MER331 – Lab 1. An Introduction to Uncertainty Analysis. Error Sources. The measurement process consists of three dis-tinct steps: calibration, data acquisition, and data reduction calibration error data acquisition error data reduction error. Uncertainty Analysis.
An Introduction to Uncertainty Analysis
The measurement process consists of three dis-tinct steps: calibration, data acquisition, and data reduction
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”
Example: Each group has a piece of paper and 2 measur-ementtools (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
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.
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)
What do we use for Rulers 1 and 2?
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
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:
Partial Derivative of R wrt Xi - A Sensitivity Coefficient
In our case:
Now calculate the partial derivatives:
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))
In our case we write this as:
Now Calculate your uncertainties in A.
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:
There are three sources of uncertainty in experimental measurements:
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.
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
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.
Nth order Uncertainty
u1 > u0
Zero- order Uncertainty
u0 = ±1/2 resolution
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
You will measure density by measuring the volume and mass of the fluid.
Density will be calculated as: rfluid= (mf-me)/V
Calculate the uncertainty in r:
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:
. Calculate the uncertainty in m: