Download Uncertainty Worksheet

EM375 Mechanical Engineering Experimentation
Measurement of Viscosity
Uncertainty Worksheet
CAUTION: Be extraordinarily careful with units. It is recommended that ALL numbers are
converted to SI before they are used in any calculations.
Use this worksheet as a guide to help you develop the uncertainty analysis. The worksheet does
not give every tiny step of the calculations. You need to SHOW ALL WORK. If you need more
space, edit the document to make it.
UNCERTAINTY OF TRANSDUCERS
First we separately inspect each transducer and identify its elemental uncertainties. We then
combine them to determine the transducer’s total standard uncertainty.
We then multiply the transducer’s standard uncertainty by a k-factor to determine the
transducer expanded uncertainty of measurement.
Calipers:
Record the resolution (scale size) of the digital calipers that you used.
Resolution =
Calculate the standard uncertainty due to the caliper resolution. Do you think the calipers
truncate or round the values they give?
𝑢𝑅𝐸𝑆𝑂𝐿𝑈𝑇𝐼𝑂𝑁,𝑐 =
=
Previous calibration of the calipers has determined that the combined standard uncertainty for
all other elemental uncertainties is 0.003 mm (for the 0.01 mm resolution calipers) and
0.0003 mm (for the 0.001 mm resolution calipers). Combine this value with the standard
uncertainty due to resolution to determine the standard uncertainty of measurement for the
calipers
𝑢𝐶𝐴𝐿𝐼𝑃𝐸𝑅𝑆 = √(
)2 + (
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)2 =
Yardstick:
Assume that the only significant elemental uncertainty for the yardstick is that due to its
resolution. From the resolution determine the standard uncertainty of measurement. Take care
with the conversion from inches to meters.
𝑢𝑌𝐴𝑅𝐷𝑆𝑇𝐼𝐶𝐾 =
Stopwatch:
Assume that the only significant elemental uncertainty for the stopwatch itself is due to its
resolution. From the resolution determine the standard uncertainty.
𝑢𝑊𝐴𝑇𝐶𝐻 =
Balance:
Record the resolution of the balance.
Resolution =
Calculate the standard uncertainty due to the resolution. Should you assume rounding or
truncation?
𝑢𝑅𝐸𝑆𝑂𝐿𝑈𝑇𝐼𝑂𝑁,𝑚 =
=
Previous calibration of the balance has determined that the combined standard uncertainty for
all other elemental uncertainties is 0.008 g. Use an RSS calculation to combine this with the
standard uncertainty due to resolution to determine the standard uncertainty of measurement
for the balance.
𝑢𝐵𝐴𝐿𝐴𝑁𝐶𝐸 =
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SUMMARY OF TRANSDUCER UNCERTAINTIES:
Complete the table of standard uncertainties. Then expand them to uncertainties at 95%
confidence (k-factor = 2). Remember to include units.
Standard uncertainty of
measurement
u
95% confidence expanded
uncertainty of measurement
U
Transducer
Calipers
Yardstick
Stopwatch
Balance
UNCERTAINTY OF EACH MEASUREMENT
For each measurement that you took you need to determine its expanded uncertainty. This is a
combination of the transducer uncertainty and the variability of each measurand.
Rotating drum diameter, 𝑎.
There is spatial variation in the drum’s diameter. Calculate the mean measured diameter and
the 95% confidence interval of the mean. The expanded uncertainty due to spatial variation is
the confidence interval.
𝑎̅ =
𝑈𝑎,𝑆𝑃𝐴𝑇𝐼𝐴𝐿 = 𝐶𝐼𝑎,95% =
Combine the expanded uncertainty of measurement for the calipers and the expanded
uncertainty for the spatial variation to obtain the expanded uncertainty of the drum’s diameter.
2
expanded uncertainty in drum diameter = 𝑈𝑎 = √(𝑈𝐶𝐴𝐿𝐼𝑃𝐸𝑅𝑆 )2 + (𝑈𝑎,𝑆𝑃𝐴𝑇𝐼𝐴𝐿 ) =
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Rotating drum length, 𝐿.
The uncertainty in the drum’s length is due to both the uncertainty in the calipers and the
spatial variation of the drum itself. Calculate the mean measured length and the 95%
confidence interval of the mean.
𝐿̅ =
𝑈𝐿,𝑆𝑃𝐴𝑇𝐼𝐴𝐿 = 𝐶𝐼𝐿 =
Combine the expanded uncertainty of measurement for the calipers and the expanded
uncertainty for the spatial variation to obtain the standard uncertainty of the drum’s length.
2
expanded uncertainty in drum length = 𝑈𝐿 = √(𝑈𝐶𝐴𝐿𝐼𝑃𝐸𝑅𝑆 )2 + (𝑈𝐿,𝑆𝑃𝐴𝑇𝐼𝐴𝐿 ) =
Cylinder inside diameter, 𝑏.
You only took one measurement of this diameter, so we have to decide a way of estimating its
uncertainty. For now let us assume the 95% confidence interval for 𝑏 is the same as the 95%
confidence interval for the drum’s diameter 𝑎. Later when you have competed the uncertainty
budget, if diameter 𝒃 is a significant contributor to the overall uncertainty in viscosity we use
this as an indicator that we need to revisit the way we measure this quantity.
Combine the expanded uncertainty for the calipers and the assumed confidence interval for the
diameter to obtain the expanded uncertainty of the cylinder’s diameter.
expanded uncertainty in cylinder diameter = 𝑈𝑏 = √(𝑈𝐶𝐴𝐿𝐼𝑃𝐸𝑅𝑆 )2 + (𝐶𝐼𝑏 )2 =
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Pulley diameter, 𝑐.
This measurement depends on the number of turns of string on the pulley. Assume that the
95% (k-factor of 2) expanded uncertainty caused by the varying thickness of the string as it
unwinds is 0.2 mm. Combine this with the standard uncertainty of measurement for the
calipers to determine the standard uncertainty of measurement.
𝑈𝑐 = √(𝑈𝐶𝐴𝐿𝐼𝑃𝐸𝑅𝑆 )2 + (
0.2 2
)
1000
Distance fallen, ℎ.
Assume that the only significant uncertainty in distance fallen is that due to the expanded
uncertainty of measurement for the yardstick.
𝑈ℎ = 𝑈𝑌𝐴𝑅𝐷𝑆𝑇𝐼𝐶𝐾 =
Time taken, 𝑡.
We will assume that in addition to the measurement uncertainty of the stopwatch, the
uncertainty in measured time is also due to your human reaction time. Using the drop time
data you obtained in the shuttlecock drop lab, calculate the standard deviation (use ALL the
data, NOT the data set after outlier elimination). Since you had a large data set, this standard
deviation is also the standard uncertainty due to reaction time, 𝑢𝑅𝐸𝐴𝐶𝑇𝐼𝑂𝑁 . Then calculate the
95% confidence expanded uncertainty for reaction time.
standard deviation = 𝑢𝑅𝐸𝐴𝐶𝑇𝐼𝑂𝑁 =
𝑈𝑅𝐸𝐴𝐶𝑇𝐼𝑂𝑁 = 𝑘. 𝑢𝑅𝐸𝐴𝐶𝑇𝐼𝑂𝑁 =
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Combine this with the expanded uncertainty of measurement for the stopwatch to determine
the expanded uncertainty in a single measured time.
𝑈𝑆𝐼𝑁𝐺𝐿𝐸 𝑡 = √(𝑈𝑊𝐴𝑇𝐶𝐻 )2 + (𝑈𝑅𝐸𝐴𝐶𝑇𝐼𝑂𝑁 )2
Returning to the viscometer lab, for your measured drop times calculate their mean and 95%
confidence interval of the mean.
𝑡̅ =
𝐶𝐼𝑡 = 𝑈𝑣𝑎𝑟𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
Combine this uncertainty for the average drop time with the expanded uncertainty for a single
measurement.
expanded uncertainty in average drop time = 𝑈𝑡 = √(𝑈𝑆𝐼𝑁𝐺𝐿𝐸 𝑡 )2 + (𝑈𝑣𝑎𝑟𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 )
2
Mass, 𝑚.
Assume the uncertainty in mass is solely due to the measurement uncertainty in the balance.
𝑈𝑚 = 𝑈𝐵𝐴𝐿𝐴𝑁𝐶𝐸 =
Acceleration due to gravity, 𝑔:
The actual acceleration due to gravity depends upon a number of factors including location,
height above sea level and local geological features such as the density of bedrock. One method
of calculating the local acceleration due to gravity uses the International Gravity Formula (IGF)
of 1980 and the World Geodetic System cartographic model WGS841, which take into account
1 WGS84 is the coordinate system used by Global Positioning System, GPS.
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the rotation of the Earth, the height above sea level, and the oblate spheroidal shape of the
geoid.
𝑔 = 9.7803267714
1 + 0.00193185138639(𝑠𝑖𝑛(𝜙))
2
2
√
( 1 − 0.00669437999013(𝑠𝑖𝑛(𝜙)) )
𝑅 2
(
) m/s2
𝑅+𝑒
Where 𝜙 is the latitude, 𝑒 is the elevation above sea level of the test facility, and 𝑅 =
6,371,000 m is the nominal radius of the earth.
Using this formula the acceleration due to gravity at the Naval Academy is 9.8008 ±0.0002 m/s2
at 5% significance. Identify the 95% confidence expanded uncertainty.
𝑈𝑔 =
CALCULATE THE VISCOSITY
Using your measured quantities, calculate the viscosity. Quote your answer to at least 6
significant figures, and give appropriate units.
𝑚𝑔𝑐 2 (𝑏 − 𝑎) 𝑡
𝜇=[
] =
2𝜋𝐿𝑎2 𝑏
ℎ
𝜇=
SENSITIVITIES AND UNCERTAINTY BUDGET
The uncertainties due to each measurement now need to be propagated into the uncertainty in
the calculated quantity. Use the uncertainty budget table on the next page to perform these
calculations. Follow these directions as you complete it. Note that there are many different
styles of uncertainty budgets. The table on the next page is the one required for this lab.
1.
Analytically determine the equations for the sensitivity of viscosity to each of the
parameters in the equation. This means symbolically – do not use Matlab, Mathcad, etc. unless
you know how to get the symbolic results. It’s best if you do these partial differentiations by
hand! Enter the equations in the appropriate column in the table below.
2.
For each sensitivity equation, substitute the measured quantities and calculate the
sensitivity as a number.
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3.
Calculate the contributions.
4.
Combine the contributions using an RSS calculation to find the uncertainty in the
calculated quantity. Give the number to at least 6 significant figures.
5.
Quote the calculated viscosity with its 95% expanded uncertainty. Show correct units
and round off the numbers appropriately.
6.
What is the calculated viscosity at 99% uncertainty? Show work.
Quantity, 𝑥
95% expanded
uncertainty of
measurement
𝑈𝑥
Sensitivity (equation)
𝜕𝜇
𝜕𝑥
Mass, 𝑚
Acceleration due
to gravity, 𝑔
Pulley diameter,
𝑐
Drum diameter,
𝑎
Cylinder
diameter, 𝑏
Drum length, 𝐿
Drop time, 𝑡
Distance fallen, ℎ
From RSS calculation, the uncertainty in viscosity is 𝑈𝜇 =
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Sensitivity
Contribution
(number)
𝜕𝜇
𝜕𝜇
𝑈𝑥
𝜕𝑥
𝜕𝑥
STATEMENT: At 95% confidence the viscosity is 𝜇 =
±
STATEMENT: At 99% confidence the viscosity is 𝜇 =
±
Uncertainty calculations - 9