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Summary of Experimental Uncertainty Assessment Methodology

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Title: Summary of Experimental Uncertainty Assessment Methodology


1
Summary of Experimental Uncertainty Assessment
Methodology
  • F. Stern, M. Muste, M-L. Beninati, W.E. Eichinger

2
Table of Contents
  • Introduction
  • Test Design Philosophy
  • Definitions
  • Measurement Systems, Data-Reduction Equations,
    and Error Sources
  • Uncertainty Propagation Equation
  • Uncertainty Equations for Single and Multiple
    Tests
  • Implementation Recommendations

3
Introduction
  • Experiments are an essential and integral tool
    for engineering and science
  • Experimentation procedure for testing or
    determination of a truth, principle, or effect
  • True values are seldom known and experiments have
    errors due to instruments, data acquisition, data
    reduction, and environmental effects
  • Therefore, determination of truth requires
    estimates for experimental errors, i.e.,
    uncertainties
  • Uncertainty estimates are imperative for risk
    assessments in design both when using data
    directly or in calibrating and/or validating
    simulation methods

4
Introduction
  • Uncertainty analysis (UA) rigorous methodology
    for uncertainty assessment using statistical and
    engineering concepts
  • ASME (1998) and AIAA (1999) standards are the
    most recent updates of UA methodologies, which
    are internationally recognized
  • Presentation purpose to provide summary of EFD
    UA methodology accessible and suitable for
    student and faculty use both in classroom and
    research laboratories

5
Test design philosophy
  • Purposes for experiments
  • Science technology
  • Research development
  • Design, test, and product liability and
    acceptance
  • Instruction
  • Type of tests
  • Small- scale laboratory
  • Large-scale TT, WT
  • In-situ experiments
  • Examples of fluids engineering tests
  • Theoretical model formulation
  • Benchmark data for standardized testing and
    evaluation of facility biases
  • Simulation validation
  • Instrumentation calibration
  • Design optimization and analysis
  • Product liability and acceptance

6
Test design philosophy
  • Decisions on conducting experiments governed by
    the ability of the expected test outcome to
    achieve the test objectives within allowable
    uncertainties
  • Integration of UA into all test phases should be
    a key part of entire experimental program
  • Test description
  • Determination of error sources
  • Estimation of uncertainty
  • Documentation of the results

7
Test design philosophy
8
Definitions
  • Accuracy closeness of agreement between measured
    and true value
  • Error difference between measured and true value
  • Uncertainties (U) estimate of errors in
    measurements of individual variables Xi (Uxi) or
    results (Ur) obtained by combining Uxi
  • Estimates of U made at 95 confidence level

9
Definitions
  • Bias error b fixed, systematic
  • Bias limit B estimate of b
  • Precision error e random
  • Precision limit P estimate of e
  • Total error d b e

10
Measurement systems, data reduction equations,
error sources
  • Measurement systems for individual variables Xi
    instrumentation, data acquisition and reduction
    procedures, and operational environment
    (laboratory, large-scale facility, in situ) often
    including scale models
  • Results expressed through data-reduction
    equations
  • r r(X1, X2, X3,, Xj)
  • Estimates of errors are meaningful only when
    considered in the context of the process leading
    to the value of the quantity under consideration
  • Identification and quantification of error
    sources require considerations of
  • Steps used in the process to obtain the
    measurement of the quantity
  • The environment in which the steps were
    accomplished

11
Measurement systems and data reduction equations
  • Block diagram showing elemental error sources,
    individual measurement systems, measurement of
    individual variables, data reduction equations,
    and experimental results

12
Error sources
  • Estimation assumptions 95 confidence level,
    large-sample, statistical parent distribution

13
Uncertainty propagation equation
  • Bias and precision errors in the measurement of
    Xi propagate through the data reduction equation
    r r(X1, X2, X3,, Xj) resulting in bias and
    precision errors in the experimental result r
  • A small error (?Xi) in the measured variable
    leads to a small error in the result (?r) that
    can be approximated using Taylor series expansion
    of r(Xi) about rtrue(Xi) as
  • The derivative is referred to as sensitivity
    coefficient. The larger the derivative/slope,
    the more sensitive the value of the result is to
    a small error in a measured variable

14
Uncertainty propagation equation
  • Overview given for derivation of equation
    describing the error propagation with attention
    to assumptions and approximations used to obtain
    final uncertainty equation applicable for single
    and multiple tests
  • Two variables, kth set of measurements (xk, yk)

The total error in the kth determination of r
(1)
sensitivity coefficients
15
Uncertainty propagation equation
  • We would like to know the distribution of dr
    (called the parent distribution) for a large
    number of determinations of the result r
  • A measure of the parent distribution is its
    variance defined as

(2)
  • Substituting (1) into (2), taking the limit
    for N approaching infinity, using definitions of
    variances similar to equation (2) for b s and e
    s and their correlation, and assuming no
    correlated bias/precision errors

(3)
  • ss in equation (3) are not known estimates for
    them must be made

16
Uncertainty propagation equation
  • Defining
  • estimate for
  • estimates for the
    variances and covariances (correlated bias
    errors) of
    the bias error distributions
  • estimates for the
    variances and covariances ( correlated precision
    errors) of the
    precision error distributions

equation (3) can be written as
Valid for any type of error distribution
  • To obtain uncertainty Ur at a specified
    confidence level (C), a coverage factor (K) must
    be used for uc
  • For normal distribution, K is the t value from
    the Student t distribution.
  • For N ? 10, t 2 for 95 confidence level

17
Uncertainty propagation equation
  • Generalization for J variables in a result r
    r(X1, X2, X3,, Xj)

sensitivity coefficients
Example
18
Uncertainty equations for single and multiple
tests
  • Measurements can be made in several ways
  • Single test (for complex or expensive
    experiments) one set of measurements (X1, X2,
    , Xj) for r
  • According to the present methodology, a test is
    considered a single test if the entire test is
    performed only once, even if the measurements of
    one or more variables are made from many samples
    (e.g., LDV velocity measurements)
  • Multiple tests (ideal situations) many sets of
    measurements (X1, X2, , Xj) for r at a fixed
    test condition with the same measurement system

19
Uncertainty equations for single and multiple
tests
  • The total uncertainty of the result

(4)
  • Br same estimation procedure for single and
    multiple tests
  • Pr determined differently for single and
    multiple tests

20
Uncertainty equations for single and multiple
tests bias limits
  • Br
  • Sensitivity coefficients
  • Bi estimate of calibration, data acquisition,
    data reduction, conceptual bias errors for Xi..
    Within each category, there may be several
    elemental sources of bias. If for variable Xi
    there are J significant elemental bias errors
    estimated as (Bi)1, (Bi)2, (Bi)J, the bias
    limit for Xi is calculated as
  • Bik estimate of correlated bias limits for Xi
    and Xk

21
Uncertainty equations for single test precision
limits
  • Precision limit of the result (end to end)

t coverage factor (t 2 for N gt 10) Sr the
standard deviation for the N readings of the
result. Sr must be determined from N readings
over an appropriate/sufficient time interval
  • Precision limit of the result (individual
    variables)

the precision limits for Xi
Often is the case that the time interval is
inappropriate/insufficient and Pis or Prs must
be estimated based on previous readings or best
available information
22
Uncertainty equations for multiple tests
precision limits
  • The average result
  • Precision limit of the result (end to end)

t coverage factor (t 2 for N gt 10)
standard deviation for M readings of the result
  • The total uncertainty for the average result
  • Alternatively can be determined by RSS of
    the precision limits of the individual variables

23
Implementation
  • Define purpose of the test
  • Determine data reduction equation r r(X1, X2,
    , Xj)
  • Construct the block diagram
  • Construct data-stream diagrams from sensor to
    result
  • Identify, prioritize, and estimate bias limits at
    individual variable level
  • Uncertainty sources smaller than 1/4 or 1/5 of
    the largest sources are neglected
  • Estimate precision limits (end-to-end procedure
    recommended)
  • Computed precision limits are only applicable for
    the random error sources that were active
    during the repeated measurements
  • Ideally M ? 10, however, often this is no the
    case and for M lt 10, a coverage factor t 2 is
    still permissible if the bias and precision
    limits have similar magnitude.
  • If unacceptably large Ps are involved, the
    elemental error sources contributions must be
    examined to see which need to be (or can be)
    improved
  • Calculate total uncertainty using equation (4)
  • For each r, report total uncertainty and bias and
    precision limits

24
Recommendations
  • Recognize that uncertainty depends on entire
    testing process and that any changes in the
    process can significantly affect the uncertainty
    of the test results
  • Integrate uncertainty assessment methodology into
    all phases of the testing process (design,
    planning, calibration, execution and post-test
    analyses)
  • Simplify analyses by using prior knowledge (e.g.,
    data base), concentrate on dominant error sources
    and use end-to-end calibrations and/or bias and
    precision limit estimation
  • Document
  • test design, measurement systems, and data
    streams in block diagrams
  • equipment and procedures used
  • error sources considered
  • all estimates for bias and precision limits and
    the methods used in their estimation (e.g.,
    manufacturers specifications, comparisons against
    standards, experience, etc.)
  • detailed uncertainty assessment methodology and
    actual data uncertainty estimates

25
References
  • AIAA, 1999, Assessment of Wind Tunnel Data
    Uncertainty, AIAA S-071A-1999.
  • ASME, 1998, Test Uncertainty, ASME PTC
    19.1-1998.
  • ANSI/ASME, 1985, Measurement Uncertainty Part
    1, Instrument and Apparatus, ANSI/ASME PTC
    19.I-1985.
  • Coleman, H.W. and Steele, W.G., 1999,
    Experimentation and Uncertainty Analysis for
    Engineers, 2nd Edition, John Wiley Sons, Inc.,
    New York, NY.
  • Coleman, H.W. and Steele, W.G., 1995,
    Engineering Application of Experimental
    Uncertainty Analysis, AIAA Journal, Vol. 33,
    No.10, pp. 1888 1896.
  • ISO, 1993, Guide to the Expression of
    Uncertainty in Measurement,", 1st edition, ISBN
    92-67-10188-9.
  • ITTC, 1999, Proceedings 22nd International Towing
    Tank Conference, Resistance Committee Report,
    Seoul Korea and Shanghai China.
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