r D-A.24 .i2 ITERATIVE CURVE FIT OF A POWER LAW EQUATION TO SOLID 1/1PROPELLANT SHIFT FACTOR DATA(U) AIR FORCE ROCKETPROPULSION LAB EDWARDS AFB CA C R FARRAR ET AL. NOV 82

UNCLASSIFIED RFRPL-TR-82-878 F/G 21/9. 2 L

*uuuuminEuiEson hhhhh

'1.

la 1 6 .2-

Dii1.251 14 66

MICROCOPY RESOLUTION TEST CHART N

NATrioNAL BUREAU OF STANOARS1963-A

*-2

AFRPL TH42478 AD

Special Report Iterative Curve Fit of aJun 161toPower Law Equation toJanury 962Solid Propellant

Shift Factor Data

44 November 1962 Authors:Charles R. FarrarDurwood I. Thrasher

Approved for Public Release

Distribution unlimited. The AFRPL Technical Services Of fice hoe reviewed tis report, and It isreleasable to the National Technical information Service, where it will be available to the generalpublic. including foreign nationals.

DTICELECT

* " Air ForceRocket PropulsionLaboratory DAir Force Space Technology CenterSpace Division, Air Force Systems CommandEdwards Air Force Base,

*California 93523, 1

I. i I I _ I . _ I. . ' . _ - .

NOTICE

When U.S. Government drawings, specifications, or other data are used for any purposeother than a definitely related Government procurement qration, the fact that theGovernment may have formulated, furnished, or in any way supplied the mid drawings,specifications, or other data, is not to be regarded by implic( tin or otherwis, or in any

manner licensing the holder or any other person or corporatl,*n or conveying any riots -

or permission to manufacture, use, or sell any patented Inv, tlon that may be relatedthereto.

FOREWORD

This Technical Report was prepared by the Structural Integrity Section (MKPB) of theAir Force Rocket Propulsion Laboratory under Job Order Number 573013NB. ThisTechnical Report is approved for release and distribution in accordance with thedistribution statement on the cover and on the DD Form 1473.

Project Manager Chief, Structural Integrity Section

FOR THE DIRECTOR

THOMAS C. MEIER LT COL USAFDirector, Solid Rocket Division

-',o .o. . - .' ,,. ... - a . -... .-.- -.- . -_. -' - .- - -. , - -... . - . . ,. . -. .- .

SECURITY CASSIFICATOU OF THIS PAGE (When 00ao Entoee________________H~RAD MM-u'C-noNs

REPORT DOMENTATION PAGE BEFORE COMPLETIG FORK1. REPORT 1UMMEN 2. GOVT ACCESSION NO. 3L Ri CIPIENT'S CATALOG' NUMBER

WRPL-TR-82-078 L""Lj ______"__--._4. TITLE (id ^"Well S. TYPE OF REPORT & PERIOD COVERED

Iterative Curve Fit of a Power Law Equation to Special ReportSolid Propellant Shift Factor Data -An-, - Jan 82

6S PERFORMING ORG. REPORT NUMBER

7. AUTHORs) &. CONTRACT OR GRANT NUMBErs)

Charles R. Farrar[urwood 1. Thrasher

9. PERFORMING OANIZATION NAE AND ADOESS; Ia. PqOGRAM ELEMENT. PROJECT. TASKORO " O*A"I2 EA a WORK UNIT NUMBERSAir Force Rcket Propulsion Laboratory/KPB ProwrU , Element 62302F

Edwards AFB, CA 93523 Pr t 5730 Task 13JON: 573013NB

II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPONT DATE

Nov,3mber 1982I. N JMUER OF PAGES

14. MONITORING AGENCY NAME A ADDRESSfil dlifeotnt frm Controlling Offce) I5. SI'CURITY CLASS. (of thia repot)

Unclassified

IS&. rECLASSI FICATION/DOWNGRADINGCHEDULE

16. DISTRIBUTION STATEMENT (of this Rmporl)

Approved for public release; distribution unlimited.

"7. DISTRIEUTION STATEMENT (f the abra ent e led In Block ", It diffret bow Repos)

IS. SUPPLEMENTARY NOTES

IS. KEY WORDS (CetCmoti an fteve aide It neceeary end Identify by block number)

Solid Propellant Rocket Motors Time Temperature SuperpositionStructural Analysis WLF Shift FactorLinear ViscoelasticityEffective ModulusPreliminary Analysis Methods

WO. ADSTRACT (Cmnthme n revere. sid H nceeem and Identify by block number)

Software was developed for an HP-9815A programmable calculator which optimizesthe constants of a single-valued algebraic or transcendental equation to fita given set of experimental data. This technique was used to fit a power lawequation to solid propellant shift factor vs temperature data. The resultsware compared to optimization methods suggested by the JANNAF Solid PropellantStructural Integrit Handbook and to the Power Law apprwcimation reported in

L-TR--66 for the same data.

DD, , 1473 EDITION OF 1 NOV ss IS OOvLDTESECURITY CLASSIFICA ION OF THIS PAGE (Ihen Date Entered)

r .C r... ... ... ....... . .-il -ilmm iil ilii uiii iii

CONTENTS

Section Pace No.

I. INTRODUCTION 5

2. SUMMARY 5

3. SIGNIFICANCE OF A POWER LAW REPRESENTATI(ON OF 6LOG (aT) VS T PLOT

- 3.1 Definition of oT 6

3.2 Power Low OT Equation 6

3.3 Application of the Power Low OT Equation 7to Structural Analysis

4. PROBLEM STATEMENT 7

5. SOLUTION 8

5.I Assumptions 8

5.2 Selection of Initial Trial Values 11

5.3 Iterative Optimization of the Constants 12

6. GENERAL CURVE FITTING 16

REFERENCES 17

APPENDIX A. Sample Problem 19

APPENDIX B. Curve Fit and Minimization of E = f(Ci) 25

Accesion For

.NTIS MA&IDTIC TAB 0Unannounced E3Justirleatio

Distribution/ fAvailability Codes IWPI,," /

jAvail and/or.e~Dis Specil

,a l- ao . e lJ u mb . -. I

. ° ' ' 'e b° ', o

. W -• " " o . -" .°.. - ,( -

LIST OF FIGURES

Fkre PageN.1. Hypothetical Error Surface, E = f(Ci),

*for Two Uhtknown Constants.9

2. Variation of Error withm. 10

3. Variation of Error with To. 10

4. Least Sguares Fit of Log aT vs Log [(TR - To)/(T - Ta)j for Sample Problem in Appnix A. II

5. Optimization of the Const nts, Cycle I. 14

6. Optimization of the Cotantst Cycle 2. 15

A-I Curve Fitted to Log aT vs T Data. 24

B-I Curve Fit and Minimization of E = f(Ci). 27

I.2

),2

LIST OF SYMBOLS

OT time - temperature shift factor (dimensionless)

b user specified number of cycles which iterative proc m is run

Ci arbitrary unknown constant

E sum of the squared errors

E root mean squared error

* E(t) relaxation modulus

m positive exponent in power law aT equation (dimensit-nless)

p user specified multiple to generate search internal

T temperature, degrees Fahrenheit (OF)

TR reference temperature, OF

Ta constant in power law aT equation, OF

Tg glass transition temperature, OF

t time, minutes (min)

S constant in interval reducing function

B constant in interval reducing function

reduced time

.1',

-o3

o . . . . . . .

777:77 .~~~ ~ ~ ~ -'- : ' ,--- --. 7- . . 77.

'I.%

ITERATIVE CURVE FIT OF A POWER LAW EQL ATION TO

SOLID PROPELLANT SHIFT FACTOR D, %TA

I. INTRODUCTION

The JANNAF Solid Propellant Structural Integrity Hanlbook (Ref. I) suggests an

empirical power low relationship between temperature (T) tind the time-temperature

shift factor (aT). To supplement existing solid propellant the-mal transient viscoelastic

response software (Ref. 2) an iterative method of fitting th ! transcendental equationrelating T and aT to a given set of experimental data was de/eloped. This method was

implemented in software formulated for use on a Hewlett-Paccard 981 5A programmable

calculator (HP 9815A). It was decided that with minor charues the software could bemodified to fit an arbitrary, single-valued, algebraic or trinscendental equation to

experimental data with the number of unknown constants 1i nited only by the storage

capacity of the calculator.

2. SUMMARY

The computer program developed for the HP 9815A successfully determines values

for the unknown constants of the power law aT equation in ten cycles or less, with atotal run time of approximately ten minutes. These constants )roduce a curve which fits

a given set of lag OT vs T data with less error than the trial aid error method suggestedby the JANNAF Solid Propellant Structural Integrity Handbook (Ref. I). It also produced Ia better fit than the direct approximate solution suggested in A FRPL-TR-81-80 (Ref. 2).

This program was also expanded to fit a general fun-tion (either algebraic or

transcendental, and having no more than three undetermined c, nstants) to a given set of 4data. Due to lack of knowledge in making initial guesses for tl e constants, this curve fit

does not converge as rapidly as the specific curve fit for the F ower law aT equation. In

fact, if the constants chosen produce too large an error, the iterative process will fail

and new trial values must be selected. Modification of the storage capacity of this 6

program would allow equations with more than three unknown c ,nstants to be evaluated.

5......................................

, • _ . ° . .j " • _, - ° . . v •, - - - • - - '-. .- . . , - . .-

3. SIGNIFICANCE OF A POWER LAW REPRESENTATION OF LOG (aT) VS T PLOT

3.1 Definition of (aT)

J.

When a viscoelastic tensile test specimen is stret- hed (at a constant speedunder uniform ambient temperature) to a specific strain Ik el, the force required to

- maintain that strain level thereafter decreases with time. Th, modulus of elasticity willalso be time dependent, and is denoted as the relaxation rr odulus E(t). This test isreferred to as a relaxation test, and if it Is performed at a seri s of ambient temperaturelevels, a group Of similar shaped modulus curves plotted as log E(t) vs log time (t) may be

* obtained.

The assumption is now made that solid propellanis are thermorheologicallysimple materials (Ref. 3) and that the time-temperature mperposition principle isapplicable. This Implies that the group of curves generated fr, -m the relaxation tests areportions of a single master relaxation curve displaced (shifted along the log (t) axis dueto the different temperature levels at which the tests were rui.

The time-temperature shift factor (aT) is defir ed as a non-dimensionalmeasure of the shift in the log E(t) vs log (t) curve due to a c xinge in the temperatureof the relaxation test. Quantitatively, aT is the distance along the log (t) axis between avalue of log E(t) on a reference modulus curve and on equal value of log E(t) on themodulus curve for the temperature of interest. In practice, the relaxation modulus

curves are treated as a single "master curve" of E(t) vs log where F taT is definedas the "reduced time" variable.'

3.2 Power Law aT Equation

An empirical power law equation has been develop .d (Ref. 2) to determine avalue of OT for a relaxation test rin at a temperature abo 'e or below an arbitraryreference temperature (TR) with the restriction that the ten perature of interest must

*II be greater than the glass transition temperature of the propell( nt. The relationship is asfollows:

4 6

4

. . . . . . . . . . . .*.

aT (1)

To is a positive or negative constant with a 3ugges ted value of I OOF to 20oFbelow the glass transition temperature, and m is a positive constant (Ref. 2). The valueof TR is usually in the range of 70OF to 80F (Ref. 2).

3.3 Application of the Power Law aT Equation to Struc ural Analysis

The reduction of a thermal transient problem to an "ordinary linearviscoelastic" problem (Ref. 3) is based on the Moreland -Lee hypothesis used inconjunction with the previous assumption that the materi il is thermorheologically

simple. The Moreland-Lee hypothesis asserts that the red, oed time under variable

temperature conditions is given by

(2)TaT

The power law representation of aT(T) was developed to provide ananalytically integrable function for the case of a linear temprature-time history and arestricted class of nonlinear temperature-time histories.

4. PROBLEM STATEMENT

From the preceding section, it becomes evident that the cnalyst needs a method to.. determine the values of the arbitrary constants Ta and m in the power law aT equation in

order to apply the Moreland-Lee hypothesis. Data relating a- and T may be generatedfrom a group of relaxation tests performed over a range of temperatures. The leastsquares method of fitting a curve to determine the values of the unknown constantsresults in a set of normal equations non-linear in the constant. Ta and m. Therefore, amethod is needed to generate the values of To and m for al vs T data such that the

• 4' power low aT equation fits this data with minimum error; I.( . to fit a transcendentalequation with two or more unknown constants to a given set of Iota.

7

The term *error" will have two definitions in the folloing discussion. The rootmen sure error, E, is defined a.

n (, y) 2

=4: "- n (3)

where n is the number of data points and AYi Is the distance in the y direction between a

data point (Xi, Yi) and the point on the curve corresponding to :(i.

The sum of the squared errors, E, Is defined as:

nE = .- (4)

5. SOLUTION

S. I Assumptions

The following method of transcendental curve f ttlng was based on two

assumptions:

() The n unknown constants CI, I I...n produce at (n + I) dimensional errorsurface which represents the error, E, in the curve fit du! to a particular set ofconstants. When the Intersection of level planes with this sur ace are projected into n-

space, they are assumed to produce a group of Inscribed contour lines (see Figure I).

8

E

-42

.-°

Figure 1. HIypthetical Error Surface. E f(C). for Two Unknom Constants.

(2) The function describing the error surface, E = f(Ci), with respect to any

constant CI may be accurately approximated over a small interval by a quadratic

equation, f(Ci) = A(CiR + 6(Ci) + D, and a minimum value for 'his quadratic exists in the

- neighborhood of that small interval. The plots in Figures and 3 are presented as

examples of the curve representing E = f(C i) developed by fitti; ig a quadratic through the

three points marked ".." The points marked "o" are actual err or points; it can be seenthat over the interval used to develop the curve, the quadrati( accurately approximates

the actual error curve and that a minimum value exists in or ner ir this interval.

9

* S § -.. . .... . .: ., ..",." ," "".-"....'.'-.....-'.'.. ..'.' " "'-- -.-- - -..-. .-.- -.- "-'- -. .'-- -.-- -..-- -.- -. " - .. -. ..

T,= 75 EFIED

T= -W EFIXED3

M 13.111117 E INITIaL VfUE3

M IINTS USE TO S~E

0- RCfUl Et.

MIT USE TO SE-I UV

Figure . Vrato of Error with..

- T~ 75 CIX10

4~rNTN EK

The data for Figures 2 and 3 were generated fr. the log aT vs T data usedin the sample problem of Appendix A.

If the trial values of the constants are not relative y close to the values thatminimize the error in the fit, the second assumption will not be valid. Also, Figure 3shows that points on the actual error curve not in the imme Hate neighborhood of thepoints used to generate the quadratic are not necessarily well p -edicted by the quadratic.

-4

5.2 Selection of Initial Trial Values

The curve fit technique developed is an iterative pi ocess that requires initialguesses at the unknown constants. The technique is very ensitive to these initialguesses. Therefore, a method was devised to develop initial gjesses for Ta and m fromaT vs T data.

If a value of Ta equal to Tg is substituted into equc tion (I), a value of m maybe determined from a least squares fit of the linear relationshlF (see Figure 4):

log a T M1og (5)

*eq

,q -2

-m.u n.m Ilnn..n nEli

LO Et TIR - Tl / C T - T433

Figure 4. Least Squares it of Log AT vs L for W1. 1 101. *ie A.

4. 11

.... *... . .... .. - ' " " " . . .-: ' - . . . . - , * -,._, . . . - -

-.. o-7

A user knowing the values of TR and Tg can thus obtain fairly accurate

qimsuKs at To and m. If TR is unknown, a value between 700! and 80OF should be usedbased on the definition of TR in Section 3.2. If Tg is ur-known, a trial and error

procedure of guessing values for Tg and plotting

~Talog aT = log ( (6)

T T a =T

should be used until a nearly straight line plot is obtained. The guess of Tq that yields

this straight line plot should then be used as the initial guess fo- Ta.

- The JANNAF Solid Propellant Structural Integrity f Iandbook (Ref. I) suggests.- the above-described trial and error procedure to obtain the fin Al OT equation assuming a

value of Ta I OOF to 20oF below Tg (Ref. 3). However, it was found in the present studythat the best fits were obtained using the above method to (enerate guesses only and

then iteratively operating on these guesses to obtain values t iat minimize the error of

the fit.

5.3 Iterative Optimization of the Constants

The initial guesses of the unknown constants descri )e a point A (see Figure 5)in the Ta - m plane. The value of m is temporarily held consta it, and points B and C aregenerated by adding and subtracting a user-specified multiple p) of the present value ofTa to or from Ta, respectively. This search interval is labeled .\Ta, and is equal to C =

p Ci where Ci equals Ta in Figure 5. In subsequent cycles, t'is search interval will be

multiplied by a decreasing factor generated from the function

4-'| i (1 -) + 8e(1b)c (7)

12

2

where b is the number of the current iteration cycle, and a and 0 are user specified

constants that determine the type (linear, exponential, etc.) and rate of interval

reduction. This reduction in the search interval allows more a,.:curate fits of a quadratic

approximation to the error curve, producing a more accurcte approximation of the

minimizing value of C i as the iterative process gets closer to ti ,e actual minimum values.

In a more general sense, then, ACi may be defined as

AC. = (P)(C i ) I-B) + 8e( l b)aAci p)(Y(8)1 j -P + se

To assure that the quadratic approximation to the error has a minimum, the

search interval will be expanded first in the negative Ci drection and then in thepositive Ci direction until the values of E associated with the )ints B and C (E(B), E(C))

nre qreater than E(A). In the example shown, Ta is expcnded in the negative Tn

direction until the point C is located such that E(C) > E(A).

A quadratic is then fitted through C, A, and 3, and the first partial3E(Ta, m)derivative, a , is set equal to zero to find a new v due of T a (point D) which

aTa

minimizes the -error. See Apoendix B for the methods used to -it the quadratic and take

the partial derivative of the error.

Since the quadratic fit is only an approximation to the actual error function,

a value of E corresponding to point D, E(D), is calculated and ci mpared with the starting

points E(A). If E(D) is greater than E(A), the new value T' a will be rejected. If E(D) is

less than or equal to E(A), the value T'a will replace To as shown in Figure 5.

13

Ti contours are a projectionof the E surface onto theTo a plane.

+ ienftas te mininme value of E.

(T a m

-. Ta

Figre $. Iteretlve Optlil-atlo of the Contants. Cycle 1.

An identical process now operates on m (Ta ten porarily fixed), with the

quadratic fitted through points D, F, and G. A minimum volut- of the error is found at

point H (T'a, m'). The value of E corresponding to point H is c wmlpred to E(D), and the

some acceptance criterion for this new point is applied. In Figure 6, E(H) was less than

E(D).

One cycle of the procedure is now complete; and due to the comparisons, first ofE(A) and E(D) and then E(H) and E(D), it is evident that the i rocess has moved in the

proper direction to reduce the error in the curve fit.

A likely starting point for the second cycle mi( ht be point H since it

corresponds to the minimum E at present. However, to avid the possibility of the

process becoming "stuck" in a local volley of the error surf ice, a point A', midway. between the newly calculated minimum point H and the star ting point of that cycle

14

p .-'. .

(ioint A), is determined and the process is started over (see Figure 6) at this

intermediate point. The next cycle produces a new minimum at point H' and a new

intermediate point A. Since the minimum value of E and the corresponding values of Ta

a -a m are always stored in memory, these values can be (and are) selected on the final

c cle rather than using the constants associated with A' for tw final values.

-f The co; tours are a projectieof the E surface onto the

Ta. plane.

+ DoIbt 1S the Ninimum value of E

>FT

Ftre G. Iterative Otudiation of the Constants. CYcle 2.

This process is repeated for a user-specified nurner of cycles or until thevclue of E at any point in the process becomes less than a user- pecified minimum E.

It should be noted that TR is considered a known co-istant, and the power-law

a-- equation should ideally pass through the point (TR, 0) on a Ic g aT vs T plot since

log aT ( To a) 0 (9)T R

However, the log OT vs T data will, most likely, ha' e inherent error, and the

ct rye which best fits this data may not necessarily pass thro xgh (TR, 0), requiring the

procedure to iterate on a third constant, TR. Since both the a ;sumptions of Section 5.1hcld for an n-dimensional problem, and since the equation f v distance between twopoints may be extended to n dimensional space, this third co- tant presents no problem

even though it produces a four dimensional error surface.

15

I .. .. . . .. . , . , - . ~ . . ./ . . . . . . , . .-. ,. ,., . ., ,.-.•.*.-.-"

Appendix A illustrates a sample problem that uses this iterative process.

Figure A-I shows a plot of the curve generated by the iterative process along with the

data used in the problem.

6. GENERAL CURVE FITTING

The method of optimizing the constants of a function de!: :ribed in Section 5 may beapplied to any single-valued equation, either algebraic or tra, scendental. The problem

again arises of picking initial values for the unknown conston-s. If arbitrary values are

selected, the user must take care in choosing thoe values :uch that the equation of

interest can be evaluated over a specified range when calculiting the error. Also, the

program must be modified to allow for the possible nonexistence of a minimum durinq

any particular variable search.

Three modifications were made to the program previc isly developed to fit the

power law OT equation. First, the portion of the program that loads the data was

modified to receive guesses rather than generate initial valu -s from the data entered.

Second, the user must program the equation of interest nto the subroutine that

calculates the sum of the squared errors. Third, the portion -f the program that fits a

quadratic through the error values and minimizes the function was modified to take into

account the possibility that the curve has no minimum. I the curve is seeking a

maximum, the program defaults to the one of three poini ; used to determine the

quadratic with the smallest E value and then proceeds to iterat -on the next constant. Ifthe curve is seeking a minimum outside the interval (Ci-nCi, ( i + nCi), the interval over

which Ci is expanded is doubled both in the plus and min,1s Ci direction until the

minimum value of Ci lies within the interval of (Ci - nCi, Ci # nCi) with the increased

value of n.

Since arbitrary initial guesses are being used, the conver( ence of the general curve

fit is not as rapid as the power law aT fit. However, successi% i use of the program will

allow a user to narrow down the choices for the constants ar d finally arrive at values

that are accurate enough for assumption (2) of Section 5.1 to ac An be valid.

16

--------

REFERENCES

1. JANNAF Solid Propellant Structural IntegrityHnbok Chemical PropulsionInformation Agency (CPIA) Publication No. 230, September 1972.

2. Leighton, Russell A., "Quick Look Structural Analysis Ttchniques for Solid RocketPropellant Grains," Air Force Rocket Propulsion Laboratory, Edwards Air Force Base,California. Report No. AFRPL-TR-8 1-80, May 1982.

3. Handbook for the Engineering Structural Analysis of clid Propellants, ChemicalPropulsion Information Agency (CPIA) Publication No. 214, Ma) 197 1.

71

.................................

APPENDIX A

SAMPLE PROBLEM

19

APPENDIX A. SAMPLE PROBLEM

This Appendix demonstrates a sample curve fit on the HP-9815A for the aT vs T

data tabulated at the end of the Appendix. Statements in quotations are user cues

printed by the program. Numbers and words shown in boxes correspond to keystrokes

needed to run the program.

1 . Subroutine to Load Data

A. User enters the following to start the program:

[0] nter [fl cad lear d [ E-&j]

R. "PROGRAM TO LOAD DATA FOR A(T) VS T CUR\ E FIT"

"NO. OF DATA PTS?"

(User enters the number of data points (up to twent',). If a number greater

than twenty is entered, the message will be repeate I.)

C. "ENTER DATA (X,Y)"

Enter coordinates of data points (X is T in OF; Y is log aT).

[A]5J 1 (actual by stroke sequence is [ [ E j'7)-. .-4511

197.520"

15 2 jR-7-j"152"-2.301 MI

°.- ="-2.30"

D. "ENTER REF. TEMP. DEGREES F"

(If this value is unknown, see Section 5.2 for a range of values.)

076" 20

- ... ., ..2. . .*.-.

2

E. "ENTER GLASS TRANSITION TEMP., DEGREES F"

(This is used as an initial guess for Ta. If Tg is unknown, see Setinn 5.2 for amethod of generating this value.)

-92 M

"-92"A trial value of m is now computed by a least squares fit of log OT vs

log (TR - T) data, m being the slope of this line:

"m = 13.087"

F. "STORE ON TAPE?"

S(For yes; -nmc ke theni j/-_ for no.)

This records the data points as well as the constants on tape.

G. "LOAD CURVE FIT?"

I (For yes;Wy- Lthen M for no.)

This loads the subroutine that iteratively optimizes the constants.

II. Iterative Optimization of Constants

A. If this subroutine has not been loaded by the subroutine to load data, enter:

(otherwise begin with Step B.)

B. "NO OF CONSTANTS 3 MAX"

Enter number of constants in the equation.

0310

C. I"ENTER SEARCH INTERVAL"

(Since OT = ml /TR-T.-T must be evaluated at points generated by thisconstant (0), care must be taken in selecting this constant so the equation

may be evaluated over the interval. In this problem, p is limited to .489. A

value of p = .489 would cause the program to take the log of a negative

number when iterating on Ta, producing an error condition. A value of 0.1 or

smaller is recommended.)

21

-S *4,* * . - . . . . . . . . . . . . - .

D. 'FNTER B"

E. "ENTER ALPHA"-":- |0.2 1/S

(These values, ALPHA and BETA (B), control the reduction of the interval

with each cycle. B = .9, ALPHA = .2 give an exponential decay to a constant

value.)

F. "ENTER TOLERANCE"

.o A E M(This value corresponds to a minimum E that the iterative process is trying to

achieve. If this value is reached, the program will stop and print the

constants associated with it.)

G. "NO OF ITERATIONS"

(This corresponds to the number of cycles to be riom. Convergence to three

decimal places is usually achieved in 10 cycles or I,.s with each cycle taking

approximately 15 seconds if all three constants are iterated on.)

H. "SET FLAG I TO CONSTRAIN C1"

RSSetting Constrain s

Flog I TaFlog 2 TR

Flog 3 m

The best fit is obtained by leaving all three constants unconstrained. If TR

was to be constrained, the user would press SFG [ i

1.. "INITIAL VALUE OF CONSTANTS"

"92" (NOTE: The program assumes Ta is nec ative and only uses the

magnitude of this number.)

"76"

22

b; -I~ % iq .u . . a . . .- . ° . .• .. . . .

-RA

"13.087"

Each constant is now redisplayed on the screen. If that constant is to be

used, press E If a different value is to be used, enter the value Ci and

pressa

NOTE: When the final constant is printed, and HN is hit, the jSFJ key should be hit

immediately if the user wishes to have the error values printed out. Re-hittingISF-Gwill

s-op the values from being printed.

J. Output and User Termination

The value of E for each cycle is printed, followed by the value of E after aniteration on one of the constants is performed, assuming SF0 is pressed in Step I. The

program will run the specified number of cycles unless E at any point becomes less than

the tolerance, or the user terminates the program by pressing RNJ. If the program

runs its full number of cycles, the values of the final E and corresponding constants, as

well as the minimum E and the corresponding constants, will be printed. If the user

terminates the program at some intermediate point, the number of the current iteration

cycle is printed along with current E and constants as well as minimum E and constants.

Finally, if the tolerance is met, the value of E and constants which met or exceeded the

tolerance will be printed. The number which flashes on the display indicates the current

iteration cycle. Figure A-I shows a plot of the data used for the problem and the curve

fit obtained by the iterative method. The input data is also tabulated below.

* Data used in sample problem

Log OT T (OF)

7.52 -45

6.60 -42

5.19 -29

3.84 -2

2.20 19

0.00 76-1.23 118-2.30 152

23

. . . . .. . .% % . .... ....* .•. - . .. . . .- .- - -.. - .'.'- ' " - 'i" ' "v i:. :--.----'-

7-. - -7-77 -7%* ' ". - . ' -- - - ".. ......... ". . . . ..... - .-- 1

CURVE FIT:

1=3.0468T p 74.581"9

., (-T) -92.0 --,

L 2U9O

-2

- -- w ,.,.,...

TEIEIFJf JW F

Figure A-]. Curve Fitted to Log aT vs T Data

24

. .. . .

. . . .. . . . . . . . . . . . . . . . .

APPENDIX B

CURVE FIT AND MINIMIZATION OF ERI OR

25

APPENDIX B. CURVE FIT AND MINIMIZATION OF ERROR

The following discussion outlines the procedure used to simultaneously fit aquadratic through the three points used to generate the error - urve and find a value of C

that minimizes this function (see Figure B-I). In this discussi )n, only one of the Ci's is

considered to vary.

H = A(CI) 2 + B(CI) + D = E(CI) (B-I)

I= A(C2)2 + B(C2) + D = E(C2) (B-2)

J =A(C3) 2 + B(C 3) + D E(C3) (B-3)

Simultaneous solution of equations (B-I) through (B-3) yi !Ids a quadratic fit of the

squared error E(C) = Z (A Yi) 2 through the three points.

C1 (J- 1) + C2 (H - J) + C3 (I - H) (B-4)A=2 322

C 1 (C 3 C2 + C2 (Cl -c 3 )+ 3 (C2 C1)

(I -J) -A(C 22 -C 32) (-5)

(c2 -c3)

The function A(Ci)2 + B(Ci) + D describes the squared err, r, E(Ci). Taking the first

partial derivative of E with respect to Ci and setting it equr I to zero determines the

minimum value of Ci

a(AC12 + BCi + 0)43E - ______ O

ac.

26

..................... . .

or

0 = 2AC i + B

and Ci mi--12A (-)

Direct substitution of equations (B-4) and (B-5) into equation (B-) yields the value

of C i that minimizes the squared error E. Due to the location of the three points ClI,

C2, and C3, the value computed in equation (B-6) is assured to be a minimum. SeeSection 5.3 for an explanation of the process of locating Cl, C2, and C3.

E

+. C

_________ ae 4 nAi l...L..... C1

CI C2 C3

Figure S-1. Curve Pit awl imllutlion of E f(C)

27

REFERENCES

I.JAI#IAF Solid Propellant Structural Integrity Handbook Chemical Propulsion

Information Agency (CPIA) Publication No. 230, September 1972.

2. Lelgton, Ruumll A., "Quick Look Structural Analysis Techniques for Solid RocketPropellant Grains, Air Forc Rocket Propulsion Laboratory, Edwards Air Force Base,California. AFRPL-TR-8 141), May I982.

3. Handbook for the Enaineerina Structural Analysis of Solid Propelants, Chemical

Propulsion Information Agency (CPIA) Pub~lication No. 214&, May 197 1.

28

II p