MEASUREMENT AND INSTRUMENTATIONBMCC 3743 LECTURE 4: EXPERIMENTAL UNCERTAINTY ANALYSIS Mochamad Safarudin
Faculty of Mechanical Engineering, UTeM
2010

Contents :

andims - FKM - UTeM 2 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

Propagation of uncertainties :

andims - FKM - UTeM 3 Propagation of uncertainties Uncertainty analysis is important to identify corrective measures while validating and performing experiments.
Propagation of uncertainties => ‘total uncertainties’, 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

General consideration :

andims - FKM - UTeM 4 General consideration If R is a function of n measured variables x1, x2, …. xn, i.e.
Then a small change in is due to small changes in s in xi’s via the differential equations: Sensitivity coefficient (1) (2)

General consideration :

andims - FKM - UTeM 5 General consideration For calculated result based on measured xi’s, Eq. (2) can be rewritten as
where | | is to make sure we don’t get zero uncertainty in R.
However, this can produce high estimate for wR. (3) Uncertainty in result Uncertainty in variables

General consideration :

andims - FKM - UTeM 6 General consideration Hence Eq. (3) is better represented by
=>root of the sum of the squares (RSS)
In this case, the confidence level must be the same for all uncertainties (typically 95%).
Assumption is made that each measured variables (hence, error) are independent of each other. (4)

Exercise :

andims - FKM - UTeM 7 Exercise To calculate the power consumption of an electric circuit, we have P = VI where
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) and Eq. (4) respectively.

Answer to Exercise :

andims - FKM - UTeM 8 Answer to Exercise Because P=VI
dP/dV=I=10.0 A , dP/di=V=100.V then

Contents :

andims - FKM - UTeM 9 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

Consideration of systematic and random components of uncertainty :

andims - FKM - UTeM 10 Consideration of systematic and random components of uncertainty Random uncertainty depends on sample size (usually large, n>30)
Systematic uncertainty is independent of sample size & does not vary during repeated reading
Need to separate for detailed uncertainty analysis

Random uncertainty :

andims - FKM - UTeM 11 Random uncertainty Using t-distribution, the random uncertainty for all measurements is given by
where Sx is the standard deviation of the sample
For a single measurement (also for each individual measurement), the random uncertainty is (5) (6)

Systematic uncertainty :

andims - FKM - UTeM 12 Systematic uncertainty Sometimes assumed as level of accuracy
Depends on manufacturer’s specification, calibration tests, mathematical modelling, considerable judgement as well as comparisons between independent measurements.

Systematic uncertainty – some examples :

andims - FKM - UTeM 13 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

Combining random & systematic uncertainties :

andims - FKM - UTeM 14 Combining random & systematic uncertainties Total uncertainty is obtained, using RSS (Eq. 4) for all measurements, is given by
For a single measurement of x, (7) (8)

Contents :

andims - FKM - UTeM 15 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

Sources of elemental error :

andims - FKM - UTeM 16 Sources of elemental error ‘Chain of uncertainties’, e.g. A/D converter would have quantisation errors, sensitivity errors and linearity errors. Each of these components contribute to further errors.
Can be random or systematic error.

Estimation of uncertainty :

andims - FKM - UTeM 17 Estimation of uncertainty Systematic uncertainty: just combine all elemental uncertainties
Random uncertainty: 3 approaches to determine Sx
Run entire test in a sufficient number of times
Run auxiliary tests for each measured variable x.
Combine elemental random uncertainties
=> Based on experiment requirement.

5 categories of elemental errors :

andims - FKM - UTeM 18 5 categories of elemental errors Calibration Uncertainties: residual systematic errors due to; uncertainty in standards, uncertainty in calibration process, randomness in the process
Data-Acquisition Uncertainties: during measurement due to; random variation of measurand, A/D conversion uncertainties, uncertainties in recording devices
Data-Reduction Uncertainties: due to interpolation, curve fitting and differentiating data curves
Uncertainties Due to Methods: due to assumptions/constant in calculation, spatial effects and uncertainties due to hysterisis, instability, etc.
Other Uncertainties

Combining elemental systematic & random uncertainties (RSS) :

andims - FKM - UTeM 19 Combining elemental systematic & random uncertainties (RSS) Calibration Uncertainties Data-Acquisition Uncertainties Data-Reduction Uncertainties Uncertainties Due to Methods Other Uncertainties Reproduced from Wheeler’s book: ASME 1998

Degrees of freedom, vx :

andims - FKM - UTeM 20 Degrees of freedom, vx When sample size is large, vx is simply number of sample, n, minus 1.
When sample size is small, then vx is given by
=> Welch-Satterthwaite formulation (ASME 1998) (9) Degrees of freedom of individual elemental error

Contents :

andims - FKM - UTeM 21 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

Uncertainty of the final result (Multiple measurement) :

andims - FKM - UTeM 22 Uncertainty of the final result (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. (10)

Uncertainty of the final result (Multiple measurement) :

andims - FKM - UTeM 23 Uncertainty of the final result (Multiple measurement) Rearranging Eq. 4 (RSS), we get the systematic uncertainty in terms of the combination of elemental systematic uncertainties, given by (11)

Uncertainty of the final result (Multiple measurement) :

andims - FKM - UTeM 24 Uncertainty of the final result (Multiple measurement) Therefore, the total uncertainty estimate of the mean value of R is
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 (12)

Uncertainty of the final result (Single measurement) :

andims - FKM - UTeM 25 Uncertainty of the final result (Single measurement) To deal with uncertainty of a single test result only
Practical applications: measuring blood pressure/ heartbeat, speed of car, etc
To estimate random uncertainty of the result, must use or combine auxiliary tests and elemental random uncertainties.

Uncertainty of the final result (Single measurement) :

andims - FKM - UTeM 26 Uncertainty of the final result (Single measurement) Similar to Eq. 11, standard deviation of the result is given by
Hence, the total uncertainty in the final result is given by (13) (14)

Uncertainty of the final result (Single measurement) :

andims - FKM - UTeM 27 Uncertainty of the final result (Single measurement) For a large n, then t is 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 (15)

Slide 28:

andims - FKM - UTeM 28 Example The manufacturer of plastic pipes uses a scale with an
Accuracy of 1.5% of its range of 5 kg to measure the
Mass 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
The mean mass of the sample
The standar deviation of the sample and the standar deviation
of the mean
c. The total uncertainty of the mass of a single product at
a 95% confidence level
The total uncertainty of the average mass of the product at a 95%
confidence level

Slide 29:

andims - FKM - UTeM 29 Solution:

Slide 30:

andims - FKM - UTeM 30

Contents :

andims - FKM - UTeM 31 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

Design-stage uncertainty analysis (Based on ASME 1998) :

andims - FKM - UTeM 32 Design-stage uncertainty analysis (Based on ASME 1998) Define the measurement process
State test objectives, identify independent parameters and their nominal values, etc
List all elemental error sources
To do a complete list of possible error sources for each measured parameter.
Estimate the elemental errors
Estimate the systematic uncertainties and standard deviations. If error is random in nature and/or data is available to estimate the std dev. of a parameter, then classify it as random uncertainties, which must have the same confidence level. For small samples, to determine degrees of freedom. Refer Table 1.

Guideline to assign elemental error (Table 1), from Wheeler :

andims - FKM - UTeM 33 Guideline to assign elemental error (Table 1), from Wheeler *assume no. of samples > 30

Design-stage uncertainty analysis (Based on ASME 1998) :

andims - FKM - UTeM 34 Design-stage uncertainty analysis (Based on ASME 1998) Calculate the systematic and random uncertainty for each measured variable
Use the RSS formulation with data & procedure in Step 3.
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.
Calculate the total uncertainty of the results
Use the RSS formulation to find the total uncertainty of the result(s).

Contents :

andims - FKM - UTeM 35 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

Applying uncertainty-analysis in digital data acquisition system :

andims - FKM - UTeM 36 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 reduction and analysis
Problem may occur due to sequential components which may have different range from adjacent components.
So, adjustment to uncertainty data must be done.

Slide 37:

andims - FKM - UTeM 37 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 ±0.050C
Zero 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

Slide 38:

andims - FKM - UTeM 38 Assuming that the random errors have been determined with samples>30, So total uncertainty hysteresis systematic Lineariz.error systematic Resolution error 0.05C random zero off set 0.1C systematic repeatability random

Slide 39:

andims - FKM - UTeM 39 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 resultants
circuits. What is the maximum possible error in each case?

Slide 40:

andims - FKM - UTeM 40 (a) In series (b) In parallel

Slide 41:

andims - FKM - UTeM 41 Another example: A mechanical speed control system works on the basis of
centrifugal force, which is related to angular velocity
through the formula:
F=mrw2
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
w :
r=20± 0.02 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%.

Slide 42:

andims - FKM - UTeM 42 Solution

Next Lecture :

andims - FKM - UTeM 43 Next Lecture Signal Conditioning
End of Lecture 4

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