MER331 Lab 2

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MER331 Lab 2

• Uncertainty Analysis

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Error Sources

• The measurement process consists of three
  dis-tinct steps calibration, data acquisition,
  and data reduction
• calibration error
• data acquisition error
• data reduction error

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Uncertainty Analysis

• Error - difference between true value and
  measured value.
• Two general categories of error (exclude gross
  blunders)
• Fixed error remove by calibration
• Random error quantify by uncertainty analysis
• The term uncertainty is used to refer to a
  possible value that an error may have

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Stages of Uncertainty Analysis

• Design Stage Uncertainty Analysis
• Uncertainty analysis can be used to assist in the
  selection of equipment and procedures based on
  their relative performance and cost.
• Advanced-Stage and Single Measurement Uncertainty
  Analysis
• Consider procedural and test control errors that
  affect the measurement

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Uncertainty Analysis

• m Kt(rball-rfluid)
• How do I find uncertainty in m if I have
  uncertainty in K, t, rball,and rfluid.?
• Three steps
• Estimate uncertainty interval for each measured
  quantity
• State the confidence limit on each measurement
• Analyze the propagation of uncertainty into
  results calculated from experimental data.

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Estimating Uncertainty Interval

• If you have a statistically significant sample
  use 2s for a 95 confidence interval. (1st
  order analysis)
• General Practice in engineering is 95
  confidence of 20 to 1 odds.
• Otherwise use ½ smallest scale division as an
  estimate. (0th order analysis)
• Technique due to Kline, S.J., and McClintock,
  F.A., Describing Uncertainties in Single Sample
  Experiments, Mechanical Engineering, 75, 1953.

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Propagation of Uncertainty -The Basic Mathematics

• The value of dxi represents 2s for a single
  sample analysis. The result, R of an experiment
  is assumed to be calculated from a set of
  measurements
• R R(X1, X2, X3,, XN)
• The effect of the uncertainty in a single
  measurement (i.e. one of the Xs) on the
  calculated result, R, if only that one X were in
  error is

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The Basic Mathematics

• When several independent variables (Xs) are used
  in calculating the Result, R, the individual
  terms are combined by a root-sum-square method
  (Method due to Kline and McClintock (1953))
• In most situations the overall uncertainty in a
  given result is dominated by only a few of its
  terms. Ignore terms that are smaller than the
  largest term by a factor of 3 or more.

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Example 1

• For a displacement transducer having a
  calibration curve y KE2, estimate the
  uncertainty in displacement y for
• E 5.00 V, dE 0.01 V
• K 10.10 mm/V2, dk 0.10 mm/V2
• (95 confidence).

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Example 1

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Uncertainty as a Percentage

• In most situations the overall uncertainty in a
  given result is dominated by only a few of its
  terms. Ignore terms that are smaller than the
  largest term by a factor of 3 or more.
• It is difficult (impossible) to compare errors
  with different units associated with them (e.g.
  how big is a 2 gram error compared to a 2 second
  error?)
• To solve this we nondimensionalize the errors

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So, as a percentage
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Uncertainty Analysis (Product form Equations)

• .
• If
• Then
• The exponent on Xi becomes its sensitivity
  coefficient

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Example 2

• For a displacement transducer having a
  calibration curve y KE2, estimate the PERCENT
  uncertainty in displacement y for
• E 5.00 V, dE 0.01 V
• K 10.10 mm/V2, dk 0.10 mm/V2
• (95 confidence).

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Example 2

• If
• Then

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Zero-Order Uncertainty

• At zero-order uncertainty, all variables and
  parameters that affect the outcome of the
  measurement, including time, are assumed to be
  fixed except for the physical act of observation
  itself.
• Any data scatter is the results of instrument
  resolution alone uo.

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Higher-Order Uncertainty

• Higher order uncertainty estimates consider the
  controllability of the test operating conditions.
• For a first order estimate we might make a
  series of measurements over time and calculate
  the variation in that measurement. The first
  order uncertainty of that measurand is then
• u1 2s at (95)
• Note Assuming we make enough measurements

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Nth-Order Uncertainty

• As the final estimate, instrument calibration
  characteristics are entered into the scheme
  through the instrument uncertainty, uc. A
  practical estimate of the Nth order uncertainty
  uN is
• Uncertainty analyses at the Nth order allow for
  the direct comparison between results of similar
  tests obtained using different instruments or at
  different test facilities.

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Uncertainty Analysis

• rfluid m/V
• How do I find uncertainty in r?
• Three steps
• Estimate uncertainty interval for each measured
  quantity, (m and V)
• State the confidence limit on each measurement
• Analyze the propagation of uncertainty into
  results calculated from experimental data.

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Report Results at uN level
Nth order Uncertainty
First-order Uncertainty u1 gt u0
Zero- order Uncertainty u0 1/2 resolution