1. ME 322: Instrumentation Lecture 9 February 6, 2015 Professor Miles Greiner Lab 4 and 5, beam in bending, Elastic modulus calculation

2. 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 shanelles@unr.edu

3. Lab 4: Calculate Beam Density W L T LTLT

4. 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

5. Beam Thickness T and Width W

6. Best Estimate Uncertainty Confidence Level 95%-Confidence Level Uncertainty 95%-Confidence Level Fractional Uncertainty Uncertainty Found From Width, W [in] 0.99440.00120.680.0025 Multiple Measurement Thickness, T [in] 0.18620.00070.680.00130.007 Multiple Measurement Length, L [in] 110.03130.9970.02080.0019 Smallest Instrumental Increment Total Length, L T [in] 25.1250.03130.9970.02080.0008 Smallest Instrumental Increment Gage Resistance, R [Ω] 120.20.10.9970.06670.0006 Manufacture Specified Value Gage Factor 2.08 (1%) 0.02 0.680.040.02 Manufacture Specified Value Mass, m [g] 207.40.10.950.10.0005 Smallest Instrumental Increment Table 3 Aluminum Beam Measurements and Uncertainties

7. 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. 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. AluminumSteel Calculated Density [kg/m 3 ] 27207948 95%-Confidence- Level Interval [kg/m 3 ] 2160 Cited Density* [kg/m 3 ] 27027854

8. 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

9. 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. 2.07 ± 1% R3R3 33  2 = -  3

10. 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

11. Bridge Output = 1 ± 0.01

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

13. Indicated Reading Slope, a

14. Calculate value and uncertainty of E

15. Strain Gage Factor Uncertainty

16. 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

17. Beam Thickness T and Width W

18. Uncertainty of the Slope, a

19. Uncertainty of Slope and Intercept “it can be shown”

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

22. Propagation of Uncertainty

23. Statistical Analysis Shows

24. General Power Product Uncertainty

25. 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

26. Will everyone in the class get the same value as