1. ME 322: Instrumentation Lecture 13
February 17, 2016
Professor Miles Greiner
Midterm I Review

2. Announcements/Reminders
HW due now
Labs
This week:
Lab 6 Elastic Modulus Measurement
Next Week:
Monday only: Lab 6
No lab on the other days
In two weeks: Lab 6 Wind Tunnel Flow Rate and Speed
Only 4 wind tunnels (currently constructing one more)
Sign-up for 1.5 hour slots with your partner this week in lab
Section 2 students will take exam next-door (OB 101)

3. This week

4. Midterm 1 Friday Open book, plus bookmarks, plus one page of notes
If you have an e-book, you must turn off internet
4 problems, some have parts
Each part like HW or Lab calculations
Remember significant figures, uncertainty, units, confidence level
Be able to use your calculator to find
Sample average and sample standard deviation
Linear regression YFIT = aX + b
Review Session
Marissa Tsugawa, Thurs., Feb. 18, starting at 7:30pm (location?)
Handout: last year’s midterm problems
These problems will not be on the exam
Neither Marissa nor I will not provide answers or solutions for this
See me after class today regarding special needs

5. Multiple Measurements of a Quantity
Quad Area [m2]
Do not always give the same results.
Affected by Measurand and Uncontrolled (random) and Calibration (systematic) errors.
Patterns are observed if enough measurements are acquired
Bell-shaped probability distribution function
The sample may exhibit a center (mean) and spread (standard deviation)
Statistical Analysis can be applied to any “randomly varying” processes

6. Statistics
Find properties of an entire population of size N (which can be ∞) using a smaller sample of size n < N.
Sample Mean
𝑋 = 1 𝑛 𝑖=1 𝑛 𝑋 𝑖 ≈𝜇= 1 𝑁 𝑖=1 𝑁 𝑋 𝑖
Sample Standard Deviation
𝑠= 1 𝑛−1 𝑖=1 𝑛 ( 𝑋 𝑖 − 𝑋 ) 2 ≈𝜎= 1 𝑁 𝑖=1 𝑁 ( 𝑋 𝑖 −𝜇) 2
How can we use these statistics?
The mean characterizes the best estimate of the measurand
The standard deviation characterizes the measurement imprecision (repeatability)
𝑋= 𝑋 ±𝑠 units 68%
Confidence Interval
Confidence Level

7. Example Problem
Find the probability that the next sample will be within the range 𝑥 1 ≤ 𝑥 ≤ 𝑥 2
Let 𝑧 𝑖 = 𝑥 𝑖 −𝜇 𝜎 ≈ 𝑥 𝑖 − 𝑋 𝑠 (# of SDs from mean)
𝑃 𝑧 1 < 𝑧 𝑖 < 𝑧 2 =𝐼 𝑧 2 −𝐼 𝑧 1
I(z) on Page 146 *Bookmark*
Useful facts: I(0) = 0, I(∞) = 0.5, I(-z) = -I(z)

8. “Typical” Problems
Find the probability the next value is within a certain amount of the mean (“symmetric”)
Find the probability the next value is below (or above) a certain value (one sided)
If one more value is acquired, what is the likelihood it is above the mean?
How much must be added to a measurement so the sum will have a specified-likelihood to be above the mean?

9. Instrument Calibration
Experimental determination of instrument transfer function
Record instrument reading y for a range of measurands x (determined by a standard)
Use least-squares method to fit line yF = ax + b (or some other function) to the data.
Hint: Use calculator to find a and b unless told (remember Units)
Determine the standard error of the estimate of the Reading for a given Measurand
𝑠 𝑦,𝑥 = 𝑖=1 𝑛 ( 𝑦 𝑖 −𝑎 𝑥 𝑖 −𝑏) 2 𝑛−2
Hint: Lean to calculate this efficiently (use table format)
For partial credit, you may want to write this formula even if you use a calculator to find the quantity.

10. To use the calibration
Make a measurement and record instrument reading, 𝑦
Invert the transfer function to find the best estimate of the measurand
𝑥 =( 𝑦 −𝑏)/𝑎
Determine standard error of the estimate of the Measurand for a given Reading
sx,y = sy,x/a (Units!)
Confidence interval
𝑥= 𝑥 ± 𝑠 𝑥,𝑦 (68%) (Units and significant figures!)
or 𝑥= 𝑥 ± 2𝑠 𝑥,𝑦 95% …
Calibration
Removes calibration (bias, systematic) error
Quantifies imprecision (random error) but does not remove it

11. What is the likelihood 𝑥 will be above the true value of the measurand
How much do you need to add to 𝑥 to be 75% sure it is above the true value?
𝑥= 𝑥 +𝑧 𝑠 𝑥,𝑦 , where
𝐼=0.75−0.5=0.25, 𝑠𝑜 𝑧=0.675

12. Stand. Dev. of Best-Fit Slope and Intercept
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 (write it in)
Hint: Learn to calculate this efficiently
(use table format)
wa = ?sa (95%)

13. Propagation of Uncertainty
Consider a calculation based on uncertain inputs
R = fn(x1, x2, x3, …, xn)
For each input 𝑥 𝑖 find the best estimate for its value 𝑥 𝑖 , and 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 )
The confidence interval for the result is
𝑅= 𝑅 ± 𝑤 𝑅 ( 𝑝 𝑅 ) units
Find 𝑤 𝑅 𝑎𝑛𝑑 𝑝 𝑅 𝑥

14. 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 general 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

15. General Power Product Uncertainty
If 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 ( 𝑝 𝑅 )
Square of fractional error in the result is the sum of the squares of fractional errors in the inputs, multiplied by their exponent.
If not a power product, use general formula (previous slide)
The maximum fractional uncertainty in the result is
𝑊 𝑅,𝑀𝑎𝑥 𝑅 = 𝑖=1 𝑛 𝑒 𝑖 𝑊 𝑖 𝑥 𝑖 (100%)
Don’t use this unless told to.

16. Instruments

17. U-Tube Manometer ∆𝑃= 𝑃 1 − 𝑃 2 = 𝜌 𝑚 − 𝜌 𝑠 𝑔ℎ 𝐼𝑓 𝜌 𝑠 << 𝜌 𝑚
∆𝑃= 𝑃 1 − 𝑃 2 = 𝜌 𝑚 − 𝜌 𝑠 𝑔ℎ
𝐼𝑓 𝜌 𝑠 << 𝜌 𝑚
𝑇ℎ𝑒𝑛: ∆𝑃= 𝜌 𝑚 𝑔ℎ
DP = 0
Measurand
Reading
Fluid
Air (1 ATM, 27°C)
Water (30°C)
Hg (27°C)
𝝆 𝒌𝒈 𝒎 𝟑
1.774
995.7
13,565
Power product?

18. Inclined-Well Manometer
If 𝑠𝑖𝑛𝜃≫ 𝐴 𝑡 𝐴 𝑤 and 𝜌 𝑚 ≫ 𝜌 𝑠
𝑇ℎ𝑒𝑛: ∆𝑃= 𝜌 𝑚 𝑔𝑅𝑠𝑖𝑛𝜃
𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛: 𝑅= 1 𝜌 𝑚 𝑔𝑠𝑖𝑛𝜃 ∆𝑃

19. Strain Gages 𝑑 𝑅 𝑖 𝑅 =𝑆𝜀+ 𝑆 𝑇 ∆𝑇
𝑑 𝑅 𝑖 𝑅 =𝑆𝜀+ 𝑆 𝑇 ∆𝑇
Electrical resistance changes by small amounts when
They are strained (desired sensitivity)
Strain Gage Factor: S=1+2υ+ C strain
Their temperature changes (undesired sensitivity)
Solution:
Subject “identical” gages to the same environment so they experience the same temperature change and the same temperature-associated resistance change.
Incorporate gages into a Wheatstone bridge circuit that cancels-out the temperature effect

20. Wheatstone Bridge Output Voltage, VO- + - +
When R1 ≈ R3 ≈ R2 ≈ R4, then 𝑉 0 ≈0
Small changes in Ri cause small changes in 𝑉 0
𝑉 0 𝑉 𝑆 = 𝑑 𝑅 1 𝑅 1 − 𝑑 𝑅 2 𝑅 𝑑 𝑅 3 𝑅 3 − 𝑑 𝑅 4 𝑅 4
If gages are in all 4 legs
with 𝑑 𝑅 𝑖 𝑅 𝑖 =𝑆 𝜀 𝑖 + 𝑆 𝑇 ∆ 𝑇 𝑖 (S and ST same)
𝑉 0 𝑉 𝑠 = 1 4 𝑆 𝜀 1 − 𝜀 2 + 𝜀 3 − 𝜀 4 + 𝑆 𝑇 ∆ 𝑇 1 −∆ 𝑇 2 +∆ 𝑇 3 −∆ 𝑇 4

21. Quarter Bridge Only one leg (R3) has a strain gauge+ - - +
Only one leg (R3) has a strain gauge
𝑑 𝑅 3 𝑅 3 = 𝑆 3 𝜀 3 + 𝑆 𝑇3 ∆ 𝑇 3
Other legs are fixed resistors
𝑑 𝑅 1 𝑅 1 = 𝑑 𝑅 2 𝑅 2 = 𝑑 𝑅 4 𝑅 4 =0
𝑉 0 𝑉 𝑆 = 1 4 𝑆 𝜀 3 + 𝑆 𝑇 ∆ 𝑇 3
Undesired Sensitivity

22. Half Bridge + - - + Use gages for R2 (-) and R3 (+) R3
Place R3 on deform specimen; ε3, ΔT3
Place R2 on identical but un-deformed; ε2=0, ΔT2 =ΔT3
𝑉 0 𝑉 𝑆 = 1 4 𝑆 𝜀 3 − 𝜀 2 + 𝑆 𝑇 ∆ 𝑇 3 −∆ 𝑇 2 = 1 4 𝑆 𝜀 3
Automatic temperature compensation

23. Beam in Bending: Half Bridgeε3
ε2 = -ε3
ε2 = -ε3
𝑉 0 𝑉 𝑆 = 1 4 𝑆 𝜀 3 − 𝜀 2 + 𝑆 𝑇 ∆ 𝑇 3 −∆ 𝑇 2 = 1 4 𝑆 (2𝜀 3 )
𝑉 0 𝑉 𝑆 = 1 2 𝑆 𝜀 3
Twice the output amplitude as quarter bring, with temperature compensation

24. Beam in Bending: Full Bridge + - - +
𝑉 0 𝑉 𝑠 = 1 4 𝑆 𝜀 1 − 𝜀 2 + 𝜀 3 − 𝜀 4 + 𝑆 𝑇 ∆ 𝑇 1 −∆ 𝑇 2 +∆ 𝑇 3 −∆ 𝑇 4
𝑉 0 𝑉 𝑠 = 1 4 𝑆 4 𝜀 3 + 𝑆 𝑇 0
V0 is 4 times larger than quarter bridge
And has temperature compensation.
= e3
= -e3
= -e3
= DT3
= DT3
= DT3

25. Tension Configuration (HW)2 3 + - - + 4 1
ε1 = ε3 ε4 = ε2 = -υ ε3
What would happen if all four were parallel?

26. Beam Surface Strain
Bending: 𝜀 3 = 𝜎 3 𝐸 = 1 𝐸 𝑀𝑦 𝐼 = 𝑚𝑔𝐿( 𝑇 2 ) 𝐸 𝑇 3 𝑊 12 = 6𝐿𝑔 𝐸 𝑇 2 𝑊 𝑚 y
F = mg
Neutral
Axis W L T σ
Tension:𝜀= 𝜎 𝐸 = 𝐹/𝐴 𝐸 = 𝐹 𝐴𝐸
Could be used for force-measuring devices F

27. Fluid Speed V (Pressure Method)
PT > PS
PT > PS PS PS V
Pitot Tube Transfer function: ∆𝑃= 1 2 𝜌 𝑉 2
To use: 𝑉=𝐶 2∆𝑃 𝜌 (Power product?)
C accounts for viscous effects, which are small
Assume C = 1 unless told otherwise
Less uncertainty for larger 𝑉 than for small ones

28. How to Find Density Ideal Gases Liquids
𝜌= 𝑃 𝑅𝑇 = 𝑃 𝑀𝑀 𝑅 𝑈 𝑇 ; 𝑉=𝐶 2∆𝑃 𝜌 =𝐶 2 ∆𝑃 𝑅 𝑇 𝑃
P = PS = Static Pressure
R = Gas Constant = RU/MM
Ru = Universal Gas Constant = kJ/kmol K
MM = Molar Mass of the flowing Gas
For air: R = kP*m3/kg*K
T = Absolute Temperature = T[°C]
Can plug this into speed formula
Liquids
𝜌=𝑓𝑛 𝑇 ≠𝑓𝑛(𝑃)
Tables

29. Water Properties (Appendix B of Text)
Be careful with header and units

30. Volume Flow Rate, Q Variable-Area Meters
Venturi Tube
Nozzle
Orifice Plate
Measure pressure drop Δ𝑃 at specified locations
Diameter in pipe D, at throat d
Diameter Ratio: b = d/D < 1
Ideal (inviscid) transfer function:
Δ𝑃= 𝜌 𝑄 𝐴 −b4
Less uncertainty for larger 𝑄 than for small ones

31. To use Invert the transfer function: C = Discharge Coefficient
𝑄=𝐶 𝐴 Δ𝑃 𝜌 1−β4 = 𝐶(pd2/4) 1−β Δ𝑃 𝜌
C = Discharge Coefficient
C = fn(ReD, b = d/D, exact geometry and port locations)
𝑅𝑒 𝐷 = 𝑉 1 𝐷𝜌 𝜇 = 𝑚 𝜌 𝜋 4 𝐷 𝐷𝜌 𝜇 = 4 𝑚 𝜋𝐷𝜇 = 4𝜌𝑄 𝜋𝐷𝜇
Need to know Q to find Q, so iterate
Assume C ~ 1, find Q, then Re, then C and check…

32. Discharge Coefficient Data from Text
Nozzle: page 344, Eqn
C = – 𝛽 𝑅𝑒 𝐷 (see restrictions in Text)
Orifice: page 349, Eqn
C = b b 𝛽 𝑅𝑒 𝐷 (0.3 < b < 0.7)

34. Correlation Coefficient
Student T
If N >30 use student t