ME 322: Instrumentation Lecture 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, ,

Lab 4: Calculate Beam Density W L T LTLT

Beam Length, L T 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% (3  ) – The uncertainty with a 68% (1  ) confidence level (1/3)(1/32) inch – The uncertainty with a 95% (2  ) confidence level (2/3)(1/32) = 1/48 inch

Beam Thickness T and Width W

Best Estimate Uncertainty Confidence Level 95%-Confidence Level Uncertainty 95%-Confidence Level Fractional Uncertainty Uncertainty Found From Width, W [in] Multiple Measurement Thickness, T [in] Multiple Measurement Length, L [in] Smallest Instrumental Increment Total Length, L T [in] Smallest Instrumental Increment Gage Resistance, R [Ω] Manufacture Specified Value Gage Factor 2.08 (1%) Manufacture Specified Value Mass, m [g] 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. 7 th ed. Wiley 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. AluminumSteel Calculated Density [kg/m 3 ] %-Confidence- Level Interval [kg/m 3 ] 2160 Cited Density* [kg/m 3 ]

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 –  2 = -  3 Adjust R 4 so make V 0I ~ 0 W L T Strain Indicator  R S INPUT ≠ S REAL From Manufacturer, i.e ± 1% R3R3 33  2 = -  3

Procedure Record  R for a range of beam end-masses, m Fit to a straight line  R,Fit = a m + b Slope a = fn(E, T, W, L, S REAL / S INPUT =1) E1E1 E 2 < E 1

Bridge Output = 1 ± 0.01

How to relate με R to m, L, T, W, and E? Neutral Axis σ y m W L T g

Indicated Reading Slope, a

Calculate value and uncertainty of E

Strain Gage Factor Uncertainty

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% (3  ) – The uncertainty with a 68% (1  ) confidence level (1/3)(1/32) inch – The uncertainty with a 95% (2  ) confidence level (2/3)(1/32) = 1/48 inch

Beam Thickness T and Width W

Uncertainty of the Slope, a

Uncertainty of Slope and Intercept “it can be shown”

Plot result and fit to a line  R,Fit = a m + b

Propagation of Uncertainty

Statistical Analysis Shows

General Power Product Uncertainty

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