ME 322: InstrumentationLecture 8

February 3, 2016Professor Miles Greiner

Lab 4, Propagation of Uncertainty, Maximum and likely, Power Product, Examples

Announcements/Reminders• All future HW: all plots must be generated and

clearly formatted by Excel, MatLab, or… – No hand-drawn plots

• This week in lab– Lab 3 Pressure Transmitter Calibration– Please bring you Excel from L3PP, and the

instructions. • For the rest of the week, groups will move to

pressure standards instead of moving standards to pairs of groups. – Should be faster for everyone.

• We will purchase more standards before next year.

Strain Gages

• Their electrical resistance changes by small amounts when – They are strained (desired sensitivity) or – 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 them in a circuit that cancels-out the temperature effect

𝑑𝑅 𝑖

𝑅𝑖=𝑆𝑖𝜀𝑖+𝑆𝑇𝑖 ∆𝑇 𝑖

Strain Gage Wheatstone Bridge

• Bridge output voltage VO will exhibit large changes compared to VOI when Ri’s changes by small amounts

• Incorporate strain gages in the bridge:

R3 +

+ -

-

• If initially, R1I = R2I = R3I = R4I – Then VOI ~ 0

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

compensation• Lab 4 Install and test stain gages on opposite sides of

aluminum and steel beams, and measure beam densities• Lab 5 Measure Aluminum and Steel Elastic Moduli

Beam in Bending: Half Bridgeε3

ε2 = -ε3

ε2 = -ε3 R3 +

+ -

-

∆ 𝑇2 ≈ ∆𝑇 3

Lab 4 Install Strain Gages on Aluminum and Steel Beams (next week)

• Install two gage on a Steel or Aluminum beams– Top and bottom – Video Instructions: http://

wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2004%20Install_Strain_Gage/Lab%2004%20Index.htm

• Measure gage electrical resistance– Checks whether wire we connected correctly– Un-deformed, under tension, compression, and when

cold

W

L

T

LT

Lab 4 Install Strain Gages on Aluminum and Steel Beams (next week)

• Measure beam thickness T, width W, – Partners use Vernier calipers and micrometer– Best values: and (mean) – Uncertainties: Use sample standard deviation: ,

• What are their confidence levels? __%

• Total length LT, distance between centers of gage and pin, L– Use ruler, tape measure for and – Uncertainties: half of smallest increment; ,

• Confidence levels? __%

• Measure beam mass and its uncertainty:

W

L

T

LT

Calculate Beam Density

• Will everyone in the class get the same density as:– A textbook? Each other? If they measure many times?

• Why not?– Different samples have different densities– Experimental errors in measuring lengths and masses

(due to calibration errors and imprecision)• How can we estimate the uncertainty in () from

uncertainties in– (), (), (), and ()

𝑥

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

• For each input find (measure, calculate) the best estimate for its value , its uncertainty with its confidence-level (certainty, probability) pi –

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

• Find the confidence-interval for the result– • Find

Concept

• Vary one input by a small amount () while holding all the others inputs constant (linear analysis)

• Uncertainty in due to a small change in () alone is– (why use absolute value?)

𝑥𝑖𝑥𝑖

𝑅

𝑅

𝑤𝑖

𝑤𝑅𝑖

( 𝛿𝑅𝛿𝑥𝑖 )𝑥 𝑖  

All the ’s contribute to wR

• How to combine them?• Two ideas:• If all the wi’s are the maximum possible uncertainty

in xi (100% confident the true value is within ± )• Then the maximum possible uncertainty in is:

– Use absolute values because it’s not likely that all of the derivatives will be positive

–

Likely Uncertainty• Maximum possible error is overly pessimistic and

not useful since it’s not likely – that all the actual measured values will have the

maximum possible deviation from the best estimate, or – that they will all push the result in the same direction

• A “Likely” estimate (p < 100%) would be more realistic and useful uncertainty estimate

– (pR < 100%) < (pR = 100%)

Statistical Analysis Shows(meaning: take my word for it)

• In this expression– Confidence level for all the ’s, pi (i = 1, 2,…, n) must be

the same– Confidence level of , pR = pi is the same at the ’s– All errors must be uncorrelated

• Not biased by the same calibration error𝑥

Rectangle Example

• Given

• Find • Find best estimate

– How many digits are significant? (find )• Which length’s uncertainty has a larger contribution to the area uncertainty?

𝑏

𝑎

𝑤𝑎

𝑤𝑏

A=ab

Work on the white board

• Identify: – Are we trying to find the Maximum or Likely

Uncertainty?– Are all the certainty levels the same?– n = ?

A Simple method to propagate uncertainties in Power Product calculations

• “Power Product”

• Confidence Intervals of Inputs– …

• Best Estimate

• Likely Uncertainty–

Fractional Uncertainty

• Divide Uncertainty by – –

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

Repeat Rectangle Example using Power Product Rule

• Given

• Find • Work on board

𝑏

𝑎

𝑤𝑎

𝑤𝑏

A=ab

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