ME 322: InstrumentationLecture 13: Exam Review

February 18, 2015

Professor Miles Greiner

Announcements/Reminders• 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

• Tesla– Looking to hire ~330 interns this summer–Many of them will be ME majors– http://www.teslamotors.com/careers/university

• Starts Today, Evening with Industry, next Wednesday• Career Fair next Thursday

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 – sample average and sample standard deviation

– linear regression YFIT = aX + b

• Review Session– Marissa Tsugawa, this evening, 7-9 pm in PE 113, see WebCampus

• 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

Multiple Measurements of a Quantity

• Do not always give the same results.– Affected Uncontrolled (random) and Calibration (systematic) errors, as

well as Measurand.

• 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 this “randomly varying” process

Quad Area [m2]

Statistics• Find properties of an entire population of size N (which can

be ∞) using a smaller sample of size n < N.• Sample Mean

–

• Sample Standard Deviation–

• How can we use these statistics?– The standard deviation characterizes the measurement imprecision

(repeatability)– The mean characterizes the best estimate of the measurand

• units

Example Problem• Find the probability that the next sample will be

within the range x1 ≤ x ≤ x2

• Let (# of SDs from mean)

– I(z) on Page 146 *Bookmark* • Useful facts: I(0) = 0, I(∞) = 0.5, I(-z) = -I(z)

“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• If one more value is acquired, what is the

likelihood it is above the mean? – How much must be added to the next measurement so

the sum will have a specified-likelihood to be above the mean?

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

– Hint: Lean to calculate this efficiently (use table format)

0 40.5328 6.881.0597 9.721.5617 12.482.0863 15.342.5295 17.831.9637 14.661.5483 12.350.9211 9.030.5216 6.830 4.01

0.5619 7.090.9595 9.181.4562 11.921.9927 14.842.6214 18.32.1092 15.431.6423 12.891.0696 9.860.5315 6.880 4.02

Standard Reading, hS

[in WC]

Transmitter Output, IT

[mA]

To use the calibration• Make a measurement and record instrument reading, • Invert the transfer function to find the best estimate of the

measurand –

• Determine standard error of the estimate of the Measurand for a given Reading – sx,y = sy,x/a (Units!)

• Confidence interval – (Units and significant figures!)– or

• Calibration– Removes calibration (bias, systematic) error – Quantifies imprecision (random error) but does not remove it

Stand. Dev. of Best-Fit Slope and Intercept

• = (68%)• = (68%)

– Not in the textbook– Hint: Learn to calculate this efficiently (use table format)

• wa = ?sa (95%)

Propagation of Uncertainty• Consider a calculation based on uncertain inputs– R = fn(x1, x2, x3, …, xn)

• For each input xi find the best estimate for its value , and its uncertainty with a certainty-level (probability) of pi –

– Note: pi increases with wi

• The best estimate for the results is:– …, )

• The confidence interval for the result is– units

• Find

𝑥

Statistical Analysis Shows

• 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

General Power Product Uncertainty

• If where a and ei are constants

• The likely fractional uncertainty in the result is– – Square of fractional error in the result is the sum of the

squares of fractional errors in inputs, multiplied by their exponent.

• If not a power product, use general formula (previous slide)

• The maximum fractional uncertainty in the result is– (100%)– Don’t use this unless told to.

Instruments

U-Tube Manometer

Measurand Reading

• Power product?

Fluid

Air (1 ATM, 27°C)

Water (30°C)

Hg (27°C)

1.774

995.7

13,565

𝝆 [𝒌𝒈 /𝒎𝟑 ]

DP = 0

Inclined-Well Manometer

R

If and

Strain Gages

• Electrical resistance changes by small amounts when – They are strained (desired sensitivity)

• Strain Gage Factor:

– 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

𝑑𝑅 𝑖

𝑅=𝑆𝜀+𝑆𝑇 ∆𝑇

Wheatstone Bridge Output Voltage, VO

• When R1 R3 R2 R4, then

• Small changes in Ri cause small changes in –

• If gages are in all 4 legs – with (S and ST same)

–

R3 +

+ -

-

• Only one leg (R3) has a strain gauge–

• Other legs are fixed resistors

Quarter Bridge

R3 +

+ -

-

Undesired Sensitivity

Half Bridge

• Wire gages at R2 (-) and R3 (+)

– Place R3 on deform specimen; ε3, ΔT3

– Place R2 on identical but un-deformed; ε2=0, ΔT2 =ΔT3

–

Automatic temperature compensation

R3 +

+ -

-

• – Twice the output amplitude as quarter bring, with temperature

compensation

Beam in Bending: Half Bridgeε3

ε2 = -ε3

ε2 = -ε3

Beam in Bending: Full Bridge

• V0 is 4 times larger than quarter bridge– And has temperature compensation.

3 1

2 4

R3 +

+ -

-

= DT3 = e3 = -e3 = -e3 = DT3 = DT3

Tension Configuration (HW)

ε1 = ε3

ε4 = ε2 = -υ ε3

2 3

4 1

R3 +

+ -

-

• What would happen if all four were parallel?

Beam Surface Strain• Bending:

Neutral Axis

σ

y

W

L

T

F

F

• Tension:

• Could be used for force-measuring devices

Fluid Speed V (Pressure Method)

• Pitot Tube Transfer function: • To use: (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

V

PSPS

PT > PS PT > PS

How to Find Density• Ideal Gases– • P = PS = Static Pressure

• R = Gas Constant = RU/MM

– Ru = Universal Gas Constant = 8.314 kJ/kmol K

– MM = Molar Mass of the flowing Gas

• T = Absolute Temperature = T[°C] + 273.15• Can plug this into speed formula

• Liquids

– Tables

Water Properties (Appendix B of Text)

• Be careful with header and units

Volume Flow Rate, Q Variable-Area Meters

• Measure pressure drop at specified locations • Diameter in pipe D, at throat d– Diameter Ratio: b = d/D < 1

• Ideal (inviscid) transfer function:

• Less uncertainty for larger Q than for small ones

Venturi Tube Nozzle Orifice Plate

To use

• Invert the transfer function:

• C = Discharge Coefficient – C = fn(ReD, b = d/D, exact geometry and port locations)

• Need to know Q to find Q, so iterate– Assume C ~ 1, find Q, then Re, then C and check…

Discharge Coefficient Data from Text

• Nozzle: page 344, Eqn. 10.10– C = 0.9975 – 0.00653 (see restrictions in Text)

• Orifice: page 349, Eqn. 10.13– C = 0.5959 + 0.0312b2.1 - 0.184b8+ (0.3 < b < 0.7)

Student TIf N >30 use student t

Correlation Coefficient