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Measurement Uncertainty and Measurement Deviation

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Measurement Uncertainty and Measurement Deviation Measurement uncertainty and measurement deviation Practice Dipl.-Ing. Susanne Nisch © WZL/Fraunhofer IPT Practice Mesurement uncertainty and mesurement deviation Content Definitions and Terms Influence Faktors on measurement uncertainty Temperature Messurement object Messurement device Method for estimation of measuremnet uncertainty Examples © WZL/Fraunhofer IPT Seite 2 Definition and Terms Measurement uncertainty is a parameter associated with the result of a measurement that characterises the dispersion of values which could reasonably be attributed to the measurand. Mesurement error is the deviation from the true value of a value gained from measurements and assigned to the measurand, or the measurement results minus the true value of the measurand. The limit of error is the maximum amount of measurement deviation of a measuring device. © WZL/Fraunhofer IPT Seite 3 Influence Factors on Measurement deviation Value Proceeding Example Value of the measurand Defining the measurand (non-realisable (specifying the quantity to Determining the true value) be measured, depending on thickness of a given the required measurement sheet of material Values of the uncertainty) at 20°C measurand due to incomplete definition Measuring the Measuring the realised thickness of the quantity sheet at 25 C° with a micrometer and measuring the Unadjusted observed applied pressure values Unadjusted arithmetic mean of the observed values Correcting all known Correcting the systematic influences influences of temperature and pressure Measurement result (best estimate) Residual error Complete measurement result © WZL/Fraunhofer IPT Seite 4 Cause and effect diagram of production metrology Human Measured oject Method Surface structure Measurement e.g. roughness strategy Form deviations Measur. procedure Commitment to perform Mass Measur. principle Efficiency Dimensions Evaluation Complete measurement result Contamination Internal evaluation of the device Humidity Resolution Vibrations Dynamic behaviour Temperature Strucutre Statistic behaviour Environment Measuring device © WZL/Fraunhofer IPT Seite 5 Types of measurement errors Measurement errors systematic coincidental measurement errors measurement errors known systematic unknown systematic errors errors correction residual errors measurement Measurement uncertainty result © WZL/Fraunhofer IPT Seite 6 Influence Factor Temperature „Temperature measurement is not everything in length metrology, but it cannot be done without“ Three types of temperature influences: • Deviation of the temperature level from the reference temperature • Temporal temperature fluctuations • Spatial temperature fluctuations Types of heat transfer: • Thermal conduction • Convection • Heat radiation Linear expansion characteristic: DL=L*α*Dt © WZL/Fraunhofer IPT Seite 7 Coefficient of thermal expansion for solid bodies Aluminium alloy α = 23..24 [10-6/K] Steel a = 10..12 [10-6/K] Glass α = 8..10 [10-6/K] Zerodur a = 0..0.05 [10-6/K] Example: A 100 mm in length Steelruler will stretch by more than 1µm with a temperature difference of 1K! The effect of change in length are negligible if: • α ~ 0 • αWerk = αM und tWerk = tM • tWerk = tM = 20 °C © WZL/Fraunhofer IPT Seite 8 Coefficient of thermal expansion for solid bodies When measuring, the change in length of scale must beconsidered as well as the change in length of the measured objekt: ∆L=L*(αWork*∆tWork-αM*tM) The calculable influence of temperature contains an uncertainty which results from the uncertainty of the temperature measurement and the uncertainty of the coefficients of expansion. This is computed by partially deriving and squared addition of the individual parts, where ua represents the uncertainty of the coefficient of expansion and u∆t the uncertainty of temperature measurement: u =L * (u αWerk * ∆t Werk )² + (u ∆tWerk * α Werk )² + (u αM * ∆t M )² + (u ∆tM * αM )² © WZL/Fraunhofer IPT Seite 9 Length measurement error conditional upon temperature Deviation [µm/m] Boundaries of the 25 area of uncertainty with correction 20 15 thermal correction 310 µm 3µm/m 5 Temperature 0 18 19 20 21 22 23 24 25 tWork =t M [C°] -5 -10 -15 -20 Boundaries of the -25 area of uncertainty without correction © WZL/Fraunhofer IPT Seite 10 Influence factor: measured object Measurement deviation due to features of the measured object: • Surface Structure • Reflection (optical measurement technology) • Roughness, Waviness (taktile) • Hardness 40 40 a a [µm] 10 1 Measurement error through F = 10 N deformation 0,1 F = 5 N 0,01 F = 1 N 2 4 6 r [mm] Example: ball-plane (steel) Flattening (Hertz Pressure) through measurement force © WZL/Fraunhofer IPT Seite 11 Influence factor: slim measured object e.g. deflection of slim measured objects F = m g For the deflection of uniform measured objects (pipes, rulers, plates) under dead weight: a) Support at the ends: maximum deflection d max = 100% b) Support at each 0,22 l: minimum deflection l d max = 2% 0,22 l 0,22 l © WZL/Fraunhofer IPT Seite 12 Influence Factor: measurement device Measurement errors due to inaccurate guides play an important part in measurement devices with built-in measure embodiment. The clearance, which is technically required in guides for measuring pins, touch probes or eyepieces, causes tilting. The influence of these on the measurement result are large (first order) or small (second order), depending on how the measure embodiment and the measured object have been positioned. Abbe Principle: In order to avoid errors of the first order, the scale of the measuring device must bepositioned such that the distance to be measured forms a straight-line continuation of the scale. (Ernst Abbe, 1893) © WZL/Fraunhofer IPT Seite 13 Influence factor: measurement device – Caliper Error of first order by Abbe 0 1 2 3 4 5 6 7 8 9 10 111213141516 0 L L' ∆L Meßabweichung Measurement Error: p ∆L ∆=L L‘ = –L’ L- L~p ~ϕ p ϕ ϕ L': abgelesener Meßwert LL’ : wahre : measured Länge value MeasuredMeßobjekt object parallel aligned versetzt to measure embodiment pL : Parallelversatz : true value zur Maßverkörperung => => complyingVerletzung to desthe Abbe'schenAbbe principle Prinzips P : parallel offset © WZL/Fraunhofer IPT Seite 14 Influence factor: measurement device – Capiler Breach of Law of Abbe With a parallel offset of the measuring distance and the reference distance, a small tilt already causes measurement errors which are no longer negligible! Measurement deviation: ∆L = L-L‘ = p*tan(φ) for φ << 1 with φ described in angle minutes: ∆L=p* φ L Example: L' •Tilt φ = 2‘ • parallel offset p = 30 mm ∆L -> ∆L = 17 µm p ϕ © WZL/Fraunhofer IPT Seite 15 Influence factor: measurement device – micrometer Abidance of law of Abbe Measurement object, measurement surface and lead screw form a straight line. The measurement deviation for φ<<1 can be described as: 1 ∆L = A −1 = A( 1+ tan²(ϕ) −1) ≈ A( 1+ ϕ² −1) cos(ϕ) A j L L’ Measured object aligned ∆L A to measure embodiment => MeasurementError Error: of measurement 2 j complying to the Abbe principle ∆L = L‘ – L Dj~AL /2= L' - L ~ A 2 A L‘ : L':measured read-outmeasurement value value L : Ltrue : true value length © WZL/Fraunhofer IPT Seite 16 Methods for the estimation of measurement uncertainty The basis of every procedure for estimating measurement uncertainty ist the “Guide to the Expression of Uncertainty in Measurement“ (GUM) The evaluation of a measurement can be carried out in four steps: i. Setting up a model which mathematically describes the relationships of the measurands (y1,y2,...yn) to all other quantities involved (x1,x2,...xn) y=f(x1,x2,...xn) i. Preparation of the given measurement values and other available data ii. Calculation of the measurement result and measurment uncertainty of the measurand from the prepared data iii. Specification of a complete measurement result and determination of the extended uncertainty. © WZL/Fraunhofer IPT Seite 17 Procedures for estimating measurement uncertainty ii. Determining the standard uncertainty Method Form of distribution Calculation Normal distribution Standard uncertainty of mean value s A s s u = s: standard deviation n n: number of observed values x- Normal distribution Assumption: a The estimated value lies within the s s u = boundaries a+ and a- with a 4 confidence level of 95 %. x- Uniform distribution Assumption: a B u = The estimated value lies within the boundaries a and a with a 3 + - a - a + confidence level of 100 %. Tringular distribution Assumption: a u = The estimated value lies within the boundaries a and a with a 6 + - confidence level of 100 %. a - a + © WZL/Fraunhofer IPT Seite 18 Procedures for estimating measurement uncertainty iv. Specification of the complete measurement result The measurement uncertainty of the measurand, the combined standart uncertainty uc, is determined through squared addition of the individual uncertainty components. As long as all input variables are independent from each other: df df df uc = ux1 + ux2 + ... + uxn dx1 dx2 dxn or simply: uc = u² x1 + u² x2 +...+ u² xn Multiplication with the extension factor k, shows the extended measurement uncertainty: U=k*uc k = 2 equals a range of confidence of 95% © WZL/Fraunhofer IPT Seite 19 Evaluation of measurement uncertainty e.g. Dial Gauge block Example: Dial Gauge for the determination of the hight of a gauge Dial gauge 10 0 10 20 20 30 30 40 40 Tripod Gauge block Seite 20 Plane table © WZL/Fraunhofer IPT Evaluation of measurement uncertainty e.g. Dial Gauge Main influence factors Surroundings: • The measurement is carried
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