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Today

- ISO Guide to the Expression of Uncertainty in Measurement
- More practical exercises

Uncertainty in Measurement

- Each and every measurement or evaluation comes with an associated uncertainty on its value
- Different philosophies leading to the same approach:
- A true value does not exists, or
- A true value exists but is unknown
Use statistical distributions of probabilityto describe the measurement value

Uncertainty in Measurement

Uncertainty can be assessed:

- By means of experienced evaluation (historical data, datasheet analysis, previous experience…) [B CATEGORY] hypothetical PDF
- By means of repeated measurements on the same measurand (statistical analysis of results) [A CATEGORY] measured PDF
- By means of propagation of other uncertainties in the case of an indirect measurement [combined uncertainty] PDF propagation

Uncertainty Propagation (GUM)

SIMPLIFIED FORM:

- To be used whenever parameters are actually independents(no correlated uncertainties)

GUM Approach

- IDENTIFY THE DATA REDUCTION EQUATION-check for simplified approach applicability
- IDENTIFY EVERY PARAMETER INVOLVED-category, uncertainty, influence coefficient
- COMPUTE THE RESULT ESTIMATED VALUE
- COMPUTE IS COMBINED UNCERTAINTY-propagate uncertainties
- COMPUTE EXTENDED UNCERTAINTY-select an appropriate coverage factor for the confidence level required
- WRITE THE RESULT IN THE CORRECT FORM
G=704±38 MPa (P=99%) or G=704±38 MPa (k=2.58) or G=704 MPa U99% (G)=38 MPa

Detailed Analysis: UMF

UMF: Uncertainty Magnification Factor

- Tells us how much the input uncertainty is magnified by the data reduction equation
- DEPENDS ONLY ON THE EQUATION CHOSEN
- Useful for a preliminar analysis when buying transducer for a particular task, identifying the most critical ones (UMF>1) or the less critical ones (UMF<1)

Detailed Analysis: UPC

UPC: Uncertainty Percentage Contribution

- Tells us which fraction of the combined uncertainty depends on the input uncertainty
- Accounts not only for the equation, but also for the uncertainties actually involved.
- Useful for finding which measurements need to be improved

Exercise 2: Building height

Using a surveyor’s wheel (300mm diameter, 100 division) and one inclinometer (one tenth of grade step), a building is measured with the following values:

ϑ1=61.5° ϑ2=-8.0° L=15m

h1 =Ltg|ϑ1|

h2 =Ltg|ϑ2|

H=h1+h2

H=L(tg|ϑ1|+tg|ϑ2|)

Report the height of the building as a measurement result, using an extended uncertainty with a confidence level of 95%

H

ϑ1

ϑ2

L

H=29.73 ± 0.14 m (k=1.96)

Exercise 2: Building height

- Data Reduction Equation:H=L(tg|ϑ1|+tg|ϑ2|)
- Parameters involved:-L - distance – B category uncertainty- ϑ1 – angle – B category uncertainty- ϑ2 – angle – B category uncertainty

Exercise 2: Building height

- Parametersinvolved: L=15m[m] B categoryuncertaintymeasuredusing a surveyor’s wheel(300mm diameter, 100 division)1 division=p*300mm/100=9.4mm=0.0094mwe’ll assume a uniformdistributionwithhalf-widhta equalto the leastdivision

Shouldhavebeenhalf-widthaequaltohalfof the leastdivision, but, as a ruleofthumb, is common todouble the contributionif the measurementisrough!

a

x

Exercise 2: Building height

- Parameters involved: ϑ1=61.5°=1.073rad[rad] B category uncertaintymeasured using a clinometer(1/10 grade division)1 division=0.1°=0.0017rad we’ll assume a uniform distribution with half-widht a equal to the least division
- The same goes for ϑ2=-8°=-0.140rad

a

x

Exercise 2: Building height

- InfluenceCoefficients
=2.0=66 m/rad=15 m/rad

Exercise 2: Building height

- COMBINED UNCERTAINTY,EXTENDED UNCERTAINTY and MEASUREMENTS RESULTS
- H=29.73m U95%(H)=0.14mH=29.73 ± 0.14m (k=1.96)H=29.73 ± 0.14m (P=95%)
- Critical Analysis: UMF, UPC

Uncertainty in Measurement

- Each and every measurement or evaluation comes with an associated uncertainty on its value
- Different philosophies leading to the same approach:
- A true value does not exists, or
- A true value exists but is unknown
Use statistical distributions of probabilityto describe the measurement value

Measurement Results Compatibility

- Two measurement results are compatible (at a given level of confidence) whereas their confidence intervals overlap.
E.g.speed of my car using a GPS signal:72±1km/h (P=95%)speed of my car using its speedometer:75±7km/h (P=95%)speed of my car measured by police:80±8km/h (P=95%)

are the three measurement results referring to the same measurand? Are they COMPATIBLE?

Measurement Results Compatibility

Yes, they are compatible, as there is an interval in common among the three confidence levels at 95%

At 95% of confidence level we cannot say that the three measurement result are different.

One further example:a force control system for a clamping device indicates a value of F=89N with a tolerance 95% given of 1Nthe same clamping force is measured repeatedly using a load cell, with the following results: F={89,91,90,92,89,89,91} Nare the two results COMPATIBLE at 99%?

Measurement Results Compatibility

One further example:F=89N with a tolerance 95% given of 1N1N extended uncertainty at 95%, supposing a normal distribution we can compute the standard uncertainty (k=1) by dividing for k95%=1.96 => u(F)=0.51 N => U99%(F)=k99%u(F)=2.58x0.51 N=1.3 Nconfidence interval: {87.7 N – 90.3 N}

repeated measurements F={89,91,90,92,89,89,91} Nmean – F=90.14 N standard deviation - σ=1.215 Nnumber of samples – n=7 => v=n-1=6 degrees of freedomThe student distribution can be assumed to extend uncertainty from repeated meas. using k=t99%,v=3.71 => U99%(F)= 1.7 Nconfidence interval: {88.4 N – 91.8 N}

YES, the values are compatible at 99% confidence: the intervals overlap between 88.4 N and 90.3 N

Exercise 3: Pin On Disk

We were asked to measure the load applied in a PIN-DISK contact during friction tests. The load is given by an hydraulic actuator using a pressure multiplier as shown.

Knowing the diameters shown were measured using 1/20 calipers, and considering the working pressures shown, which transducer is suitable for the task (considering the same price for both)?

Which uncertainty could be associated to the load measured?

Transducer A: 300 kPa range, overall uncertainty 2%FS Transducer B: 10 MPa range, overall uncertainty 1%FS

d2=200mm

d1=40mm

d0=10mm

p2≈200kPa

p1=p2 (d2/d1)²

Exercise 3: Pin On Disk

Hypothesis A:

- use the transducer A to measure p2 with an uncertainty of 6kPa
Hypothesis B:-use the transducer B to measure p1 with an uncertainty of 100kPa

The best optionishypothesis B, asitsuncertaintyissmallerthan the other.

Butwhatif the relative uncertaintyof A was 1%FS?

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