EFD_UA_Summary - Summary of Experimental Uncertainty...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Summary of Experimental Uncertainty Assessment Methodology F. Stern, M. Muste, M-L. Beninati, W.E. Eichinger 12/09/11 1 Table of Contents Introduction Test Design Philosophy Definitions Measurement Systems, Data-Reduction Equations, Measurement and Error Sources and Uncertainty Propagation Equation Uncertainty Equations for Single and Multiple Uncertainty Tests Tests Implementation & Recommendations Introduction Experiments are an essential and integral tool for Experiments engineering and science engineering Experimentation: procedure for testing or determination of a truth, principle, or effect truth, True values are seldom known and experiments have errors True due to instruments, data acquisition, data reduction, and environmental effects environmental Therefore, determination of truth requires estimates for Therefore, experimental errors, i.e., uncertainties experimental Uncertainty estimates are imperative for risk assessments in Uncertainty design both when using data directly or in calibrating and/or validating simulation methods validating Introduction Uncertainty analysis (UA): rigorous methodology for Uncertainty uncertainty assessment using statistical and engineering concepts 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 Presentation methodology accessible and suitable for student and faculty use both in classroom and research laboratories Test design philosophy Purposes for experiments: Purposes Type of tests: Science & technology Research & development Design, test, and product liability and acceptance Instruction 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 Decisions 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 Integration key part of entire experimental program key Test description Determination of error sources Estimation of uncertainty Documentation of the results Test design philosophy D E F IN E P U R P O S E O F T E S T A N D R E S U L T S U N C E R T A IN T Y R E Q U IR E M E N T S S E L E C T U N C E R T A IN T Y M E T H O D - D E S IG N T H E T E S T D E S IR E D P A R A M E T E R S ( C D , C R , ....) M O D E L C O N F IG U R A T IO N S (S ) T E S T T E C H N IQ U E ( S ) M E A S U R E M E N T S R E Q U IR E D S P E C IF IC IN S T R U M E N T A T IO N C O R R E C T IO N S T O B E A P P L IE D D E T E R M IN E E R R O R S O U R C E S A F F E C T IN G R E S U L T S YES E S T IM A T E E F F E C T O F T H E E R R O R S O N R E S U LTS IM P R O V E M E N T P O S S IB L E ? NO U N C E R T A IN T Y A C C E P TA BLE ? NO YES NO NO TEST IM P L E M E N T T E S T S TAR T TE S T R E S U LT S A C C E P TA B LE ? NO YES YES NO C O N T IN U E T E S T PURPOSE A C H IE V E D ? M EASUREMENT SYSTEM PRO BLEM ? S O LV E P R O B LE M YES E S T IM A T E A C TU A L D ATA U N C E R T A IN T Y - DOCUMEN REFERENC P R E C IS IO N B IA S L IM IT TO TA L U N C T R E S U LT S E C O N D IT IO N L IM IT E R T A IN T Y Definitions Accuracy: closeness of agreement between measured and true value measured Error: difference between measured and true value difference Uncertainties (U): estimate of errors in measurements of individual variables Xi (Uxi) or or results (Ur) obtained by combining Uxi results obtained Estimates of U made at 95% confidence level Estimates 95% Definitions ixed, Bias Bias error β: ffixed, systematic systematic δ k+ Bias Bias limit B: estimate 1 εk δk ε k+ 1 X X of β X t ru e X k k+ 1 ( a ) tw o r e a d in g s random random Precision limit P: estimate of ε β+ ε β ε FREQUENCY OF OCCURRENCE Precision Precision error ε: Total Total error: δ = δ = to ta l e r ro r β = b ia s e r r o r ε = p r e c is io n e r r o r β ε X X t ru e µ M A G N IT U D E O F X (b ) in fin ite n u m b e r o f re a d in g s Measurement systems, data reduction equations, & error sources Measurement systems for individual variables Xi: Measurement instrumentation, data acquisition and reduction procedures, and operational environment (laboratory, large-scale facility, in situ) often including scale models facility, Results expressed through data-reduction equations r = r(X1, X2, X3,…, Xj) ,…, Estimates of errors are meaningful only when considered Estimates in the context of the process leading to the value of the quantity under consideration quantity Identification and quantification of error sources require Identification 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 Block diagram showing elemental error sources, individual Block measurement systems, measurement of individual variables, data reduction equations, and experimental results data E LE M EN TA L ERROR SOURCES 1 2 J IN D IV ID U A L M EASUREM ENT SYSTEMS X 1 B ,P X 2 B ,P X J B ,P M EASUREM ENT O F IN D IV ID U A L V A R IA B L E S 1 1 2 2 J r = r ( X , X ,......, X ) 1 2 r B ,P r J r J D A T A R E D U C T IO N E Q U A T IO N E X P E R IM E N T A L R E S U LT Error sources Estimation assumptions: 95% confidence level, Estimation large-sample, statistical parent distribution large-sample, T E S T E N V IR O N M E N T: M O D E L F ID E L IT Y A N D TEST SETUP: - A s b u ilt g e o m e tr y - H y d ro d y n a m ic d e fo rm a tio n - S u r f a c e f in is h - M o d e l p o s itio n in g - C a l ib r a t i o n v e r s u s t e s t - S p a t i a l/ t e m p o r a l v a r i a t i o n s o f t h e f lo w - S e n s o r i n s t a l l a t io n / l o c a t i o n - W a ll i n t e r f e r e n c e - F lu i d a n d f a c i l i t y c o n d i t i o n s C O N T R IB U T IO N S T O E S T IM A T E D U N C E R T A IN IT Y S IM U L A T IO N T E C H N IQ U E S : - I n s tr u m e n ta tio n in te r fe r e n c e - S c a le e ff e c ts D A T A A C Q U IS IT IO N A N D R E D U C T IO N : - S a m p l in g , f ilt e r in g , a n d s t a t is t i c s - C u r v e fits - C a li b r a t i o n s Uncertainty propagation equation Bias and precision errors in the measurement of Xi Bias propagate through the data reduction equation r = r(X1, X2, X3,…, Xj) resulting in bias and precision errors in the ,…, resulting experimental result r A small error (δXi) in the measured variable leads to a in small error in the result (δr) that can be approximated using small that Taylor series expansion of r(Xi) about rtrue(Xi) as about δ r = r ( X i ) − rtrue ( X i ) = δ X i dr dX i The derivative is referred to as sensitivity coefficient. The The larger the derivative/slope, the more sensitive the value of the result is to a small error in a measured variable the Uncertainty propagation equation Overview given for derivation of equation describing the error Overview 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) Two r = r ( x, y ) βx xk = xtrue + β xk + ε xk x tr u e yk = ytrue + β yk + ε yk rk − rtrue = βy k εx εy k µx yk r k = r ( x k , yk ) ∂r ( xk − xtrue ) + ∂r ( yk − ytrue ) + R2 ∂x ∂y βr The total error in the kth determination of r The th δ rk = rk − rtrue = θ x ( β xk + ε xk ) + θ y ( β yk + ε yk ) θ x = ∂r ∂x; θ y = ∂r ∂y µy xk r tru e (1 ) sensitivity coefficients k k k εrk rk y tr u e Uncertainty propagation equation We would like to know the distribution of δr (called the parent We distribution) for a large number of determinations of the result r A measure of the parent distribution is its variance defined as measure σ 2 δr () 1 N 2 = lim ∑ δ rk N →∞ N k =1 (2 ) Substituting (1) into (2), taking the limit for N approaching infinity, using definitions of variances similar to equation (2) for β ’s and ε ’s and ’s their correlation, and assuming no correlated bias/precision errors their 2 2 σ δ2r = θ x2σ β x + θ y2σ β y + 2θ xθ yσ β x β y + θ x2σ ε2x + θ y2σ ε2x + 2θ xθ yσ ε xε y (3 ) n σ’s iin equation (3) are not known; estimates for them must be made Uncertainty propagation equation Defining u c2 estimate for σ δ2r 2 2 bx2 , b y , bxy 2 S xw , S y , S xy 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 22 22 uc2 = θ x2bx2 + θ y by + 2θ xθ y bxy + θ x2 S x2 + θ y S y + 2θ xθ y S xy Valid for any type of error Valid distribution distribution To obtain uncertainty Ur at a specified confidence level (C%), a %), coverage factor (K) must be used for uc: coverage must For normal distribution, U r = Kuc K is the t value from the Student t distribution. is value distribution. For N ≥ 10, t = 2 for 95% confidence level For Uncertainty propagation equation Generalization for J variables in a result r = r(X1, X2, X3,…, Xj) Generalization ,…, J J −1 U = ∑ θ B + 2∑ 2 r i =1 2 i 2 i θi = J J ∑θ θ i =1 k =i +1 ik ∂r ∂X i J −1 Bik + ∑θ P + 2∑ i =1 22 ii J ∑θ θ i =1 k = i +1 P i k ik sensitivity coefficients Example: CD = U 2 CD D = C D ( D, ρ , U , A ) 2 1 2 ρU A J J = ∑θ B + ∑θ i2 Pi 2 2 i i =1 2 2 i i =1 2 2 2 ∂C 2 ∂C 2 ∂C 2 ∂C 2 2 = D BD + PD + D Bρ + Pρ2 + D BU + PU2 + D BA + PA2 ∂ρ ∂D ∂U ∂A ( ) ( ) ( ) ( ) Uncertainty equations for single and multiple tests Measurements can be made in several ways: Single test (for complex or expensive experiments): one set Single of measurements (X1, X2, …, Xj) for r of …, for According to the present methodology, a test is considered a single test if the entire According test is performed only once, even if the measurements of one or more variables are made from many samples (e.g., LDV velocity measurements) made Multiple tests (ideal situations): many sets of Multiple measurements (X1, X2, …, Xj) for r at a fixed test condition measurements …, for with the same measurement system with Uncertainty equations for single and multiple tests The total uncertainty of the result The U r2 = B 2 + P 2 r r (4 ) Br : same estimation procedure for single and multiple tests multiple Pr : determined differently for single and multiple tests tests Uncertainty equations for single and multiple tests: bias limits J J −1 B = ∑ θ B + 2∑ 2 r Br : Sensitivity coefficients i =1 2 i 2 i J ∑θ θ i =1 k =i +1 ik Bik ∂r ∂X i θi = : estimate data reduction, conceptual estimate Berrors for Xof. calibration, data acquisition,may be several elemental bias Within each category, there Within i i. 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 (B (B (B ], 2 J B = ∑ ( Bi ) 2 i k =1 k L Bik = ∑ ( Bi ) α ( Bk ) α stimate B : eestimate of correlated bias limits for X and X ik α =1 i k Uncertainty equations for single test: precision limits Precision limit of the result (end to end): Pr = tS r t: coverage factor (t = 2 for N > 10) Sr: the standard deviation for the N readings of the result. Sr must be the determined from N readings over an appropriate/sufficient time interval Precision limit of the result (individual variables): Precision J Pr = ∑ ( θ i Pi )2 i=1 Pi = ti Si the precision limits for Xi the Often is the case that the time interval is inappropriate/insufficient and Pi’s Often ’s or Pr’s must be estimated based on previous readings or best available ’s information information Uncertainty equations for multiple tests: precision limits 1 r= M The average result: M ∑r k k =1 Precision limit of the result (end to end): Precision tS r Pr = M M ( rk − r ) 2 S r = ∑ M −1 k =1 t: coverage factor (t = 2 for N > 10) S r : standard deviation for M readings of the result The total uncertainty for the average result: The ( U = B + P = B + 2 Sr 2 r 2 r 2 r 2 r M ) 2 Alternatively Pr can be determined by RSS of the precision limits of the individual variables precision 1/ 2 Implementation Define purpose of the test Determine data reduction equation: r = r(X1, X2, …, Xj) Determine …, 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 Computed “active” during the repeated measurements “active” Ideally M ≥ 10, however, often this is no the case and for M < 10, a coverage factor t Ideally = 2 is still permissible if the bias and precision limits have similar magnitude. is If unacceptably large P’s are involved, the elemental error sources contributions must If ’s be examined to see which need to be (or can be) improved be Calculate total uncertainty using equation (4) For each r, report total uncertainty and bias and precision limits For r, Recommendations Recognize that uncertainty depends on entire testing process and that Recognize any changes in the process can significantly affect the uncertainty of the test results the Integrate uncertainty assessment methodology into all phases of the Integrate testing process (design, planning, calibration, execution and post-test analyses) analyses) Simplify analyses by using prior knowledge (e.g., data base), Simplify concentrate on dominant error sources and use end-to-end calibrations and/or bias and precision limit estimation and/or Document: 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 all (e.g., manufacturers specifications, comparisons against standards, experience, etc.) (e.g., detailed uncertainty assessment methodology and actual data uncertainty estimates References AIAA, 1999, “Assessment of Wind Tunnel Data Uncertainty,” AIAA AIAA, S-071A-1999. S-071A-1999. ASME, 1998, “Test Uncertainty,” ASME PTC 19.1-1998. ANSI/ASME, 1985, “Measurement Uncertainty: Part 1, Instrument ANSI/ASME, and Apparatus,” ANSI/ASME PTC 19.I-1985. and Coleman, H.W. and Steele, W.G., 1999, Experimentation and Coleman, Uncertainty Analysis for Engineers, 2nd Edition, John Wiley & Sons, Uncertainty Inc., New York, NY. Inc., Coleman, H.W. and Steele, W.G., 1995, “Engineering Application of Coleman, Experimental Uncertainty Analysis,” AIAA Journal, Vol. 33, No.10, pp. 1888 – 1896. pp. ISO, 1993, “Guide to the Expression of Uncertainty in Guide Measurement,", 1st edition, ISBN 92-67-10188-9. Measurement ITTC, 1999, Proceedings 22nd International Towing Tank Conference, ITTC, Proceedings “Resistance Committee Report,” Seoul Korea and Shanghai China. “Resistance ...
View Full Document

This note was uploaded on 12/08/2011 for the course MECHANICS 57:020 taught by Professor Fredrickstern during the Fall '10 term at University of Iowa.

Ask a homework question - tutors are online