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Summary of Experimental Uncertainty Assessment MethodologyPowerPoint Presentation

Summary of Experimental Uncertainty Assessment Methodology

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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
- Uncertainty Equations for Single and Multiple Tests
- Implementation & Recommendations

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

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

- 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 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

- 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

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 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 sources

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

Error sources sources

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

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 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 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 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 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 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 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

- 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 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 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 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 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 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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