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AD907326

NEW LIMITATION CHANGE

TOApproved for public release, distributionunlimited

FROMDistribution authorized to U.S. Gov't.agencies only; Administrative/OperationalUse; NOV 1972. Other requests shall bereferred to Advanced Ballistic MissileDefense Agency, Huntsville, AL 35807.

AUTHORITY

ABMDA/AL ltr, 1 May 1974

THIS PAGE IS UNCLASSIFIED

AD072 -. W~~f~l'9ieI 'Ale'JIJL A SA~Tht''ir

- Y N I

Te "'Tht 4L

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SCOMPUTE~RcRg FOR tN

AND PE--RFO ,R M NdE ANAY..~SOID PROPELN RcTMOTT'OR

November 1972-1

FV I

B ROWN ENGIN EERING ,5,-a,

ATELEDYNE

BROWN ENGINEERINGRESEARCH PARK

HUNTSVILLE, ALABAMA 35807

-(205) 536-4455JWX (810) 726-2103

February 7, 1973Ditibution limited to U.S. Gov'IN agenctes onTn-

z A1luation; 1 2 FEB 1973. Other requestsf-Ll1s documont -must be refOrrold to

Director AecAdvanced Ballistic Missile Defense AgencyDepartment of the Army

P. 0. Box 1500Huntsville, Alabama 35807

Attention: RDMH-S/Mr. Carmichael

Subject: Contract DAHC60-69-C-0037

Dear Sir:

In accordance with Sequence No. G003 -of the DD Form 1423 for thesubject -contract, Teledyne Brown Engineering is submitting seven (7)copies of its Technical Report MS-ABM.DA-1683, "Computer Programfor Sizing and Performance Analysis of Solid Propellant Rocket Motors".This report was verbally approved. Distribution is being made per theDD Form 1423.

Sincerely,

BROWN ENGINEERING COMPANY, INC.

Martha H. TeenorContract Manager LJ D

MHT/wjc !FEB 12

Distribution: See Page E2n s

i

Advanced Ballistic Missile Defense AgencyAttention: RDMH-S /Mr. CarmichaelFebruary 7, 1973Page 2

Distribution: DirectorAdvanced Ballistic Missile Defense AgencyDepartment of the ArmyCommonwealth- Building1320 Wilson BoulevardArlington, Virginia 22209Attention: RDMD-AO (1 copy)

Defense Documentation CenterCameron Station-Alexandria, Virginia 22314Attention: Documentation Center :(2 -copies)

Commanding GeneralU. -S. Army Safeguard System CommandDepartment of the ArmyP. -0- Box 1500Huntsville, Alabama 35807Attention: SSC-CS -(w/o copy)

Defense Contract Administration Services =OfficeFifth Floor, Clinton Building-210.9 West Clinton-Avenue

Huntsville, Alabama 35805Attention: Mr. D. W. VanBrunt (w/o copy)

f1I

f

, - -

TECHNICAL REPORTMS-ABMDA-1683

COMPUTER PROGRAM FOR SIZING-AND

PERFORMANCE ANALYSIS OF SOLIDPROPELLANT ROCKET MOTORS

L_ By

j J. L. Thurman

November 1-972

1Prepared For

-U.S. ARMY ADVANCED BALLISTIC MISSILE DEFENSE AGENCYDEPARTMENT OF THE ARMY

HUNTSVILLE, ALABAMA

FContract No. DAHC60-69-C-0037

fPrepared By

MILITARY SYSTEMSTELEDYNE BROWN ENGINEE RING

HUNTSVILLE, ALA-BAMA

i

ABSTRACT

IA FORTRAN digital computer program was developed which is

generally applicable for preliminary design tradeoff analyses of solid-

propellant rocket-motors for a variety of applications. The pro-

i gram computes rocket motor length, propellant and inert component

weights, mass fraction, burn-time, specific impulse, -ideal stage burn-

jout velocity, and -other performance parameters as a function of -input

motor diameter, chamber pressure, thrust, payload- mass, propellant

i ballistic properties, propellant- web fraction, port-to-throat area ratio,

and -other required-parameters. Either conical or contoured nozzle

I exit geometries -may be analyzed. Options are available for considering

a head-end propellant web and inert residual slivers.

I

SIo Approved: Approved:

G.R. Cuinn, Ph.D. -S. M. Gilbert, Ph.D.Manager ManagerMissile Requirements and System Synthesis DepartmentTechnology Branch

Approved:

. ,__ _ _ _ _ _ _ _ _ _ __.

1-. A. Matheny (Deputy-Program Management

TABLE OF CONTENTS

Page

1. INTRODUCTION ............ . . . . . . . . . . 1-1

2. PROGRAM CAPABILITIES ................. .2-1

3. DESIGN ANALYSIS.. ................. 3-1

3.1 Internal Ballistics . . . . . . ... . . . . . 3-1

1 3.2 Design of Inert Components . . . . . . . . . 3-9

1 3.3 Rocket Motor Performance Charactemization 3-27

4. PROGRAM APPLICATION .............. 4-1

4.1 Methodology ..................... 4-1

4.2 Sample Problem . . . . . . .......... 4-4

5. PROGRAM OPERATING INSTRUCTIONS .......... 5-1

5.1 Input Data .................... 5-1

5.2 Stored Data 5-5

6. REFERENCES ................... 6-I

APPENDIX A - FORTRAN LISTING Or SIZINGPROGRAM . ............... .. A-1

APPENDIX B - DESCRIPTION OF MISSILE OPTIMIZATIONj PROGRAM (MOP) . ............. B-i

APPENDIX C - FORTRAN-LISTING OF MISSILEOPTIMIZATION PROGRAM (MOP)- C-1-

II

- U ' iii

LIST OF ILLUSTRATIONS

Figure Title Page

3-1 Motor Case Nomenclature .......... .. 3-10

3-2 Conical Nozzle Nomenclature ......... . . 3-1l

3-3 Contoured Nozzle Nomenclature . . . . . . 3-12

3-4 Equivalent Conical Nozzle Expansion RatioIas a Function of Contoured Nozzle Expansion|Ratio . . . o. . . . . . . . . . . . .. . . . 3-23

3-5 Lcont/Lcon as a Function of ContouredNozzle Expansion Ratio . ...... . . . . . . 3-24

-4-1 General Arrangement of Loaded TX-354-3Rocket Motor Chamber ............. 4-5

[ 4-2 TX-354-3 Rocket Nozle Assembly 4-6

-4-3 Sample Problem Solution .......... . 4-7

-5-1 Sample Input Coding Sheet ............. . . . 5-6

LIST OF TABLES

i Table Title Page

4-1 Program Application to Typicalj Design Problems ............. ......... 4-2

5-1 Input Variable Definition for Cards 2f through 28. . . . . . . . . . . . . . . . . 5-2

5-2 Typical M aterial Physical Properties andEquivalent Program Variables-. ......... 5-8

1 iv

LIST OF SYMBOLS

Symbol Definition

Ab Average burning surface area, ir4

Abc Average burning -surface area within cylindricallength, in2

Abh Combined average burning surface area of forwardand aft head-end-propellant, in2

Abo Thrust-to-weight ratio at burnout

SAe, Nozzle exit area_(inside), in2

Af Cross sectional area corresponding to outsidediameter of propellant, -in2

Aign Thrust-to-weight ratio at ignition

Ap PInitial port area, in?

I Apt Initial port-to-throat area ratio-

As Cross sectional area of residual sliver(active or inert), -inz

At Nozzle throat area, in

Aw- Initial cross sectional area of propellantweb, in2

a Propellant burning rate coefficient

C1 Constant; 0. 0163 for upper stage; 0.0055for remaining stages

, CD Mass discharge coefficient, lbm/lbf sec

f Cf Thrust coefficient

Ii: V

LIST OF SYMBOLS (Continued)

Symbol Definition

c i -Specific heat of insulation, Btu/lbm- *R

c s -Specific -heat of structural material, Btu/lbm- R

Dc, Motor case inside -diameter, in.

j Dei Nozzle exit inside diameter, in.

De Maximum- nozzle exit outside diameter, in.

Deo -Nozzle exit outside diameter, in.

-Df Outside diameter of propellant charge, in.

Dm Motor case outside diameter, in.

Dhi Inside major axis -of ellipsoidal forward head,- -in.

Ec Young's modulus of motor case material, psi

Eint Young'. modulus of interstage structure, psi

E S Young's modulus of nozzle thi oat structural

shell, psi

I Et Young s modulus of throat insert material, psi

{ F Required- thrust (input), lbf

Fp Fraction of propellant burned at web -burn-L through

Fsm Motor case design safety factor

Fsn Nozzle design safety factor

I Propellant web fraction (ZTW/Df)

1 gc -Conversion constant, 32. 174 lbm ft/lbf sec,

LIST OF SYMBOLS (Continued)-

j Symbol Definition

gmaxi Maximum longitudinal acceleration of stage i, g's

gmaxi 1 Maximum longitudinal acceleration of stage -i- 1, g's

is SPd Delivered specific impulse, lbf sec/lbm

kt Ratio of nozzle throat radius of curvature tothroat radius

Lah Length of aft head, in.

Lc Rocket motor cylindrical length, in.

Lcont/Lcon Ratio of contoured nozzle length to conical nozzlelength

Lh Length of forward -head, in.

Lhi Inside semiminor axis of ellipsoidal forwardhead, in.

-Outside semiminor axis of ellipsoidal forward--head, in.

Lint Interstage clearance between forward head of-stage i and nozzle -exit plane of stage -(i + 1), -in.

I Lm Rocket motor length, in.

Ln -Nozzle length, in.

I Lni +1 Nozzle length of stage (i + 1), in.

rmn Mass flowrate, lbm/sec

n Propellant burning rate exponent

Pamb Ambient pressure, psia

A , vii

LIST OF SYMBOLS (Continued)

Symbol Definition

Pb Average prope]lant burning perimeter, in.

c PC Average chamber pressure, psia

P PD Design chamber pressure for structural

analysis, psia

SPDx Local design pressure, psia

Pe Nozzle exit pressure, psia

R Specific-gas constant, ft lbf/lbm°R

rat Nozzle -inside radius -at exit cone tangentpoint (see Figure 3-2), in.

rb- Propellant burning rate, in/sec

r€ Throat insert radius -of curvature, in.

re Inside radius at nozzle exit plane, in.

j res Inside radius of nozzle structure at aft end

of throat insert, in.

rop Radius of motor case aft head opening, in.

ros- Inside radius of nozzle structure at throat, in.

rot Inside radius at nozzle entrance cone tangentpoint, in.

rt Nozzle throat radius,- in.

Tc Propellant flame temperature, 'R

I t Insulation- exposure -time, sec

VIM

LIST OF SYMBOLS (Continued)

Symbo Definition

tb Elapsed time between-ignition and web burn--through, sec

Vpc w -Volume of propellant within cylindrical lengthwhich is burned prior-to web -burnthrough, in3

Vph Propellant volume in forward and aft heads, in 3

V: Vps c -Total volume of propellant charge in- cylindricalchamber (including slivers), -in3

V p Total volume of propellant which is burned priorto web=-burnthrough, in3

Vs Volume -of residual slivers, -in3

Wab Weight of motor case aft head attachment boss, lbm

Wahi Weight -of aft head insulation,- lbm

Wahs Weight -of aft head membrane -structure, lbm

[ Was Weight of aft head attachment skirt, lbm

Wbo Stage burnout Weight, lbm

Wcl Liner weight in- cylindrical chamber, lbm

Wcs Weight of cylindrical shell, lbm

f Wei Weight of nozzle entrance insulation,. lbm

Wes Weight of structural material-in nozzleentrance cone, lbm

Wfhs Weight of forward head membrane structure, lbm

3ix

LIST OF SYMBOLS (Continued)

Symbol Definition

Wfs Weight of forward- head attachment skirt, lbm

Whi Weight of forward head insulation, Ibm

Wib Weight-of ignitor -boss, ibm

Wign Ignitor weight, ibm

Wint Weight of interstage structure, ibm

Wip Total weight of inert parts, lbm

Wn Total nozzle weight, ibm

Wnb Weight of nozzle attachment boss (nozzleportion),_ ibm

Wo Gross stage weight, lbm

p Propellant weight, Ibm

WPL Stage payload weight, lbm=

L Wt Weight of nozzle -throat inset, lbm

! Wti Weight of nozzle -throat insulation, lbin

Wts Weight of throat structural shell, ibm

Wxi Weight of nozzle exit cone insulation, ibm

1 Wxs Weight of nozzle exit cone structure, lbm

a Nozzle exit cone-half-angle, radians

eae Contoured- nozzle half-angle at exit plane, radians

i

V xC

LIST OF SYMBOLS (Continued)

I Symbol Definition

I l Thermal diffusivity of nozzle insulation, in2 /sec

a ~Contoured nozzle half -angle immediately down-stream of throat, radians

Forward and aft head ellipse ratio

I - Specific heat ratio of combustion gas

AT Predicted temperature rise of protected-structural material, 'R

AVI Ideal velocity increment, ft/sec

F :e' Nozzle expansion ratio

- _ e Equivalent conical nozzle expansion ratio forS-contoured nozzle analysis

11 q D Mass discharge correction factor for nonideal flow

7 v Exit velocity correction factor -for nonideal flow

j Rocket motor mass fraction

Nozzle divergence loss factor

1± Stage mass fraction

t ts Poisson's ratio of throat structural shell

±Lt Poisson's ratio of throatinsert material

Nozzle -entrance cone half angle, radians

SPj Liner density, lbmjin3

f, xi

LIST OF SYMBOLS (Continued)

t Symbol Definition

Pi Insulation density, ibm/in

Pm Motor case density, lbm/in3

1 pP Propellant density, lbm/in3

i ps Density-of structural material, lbm/in3

Pt Density of throat -insert, lbm/in3

m Yield strength ofmotor case, psi-

(rn Yield strength of nozzle structural material, psi

-Te Wall thickness of nozzle entrance cone, in.

" C Liner thickness of cylindrical chamber, in,

f Th Forward- head membrane thickness, in.

Ti Insulation thickness, in.

Tie Insulation thickness in nozzle entrance cone-, in.

T • Insulation thickness at nozzle exit plane, in.Tx

Tm Motor case wall thickness,- -in.

Tmn Minimum wall thickness due to fabrication

1 constraints, in.

Tn Nozzle wall thickness at exit plane-,- -in.e

I Ts Structural material thickness, in.

Tts Thickness of throat structural shell, in.

xii

LIST OF SYMBOLS (Concluded)

Symbol Definition

Tw Propellant web thickness, in.

'xs Thickness of nozzle structure at aft end ofthroat insert, in.

0 Time constant ("i /ra ai), sec

ii

IL

Ii

1.

I

1. INTRODUCTION

Preliminary sizing analyses of aerospace vehicles which utilize

solid propellant rocket motors usually require rather extensive trade-

off analyses to ensure the near-optimal selection of pertinent motor

design parameters. For multistage vehicles, the motor design

analyses usually follow energy management studies (e.-g. , Ref. 1) which

j apportion propellant masses optimally among the constituent stages to

achieve prescribed vehicle performance characteristics; e. g. , to

j minimize gross weight for a required velocity increment or to-maximize

velocity for a fixed gross weight. The objective of this investigation

J was -to develop a -digital computer program which would permit rapid,

accurate, preliminary design tradeoff analyses of solid propellant

Irocket motor concepts.

I

I_

I-

41

:1-1

2. PROGRAM CAPABILITIES

The program described in this report computes rocket motor

length, propellant and inert component weights, mass fraction, burn

time, specific impulse, stage burnout velocity, and other performance

parameters as a function-of input motor diameter, chamber pressure,

thrust, payload mass, propellant ballistic properties, propellant web

fraction, port-to-throat area ratio, residual sliver fraction, and other

required parameters.

-The generic representation of a selected propellant grain con-

figuration by its web fraction, Fw, and residual sliver -fraction, Fp, and

a prescribed port-to-throat ratio, Apt, permits evaluation of the applica-

bility of many candidate grain designs without requiring detailed surface-

web calculations. This feature simplifies -input preparation and enhances

program flexibility for its-intended use in- conceptual design tradeoff

I analyses. Realistic values of Fw and Pp may be derived-from known

geometric characteristics of previously configured grain cross sections

I (Refs. 2 and 3). All star, wagonwheel, -cylindrical port, and slotted

tube grain -configurations wherein the propellant perforations traverse

the entire cylindrical motor -length are -ideally suited for analysis using

this technique. Tapered grain configurations and those which employ

radial slots, or longitudinal slots which-traverse only-a raction of the-

cylindrical motor length, -may be analyzed provided a length-averaged-

propellant web fraction is-entered. Ellipsoidal head-end -propellant

webs and -inert propellant slivers may -be -included in the -design analysis

simply through the proper -selection of input code words,, as described-

later.

Dimensions and/or weights of inert components, -including the

motor case, head closures, nozzle, attachment skirts, interstage

{ 2-1

I1

structure, liner, insulation, and igniter, are computed using proven

theoretical and empirical relationships in conjunction with input

) material physical properties. Either -conical or contoured nozzle

designs may be analyzed. The nozzle expansion ratio is sized to

provide as near optimum expansion as possible while maintaining -the

nozzle exit diameter less than the input maximum allowable value.

j (Optimum expansion occurs when the -nozzle exit pressure is equivalent

to the input ambient pressure. ) The validity of the computer program

J was confirmed through the analytical reproduction of various rocket

motor designs which are documented in Reference 4.

-

1

IIII

I

3. DESIGN ANALYSIS

3.1 INTERNAL BALLISTICS

The internal ballistics analysis described in this section isbased on the assumption of constant mass flow rate throughout motor

-operation. The appropriate gas dynamic relationships for combustion

gas flow within the motor port and nozzle are based on the assumption

of one-dimensional flow of a perfect gas. Most of the applicable

thermodynamic relationships may be found in standard propulsion

textbooks; for example, Reference 5. Following program initialization

and the reading of appropriate input parameters, the rocket motor

-design analysis proceeds as described- in the following paragraphs.

The required thickness of the motor case wall is -sized using the

K F-well- known-hoop stress relationship

Pc DmFsm (3-1)Tm

The independent variables Pc, Dim, and Fsm are -program inputs. The

yield strength of the motor case, 0 m and most of the other required

inert component physical properties discussed later are -specified

j within the program in well-defined DATA statements, as -described

in Section 5.2. This method of specifying material properties provides

flexibility for analyzing various candidate materials (at the expense of

program recompilation for each DATA statement alteration) while

reducing the number of inputs required for normal-design analyses.

The case wall thickness computed from Equation 3-1 is -compared

with a specified minimum wall -thickness based onfabrication limita-

tions, and the larger value is employed in further -calculations.

3-1

Following an initial estimation of the required cylindrical case

liner thickness, Tc the outer diameter of the propellant charge is

computed from

Df D m - 2 (Tm +I rj c ) (3Z)

The final design value of liner thickness is determined through iteration

using an empirical relationship which is described later.

Equation 3-2 -begins an iteration wherein- the nozzle expansion

ratio, cylindrical case liner thickness, and propellant geometrical

Icharacteristics are established. An attenipt is first made to size the

nozzle for optimum expansion such that the nozzle exit pressure is

equivalent to the input ambient pressure. If this -is not feasible within

the constraints of the input maximum nozzle exit diameter, Dem , and -the

thrust-dictated throat area requirement, the nozzle design which most

[ nearly approaches optimum- expansion within these constraints is

established. In the latter situation,- the nozzle exit pressure is always

[greater -than ambient;--i. e. , the nozzle is underexpanded. As a first

trial estimate, -the nozzle exit pressure, Pe, is set equal to the -nput

ambient pressure Pamb' The corresponding nozzle expansion-ra'io, - ,

is then determined from the relationship

11 1 Y -l!

I 1 1

3-

The -thrust coefficient, Cf, is computed from

Y + 1 _Y-

Cf XDn'1D v 2 Y T-1 [ C)

(Pe Pamb)c+

PC

The divergence loss factor, Xn , is defined as

1)In = - (1 + -Cos a) (35)

J For analysis of contoured nozzles using-this program, aneffective exit half-angle of 17.5 degrees should- be input, as described

later. The nozzle discharge correction factor, 'qD' is usually greater

than unity, as discussed in Reference -6, but may range from 0. 98 to

1 1. 15. (A value of 1. 02 is typical. ) The velocity correction factor,

i]v' ranges between 0.-85 and 0.98, with an average near 0. 92 (Ref. 6).

The required nozzle throat area is established by

At F1(Pc Cf) . (3-6)

The corresponding nozzle exit area, Aei, is computed-from

-Aei AtE. (37)

3-3-I1*

The required- nozzle wall thickness at the exit plane (-ne) is sized

from the hoop stress -elation

"Pe Dei Fun (3-8)m Tn e -2 0n

The variable Fsn is a program input,- and arn is specified within the

program by a DATA statement. Again, the computed wall thickness

from Equation 3-8 is compared with-a specified minimum thickness

based on fabrication constraifits, and--the larger value is selected for

further calculations. The outside diameter at the nozzle exit plane is

then -computed- from

IF Deo= Dei + 2(0 ne + Tix) (3-9)

where -ix, the insulation thickness at-the nozzle exit plane, is

computed using procedure!; described later in-this section (Equation 37).

If the -computed outside exit diameter is less -than the -input maximum

value, the design analysis continues with the computed value. More-

over,- -if this test is satisfied on the first iteration, with Pe = Pamb'

Ithe nozzle is designed for optimum expansion. If, however, the com-

puted -outside -exit diameter exceeds the input maximum value, the exit

I pressure estimate is increased- slightly and the analysis returns to

Equation 3-3. The iteration continues- until a satisfactory exit diamete

Iand corresponding exit pressure and expansion ratio are obtained.

I3

IJ

Following the establishment of the nozzle throat and exit areas,

the -internal -ballistics analysis and propellant geometrical characteriza-

tion proceed. The mass discharge coefficient is computed from

cD = ~ (2-i 10.5l ] • (3-0)

The independent variables y, R, Tc are program inputs. The average

mass flow rate, rr, discharged by the nozzle is computed- from

MT= CDPcAt . (3-11U)ftThis discharged mass flowrate must equal the rate of mass- generation

at the propellant burning surface, which is determined from

t = rbAbPp (3-I2)

where the average propellant burning rate is governed, for nonerosive

flow conditions, by the well-known relation-Irb = .ap

n • (3-13)

Equating the mass generation rate -(Equation 3-1 1) to- the mass discharge

rate (Equation 3-10) yields, after rearrangement, the following relation-

ship for computing the required average burning surface area:

Ab GDPcAt (3-14)rbPp

3--5t

The initial motor port area is established by the input port-to-

-throat area ratio-and the previously computed throat area

p= AtAt • (3-15)

The initial port-to-throat area ratio, Apt) must be input as unity or

greater to assure a nonerosive flow -condition inside the motor port.

For tapered grains, Apt is the length-averaged value. The cross

-sectional area, Aw , of the propellant web at ignition is computed from

the geometric relationship

(Af - Ap) Fp (3-16)-I-where F p is the-fraction of propellant -burned prior to-web burnthrough.

I Values of Fp usually range from 0.9 to 1.0 (Refs. 2 and- 3). The

-corresponding area, As, of residual slivers (active- or inert) remaining

I after web burnthrough is established -by-the expression

I As =Af Aw -Ap . (3-17)-

The propellant web thickness-is determined from

fw w (3-18)

As described previously, .Fw represents the input propellant web

fraction. Values of Fw typically range from 0. 2 to 0. 5 for star or

wagonwheel designs and from 0. 28 tO 0. 9 for slotted-tube designs.

-Star and wagonwheel configurations are usually employed with slow or

-medium burning rate propellants where relatively small burn times and

=I [ 3-6

large thrust levels are required. Slotted tube configurations yield

large motor mass fractions; they are usually employed with relatively

slow burning propellants for long burn-time, low thrust applications or

with fast burning propellants for short burn time, high thrust applica-

- tions.

The motor burn time is computed from

tb = (3-19)

At this point in the analysis, the required liner thickness adjacent to the

cylindrical case wall, Ti , may be recalculated more accurately

using the -proven empirical expression developed by Thiokol Chemical

Corporation (Ref. 7).

11r 1 c = 0.02 Dm' 5 (tb + 1)0.1 (1- Fp)0 2 5 - (3-20)

The liner thickness computed from Equation 3-20 is compared-with

the previously estimated value used in Equation- 3-2. if the computed-

liner thickness differs by more- than a specified amount from the

previous estimate, the analysis returns-to Equation 3-2, with the

computed value used as the new estimate. This iteration continues

until the computed and estimated values converge.

The next step in the analysis is the determination of the required

propellant volume and corresponding motor length. Before motor

length may be computed, -the amount of propellant must be evaluated

which may be loaded into -the available forward- and aft -head-end

volumes, if head-end webs are desirable. (The-computer program is

formuulated- such that the analysis of head-end webs is optional.) For

cases in which a forward head-end web analysis is requested, a

* , V 3-'7

fractional ellipsoidal propellant volume, conforming to the head

closure shape, is considered. A cylindrical perforation is allowed

for protrusion of the ignitor through the head-end web. The analysis

of the propellant in -the propellant volume in the aft head is similar -to

that for the forward head, except the cross-sectional area of the center

perforation is equivalent to -the port area.

Following evaluation of the combined forward and aft head-end

propellant volume, Vph, and corresponding average burning surface,Abh, if applicable, the required average burning surface, Abc, within

-the cylindrical motor length is computed from

Abc = Ab - Abh (3-2T)

The required total volume of the propellant cha ge, VP, includingep noresidual slivers (active or -inert), -in the cylindrical portion -may now

-be computed from the geometrical: relationship

-rw Ab (3.-22)[ YP~c- Fp_

L Likewise, the volume of propellant, V cw , within the cylindrical

length which is burned before web burnthrough is determined fromIV ~V F(3-Z3)Pcw Psc p

The total volume of propellant, VPw ' which is burned before web

burnthrough is -determined by the sum

IL 3-8

tt

Vw= Vc w - + VPh(3-24)

and the volume of residual slivers, Vs, is computed-from the-geometri-

cal relationship

Vs V (1 - Fp) • (3-25)

Because of the simple propellant geometry selected for the head-end

web, residual slivers-there will be negligible,

The required c-ylindrical length of the motor, Lc, may now be

computed from

L c Vpcw/Aw . (3-26)

The propellant and residual sliver weights are evaluated by multiplying

the previously computed volumes by their respective densities. For

active slivers,- the propellant density is used, whereas for inert slivers

j a density of 0.-02 lbmin3 , which is typical of conventional phenolic/

microballoon--filler material is used.

3. 2 DESIGN OF INERT COMPONENTS

Sufficient information is now available -to permit computation of

weights and dimensions of the remaining constituent inert ccmponents

of the rocket -motor. These calculations are performed within-the

j computer program by Subroutines INERT, NOZZ, and INSUL. In the

interest of brevity, only the final formulations of the pertinent working

equations will-be presented herein. The reader is referred to the cited

references for the appropriate derivations. The nomenclature for the

type of motor case considered herein-is defined in Figure 3-1. Also,

the nomenclature which applies to the conical and contoured nozzle

analyses is defined in Figures 3-2 and 3-3, respectively.

I3-9-

INI

LLi

S.--

coc

LL

3-10.

AA

I.L

ILX-C

Lii

3- 4-)

4--) L)Ito- -

r- 4- C C~)

S- 00

J .,4 1 1 -.J

4--)

LLL)

3-1-4

-Li4-tC

N J C)

4-1 4- CD.

4-) 4-)

4-) (aIt 0

PI I

4 4J CA

tC S- L

3-12)

3.2. 1 Forward- Head Membrane

The required thichness, Th, of the forward- head membrane is-

-sized by the resultant of the meridional and tangential stresses as

-discussed in Reference 8. Thus,

O. 177 P Dm2

Lhm (3-27)-

The design chamber pressure,- PD' in taken as 1.5 times the average

chamber pressure. The value of Lh o is dictated--by-the input-motor

-diameter and the head-end ellipse ratio, -P, which typically ranges

from 1. 0 to 2. 0. The values of the inside major axis, Dhi, and semi-

-minor axis, Lhi, are determined from the input motor diameter,

-ellipse ratio, and membrane thickness. The weight of the forward head

S-membrane structure -is then computed from

Wfh§- 0. 167 Pmir (Lh0 Din2 - Lh i Dh.2 ) • (3-28)

3.2. 2 Forward Head Insulation-

The weight of the forward head insulation is computed-from the-

following empirical relationship (Ref. 9):

Whi =1. 93-(100) Di

(3-29)-

0.7854 + 03925 .n PD-" tb CD'

where

3-13

3.2.3 Ignitor Boss

The weight of the -ignitor boss, W ib on the forward head is

estimated from the empirical expression (Ref. 9):

mWib 1. 13 PD D At-30)

f 3. 2.4 Ignitor

The weight of the ignitor, Wign, is estimated from the follow-

I ing empirical relationship which was derived through correlation ofreported ignitor Weights (Ref. 4)-for previously designed-and fabricated

11 rocket motors:

ign b]

Equation 3-31 was found to produce an excellentfit of reported igix,4-rLweight data for rocket motors with an average burning surface area

ranging between 35 00 and- 44000 in2z.

3.2.5 Cylindrical Case Structure

f The weight-of the cylindrical portion of the motor case strtic-

ture, Wcs-,- is computed from the relationship

Wc _ L D2- D (3-32)iCS 4 m 4M c

3-14

3.2.6 Cylindrical Case Liner

The weight of the liner adjacent to the cylindrical motor c ,

Wcl, -is determined from

W D L TI c P1 (3-33)

For the computer -program described herein, a typical liner density,

P -of 0.06 Ibm/ins is employed

3.2.7 AftHead Membrane

The weight--of the aft head membrane- structure, Wahs' is

estimated from the following empirical relationship (Ref. 9):

Ii

I Wahs 0.25?l -,Wahs -o , D - m + +"

0.3925 ____

X. XDM' [0O.7854 + In-1 2At13.z :(3-34)

} 13.2.8 Aft Head Insulation

The weight of the aft head insulation, Whi is scaled as 1. 54

times that of the previously computed forward head insulation as

+ recommended by Reference 9.

* f 3-15

3. 2. 9 Aft Head Nozzle Attachment

The weight of the aft head nozzle attachment boss, Wab, is

estimated using -the following relationship (Ref. 9):

W 10.z6 P D Z (3-35)wab D M (

3.2. 10 Nozzle

Either conical or contoured nozzle designs may be analyzed

using this -program. Conical nozzles of the type illustrated in Figure

3-2 are designed using -nondocumented procedures developed by

Mr. A. R. Maykut of the U.-S. Army Missile Command (MICOM) at

Redstone Arsenal The validity of these procedures -has been proven at

Redstone Arsenal through numerous design analyses and reproductions

of previous operationalnozzle designs. Contoured nozzle length and

weight, when applicable, are scaled from a corresponding design of

a conical nozzle with equivalent throat area and fixed exit cone half-

angle of 17.5 degrees using procedures described in Reference 10. A

more detailed description of the contoured- nozzle scaling procedure is

[i presented later.

Discussing first the conical -nozzle analysis, the entrance wall

thickness, Te, of the nozzle at the aft head attachment radius is compu-

ted fromI ro PDop D (3-36)

e 0o cos4n-

The parameter r usually is governed by the size of the aft head

] opening which is required for insertion and extraction of the propellant

mandrel. For this computer program, the nozzle entrance cone half-

angle, 4, is fixed at 60 degrees.

3-16

- ,

The thickness computed by Equation 3-36 is compared with a specified

minimum value which is dictated by fabrication constraints, and the

larger of the two values is used in further calculations.

The next step in the nozzle analysis is the determination of the

required insulation thickness at the nozzle entrance. The procedure

which is employed in sizing-the insulation thickness is described in

Reference 5. A thermally thin structural wall is assumed to be pro-

tected from the combustion- 'gases by a refractory insulation. The

temperature of the exposed insulation surface is assumed -to be equivalent

to that of the adjacent combustion gases, T c , and the temperature rise

of the protected structure during motor operation is assumed to be

i much smaller than T . The -predicted temperature rise ofithe struc-

tural wall during time t is computed from the -general expression

c. P. T.

AT T . Tf- (3-37)c P c

where 0 is--the time constant (_Ti 2 / 2 ai). The function f(t/e) is reported

in Reference 5 as

t -t 1 Z (-I)n + l

t + + (exp nz (3-38)r6 7r n=l n e

Equation 3-37 cannot be solved directly for Ti because of the implicit

relationship-of ri with the series function f(t/8). Therefore, an itera-

tive solution is accomplished -in which Equation 3-37 is solved- repeti-

tively to compute values of predicted temperature rise for progressively

improved estimates of 'ri. The least value of Ti for which the predicted

structural temperature rise is maintained below a specified limit of

10°R is selected for design application.

3-17

The first 20 terms are utilized in the series evaluation of f(t/0).

Ablation -of the insulation is not accounted for in the determination of

the required insulation- thickness. Therefore, the selected thickness

should be conservative. An approximate, but nonconservative, means

of accounting for ablation would be to substitute the insulation ablation

temperature for T in Equation 3-37.c

The weight of insulation in the nozzle entrance cone, Wei, is com-

puted from

W ( T o, r-rt T. . -(3 -39)i .COS + r ° lro - rwei sin o Le otr eop e

The weight of the nozzle entrance cone structural material, Wes, is

computed from-

r )-cos + -r p

Yes (Te 2(3-40)We sin op +(ie

The weight of the nozzle throat insulation is computed from the relation

Wti IT-p Te rt + Tie (cos=-+ cos a) + r a t](2ktrt) sintea (3-41)L(N)The thickness of the throat structural shell, Tts, is sized-from the

Sfollowing expression which accounts for the reductioninpressure load-

ing of the structural shell due- to the load carried by the throat insert

material:

I r D- Et ___

T -s r - (3-42)-ts rn sE t

n--s

; ) 3-18

The design pressure PD in Equation 3-42 should rigorously be based

upon a gas pressure whose value lies between the expected values at

the throat and in the combustion chamber, since the inner surface of

the throat insert is exposed to pressures in this range. However, for

design conservatism, the value of PD used in this program-is computed

as- 1.5 times the chamber pressure. The corresponding weight of the

throat structural shell, Wts, is determined from

[ro ~t o)]Wt - Tt r + T. (cosp+ cos-a)+ ra tste ra + (Cos b + -co sc

S2 k rt sin ((-43t t ( - 3

From -geometrical considerations, the weight of the -throat

insert, Wt, is established as

W irp kt--r [ -sin(+at t

-0.667k sin3 ) 1+a - 5sin (:+a) (3-44)

The thickness -of the exit -cone structure at the downstream end of thenozzle throat insert, Txs, is sized from

P rD es~X

T • (3-45)xs oCos a;n

The design pressure PDx is defined-as 1.5 times -the local pressure in

the nozzle exit cone. The structural thickness computed from Equa-

tion 3-45 is compared with a specified =minimum value which is dictated-

by fabrication- constraints,_ and the larger of the-two values is selected-

for design application. The corresponding weight of the exit cone

structure, Wxs, is computed from

: 3-19-

'77

w iT ct oxs_ s 3 n(8 (erat)F rat +(Ti +7

XS S) e XS

4 ~t+r.+TS~os£KIr+ rie co )+ r Ti+Tmn)

-(rat + Ti COS-az (rat +le- cs)-( +.e) -(re I(3

The-weight of the nozzle exit-cone insulation, W.,i is- determined from

W. ITeT ictnra + Te +re) rrt (3-47)

The-weight -of the attachment- -boss, VlI which is an.-integral part of

7'the nozzle structure,- -is estimated -from

W M= 1rr T -p- .(3 -48)Ynb op e-s

kFrom thje previously-computed- nozzle -component weights, the -total

nozzle weight, W -is- computed as the- sumn

W=W .+W- +W + W +W +W + W.+W -(3-49)-n ei es ti ts t xs- xI_ ib-

The-nozzle length, L ,for conical entrance and- exit geometries,

may be computed from

Entrance Throat Exit

L (r -rctn + r k(s in4 + sina +( - r ~Itn a . (5)n opot0 t t \re at)J

Equations 3-36 -through 3.-5O were derived for nozzles with

conical entrance and- exit geometries.

3-20-

3am

However, most modern-day rocket motors which are designed .or

flight operation employ nonconical contours which are optimized for

improved performance and reduced length. Because of the difficulty

associated with detailed design of a contoured nozzle, the simplified

empirical procedure described in Reference 10 for approximating

contoured nozzle weight and length from a corresponding conical

nozzle design was employed in this program,

To calculate contoured nozzle weight using t',e previously

described approximate method, the following assumptions are made:

* The weights of conical and contoured nozzles areidentical from the forward attachment boss to anexpansionr~atio downstream of the throat of 3.-0 for

nozzles of equal throat area. (Nozzle shape withinthis region is relatively independent of downstreamshape.)

0 The nozzle weights downstream of an exit expansionratio of 3A-are identical for contoured and conicalnozzles of equal surface area,

The two types of nozzles are assumed to have identical throat areas,

chamber presures, -burning durations, and propellant flame tempera-

ture. The family of contoured nozzles for which this procedure applies

is described by O. J. Demuth in Reference 11.

To determine the conical nozzle surface area which is equiva-

1 lent to that of a contoured nozzle having the same throat area, anequivalent conical nozzle expansion ratio, Ee , is derived from prei-

ously computed contoured nozzle data. This equivalent expansion

ratio is defined as -that of a conical nozzle with a 17. 5-degree exit

cone half-angle which -has the same surface area as a particular con-

toured nozzle of the same throat area. Contoured nozzle weights are

then computed from-the previously described conical nozzle equations

(Equations 3-36 through 3-50) when the applied value of the inside radius

{ t3-21

..........

of the throat exit plane, re, is based on the equivalent expansion ratio,

ce. and the input nozzle exit cone half-angle is taken as 17. 5 degrees.

The curve of Fe as a function of c (Ref. 10), which is incorporated

into this computer program, is presented in Figure 3-4. Figure 3-4

applies for a Demuth-type contoured nozzle with an initial divergence

half-angle (immediately downstream of the throat) of 32. 5 degrees

and an expansion ratio at Lhe point of parallel flow of 30. A review of

previously designed contoured nozzles (Ref. 4) revealed that this

particular exit shape is generally applicable for relatively large tactic-

cal rocket motors. Of course, flight nozzles are truncated at expan-

sion ratios slightly less than that required for parallel flow with little

performance penalty. Additionally, Figure 3-4 is based on a ratio of

combustion gas specific heats of -1. 2 and a ratio of throat radius of

curvature to throat radius of 1.2.

Direct calculation of the length of a contoured nozzle is again

quite difficult. To simplify the calculations in this program, contoured

nozzle length is related empirically to a 17.- 5-degree conical nozzle of

the same throat area and expansion ratio. In determining contoured

nozzle length, the length-from the -throat -to -the exit plane is first cal-

culated as for a 17. 5-degree conical nozzle. This length is then multi-

plied by an empirically derived factor, Lcont/Lcon, for the appropriate

contoured nozzle expansion ratio. The curve of Lcont/Lcon as a func-

tion of e (Ref. 10), which is employed in the computer program de-

scribed herein, is presented in Figure 3-5. The curve shown in

Figure 3-5 is for the same Demuth-type nozzle shape as discussed

previously for nozzle weight calculations. The corrected length between

the throat and the nozzle exit plane is then added to the entrance length,

between the forward attachment boss and the throat, to determine the

total contoured nozzle length.

3f, 3-22

40

36 _

32

28

24

oN 20

U

4 16C4J

c"

-V ~ ~ ~ ~ L 12 - ______

= 1.2

r /rt 1.24- 0 a = 32--.55*

c = 30 for Parallelflow-

32550 4812 16 20 :24 :28

Contoured-Nozzle Expansion Ratio, 0

FIGURE 3-4. EQUIVALENT CONICAL EXPASION RATIO AS A FUNCTIONj OF CONTOURED -NOZZLE EXPANSION RATIO

3-23

14 ~- ----

'101

0 _

00

.1.2---- C

J. 1Oo -___ 32.550_

c00 o

43.2

The -previously described procedure could be extended for

analyzing other Demuth-type exit shapes, with different values of

a. and 4 for parallel flow, by incorporating additional curves of feI/

and L /L as a function of 4 from Reference 10.cont/ -con

3.2.11 Motor Attachment Skirts

The weight of the forward attachment skirt, Wfs,- is estimated

from the following relationship recommended by Reference 9:g max W 021-(L C D/Pi PL m +1 (3-51)

W [0. Dcfsm- E DEc D_

The maximum longitudinal acceleration--of the stage under considera-

tion, gmax, is -assumed to- Qccur at stage burnout:

g = F/Wbo (3-52)

Now, at this stage of the analysis, W is unknown, since its valuebo -

depends upon-the undetermined weights of the attachment skirts and

interstage structure. However, these weights are usually relatively

small in comparison with the remaining inert component weights.

Therefore, an iterative solution is accomplished in which gm isax.

first estimated using a value of Wbo which includes only the previously

computed weights of the motor case, nozzle, ignitor, liner-, insula-

-tion, and the -tage payload. This value of gmaxi is utilized- in-

the calculation of the attachment skirt and interstage weights. The

value of g -is then recalculated using the estimated skirt and inter-1.

-stage weights included in W bo This iteration- is continued until the

3-25

computed values of Wbo (or gmaxi) on successive -iterations are

equivalent within a specified tolerance. At this point, the correct values

of the attachment skirt and interstage weights are considered to be

established.

The weight of the aft attachment skirt, Was, is estimated from-

/gmaxi 1 W °SWas 0. 0055 Dm nE

-(3 -53)

05 LC ++110)

Dm + "

In the -evaluation of Was, -the value of gmaxi.i1 must -be estimated from-

a separate analysis of stage i-l, and Wois estimated as part of the

previously described iteration on Wbo.

S3. 2. 12 InterstageStructure

The interstage structural weight, Wint, is estimated, when re-

quired:, from-the following relationship which is recommended by

i Referen,,e 9:Dm

Wint =0. 155 Dm + Lni+i + Lint)

maxi WPL _215 (LC + Dm+ 05(354)

Eint-D + 11

IThe evaluation of gmaxi and Wint is included in the previously described

-iteration on Wbo.

-- 3-26

3. 2. 13 Rocket Motor Length

Rocket motor length between the nozzle exit plane and the for-

ward head, Lm, is computed by summing the constituent lengths.

Lm= L h + Ic + Lah + Ln -(3-55)

The length of the forward head, Lfh, is determined from

14-h Dm/2P (3-56)

and the length of the aft head, Lah, is computed- from

Pm (12rop- 2 0.5

Lah ~ - Dm(3-57)i-

3.3 ROCKET MOTOR PERFORMANCECHARACTERIZATION

Sufficient preliminary design information has now-been computed

-from -the previously described analysis to -permit prediction of rocket

motor performance. After setting the sum of the-previously computed

-inert component weights equal to Wjp, the gross stage weight, Wo, is

computed from

Wo WIP + WpL+ p . (3-58

j. 3-27

- - -_ _ _ _ _ - -- ,- --- ' - - - - -. . . . "-9

Stage mass fraction, p., is defined as

Wp_ (3-59)IL Wo - WPL

Rocket motor mass fraction, X, is defined as

Wp. (3-60)Wo - Win t - WpL

The delivered specific impulse, Ispd, is computed from

sPd =Cf (3-61)

The predicted ideal velocity increment produced by the stage, neglect-

ing drag and gravity losses, is computed from

In Ebo (3-62)AV I e-Isp d c nWb-o

Thrust-to-weight ratios at ignition, Aign , and burnout, Abo, are com-

puted from Equations 3-63 and 3-64, respectively:

Aign F/W 0 (3-63)

Abo = F/Wbo . (3-64)

This completes the description of the rocket motor design

J analysis -as it is- programmed for the digital computer. The program

is formulated in such a manner that parametric tradeoff analyses of

{ pertinent design variables may be readily accomplished by stacking

input cases back-to-back. Moreover, an automatic variable adjusting

procedure, which would determine the required value of a specific

3-28

independent variables (e.-g., motor diameter, chamber pressure,

propellant web fraction, etc. ) to produce a desired motor performance

or design characteristic (e.g., mass fraction, motor length, or

velocity increment) may be easily incorporated into the program if

desired.

3U

i '. 3=29-

4. PROGRAM APPLICATION

4.1 METHODOLOGY

The computer program described herein may be utilized to

generate both point and optimized preliminary designs of solid propellant

rocket motors through the proper selection and manipulation of appropri-

ate input variables. Typical design problems and corresponding methods

for using the program to solve these problems are presented in Table 4-1.

The computer program contains no automated variable optimization

scheme. Therefore, the optimization- of pertinent design variables must

be accomplished using "brute force" techniques wherein graphs are

plotted of computed performance and/or weight characteristics which

result from manual perturbation of the variables of interest. Optimum

values of design variables -which yield either maximum performance or

minimum weight characteristics are then selected =from these graphs.

For example, Problem 1 of Table 4-1 Illustrates -the technique which

would- be employed to establish the optimum chamber pressure for a

minimum weight-design, assuming all other input variables are fixed.

j Problem- Za of Table 4-1 illustrates how the designer would

determine the optimum motor diameter and corresponding chamber

pressure which would be required to provide a given motor length.

Similarly, Problem Zb illustrates the technique which would -be used to

determine the optimum motor length and corresponding chamber pressure

which would yield a given motor diameter. The latter two examples are

frequently encountered in -the design of tactical and- strategic -missiles

which must be contained within fixed- envelope dimensions.

Problem 3 illustrates how the program would be employed to

evaluate the variation of stage impulsive velocity with motor diameter,

expansion ratio and payload weight.

44-1i

U_ .0

OCLc4. 4-b 41 4.1 41t2

5 s z:; a,= 4.L

4J- - i 4534 -a-4.a 4C2 ;I0 0.1 .u 0)a.~ .00 ~s.a.V 0454. LL -a 0- ' a S.- E

4- -. - S. a4j-2-

v5', 0 451.4n0. CO u - q on - u -- V1 -A. 0- U

-h j

coo xM

x- I- "

E w

-Z.

4. WCom 0 o_ _J. .'. .l 01 *-.

10 =-I zS'U 2,~a) F ) u,31-1'E 4'u Z-,- 1C- ' O -O ' 4% mL +- >'-I 'd

x,4 x -4 OJp CV - -L '0--Ia. ~ L c.-l- > .Cr d

4104 - .C - -a EW0.W - 2-_ (U- -.-2 >1 .. E >lS *- 0j I-I *.aID' - 4. 0~ W X'. 4seo- mn (V in C 4'S wc 's o ~ S4o L- .- n~n-W'' - 4EC ' u

t.J- -'04. ST '-'li ULA':~ ;2 E -' 0~ 0S. a- -v On 0 L4-A 4'~ 4L 0. S_ Rs- (DC n = ; 3 C_ '0,;--14-2 om 0 (1 --0 41 U~ s_ W40.C. _o.C LA

*.j 0 I-in Lu ~ 2Oi wE~' -0. 4j vAWW E,4-s. 4jich.-WS.-- 'i -W * V a-.-- dLiZ ~ ~ 4 4J5> Xs W6 4LjUC'l C60.5 ..- -u s_ (D - -,-4J-a.- 4UC Ua.-C O D Cn*-U C. e5 *U.. 5 2 -L.-. > = -a in- a) -'d u s. 4j 0o " 4)Cl 1 411us0 V - o in *. Q0 Ms~J' 4 m W -- a.WcouW 'u 0

E r . .- WWW0 C.- 2''u-in. 1, C .:C'. S E *.,u l0in in >L in' 4-. 1. -U. :C UA.CV i 4 ' - '4U 4 >4 3: 4' nn~i LA4UJ1 0Li 'a

uJ W-0 W . 0 5 s.-Cn --U WE a0iW. cN00..Z0 L0U . 10~.- 4-' Mi. -0- 39- O L.- J_. os>' 4-.,F . sri- Ci . - iv L.Oiu tA M_ e 45 C - 4O-civi o L0 ~ ~ ~ ~ I a.1 It>-1 a. a= L..-- >-0 a. 1 - *-,0,i a '0L 6 _ _ _o _ _ _ _ _-r c

4-': : LoM(D4-QC C)>u ( 4. q - 03

2u-' w- -o w 0

mI 417

r 0,0-aoi i 0,.C1

io 2W -WC IC.- oU 4j Wa -> Ia=U ) z .. ( c'4' 0 D. - >1 o -I-a.-q* C '-' rA'4 01 0 -,d-

o: -. C _3 CL 'a 20' (A"0 .'' L: 4-64 2. aI a.O s_~ U,

o 5 44~~. -4 ~ 4) W 4. U 'UI--.o *iU.;- 2 L I.n 245 .on, *0 0 5

cc u. 4O s-S W 0 C. EN -2 4' ' 0~ 4."W

LE .0 E m 4-c IV 5.M ~ W4'24' W +A 0 M- 0 .- - W 0 4

.02~~ t- -v-D o. 444 - 00 W _ I.-0. 4 W 42-.- 0L )~ 20. M .'- "4 4.D'. -4.~ 44 - 0 0 C

04C d)> .'U L' m 347, 4f MA 4 0 4 . IW 1 c L -- iEMcL-XmML ML W CL 4-POL.L WLM a

cu' wL. 0. 0_ C 22 0a).L4xOL440. 32' W 'v- EC 04o 2 4' -- Li E.- 0 - J- U c >,2 4-O C .*4 IO U ~ L-~-- J r4~W 'W 4S 4-' W 3C-. 4 >La IA 'WW.- 5 .UxW

41W Wi 2-> r= 4LA> E5 G~' '-- Li WO.C =U2 .0 '4 1n2.o

C W'U. -3: L UWC 14) .U LOWO C W'. 1.n'uC4V 4CA'u4D oL= 0+.- a, 2.- .W 4 *--)( W.- U 0.' 4.'- Z-C. (; 4JCW

wL d)-i 4) wL 4.n.VL-- .UV W > F UV 5.3 = ~ 20 3O >n 20 UL 0 UW 45

g, . _-1oJ1

Problem 4 illustrates how the sizing program described herein

may be used in conjunction with a stage optimization routine to develop

a near-optimum preliminary design of the propulsion components of a

multistage vehicle. A typical stage optimization routine, which was

computerized by Mr. J-. H. Dobkins of Teledyne Brown Engineering, is

described in Appendix B. Results from this routine, which is entitled

the Missile Optimization Program (MOP), have been demonstrated to

agree closely with those from other more elaborate analyses. Since

MOP requires ideal impulsive velocity as input, the designer must

estimate velocity losses which will result from drag and gravitational

effects. Additionally, input values of mass fraction and specific -impulse

for each stage must be specified. Minimized gross vehicle weight and

* optimized propellant weight distributions for a prescribed number of

stages are then computed using MOP for the ranges of-payload weights

and ideal impulsive velocities of interest. For selected point values of

ideal impulsive velocity and payload weight, the designer computes

corresponding values of required total impulse and thrust (for a pre-

scribed burn time) for each stage. The sizing program may then be

utilized to establish optimum diameters (or lengths), -chamber pressures

and expansion ratios for all of the stages as- discussed previously.

Following the preliminary design of a single or multistage ,ehicle,

the evaluation of its predicted flight performance is usually desireG.

The -trajectory analysis which is 'equired to evaluate flight performance

is beyond the -scope of this investigation. However, vehicle design

characteristics, e.g.-, weights, -mass flow -rates, thrust levels,J [envelope dimensions, which are desired from the sizing program may

be input into a trajectory simulation computer program to predict

flight performance. The trajectory simulation program described in

:Reference 12 is typical of many applicable programs which are available.

4-3

4. z SAMPLE PROBLEM

To demonstrate the validity of the computer program, a sample

problem was processed which consisted of an analytical reproduction of

the Thiokol TX-354-3 (Castor II-A) rocket motor design described in

Reference 13. This motor was chosen for discussion herein because

of the ready availablity of Unclassified documentation describing its

design-and performance characteristics in detail. Other rocket motor

designs described in Reference 4, e.g., Spartan, were also analyzed

successfully using this program, but the classified nature of their design

characteristics precluded their discussion herein.

The cross-sectional arrangement of the propellant grain in the!TX-354-3 rocket-motor is illustrated in Figure 4-1. The grain design

is basically a cylindrical port type which is segmented by two radial

slots, as shown in Figure 4-1. A propellant web fraction, Fw, of 0. 76-was

deduced- from the published geometric characteristics of the propellant

charge. The TX-354-3 nozzle design (Figure 4-2) utilizes a conical

expansion geometry, a graphite -throat insert, and requires no entrance

structure. The nozzle is designed for operation-at near-vaccum condi-

tions. AISI 4130 steel is utilized as the structural material for the

rocket motor chamber and nozzle body.

The required-input parameters corresponding to the TX-354-3

rocket-motor were -translated into a set of motor design- and- performance

characteristics using the computer program described herein. The

computer printout of the input and- output parameters of the sample

problem analysis -is--presented in Figure 4-3. For comparison, published

values (Ref. 13) of pertinent design and performance parameters of

the TX-354-3 motor are included in parentheses adjacent -to the corre-

sponding computer values. Comparison of the published and computed'

[ ,44,

-il "P O

Ib

u-C)

JU,

IiC s/TIZ

4

_______________ _____________________________ I______________ A ~;4 ___________________________

-~ 1_ .1 _________________ *1 -

w(/.)

w-Jr-4"4

4.- 0

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'4' 4~ -

3 .4' .4 _____LACV)

L 3 IIS I *

I I aI I . ~ 0 .~j.Ii *' I 1 ~ {,IJ w

4 2~.C,,I II * b Li.

___ i~______ ~1

________ -

L 4-6

.. . . .. 0 . . N 04 0

. . . .40 . . . . N .' ..- * .0

-- a3 , -- a 1

0 w~n nlen I7t. 0u1 4*000 u14.4 W.. 42nf to .3; n

a 4S~l141 407, 0'.44 IV0 r -1.0 0200 *4 .q.001 44 000 -44,~ o r* 0 3 4 9 744 . 4 1 * 4. 4 ' . -

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z* Zw c 0 Z.- -t 'a*l 0r~ .0'.F - -l-' .

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02 04 .44V *.. . 024 .4 0

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r, z42.4 Z-2 I 3 W r Z: 9.4s 44,0 -V. 7 0

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,4

4444.2, .40 0.

0 5c4

.1 I

Or 44a.44i. 4.470 0 0TV0 - *

, 0 0114

a 100 10 .. 1%. 04. 0 m.,-. -&; 0

04 .Ofr. 44 OL0 0 :ILr' 2

j 43 'O0.0.', N4'4.41.00 4 *744-7

values reveals certain discrepancies which must be discussed. For

example, the computer nozzle weight of 170. 5 Ibm is much less than

the published value of 539 Ibm. This large discrepancy is difficult to

explain without a knowledge of the details of the original structural

analysis from which the nozzle was designed. However, it may be

speculated that bending stresses from predicted lateral flight loads

could have dictated larger required structural thicknesses than those

which resulted from the pure hoop stress equations employed in this

computer program. (The credibility of the nozzle design procedures

employed herein was demonstrated through the almost exact analytical

reproduction of the existing Spartan booster nozzle.) The only other

conceivable explanation for this discrepancy is that relatively large

design safety factors (>> 1.5) could have been employed in the original

-design analysis of the TX-354-3 nozzle.

The only other significant discrepancies between published and

computed motor design parameters are in total motor length, total

inert parts weight, and mass fraction. The discrepancy between the

published motor length of 245 inches and the computed value of 242. 6 inches

results primarily from the neglect in the computer program of the reported

lengths associated with the pyrogen and nozzle attachment bosses (Figure

4-1). The discrepancy between the published total inert parts weight of1523 Ibm and the computed value of 811.4 ibm results primarily from

the nozzle weight deficit (368. 5 lbm), the chamber weight deficit (92 ibm),

and the weight deficit caused by neglect of the additional insulation weight

required by the two radial slots in-the propellant. Additionally, the

discrepancy between the published- mass fraction of 0.-84 and the computed

value of 0. 91 results directly from the previously described discrepancy

in total inert parts weight.

L -4-8

Additional test cases, covering a variety of motor configurations,

should be processed in the future to clearly establish the accuracy and

general applicability of the computer program.

4-9

£

5. PROGRAM OPERATING INSTRUCTIONS

The previously described design equations were coded into a

FORTRAN computer program consisting of approximately 530-cards.

Multiple cases may be submitted simultaneously by stacking the- input

cards back-to-back. Only the -input parameters which change need be

added to complete each new case. The remaining input parameters

always revert to the values used in-the preceding case.

5.1 INPUT DATA

All program input variables are read into the array A(i), i =

1, 27. A card-by-card description of the inputs, including symbol

definition-and format, is presented--below.

S Card 1, Format (I2)

A Code word KTR

A KTR is the number of input variables (one per -card) tobe read following this card. At the end of the -last case,input KTR as 99 to end-the job.

-0- Cards 2 through 28, Format (IZ, El5. 8)

A These 27 cards contain the 27 input variables which are

read into the array A(i). On each of the 27 cards, theproper index, i, is punched into columns 1 and--2, andthe corresponding element A(i) is punched into columns3 through 17, using the format given above.

A The 27 inputs, A(i), i = 1, 27, and the equivalentprogram variables are defined in Table 5-1.

* Card 29, Format (I2)

A Code word KTR

A This is the same code word as described above. -Ifanother case is to-follow, KTR is the number of inputelements A(i) to be -changed from the values used-inthe preceding case. If no more cases follow, KTRshould be punched as 99.

5-L

TABLE 5-1. INPUT VARIABLE DEFINITIONFOR CARDS-2 THROUGH 28

EQUIVALENTPROGRAM

ELEMENT VARIABLE DEFINITION UNITS

A(l) DM Rocket motor outs.ide diameter in.

A(2) PC Chamber pressure psia

A(3) F Required thrust F

A(4) FW Pronellant web fraction --(Ratio of web thickness topropellant outside radius)

A(5) APT Port-to-throat area ratio --

A(6) PAM Ambient pressure psia

A(7) TC Propellant flame temperature OR

A(8) RHOP -Propellant density lbm/in 3

A(9) GAM -Combustion gas ratio of specific ---heats

A(lO) ATB Pressure-coefficient in pro-j pellant-burning rate law

A(I1) AMW Molecul-ar weight of combustion l bm/moleL gas

A(12) AN Pressure -exponent in propel-lant ---burning rate law

A(13) ETAD- -Mass discharge correction factorfor nonideal flow

A(14) ETAV Exit velocity correction factorfor nonideal flow

A(15) ALF -Nozzle exit cone half-angle (for degcontoured nozzle analysis, -input-effective value of ALF of 17.5-degrees)

TABLE 5-1. INPUT VARIABLE DEFINITIONFOR CARDS 2 THROUGH 28 (Continued)

EQUIVALENTPROGRAM

ELEMENT VARIABLE DEFINITION UNITS

A(16) DEM Maximum permissible nozzle exit in."plane diameter

A(l7) FP Fraction of propellant burned

before -web burn through

A(18) SFM Motor case design safety factor

A(19) SFN Nozzle -design safety factor --

A(20) WPL Stage payload weight (includes Ibmterminal -payload and upperstage-weight, where applicable)

A(21) WEX Miscellaneous weights to be lbmadded to stage weight (e.gthrust vector control hardware)

A(22) ZLN- Nozzle length of next higher in.stage-

A(23) ZLI Interstage clearance between in.forward head of stage beingdesigned and nozzle exit planeof next higher stage

A(24) CODE Code designating which stageis being designed (CODE = 2.0

]' for terminal stage of multiplevehicle or for single-stagevehicle; CODE = 1.0 forremaining stages)

A(25) FLAG Set FLAG = 0.0 when head end --web(s)are not present;-FLAG =1.0 forwardlead web only;

FLAG --2.0 for forward and afthead webs; FLAG = 3.0 for afthead web only.

5

[ 5-3

TABLE 5-1. INPUT VARIABLE DEFINITION-FOR CARDS 2 THROUGH 28 (Concluded)

EQUIVALENTPROGRAM

ELEMENT VARIABLE DEFINITION UNITS

A(26) NOZ -Set NOZ = 0.0 for conical --

nozzle analysis; NOZ = 1.0for contoured nozzle analysis

A(27) ISLIV For cases where FP < 1.0, set --

ISLIV = 0.0 for active slivers;ISLIV > 0.0 for inert slivers

-

5-

In preparation of the program inputs, FORTRAN coding sheets

are usually completed to serve as a -guide in punching the appropriate

values onto input cards. Example input coding sheets, including input

variable-definitions and typical ranges and/or recommended values of

the inputs, are presented in Figure 5-1.

5

!_

F

[ -

FIGURE 5-1. S-IPLE INIPUT CODING SHEET

PROGRAM TYPICAL RANIGES ORSAMPLE INPUTS* VARIABLES** RECO"MENDED VALUES

1 567 10 15 20.

" RIumDer ot inputs to tot tow or. .. I . KTR Q tn n inh -

Constrained by envelope1 E+I02. r DM limitations

.2+ 9 E!1o9 k PC 300 to 2000 psia__ - Dictated by total impulse,

. ±, L 76.9O I E.t4O F hurn tinm rduir~emnts-a S[far. wagonwh.eeli P,20t 0.5

.- 127 r +ol .... APT 1.1 to 3.5

I EAoo . . PAM 0.0 to 14.7 psia

9-112 I E+0.4 j TC 4000 to-6600 OR

- t . E.,,,.o + RHOP 0.053 to 0.07 ibM/in 3

p 1 LGAM 1.12 tol.2

S. O .. .. O.... . ATB 0. 002 to 0.1_

E Lof~z.. f Mip 25 to 29--

L.. . 27 . Ei.O. AN 0.1 to-0.

ETAD 0.98 tol1.15 (1.02 typ.)

I_____r ______ ETAV- 0.85 to-0.98 (0.92 typ.) -

Sj E.+iaz ALF 10 to 25-de(. (15 typ. )Constrained by e yeloe

DexpanU ratjo irntatoas

+VO.. FP- 0.9 to-1.0

1-. - __151__ E._io ___V SFM 1.5

SE I -- SFN 1.5

12.. +1 4kO, WPL Dictated-by mission obiectivesAdditional-miscellaneous-eight

2--0. ...<..._._,_,, ,WEX (Fins.-TVCHardware. etc.1ero-for single or upper stages;

ZLN Appropriate value for lower stages.

"_.o. AJ±1 o___ ZLI 3 - 10 in.

FLAG **

E+OO.. NOZ **

.. . I OO, ,,IISLIV **

, .. KTR End of job

-Values listed are inputs for Sample Problem (Figure 4-3)** Refer to page 5-1 and Table 5-1 for Definitions

5-6

5.2 STORED DATA

Most of the -material physical properties which are requircd in

the motor design are defined through the use of FORTRAN DATA

statements within the main segment of the program. This procedure

was adopted to reduce the number of inputs which would be required

in parametric analyses once -the motor case, insulation, and nozzle

materials have been- selected for a particular rocket motor design.

Program -flexibility -is not impaired significantly,- because various

structural materials may be readily analyzed by simply modifying

-the appropriate property values in the preyiously mentioned DATA

statements before program compilation. These property values are

then employed by the program- until further changes in the DATA

statements are made.

The program variables which represent the various -physical

properties are well'defined in the main program listing in the Appendix

-by several COMMENT cards. Program variable definitions and

corresponding typical property values are presented in Table 5-2.

The values shown in Table 5-2 are those incorporated in the program-

-deck which -is listed-in the Appendix.

J5* -

TABLE 5-2. TYPICAL MATERIAL PHYSICAL PROPERTIESAND EQUIVALENT PROGRAM VARIABLES

PROGRAM

VARIABLE DEFINITION TYPICAL VALUE

RHOM Motor case density 0.283 ibm/in 3

SIGM Allowable tensile 150,000 psistress

Motor'Case EC Young's modulus of 30 (106) psi

case

EI Young's modulus of 30 (106) psiinterstage structure

BETA Forward and aft head 1.0 - 1.6ellipse ratio

RHOS Density 0.283 ibm/in 3

SIGN Allowable tensile 150,000 psij- Nozzle' stress

Structure ESUBS Young's modulus 30 (106) psiMUS Poi-sson's ratio 0-.3

CSUB2 Specific heat 0.11 Btu/lbm-OR

-RHOI Density 0.062 lbm/in 3Nozzle2Insulation KSUBI Thermal- conductivity -0.5 (10") Btu/in.-sec-°R

CSUB Specific heat 0.21 Btu/lbm-°R

RHOT Density 0.-062 lbm/in3

L Throat 3 )ESUBT Young's Modulus 1.7 (10) psi-

Insert -MUT Poi sson-'s ratio -0.12SIGT Allowable tensile 3,355 psi

stress

Insert 4 ( RHOSL Density 0.02 lbm/in 3~Sl ivers

NOTES:Typical values for 4130 steel.

j 2 Typical values for FM 5048 silica/phenolic.

3 Typical values for ATJ graphite.

4 Typical value for phenolic microballoon.

5-8

6. REFERENCES

1. Weisbord, L., "A Generalized Optimization Procedurefor N-Staged Missiles", Jet Propulsion, March 1958

2. Stone, M. W., "Grain Design Handbook", Internal Memo-

randum, Rohm and Haas Company, Redstone Arsenal-,Alabama, October 4, 1956

3. Stone, M.W., "The Slotted-Tube Grain Design", Re-port No. S-27, Rohm and Haas Company, Redstone Arsenal,Alabama, December, 1960

4. "Rocket Motor Manual", CPIA/M1, Chemical PropulsionInformation Agency, Johns Hopkins University, SilverSpring, Maryland, May 1969

5. Barrere, M., et. al., Rocket Propulsion, Elsevier Pub-lishing Company, New York, N. Y.-, 1960

6. Sutton, G. P., Rocket Propulsion Elements, John Wiley andSons, New York, N. Y., Second Edition, June 1958

7. "Program for Preliminary Design of Internal BurningSolid Propellant Rocket Motors", Thiokol Chemical Cor-poration, !Huntsville, Alabama

8. Bruhn, E.:F. , Analysis and Design of Flight VehicleStructures, Tri-State Offset Company, Cincinnati, Ohio,1965

9. Alexander, R. V., "Summary Report-of Program Beta, ADigital Computer Program for High-Speed Analysis ofPerformance, Preliminary Design, and Optimization-ofU Solid Rocket Vehicles", Technical Memorandum No.176-SRP, Aerojet General Corporation, Sacramento,California

10. Threewit, T. R., et. al., "The Integrated Design ComputerProgram and ACP--103 Interior Ballistics ComputerProgram", STM-180, Aerojet General Corporation,Sacramento, California, December 1964

11. Demuth, O.J. , and M.J. Ditore, "Graphical Methods forSelection of Nozzle Contours", presented at the SolidPropellant Rocket Research- Conference, Princeton Univer-sity, Princeton, N.-J. 28-29 January 1960

FII

6-I

12. Boehni, B. W., Rand's Ommibus -Calculator of the Kinematicsof Earth Trajectories, Prentice-Hall, Inc., Englewood Clifts,N. J., :1964

13. "Technical Description of the TX-354-3 Rocket Motor,Report No. 57A-66, Thiokol Chemical Corporation,Huntsville, Alabama, April 7, 1967

-V

11

Ii-

k [A 6-2

APPENDIX A

FORTRAN LISTING OF SIZING PROGRAM

A complete listing of the FORTRAN statements is furnished inthis appendix as a reference for making any desired changes to thesource deck of the solid rocket motor sizing program.

t

11

I

r ;. A..l

4h4 - - - -- -- - -

S U.

3a r r - m4 o

') 4 - .44: - .4 It'f J 1 04 C 0 44 N Ml

'.4 .1 4 x 44 4 u .- X-0

44W: -4.4 X4. -. 4, 4 4 4

01. X- . w &Z9 C

4f 2%0 2=1 -a1 . 0 4 -

44. tZ - - -a mD .4 Ia wi Ii 4'a ~ ~ ~ ~ w wo% 4 . 4 .-

0.o %. 4 -44 . . tI

&m taw.414 -. . &r- . wn0 4.o

o. - 4 &4 . w 1. o

5,- ow- 4.4 0 -.. N"4.1w L- z V;4 . o0 0 N~ 0' .2m Z0 a% 7

r.0 I N .a -o ' . o oza 4 l zo44. "144 43 ; 40au

0 z I,41 =4 05. T-4 ' .

w. 44 w4 . 04 .. 4 4wm W4 3t.44a wu - .4.4/ - m 444. y4*% 2 40 *

u* ooo Quuuo o.o

0C44C44~ ~ v,,4. ' -3

:4- *4*''* I4~144 . .04 -4 4 4

I- li

'aN

ILA

t NN

4w 40

0 .

44 I I4

IL 20 a, a,

A-3 A4 r f

itw1zIw

kz

Q wx' o I~ II

pam

14 1% 4

2.4 ulzTo4

o 4

I *IAa 4 N

4204 Is." I 0, . . -'.J1

N uo WOOt-, al Nw . '. moo. I* C* 4wo' 42 ..4 o .. Ng4 JS 4

* ~ ~ ~ ~ z 0. %1W3IA 0 . m WN43 33xuM N o &L 'a w . 03.4o h cc.4 0*414044 o00 ! r-. o 0 3 40 04 . 4.. x 2-0 *aN0w4 4.

ww o tt7 *2 -Z24 330 0 W t4 S 4 .. 4

.4 ~ ~ j 144 I.'0

0 .40. 0. N O.40 40 N& 7 0> 0

- ow .So U .0 N ,. 0 VI *4t0,.4 sIs0r'44s-

5.4 0Z .I *4~3UJ4 230.71 * N.-. 241 .3s7 .... 7

We &'.0.. N0 - -4003 * N.sS ** 4 47~.).3

a& w

taaL IL

-. O f -IU

2"0 0

I , ~ a - .-- IOL. OINJ N%I ~~~ m 42N K.0 N -*~" ILN~ ul; 4N4 N

0.0j U ... Z4

" ;.11."1.: tNS10LJ 0 z a

0 ao T0 z" 3 a .5 % .W O.WIflt .- 0 wo j xx00. -X ' 4 .& m .

.4.1401 .4

.100.0-0

.4 . . . .C~= " z .i .*' ...

0 03 a.,0 N4L .0 .42 40,a,0 1!402 12 I4 IT(4 U. O. 5 010 :

W W N IX 'c.4*3.' 0'.

W9 . *A I1 ;:w.-.4 00

CA2 0 a94 CI I- r 5.4_l2 l . x

Z,-N 10 0

II 0.W OW 0A-5

v x

41 U. U 4 Z4w

N w ec 5 0r -mN 0

Nz w

w* .41.N w .4 4

4.1. -0 4 w4 '4w-p'

. .W o4 0. -4 4 00 0

- 4X0 1 V .ft~ft OW 0. f I . I

.0~~~~ ~ ~ ~. .WOw. NC 0 .%. '..

41 a1 ox N N N 41 0 *01441 *NNIX 440. r4I

'0 Qz o 00 1 N .0 I0 .C .* 0 . . . . . . .Q 0 D 0

N-O N N" . A N II. Nm..4 - . .4 o ' 0. w 2 -NI -

*.4tm 141 *N N .1 N 244 4 0.

N 4j4. 1 **...~-I4 . . . . . . . . . . 2-.J 21.A-644.

IL %%

41 1..aI 0L

a 0I~~ ~ * - ta.4.423 3M . 3*3 *0

I ~ Cs- - if s-

09 I..-3I~~6 z ~f.

zw.~

a KW2 .0 1.0

a ago ON M.o- M 4

.z EW4 e - 3-t __Ok .4 -0 &1 . . W

N a zW - z &Z nw00.3 to - ,

.0 w w34 0 OU'" .. aMWWh ;- . .4 *0 1N3 11 . 44 0 x O . D

Z .a.Z-..Z.33.0 O. - .W aw - X. W4N -C .~4 -. :1 0w4. a m 2I 30W3. 05. - wwK*..4.'( Z .3ZOm OZ zo.W.

0023- u3.4IfL034420 a. .N.-3..n Z0if. Zf43.a 2 4 *4

.434 ty

1) 0 U ow 0 wo U 0W W U 00 U U0.

z

0LIb - W 1 WI- W, IA

I A-7

'A

LI1- x b-

--Ii

TW2 A A IA 0 0 Wo

M- '3 1. ADJ

U. hi = j I M64 i 4 01 Md6 w O D W

e xx x 2 0 0 sC

2 0

vi hi

.0 '3 A-8

N 4-o

IIha-*1

o~ 0 00 at0'0 00 0 0 0N N N NNN N N N N N N

6C I0- X A 0 -0. At- -. u N

03- C. 0- -1- 0--MW4 W 0 %

I 0 OX -0 -60 --

.U - CO*I I CN *C

V. z * -40 .4 ZK W- W_ -. - MU

* 1 C. IC * ff- RI'D IgS IJ AN%" -ee "W C9 0 S OW I t 2 "

C; 0#6 0.*CU0 -ON.

LN RU .. 46 VIN 4:w .02 a- xi -a do

-Uz .*. . oN. 0oo meet*N %I __ =0 a *1 *4W 11 * NB4;

00 WNm_ ' 4.. UgI -b. MO 0 9-N.3- 6. *NCJ- .0% C*W%% R B SS

c .W*NCR 0 NU. wB _00.Z I- awlCI- .~4e3N UN0 I- !Z N 4 -*.13 B

x%, .N . *I. . Za0%- y W4 B, -0W- X, - ' -- I erNO;;.

N:)4 *XC. .N OW oN I-40C IL L 0 3 0 . I * W

W RN .D .1.'. I-N WI- 14.3 W0~ 4m:I-O INICBBN O W W _ . It "0 0 S" CO". NN ZN 0IW IC.%- *U

WoC 1- 0 0 be40 0B N~~~~m*k 4BNJ~* 0% 000W&B O LX-- ~ -*IR *. IN~%Z jl "' IN'. rW *N U C R .C B . U I-0 N o -0 tWRC~ I N.IN 3- RZ N 4= .N.814% .J I- .J R CON R -R0 .. C4I~~~~~~~~~~~~O BR.JCf 0.I.I w xR. N U.~4 N O N N 400.%J4 . S

I- W a0N 003 U . .W-rw .. W4 C A *'. W N R of*x o N LLR O.N if0.-.B42 B RW _

W N BU .NU .3 0ON I--O .R IR N Iy 90 I I0Z N 444114 IBei R N C.NIe .*O

V I, eg I W 414 0WI. I e 1,NO 61 e In.

BN3IBB.CBZROUI .JJUBIIS..B A OIU9I

00 00

4

I--v .t

* Cj 0

Irv 4 -

- I- Us-VS

A- 1~m- .

w ow

s-sww I* -

.z s -- M7.,

mw~ "5- 46-aUN....*. I-a U

5*A-1

0

P4 ~ ~ f .N N0 ly-*0.

A III

ZO4 -0 L)

0 0 -P

M. 0 % .4

t. 0 1 1 4.1 pi

4.w Ito. Z 0.fi 4 . 6 u* leg! so

0 z4 t.. "m. y4 "m T

e. 00.0 = . 0 w 0z 0).- W X*! to*. OW 4 4

I. -. ~ U.4 II W0 . UV- x AW.2 P . 2 ,4 ft- . 0:44 50. 40 s .

IL %0 414 ft.4.4I..4

0 . 0. 0 *Zo Z 40444 404. ~ ~ ~ ~ ~ ~ ~ I P. 40 944~JWM-~-OP .

S4..441 244401.4 4.4W.S.4.4 4012 Ub-

-. 5 4 0445) 44. .

or 4 Z K 0 O O 1 WI . . Z Z 2 Z W

4404004444.404. .. 4 U4.44..4144 4.04U4.44444 *.4.4 .4.4 41 I

44A-11

4, j

- 1.0

ly N

-A 12

IL7

IV . 0 . 0

-0 0 --

0

6- W - -44

% zN ZR - 0

W 4* - C

W 1 1-, 10ko -0 W IL

a. a. .- 0 W&

0 4 0 MWs-

U W * N O a.CW

x AL Y . WW -0,4N W 3; 10.4Z N 0.4 0

0 z

P~~ -lo~ N~ w&

a Z EWNNC= O 04. JU 4

a U.C Z 2: :z IL

4-0-5 W~O ofJ 4-Is .

W4 ZRt .4NN*B. U U

V.B

0. N NZX* N ON:

4- ! .-. go 4 s

Z"2 I4ZvIOOZ% . N."o. 0 04.0 ZS 0 o L) 4s 9WC~I4~N54 4*N M044 4*aO UW cocaU.4-j O54 N m4 a a 4 N ly 400 so. 905W

0

A--13

APPENDIX B

DESCRIPTION OF MISSILE OPTIMIZATION PROGRAM (MOP

B. 1 ANALYSIS

I The following analysis is based primarily on the work of

Weisbord (Ref. 1) with modifications to simplify automation and nn.: .. 1

changes to fit current usage.

The ideal impulsive velocity, AV I , achievable for any roclkc

with constant specific impulse for each stage is given in Equation 1.

wol WozAV = C In + C2 In .bo---- (1)

Wbol WboZ

where

Wo, - gross start weight for the ith stage, ibm

Wboi - burnout gross weight for the ith stage, ibm

I C, - gcIsp, ft/sec

I For any given mission, AV I can be estimated from Equation 2.

j tb

AV I = AVact + g sin 0 dt + AVD (2)

I 0

4 The Missile Optimization Program was developed by Mr. J. H.

Dobkins of Teledyne Brown Engineering.

B-1 .

where

AVact - actual impulsive velocity, ft/sec

0 - angle of velocity vector above local horizon, rad

AVD - drag losses, ft/sec

t - time, sec

tb - burn time, sec

For large missiles drag losses usually amount to less than 5 percent

and are not considered important for preliminary analyses. Howc ,c2-

for smaller, high-velocity vehicles this factor becomes more c~r1,

and must be estimated.

The gravity loss -term is zero for horizontal launch and is (gt) io,

a vertical launch. Thus it-is strongly trajectory dependent and son~e

approximation -between the limits must be -made.

Each stage consists of motors, propellant tanks, misceli c n

hardware, guidance, etc. The mass fraction (XMFi) and structu:7

fraction (XMSi ) of stage i are defined in Equations 3 and 4.

XMF i = Wp/(Woi - WPL i ) (3)

XMSi = WS/(Wo i - WPL, ) (4)

where

WPi - weight of propellant, lbm

Wsi - weight of stage - empty, Ibm

WPL i - weight of stage payload (upper stagesplus payload), Ibm.

B-Z

Mass fraction data can be estimated from past experience. Structural

fraction is calculated within the program from the mass fraction and

thus need not be input.

The burnoixt weight of any stage may be written in the form of

Equation 5.

Wbo i =Woi+ + XMS i ( W o i - oi+ 1

= XMSi W0 i + (1 - XMSi) Wo i + l (5)

and Equation-1 -becomes

N Woii AV I = Ci In (6a)i= 1 XMSi W0 i + (1 - XMSi) Woi+l

or

0 ~I~n Fwi0+I AV1 (6b):i= 1XMSi W0 i + (1 - XMSi) W

The problem requires that Wo. be minimized for a given AV,.TI

- Therefore, the Wo, become variables and the maximum or minimum

take-off weight is achieved when

aWo,o 0 i=2, N (7)

-i B-

B-3

Since it is difficult to explicitly write these differentials, note

that Equation 6 is of the form

f (Woi) = o(8)

and the partials can be written

awo- af/aw0 ia Woi af/aWo I

from the chain rule for differentiation. Equation 9 implies that if

t Equation 7 is to be satisfied then

-= 0 (10)-aWoi aw0 iI

must also be satisfied. The process yields for i 2, 3

af C1 (I - XMS 1 ) Cz Cz XMSz0 = - = - "- "- -_-

aWo XMS, Wo0 + (I - XMS1 ) Wo2 Woz XMSz Wo' + (1 - XMSZ-) W03-

Of C2 (1 - XMS 2 ) C3 C, XMS3 (11)i 0 "- + --

aWo, XMS2 Wo, + (1 - XMSZ ) Wo3 XMS3 Wo3 + (1 - XMS,) Wo4

This can be algebraically manipulated to -become

C1 XMS 2 (1- XMS1

WOz (1 - XMS)(1 - XMS2 ) (Cz - C 1 ) + Cz XMS, (Wo/Wo) (I - XMSz

W04 Cz XMS3 (1 - XMS 2 )

Wo, (1 - XMS 2 ) (0 - XMS3.) (C3 - Cz ) + CS XMS (Wo/Wo3) (0 - XMS3

B-4

and can be generally stated as

W Ci.1 XMSi 1 - XMS (i- 1)]

W0i (I - XMSi~1 ) (I- - XMSi) (Ci - Ci1 1 ) + Ci XMS i 1 (Wil/Woi) (1 - XMS i ) (13)

i =Z,N

Note that difficulties will occur when i = 1. An iterative process with

an estimated Wo? /Wo, ratio is required until the required AVI is availa-

ble.

It has not been shown in the previous discussion that the resulting

data will represent a minimum weight vehicle (rather than maximum).

A simple test is to try any other weight ratio between stages that pro-

L duces the same performance and compare the resulting total vehicle

weight, Wo, . Other weight ratios invariably produce greater values of

Woz , thus proving that stage weight ratios determined by the previously

described procedures represent a minimum Wo.

The terminal payload weight is finally used to calculate the Woi

for the terminal stage. The vehicle is then specified totally from the

following equations.

Wsti o i+l/(Woi+l " Woi+l (14)

W = Wsti XMF i (15)I- Pi

Wsi Wsti - wpi (16)

Wboi S Wji + Wsi (17)

AVI = Ci In Woi/(Wo, - Wp,) (18)

where Wst. is the total weight of stage i.

B-5

B. 2 PROGRAM APPLICATION

The Missile Optimization Program -(MOP) will determine the

optimum weight distribution between stages for up to nine stages. If

for some reason not enough stages were input, the system will assume

extra stages with parameters equal to the top stage until it is possible

to build the system. Conversely, when the number of stages is too

great the last stage is ignored until the system is feasible. This does

not mean however, that the numbe--r of stages has been optimized.

The input data required is given in Table B-I. Provision has

been made to study a spectrum of velocities and payloads and obtain

vehicles optimized for each combination. Appendix C contains a

~ FORTRAN listing of the program.

I

I.

1B-6

TABLE B-l. INPUT DESCRIPTION FOR MOP

INPUTCARDTYPE VARIABLE FORMAT DEFINITION

1 DV F1O.2 Ideal Velocity, ft/sec

WPL FlO.2 Payload, Ibm

N 12 Number of stages, 9 max.

LAST 12 Last case 0 No1 Yes

2 (N cards) XISP(I) FlO.2 Specific Impulse, sec

XMF(I) FIO.2 Mass Fraction

3 VST FlO.2 Velocity Step Desired

j WST F1O.2 Weight Step Desired

JVS 12 Number of Velocity Steps Desired

JWST 12 Number of Weight Steps Desired

-IBI

I

j B-?

APPEND IX C

FORTRAN LISTING OF MISS-ILE OPTIMIZATION PROGRAM (,,P)

A complete listing of the FORTRAN statements is furnished in

this appendix as a reference for making any desired changes to the

source deck of the missile optimization program (MOP).

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