1. ME 322: Instrumentation Lecture 9
February 5, 2016
Professor Miles Greiner
Lab 4 and 5, beam in bending, Elastic modulus calculation

2. 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

3. Lab 4: Calculate Beam DensityW L T LT
𝜌= 𝑚 𝑉 = 𝑚 𝑊𝑇 𝐿 𝑇
Measure and estimate 95%-confidence-level uncertainties of
𝑚= 𝑚 ± 𝑤 𝑚 𝑔𝑚 95%
𝑊= 𝑊 ± 𝑤 𝑊 𝑖𝑛𝑐ℎ 95%
𝑇 = 𝑇 ± 𝑤 𝑇 𝑖𝑛𝑐ℎ 95%
𝐿 𝑇 = 𝐿 𝑇 ± 𝑤 𝐿 𝑇 𝑖𝑛𝑐ℎ 95%
Best estimate
𝜌 = 𝑚 𝑊 𝑇 𝐿 𝑇 Power product? (yes or no)
𝑤 𝜌 𝜌 2 = Fill in blank
If all the 𝑝 𝑖 =0.95, then 𝑝 𝜌 = ?
How to find 𝑤 𝑚 , 𝑤 𝑊 , 𝑤 𝑇 and 𝑤 𝐿 𝑇 , all with 𝑝 𝑖 =0.95?
Estimating uncertainties is usually not a well defined process!

4. 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 = inch

5. 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 𝑠 𝑇 and 𝑠 𝑊 for the 68%-confidence-level uncertainty
The 95%-confidence-level uncertainties are
𝑤 𝑇 = 2 𝑠 𝑇
𝑤 𝑊 = 2 𝑠 𝑊
Manufacturer Stated Analytical balance uncertainty: 0.1 gm (p = 0.95?)

6. Table 3 Aluminum Beam Measurements and Uncertainties
𝑤 𝑊 𝑊
𝑤 𝑇 𝑇
𝑤 𝐿 𝑇 𝐿 𝑇
𝑤 𝑚 𝑚
𝑤 𝜌 𝜌 2 = 𝑤 𝑊 𝑊 𝑤 𝑇 𝑇 𝑤 𝐿 𝑇 𝐿 𝑇 𝑤 𝑚 𝑚 2 =5.61∗ 10 −5

7. Calculated Density [kg/m3] 95%-Confidence-Level Interval [kg/m3]
Example: Show how to calculate densities and uncertainties from measurements
Aluminum
Steel
Calculated Density [kg/m3]
2721
7948
95%-Confidence-Level Interval [kg/m3] 20 60
Cited Density* [kg/m3]
2702
7854
*Bergman, T.L., Lavine, A., Incropera, F.P., and Dewitt, D.P., 2011: Fundamentals of Heat and Mass Transfer. 7th ed. Wiley 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

8. 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

9. Set-Up Wire gages into positions 3 and 2 of a half bridge
e2 = -e3
From Manufacturer, i.e ± 1%
Strain Indicator
meR
SINPUT ≠ SREAL
Wire gages into positions 3 and 2 of a half bridge
e2 = -e3
Adjust R4 so that V0I ~ 0
Enter Sinput (from manufacturer) R3

10. Procedure Record meR for a range of beam end-masses, m
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 )

11. Bridge Output 𝑉 0 𝑉 𝑆 = 1 4 𝑆 real 𝜀 3 − 𝜀 2 + 𝑆 𝑇 ∆ 𝑇 3 −∆ 𝑇 2
𝜀 2 =− 𝜀 3
𝑉 0 𝑉 𝑆 = 1 4 𝑆 real 2 𝜀 3 = 𝑆 real 𝜀 3 2
How does indicator interpret VO?
It assumes a quarter bridge and Sinput
𝑉 0 𝑉 𝑆 = 𝑆 input 𝜀 𝑅 = 1 4 𝑆 input 𝜇 𝜀 𝑅 𝜇𝑚 𝑚
Bridge Transfer Function; let 𝑅 𝑆 = 𝑆 𝑅𝑒𝑎𝑙 𝑆 𝐼𝑛𝑝𝑢𝑡 = 1 ± 0.01
𝜇 𝜀 𝑅 = 𝑆 real 𝑆 input 𝜀 × 𝜇𝑚 𝑚 = 𝑅 𝑆 2× 𝜇𝑚 𝑚 𝜀 3
1 ± 0.01

12. How to relate 𝜀 3 to m, L, T, W, and E?y g
Neutral
Axis m W L T σ
Bending Stress: 𝜎 3 = 𝑀𝑦 𝐼
M = bending moment = FL = mgL
Beam cross-section moment of inertia
Rectangle: 𝐼= 𝑇 3 𝑊 12
Measure strain at upper surface, y = T/2
Strain: 𝜀 3 = 𝜎 3 𝐸 = 1 𝐸 𝑀𝑦 𝐼 = 𝑚𝑔𝐿 𝑇 2 𝐸 𝑇 3 𝑊 12 = 6𝑔𝐿 𝐸 𝑇 2 𝑊 𝑚

13. Indicated Reading Best estimate of modulus, E
𝜇 𝜀 𝑅 = 2× 𝑅 𝑆 𝜀 3
= 2× 𝜇𝑚 𝑚 𝑅 𝑆 6𝑔𝐿 𝐸 𝑇 2 𝑊 𝑚
𝑎= 12× 𝜇𝑚 𝑚 𝑅 𝑆 𝑔𝐿 𝐸 𝑇 2 𝑊
Units 𝑎= 𝜇𝑚 𝑚 𝑘𝑔
Best estimate of modulus, E
𝐸 = 12× 𝜇𝑚 𝑚 𝑔 𝐿 𝑅 𝑆 𝑎 𝑇 2 𝑊
= best estimate of measured or calculated value
Slope, a

14. Calculate value and uncertainty of E
𝐸 = 12× 𝜇𝑚 𝑚 𝑔 𝐿 𝑅 𝑆 𝑎 𝑇 2 𝑊
Is this a Power Product? (yes or no?)
𝑤 𝐸 𝐸 2 = Fill in blank (FIB)
Find 95% (2σ) confidence level uncertainty in E
Find ?% confidence level (? σ) uncertainties in each input value

15. Strain Gage Factor Uncertainty
𝑅 𝑆 = 𝑆 𝑅𝑒𝑎𝑙 𝑆 𝐼𝑛𝑝𝑢𝑡
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 𝑅 𝑆 = 𝑆 𝑅𝑒𝑎𝑙 𝑆 𝐼𝑛𝑝𝑢𝑡 =1±0.02 (95%)

16. Uncertainty of the Slope, a
𝑠 𝑦,𝑥
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)
𝑠 𝑦,𝑥 = 𝑖=1 𝑛 ( 𝑦 𝑖 −𝑎 𝑥 𝑖 −𝑏) 2 𝑛−2

17. Uncertainty of Slope and Intercept “it can be shown”
𝑠 𝑎 = 𝑠 𝑦,𝑥 𝑛 𝐷𝑒𝑛𝑜 (68%)
𝑠 𝑏 = 𝑠 𝑦,𝑥 ( 𝑥 𝑖 ) 2 𝐷𝑒𝑛𝑜 (68%)
where Deno=𝑛 𝑥 𝑖 2 − 𝑥 𝑖 2
Not in the textbook
wa = ?sa (95%)
Show how to calculate this next time

18. End 2015

19. 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

20. 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 𝑠 𝑇 and 𝑠 𝑊 for the 68%-confidence-level uncertainty
The 95%-confidence-level uncertainties are
𝑤 𝑇 = 2 𝑠 𝑇
𝑤 𝑊 = 2 𝑠 𝑊

22. Plot result and fit to a line meR,Fit = a m + b
Last lecture we found:
𝐸 = 12× 𝑆 real 𝑆 input 𝑔 𝐿 𝑎 𝑇 2 𝑊 = 12× 𝑔 𝐿 𝑅 𝑆 𝑎 𝑇 2 𝑊
where 𝑅 𝑆 = 𝑆 real 𝑆 input

23. 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 𝑥 𝑖 , its uncertainty 𝑤 𝑥 𝑖 = 𝑤 𝑖 with a certainty-level (probability) of pi
𝑥 𝑖 = 𝑥 𝑖 ± 𝑤 𝑖 𝑝 𝑖 𝑖=1,2,…𝑛
Note: pi increases with wi
The best estimate for the results is:
𝑅 =𝑓𝑛( 𝑥 1 , 𝑥 2 , 𝑥 3 ,…, 𝑥 𝑛 )
Find the confidence interval for the result
𝑅= 𝑅 ± 𝑤 𝑅 ( 𝑝 𝑅 )
Find 𝑤 𝑅 𝑎𝑛𝑑 𝑝 𝑅 𝑥

24. Statistical Analysis Shows
𝑤 𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 = 𝑖=1 𝑛 𝑤 𝑅 𝑖 = 𝑖=1 𝑛 𝛿𝑅 𝛿 𝑥 𝑖 𝑥 𝑖 𝑤 𝑖 2
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

25. General Power Product Uncertainty
𝑅=𝑎 𝑖=1 𝑛 𝑥 𝑖 𝑒 𝑖 where a and ei are constants
The likely fractional uncertainty in the result is
𝑊 𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 𝑅 2 = 𝑖=1 𝑛 𝑒 𝑖 𝑊 𝑖 𝑥 𝑖 2
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
𝑊 𝑅,𝑀𝑎𝑥 𝑅 = 𝑖=1 𝑛 𝑒 𝑖 𝑊 𝑖 𝑥 𝑖 (100%)
We don’t use maximum errors much in this class

26. 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

27. 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?