measurement and instrumentation_lecture4

8/12/2019 Measurement and Instrumentation_lecture4

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MEASUREMENT AND INSTRUMENTATION

BMCC 3743

LECTURE 4: EXPERIMENTALUNCERTAINTY ANALYSIS

Mochamad Safarudin

Faculty of Mechanical Engineering, UTeM

2010


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Contents

Propagation of uncertainties

Consideration of systematic and random

components of uncertainty

Sources of elemental error

Uncertainty of the final result

Design-stage uncertainty analysis

Applying uncertainty-analysis in digital data

acquisition system


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Uncertainty analysis is important to identifycorrective measures while validating andperforming experiments.

Propagation of uncertainties => totaluncertainties, e.g. P= VI= n

Two important factors in uncertainty:

Random uncertainty(or precision uncertainty) :imprecision in measurements

Systematic uncertainty(or bias uncertainty):estimated maximum fixed error

Pw


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General consideration

If Ris a function of nmeasured variablesx1,x2, .xn, i.e.

Then a small change in is due to smallchanges in s inxis via the differential

equations:

nxxxfR

,, 21

R'

ix

n

i i

ix

RxR

1

Sensitivity

coefficient

(1)

(2)


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For calculated result based onmeasuredxis, Eq. (2) can be rewritten

as

where | | is to make sure we dont get

zero uncertainty in R. However, this can produce high

estimate for wR.

(3)

n

i i

xRx

Rww

i

1

Uncertainty

in result

Uncertaintyin variables


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Hence Eq. (3) is better represented by

=>root of the sum of the squares(RSS)

In this case, the confidence level must be thesame for all uncertainties (typically 95%).

Assumption is made that each measuredvariables (hence, error) are independent ofeach other.

(4)

2/1

1

2

n

i i

xR

x

Rww

i


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Exercise

To calculate the power consumption of

an electric circuit, we have P= VIwhere

V= 100 2 V and I= 10 0.2 A

Calculate the maximum possible error

(uncertainty) and best-estimate

uncertainty (RSS). Hint: Use Eq. (3)andEq. (4)respectively.


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Answer to Exercise

Wxxi

P

V

P

w iVp 402.0100210max

Because P=VIdP/dV=I=10.0 A , dP/di=V=100.V then

Wxxi

P

V

Pw iVp 3.282.0100210

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Contents


Consideration of systematic andrandom components of uncertainty




Applying uncertainty-analysis in digital dataacquisition system


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Consideration of systematic and

random components of uncertainty Random uncertaintydepends on

sample size (usually large, n>30)

Systematic uncertaintyis independentof sample size & does not vary during

repeated reading

Need to separate for detaileduncertainty analysis


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Random uncertainty

Using t-distribution, the random uncertaintyfor all measurements is given by

where Sxis the standard deviation of thesample

For a single measurement (also for eachindividual measurement), the randomuncertainty is

M

S

tP x

x (5)

xx tSP (6)


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Systematic uncertainty

Sometimes assumed as level of

accuracy

Depends on manufacturersspecification, calibration tests,

mathematical modelling, considerable

judgement as well as comparisonsbetween independent measurements.


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Systematic uncertainty some

examples Radiation heat transfer => lower

measured value

Instrument location => spatial error, e.g.a single thermometer measures

temperature in a box oven

Dynamic errors


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Combining random & systematic

uncertainties

Total uncertainty is obtained, using RSS

(Eq. 4) for all measurements, is given by

For a single measurement ofx,

2/122 xxx PBW (7)

2/122 xxx PBW (8)


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Contents




Sources of elemental error Uncertainty of the final result



acquisition system


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Estimation of uncertainty

Systematic uncertainty: just combine all

elemental uncertainties

Random uncertainty: 3 approaches todetermine Sx

1. Run entire test in a sufficient number of times

2. Run auxiliary tests for each measured variablex.

3. Combine elemental random uncertainties

=> Based on experiment requirement.


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5 categories of elemental errors

Calibration Uncertainties: residual systematic errorsdue to; uncertainty in standards, uncertainty incalibration process, randomness in the process

Data-Acquisition Uncertainties: during measurementdue to; random variation of measurand, A/Dconversion uncertainties, uncertainties in recordingdevices

Data-Reduction Uncertainties: due to interpolation,curve fitting and differentiating data curves

Uncertainties Due to Methods: due toassumptions/constant in calculation, spatial effectsand uncertainties due to hysterisis, instability, etc.

Other Uncertainties


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Combining elemental systematic &

random uncertainties (RSS)

m

i

ix

k

i

ix

SS

BB

1

22

1

22

Calibration

Uncertainties

Data-Acquisition

Uncertainties

Data-Reduction

Uncertainties

Uncertainties Due

to Methods

Other

Uncertainties

2/122yuncertaint

Variable

xxx tSBw

x

Reproduced from Wheelers book:

ASME 1998


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Degrees of freedom, vx

When sample size is large, vxis simplynumber of sample, n, minus 1.

When sample size is small, then vxis givenby

=> Welch-Satterthwaiteformulation (ASME1998)

m

i

ii

m

i

i

x

vS

S

v

1

4

2

1

2

/

(9)Degrees of freedomof individual

elemental error


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Contents








acquisition system


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(Multiple measurement)

Referring to Eq. 1, then for multiple

measurements, M, the mean results is given

by

Little exercise:

Derive the standard deviation (SR) and

random uncertainty ( ) of R.

M

j

jRM

R1

1(10)

RP


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Rearranging Eq. 4 (RSS), we get the

systematic uncertainty in terms of the

combination of elemental systematicuncertainties, given by

2/1

1

2

n

i i

iRx

RBB

(11)


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Therefore, the total uncertainty estimate of the mean

value of Ris

To estimate random uncertainty for multiple

measurements, results are more reliable using the

test results themselves, compared to auxiliary tests

or combination of elemental uncertainties.

Practical applications: The life of a light bulb, the life

span of a certain brand of tyre or car engine

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RRR PBW (12)


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(Single measurement)

To deal with uncertainty of a single testresult only

Practical applications: measuring bloodpressure/ heartbeat, speed of car, etc

To estimate random uncertainty of theresult, must use or combine auxiliarytests and elemental randomuncertainties.


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(Single measurement)

For a large n, then tis independent of v,

the degree of freedom, (and has a value

of 2.0 for a 95% confidence level). For a small n, again using Welch-

Satterthwaite formulation, we get

n

i

i

ii

RR

Sx

R

v

Sv

1

2

22

1

(15)


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Example

The manufacturer of plastic pipes uses a scale with an

Accuracy of 1.5% of its range of 5 kg to measure theMass of each pipe the company produces in order to

Calculate the uncertainty in mass of the pipe. In one batch

Of 10 parts, the measurements are as follows:

1.93, 1.95, 1.96, 1.93, 1.95, 1.94, 1.96, 1.97, 1.92, 1.93 (kg)

Calculate

a. The mean mass of the sample

b. The standar deviation of the sample and the standar deviation

of the mean

c. The total uncertainty of the mass of a single product ata 95% confidence level

d. The total uncertainty of the average mass of the product at a 95%

confidence level


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Solution:

10

93.192.197.196.194.195.193.196.195.193.1

n

mm

i

av

kg94.1

(a)

(b)

2/1

2

]1

)([

n

mmS

avi

samp

kgn

SS

kg

Samp

mean

22

2

2/1

222

1052.010

10*70.1

10*65.1

]9

..........02.001.001.0[

)Confidence(95%k10*73.32 gtSP sampSingle

)%95(084.0)(:yuncertaintTotal

075.05015.0015.0B

6.6Tablet-Studentfromfreedomofdegrees91-nforis2.262t

2/12

sin

2

sinsingle

single

confidenceBPw

kgkgRange

glegle

(c)


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)%95(076.0)(:yuncertaintTotal075.0B

)Confidence(95%1018.1

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mean

singlemean

2

mean

confidenceBPwkgB

kgtSP

meanmean

mean

(d)

As can be seen, the dominant factor is the systematic uncertainty.


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Contents








acquisition system


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(Based on ASME 1998)1. Define the measurement process State test objectives, identify independent parameters and

their nominal values, etc

2. List all elemental error sources To do a complete list of possible error sources for each

measured parameter.

3. Estimate the elemental errors Estimate the systematic uncertainties and standard

deviations. If error is random in nature and/or data isavailable to estimate the std dev. of a parameter, thenclassify it as random uncertainties, which must have thesame confidence level. For small samples, to determine

degrees of freedom. Refer Table 1.


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Guideline to assign elemental

error (Table 1), from WheelerERROR ERROR TYPEAccuracy

Common-mode volt

Hysterisis

Installation

Linearity

Loading

Noise

RepeatabilityResolution/scale/quantisation

Spatial variation

Thermal stability (gain, zero, etc.)

Systematic

Systematic

Systematic

Systematic

Systematic

Systematic

Random*

Random*Random*

Systematic

Random*

*assume no. of samples > 30


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(Based on ASME 1998)4. Calculate the systematic and random uncertainty

for each measured variable

Use the RSS formulation with data & procedure in Step 3.

5. Propagate the systematic uncertainties and

standard deviations all the way to the result(s)

Use the RSS formulation to find the final test results, with

the same confidence level in all calculations.

6. Calculate the total uncertainty of the results Use the RSS formulation to find the total uncertainty of the

result(s).


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Contents


Consideration of systematic and randomcomponents of uncertainty

Sources of elemental error Uncertainty of the final result


Applying uncertainty-analysis indigital data acquisition system


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Applying uncertainty-analysis in

digital data acquisition system A digital DAS typically consists of sensor,

sensor signal conditioner, amplifier, filter,

multiplexer, A/D converter, Data reductionand analysis

Problem may occur due to sequential

componentswhich may have different range

from adjacent components. So, adjustment to uncertainty data must be

done.


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Another example

In using a temperature probe, the following uncertainties

were determined:

Hysteresis 0.10C

Linearization error 0.2% of the reading

Repeatability 0.20C

Resolution error 0.050CZero offset 0.10C

Determine the type of these error (random or systematic) and

the total uncertainty due to these effects for a temperature

reading of 1200C


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C

B

28.0

)1.0)]120)(002[(.1.0( 2/1222

CP 21.0)2.05(. 2/122

Cw

w

tSBpBw

35.0

])21.0()28.0[(

])([][

2/122

2/1222/122

Assuming that the random errors have been determined with samples>30,

So total uncertainty

hysteresis 0 1. C

systematic

Lineariz.error 0 2%. of reading

systematic

Resolution error 0.05C random

zero off set 0.1C systematic

repeatability 0 2. C random


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Two resistors, R1=100.0 0.2 and R2=60.0 0.1

are connected (a) in series and (b) in parallel.

Calculate the uncertainty in the resistance of the resultantscircuits. What is the maximum possible error in each case?


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R R R R

R

R

R

wR

RR

wR

RR

wR

wR

R

Rw

R

R

Rw

R

1 21

1

2

1

112

12

2 1 2

0 2 12

01 12 1 2

0 22

11

12

0 2 01 0 3

,

[( ) ( ) ] /

[( . * ) ( . * ) ]/

.

,max

. . .

R R R

R R

R

R

R

R R

R

R

R

R R

w

w

R

R

1 2

1 2 1

22

1 2

2

2

12

1 2

2

2

2

2

2

2 2 1 2

50

150

100

150

0 11 0 44

0 11 0 2 0 44 0 1 0 05

0 11 2 0 44 0 1 0 07

( ),

( )

( )

. .

[( . * . ) ( . * . ) ] .

. *. . * . .

/

,max

(a) In series

(b) In parallel


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Another example:

A mechanical speed control system works on the basis of

centrifugal force, which is related to angular velocity

through the formula:

F=mr2

where F is the force, m is the mass of the rotating weights,r is the radius of rotation, and w is the angular velocity of

the system. The following values are measured to determine

:

r=200.02 mm, m=100 0.5 g and F=500 0.1%N

Find the rotational speed in rpm and its uncertainty. All

measured values have a confidence level of 95%.


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)%95(sec/30.1

)%95(%26.0,10*60.2

]101010*25[2

1

])

500

5.()

20

02.()

100

5.[(

2

1

])()()[(2

1

6.7.

sec/500,000,250,)(

3

2/1666

2/1222

2/1222

22/1

levelconfidenceradw

levelconfidencew

w

F

w

r

w

m

ww

Eq

rad

mr

F

Frm

Solution


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SignalConditioning

End of Lecture 4

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