The Analysis of Solid Rocket Motor Performance Errors by Monte Carlo Techniques

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THE ANALYSIS OF SOLID ROCKET MO1O PERFORMANCE ERRORS BY MONT1E CARLO TECHNI(JES

A. W. Langill, Jr. Aerojet-General Corporation Sacramento, California

influenced by the overall accuracies attributed to the basic data acquisition system. Major solid rocket motor development program requirements invariably include a maxim acceptable motor-to-motor product variability specification (with associated confidence intervals) as reflected in total and specific impulse parameters. Frther, it is required that the ballistic parameter repeatability bands be demonstrated by means of static testing. It can be easily shown that, as the data acquisition system accuracy increases, the total number of motor firings required to demonstrate the motor variability specification decreases markedly; thus a significant saving in both facilities and total test cost result.

ABST[RACT In the captive testing of conventional solid rocket motors, a large number of fundamental physical measurements are normally implemented. These include the determination of instantaneous motor force, pressure, temperature, deflection, acceleration and electrical power data associated with the particular motor configuration. In the majority of applications, however, the motor pwformance indicators of most general interest are not measured directly, but are calculated from these basic data. Thus, for example, motor specific impulse is computed from given instantaneous values of measured thrust, pressure and weight parameters.

All physical measurements are subject to inaccuracies, i.e., even the most sophisticated data acquisition system will necessarily inject discrepancies into the measured data. Hance, the mechanisms through which errors inherent in the measured quantities affect the accuracy of the calculated ballistic parameters are of extreme practical importance and must be quantitatively defined. At the Aerojet-Gereral Corporation, an intensive and continming analysis of overall measurement system errors (through rigorous analytical and experimental techniques) has permitted the statistical characterization of the error parameters within the framewcrk of each current large scale motor development program. Based upon these data and for a specific ballistic parameter computational routine., an extensive error analysis is required to deduce the errors associated with the computed performance parameters. Because of the nonlinearities and interactions inherent within the ballistic computationss, a Monte Carlo technique is invoked.

In the final analysis, however, it is necessary that the instrumentation variability be separated from the overall test variability in order to obtain the true product variability. This extraction procedure requires a knowledge of the following quantities: (1) the statistical data acquisition inaccuracies, (2) the processing routine tUrough which the final performance parameters are computed from basic motor measurements, and (3) the specific manner in which the measurement errors interact through the processing routine to generate inaccuracies in the computed

parameters.

A discussion of the development and interpretation of the tiree major criteria listed above form the basis for this paper. In particular, the techniques employed in deducing the measurement system inaccuracies are first detailed. A nominal set of equations for the ballis-tic evaluation of solid propellants is next discussed; this data processing routine is characteristic of the type normally employed at the Aerojet-General Corporation. Based upon the computational routine, a set of error expressions are detailed which describe the interactions of

INTRODUCTION As suggested by Boulton1 and Dow2, the economics of solid rocket motor testing are greatly

error parameters and ballistic performance pro-

cessing. And finally, the error equations are programmed for digital computer simulation. Invoking a conventional Monte Carlo error analysis technique, the digital computer output defines a set of ballistic parameter error histograms suitable for plotting, i.e., for a given set of statistical inputs, the computer generates the resulting error spectrum which would be expected in the computed performance indicators.

lBoulton,

V.R., "Economics of Instrumentation Precision for Space Vehicle Development," Institote of the Aerospace Sciences, Vol. 20, No. 3,

1961 2Dow, R.H., "The Cost of Instrumentation Error," ISA 6th National Flight Test Symposium, April 3960 99

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0-10,000 pounds force, all statistical parameters are referenced to 7,500 pounds force. This procedure is in keeping with the general philosophy of transducer usage at the Aerojet-General Corporation whereby the physical stimulus, at the anticipated operating point, corresponds to approximately 75% of the associated transducer range.

MEASUREMENT S7STEM ERRORS

This section describes typical procedures for determining accuracy and repeatability characteristics of solid rocket motor perfomance data. All measurement system accuracy and repeatability values as stated, are consistent with and based upon the particular sensors and related data acquisition systems currently employed in the Aerojet-General Corporation solid rocket motor

The repeatability values for measured parameters are obtained by listing the maxim error (30(T) of each of the variables affecting the given system as a percentage of the operating point. These values are then combined by RSS techniques (square root of tin sum of the squares) this operation is legitimate in that each of the values listed is functionally independent. The expected standard deviation of the parameters is assumed to be one-third of the maximum derived

testing facility.

The terms repeatability, bias and accuracy, as employed Nwithin the context of the subsequent discussion are defined as follows:

Repeatability is the deviation from the averag o ipoints obtained from repeated tests under identical invariant conditions for each particular test, i.e., the degree with which test results agree on a run-to-run basis, with all test parameters held constant. Repeatability is conventionally defined in terms of confidence intervals, i.e., t10 implies that 68.26% of all samples fall between the stated values.

error.

In the sections to follow, the measurement systems associated with the acquisition of basic pressure, force and weight data, are suggested. In particular the approaches invoked by Armstrong & Kapandritis3 to statistically characterize each measurement error bandwidth are detailed.

Bias is the variation between the mean value of a particular group of samples and the corresponding true or reference value. As such, the term represents a form of "average attemation or amplification" exhibited by a particular measurement system.

Pressure The pressre measuring system consists of a strain gage transducer, power supply, electrical calibration network, dc amplifier, and Beckman 210 analog-to-digital conversion and recording system. The maximum error attributable to the transducer was determined by calibrating the pressure transducers with a laboratory deadweight standard. Errors introduced by the strain gage channel and Beckman 210 end recorder were determined from repetitive calibrations employing a simulated pressure transducer.

Accuray is the overall ability of a measuredefine a known physical excitation (standard) to within a stated deviation; as such, accuracy includes both repeatability and bias.

ment

sytemFto

To clarify these concepts, consider the folloving simplified example. Assume that the temperature corresponding to the triple point of water is to be measured. This physical temperature envirorent is, by definition, 0°C. A great number of individual measurements are now recorded and a subsequent data analysis yields the following results: The mean value of all measurements is +l.0C, and 68.26% of all measurements are within a band of +0.5° about the mean. Under these conditiors, the one standard deviation repeatability is t0.5° and the measurement system bias is +1.0O. This, the accuracy, based upon a 10- confidence interval is defined between the limits of +0.5 to +1.5°C. If a 30 repeatability were computed from the temperature data, an associated measurement system accuracy based upon this confidence interval could then be stated. Within the framework of these definitions, it is evident that the accuracy of a given measurement cannot be better than the repeatability.

The maxim errors of the individual components of a typical pressure measuring system are as follows: Transducer Calibration, variables

Percent

of the Point

(1) Linearity, hysteresis and reproducibility (3 x RSS of 10 of each variable)

+ 0.26

(2) Electrical shunt to pressure relationship through the temperature range of +30 to +130*F (max deviation)

(3) Shift in calibration caused + by use (3 x max AO-)

In the above example, the statistical parameters were expressed in absolute physical units (*C). It is normally more convenient, however, to normalize the parameters to a predetermined fixed value of the physical excitation. Thus, all bias, repeatability and accuracy values are often quoted as a percentage of the 75% transducer value, i.e., if a particular force transducer is rated at

(4) Dead-weight standard

0.25

0.15 + 0.05

3Armstrong, R.W. & G.N. Kaparxiritia, "Instrumentation Accuracy and Measurement Precision for Performance Calculations of Solid Rocket Motors, "Aerojet-General Corporation T1-156 SRP, 1961 100

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Percent of the Point

Chanel, variables

The environmental residual error is defined as the difference between the maxium repeatability error as determined by redundant measurements, and the maximum channel-transducer error

(1) Electrical Calibration d standard

combiemd in root-square-summation. The axiom that the measurement system accuracy cannot exceed the measurement system repeatability dictates the foregoing analysis. As a result of the above analysis, this error can be shown to be + 0.48%.

± 0.10

(2) ADC system

(3) Amplifier Environment residual error

Root-square-summation (RSS) One standard deviation (10-) -

t 0.48 + +

Thrust

0.63 0.21

The most accurate method of measuring rocket motor thrust is that of determining the net reactive force of a highly linear and repeatable spring. Although the true force measurement is not theoretically complete until the kinematic force components resulting from acceleration and velocity are summed with the linear deflection term, the total dynamic force integral error averaged over a long duration firing has been proven to be negligible. The loss of high frequency transient data is recognized, however, and its importance is evaluated for each type of rocket motor.

Pressure transducers mamfactured by the Taber Manufacturing Company (Model 206), with a range of 0 to 750 psig are used for most precise chamber pressure measurements. A root-squaresummation (RSS) of the individual errors result in a maximum error of t 0.63% of the point, assuming a normal operating range of 75% of full scale for the transducer and Beckman converter. The resulting standard deviation (10-) for basic chamber pressure measurements is t 0.21J of the point.

The thrust measuring system consists of a tbrust stand together with a strain gage loadmeasuring transducer and channel components identical to those of the pressure measuring system. For force cells within the range 0-60,000 pounds, maximum errors attributed to the load cells are determined from calibrations employing dead weight, certified by the National Bureau of Standards. Ths load cells and the pressure transducers are temperature compensated and trimmed to provide a standardized electrical shunt output through temperature extremes of +30 to +130°F.

Data substantiating individual components (1) (2) and (3) of the transducer calibration variables were obtained by comparing a large number of pressure transducer calibrations over an extended temperature range before and after being subjected

to the actual static motor test enviroment. Individual component (4) is derived from National Bureau of Standards dead weights data. Components (1), (2) and (3) of the channel variable were deduced by substituting a known standard in place of the transducer and comparing the excitation and response variables under controlled invariant conditions.

In-place thrust stand calibration is considered of paramount importance in the process of obtaining a high thrust measurement accuracy. Although load cells are calibrated in the laboratory, this form of controlled calibration is not considered a substitute for the complete measuring system in-place calibrations in determining a characteristic calibration factor for each stand. A complete study of all loads occurring during a normal test firing, such as stiffness effects and deflections under load, is conducted during thrust stand evaluation. In-place end-to-end calibrations provide information including the thrust stand characteristics in conjunction with overall characteristics of the data acquisition system. As a result of studies previously conducted, it is concluded that once the calibration factors have been determined for a conventional test stand., further end-to-end calibration activities are not

Redundant pressure measurements are normally acquired by implementing two independent measurement systems to monitor chamber pressure from an identical source. A statistical comparison, comprised of the indicated difference in measured pressure from the two transducers at the normal operating point, for a large data sample, illustrated a t 0.30% standard deviation. These data are indicative of the system reproducibility during actual motor operating conditions. The effects of measurement repeatability to the variance of the indicated comparison difference is t 0.21% of the po'nt, as shown by the relationship O-p2 - 0-p1 + TP2. When 0-p is the standard deviation of the comparison difference and 0-pland 0-P2 are the standard deviation of the individual pressure measurements. Both C-Pl and 0-P2 are considered to be equal in mag-

nitude, therefore: 0.3 -\ p2 + p2P

2 0.3

required.

Static and dynamic thrust calibration measurements were obtained using the end-to-end systen concept. The high precision U3XXA BaldwinLima-Hamilton load cells, manufactured to Aerojet specifications, were connected in tandem on either side of a motor and thrust stand assembly. A calibration motor was then dynamically loaded to

2p2

P -

0.21 101

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accurately simulate actual motor firing cornitions. A comparison was made of the output load cell to

load ceUl variable items (1), (2) and (3) are verified by means of laboratory calibrations.

the input load cell (base measurement) and, on the basis of these repeated calibrations, a system bias was established. System repeatability was determined during this analysis for a singlecomponent tension-flexure test stand of Aerojet design.

Propellant Weight The determination of motor propellant weight is accomplished by means of multi-beam platform scales equipped with extremely precise knife-edge pivots. The measurement procedure consists of weighing the empty motor case; the chamber insulation is then inserted and the assembly reweighed. After the propellant is cast and trimmed, the chamber is again weighed and the net propellant weight determined. This propellant weight is then used for all ballistic calculations.

The maximum errors for all components of the thrust measurement system are as follows:

Thrust stand calibration, variable

Percent of the Point

(1) Dead weight calibration technique

Scale calibrations are accomplished in accordance with established National Bureau of Standards procedures. All scales used for performance data are calibrated at 30 day intervals or less, depending upon usage and instances of accidental overload. The weights employed as standards for purposes of calibration are periodically checked by the State of California Laboratories and have a certification accuracy of + 0.025%.

t 0.1

(2) AGC working standard to dead weight

+ 0.15

(3) AGC working standard to measuring cell

!: 0.15

(4) Bias repeatability of the

Thrust Stand & Electrical System

The largest error associated with the determination of propellant weight is the human technique or systematic error, i.e., the scale resolution and the calibration standard errors are relatively insignificant. To establish the systematic error, a series of tests were conducted with a group of observers and a large number of different weights. The results of this experiment demonstrated a purely hman error of ± 0.20%, 3Cr

t 0.1

Measuring load cell

(1) Iinearity, hysteresis, repeatability

± 0.12

(2) Shift in calibration due to use, (3 x max /0\()

± 0.24

(3) Shunt to force relationship through temperature +30 to +130 °F

Variable

± 0.10

(1) Calibration weight

Percent of the Point

t 0.025

error

Channel

(2)

Scale resolution

+ 0.050

(1) Electrical calibration

(3) Systematic error

+ 0.200

(2) ADC system

+ 0.10

RSS

-

--

(3) Amplifier

The combination of the individual errors results in a maxim error of + 0.210 of the point at the normal operating range of the scale used, and produces a one standlard deviation of + 0.07% for weight measurement.

RSS - + 0.40%

10

- +

t 0.210 0.07%

0.13%

Combining the individual errcrs by rootsquare-summation results in a maximum error of

0.40% of the point at a normal operatirg range (75% full scale). The one standard deviation for thrust measurement is thus ± 0.13% of the point.

+

Variable (1) was verified by NBS certification; Variable (2) was determined by the least count graduation of each respective scale; VariAble (3) was verified by actual experimentation.

Items (1) and (2) of the thrust stand calibration variable are obtained from NBS certification and calibration standards. Item (4), test stand system-bias repeatability, is based upon the results of an extensive in-place static and dynamic thrust stand calibration4. The measuring

As illustrated in the previous discussion, no systematic bias errors were apparent in eitber pressure, force or weight parameters. It should be noted that this statement would not be true had a multi-component test fixture been involved in lieu of the ballistic stand.

4Simonsen, R.S., "The Static and Dynamic Timst Calibration of a PUll-Scale Solid Rocket Engine Test Firing System, ' JANAF Pub. #SPSTP/9, Sept 1960 102

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A TYPICAL BALLISTIC PARAMETER FROCESSING ROUTINE

Equations (1), (2) and (3) can be solved simultaneously for the parameters Pe, At and Cf. The initial computed values of At, corresponding to the two Kf choices, are related to the true prefire measured throat area and the true value of Kf determined by linear interpolation.

The primary objective of any ballistic computational program is the conversion of pressure and thrust data into meaningful ballistic parameters, indicative of the propellant potential energy content. Instantaneous values of motor thrust and pressure are repetitively sampled by the data acquisition analog-to-digital converter described previously. These sampled-data sequences appear as inpIus to the bal4istic program. As discussed by Sutton- and Krimsky0, a set of three equations can be employed to describe the instantaneous state of the nozzle exhaust gas for conditions of isentropic, supersonic, onedimensional, quasi-static flow in thermodynamic equilibrium. F - Kf Cf

A6 At

Pan At 2

2

__Y+l

(Pe/psn)

[f

-

computed.

Cw -

/PsnAt

-yl+l

A-

.F

E[l Where:

-

(Pe/Pan) "]

Where:

r ]l/2

( Psn

Cw is the motor discharge coefficient

)At

(3)

I5p(std)

weight

is specific impulse at motor

conditions

is specific impulse at stan-

dard conditions

Cf(std) is the thrust coefficient

com-

puted for standard conditions

Kf is the thrust coefficient efficiency factor

"Standard conditions" consist of an operational envirorment such that Psn a 1000 psia, Pe Pa - l4.7 psia and /\= 0.98296 (half angle -

the measured stagna'cion pressure

15).

At is the calculated effective nozzle throat area

ERROR ANALYSIS FUNDAMENTALS7

Ae is the measured nozzle exit area

I'

is total measured propellant

Isp(mc)

F is the measured longitudinal thrust

Pgn is

(7)

m is the motor mass flow rate

)1/2 +

(6)

Cf(std) Isp(mc) Ispst' spi.td,

(2)

)2(y-l

1/2

(5)

m

Wp Cf

(4)

dt

Cw Psn At

Isp(Mc)

(-Y-l'7

(Pe/Psn)

are

m -

(1)

,\(Y-)1/2 ( 2

Based upon the results of the preliminary calculations and one additional input (total propellant weight), the final ballistic parameters

If A - f(x, y, z) and if x, y, z are independently measured physical quantities, (e.g.., pressure, thrust, propellant weight) then the absolute error in A produced by errors in x, y, z is expressed in general as,

is the ratio of specific heats

Pe is the calculated exit pressure Pa is the atmospheric pressure

/\AAindfAx+ dx

Cf is the thrust coefficient

dfAy+ dyy

d 2f + d

2 (Ay)2 + 2f(A,Z)2 y29?

A is the Tsien correction factor (nozzle half angle correction)

f"Az z +

+ d xf

.............. g91fj (AiX)'

Assuming two values of Kf (which bracket the true thrust coefficient efficiency factor),

+ 7 Jf

5Sutton, G.P., "Rocket Propulsion Elements,," John Wiley and Sons, Inc., 1956 6Krimsky, S.H., "A System for the Ballistic Evaluation of Solid Propollants," Aerojet General Corporation 24-135, 1960

Where i, j , k need not be of equal order.

(Ly)

+

(fz)k

7Nelson, A.L., K.W. Folley and W.M. Borgman, "Calculus," D.C. IHath & Co., 1946 103

(8)

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If the function cannot be solved explicitly for A, then the absolute error in A produced by errors in x, y, z can be expressed in implicit form,

_ d2Af--AA

o +

2 + A2

+ -Ax + - f

(AlA)2

+

+

9x2

(A)2

+

0**

kf

(x)2 +

y g

2f

y2

APe [D Psn B

z

(L

Ds

)2

+ (A)i (ALx)J CLA~~9x

(Ay)k + -°f (,z))

(9)

4

.

sults are tabulated for convenience as,

f AZ

+

-

z2 +

d9y

$x

derivations, together with a complete list of symbols are presented in the Appendix. The re-

c9z

+ d Lx 9A

+

9

ed Az

y

A-r D +

Ax-~-Ex x

x

1

e~

s

<

m

2DB

mL 1.. Cl

A

+

.

(10)

relative errors in A, x, y and s. The relative error is symbolized by the greek letter epsilon, e.g.,

E A'

e

-

r C<

Ac A

>

] AI.<

c

'(

>

A6-1/r i

(13)

_CsI,-1 -1/2

A Cr4aAPe Psn

-Pa

Psn

APsn Pe psn Psn

APsn Psn

ANALYSIS OF BALLISTIC PARAMETER ROUTINE8 As a preliminary step, Equations (1), (2) and (3) are combined into a single expression 1ontaining only the unknown normalized parameter sn i.e., F

D Psn B

B D

Where:

D B

1)

(Pe)l/r Psn

(14t)

Cf

D

EAt - IMI 6pse Pn Where:

(11)

t-1 2r

(Cw S (1

m

n

-

p

sn

(15)

on

-., Kf, expansion ratio, nozzle exit area, total propellant weight and ambient pressure. In addition, the statistical error distributions in measured force, pressure and weight are introduced in terms of measurement system bias and repeatability (1() variables. Based upon these data, the computer solves for D, B, Ck

t

_1

+

-1r

152c><

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C><

2 C