ME 322: Instrumentation
Lecture 9
February 5, 2016
Professor Miles Greiner
Lab 4 and 5, beam in bending, Elastic
modulus calculation
Announcements/Reminders
• HW 3 Due Monday
– Joseph Young will hold office hour in PE 2 after class
today
– Marissa Tsugawa will give a Lab 4 Excel Tutorial at 6
pm in PE 2
• Midterm 1, February 19, 2016
– two weeks from today
Lab 4: Calculate Beam Density
LT
W
T
L
• 𝜌=
𝑚
𝑉
=
𝑚
𝑊𝑇𝐿𝑇
• Measure and estimate 95%-confidence-level uncertainties of
–
–
–
–
𝑚 = 𝑚 ± 𝑤𝑚 𝑔𝑚 95%
𝑊 = 𝑊 ± 𝑤𝑊 𝑖𝑛𝑐ℎ 95%
𝑇 = 𝑇 ± 𝑤𝑇 𝑖𝑛𝑐ℎ 95%
𝐿 𝑇 = 𝐿 𝑇 ± 𝑤𝐿𝑇 𝑖𝑛𝑐ℎ 95%
• Best estimate
– 𝜌=
–
𝑤𝜌
𝜌
𝑚
𝑊𝑇𝐿𝑇
2
Power product? (yes or no)
= Fill in blank
– If all the 𝑝𝑖 = 0.95, then 𝑝𝜌 = ?
• How to find 𝑤𝑚 , 𝑤𝑊 , 𝑤𝑇 and 𝑤𝐿𝑇 , all with 𝑝𝑖 = 0.95?
– Estimating uncertainties is usually not a well defined process!
Beam Length, LT
• Measure using a ruler or tape measure
– In L4PP, ruler’s smallest increment is 1/16 inch
• Uncertainty is 1/32 inch (half smallest increment)
– In Lab 4 – depends on the ruler you are issued
• May be different
• Assume the confidence-level for this uncertainty is
99.7% (3s)
– The uncertainty with a 68% (1s) confidence level
• (1/3)(1/32) inch
– The uncertainty with a 95% (2s) confidence level
• (2/3)(1/32) = 1/48 inch = 0.0208 inch
Beam Thickness T, Width W and Mass m
• Both lengths are measured multiple times
using different instruments
– Use sample mean for the best value, 𝑇 𝑎𝑛𝑑 𝑊
– Use sample standard deviations 𝑠𝑇 and 𝑠𝑊 for the
68%-confidence-level uncertainty
• The 95%-confidence-level uncertainties are
– 𝑤𝑇 = 2𝑠𝑇
– 𝑤𝑊 = 2𝑠𝑊
• Manufacturer Stated Analytical balance
uncertainty: 0.1 gm (p = 0.95?)
Table 3 Aluminum Beam
Measurements and Uncertainties
𝑤𝑊
𝑊
𝑤𝑇
𝑇
𝑤𝐿𝑇
𝐿𝑇
𝑤𝑚
𝑚
•
𝑤𝜌 2
𝜌
=
𝑤𝑊 2
𝑊
+
𝑤𝑇 2
𝑇
+
𝑤𝐿𝑇 2
𝐿𝑇
+
𝑤𝑚 2
𝑚
= 5.61 ∗ 10−5
Example: Show how to calculate densities
and uncertainties from measurements
Aluminum Steel
Calculated Density
[kg/m3]
95%-ConfidenceLevel Interval [kg/m3]
Cited Density*
[kg/m3]
2721 7948
20
60
2702 7854
• *Bergman, T.L., Lavine, A., Incropera, F.P., and Dewitt, D.P.,
2011: Fundamentals of Heat and Mass Transfer. 7th ed.
Wiley. 1048 pp.
• The cited aluminum density is within the 95%−confidence
level interval of the measured value, but the cited steel
density is not within that interval for its measure value
Lab 5 Measure Elastic Modulus of Steel
and Aluminum Beams (week after next)
• Incorporate top and bottom gages into a half bridge of a Strain
Indicator
– Power supply, Wheatstone bridge connections, Voltmeter, Scaled output
• Measure micro-strain for a range of end weights
• Knowing geometry, and strain versus weight, find Elastic Modulus
E of steel and aluminum beams
• Compare to textbook values
Set-Up
e3
W
e2 = -e3
T
L
From Manufacturer, i.e. 2.07 ± 1%
Strain Indicator
meR
SINPUT ≠ SREAL
• Wire gages into positions 3 and 2
of a half bridge
– e2 = -e3
R3
• Adjust R4 so that V0I ~ 0
• Enter Sinput (from manufacturer)
Procedure
EAl < ESteel
• Record meR for a range of beam end-masses, m
• Fit to a straight line meR,Fit = a m + b
• Slope a = fn(E, T, W, L, Sreal/ Sinput )
Bridge Output
•
•
𝑉0
𝑉𝑆
=
1
4
𝑆real 𝜀3 − 𝜀2 + 𝑆𝑇 ∆𝑇3 − ∆𝑇2
–
𝜀2 = −𝜀3
𝑉0
𝑉𝑆
1
𝑆real
4
=
2𝜀3 =
𝑆real 𝜀3
2
• How does indicator interpret VO?
– It assumes a quarter bridge and Sinput
–
𝑉0
𝑉𝑆
=
1
4
𝑆input 𝜀𝑅 =
1
𝜇𝜀𝑅
𝑆
𝜇𝑚
4 input 106
𝑚
• Bridge Transfer Function; let 𝑅𝑆 =
– 𝜇𝜀𝑅 =
𝑆real 𝜀3
𝑆input 2
4×
𝜇𝑚
6
10
𝑚
𝑆𝑅𝑒𝑎𝑙
=
𝑆𝐼𝑛𝑝𝑢𝑡
1 ± 0.01
= 𝑅𝑆 2 ×
1 ± 0.01
𝜇𝑚
6
10
𝑚
𝜀3
How to relate 𝜀3 to m, L, T, W, and E?
y
g
Neutral
Axis
W
m
σ
L
• Bending Stress: 𝜎3 =
𝑀𝑦
𝐼
– M = bending moment = FL = mgL
– Beam cross-section moment of inertia
• Rectangle: 𝐼 =
𝑇3𝑊
12
• Measure strain at upper surface, y = T/2
• Strain: 𝜀3 =
𝜎3
𝐸
=
1 𝑀𝑦
𝐸 𝐼
=
𝑇
𝑚𝑔𝐿
2
𝑇3 𝑊
𝐸
12
=
6𝑔𝐿
𝐸𝑇 2 𝑊
𝑚
T
Indicated Reading
• 𝜇𝜀𝑅 = 2 × 106 𝑅𝑆 𝜀3
= 2×
𝜇𝑚
6
10
𝑚
6𝑔𝐿
𝑅𝑆
𝐸𝑇 2 𝑊
𝑚
Slope, a
– Units 𝑎 =
𝑔𝐿
𝐸𝑇 2 𝑊
𝜇𝑚
𝑚 𝑘𝑔
• Best estimate of modulus, E
– 𝐸 = 12 ×
–
𝜇𝑚
6
10
𝑚
𝑔𝐿𝑅𝑆
𝑎𝑇 2 𝑊
1000
Microstrain Reading me R [mm/m]
𝜇𝑚
• 𝑎 = 12 × 106
𝑅𝑆
𝑚
800
600
400
meFit = 921.3[mm/(m*kg)]m - 2.1283[mm/m]
200
0
-200
0
0.2
0.4
0.6
Mass, m [kg]
= best estimate of measured or calculated value
0.8
1
1.2
Calculate value and uncertainty of E
6 𝜇𝑚
• 𝐸 = 12 × 10
𝑚
𝑔 𝐿 𝑅𝑆
𝑎𝑇 2 𝑊
• Is this a Power Product? (yes or no?)
–
𝑤𝐸 2
𝐸
=
Fill in blank (FIB)
• Find 95% (2σ) confidence level uncertainty
in E
– Find ?% confidence level (? σ) uncertainties in
each input value
Strain Gage Factor Uncertainty
• 𝑅𝑆 =
𝑆𝑅𝑒𝑎𝑙
𝑆𝐼𝑛𝑝𝑢𝑡
• In L5PP, manufacturer states
• S = 2.08 ± 1% (pS not given)
– In Lab 4 and 5, the values of 𝑆 and wS may be different!
• In L5PP and Lab 5, assume pS = 68% (1s)
– So assume the 95%-confidence-level uncertainty is twice
the manufacturer stated uncertainty
• S = 2.08 ± 2% (95%) = 2.08 ± .04 (95%)
• So 𝑅𝑆 =
𝑆𝑅𝑒𝑎𝑙
𝑆𝐼𝑛𝑝𝑢𝑡
= 1 ± 0.02 (95%)
Uncertainty of the Slope, a
Microstrain Reading me R [mm/m]
1000
800
𝑠𝑦,𝑥
600
400
meFit = 921.3[mm/(m*kg)]m - 2.1283[mm/m]
200
0
-200
0
0.2
0.4
0.6
0.8
1
1.2
Mass, m [kg]
• Fit data to yFit = ax + b using least-squares method
• Uncertainty in a and b increases with standard error
of the estimate (scatter of date from line)
–
𝑠𝑦,𝑥 =
𝑛 (𝑦 −𝑎𝑥 −𝑏)2
𝑖
𝑖=1 𝑖
𝑛−2
Uncertainty of Slope and Intercept
“it can be shown”
• 𝑠𝑎 = 𝑠𝑦,𝑥
𝑛
𝐷𝑒𝑛𝑜
(68%)
• 𝑠𝑏 = 𝑠𝑦,𝑥
( 𝑥𝑖 )2
𝐷𝑒𝑛𝑜
(68%)
– where Deno = 𝑛 𝑥𝑖2 −
– Not in the textbook
𝑥𝑖
2
• wa = ?sa (95%)
• Show how to calculate this next time
End 2015
L, Between Gage and Mass Centers
• Measure using a ruler
– In L5PP, ruler’s smallest increment is 1/16 inch
• Uncertainty is 1/32 inch (half smallest increment)
– Lab 5 – depends on the ruler you are issued
• may be different
• Assume the confidence-level for this uncertainty is
99.7% (3s)
– The uncertainty with a 68% (1s) confidence level
• (1/3)(1/32) inch
– The uncertainty with a 95% (2s) confidence level
• (2/3)(1/32) = 1/48 inch
Beam Thickness T and Width W
• Each are measured multiple times using
different instruments
– Use sample mean for the best value, 𝑇 𝑎𝑛𝑑 𝑊
– Use sample standard deviations 𝑠𝑇 and 𝑠𝑊 for the
68%-confidence-level uncertainty
• The 95%-confidence-level uncertainties are
– 𝑤𝑇 = 2𝑠𝑇
– 𝑤𝑊 = 2𝑠𝑊
Plot result and fit to a line meR,Fit = a m + b
Microstrain Reading me R [mm/m]
1000
800
600
400
meFit = 921.3[mm/(m*kg)]m - 2.1283[mm/m]
200
0
-200
0
0.2
0.4
0.6
Mass, m [kg]
•
Last lecture we found:
– 𝐸 = 12 × 106
– where 𝑅𝑆 =
𝑆real
𝑆input
𝑆real
𝑆input
𝑔𝐿
𝑎𝑇 2 𝑊
= 12 × 106
𝑔𝐿𝑅𝑆
𝑎𝑇 2 𝑊
0.8
1
1.2
Propagation of Uncertainty
• A calculation based on uncertain inputs
– R = fn(x1, x2, x3, …, xn)
• For each input xi find (measure, calculate) the best
estimate for its value 𝑥𝑖 , its uncertainty 𝑤𝑥𝑖 = 𝑤𝑖
with a certainty-level (probability) of pi
– 𝑥𝑖 = 𝑥𝑖 ± 𝑤𝑖 𝑝𝑖 𝑖 = 1,2, … 𝑛
– Note: pi increases with wi
• The best estimate for the results is:
– 𝑅 = 𝑓𝑛(𝑥1 , 𝑥2 , 𝑥3 ,…, 𝑥𝑛 )
• Find the confidence interval for the result
– 𝑅 = 𝑅 ± 𝑤𝑅 (𝑝𝑅 )
• Find 𝑤𝑅 𝑎𝑛𝑑 𝑝𝑅
𝑥
Statistical Analysis Shows
• 𝑤𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 =
𝑛
𝑖=1
𝑤𝑅𝑖
2
=
𝑛
𝑖=1
𝛿𝑅
𝛿𝑥𝑖 𝑥
𝑖
2
𝑤𝑖
• In this expression
– Confidence-level for all the wi’s, pi (i = 1, 2,…, n) must be the same
– Confidence level of wR,Likely, pR = pi is the same at the wi’s
– All errors must be uncorrelated
• Not biased by the same calibration error
General Power Product Uncertainty
𝑛
𝑒𝑖
𝑥
𝑖=1 𝑖
• 𝑅=𝑎
where a and ei are constants
• The likely fractional uncertainty in the result is
–
𝑊𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 2
𝑅
=
𝑛
𝑖=1
𝑊𝑖 2
𝑒𝑖
𝑥𝑖
– Square of fractional error in the result is the sum of the
squares of fractional errors in inputs, multiplied by their
exponent.
• The maximum fractional uncertainty in the result is
–
𝑊𝑅,𝑀𝑎𝑥
𝑅
=
𝑛
𝑖=1
𝑊𝑖
𝑒𝑖
𝑥𝑖
(100%)
– We don’t use maximum errors much in this class
Lab 5 Measure Elastic Modulus of Steel
and Aluminum Beams (week after next)
• Incorporate top and bottom
gages into a half bridge of a
Strain Indicator
• Record micro-strain reading for
a range of end weights
Will everyone in the class get the same
value as
• A textbook?
• Each other?
• Why not?
– Different samples have different moduli
– Experimental errors in measuring lengths and masses
(due to calibration errors and imprecision)
• How can we estimate the uncertainty in 𝐸 (wE) from
uncertainties in 𝐿 (wL), 𝑇 (wT), 𝑊 (wW), 𝑆 (wS), and
𝑎 (wa)?
– How do we even find these uncertainties?