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Summary of Experimental Uncertainty Assessment Methodology. F. Stern, M. Muste, M-L. Beninati, W.E. Eichinger. Table of Contents. Introduction Test Design Philosophy Definitions Measurement Systems, Data-Reduction Equations, and Error Sources Uncertainty Propagation Equation

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Summary of experimental uncertainty assessment methodology

Summary of Experimental Uncertainty Assessment Methodology

F. Stern, M. Muste, M-L. Beninati, W.E. Eichinger


Table of contents
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


Introduction
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


Introduction1
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


Test design philosophy
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


Test design philosophy1
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



Definitions
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 at95% confidence level


Definitions1
Definitions

  • Bias errorb: fixed, systematic

  • Bias limitB: estimate of b

  • Precision errore: random

  • Precision limit P: estimate of e

  • Total error:d = b + e


Measurement systems data reduction equations error sources
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


Measurement systems and data reduction equations
Measurement systems and data reduction equations sources

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


Error sources
Error sources sources

  • Estimation assumptions: 95% confidence level, large-sample, statistical parent distribution


Uncertainty propagation equation
Uncertainty propagation equation sources

  • 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


Uncertainty propagation equation1
Uncertainty propagation equation sources

  • 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


Uncertainty propagation equation2
Uncertainty propagation equation sources

  • 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 ’sand their correlation, and assuming no correlated bias/precision errors

(3)

  • s’s in equation (3) are not known; estimates for them must be made


Uncertainty propagation equation3
Uncertainty propagation equation sources

  • 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


Uncertainty propagation equation4
Uncertainty propagation equation sources

  • Generalization for J variables in a result r = r(X1, X2, X3,…, Xj)

sensitivity coefficients

Example:


Uncertainty equations for single and multiple tests
Uncertainty equations for single and multiple tests sources

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


Uncertainty equations for single and multiple tests1
Uncertainty equations for single and multiple tests sources

  • The total uncertainty of the result

(4)

  • Br : same estimation procedure for single and multiple tests

  • Pr : determined differently for single and multiple tests


Uncertainty equations for single and multiple tests bias limits
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


Uncertainty equations for single test precision limits
Uncertainty equations for single test: precision limits limits

  • Precision limit of the result (end to end):

t: coverage factor (t = 2 for N > 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 Pi’s or Pr’s must be estimated based on previous readings or best available information


Uncertainty equations for multiple tests precision limits
Uncertainty equations for multiple tests: precision limits limits

  • The average result:

  • Precision limit of the result (end to end):

t: coverage factor (t = 2 for N > 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


Implementation
Implementation limits

  • 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 < 10, a coverage factor t = 2 is still permissible if the bias and precision limits have similar magnitude.

    • If unacceptably large P’s 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


Recommendations
Recommendations limits

  • 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


References
References limits

  • 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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