ITTC Recommended Procedures

Transcription

1 Page 1 of 17 Table of Contents 1. PURPOSE OF PROCEDURE.... SCOPE GEERAL SYMBOLS AD DEFIITIOS Result of a measurement Measurement equaton UCERTAITY CLASSIFICATIO Standard uncertanty (u) Combned standard uncertanty (u c ) Expanded uncertanty (U) EVALUATIO OF STADARD UCERTAITY Evaluaton of uncertanty by Type A method Evaluaton of uncertanty by Type B method EVALUATIO OF COMBIED UCERTAITY EVALUATIO OF EXPADED UCERTAITY RELATIVE UCERTAITY Propeller equatons Resstance equaton SIGIFICAT DIGITS OUTLIERS Hypothess t-test Chauvenet s crteron Hgher-order central moments ITER-LABORATORY COMPARISOS SPECIAL CASES UA for mass measurements UA for nstrument calbraton Repeat tests PRE-TEST AD POST-TEST UCERTAITY AALYSIS REPORTIG UCERTAITY LIST OF SYMBOLS REFERECES SESITIVITY COEFFICIETS. 8 Updated / Edted by Approved Qualty Systems Group of the 7 th ITTC 7 th ITTC Date 03/ Date 09/

2 Page of PURPOSE OF PROCEDURE Ths procedure s a summary of the gudelnes for evaluaton and expresson of uncertanty n measurements for naval archtecture expermental measurements, offshore technology testng, and expermental hydrodynamcs. It s based on the comprehensve Internatonal Organzaton for Standardzaton (ISO) Gude to the Expresson of Uncertanty n Measurement, also called GUM, now JCGM (008a). Other relevant references nclude Taylor and Kuyatt (1994), AIAA S-071A-1999, AIAA G , and ASME PTC Kacker et al. (007) s a recent descrpton of the evoluton of the ISO GUM. The Internatonal Vocabulary of Basc and General Terms n Metrology or VIM (JCGM, 008b) gves the defntons of terms relevant to the feld of uncertanty n measurements. Four procedures for Resstance Testng (ITTC Procedure -01-, 008a), Calbraton Uncertanty (ITTC Procedure, 008b), Laser Doppler Velocmetry (ITTC Procedure , 008c), and Partcle Imagng Velocmetry (ITTC Procedure , 008d) are examples for drect applcaton of the gudelnes outlned n ths procedure.. SCOPE Ths procedure s concerned manly wth the expresson of uncertanty n the measurement of a well-defned physcal quantty (called the measurand 1 ) that can be characterzed by a unque value. If the measurement of nterest can be represented only as a dstrbuton of values or t depends on other parameters, such as tme, then the defnton of measurand should nclude a set of quanttes, whch descrbes that dstrbuton or that dependence. In addton to uncertanty n measurements, ths procedure s applcable to evaluaton and expresson of uncertantes assocated wth conceptual desgn, set up of actual experments, methods of measurements, nstruments calbratons, and Data Acquston Systems (DAS). A general gudelne s provded for the evaluaton and expresson of uncertanty n measurements, rather than a descrpton of the detals of a specfc experment. Therefore, development of procedures from ths general gudelne s necessary where the uncertanty n specfc experments s evaluated. Examples nclude the specfc procedures for LDV and PIV measurements ITTC (008b, c). Ths procedure does not dscuss how the uncertanty of a partcular measurement result may be used for dfferent purposes, such as drawng conclusons about the compatblty of the measurement result wth other smlar results, establsh the tolerance lmts n a gven manufacturng process, or decde f a certan course of acton may be safely taken. The use of uncertanty results to those ends s not wthn the scope of ths procedure. 3. GEERAL The word uncertanty means doubt, and therefore n ts broadest sense uncertanty of a measurement means a doubt about the valdty of the result of that measurement. The concept 1 The defnton of measurand s gven n the followng sectons

3 Page 3 of 17 of uncertanty as a quantfable attrbute s relatvely new n the hstory of measurement. However, concepts of error and error analyss have long been a part of measurements n scences, engneerng, and metrology. When all of the known or suspected components of an error have been evaluated, and the approprate correctons have been appled, an uncertanty stll remans about the truthfulness of the stated result that s a doubt about how well the result of the measurement represents the value of the quantty beng measured. The expresson true value s not used n ths procedure snce the true value of a measurement may never be known. 4. SYMBOLS AD DEFIITIOS The symbols used n ths procedure are the same as those used n Annex J of JCGM (008a). The basc and general defntons of metrology terms relevant to ths procedure are gven n the Internatonal Vocabulary for Metrology (JCGM, 008b). Among these are defntons for terms such as measurand, error, uncertanty, and other expressons used routnely when performng uncertanty analyss on a measurement. The dfference between the defnton of repeatablty of measurement results and that of reproducblty of measurement results s mportant. The condtons for repeatablty are: a) The same measurement procedure b) The same measurng nstrument used under the same test envronmental condtons c) The same locaton (laboratory or feld locaton) d) Repetton over a short perod of tme, roughly, tests are performed n the same day The term reproducblty of measurement results s used when one or more of the above four repeatablty condtons are not met. Examples nclude a dfferent observer, a dfferent test crew, a dfferent laboratory, dfferent envronment such as laboratory room temperature, dfferent test condtons, or dfferent day. Usually, reproducblty has a hgher uncertanty than repeatablty. 4.1 Result of a measurement The objectve of a measurement s to determne the value of the measurand that s the value of the partcular quantty to be measured. A measurement begns wth an approprate specfcaton of the measurand, the method of measurement, and the measurement procedure. The result of a measurement s only an approxmaton or an estmate of the value of the true quantty to be measured, the measurand. Thus, the result of a measurement s complete only when accompaned by a quanttatve statement of ts uncertanty. 4. Measurement equaton The quantty Y beng measured, defned as the measurand, s not measured drectly, but t s determned from other measured quanttes X 1, X, X.. Thus, the measurement equaton or data reducton equaton s Y f X,X,X, X ) (1) ( 1 3 The functon f ncludes along wth the quanttes X(, = 1,,, ) correctons (or correcton factors), as well as quanttes that take nto account other sources of varablty, such as dfferent observers, nstrument calbratons, dfferent laboratores, and tmes at whch observatons were made. Thus, the functon f should express not only the physcal law but also the measurement process, and n partcular, t should contan all quanttes that can contrbute to the uncertanty of the measurand Y.

4 Page 4 of 17 An estmate of the measurand (Y) s denoted by (y) and s obtaned from equaton (1) wth the estmates x 1, x,, x for the values of the quanttes X 1, X,, X.. Therefore, the output estmate (y) becomes the result of the measurements: y f x,x,x, 1 3 x n () As an example, typcal data reducton equatons for propulson performance from ITTC Procedure (0) are as follows: Reynolds number: Re D f (,V,D, ) VD / Advance rato: (3) J f ( V,n,D) V /( nd) (4) Thrust coeffcent: K T f ( T,ρ,D,n) T /( ρ D Torque coeffcent: 4 n 5 K Q f ( Q,ρ,D,n) Q / ( ρ D n ) ) (5) (6) where Q, T, ρ,, D, and n are torque (.m), thrust (), mass densty of water (kg/m 3 ), vscosty of water (kg/m-s), propeller dameter (m), and rotatonal rate (1/s), respectvely, and densty and vscosty are functons of the temperature, t. Therefore, an estmate for KQ s obtaned from estmates of the quanttes Q, ρ, D, and n, whle the estmates for KT are obtaned from quanttes T, ρ, D, and n. The estmates for each quantty Q, T, ρ, D can be obtaned from drect measurements or can be functon of other quanttes. The uncertanty n a measurement y, denoted by u(y), arses from the uncertantes u(x ) n the nput estmates x n equaton (). For example n equatons (5) and (6), the uncertantes n KQ and KT are due to uncertantes n the estmatons of Q, T, ρ, D, and n. 5. UCERTAITY CLASSIFICA- TIO JCGM (008a) classfes uncertantes nto three categores: Standard Uncertanty, Combned Uncertanty, and Expanded Uncertanty. 5.1 Standard uncertanty (u) Uncertanty, however evaluated, s to be represented by an estmated standard devaton. Ths s defned as standard uncertanty wth the symbol u and equal to the postve square root of the estmated varance. The standard uncertanty of the result of a measurement conssts of several components, whch as per le Comté Internatonal des Pods et Mesures (Gacomo, 1981) can be grouped nto two types. They are: Type A uncertantes and Type B uncertantes. Ether type depends on the method for estmaton of uncertanty. Type A: Uncertanty components obtaned usng a method based on statstcal analyss of a seres of observatons. Type B: Uncertanty component obtaned by other means (other than statstcal analyss). Pror experence and professonal judgements are part of type B uncertantes. The purpose of Type A and Type B classfcaton s a convenence for the dstncton between the two dfferent methods for uncertanty evaluaton. o dfference exsts n the nature of

5 Page 5 of 17 each component resultng from ether type of evaluaton. Both types of uncertantes are based on probablty dstrbutons and the uncertanty components resultng from both types are quantfed by standard devatons. 5. Combned standard uncertanty (u c ) Combned standard uncertanty of the result of a measurement s obtaned from the uncertantes of a number of other quanttes. The combned uncertanty s computed va the law of propagaton of uncertanty, whch wll be descrbed n detal later n ths procedure. The result s dfferent f the quanttes are correlated or uncorrelated (ndependent). 5.3 Expanded uncertanty (U) Mathematcally, expanded uncertanty s calculated as the combned uncertanty multpled by a coverage factor, k. The coverage factor, k, ncludes an nterval about the result of a measurement that may be expected to encompass a large fracton of the dstrbuton of values that could reasonably be attrbuted to the measurand. Thus, the numercal value for the coverage factor k should be chosen so that t would provde an nterval Y = y ± U correspondng to a partcular level of confdence. In expermental hydrodynamcs, k corresponds usually to 95% confdence. All ITTC results wll be reported wth an expanded uncertanty at the 95 % confdence level. The GUM ndcates that a smpler approach s often adequate n measurement stuatons, where the probablty dstrbuton of measurements s approxmately normal or Gaussan. If the number of degrees of freedom s sgnfcant ( > 30), the dstrbuton may be assumed to be Gaussan, and k wll be evaluated as. Ths assumpton produces an nterval (Y = y ± U) havng a level of confdence of approxmately 95%. For a small number of samples, the nverse Student t at the 95 % confdence level s recommended. The Student t at the 95 % confdence level s shown n Fgure 1, where the number of degrees of freedom s = n - 1. t Student t Gaussan: , Degrees of Freedom Fgure 1: Inverse Student t at 95 % confdence level. 6. EVALUATIO OF STADARD UCERTAITY 6.1 Evaluaton of uncertanty by Type A method The best avalable estmate of the expected value of a quantty q that vares randomly and for whch n observatons have been obtaned under the same condtons of repeatablty s the arthmetc mean or average: n q k k1 q ( 1/ n) (7)

6 Page 6 of 17 Each ndvdual observaton has a dfferent value from other observatons due to the random varatons of the nfluence quanttes, or random effects. For a DAS, the data, q, s collected as a tme seres of a unform sample nterval of n samples. The mean value of the tme seres s then computed from equaton (7). The expermental varance of the observatons, whch estmates the varance of the normal probablty dstrbuton of q s: s n [ 1/ ( n 1)] ( q k q) (8) k1 Ths estmate of varance and ts postve square root (s), termed the expermental standard devaton, characterze the varablty of the observed values of q, or more specfcally the dsperson of the values (qk) about ther mean. For a statonary tme seres, the uncertanty of the mean value s dependent on the correlaton n the sgnal. If there s no correlaton (whte nose) then the standard uncertanty can be estmated wth: u( q) s/ n (9) where n s the number of repeated observatons, for a sngle measurement n = 1. In a calbraton test set-up, the varaton n the measurement sgnal s manly determned by the nose level of the DAS, whch can be consdered as whte nose. For n = 100, the standard devaton of the mean from equaton (9) s then a factor 10 smaller than the sample standard devaton. Consequently for hgh qualty nstrumentaton n well controlled condtons, the Type A uncertanty s usually small n comparson to the Type B uncertanty. In many experments, however, measurement sgnals are oscllatory and thus contan correlaton. Brouwer et al. (013) gve two methods to determne the uncertanty of the mean value for oscllatory tme seres. The frst s a more elaborate method, based on the autocorrelaton functon of the sgnal. The second s a more easy and transparent method, based on splttng the sgnal nto a number of equallyszed segments. 6. Evaluaton of uncertanty by Type B method Type B evaluaton of standard uncertanty s usually based on judgment from all relevant nformaton avalable, whch may nclude: Prevous measurement data, Experence and knowledge of the behavour of relevant materals and/or nstruments, Manufacturer s specfcatons, Data provded n calbraton and other reports, whch must be traceable to atonal Metrology Insttutes (MI), and Uncertantes assgned to reference data taken from handbooks. Typcal examples n naval hydrodynamcs nclude values obtaned from equatons for water mass densty, vscosty, and vapour pressure from ITTC Procedure (011). The proper use of the pool of avalable data and nformaton for a Type B uncertanty requres an nsght based on experence and general knowledge. It s skll that can be learned wth practce. The Type B evaluaton of standard uncertanty may be as relable as a Type A uncertanty, especally n a measurement stuaton where a Type A evaluaton s based on a comparatvely small number of statstcally ndependent observatons. In general, all fnal results by the Type B method should be traceable

7 Page 7 of 17 to an MI. Other methods may be appled n the desgn stages of a test or experment. 7. EVALUATIO OF COMBIED UCERTAITY Combned uncertanty s evaluated by the law of propagaton of uncertanty. The general equaton for combned standard uncertanty of a measurement result y, desgnated by u c (y), s from the JCGM (008a): c 1 u ( y) ( f / x ) u ( x ) 1 ( f / x )( f / x ) u( x, x ) 1 j1 j j (10a) where s the total number of nput quanttes from observatons. Equaton (10a) s based on a frst-order Taylor seres approxmaton of the measurement equaton and ts estmate. The partal dervatves of f wth respect to x and xj are called senstvty coeffcents c and cj: c f / x, c j f / x j (10b) The general equaton accounts for standard uncertantes n both uncorrelated (ndependent) and correlated measurement quanttes (x and xj). If the nput quanttes are correlated or dependent on each other, ther degree of correlaton s represented by the correlaton coeffcent r(x, xj): r x,x ) u( x,x )/[ u( x ) u( x )] (10c) ( j j j The values for the correlaton coeffcent are symmetrc r(x, xj) = r(xj, x), ther values range s: -1 r (x, xj) +1. When equaton (10a) s re-wrtten n terms of senstvty and correlaton coeffcents, t becomes: c 1 u ( y) c u ( x ) 1 c c u( x ) u( x ) r( x, x ) j j j 1 j1 (11) When the nput quanttes x and xj are uncorrelated (ndependent), then r(x, xj) = 0, and the total combned standard uncertanty s the square root of the sum of the squares of standard uncertantes: c 1 u ( y) c u ( x ) (1) Essentally, equaton (1) s the estmated standard devaton of the result for perfectly uncorrelated nput quanttes. Equaton (1) s the most commonly appled verson of the law of propagaton of uncertanty. If the nput quanttes x and xj are fully correlated, then r(x, xj) = 1 and the total combned standard uncertanty s smply the lnear sum of the standard uncertantes. u ( y) c u( x ) (13) c 1 The most common applcaton of ths equaton n expermental hydrodynamcs s the calbraton of force wth mass. Snce the masses are calbrated aganst the same reference standard, the uncertantes of the masses are correlated. Therefore the combned uncertanty s the sum of the uncertantes of the ndvdual masses. 8. EVALUATIO OF EXPADED UCERTAITY The combned standard uncertanty uc(y) s unversally appled n the expresson of the un-

8 Page 8 of 17 certanty of a measurement result. Expanded uncertanty, U, from the combned uncertanty uc(y) multpled by a coverage factor, k, s: U k u c ( y) (14) The result of a measurement should be expressed as Y = y ± U, or the best estmate of the value attrbutable to the measurand Y s between (y - U and y + U). The nterval y ± U may be expected to encompass a large fracton of the dstrbuton of values that could reasonably be attrbuted to Y. From practcal vewpont, n expermental hydrodynamcs and flow measurements, an nterval wth a level of confdence of 95% (1 chance n 0) s justfable. If a normal probablty densty functon (pdf) for the measurement result s assumed, then the value of for the coverage factor s appled for the 95% confdence level for an acceptable number of repeated observatons. allows the mean value of Y to be approxmated by a normal dstrbuton from IS0 GUM (1995). 9. SESITIVITY COEFFICIETS A numercal method (or computer routne) based on the functonal central dfferencng scheme was proposed by Moffat (198) for calculaton of the senstvty coeffcents c n equaton (10). The method also s ncluded n the JCGM (008a). If the uncertanty u(y) s represented by the functonal dfference, Z: u ( y ) c u( x ) Z ( 1/ )[ f x, x, x u( x ), x 1 1 f x, x, x u( x ), x ] Then, the senstvty coeffcents, c, are: c Z /u x ) ( (15a) (15b) Theoretcally for specfcaton of the value for the coverage factor for a specfc level of confdence, detaled knowledge of the probablty dstrbuton functon of the measurement result and ts combned standard uncertanty are needed. In most towng tank and water tank experments, the t-dstrbuton may be assumed for a small number of observatons, and the value for coverage factor can be obtaned drectly from the plot n Fgure 1. If the number of degrees of freedom s hgh enough ( > 30), the Student t becomes very close to Gaussan. For =30, k = t95 =.04, n comparson to the Gaussan value of If the probablty dstrbuton functons of the nput quanttes X(, = 1,,, ) upon whch the measurand Y depends are normal, then the resultng dstrbuton of Y wll also be normal. If the dstrbuton of the nput quanttes X, s not normal dstrbutons, the Central Lmt Theorem

9 Page 9 of 17 Start Yes Read x, x Subroutne f = f(x 1, x,, x ) = 1 g = 0 Subroutne c = f/ x e = ( x f/ x ) g = g + e = + 1 < o f = (g) 1/ Prnt f, f Stop Subroutne c = f/ x Subroutne F +u = f(x 1,...,x +u,, x ) F -u = f(x 1,...,x -u,, x ) f/x = (F +u - F -u )/(u ) Return Fgure : Flow chart for numercal determnaton of senstvty coeffcents (Moffat, 198)

10 Page 10 of 17 A flow chart for the central dfferencng method s gven n Fgure. The detals of the central dfferencng method were gven by Moffat (198). The method n the program, whose flow chart s n Fgure, may be descrbed as Central Dfferencng for Evaluaton of Senstvty Coeffcents or Jtter Program per Moffat, (198). It elmnates the need for development lengthy tables of partal dervatves for parameters n data reducton equatons. 10. RELATIVE UCERTAITY In hydrodynamcs, the data reducton equatons are typcally a product of terms of the form Y cx X X (16) p1 1 p p Then, the relatve combned uncertanty from equaton (1) s [ u c ( y) / y] [ pu( x ) / x ] 1 where y 0 and x 0. (17) The followng are examples for relatve uncertantes of well defned equatons n expermental hydrodynamcs Propeller equatons The relatve uncertantes n the propeller equatons are obtaned from equaton (17) and equatons (3) to (6): Reynolds number: [ u c ( Re D )/ Re D Advance rato: ] ( u ρ ( u / ) D /D) ( u V /V ) ( u μ / ) (18) [ uc ( J) /J ] ( uv /V ) ( un /n) ( ud/d) (19) Thrust coeffcent: [ u c ( K Torque coeffcent: [ u c ( K 10. Resstance equaton (0) (1) The data reducton equaton for total resstance s from ITTC Procedure -01- (008a) C T T ) / K T ) / K R / ( SV ) () T ] ( u / T) ( / t) ( u ] The relatve uncertanty n CT s then: [ u c ( C Q T ) / C Q 11. SIGIFICAT DIGITS T ( u Q [( / t)( u T ] (u V [( / t)( u t (3) For a measurement result, the number of dgts after the decmal pont should be the same as those after the decmal pont reported for ts assocated combned uncertanty uc. In general, the uncertanty should be reported to sgnfcant dgts. For example: Consder a mass (m = ± ) kg, where the number of dgts after the symbol ± s the numercal value of expanded uncertanty (U). The expanded uncertanty s computed from value of the combned uncertanty, uc = kg and a coverage factor, k =.6 where k s / Q) t 4( u / ) / )] / V ) t n / n) 16( u 4( u / )] n 5( u ( u RT ( u D / n) S D / D) / R / S) / D) T )

11 Page 11 of 17 based on the t-dstrbuton for = n 1 = 9 degrees of freedom and an nterval estmated wth a level of confdence of 95 percent. The number of dgts after the decmal pont s 5, n both the estmated value for m and ts assocated combned uncertanty (uc = kg). 1. OUTLIERS Sometmes data occurs outsde the expected range of values and should be excluded from the calculaton of the mean value and estmated uncertanty. Such data are referred to as outlers. If an outler s detected, the specfc cause should be dentfed before t s excluded. Several methods may be appled n the determnaton of outlers. Addtonal nformaton on outlers as appled to calbraton s contaned n the procedure on Calbraton Uncertanty n ITTC (008b). 1.1 Hypothess t-test The conventonal method for outlers s the t- test from hypothess testng. The detals of the methodology may be found n a standard statstcs text such as Ross (004). Then the T statstc s defned as: T ( q q ) / s (4) Accept as vald f T t95, n1 Reject as outler f T t95, n1 That s, q s an outler, where t95,n-1 s the nverse Student t for a -taled probablty densty functon (pdf) at the 95 % confdence level and the cumulatve probablty s p > In practcal terms, any T that exceeds may be consdered as an outler at the 95 % confdence level. 1. Chauvenet s crteron A less strngent test s gven by Chauvenet s crteron from Coleman and Steele (1999). By ths crteron a data pont s rejected as an outler f the nverse Gaussan, Z, for a -taled pdf s Z ( q q) / s (6) Reject f Z z1 1/(4 n) As an example for n = 10, then p > and z = A plot of Chauvenet s crteron s presented n Fgure 3. z(p > 1-1/(4n)) n, umber of Data Ponts Fgure 3: Chauvenet s rejecton crteron 1.3 Hgher-order central moments Another useful concept for outlers s the hgher-order central moments, whch are defned from Papouls (1965) as m p / s p [1 / ( ns p )] n 1 ( q q) p (7)

12 Page 1 of 17 where they are non-dmensonalzed wth the standard devaton. The central moments may also be appled as a measure of how close to Gaussan a process s. For a Gaussan pdf, the hgher-order central moments are as follows: For p odd: j1 m / s 0 (8) j1 For p even m s j (9) j j / ( 1) The commonly appled hgher-order central moments are the thrd-order, defned as skewness factor (S) and the fourth-order defned as flatness factor (F). Thus for a Gaussan pdf, S = 0 and F = 3. The fourth-order moment has also been defned as kurtoss, K, where K = F 3 = 0. For sgnfcant devatons from these values, ether the pdf s non-gaussan or contans outlers. Any tme seres wth on the order of F > 5 to 10 should be nvestgated for outlers. Wth a DAS, the mean, standard devaton, skewness factor, and flatness factor may be computed routnely n almost real tme for a tme seres. 13. ITER-LABORATORY COMPARI- SOS As a better measure of a laboratory s uncertanty estmates, nter-laboratory comparsons are routnely performed. The method adopted by the MIs s the Youden plot (1959). The method requres the measurement of smlar test artcles, A and B, by several laboratores and then plottng the results of A versus B. For a naval hydrodynamcs test, the test models (artcles) may be propellers n a propeller performance test or shp hulls n a resstance towng test. A schematc of a Youden plot for flowmeters for the results from 5 laboratores s shown n Fgure 4 from Mattngly (001). In the method, vertcal and horzontal dashed lnes are drawn through mean values of all laboratores. Then, a sold lne s drawn at 45 through the crossng pont of the dashed lnes. The data pattern s then as follows: E and SW quadrants, systematc hgh and low values W and SE quadrants random values both hgh and low Usually ellptc n shape wth random values along mnor axs and systematc errors along major axs Ideally, the pattern should be crcular. The varance of n laboratores normal to the 45 axs s gven by s n r 1/ ( n 1)] 1 [ (30a) and parallel to the axs s n s 1/ ( n 1)] P 1 [ (30b) where sr and ss may be nterpreted as the random and systematc devatons of the data, respectvely, and and P are the respectve normal and parallel components of the data projected onto the lne wth the slope of +1. The rato of these two quanttes s then the crcularty of the data: c ss / s r (30c)

13 Page 13 of 17 The expanded uncertanty for the calbrated mass s requred by OIML (004) to be: U m m/3 (31c) where m s the rated tolerance. For many naval hydrodynamcs laboratores, the masses have a tolerance of 0.01 %. 14. UA for nstrument calbraton Fgure 4: Youden plot for flow meter test 14. SPECIAL CASES 14.1 UA for mass measurements Durng calbraton of force nstruments, such as load cells and dynamometers, the force s changed by addton or removal of weghts from the calbraton fxture. The total mass s the sum of the ndvdual masses: m m (31a) 1 The weght set s usually calbrated as a set at the same tme aganst the same reference standard. OIML (004) and ASTM E740- performance specfcatons recommend that the uncertanty n weghts s perfectly correlated. The standard uncertanty n the total mass s from equaton (13): Electronc nstruments must be calbrated by a reference standard that s traceable to an MI. Such calbraton s necessary for converson of voltage unts to physcal unts. Most nstruments n expermental hydrodynamcs are hghly lnear. Consequently, the calbraton ncludes a lnear ft of the data. Two types of nstrument calbratons exst. These are end-to-end calbraton or bench ndvdual nstrument calbratons. For example, end to end means that both the measurement sensor (such as load cell) and Data Acquston Card (DAC) are calbrated together as one unt. Otherwse, the load sensor and DAC are calbrated separately. Usually the uncertanty n nstrument calbraton s assocated wth the data scatter n the regresson ft. The MI traceable reference standard for the calbraton should have an uncertanty that s small n comparson to the uncertanty from the curve ft. A separate Calbraton Uncertanty procedure (ITTC, 008b) descrbes n detal the uncertantes assocated wth both lnear and non-lnear curve fttng Repeat tests u m u 1 (31b) In some cases, the methodology outlned n ths procedure does not adequately defne the uncertanty of a test. Frequently, tests n naval hydrodynamcs contan an uncontrolled element

14 Page 14 of 17 that s not ncluded n the uncertanty estmate. Consequently, repeat tests, at least 10, are suggested for a representatve condton as a better estmate of the uncertanty. Ten tests should provde a reasonable estmate of the standard devaton. The standard devaton s computed from equaton (8). Snce ths wll provde an estmate for tests whch are performed only once, equaton (9) should not be appled. Forgach (0) provdes such an example. In hs report, the expanded uncertanty estmate for carrage speed based upon rotaton of a metal wheel was ± m/s. However, the expanded uncertanty from 3 repeat runs ( standard devatons) was ± m/s or 3 tmes the uncertanty estmate from the wheel speed. The speed for ths case was.036 ± m/s (± 0.08 %) for the expanded uncertanty ncludng both the uncertanty n repeat runs and the wheel speed. In ths example, the uncontrolled varable was an estmate of the uncertanty contrbuton from the carrage speed controller, whch ncluded a manual settng by a carrage operator. 15. PRE-TEST AD POST-TEST U- CERTAITY AALYSIS Before the frst data pont s taken n a test, the data reducton equatons should be known. A data reducton program for the DAS should nclude the measurement equatons, data for converson of the dgtally acqured data to physcal unts from calbratons (traceable to an MI), and fnally uncertanty analyss should be ncluded n the data processng codes. The codes should nclude the detals of the uncertanty analyss: Elemental uncertantes, u(y) = cu(x) and ther relatve mportance to the combned uncertanty, uc(y) Combned and expanded uncertanty, uc and U Calbraton factors for converson from dgtal unts to physcal unts Contrbutons to the uncertanty by Type A and Type B methods. A pre-test uncertanty analyss should be performed durng the plannng and desgnng phases of the test wth the same computer code appled durng the test. The pre-test uncertanty wll only nclude Type B uncertantes. In ths phase, all elements of the Type B uncertanty should be appled. In partcular, manufacturer s specfcatons may be ncluded for an assessment of adequacy of a partcular nstrument for the test before the devce s purchased. Selecton of an nstrument may nvolve economc tradeoffs between cost and performance. For the post-test uncertanty analyss after the data are acqured, the post-processng code should provde suffcent data on uncertanty analyss for the fnal report of the test. In ths case, data wll nclude results from both the Type A and Type B methods. All of the elemental uncertantes should be based upon measurements that are traceable to an MI. That s, all measurements should be accompaned by documented uncertantes. These should contan no guesses or manufacturer s specfcatons unless the manufacturer supples a calbraton certfcate that s MI traceable. Fnally, the contrbutons of the elemental uncertantes u(y) should be compared to the combned uncertanty, uc(y). Such comparson wll dentfy the mportant contrbutors to the combned uncertanty. These results should be compared to the pre-test uncertanty analyss. In ths manner, the expected performance should be verfed. Are the results of the pre-test and post-test uncertanty analyss consstent? Fnally, the results should be revewed for potental mprovements or reducton n the uncertanty for future tests.

15 Page 15 of REPORTIG UCERTAITY The man drectve for reportng uncertantes s that all nformaton necessary for a reevaluaton of the measurement should be avalable to others when and f needed. When uncertanty of a result s evaluated on the bass of publshed documents, such as the case of nstrument calbraton results that are reported on a manufacturer certfcate, these publcatons should be referenced, and nsure that they are consstent wth the measurement procedure actually used. If experments are performed wth nstruments that are subjected to perodc calbraton and/or legal nspectons, the nstruments should conform to the specfcatons that apply. In practce, the amount of nformaton necessary to document uncertantes n a measurement result depends on ts ntended use. The followng s a lst for the base gudelne n reportng uncertanty. uncertantes, combned standard uncertanty, and expanded uncertanty. 17. LIST OF SYMBOLS c Senstvty coeffcent, c = f/ x CT Total resstance coeffcent, equaton () D f Dameter of propeller m Functon of measurement varables or data reducton equaton J Advance rato, equaton (4) 1 k Coverage factor, usually 1 KQ Torque coeffcent, equaton (6) 1 KT Thrust coeffcent, equaton (5) 1 Descrbe clearly the method used to obtan the measurement result and ts uncertanty. Lst all uncertanty components and document fully how they were evaluated: these are standard uncertanty, combned uncertanty, and expanded uncertanty. Expanded uncertanty should be reported at the 95 % level and the bass of the coverage factor, k, documented. The fnal measured values should be documented as y ± U (U/y n percent, y 0 ). Present the data and uncertanty analyss n such a way that each of ts mportant steps can be readly followed. The calculaton of the reported result can be ndependently repeated f necessary. Gve all correctons and constants used n the analyss and ther sources. JCGM (008a) gves specfc gudance on how to report the numercal values of a measurement result (y) and ts assocated standard n umber of samples or observatons 1 n Also, propeller rotatonal frequency Hz umber of nput quantttes 1 p Probablty 1 Q Torque m r Correlaton coeffcent, eq. (10c) 1 RT Total resstance Re Reynolds number, equaton (3) 1 s Standard devaton, equaton (8) S Surface area m t Water temperature C tp, Inverse Student t 1

16 Page 16 of 17 T t-value for hypothess test 1 T Also, thrust u uc U Standard uncertanty, u s / Combned standard uncertanty Expanded uncertanty, U = kuc V Velocty m/s Absolute vscosty kg/(m s) Degrees of freedom 1 Also, knematc vscosty, =/ m /s Water densty kg/m REFERECES AIAA S-071A-1999, Assessment of Expermental Uncertanty Wth Applcaton to Wnd Tunnel Testng, Amercan Insttute of Aeronautcs and Astronautcs, Reston, Vrgna, USA. AIAA G , Assessng Expermental Uncertanty Supplement to AIAA S- 071A-1999, Amercan Insttute of Aeronautcs and Astronautcs, Reston, Vrgna, USA. ASME PTC , Test Uncertanty, Amercan Socety of Mechancal Engneers, ew York. ASTM E617-97, 1997, Standard Specfcaton for Laboratory Weghts and Precson Mass Standards, Amercan Socety for Testng and Materals, West Conshohocken, Pennsylvana, USA. n Coleman, H. W. and Steele, Jr., W. G., 1999, Expermentaton and Uncertanty Analyss for Engneers, Second Edton, John Wley, and Sons, Inc., ew York. Brouwer, J., Tukker, J., van Rjsbergen, M., 013, Uncertanty Analyss of Fnte Length Measurement Sgnals, 3 rd Internatonal Conference on Advanced Model Measurement Technologes for the Martme Industry, Gdansk, Poland. Forgach, K. M., 0, Measurement Uncertanty Analyss of Shp Model Resstance and Self Propulson Tests, Techncal Report SWCCD-50-TR-0/064, aval Surface Warfare Center Carderock Dvson, West Bethesda, Maryland, USA. Gacomo, P., 1981, ews from the BIPM, Metrologa, Vol. 17, o., pp ITTC, 011, Fresh Water and Seawater Propertes, ITTC Procedure ,. ITTC, 008a, Gudelnes for Uncertanty Analyss n Resstance Towng Tank Tests, ITTC Procedure -01-,. ITTC, 008b, Uncertanty Analyss: Instrument Calbraton, ITTC Procedure ITTC 008c, Uncertanty Analyss: Laser Doppler Velocmetry Calbraton, ITTC Procedure ITTC 008d, Uncertanty Analyss: Partcle Imagng Velocmetry, ITTC Procedure ITTC 0, Propulson Test, ITTC Procedure

17 Page 17 of 17 JCGM, 008a, Evaluaton of measurement data Gude to the expresson of uncertanty n measurement, JCGM 100:008 GUM 1995 wth mnor correctons, Jont Commttee for Gudes n Metrology, Bureau Internatonal des Pods Mesures (BIPM), Sèvres, France. JCGM, 008b, Internatonal vocabulary of metrology Basc and general concepts and assocated terms (VIM) JCGM 00:008 VIM, Jont Commttee for Gudes n Metrology, Bureau Internatonal des Pods Mesures (BIPM), Sèvres, France. Kacker, R., Sommer, K-D., and Kessel, R., 007. Evoluton of modern approaches to express uncertanty n measurement, Metrologa, Vol. 44, pp Mattngly, G. E., 001, Flow Measurement Profcency Testng for Key Comparsons of Flow Standards among atonal Measurement Insttutes and for Establshng Traceablty to atonal Flow Standards, Proceedngs of the ISA 001 Conference, Houston, Texas, USA. Moffat, R. J., 198, Contrbutons to the Theory of Sngle-Sample Uncertanty Analyss, Journal of Fluds Engneerng, Vol. 104, o., pp OIML R 111-1, 004, Weghts of Classes E1, E, F1, F, M1, M1-, M, M-3, and M3, Part 1: Metrologcal and techncal requrements, Organsaton Internatonale de Métrologe Légale, Pars, France. Papouls, A., 1965, Probablty, Random Varables, and Stochastc Processes, McGraw- Hll Book Company, ew York. Ross, S. M., 004, Introducton to Probablty and Statstcs for Engneers and Scentsts, Thrd Edton, Elsever Academc Press, Amsterdam. Taylor, B.. and Kuyatt, C., 1994, Gudelnes for Evaluatng and Expressng the Uncertanty of IST Measurement Results, IST Techncal ote 197, atonal Insttute of Standards and Scence, Gathersburg, Maryland, USA. Youden, W. J., 1959, Graphcal Dagnoss of Interlaboratory Test Results, Industral Qualty Control, Vol. XV, o. 11, pp to