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

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

Lab 4: Calculate Beam Density W 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!

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 = 0.0208 inch

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?)

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 2 + 𝑤 𝑇 𝑇 2 + 𝑤 𝐿 𝑇 𝐿 𝑇 2 + 𝑤 𝑚 𝑚 2 =5.61∗ 10 −5

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. 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

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 From Manufacturer, i.e. 2.07 ± 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

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 )

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 = 1 4 𝑆 input 𝜀 𝑅 = 1 4 𝑆 input 𝜇 𝜀 𝑅 10 6 𝜇𝑚 𝑚 Bridge Transfer Function; let 𝑅 𝑆 = 𝑆 𝑅𝑒𝑎𝑙 𝑆 𝐼𝑛𝑝𝑢𝑡 = 1 ± 0.01 𝜇 𝜀 𝑅 = 𝑆 real 𝑆 input 𝜀 3 2 4× 10 6 𝜇𝑚 𝑚 = 𝑅 𝑆 2× 10 6 𝜇𝑚 𝑚 𝜀 3 1 ± 0.01

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

Indicated Reading Best estimate of modulus, E 𝜇 𝜀 𝑅 = 2× 10 6 𝑅 𝑆 𝜀 3 = 2× 10 6 𝜇𝑚 𝑚 𝑅 𝑆 6𝑔𝐿 𝐸 𝑇 2 𝑊 𝑚 𝑎= 12× 10 6 𝜇𝑚 𝑚 𝑅 𝑆 𝑔𝐿 𝐸 𝑇 2 𝑊 Units 𝑎= 𝜇𝑚 𝑚 𝑘𝑔 Best estimate of modulus, E 𝐸 = 12× 10 6 𝜇𝑚 𝑚 𝑔 𝐿 𝑅 𝑆 𝑎 𝑇 2 𝑊 = best estimate of measured or calculated value Slope, a

Calculate value and uncertainty of E 𝐸 = 12× 10 6 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

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%)

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

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

End 2015

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, 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

Plot result and fit to a line meR,Fit = a m + b Last lecture we found: 𝐸 = 12× 10 6 𝑆 real 𝑆 input 𝑔 𝐿 𝑎 𝑇 2 𝑊 = 12× 10 6 𝑔 𝐿 𝑅 𝑆 𝑎 𝑇 2 𝑊 where 𝑅 𝑆 = 𝑆 real 𝑆 input

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

Statistical Analysis Shows 𝑤 𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 = 𝑖=1 𝑛 𝑤 𝑅 𝑖 2 = 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

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

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 𝐸 (wE) from uncertainties in 𝐿 (wL), 𝑇 (wT), 𝑊 (wW), 𝑆 (wS), and 𝑎 (wa)? How do we even find these uncertainties?