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**An Introduction to Uncertainty Analysis**

MER331 – Lab 1 An Introduction to Uncertainty Analysis

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**Error Sources calibration error data acquisition error**

The measurement process consists of three dis-tinct steps: calibration, data acquisition, and data reduction calibration error data acquisition error data reduction error

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Uncertainty Analysis Error - difference between true value and measured value. Two general categories of error (exclude gross blunders) Fixed error – remove by calibration Random error – quantify by uncertainty analysis The term uncertainty is used to refer to “a possible value that an error may have”

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**Stages of Uncertainty Analysis**

Design Stage Uncertainty Analysis Uncertainty analysis can be used to assist in the selection of equipment and procedures based on their relative performance and cost. Advanced-Stage and Single Measurement Uncertainty Analysis Consider procedural and test control errors that affect the measurement

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**A = H*W (H is the longer side)**

Uncertainty Analysis Example: Each group has a piece of paper and 2 measur- ement tools (labeled 1, 2). You must use one measurement tool to measure the height (H) and the other to measure width (W). Does it matter which tool you use to measure H? A = H*W (H is the longer side) How do we estimate the uncertainty in our measurement of A? Step 1: Figure out the uncertainty in H and W. Step 2: Figure out how the uncertainty in H and W affects the uncertainty in A

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**Step 1 - Estimating Uncertainty Interval**

If you have a statistically significant sample use: ±2s for a 95% confidence interval. The general practice in engineering is 95% confidence of 20 to 1 odds.

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**Step 1 - Estimating Uncertainty Interval**

What do you do if you DON’T have a statistically significant sample? Use ± ½ smallest scale division as an estimate. ( this is called a 0th order analysis)

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**Step 1 - Estimating Uncertainty Interval**

What do we use for Rulers 1 and 2? Some nomenclature: H = height, W = Width dH = uncertainty in H in units of H uH = percent uncertainty = dH/H dW = uncertainty in W in units of W uW = percent uncertainty = dW/W

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**Step 2: How does the uncertainty in H and W affect uncertainty in A?**

The calculated result, R is assumed to be calculated from a set of measurements: R = f(X1, X2, X3,…, XN) In our case we would write this as: R = A, X1 = H, X2 = W or A = f(H,W) where f = H*W The effect of the uncertainty in a single measurement (i.e. one of the X’s) on the calculated result, R, if only that one X were in error is: The partial derivative of R with respect to Xi is the sensitivity coefficient or the result R with respect to the measurement Xi Partial Derivative of R wrt Xi - A Sensitivity Coefficient

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**Step 2: How does the uncertainty in H and W affect uncertainty in A?**

In our case: Now calculate the partial derivatives: The partial derivative of R with respect to Xi is the sensitivity coefficient or the result R with respect to the measurement Xi

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**Step 2: How does the uncertainty in H and W affect uncertainty in A?**

When several independent variables (X’s) are used in calculating the Result, R, the individual terms are combined by a root-sum-square method (Method due to Kline and McClintock (1953))

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**Step 2: How does the uncertainty in H and W affect uncertainty in A?**

In our case we write this as: Now Calculate your uncertainties in A.

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**Uncertainty as a Percentage**

In most situations the overall uncertainty in a given result is dominated by only a few of its terms. Ignore terms that are smaller than the largest term by a factor of 3 or more. It is difficult (impossible) to compare errors with different units associated with them (e.g. how big is a 2 gram error compared to a 2 second error?) To solve this we “nondimensionalize” the errors:

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So, as a percentage…

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Uncertainty Analysis There are three sources of uncertainty in experimental measurements: Interpolation uncertainty (zeroth order) Unsteadiness (1st order) Instrument Calibration (Nth order)

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**Zero-Order Uncertainty**

At zero-order uncertainty, all variables and parameters that affect the outcome of the measurement, including time, are assumed to be fixed except for the physical act of observation itself. Any data scatter is the results of instrument resolution alone uo.

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**Higher-Order Uncertainty**

Higher order uncertainty estimates consider the controllability of the test operating conditions. For a first order estimate we might make a series of measurements over time and calculate the variation in that measurement. The first order uncertainty of that measurand is then: u1 = ± 2s* at (95%) * Note: Assuming we make enough measurements

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**Nth-Order Uncertainty**

As the final estimate, instrument calibration characteristics are entered into the scheme through the instrument uncertainty, uc. A practical estimate of the Nth order uncertainty uN is: Uncertainty analyses at the Nth order allow for the direct comparison between results of similar tests obtained using different instruments or at different test facilities.

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**Zero- order Uncertainty**

Report Results at uN level Nth order Uncertainty First-order Uncertainty u1 > u0 Zero- order Uncertainty u0 = ±1/2 resolution

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**Your Homework for Lab next week**

Next week in lab we will be measuring viscosity and density. You are to develop a tool* to calculate the uncertainty in your measurement of both. * tool can be in Matlab or Excel

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Density Measurement You will measure density by measuring the volume and mass of the fluid. The volume, V, will be measured in a graduated cylinder. The mass, m, will be measured on a balance as the difference between the full mf and empty mass me of the graduated cylinder. Density will be calculated as: rfluid = (mf-me)/V

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**Density Measurement Calculate the uncertainty in r:**

Estimate uncertainty interval for each measured quantity, (mf, me and V) Analyze the propagation of uncertainty into results calculated from experimental data. Check your results using me = 100 ± 1g, mf = 150 ± 1g, and V = 94 ± 0.5 mL (I will ask for your answer in lab next week).

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**Viscosity Measurement**

You will estimate the viscosity of a fluid using a falling ball viscometer. The viscosity, m, (in units of cp) is calculated as the product of the calibration constant, K, the time, t, and the difference between the falling ball and fluid density: = Kt(rball-rfluid) You will measure time using a stopwatch. The values of K and rball will be given to you.

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**Viscosity measurement**

. Calculate the uncertainty in m: Estimate uncertainty interval for each measured quantity (t, , rfluid). Analyze the propagation of uncertainty into results calculated from experimental data. Test your calculations for K = 5 ± 1 cp-mL/g-s, t = 30 ± .01s, rball= 5.2 ± 2 g/mL, and rfluid= 1 ± 1g/mL

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