3^0 (5a) W z IT R b s 1 + R, when S a 0 (5b)

15 Althugh this utage functin f S is nnlinear because f its discntinuus slpe at S =0, the tw parts f it are assumed (Assumptin c, Sectin 1) t be individually linear within the ranges f the errrs. The slpes at S = 0 are chsen as the slpes f the linear segments f the functin. These segments are btained by assuming the laded rati R. and the burning rati R, are bth cnstants in Eq. (5) and that they are equal. They are als assumed t be equal t the nminal value f the burning rati, which is dented by R. Actually, bth R T and R, vary statistically, and their average values are functins f S. Hwever, their variatins frm R are small and result in crrespndingly small errrs in utage. Thus the fractinal utage functin may be written Z f = - i S when SSO (6a) n R Z =. " S when S s 0 (6b) 1 + R A plt f a typical utage functin is shwn in Figure 1, represented by the slid lines. In the exhaustin shutdwn f a stage using pump-fed engines, an addi- tinal effect may be encuntered as described in detail in Ref. 6. As ne cmpnent f the prpellant is exhausted, the ther cmpnent is pumped t the engine at an increased rate, s that an extra quantity f the excess cmpnent is cnsumed. this extra quantity. Thus the utage is reduced by an amunt equal t An utage that wuld be less than the extra quantity if the effect were nt present, then becmes zer. A cmpensatin is intr- duced int the analysis t allw fr the burning rati being s far ff during the shutdwn transient that the specific impulse is significantly reduced. An apprximatin t the ver-all result is shwn by the dashed lines f Ficure 1. In Ref. 6, fr a Titan III cre stage i, 6, and 6 were fund t be 6 e f equal, bat in the present study they are treated as separate quantities t accmmdate situatins in which they are unequal.

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17 The analytical expressin in exhaustin shutdwn is a mdificatin f Eq. (6) as Z = - ^ ^ n (S + 6 f ) when S S - 6 f (7a) Z = T+^-( S - 6 > whens^6 (7b) Z f = Z = 0 when - 6 f s s s 6 (7c) Fr cnvenience, Eq. (7) will be used fr all stages whether the exhaustin effect ccurs r nt. When the pump-fed exhaustin phenmenn des nt ccur, 6, and 6 will simply be set equal t zer, which will make the equatin the same as Eq. (6). 7-

18 SECTION III THE OUTAGE DISTRIBUTION If S is characterized by a prbability distributin which has a density functin f (S), then the quantity Z which is a mntne functin f S in a given interval will be characterized by a density functin in that interval wherever the derivative exists accrding t (see Figure 2) f z (Z) dz = f a (S) ds (8a) f z <z ) = lall f s< S ) < 8b ) Multiplicatin by the differentials simply cmpensates fr the change in scale s that the integral f f (Z) ver any subinterval in Z will be equal t the integral f f (S) ver the crrespnding subinterval in S. Thus, frm Eqs. (7) and (8) f f (Z) = (1 + R n ) f 8 (S) when S < -6 f (9a) f 0 (Z) = 1 + R R - f s (S) when S > 6 (9b) n where f, and f represent the cntributins t the density functin frm fuel utage and xidizer utage. A density functin des nt exist fr Z crrespnding t the regin between -6, and 6 fr S. Instead, there is a discrete prbability that Z = 0. It is equal t the prbability that S lies between -6, and 6. Slving Eq. (7) fr S and substituting int Eq. (9) gives IJZ) = (1 + R ) f i-s - (1 + R ) z,] when Z r > 0 (10a) f n s f n' f f * ' 1 + R f < z ) = -TT-^'s 1 + R = Z when Z > 0 (lob) R I ' ' The restrictins n Z crrespnd t the restrictins n S in Eq. (9). -9-

19 dz Figure 2. Transfrmatin f Density Functin -10-

20 Nw it is pssible t write an analytical expressin fr the density functin f Z in terms f the density functin f S. It is a hybrid cntinuus and discrete density functin. f 2 (Z) = f f (Z) + f (Z) when Z > 0 (11a) 6 f (S) ds when Z = 0 (lib) f (Z) = 0 when Z < 0 (lie) where the functins f, and f are given in Eq. (10). Since bth these expressins cver the same regin f Z, their prbability densities are additive in that regin. F is the discrete prbability that Z = 0. Eq. (lie) is cnsistent with negative utage being meaningless. The density functin f (S) is btained by assuming that the errrs in W. W-, and R. are all nrmally distributed. If the.lgarithms f bth sides f Eq. (1) are differentiated at specific values f the variables, the result is dw dr. dw, IT = Iff RT " "W t 12 ' b f Since the fractinal errrs are thus additive, and since they are independent accrding t Assumptin b, the means and variances f the fractinal errrs are als additive (Ref. 7). And since X = 1 at zer errr, the abslute errr in X is the same as the fractinal errr in X. This means that S, which is equal t the abslute errr in X, is als equal t the fractinal errr in X. Therefre, the variance f S is equal t the sum f the variances f the fractinal cntributing errrs. 2 = 2 2 2,.,. ''W + + "K ^W, (,3 ' b f "s -11-

21 Als, the distributins f the cntributing errrs are nrmal accrding t Assumptin a. Since the sum f independent nrmally distributed variables is nrmally distributed, S is nrmally distributed.. (S-ß) 2 f a (S) = i e 2<rf (14) s v Abve, p is an intentinal bias n the quantity S t make ne type f utage mre likely than the ther. Nrmally the weight f ne cmpnent f the prpellant is much less than the crrespnding weight f the ther cmpnent with which it cmbines. Thus an utage in ne directin causes mre penalty t the system than an utage in the ther directin caused by the same magnitude f errr. The errr is intentinally biased t favr an utage f ne f the cmpnents, usually the fuel, accrding t sme criterin. If pssible this criterin will minimize utage effects. Methds f arriving at a desirable bias are discussed in Sectin VI. If the expressin in Eq. (14) is substituted int Kqs. (10) and (11), an analytical expressin is btained fr the distributin f utage. It is a highly skewed distributin which lks smething like that shwn in Figure 3. Its mean and variance may be cmputed by taking the first and secnd mments f the density functin as CD "Z = y Z f z (Z) dz (15) 0 2 -s-z = Z 2 F z (0) + y (Z - Z) 2 f z (Z) dz (16) 12-

22 - f z (Z) F t io) Figure 3. Outage Distributin 13-

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24 SECTION IV THE COMBINED DISTRIBUTION FOR ONE STAGE The effects f cntributing errrs n rcket perfrmance are determined as perturbatins in a trajectry simulatin. The results are a variatin in sme perfrmance parameter caused by each cntributing errr. The perfrmance parameters mst cmmnly chsen are (1) prpellant margin, which is the excess prpellant at burnut f the last stage when the missin cnditins have been met, and (2) velcity margin, which is the excess velcity ver the missin cnditins, which culd be attained if all f the prpellant were burned. Since the transfer functins used in the trajectry simulatin are assumed t be linear within the ranges f the errrs (Assumptin c), the perfrmance parameter variatins have distributins that are the same, except in scale, as the distributins f the errrs which caused them. If utage is treated as a cntributing errr in place f the three errrs which cntribute t it, this situatin then hlds fr all cntributing errrs. Since the errrs ther than utage are independent and nrmally distributed, their cmbined effect n the perfrmance parameter is als nrmally distributed with a variance that is the sum f the variances f the cntributing effects n the perfrmance parameter. The means f the cntributing effects may be cnsidered zer relative t nminal perfrmance. If the effect n the perfrmanca parameter by the ther errrs is called W, then tha frequency functin f W is W 2 f w (W) = i -==re /Z 2 a w 2 (17) - v2n ' ' -15-

25 If X is the ttal effect f all the errrs in a single stage n the perfrmance parameter f that stage, then the frequency functin f X is CD f x (X) = /" f w (X-u) f z (u)dv (18a) - f x (X) = F Z (0) f w (x) + / f w {x-ü) f z (y) du (i8b) where u is a dumnny variable. The first frm f this equatin crrespnds t Eq. ( ) f Ref. 7. The secnd frm breaks f int its three cmpnent parts. The first term n the right hand side accunts fr part f the distributin f Z being discrete. It is simply the prbability that 2.-0 multiplied by the distributin f X when Z = 0, which is the distributin f W. The lwer limit f zer in the integral accunts fr the f functin being zer when the argument is negative. Since the utage effect is independent f the ther effects i and since the mean f the ther effects is taken as zer X = "Z (19) X = a z + (r w (20) ' ' The imprtant characteristics f the distributin f X are thse which determine the prbability f nt exceeding a certain adverse errr r the adverse errr which will nt be exceeded with a certain prbability. Only the regins f high prbability, in which the errrs are several times the standard deviatin f the distributin, are f interest. A plt f the entire density curve wuld prvide little useful infrmatin. Anther functin, P, is defined as f (X) dx (21) '-/ * +a^x X -16-

26 In ther wrds, P shws the prbability that the errr differs frm the mean errr by mre than Of times the standard deviatin. A cmputer prgram r a has been written t cmpute ^ P a as a functin f R nsf, -, 3, 6,, 6, <r, and r. An example f a set f curves f P versus a is shwn w a in Figure 5. Althugh it is pssible t integrate Eq. (18) analytically, as shwn in Refs. 5 and 8, the integrand must be split int several parts and the limits becme very cmplicated functins f the parameters. If hand cmputatin is necessary, this methd may be easier because the resulting functins are f the frm f the nrmal distributin functin and can be fund in tables. Hwever, fr a machine, numerical integratin is easier. Even Eq. (lib), which can be evaluated simply by lking up the limits in the tables f the nrmal distributin functin, can be evaluated mre easily by numerical integratin when a machine is used. In the cmputer prgram, each value f the integrand in the numerical integratin f Eq. (21) is evaluated by numerically integrating Eq. (18). and each value f the integrand in that numerical integratin is evaluated by using Eqs. (17), (11) and (10). The values f R and <r are available frm the errr analysis f the n s ' vehicle, such as in Ref. 9. In this case, the criterin fr chsing ß has already been selected, s that ß is als available frm the reprt. Thus far, values f 6 and 6 have been determined nly fr Titan III Stage 1 as given in Ref. 6. The infrmatin is nt included in the fficial errr analysis f any f the Titan III series vehicles, and s the figures given in Ref. 6 are just assumed t be representative f all Titan III first stages. Fr Titan III, the value f a may be fund as a multiple f - by nting the relative effects f the tw types f errrs n the final perfrmance f the vehicle. This infrmatin may be btained frm the errr perturbatins in a trajectry analysis such as thse presented in Ref. 10, Table III. Thus, the value f the rati f cr t- will nrmally be entered int the w z cmputer prgram t determine the errr distributin when trajectry analysis 7 is available. Once the value f <r has been determined, w 17-

27 the values f the ther parameters may then be varied with - held cnstant. The remaining parameter a is used as the independent variable fr pltting P. Usually values f t frm 1.5 t 5 will cver the useful range. Ntice that the abve analysis fr a single stage is useful nly when a particular stage has a requirement t meet certain perfrmance standards independently. Otherwise, the stages in a multistage vehicle must be cnsidered tgether as shwn in Sectin V. 18-

28 SECTION V THE MULTISTAGE CASE When mre than ne biprpellant stage is used in a rcket, each such stage has its wn utage distributin which affects the perfrmance parameter. In additin fr range safety, it is smetimes necessary t cmmand a stage shutdwn at a certain maximum level f perfrmance. Therefre, the distributin f perfrmance errrs caused by each stage must be determined separately and then cmbined with thse caused by the previus stages. T cmbine the distributins f errrs frm the different stages, the errr variables must be cnverted t a cmmn scale in the final-stage perfrmance parameter which they affect. Eq. (18) must be mdified accrdingly as f.(x.) = F.(O)K.f.(K.X.)+K 2 f. [K.(X.-u)]f.(K.v)dv (22) XI 1 Zl' ' l Wl' I l' I / Wl l' I ' zi I ' where X. is the cntributin t the perfrmance errr by the i stage. This mdificatin simply scales the f. and f. functins s that their frms remain the same except the scaling; their standard deviatins becme <r wi./k. i and a zi./k.; i and the mean f the f zi. functin becmes ^i/k.. i The mean f the f. functin remains zer. K. is btained by dividing cr. by wi i ' 0 zi ' the standard deviatin f the final-stage perfrmance errr caused by utage in the i stage. This figure is available frm the trajectry perturbatin analysis. If any stage is preceded by stages nt subject t utage, the perfrmance errrs caused by these preceding stages may simply be cmbined with thse caused by nn-utage effects in the biprpellant stage, fr they are assumed t be nrmally distributed. -19-

29 First, Eq. (22) determines the distributin f errrs caused by the first biprpellant stage. If this stage is t have a cmmand shutdwn, the distributin f perfrmance errrs caused by this stage is truncated at a pint which crrespnds t the level f perfrmance 5i at which the stage is shut dwn. The density functin beynd that pint in perfrmance is integrated t determine the prbability that the shutdwn perfrmance r better culd be achieved if shutdwn did nt ccur. This number then represents the discrete prbability f ccurrence f 5j when shutdwn des ccur. It is cmbined with the prtin f the density functin n the lw perfrmance side f 5, t frm a new hybrid cntinuus and discrete density * functin, f, (X,). xl 1 Cl lx l> = f xl ^l' When X l > 5 1 (23a,? F^Xj) = f - f xl (X x ) dxj when Xj = Ij (23b) Ci^i' = 0 when X l <? 1 (23c, v/here f, (X,) is given by Eq. (22) and F, is the discrete prbability that X, =?,. The shutdwn perfrmance may be cnveniently expressed as a deviatin C, frm the mean f X. in multiples f the standard deviatin f X,. h = 3 S + C l -xl < 24 ' where lc. and tr, are fund by mdifyinfj; Eqs. (19) and (20) as X l = KT" < 25 ) zl wl, <r = 2- + r (26) -20-

30 The deviatin f shutdwn perfrmance frm mean perfrmance is given as a multiple f the standard deviatin fr cnvenience if the infrmatin is available in that frm and als t prvide an indicatin f hw much f the density curve is being cut ff. Usually, the available infrmatin is in the frm f the actud deviatin, which is the full quantity C.<r.. The next functin that must be defined is the distributin f cmbined perfrmance errrs caused by the first i stages. It is called f. (T.), where ti i,. T. is the ttal perfrmance errr caused by the first i stages. If the i stage is t have a cmmand shutdwn, the f ti. functin, is truncated as the f xl. functin was truncated s that a new hybrid 7 functin T. is frmed. Ntice ti that f t*i (T i ) ' Ci&O fr a11 T i H X l (27) Eq. (23) may nw be written as a frm that applies t the sum f the errrs caused by the first i stages. f* (T.) = f. (T.) when T. > 5- (28a) ti i' ti i' ii * ' F \(T). ; (T:) = f f^(t.)dt. f.(t.)dt. when T. =.?. (28b) ti i / ti' i' i ii f *i (T i' = 0 when T i <?. (28c) The expressin f. (T.) is btained by cmbining the errr distributin fr the i stage with the truncated distributin fr the sum f the previus stages. f.(t.) = F*,.(..).{T. -..)+ f.(t.-i')f*...(u) du tl* l' t (l-l)' l-l' XI 1 l-l' / XI 1 ' t (i-l) 5 i-l (29) -21-

31 This equatin has the same frm as Eq. (18), which was used t cmbine the tw distributins f errrs within a stage. As befre, the lwer limit f * integratin simply accunts fr the f. functin being zer when the argument is less than.. Als, since the distributins are independent i T. = T*.! + 3f. (30) 2 ^ti = * i*i\,r t(i-l) + <r xl (31) and, as in Eqs. (15) and (16) T* i = I. i F* ti (I.) ' i' + I f T. i f,.(t.) ti i dt. i \ CO (32) ^T Z - (Tf * - l/ F* (?.) + f (T.-T*) 2 f ti (T.) dt. (33) h If the rcket has N stages, f t»j(t N ) is the distributin f all the cmbined errrs fr the entire rcket. >cket. As in single stages, P is defined as P * = X f tn (T N) dt N (34) 'N tn The truncated frm f the errr distributin is used in the general case fr every stage, and when a cmmand shutdwn is nt used, is simply set at - r the cmputer is tld t ignre the truncatin prcess. Nte that the exhaustin shutdwn effect described in the latter part f Sectin II still applies t the case f a cmmand shutdwn. Exha--«tin ccurs whenever -22-

32 shutdwn perfrmance is nt achieved, and the perfrmance btained in this situatin can be significantly affected by the exhaustin effect ccurring in sme engines. Hwever, data n the exhaustin effect may smetimes nt be available fr engines shut dwn by cmmand. In the multistage case, the nly feasible methd f cmputatin is numerical integratin. The prcess is similar t that f the single stage except there are many mre levels f numerical integratin. Fr example, in evaluating the expressin fr f. (T.), each value f the integrand is evaluated by numerically integrating the f. functin fr the same stage and the f. functin fr the previus stage. With tw additins, the multistage parameters are the same fr each stage as thse used in the single stage. The scale factrs K. are included by entering the rati f <r. t K. int the prgram. This rati is btained frm the trajectry perturbatin analysis. The shutdwn perfrmance level is als included in the trajectry analysis, s that the difference between nminal perfrmance and shutdwn perfrmance is easily determined. Nw that the multistage cmputer prgram is available, it may be used fr the single stage by setting N =

33 f f * 1 1 BLANK PAGE

34 SECTION VI THE CHOICE OF BIAS Bias is an intentinal deviatin f the laded rati frm the nminal value f the burning rati, s that the mean value f X is different frm 1 and the mean value f S is different frm zer. A desirable bias increases the expected utage f the lighter cmpnent f prpellant and diminishes the expected utage f the heavier cmpnent. The fllwing criteria fr ptimizing bias have been suggested in previus studies, and methds fr using them t ptimize bias have been frmulated: a. Minimum mean utage b. Minimum mean square utage (minimum variance abut Z = 0) c. Maximum prbability that utage is less than a given value d. Minimum value f utage which is nt exceeded with a given prbability e. Maximum prbability that perfrmance is greater than a given value f. Maximum value f perfrmance which is exceeded with a given prbability Criterin a was used nly fr cmparisn with ther criteria in Refs. 2 and 4. Criterin b has had the widest use thus far. It is described in Refs. 1, 2, and 6. Bth f these criteria have the advantage that they carry ver frm the utage variable t the perfrmance variable in a single sta^e. Eq. (19) shws that minimum mean utage is equivalent t minimum mean perfrmance degradatin r maximum mean perfrmance. Als, the variance f X abut zer is equal t the square f the mean plus the variance abut the mean. X 2 = If (35) x ' ' -25-

35 Thus, frm Eqs. (19) and (20) P = Z 2 + «^ + <r^ = Z 2 + <r^ (36) Since the variance f W is independent f the bias, <r is a cnstant in Eq. (36) when the bias is varied. Therefre, when Z^ is minimized, X^ is minimized, and the mean square f the perfrmance variable is maximized. In spite f this advantage, Criterin b is nt a desirable criterin because it des nt necessarily maximize the prbability f missin success. Criteria c and d are related nly t utage and d nt cnsider perfrmance t ptimize bias. They may be used where the utage errr alne must be held belw a certain level. They are als useful as bunds in Criteria e and f as shwn belw. Since the assurance f missin success is why errrs and penalties are investigated in the first place, the nly really meaningful criteria are e and f. Usually, and especially in a man-rated missin, an acceptable prbability f success is chsen, and the paylad and missin requirements are tailred t fit that prbability. In this case, if we use the specified prbability f success. Criterin f wuld ptimize the bias. Hwever, smetimes perfrmance fr a specific missin may be critical and the decisin may be t maximize the prbability f achieving the required perfrmance. In this case, we emply Criterin e t ptimize the bias and use the assigned perfrmance level. Nte that if the prbability btained by Criterin e is equal t that assigned fr Criterin f, the tw criteria are equivalent, and the ptimum bias is the same. Hwever, if these prbabilities differ significantly, then the ptimum biases may als cnsiderably differ. Criteria c and d are similar t e and f but cncern themselves nly with utage. Hwever, they d have a useful applicatin, and they have the advantage that the ptimum bias is very easy t cmpute. Frmulas are derived in Ref. 5, but the exhaustin effects (6 terms) are nt -26-

36 included. Fr an alternate derivatin, nte in Figure 4 that when the utage ia less than snne specific value Z the value f S lies within a crrespnding interval frm Sj t S,. The prbability that the utage is less than the indicated value is equal t the integral f f (S) frm S. t S2- Since f s (S) is a nrmal density functin, this integral is a maximum when the mean lies halfway between S, and S?. This situatin must hld fr either Criterin c, when the value f Z is specified, r fr Criterin d, when the value f the integral is specified. Thus, if ne uses values f Sj and S-, btained frm Eq. (7) P = 4- (S.+S,) = T I - 6, - (1+R ) Z ' 2 12' 2 1 f ' n' m P ^» 421R /B-- R n/m )Z ( 6-6 J i' 1 + R n "R "" Z m n (37) where P is the ptimum ^ bias. If Criterin c is used, s that Z m is specified, r the final expressin f Eq. (37) is all that is needed t find the ptimum bias. When 5 and 6, are equal, the last term f the expressin is zer. If Criterin d is used, s that the prbability f success is specified, the length f the interval frm S. t S-, will be determined as a functin f the prbability f success. Frm the tables f the nrmal distributin, a multiple A f the standard deviatin f f (S) can be fund such that the integral f the functin frm -Acr t +Acr s will be equal t the required prbability. Thus the prbability is specified by specifying A. Then 1 + R 2A<r s = S, 21 - S. = 6 + = R Z mf' (1 + R nm ) Z (38) n Slving B fr Z m 2A(r Z = 2^ -R (39) m (1 + R n ) 2 27-

37 Figure 4. Distributin f S -28-

38 and substituting int Eq. (37) P = -rnr (A<r s- 6) + r (6-6 f) n when A(r s a 6 (40) P 0 = j ( f) whe " A- s ^ 6 where 6 is the average f 6 and 6. Once again, when the deltas are equal the last term is zer. When A is specified, this expressin determines the ptimum bias. One useful applicatin f Criteria c and d is the lcatin f bunds fr determining the bias by Criteria e and f. First we nte the tw extremes f the shape f the prbability density functin f the ttal errr f (X) [determined by Eq. (18)]. One extreme ccurs when cr =0, and the entire errr is just the utage errr. In this case, the density functin is highly skewed, lking like the example in Figure 3, and Criteria c and d are the same as Criteria e and f fr ptimizing the bias. At the ther extreme, the utage makes a negligible cntributin tward the ttal errr and the density functin is characterized by a nrmal distributin with zer mean. As this limit is apprached, the cntributin f utage t the variance f the ttal errr appraches the mean f the square f utage (frm Eqs. (35) and (36) with X appraching zer). Thus minimizing the mean square utage (Criterin b) appraches equivalence with minimizing the variance (and thus the standard deviatin) f a nrmally distributed ttal errr. This situatin is exactly equivalent t minimizing any given multiple f the standard deviatin f this same ttal errr and thus minimizing the allwable errr which is nt exceeded with any given prbability (as lng as the prbability exceeds 0.5). Therefre, in this extreme. Criterin b appraches equivalence with Criterin f, and thus als with Criterin e, fr any assigned prbability r perfrmance level. -29-

39 Actually, the shape f the ttal errr density functin lies smewhere between the extremes mentined abve. Fr example, the P curves in Figure 5 shw a gradual variatin as tr is varied fr a cnstant <r. Thus the ptimum bias accrding t Criterin e r f lies smewhere between that btained by Criterin b and that btained by Criterin c r d. prbability used fr Criterin d must f curse be the same as that t be used fr Criterin f. The If Criterin e is t be used, an assigned value f utage fr Criterin c is mre difficult t chse. The value shuld be ne which results in abut the same prbability f success as that expected frm Criterin e. guess shuld be adequate. Since accuracy is nt imprtant fr these bunds, a If the prbability f success invlved in these criteria is greater than abut 0.7, Criterin b will prvide the lwer bund and Criterin c r d will furnish the upper bund fr the ptimum bias. Criterin e has been discussed in Ref. 5 fr a single stage, including the derivatin f an implicit expressin fr determining the ptimum bias. Hwever, the expressin is very cmplex and must be slved by trial and errr fr each assigned value f the ttal errr. exhaustin effects and is useful nly in a single stage. Als, it des nt include The straightfrward apprach t bias ptimizatin is t try different values f the bias and see which value best meets the desired criterin. The cmputer prgram which calculates P Q has been mdified s that it can be used fr this purpse. Only the multistage prgram has been mdified because it is mre cnvenient even fr the single stage case (N = 1). In the frm used fr the multistage analysis, 7 scale factrs are intrduced which make the ttal errr X + Qfcr x equal t the actual decrement in the chsen measure f perfrmance. The riginal single-stage prgram did nt prvide fr these scale factrs. T ptimize the bias in a single stage, a set f pssible bias values spanning the regin between the bunds determined as described abve, is set int the prgram. Additinal inputs are a set f levels f the perfrmance 30-

40 decrement X -I- eta-,r a set f values f a which will generate a set f x' B values f X + /a if the first value f ß in the set is used. In either case, x the perfrmance decrement is held fixed at each value and used as the lwer limit f integratin in Eq. (34) t cmpute P fr all values f ß. Thus a set f curves f P versus ß may be pltted fr the different values f the perfrmance decrement. An example is shwn in Figure 7. On each curve, the minimum P the perfrmance represented by that curve. represents the maximum prbability f achieving If the ptimum ß fr each curve is pltted against the perfrmance decrement fr that curve, as in Figure 8, the ptimum bias fr any assigned level f perfrmance (Criterin e) can be fund by simply reading it ff the curve. Criterin f minimizes the perfrmance decrement that is exceeded with the prbability P a, fr P level f adverse errr. fr a given P is the prbability f being beynd a certain If ne bserves the curves f Figure 7, he can see the minimum perfrmance decrement is the ne whse curve is just tangent t the hrizntal line representing that P. Fr a lesser perfrmance decrement, the given P cannt be achieved. Since the pint f tangency is the minimum pint, the ß at the minimum pint minimizes the perfrmance decrement fr the P a at the same minimum pint. the P Therefre, at the mininnum pints are pltted against the ß at the same minimum pints as shwn by the brken line in Figure 7. The ptimum bias fr any assigned prbability (Criterin f) can be fund by simply reading it ff this curve. At the time f the analysis, if it is nt knwn whether Criterin e r Criterin f is t be used, bth curves (Figure 8 and the brken line in Figure 7) may be pltted. In multistage rckets the prblem is mre cmplex. each stage is nt independent f the ther stages. The ptimum bias fr An ptimum set f biases must be fund by cmparing hw different sets f biases affect rcket per^ frmance. Gemini Launch Vehicle. Tw versins f this apprach have been used in the tw-stage In the earlier methd (Ref. 4), the means and 31-

41 variances f the nnnrmal utage distributins are cmputed and then simply cmbined with thse f the nrmal distributin f ther errrs. The cmbined distributin is assumed t be nrmal. The bias is ptimized by minimizing the value f the mean plus three sigma with n knwledge f the actual prbability f success. In the later methd, effectively the same thing is dne, but the actual prbability is fund by cmbining the distributins. The cmbinatin is dne by a methd similar t that described in this reprt, but the numerical integratins are dne by hand, and the results are nt very accurate. Als, the criterin thus far is nt a valid ne, fr neither the prbability nr the perfrmance is set at a desired level. The result is still three si'rma, but the prbability f success is nt that assciated with three sigma fr a nrmal distributin ( ). Minimum perfrmance and the prbability assciated with it are bth accepted at whatever values they turn ut t be. The prcess wuld have t be repeated fr different multiples f sigma t use Criterin e r f. Als, hw exhaustin shutdwn affects the utage distributin has nt been cnsidered in the Gemini Launch Vehicle. Since the first stage is essentially the same as that f Titan III series, the effects described in Ref. 6 shuld still be present. In the slutin f the multistage prblem, the bunds cmputed fr the individual stages will still be applicable if nne f the stages uses a cmmand shutdwn. The bund generated by Criterin b still hlds because it maximizes the mean square perfrti-rnce fr the entire vehicle when applied t each stage, s that the effects c, ihe individual biases are independent in this case. This fact can be seen frm Eqs. (30) and (31), with the asterisks nt being applicable because there is n cmmand shutdwn. As the number f stages increases, the cmbined distributin becmes mre like a nrmal distributin accrding t the central limit therem. Therefre, the ptimum biases becme clser t thse btained by Criterin b than they were fr the single stages. This puts them even further inside the bunds generated by Criterin c r d than fr the single stages. But a cmmand shutdwndistrts the -32-

42 cmbined distributin f all stages up t the shutdwn. distrtin, the generalizatins frm a single stage d nt hld. Because f this Hwever, the "bunds" cmputed fr the single stages will still prvide a rugh indicatin f the regins within which the biases shuld be investigated. The prcedure fr ptimizing the multistage biases is simply an extensin f that used in the single stage. A set f biases spanning the regin f interest is inserted int the prgram fr each stage, and P is cmputed fr all cmbinatins f the biases at each level f perfrmance decrement. The results are pltted in whatever way is mst cnvenient t find the ver- all minimum P, and the cmbinatin f biases that resulted in the minimum r P fr each level f perfrmance decrement. An example fr a tw-stage vehicle at a single level f perfrmance decrement is shwn in Figure 9. A curve f P versus p. is pltted fr each p?. Then the minimum values f P lr these curves are pltted versus fl, in Figure 10. The minimum P r r ^2 s a in this curve is the ver-all minimum, and the crrespnding abcissa is the ptimum ß,. As shwn, the ptimum ß. is apparent frm Figure 9. fr it is almst independent f ß, within the regin f values cnsidered. Where the dependence is greater, a curve thrugh the minimum pints f the curves f Figure 9 culd be drawn. an abcissa equal t the ptimum ß.. Its minimum pint wuld ccur at As an aid in drawing this curve, ntice that its minimum value is the same as that f the curve f Figure 10. P,. In a three-stage vehicle, this prcedure wuld be carried ut fr each Thus a set f sets f curves like thse in Figure 9 and a set f curves like the ne in Figure 10 wuld be pltted. The minimum values f P fr the Figure 10 curves wuld then be pltted versus ß, n a separate page. The minimum pint f this new plt wuld be at the ptimum ß,. P, wuld be fund frm the Figure 10 curves, as the ptimum p. was fund frm the Figure 9 curves in the tw-stage case. If the ptimum The ptimum p. is sensitive t p, and p,, a curve thrugh the minimum pints f each set f Figure 9 curves culd he drawn, and the minimum pints f the -33-

43 resulting curves culd then be cnnected by a single curve. The abcissa f the minimum pint f the single curve wuld be the ptimum p.. The methd may similarly be extended t mre than three stages. If mre than ne level f perfrmance decrement is cnsidered in the bias ptimizatin, as it frequently will be, the prcedure described abve is carried ut fr each f a set f levels cvering the regin f interest. The ptimum bias fr each stage may then be pltted versus the perfrmance decrement as was dne in the single stage. Frm these curves, if ne uses Criterin e, all f the ptimum biases may be fund fr any desired level f perfrmance. Als, the ver-all minimum value f P fr each perfrmance level, btained mst accurately frm the single curve used t ptimize the bias f the last stage, may be pltted against the ptimum bias fr each stage. Frm this set f curves, if ne uses Criterin f, all f the ptimum biases may be fund fr any desired prbability f success. -34-

44 SECTION VII EXAMPLES OF RESULTS A. SINGLE STAGE CASE The -jrrr distributin in a single stage is investigated fr a Titan III first stage. A nminal set f parameters is taken frm Ref. 9, except the deltas are taken frm Ref. 6. The bias in Ref. 9 was chsen n the basis f minimum mean square utage (Criterin b f Sectin VI). mre realistic bias will be shwn in later examples f results. The effects f a Plts f P versus a are shwn in Figure 5 fr a number f values f -. The curve e w labeled - =00 represents a pure nrmal distributin. The pints n the curve are taken frm the tables f the nrmal distributin functin. cmputer prgram described in Sectin IV was used t cmpute the pints n the ther curves. The curve labeled a =0 represents a pure utage distri- butin with n ther cntributing errrs. The value f cr turns ut t be The value f a will be different fr different versins f Titan III, w but will generally be abut the same as - w'.thin a factr f tw. Thus, the curve labeled - = is a typical example. The curves f Figure 5 can be interpreted tw ways. The The errr distri- butins may be cmpared with a nrmal distributin either by chsing a fixed multiple f the standard deviatin and cmparing the crrespnding prbabilities f failure r by chsing a fixed prbability f failure and cm- paring the multiples f the standard deviatin that crrespnd t that prba- bility. Since the errr analyses t date have almst always used 3<r(a = 3) as the maximum allwable errr under the assumptin that the distributin is nrmal, cmparisns with this pint n the nrmal distributin curve are taken as a useful example. bility P Nte that fr the nrmal distributin, the prba- that the errr will exceed the mean plus 3cr in the adverse directin is ( percent chance). This value appears t be the prbability cunted n when a nrmal distributin is assumed. Hwever, fr a nnnrmal -35-

45 Figure 5. Prbability (P ) Versus Deviatin Frm Mean Errr; Titan III, Stage 1 36-

46 distributin, the prbability is different. Fr example, if <r were equal t , the prbability f exceeding the 3- level wuld actually be (0.59 percent chance). By ur lking at it anther way, we see that if the prbability usually identified with a Scr errr is required, then the allwable errr which crrespnds t that prbability is actually 3. Sa. The situatin may be better r wrse than this example depending n the actual value f O". In any event, the use f 3(r as a specified minimum perfrmance level w ^ is nt justified. A detailed analysis is required in each case t determine the maximum allwable errr as a functin f prbability f failure. B. MULTISTAGE CASE The multistage analysis has been used fr Titan IIIC, Vehicle 11. Sme f the parameters were btained frm the Titan III prgram ffice, and the thers were taken frm Ref. 10. The plt f P versus * is shwn r Of in Figure 6. Because the secnd stage is shut dwn by cmmand at a level f perfrmance far belw the nminal level s that the prbability f achieving the shutdwn level f perfrmance is very high, a single-stage cmputatin fr the third stage nly was made fr cmparisn with the multistage case. The resulting curve is indistinguishable frm that repre- senting the multistage case, which shws that the shape f the distributin f ttal perfrmance errr is the same. Als, the means and variances f the tw distributins turn ut t be equal within a little mre than ne-tenth f ne percent. Therefre, the perfrmance f the ver-all vehicle may be assumed t be independent f the perfrmance f the first tw stages when the secnd stage is shut dwn at the chsen level f perfrmance. Since the errrs in stage three can be cnsidered independently, the bias fr this stage may be ptimized separately using the single-stage pti- mizatin prcedure. first tw stages. stages is nt a matter f chice. Then a tw-stage ptimizatin may be used fr the Als, the criterin fr bias ptimizatin f the first tw One simply maximizes the prbability f achieving the shutdwn level f perfrmance (Criterin e). -37-

47 Figure 6. Prbability (P ) Versus Deviatin Frm Mean Errr; Titan IIIC, Vehicle

48 Fr the third-stage bias ptimizatin, the single-stage prcedure utlined in Sectin VI is used. Fr the lwer bund f the magnitude f ß, is used. It is the value cmputed by the cntractr using Criterin b. Fr the upper bund, if we assume a maximum allwable utage f 1.59 percent (taken frm the figures f Ref. 10). Criterin c [Eq. (37)] yields a bias f abut If we assume A = 3.2 in rder t crrespnd apprximately t the prbability f being utside the interval f S, Criterin d [Eq. (40)] yields a bias f abut On the basis f these estimates, the biases chsen fr the ptimizatin prcedure are , , , , , and The perfrmance decrements chsen are 400, 500, 600, and 700 fps, which crrespnd t values f a frm less than 2 t almst 4. Curves f P versus ß fr the different perfrmance decrements are shwn in Figure 7. A curve f ptimum bias versus perfrmance decrement, pltted frm the minimum pints f Figure 7, is shwn in Figure 8. Frm this curve, the maximum prbability f achieving any specified level f perfrmance (minimum P -- Criterin e) can be fund. The brken line in Figure 7 shws a curve f minimum P versus bias, als pltted thrugh the minimum pints. Frm this curve, the maximum perfrmance (minimum perfrmance decrement) that can be achieved with any specified prbability f success (1 - P ) can be fund. The fllwing examples f the uses f Criteria e and f with reference t Figures 7 and 8 are illustrative, thugh nt necessarily realistic. Suppse first that a critical perfrmance level 550 fps less than nminal must be achieved with maximum prbability f success (Criterin e). Frm Figure 8, the ptimum bias is seen fr this situatin t be abut Nw suppse instead that a certain prbability f success is required and that the perfrmance t be achieved with that prbability is t be maximized (Criterin f). By ur referring t the dashed line in Figure 7, we see if the required prbability is t be percent (P = , crrespnding t three sigma fr a nrmal distributin), then the ptimum bias is abut On the ther hand, if nly 99 percent cnfidence is required (P = 0,01), the ptimum bias is abut

49 Figure 7. Prbability (P ) Versus Bias; Titan IIIC, Stage 3-40-

50 ß PERFORMANCE DECREMENT X + dff,,, fps Figure 8. Optimum Bias Versus Perfrmance Decrement; Titan I1IC, Stage 3-41

51 In general, as a higher prbability f success is needed, a larger negative bias is required. Hwever, the maximum perfrmance achievable with the required prbability is lwer. Where hard requirennents fr either perfrmance r prbability are nt present, a tradeff may be cnducted t determine the best cmprmise between perfrmance and prbability. Figures 7 and 8 prvide all the infrmatin needed fr such a tradeff. Remember that nne f these ptima is critical in either perfrmance r prbability. Althugh the ptima may be fund t the fifth decimal place in this prblem, they are nt accurate beynd the furth because f the graphical slutin. Only the first three decimal places are imprtant, as can be seen by the flatness f the curves in Figure 7 near the minima. Optimizatin f the biases in the firf t tw stages is less imprtant than fr the third stage because f the high prbability f achieving shutdwn velcity at the end f secnd-stage burning. Hwever, the pr^ ibility f achieving this shutdwn velcity can be maximized by chsing the best cmbinatin f biases. The shutdwn velcity represents a perfrmance decrement f fps with respect t nminal perfrmance at stage-tw burnut. This value is the nly ne used. The lwer and upper bunds fr the bias, based n the same criteria as in the third stage are the fllwing: and fr the first stage, and and fr the secnd stage. Frm these figures, the values chsen fr ß. are , , , and Thse chsen fr ß 2 are , , , and Curves f P a versus ß, fr the fur values f ß? are shwn in Figure 9. The minimum pints f these curves, which are essentially the values at ß. = , are pltted versus ß 2 in Figure 10. The ptimum value f ß? crrespnds t the minimum pint f this curve, which is abut The ptimum value f ß. is seen frm Figure 9 t be almst independent f ß in the regin f interest. Since all the minima ccur at apprximately , it is the ptimum value. -42-

52 ITEM STAGE 1 STAGE 2 R <r , <Tm K 5.502XIO XK)" 5 T + Off, ß VARIABL E VARI ABLE 0.02 ßi* ^ ^ ^^ ^z -- zsss^^ ^ ßs Figure 9. Prbability (P^) Versus First Stage Bias Titan IIIC, Stages 1 and 2-43-

53 8 S s V 01 «tn 1 O rj t 3 ^ in u V im > rt 0? OT g U 1 ( d» 1 fc c.s I«2 H d Ö M d CM a I _ a -44-

54 By ur using the ptimum values fr the biases in the first tw stages, a P abut , r a prbability f success abut percent, can be achieved. If we use the biases chsen by the cntractr accrding t Criterin b, P is , r the prbability f success is abut percent. Therefre, in this particular case, the mre sphisticated bias ptimizatin is nt very helpful. Hwever, where higher prbabilities are invlved, significant imprvement ver Criterin b can be achieved. Ntice hw bias affects the frm f the errr distributin. As the magnitude f the bias increases, the distributin becmes mre like a nrmal distributin because a greater prtin f the f (S) functin lies cmpletely t the left f the nnlinear prtin f the utage functin (see Figure 4). Figure 11 repeats the curve f Figure 6, wmch shws the errr distributin fr Titan II1C, Vehicle 11, with a third-stage bias f This value used by the cntractr is based n Criterin b. Fr cmparisn, tw additinal curves are pltted using different third-stage biases. As discussed previusly in this sectin, the bias f represents an ptimizatin based n a required prbability f success f 99 percent and the bias f represents an ptimizatin based n a required prbability f percent. This prbability is assciated with the three-sigma level f a nrmal distributin. The mst significant fact t be learned frm Figure 11 is that when the bias is ptimized fr a high prbability f success s that a large (negative) bias is used, the cmbined errr distributin is clser t a nrmal distributin than when less bias is used. The distributin is still significantly nnnrmal, having a prbability f exceeding three sigma abut as cmpared with fr the nrmal distributin. But when the bias based n minimum mean square utage is usi_d, the prbability f exceeding three sigma is abut If we lk at it the ther way, with the high-prbability bias it takes abut sigma t give the same prbability f failure that three sigma gives fr a nrmal distributin; hwever, with the minimum mean square utage bias it takes abut 4.23 sigma t give that prbability. -45-

55 Figure 11. Prbability (P ) Versus Deviatin Frm Mean Krrr; Titan IIIC, Vehicle

56 SECTION VIII CONCLUSIONS AND RECOMMENDATIONS A. CONCLUSIONS The fllwing three cnclusins are reached frm this investigatin f the effects f utage n perfrmance statistics fr biprpellant rckets: a. The ver-all distributin f perfrmance errrs f a biprpellant rcket can be significantly nnnrmal. b. The prbability assciated with the achievement f a given perfrmance fr a biprpellant rcket can be significantly less than that btained by assuming the distributin t be nrmal. c. If a high prbability f success is required, similar t that assciated with three sigma fr a nrmal distributin, a prpellant lading bias significantly greater than that based n maximum mean square utage may be needed. B. RECOMMENDATIONS After cnsideratin f the reprt text and cnclusins, these three recmmendatins are suggested: a. Perfrmance requirements fr biprpellant rckets shuld be specified in actual prbability f success rather than as a multiple f sigma. b. Fr each nminal vehicle, an analysis f the ttal errr distributin shuld be made using the multistage cmputer prgram described in the Appendix r its equivalent. c. Prpellant lading bias shuld be chsen accrding t prcedures similar t thse described in this reprt t insure maximum perfrmance fr a required prbability f success r maximum prbability f success fr a required perfrmance. -47-

57 V BLANK PAGE

58 APPENDIX MULTISTAGE COMPUTER PROGRAM by Philip H. Yung Cmputing and Data Prcessing Center Aerspace Crpratir. A. GENERAL DESCRIPTION The cmputer prgram carries ut the numerical implementatin f the analysis in the preceding sectins. The main task is t perfrm accurately and efficiently the numerical integratins necessary t btain P. T d this, the Legendre-Gauss numerical integratin -was determined t be the mst apprpriate methd (see Ref. 11). T d an N-stage analysis an (N +1) level multiple integratin must be dne (tgether with several integrals f lwer level). The time required increases an rder f magnitude fr each level f integratin. It was therefre thught best t vary the number f pints in the integratin technique at each level f integratin. This enables the us«" t select the rder f apprximatin that gives the accuracy needed cmmensurate with the number f stages and the amunt f cmputer time available. B. NUMERICAL INTEGRATION Since the Legendre-Gauss numerical integratin methd requires that the limits f integratin be finite and that the upper limit be +1 and the lwer limit be - 1, all the infinite integrals invlved had t be truncated at sme finite value, and all the integrals transfrmed t the interval (-1, 1). Because the integrands invlved were prbability frequency functins they culd be truncated at sme psitive r negative multiple f cr frm the mean, where the area utside that pint culd be assumed t be negligible. The multiple was determined quickly by trial and errr t be abut nine fr the functins that have been tried. -49-

59 At each level in the numerical integratin prcess a different rder f apprximatin can be used. The rder here is the number f integrand values used in the numerical integratin. The chices f the rder are 2, 4, 8, 16, 20, 24, 32, 40, and 48 and the values f the abscissas and weights fr all these rders were btained frm Ref. 11. A cnvenient check n the chice f the multiples f tr and the rder f apprximatin is t calculate P fr a = - which shuld be 1, because P is a prbability functin. This calculatin als gives a rather lse but useful check n the accuracy that can be expected fr finite values f a. Clearly, the area under the ttal curve can be n mre accurate than the area under sme subset f the curve fr the functins invlved and fr a given rder f apprximatin. The prgram autmatically calculates P_, and prints the value btained. C. RESULTS The greatest number f stages attempted with the prgram was three. Therefre a quadruple integral was the mst cmplicated evaluatin t be dne. Satisfactry results were btained using a multiple f nine fr all infinite limits. The rder f apprximatin at each level f integratin was the fllwing: Level Integral 1 48 (innermst) P 16 (utermst) These rders f apprximatin gave P accurate t tw significant figures. They gave P. - P^ accurate t n wrse than fur significant figures. This accuracy was determined by perfrming the fllwing apprximatin: -50-

60 Level Integral 48 (innermst) (utermst) The tw results are then cmpared fr r = 1.5. The accuracy f P fr a = 2,.. 5 will be n wrse than the accuracy f P fr r = 1.5. D. DESCRIPTION OF DATA Table I describes the inputs t the cntrl card. Table I. Cntrl Card, Data Set 1 Prgram Symbl Mathematical Symbl Definitin NZ N Number f stages 8 z NA N Number f entries in ALPHA vectr a s I 0 Order f apprximatin t first level integrals 11 l i Order f apprximatin t secnd level integrals 12 I 2 Order f apprximatin t third level integrals 13 I Order f apprximatin t furth level 3 integrals 14 h Order f apprximatin t fifth level integrals 15 I Order f apprximatin t sixth level 5 integrals 16 h Order f apprximatin t seventh level integrals 17 I 7 Order f apprximatin t eighth level integrals 18 I 8 Order f apprximatin t P integral level -51-

61 Specific values f IJ frm 1 t 9 in the rder given in Table I are furnished in Table II. Table II. Values f IJ (J =0,, 8), Data Set 1 IJ Order ] The infrmatin in Table 111 shws the case t be run has three stages, 10 ALPHA'S; the rder f integratin f level ne integrals is 48, f level tw integrals is 24, f level three integrals is 24, and f the P level integral is 16. Table III. Frmat f Card, Data Set 1 Clumn Item Example 1-3 NZ NA j

62 Table III. Frmat f Card, Data Set 1 (Cntinued) Clumn Item Example ^ nt used when NZ = Table IV. Vectr Data, Data Set 2 i = 1,, N and j = 1, z. N. Prgram Symbl NT Mathematical Symbl ^ Definitin Multiple f -'p. fr cmputing infinite limits fr T. integrals. If N T. SO, ci = Ciin«^. If N T < 0, Ci = c t input 1 /a-r,. 1 N T f fs always used in the calculatins. NX s Multiple f cr fr cmputing infinite limit fr x. integrals, i BETA SIGL Pi '\ Bias n a randm variable. Standard deviatin f s. DEL0 6. i Oxidizer utage limit n s. DELF 6 F. i Fuel utage limit n s. RB V Burning rati. -53-

63 Table IV. Vectr Data, Data Set 2 (Cntinued) Prgram Mathematical Symbl Symbl Definitin NL sw c N \ X c i Multiple f (rj^. fr cmputing infinite limit fr Z. integrals. Standard deviatin f w.. If a^. ^ 0, - w. = - ifcr <0,- l = - l l.(r. w.. t w. w. w. z. iinput i ii i Multiple f -'p- fr cmputing cutff pint fr T. integrals. SCAL1 k Input scale factr. Ifki. > 0, k? =ki./l. w.; i Ifkt. =0, k 2. = 1; ifkt.^0, k l l 2i = ik^i. ALPHA Of r L If ALPHA a 0, magnitude f ALPHA = a.; if J ^j ALPHA < 0. magnitude f ALPHA = L^ J. j The card frmat fr each f the quantities in Table IV is as fllws (each vectr may be cntinued n mre than ne card): Table V. Frmat f Card, Data Set 2 Card Clumn Entry {all quantities are right justified) ±.XXXXXXXXE±XX ±.XXXXXXXXE±XX ±.XXXXXXXXE±XX ±.XXXXXXXXE±XX The rder f the deck f vectrs shwn in Table VI is the same as the rder f the defining table (Table IV). -54-

64 Table VI. Number Data Deck Setup (ne case) Data Item 1 Cntrl Card 2 N T Vectr 3 N z Vectr 4 BETA Vectr 5 SIGL Vectr 6 DEL«!) Vectr 7 DELF Vectr 8 RB Vectr 9 NL Vectr 10 SW Vectr 11 C Vectr 12 SCAL1 Vectr 13 ALPHA Vectr Fr the CDC 6600 Chippewa versin 1. 1 FORTRAN IV Cmpiler the deck setup is shwn in Table VII. Table VII. Number Prductin Deck Setup Item System accunting cards (Aerspace nly) RUN(S) Card (0TAN2. Card 7 8 Card 9 55-

65 Table VII. Prductin Deck Setup (Cntinued) Number Item FORTRAN Surce Psck 7 8 Card 9 Data deck Card 9 As many cases as desired can be run with ne pass n the cmputer. E. DFSCRIPTION OF OUTPUTS (see Table IX fr sample utputs) All vectr quantities rü r- printed in the rder left t right, tp t bttm alng the page. Output set 1 - Inputs All the input cards are printed directly under the wrk INPUTS n the first utput page. The rder and frmat f these quantities is the same as the card rder and frmat. The utput quantities in Table VIII are printed directly under the wrd OUTPUTS n the secnd utput page. 56-

66 Ttble VIII. Outputs, Output Set 2, N and j = 1. z. KL Mathematical Data Item Symbl Definitin 1 Mean f z. \ i 2 Standard deviatin f z. ^ 3 k. Scale factr fr z. i X«Ti I ^ Area under ttal f (z) curve + F (0) Z i Z i Standard deviatin f Wj Mean f T. Standard deviatin f Tj 8 Infinite upper limit fr T. integrals 9 -"T. 10 T i + c i -T. Infinite lwer limit fr T. integrals Cutff pint fr T. integrals 11 P - Area under f_(t) J J Lwer limit fr evaluating P P(T N * T N + '"T N T > Each data item is rdered frm left t right, tp t bttm. When the item cnsists f a vectr it has the same number f entries as the rder f the vectr. The first item (item number ne) is the z. vectr, etc. Table IX illustrates the utput frmat. -57-

67 I';:," a B d Cu 3 e ^ hj m ^ ui ik» e & e e e «nir>^tf)«^ire hj UJ e c e e e r» lu e UJ bj UJ c m«f)^em^il>e 0 1 UJ c 9 CD M mm t CD <>» Ui 0>m O c CD rsi» mm W 9 $ UJ m e m fw tf> # tn < UJ If) fn r» CM lo tf> «CD K (W e UJ c e Ui c e e u; IM > 9 mm 9 O UI (Si 9 CD e UJ c c UJ e & «Ui tf) fv Ui If) (Si f ōrvi fv r- in UI mm a m a u la a m mc M M rsj a- < CD # «# ri «- n CM e UI M CD (Si e (V CD m 99 O «u r»i e»- O O Ui 9 CD in 90 9 m 9 1 fs r tf. p- I Ui 9 9 in 90 9 in CD 9 1 (T) H 9 m UI UI e e e Ui Ui Ui e c c m in «^c m»i-«it e c UI UI e e Ui e «UJ U' u; e e c c,.4*ii4^ee^«^^ c ccccccc ccccc UiUJUJUil^UiUiUJi^UiUJUilUUIUJUJUiUlUl UiUJUiUi ^-lnln f^i<moo^«^^ (Ca0C< 9f^ ^ f^cdi>j OfXJO'# <y 00^^00 ^<c^i<#fm<vmnnjin <^<vj c mfn99cccv>c9 M -#in^ m cv 9 9 9f\,CVOC 9 9C C9 9 9tmm ir rtmm (t cdirit^^^c ^9 ^i^9irminif>^ ^9 IV iv ^ m c c #-^cfv««c ir*c 9 t' 9 r ^ «'iv 9 ^KCtT'# nn^ m9in9 in O 9'ir r«c C'C*C C 9 ^ IV 9t^*C < <J^»9 9 mm ^^ ^trnj^^^irir m- «ir <m mmcvicvjjfxj 99 9 f*»r* J> fm. f\jfmf*>j>fvi K r^rvrvr^ <v. < cvr^ n* «^icvo C<90 til* mtf'omcir^tr 3 a 3x O 4Wtm<e<cr~»*iMri -58-

68 REFERENCES 1. D. G. Stechert, The Effect f Outage n Missile Perfrmance, 494/ (Denver: Martin Crp., I960). 2. David W. Whitembe. Optimum Prpellant Lading and Prpellant Utilizatin Techniques, TDR-594(1114)-TN-2 (El Segund: Aerspace Crp., 1 May 1961). 3. M.C. Tangra, Truncated Nrmal Distributirs and the Prbability Distributin f Outage, Reprt (El Segund: Aerspace Crp., 10 September 1962). 4. M. C. Tangra, Fuel Bias Optimizatin fr the Gemini Launch Vehicle, TDR-169(3126)TN-1 (El Segund: Aerspace Crp., 16 Octber 1962). 5. D. G. Stechert, Launch Vehicle Perfrmance Statistics, TM0495/ (Denver"! Martin Crp., July 1964). 6. G. Presta, Effect f Prpellant Exhaustin Shutdwns n Outage - Titan III, Stage 1, TM-III-66-1 (Denver: Martin Crp., March 1966). 7. Harald Cramer, Mathematical Methds f Statistics, Sectin (Princetn, N. J."! Princetn University Press; 195 1). 8. S. Sugihara, The Statistical Characteristics f a Nn-Gaussian Perfrmance Variable, TM-66( )-30 (El Segund: Aerspace Crp., 2 May 1966). 9. Critical Perfrniance Weight Errr Analysis and Fluid Lad Reprt, Reprt SSD-CR (Denver: Martin Crp., April 19bb). 10. Flight Plans VII, VIIA. and VIIB Perfrmance Margin Analysis, TM 5141/ (Denver: Martin Crp., March 1966). 11. P. Davis and P. Rabinwitz, "Abscissas and Weights fr Gaussian Quadratures f High Order," Jur Res Nat Bur Stan, 56 (1) (January 1956). -59-

69 UNCLASSIFIED Security Classificatin DOCUMENT CONTROL DATA - R&D fsmcurtly etmmmhieatin t tttlm. bdy l mbatrmet mrtd tndmxing mnnlmtin mumt bm nffd vtbmn thm vmrmll reprt im clmsmihd) Olir.lNATIN O ACTIUI^V (Crprmlm mulhr) Aerspace Crpratin 1 Segund, Califrnia * noomt TITLE 2«. MCPORT SCCURITV CLASSIFICATION Unclassified 2b GROUP THE EFFECTS OF OUTAGE ON PERFORMANCE STATISTICS FOR BIPROPELLANT ROCKETS «DKSCniPTIVC NOTES (Typt l rprl and Inclutln dlmrn) S AUTHORfS; flmat nmmm. lint nmmm, Inltiml) Durux, Jseph W. S REPORT DATE July «CONTRACT OR GRANT NO. F PROJCC T NO. p 7«TOTAl. NO OF RACES 62 9» ORIOINATOR'S REPORT NUMBERfSj TR-0158{ )-l JO. OF REFS 11 9b. OTHER REPORT NOfSJ (A ny th*r numbmrm thmt mmy bm mmigrfd thl» rmprt) 10 A VA IL ABILITY/LIMITATION NOTICES This dcument has been apprved fr public release and sale; its distributin is unlimited. II. SUPPLEMENTARY NOTES 12 SPONSORING MILITARY ACTIVITY SPACE AND MISSII SYSTEMS ORGANIZATION AIR FORCE SYSTEMS COMMAND tg AngeleB, Califrnia II ABSTRACT The distributin f perfrmance errrs in biprpellant rckets is determined as a functin f engine and prpellant lading parameters, and cntributing errrs. The distributin is fund t be highly skewed, s that the prbability f exceeding three sigma can be significantly greater than it wuld be if the distributin were nrmal. Thus, three sigma as a minimum perfrmance limit is nt justified. A detailed analysis is required in each case t determine the relatinship be- tween perfrmance margin and prbability f missin success. methd f ptimizing the prpellant lading bias is develped. Als, a realistic The methds previusly used have resulted in fuel biases that are usually much lwer than the ptimum. Cmputer prgrams have been develped t perfrm the analyses described in this reprt, bth fr the single-stage and the multistage case, r DD F0RM 1473 (FA CSIMI l_e) UNCLASSIFIED Security Classificatin

70 UNCLASSIFIED Security Classificatin KEY WORDS Outage Errr Distributins Perfrmance Statistics Biprpellant Rckets Perfrmance Margin Skewed Distributins Prbability Cmputer Prgrams Prpellant Bias Abstract (Cntinued) UNCLASSIFIED Security Classificatin