MEASUREMENT AND INSTRUMENTATION BMCC 3743

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MEASUREMENT AND INSTRUMENTATIONBMCC 3743

• LECTURE 4 EXPERIMENTAL UNCERTAINTY 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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Propagation of uncertainties

• Uncertainty analysis is important to identify
  corrective measures while validating and
  performing experiments.
• Propagation of uncertainties gt 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

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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 xis via the differential
  equations

(1)
Sensitivity coefficient
(2)
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General consideration

• For calculated result based on measured xis, 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.

Uncertainty in variables
(3)
Uncertainty in result
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General consideration

• Hence Eq. (3) is better represented by
• gtroot 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)
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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.

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Answer to Exercise
Because PVI dP/dVI10.0 A , dP/diV100.V then
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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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Consideration of systematic and random components
of uncertainty

• Random uncertainty depends on sample size
  (usually large, ngt30)
• Systematic uncertainty is independent of sample
  size does not vary during repeated reading
• Need to separate for detailed uncertainty analysis

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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)
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Systematic uncertainty

• Sometimes assumed as level of accuracy
• Depends on manufacturers specification,
  calibration tests, mathematical modelling,
  considerable judgement as well as comparisons
  between independent measurements.

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Systematic uncertainty some examples

• Radiation heat transfer gt lower measured value
• Instrument location gt 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 of x,

(7)
(8)
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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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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.

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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
• gt Based on experiment requirement.

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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

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Combining elemental systematic random
uncertainties (RSS)
Calibration Uncertainties
Data-Acquisition Uncertainties
Data-Reduction Uncertainties
Uncertainties Due to Methods
Other Uncertainties
Reproduced from Wheelers book ASME 1998
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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
• gt Welch-Satterthwaite formulation (ASME 1998)

(9)
Degrees of freedom of individual elemental error
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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 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)
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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)
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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)
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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.

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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)
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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)
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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 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

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Solution
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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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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.

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Guideline to assign elemental error (Table 1),
from Wheeler
ERROR ERROR TYPE
Accuracy Common-mode volt Hysterisis Installation Linearity Loading Noise Repeatability Resolution/scale/quantisation Spatial variation Thermal stability (gain, zero, etc.) Systematic Systematic Systematic Systematic Systematic Systematic Random Random Random Systematic Random
assume no. of samples gt 30
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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).

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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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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.

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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
0.050C 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
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systematic
hysteresis
systematic
Lineariz.error
Resolution error 0.05C random
zero off set 0.1C systematic
repeatability
random
Assuming that the random errors have been
determined with samplesgt30,
So total uncertainty
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Two resistors, R1100.0 0.2 and R260.0 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?
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(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 Fmrw2 w
here 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
r20 0.02 mm, m100 0.5 g and F500
0.1N Find the rotational speed in rpm and its
uncertainty. All measured values have a
confidence level of 95.
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Solution
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Next Lecture

• Signal Conditioning
• End of Lecture 4