ME 322: InstrumentationLecture 9

February 6, 2015

Professor Miles Greiner

Lab 4 and 5, beam in bending, Elastic modulus calculation

Announcements/Reminders

• HW 3 Due Monday • Add L4PP problem

• Midterm 1, February 20, 2015 (two weeks)• Service Learning Extra Credit (tomorrow)

– Probably too late to sign up now– If you signed-up but don’t show-up, you will loose 1%

• If you must cancel please inform Ms. Davis , 682-7741, shanelles@unr.edu

Lab 4: Calculate Beam Density

• Measure and estimate 95%-confidence-level uncertainties of

• Best estimate

• Power product? (yes or no)– Fill in blank – If all the , then ?

• How to find with ?

W

L

T

LT

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

Beam Thickness T and Width W

• Each are measured multiple times using different instruments – Use sample mean for the best value, – Use sample standard deviations and for the 68%-

confidence-level uncertainty

• The 95%-confidence-level uncertainties are– = 2– = 2

Best Estimate Uncertainty

Confidence

Level

95%-Confidence Level Uncertainty

95%-Confidence Level Fractional

Uncertainty

Uncertainty Found From

Width, W [in]

0.9944 0.0012 0.68 0.0025 0.0025 Multiple Measurement

Thickness, T [in]

0.1862 0.0007 0.68 0.0013 0.007 Multiple Measurement

Length, L [in]

11 0.0313 0.997 0.0208 0.0019 Smallest Instrumental Increment

Total Length, LT [in] 25.125 0.0313 0.997 0.0208 0.0008 Smallest Instrumental

Increment

Gage Resistance,

R [Ω]120.2 0.1 0.997 0.0667 0.0006 Manufacture

Specified Value

Gage Factor 2.08 (1%)0.02 0.68 0.04 0.02 Manufacture

Specified Value

Mass, m [g]

207.4 0.1 0.95 0.1 0.0005 Smallest Instrumental Increment

Table 3 Aluminum Beam Measurements and Uncertainties

Show how to measure densities and uncertainties

*Bergman, T.L., Adrienne, S.L., 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.

Aluminum SteelCalculated Density

[kg/m3] 2720 794895%-Confidence-

Level Interval [kg/m3] 21 60Cited Density*

[kg/m3] 2702 7854

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

• Wire gages into positions 3 and 2 of a half bridge– e2 = -e3

• Adjust R4 so make V0I ~ 0

W

L

T

Strain IndicatormeR

SINPUT ≠ SREAL

From Manufacturer, i.e. 2.07 ± 1%

R3

e3

e2 = -e3

Procedure

• 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 =1)

meFit = 921.3[mm/(m*kg)]m - 2.1283[mm/m]

-200

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2

Mic

rost

rain

Rea

din

g m

e R[m

m/m

]

Mass, m [kg]

E1E2 < E1

Bridge Output

–

• How does indicator interpret VO?– It assumes a quarter bridge and the Inputted S

• Bridge Transfer Function; let

= 1 ± 0.01

How to relate μεR to m, L, T, W, and E?

• Bending Stress: – M = bending moment = FL = mgL– Beam cross-section moment of inertia

• Rectangle:

• Measure strain at upper surface at y = T/2• Strain:

Neutral Axis

σ

y

m W

L

T

g

meFit = 921.3[mm/(m*kg)]m - 2.1283[mm/m]

-200

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2

Mic

ros

trai

n R

ead

ing

me R

[mm

/m]

Mass, m [kg]

Indicated Reading

– Units

• Best estimate of modulus, E

– = best estimate of measured or calculated value

Slope, a

Calculate value and uncertainty of E

• Is this a Power Product? (yes or no?)

– 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

• In L5PP, manufacturer states• S = 2.08 ± 1% (pS not given)

– In Lab 4 and 5, the values of and wS may be different!

• In L5PP and Lab 5, assume pS = 68% (1 )s– So assume the 95%-confidence-level uncertainty is twice

the manufacturer stated uncertainty

• S = 2.08 ± 2% (95%) = 2.08 ± .04 (95%)• So (95%)

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, – Use sample standard deviations and for the 68%-

confidence-level uncertainty

• The 95%-confidence-level uncertainties are– = 2– = 2

Uncertainty of the Slope, a

• 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)–

meFit = 921.3[mm/(m*kg)]m - 2.1283[mm/m]

-200

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2

Mic

rost

rain

Rea

din

g m

e R[m

m/m

]

Mass, m [kg]

𝑠𝑦 , 𝑥

Uncertainty of Slope and Intercept“it can be shown”

• = (68%)• = (68%)

– Not in the textbook

• wa = ?sa (95%)

• Show how to calculate this next time

Plot result and fit to a line meR,Fit = a m + b

• Last lecture we found:

– where

meFit = 921.3[mm/(m*kg)]m - 2.1283[mm/m]

-200

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2

Mic

rost

rain

Rea

din

g m

e R[m

m/m

]

Mass, m [kg]

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 , its uncertainty with a certainty-level (probability) of pi –

– Note: pi increases with wi

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

• Find the confidence interval for the result–

• Find

𝑥

Statistical Analysis Shows

• 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

• 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.

• The maximum fractional uncertainty in the result is– (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 (wE) from uncertainties in (wL), (wT), (wW), (wS), and (wa)?– How do we even find these uncertainties?