Specialist Committee on Uncertainty Analysis Joel T. Park, Ph. D., Chairman, USA Ahmed Derradji Aouat, Ph. D., Secretary, Canada Baoshan Wu, China Shigeru Nishio, Ph. D., Japan 25 th ITTC, Fukuoka, Japan 15 September 2008, 15:00 – 16:15
Outline Procedures (5) Introduction – UA procedure – Important concepts (revised) – ISO GUM (1995) – Calibration Procedure Membership & – LDV Calibration meetings – PIV Uncertainty – Resistance Tests (revised) Summary Recommendations 2
Important Concepts Measurements traceable to a National Metrology Institute (NMI) Most uncertainty from data scatter in curve fit for conventional methods – Calibration data – Thrust coefficient versus advance ratio – Residual plot of data Most uncertainty in naval hydrodynamics in repeatability of tests – Uncontrolled element in test – Resistance tests 3
Committee Members Joel T. Park, Ph. D., Chair, NSWCCD, USA Ahmed Derradji Aouat, Ph. D. Sec., IOT, Canada Erwan Jacquin, BEC, France Baoshan Wu, CSSRC, China Shigeru Nishio, Ph. D., Kobe U., Japan Wu Derradji Jacquin Park Nisho 4
Meetings Bassin d’Essais des Carènes, Val-de-Reuil, France, March 30 - 31, 2006. China Ship Scientific Research Center, Wuxi, China, October 23 - 25, 2006. National Research Council Canada, Institute for Ocean Technology, St. John’s, Newfoundland, Canada, June 7 - 8, 2007. U. S. National Academy of Sciences, Naval Studies Board, Washington, DC, January 30 - February 1, 2008 Kobe University, Kobe, Japan, September 12, 2008 5
Objectives of General Procedure Simple and useable procedure tailored to ITTC hydrodynamics testing Specific guidance on application of uncertainty to experimental naval hydrodynamics Self-contained without need for consultation with reference documents ITTC procedure derived from ISO Guide to the Expression of Uncertainty in Measurement (1995), referred to as the ISO GUM. ITTC 7.5-02-01-01 (2008) 6
Uncertainty Analysis Fundamentals Type A evaluation of standard uncertainty n – Average or sample mean ∑ < >= x n x ( 1 / ) k = k 1 n – Sample variance ∑ = − − < > s 2 n x x 2 [ 1 /( 1 )] ( ) x k = k 1 – Standard deviation, s x – Type A standard uncertainty definition or standard = = u s s n deviation of the mean / < > A x x Conventional statistical definitions: Ross (2004) p. 203, AIAA (1999) p. 7, ASME (2005) p. 6 7
Uncertainty Analysis Fundamentals (cont.) Type B evaluation of standard uncertainty – Previous measurement data » Density, viscosity, and vapor pressure – Experience – Manufacturer’s specifications – Handbook – National Metrology Institute (NMI) traceable calibration: NIST in USA, NMIJ in Japan, NMi in The Netherlands, NPL in UK, PTB in Germany, NRC in Canada, KRISS in ROK 8
Uncertainty Analysis Fundamentals (cont.) = y f x x x K ( , , , ) Functional relationship N 1 2 n ∑ = ∂ ∂ u 2 y f x 2 u 2 x ( ) ( / ) ( ) Combined uncertainty c i i = i 1 – Law of propagation of uncertainty – Sensitivity coefficient ≡ ∂ ∂ c f x / i i N N – Independent ∑ ∑ = ≡ u 2 y c u x 2 u 2 y ( ) [ ( )] ( ) c i i i = = i i 1 1 N ∑ = – Correlated (weight set) u y c u x ( ) ( ) c i i = i 1 Expanded uncertainty = = = U ku y k ( ), 2 , % 95 – Coverage factor, k c % k = t % – Student t 9
Relative Uncertainty Propeller Re D = ρ μ Reynolds number VD / = ρ + + u Re Re u u V 2 2 2 [ ( ) / ] ( / ) ( / ) ρ c D D V + − μ u D 2 u 2 ( / ) ( / ) μ D J = V /( ND Advance ratio ) = + − + − u J 2 u V 2 u N 2 u D 2 [ / ] ( / ) ( / ) ( / ) J V N D = ρ Thrust coefficient K T T N 2 D 4 /( ) = + ∂ ρ ∂ − ρ + u K 2 u T 2 t 2 u 2 [ / ] ( / ) ( / ) ( / ) K T T T t − + − u N 2 u D 2 4 ( / ) 16 ( / ) N D 10
Numerical Evaluation of Sensitivity Coeff. Central difference for uncertainty = = + − u y Z f x x u x x K K ( ) [ 1 / 2 ][ ( , , ( ), , ) i i i i N 1 − f x x u x x K K ( , , ( ), , )] i i N 1 Numerical sensitivity coefficient c = Z u x / ( ) i i i Jitter program, Moffat (1982) 11
Reporting Uncertainty Specific for U Give full description of how measurand Y was defined State measurement result as Y = y ± U – Give units of y and U Include relative expanded uncertainty ≠ 0 U /| y|, |y| Give values of k and u c (y) Give approximate level of confidence for the interval y ± U and state how determined Provide details or reference published document 12
Specification of Numerical Results m s = (100.021 47 ± 0.000 79) g, where the number following the symbol ± is the numerical value of U = ku c (y), with U determined from u c = 0.35 mg and k = 2.26 based on the t-distribution for ν = 9 degrees of freedom, and defines an interval estimated to have a level of confidence of 95 percent. 13
Other Elements of Procedure Pre- & post-test Special cases uncertainty analysis – Mass measurements – Uncertainty analysis in – Instrument calibration data processing code » NMI traceable » End-to-end or through Outliers system calibration Inter-Laboratory – Repeat tests Comparisons – Youden plot 14
Calibration Theory Calibration value in Linear function post-processing code – Independent variable, for engineering units x , calibration value set = = − α β = + in engineering units x f ( y ) ( y ) / A By – Dependent variable, y , = − α β = β where A / , B / 1 instrument response, Uncertainty from usually in voltage units theory of Scheffe from A/D converter (1973) & Carroll, et = = α + β y f x x ( ) al. (1988) ITTC 7.5-01-03-01 (2008) 15
Instrument Calibration Procedure Calibration @ 10 approximately equal increments – System calibration with computer and software for test measurements – Calculate mean and standard deviation for each point on the order of 100 to 1000 points (5 to 50 s at 20 Hz sampling rate) – Document » cutoff frequency » sample rate » number of samples – NMI traceable reference standard with known uncertainty 16
Instrument Calibration Procedure (cont.) Regression analysis of calibration data – Slope, intercept, correlation coefficient, standard error of estimate, & outliers – Identify cause of outliers – Compute calibration uncertainty Compare results to previous calibration – Hypothesis test of slope & intercept Update computer slope & intercept 3-point calibration check @ computer with test software 17
Uncertainty Elements for Instrument Cal. Uncertainty in reference standard: NMI traceable Uncertainty in curve fit Type A uncertainty in data collection via computer for time series – Number of samples – Average – Standard deviation – Normally negligible: large values indication of problem 18
Vertical Gyroscope Calibration in Roll 10 6 Increasing Roll Angle Data 8 Linear Regression Decreasing Roll Angle Data 4 Increasing Roll Angle Standardized Residuals 6 Calibration Theory Decreasing Roll Angle 4 A/D Output (V) 2 2 0 0 -2 -2 -4 Intercept: +0.0052 V Slope: -0.09143 V/deg -6 Intercept: -0.056 deg SEE: 0.0362 V -4 Slope: -10.9370 deg/V r: 0.999968 -8 SEE: 0.396 deg -10 -6 -100-80 -60 -40 -20 0 20 40 60 80 100 -100-80 -60 -40 -20 0 20 40 60 80 100 Reference Angle (deg) Reference Angle (deg) 19
Other Elements of Calibration Hypothesis tests – Comparison with previous calibration results Outliers – Chauvenet’s criteria – Standardized residuals = − ρ / ρ F mg ( 1 ) Force calibration a w n ∑ – Mass uncertainty = u u m i Pulse count – optical encoders = i 1 – Propeller shaft speed – Carriage speed 20
Equations for LDA Calibration LDA optical calibration = θ λ = δ V f f [ 2 (sin / 2 ) / ] D D f Rotating disk calibration = π r ω V 2 Uncertainty equations = ∂ ∂ = πω = ∂ ∂ ω = π c V r c V r / 2 , / 2 1 2 = πω + π u 2 2 u 2 r 2 u 2 ( 2 ) ( 2 ) ω V r = + ω u V 2 u r 2 u 2 ( / ) ( / ) ( / ) ω V r ITTC 7.5-01-03-02 (2008) 21
LDV Velocity Uncertainty 0.030 r = 100 mm 0.025 07 May 2001 Total Uncertainty for r = 50 mm Rotational Velocity Uncertainty 0.020 Radial Uncertainty U 95 (m/s) LDA Noise r = 50 mm 0.015 0.010 Bean & Hall (NIST, 1999) 0.005 0.000 0 5 10 15 20 Axial LDA Velocity (m/s) 22
Comparison of Disk Results Element NSWC NIST NMIJ PTB 0.061 0.0074 0.0017 0.041 r f r 0.062 0.26 0.0018 0.035 δ f --- 0.16 --- --- Angle --- 0.011 0.017 0.0022 f D 0.026 0.36 0.20 0.010 Curve fit 0.043 --- --- --- Combined 0.10 0.48 0.20 0.055 Expanded Uncertainty in % @ 20 m/s 23
Particle Image Velocimetry (PIV) Example of uncertainty analysis (UA) procedure Present procedure developed from the guideline of Visualization Society of Japan recommendation (PIV-STD project, Nishio, et al., 1999) Flow speed Time interval = α + δ u Δ X/ Δ t u ( ) Magnification factor Image displacement ITTC 7.5-01-03-03 (2008) 24
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