chapter 5 notes-errors

8/22/2019 Chapter 5 Notes-Errors

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Chapter 5

Uncertainty Analysis

Figures from Theory and Design for Mechanical Measurements;Figliola,

Third Edition


Error =difference between true valueand observed

Uncertainty =we are estimating theprobable error, giving us an intervalabout the measured value in which webelieve the true value must fall.

Uncertainty Analysis =process ofidentifying and qualifying errors.

Measurement Errors

Two General Groups

1.Bias Error- shifts away from true mean

2.Precision Error- creates scatter


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Figure 5.1

Best estimate of true value:

Assumption:

1. Test objective known

2. Measurement is clearly defined, where all bias errors havebeen compensated through calibration

3. Data obtained under fixed operating conditions

4. Engineers have experience with system components

x x x'= Figliola, 2000

Design-Stage


Feasibility Assessment Figliola, 2000

Two Basic Types of

Error Considered

Zero-Order Uncertainty 0is an estimateof the expected uncertainty caused by

reading the data (interpretation error orquantization error). It is assumed that thiserror is less than the instrumentation error.

Arbitrary Rule set0 equal to instrumentresolution with 95% probability.

0 =resolution (95%)

At 95%, only 1 in 20 measurements would falloutside the interval defined by 0.


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Two Basic Types of

Error ConsideredInstrument Error - c, is a combination

of all component errors and gives an

estimate of the instrument bias error.Errors are combined by the root-sum-squares (RSS) method.

Elemental errors combine to give anincrease in uncertainty

x ke e e + + +12 2 2... x jj

k

e=

21

-must maintain consistency in units of error

-should use the same probability level in all cases (95% preferred)

Assume that the combined errorsfollow a Gausion distribution. By usingRSS, we can get a statistical estimateof error, which assumes that the worstcase element errors will not all add upto the highest error at any one time.

Design Stage

Uncertainty

Combine zero order and instrumental error- Estimates minimum uncertainty

- Assumes perfect control over test conditions andmeasurement procedures

- Can be used to chain sensors and instruments topredict overall measurement system error

Figliola, 2000


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Figliola, 2000

Error Sources

Calibration Errors

Data-acquisition errors

Data-reduction errors

Within each category, there are elementalerror sources. It is not critical to have eachelemental error listed in the right place. It issimply a way to organize your thinking. Thefinal uncertainty will come out the same.

Calibration Error

Sources:

1. Bias and precisionerror in standard used

2. Manner in whichstandard is applied

Calibration does noteliminate error; itreduces it to a moreacceptable level.Figliola, 2000


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Data-Acquisition Errors errors that arise during theactual act of measurement.

Data-Reduction Errors errors that occur whileinterpreting data.

Figliola, 2000

Figliola, 2000

Bias and Precision

Errors

In general, the classification of an errorelement as containing bias error,precision error, or both can besimplified by considering the methodsused to quantify the error.

Treat an error as a precision error if itcan be statistically estimated in somemanner; otherwise treat the error as a

bias error.

Bias Error

A bias error remains constant during agiven series of measurements under

fixed operating conditions.Thus, in a series of repeated

measurements, each measurementwould contain the same amount ofbias.

Being a fixed value, the bias error cannot be directly discerned by statisticalmeans alone.


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Bias Error

It can be difficult to estimate the valueof bias or in many cases, recognizethe presence of bias error.

A bias error may cause either high orlow estimates of the true value.

Accordingly, an estimate of the biaserror must be represented by a biaslimit, noted as B, a range within whichthe true value is expected to lie.

Bias Error

Bias can only be estimate bycomparison.

Various methods can be used:

Calibration

Concomitant methodology

Interlaboratory comparisons

Experience

Bias Error

The most direct method is bycalibration using a suitable standard or

using calibration methods of inherentlynegligible or small bias.

Another procedure is the use ofconcomitant methodology, which isusing different methods of estimatingthe same thing and comparing theresults.


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Bias Error

An elaborate but good approach isthrough interlaboratory comparisons ofsimilar measurements, an excellentreplication method.

In lieu of these methods, a valuebased on experience may have to beassigned.

Precision Error

When repeated measurements aremade under nominally fixed operatingconditions, precision errors willmanifest themselves as scatter of themeasured data.

Precision Error

Precision error is affected by: Measurement System (repeatability and

resolution) Measured Variable (temporal and spatial

variations)

Process (variations in operating andenvironmental conditions)

Measurement Procedure and Technique(repeatability)


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Error Propagation

Consider: y =f(x)

True x':

True y:

x tSx=

y y f x tSx = ( )

x is measured to

establish

its sample mean and

precision.

Figliola, 2000

Expanding as Taylor

Series

By inspection

y y f x dy dx tS d y dx tSx x x x x = + += = ( ) [( / ) / ( / ) ( ) ...]1 22 2

22

y f x= ( ) and = += y dy dx tSx x x[( / ) ...]

A linear approximation that ignores high order terms:

y dy dx tSx x x =( / )

The derivative defines sensitivity at

the width of interval defines precision interval about

y that is .From this, the uncertainty in y is dependent on uncertainty in x

( / )dy dx x x= tSx

y

u dy dx uy x x x= =( / )

x

Expanding as Taylor

Series

In multivariable applications,

L=number of independent variables

R f x x xL= 1 1 2( , ,..., )

Best estimate of true value R: (P%)R R uR' = UR=

uncertainty

in R

Sample mean: R f x x xL= 1 1 2( , ,..., )

Uncertainty in R: u f u u uR x x xL= 2 1 2( , ,..., )


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Uncertainty in R

UR represents the uncertainty of eachindependent variable as it ispropagated through to the result R.The estimate of UR came from Kline-McClintock.

Using a Taylor Series expansion ofR, a sensitivity index: i x xR xi= =/

i=1, 2, , L

Index is evaluated using either mean or expected nominal

values of x.

Uncertainty Analysis:

Error Propagation

The sensitivity index relates to howchanges in each xi affect R.

The use of the partial derivative isnecessary when R is a function ofmore than one variable.

This index is evaluated by using eitherthe mean values or, lacking theseestimates, the expected nominal

values of the variables.

Uncertainty Analysis:

Error Propagation

From a Taylor series expansion, thecontribution of the uncertainty in x to

the result, R, is estimated by the termiuxi.

The propagation of uncertainty in thevariables to the result will yield anuncertainty estimate given on the nextslide.


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The Uncertainty

Estimate

u uR i x

i

L

i= =

( )2

1

Sequential Perturbation

This is the numerical approach toestimate the propagation ofuncertainty, using finite difference. replaces the direct approach that requires

the solution of cumbersome partialdifferential equations.

The method is straightforward anduses a finite-difference method toapproximate the derivatives.

Method1. Based on measurement of independent

variables, calculate R0=f(x1, x2,, xL)

2. Increase xi by uxi uncertainty to get Ri+

R1+=f(x1 +ux1, x2,, xL)

R2+=f(x, x2+ux2,, xL)

RL+=f(x, x2,, xL+uxL)

3.Repeat by decreasing xi by uxi to get Ri-

4.Calculate difference Ri+and Ri- for i =1, 2,, L Ri+=Ri+- R0 Ri- =Ri-- R0


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Method5. Evaluate the approximation of uncertainty

contribution from each variable.

6. Uncertainty

(P%)

Ri Ri Ri i iu= + + ( ) / 2

uR Rii

L

= =

[ ( ) ] / 2 1 21

Numerical approach and partial diff eq approach give similar results.

Occasional unreasonable estimates of error propagation can be

caused by rapidly changing sensitivity indexes due to small change

in independent variables.

Advanced Stage and

Single Measurement

Uncertainty

Considers not only resolution errorsand intrinsic instrument errors, but alsoprocedural and test control errors

Method: Estimates expected uncertainty beyond

design stage

Reports results of a test program overrange of one or more parameters with no

or few replications at each test condition.

Advanced Stage and

Single Measurement

Uncertainty

In design-stage uncertainty analysis, onlyerrors caused by a measurement systems

resolution and estimated intrinsic errors areconsidered.

Single-measurement uncertainty analysispermits taking that treatment further byconsidering procedural and test controlerrors that affect the measurement.


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Advanced Stage and

Single Measurement

UncertaintyGoal:

Estimate the uncertainty in some measured valueof x or result R, based on estimation of uncertaintyin the factors that influence x or R.

Zero-Order:

All variables and parameters are fixed, except forthe physical act of observation or quantization.

Instrument resolution is the only factor.

uo=1/2 resolution

Higher Order

UncertaintyConsider controllability of test operating

conditions

First order u1 evaluates influence of time

What is variability if all factors are heldconstant and

you collect data over period of time?

Collect N30 samples, u1=tv,93sxiIf u1=u0 then time has no influence

Higher Order

Uncertainty

Higher Order

Each successive order identifies a factor thatinfluences the outcome, thus giving a more

realistic estimate of uncertaintyAt each stage, collect N30 samples,

ui=tv,93sxi

For example, at the second level it mightbe appropriate to assess the limits of theability to duplicate the exact operatingconditions and the consequences on thetest outcome.


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Higher Order

Uncertainty

Nth Order

Instrument calibration uncertainty uL is entered in

(95%)

- Allows for direct comparison between results obtained byeither different instruments or different test facilities.

- Approximate value to use in reporting results.

- Note the difference between design stage and advanced stageis u1, u2,, un-1 terms that consider the influence of time andoperating conditions

u u uN c i

i

N

= +=

( )2 21

1

Figliola, 2000

Multiple Measurement

Uncertainty

Estimation of uncertainty assigned tomeasured variable, based on set of

measurements obtained under fixedoperating conditions.

Parallels uncertainty standardsapproved by many professionalsocieties.


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Method

1. Identify elemental error in threegroups (calibration, data acquisition,and data reduction)

2. Estimate the magnitude of bias andprecision error for each elementalerror.

3. Estimate propagation through toresults

Method- consider x to have precision error Pij and

bias Bij i =1 for calibration j =1,, K elemental errors

i =2 for data acquisition

i =3 for data reduction

Then eij=P ij +Bij is the elemental error for the ithgroup and jth elemental error.

Pij=t

v,95s

x

1.Propagation of precisionerrors in the source groupis called source precisionindex P i, estimated byRSS.

2. Measurement precisionindex is a basicmeasurement of theprecision of the overall

measured variable due toall elemental errors.

3. Source bias treated similarly.

4. Measurement biasrepresents the basicmeasurement of theelemental errors that affectthe overall bias.

5. Measurement uncertainty inx can be represented as acombination of bias andprecision uncertainty.

Figliola, 2000


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Estimation of the degrees of freedom, v, inthe precision index, P, requires specialtreatment, since elements have differentdegrees of freedom.

In this case, the degrees of freedom in themeasurement precision index is estimatedby using Welch-SatterthwaiteFormula:

= =

= ==3

1 1

4

3

1 1

22

i

K

j ij

ij

i

K

j

ij

v

P

P

v

)(

Where i refers to the three source error groups and j to each

Elemental error within each source group with vij = Nij 1.

chapter 5 notes-errors

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