1 / 25

# Summary of Experimental Uncertainty Assessment Methodology - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Summary of Experimental Uncertainty Assessment Methodology' - jessenia-kianoush

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Summary of Experimental Uncertainty Assessment Methodology

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

• 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

• 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

• 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

• 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

• 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

• 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

• 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 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

• 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

• 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

• 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

• 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

• 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

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

sensitivity coefficients

Example:

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

• The total uncertainty of the result

(4)

• Br : same estimation procedure for single and multiple tests

• Pr : determined differently for single and multiple tests

• 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

• 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

• 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.