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This cover page must be submitted with the assignment ME 3057 – EXPERIMENTAL METHODOLOGY & TECHNICAL WRITING Report/Worksheet/Writing Task: Lab number 2 Full Report: ????? Partial Report: X (Please check one. ) Lab section: KGrader: _____________ NAMES : Matthew Carson, ????? , ????? , ????? Date Turned In: 02/07/2010Date Returned by TA: ____________________ The effort / participation in this laboratory and lab report is divided as follows: Name: ????? , primarily responsible for sections: ????? Name: ????? , primarily responsible for sections: ????? Name: ????? primarily responsible for sections: ????? Name: ????? , primarily responsible for sections: ????? By submitting this lab report electronically, I/we are agreeing to the following honor pledge, which is consistent with the rules described in the laboratory manual, the syllabus and in class: On my honor, I / we pledge that I / we have neither given nor received inappropriate aide in the preparation of this lab report. The only laboratory reports from prior semesters that I / we have viewed, reviewed, or used in any way were provided by the laboratory TAs during office hours.
I / we have reviewed the consequences of using prior laboratory reports in the laboratory manual. GRADE: / Grader Initials: ________ COMMENTS (grader / students) ????? LAB 2: MEASUREMENT UNCERTAINTY AND ERROR ANALYSIS ABSTRACT Important concepts used by engineers when measuring properties are explored in this lab, such as uncertainty and error. A 15mm gage block and an aluminum pendulum are measured to gain a better understanding of these concepts. In the first section of the lab, the calibrated dimension of the 15mm gage block is measured with 0. 001″ and 0. 0001″ micrometers along with 0. 001″ calipers.
In the second section of the lab, the density of the gage block is found by measuring the length and the thickness of the block to find the volume along with taking the mass of the block on a scale. In the last section of the lab, the period of Pendulum A is measured along with the geometry of the pendulum. The results of the first section of the lab lead to the conclusion that less error occurs in the instruments with higher resolutions. In the next section the results show that when finding a derived value the propagated error is less when taking error based on resolution as opposed to statistical error.
The error based on resolution for the density was 2. 23 x 10-5 compared to the error based on statistics of 0. 00101. PART I: MEASUREMENTS OF THE DIMENSIONS OF A GAGE BLOCK A. Experimental Results Table A. 1 presents measurements of the calibrated dimension of a 15 mm gage block taken with a 0. 001″ resolution caliper, a 0. 001″ resolution micrometer, and a 0. 0001″ resolution micrometer. The manufacturer specified value for the gage block is 15. 00004 mm and the specified uncertainty is 0. 075 mm. B. Analysis The data set of measured values from Table A. 1 is used to determine the mean values and standard deviation for the gage block.
The mean, X, obtained using Equation 1 and the standard deviation, ? , measures the spread of data around the mean and is obtained using Equation 2 X=1Ni=1NXi (Eq. 1) ?=1N-1i=1NXi-X2(Eq. 2) where N is the number of data points and Xi is the measurement at i. Table 1 shows the mean measurements of the calibrated side of the gage block and the associated standard deviation associated with that mean value. The standard deviation is a measure of uncertainty, and the instrument with the smallest uncertainty of can be considered the most precise. Table 1. Mean and Standard Deviation for Gage Block | | 0. 001″ Micrometer (in. )| 0. 01″ Micrometer (mm)| 0. 0001″ Micrometer (in. )| 0. 0001″ Micrometer (mm)| 0. 001″ Caliper (in. )| 0. 001″ Caliper (mm)| | | | | | | | | Mean| 0. 5898| 14. 9809| 0. 5901| 14. 9896| 0. 5902| 14. 9911| Standard Deviation| 0. 0023| 0. 0579| 0. 0026| 0. 0654| 0. 0016| 0. 0419| | | | | | | | | From Table 1, it can be determined that 0. 001″ Micrometer has the least uncertainty and the most precise results. The resolution for an instrument is the smallest increment that can be measured. The uncertainty,U, can be calculated as either half of the resolution or calculated based off a desired confidence interval and obtained from Equation 3
U=K? (Eq. 3) where K is set to 1. 96 to obtain a 95% confidence interval. In Table 2, the uncertainty based on a 95% confidence interval is compared to the uncertainty based on the resolution of the instruments used to find the measurements of the gage block. Table 2. Statistical and Resolution based Uncertainty | | 0. 001″ Micrometer (in. )| 0. 001″ Micrometer (mm)| 0. 0001″ Micrometer (in. )| 0. 0001″ Micrometer (mm)| 0. 001″ Caliper (in. )| 0. 001″ Caliper (mm)| | | | | | | | | Resolution Uncertainty| 0. 0005| 0. 0127| 0. 00005| 0. 0013| 0. 000125| 0. 0032| Statistical Uncertainty| 0. 0045| 0. 1135| 0. 0050| 0. 1282| 0. 0032| 0. 821| | | | | | | | | | | | | | | | PART II: PROPAGATION OF ERRORS A. Experimental Results The measurements for the non-calibrated dimension of the gage block are recorded in Table A. 2 along with the mass of the gage block. The measurements were taken with the 0. 001″ Calipers and the mass was found with the scale. The instruments have a 0. 00025″ and a 0. 1 g resolution respectively. B. Analysis The density for the gage block will be computed using the measured dimensions of the block along with its mass. The mean and standard deviation of the non-calibrated dimensions of the gage block are listed in Table 3 along with the mass.
The volume of the gage block can be found with the non-calibrated and calibrated dimensions are used. Table 3. The Mean and Standard Deviation of the Non-Calibrated Sides of the Gage Block | thickness (in. )| thickness (mm)| length (in. )| length (mm)| mass (g)| Mean| 0. 351| 8. 926| 1. 378| 35. 001| 36. 287| Standard Deviation| 0. 023| 0. 573| 0. 018| 0. 453| 0. 063| The thickness and length are found from the non-calibrated dimension of the gage block and the calibrated dimension gives the width of the gage block. Using the length, thickness, width and mass the density can be calculated using Equation 4. ?=massl? w? t(Eq. 4) where ? s density, l is length, w is width, and t is thickness of the gage block. The density is a derived quantity from the measured dimensions and mass of the block, which means the uncertainty associated with each measurement will influence the uncertainty associated with the density. Equation 5 shows the propagated uncertainty for the density and its derivation is shown in Equation 6 and 7 ?? =i=1n? f? xi? xi2(Eq. 5) where ?? is the uncertainty associated with the density. Equation 6 shows the uncertainty associated with ? the instrument resolution. ?? =??? l? l2+??? w? w2+??? t? t2+??? m? m2(Eq. 6) ?? =-ml2wt? l2+-mlw2t? w2+-mlwt2? t2+1lwt? 2 Equation 7 shows the uncertainty using 2? to find the uncertainty associated with a 95% confidence interval. ?? =??? l2? l2+??? w2? w2+??? t2? t2+??? m2? m2(Eq. 7) ?? =-ml2wt2? l2+-mlw2t2? w2+-mlwt22? t2+1lwt2? m2 Table 4 presents the density calculated using Equation 4 and the resolution and the statistical uncertainties using Equations 6 and 7. Table 4. Density and Propagated Uncertainty of Gage Block Density (g/mm3)| Resolution Uncertainty| Statistical Uncertainty| 0. 007747375| 2. 23374E-05| 0. 001015741| The large difference between the statistical uncertainty and the resolution uncertainty could be caused by operator error.
Outliers in the measurement data set could explain why the resolution uncertainty is much smaller than statistical uncertainty. PART II: MEASUREMENTS OF THE PERIOD OF A PENDULUM A. Experimental Results In this experiment, Pendulum A was chosen to test the freely oscillating period. Pendulum A is a T6061 aluminum block with a hole. T6061 aluminum has a density of 0. 0027g/mm3. 1 The geometry of the pendulum was measured using a ruler with a resolution of 1 mm and recorded in Table 5. Table 5. Measured Dimensions for Pendulum A Dimension| Value (mm)| Diameter (d)| 22| Height (h)| 130| Length (l)| 300|
Width (w)| 50| Thickness (t)| 9| D w h dh l Figure 1 presents the geometry of Pendulum A. The labeled dimensions were found by measurement with a ruler and presented in Table 5. Figure 1. Aluminum Pendulum A Table A. 3 presents the period of the pendulum measured with an optical sensor and measured with a stop watch. The time for the pendulum to complete one cycle, three cycles, and twenty-five cycles was found using the stop watch. B. Analysis The period of a freely oscillating pendulum is measured and then the theoretical period for the pendulum is calculated using Equation 8 T=2?? n(Eq. ) where T is the period of the pendulum and ? n is the natural frequency. The natural frequency can be calculated using Equation 9 ? n=mgh’I(Eq. 9) where h’ is distance between the pivot point and the mass center of the pendulum and I is the moment of inertia of the pendulum about the pivot point. Equation 10 and 11 are used to calculate h’ and I for the pendulum h’=d2+h+dh (Eq. 10) I=12mbl2+w2+mbh+d22-38mhd2 (Eq. 10) where mb is the mass of the pendulum without the hole and mh is the mass of the hole. mb and mh are found using density and volume in Equations 11 and 12. mb=? lwt(Eq. 11) h=?? d24t(Eq. 12) Using Equation 8 the theoretical period is found 0. 886 s. The mean and the 95 % confidence interval of the period for the optical sensor and the stopwatch for one cycle, three cycles, and twenty-five cycles are displayed in Table 6. Table 6. Mean and 95% Confidence Interval of the period | Mean (s)| 95% Confidence Interval (s)| Optical Sensor| 0. 8911| 0. 0040| 1 Cycle| 0. 81| 0. 22| 3 Cycle| 0. 87| 0. 22| 25 Cycle| 0. 90| 0. 13| The mean periods and associated errors found in Table 6 are compared in Figure 2. Figure 2. Theoretical Period compared to the Experimental Period
Table A. 4 records the absolute error and the relative error for the data obtained using the stopwatch. The mean period for each set of values is assumed to be the “true” period. Equations 13 and 14 are used to determine the absolute error and the relative error. Absolute error=measured value-“true “value(Eq. 13) Relative error=Absolute error”true “value(Eq. 14) If only twenty-four cycles have been measured for the calculation of the period which uses twenty-five cycles, then an error of -0. 812 is introduced into the calculation. This error can be calculated by Equation 15.
Error Introduced=measured value? 24-theory value? 25(Eq. 15) Figure 3 shows the running average of “1 Cycle” data vs. the number of data points comparing that data in the order it was taken to the same data plotted in random order. The figure shows shows a saturation of the data between 0. 8 and 0. 9. Figure 3. Comparison of Data Points to the Running Mean BIBLIOGRAPHY MatWeb Material Property Data. “Data sheets for over 70,000 metals, plastics, ceramics, and composites. ” 2000. 02/06/10 <http://www. matweb. com/search/DataSheet. aspx? MatGUID=0cd1edf33ac145ee93a0aa6fc666c0e0> Appendix Table A. . Measured Width of Gage Block (Calibrated Dimension) 0. 001″ Micrometer (in. )| 0. 001″ Micrometer (mm)| 0. 0001″ Micrometer (in. )| 0. 0001″ Micrometer (mm)| 0. 001″ Caliper (in. )| 0. 001″ Caliper (mm)| | | | | | | 0. 591| 15. 0114| 0. 5972| 15. 16888| 0. 592| 15. 0368| 0. 591| 15. 0114| 0. 5877| 14. 92758| 0. 588| 14. 9352| 0. 590| 14. 986| 0. 5904| 14. 99616| 0. 590| 14. 986| 0. 584| 14. 8336| 0. 5907| 15. 00378| 0. 585| 14. 859| 0. 589| 14. 9606| 0. 5884| 14. 94536| 0. 595| 15. 113| 0. 590| 14. 986| 0. 5901| 14. 98854| 0. 592| 15. 0368| 0. 590| 14. 986| 0. 5892| 14. 96568| 0. 590| 14. 86| 0. 590| 14. 986| 0. 5906| 15. 00124| 0. 589| 14. 9606| 0. 590| 14. 986| 0. 5906| 15. 00124| 0. 590| 14. 986| 0. 592| 15. 0368| 0. 5934| 15. 07236| 0. 589| 14. 9606| 0. 590| 14. 986| 0. 5898| 14. 98092| 0. 590| 14. 986| 0. 589| 14. 9606| 0. 5895| 14. 9733| 0. 590| 14. 986| 0. 590| 14. 986| 0. 5899| 14. 98346| 0. 593| 15. 0622| 0. 590| 14. 986| 0. 5907| 15. 00378| 0. 592| 15. 0368| 0. 586| 14. 8844| 0. 5834| 14. 81836| 0. 591| 15. 0114| 0. 590| 14. 986| 0. 5923| 15. 04442| 0. 590| 14. 986| 0. 595| 15. 113| 0. 5901| 14. 98854| 0. 590| 14. 986| 0. 590| 14. 986| 0. 5905| 14. 9987| 0. 590| 14. 86| 0. 585| 14. 859| 0. 5852| 14. 86408| 0. 590| 14. 986| 0. 590| 14. 986| 0. 5902| 14. 99108| 0. 590| 14. 986| 0. 590| 14. 986| 0. 5901| 14. 98854| 0. 590| 14. 986| 0. 590| 14. 986| 0. 5898| 14. 98092| 0. 591| 15. 0114| 0. 594| 15. 0876| 0. 5950| 15. 113| 0. 590| 14. 986| 0. 588| 14. 9352| 0. 5859| 14. 88186| 0. 590| 14. 986| 0. 594| 15. 0876| 0. 5904| 14. 99616| 0. 590| 14. 986| 0. 590| 14. 986| 0. 5921| 15. 03934| 0. 590| 14. 986| 0. 589| 14. 9606| 0. 5903| 14. 99362| 0. 590| 14. 986| 0. 588| 14. 9352| 0. 5903| 14. 99362| 0. 589| 14. 9606| 0. 590| 14. 986| 0. 5902| 14. 99108| 0. 590| 14. 986| . 589| 14. 9606| 0. 5903| 14. 99362| 0. 590| 14. 986| Table A. 2. Measured Values of Length, Thickness, and Mass thickness (in. )| thickness (mm)| length (in. )| length (mm)| width (mm)| mass (g)| 0. 350| 8. 89| 1. 382| 35. 1028| 15. 0368| 36. 3| 0. 350| 8. 89| 1. 381| 35. 0774| 14. 9352| 36. 3| 0. 351| 8. 9154| 1. 381| 35. 0774| 14. 986| 36. 3| 0. 356| 9. 0424| 1. 381| 35. 0774| 14. 859| 36. 2| 0. 360| 9. 144| 1. 380| 35. 052| 15. 113| 36. 2| 0. 350| 8. 89| 1. 382| 35. 1028| 15. 0368| 36. 3| 0. 350| 8. 89| 1. 380| 35. 052| 14. 986| 36. 2| 0. 351| 8. 9154| 1. 381| 35. 0774| 14. 9606| 36. 3| 0. 53| 8. 9662| 1. 381| 35. 0774| 14. 986| 36. 3| 0. 351| 8. 9154| 1. 381| 35. 0774| 14. 9606| 36. 3| 0. 350| 8. 89| 1. 382| 35. 1028| 14. 986| 36. 3| 0. 351| 8. 9154| 1. 381| 35. 0774| 14. 986| 36. 3| 0. 350| 8. 89| 1. 382| 35. 1028| 15. 0622| 36. 3| 0. 351| 8. 9154| 1. 381| 35. 0774| 15. 0368| 36. 3| 0. 400| 10. 16| 1. 381| 35. 0774| 15. 0114| 36. 2| 0. 350| 8. 89| 1. 381| 35. 0774| 14. 986| 36. 3| 0. 351| 8. 9154| 1. 380| 35. 052| 14. 986| 36. 2| 0. 350| 8. 89| 1. 382| 35. 1028| 14. 986| 36. 2| 0. 252| 6. 4008| 1. 284| 32. 6136| 14. 986| 36. 2| 0. 355| 9. 017| 1. 381| 35. 0774| 14. 986| 36. 4| 0. 53| 8. 9662| 1. 381| 35. 0774| 14. 986| 36. 4| 0. 351| 8. 9154| 1. 383| 35. 1282| 15. 0114| 36. 3| 0. 350| 8. 89| 1. 384| 35. 1536| 14. 986| 36. 2| 0. 400| 10. 16| 1. 380| 35. 052| 14. 986| 36. 3| 0. 350| 8. 89| 1. 383| 35. 1282| 14. 986| 36. 4| 0. 350| 8. 89| 1. 375| 34. 925| 14. 986| 36. 4| 0. 350| 8. 89| 1. 380| 35. 052| 14. 986| 36. 3| 0. 353| 8. 9662| 1. 383| 35. 1282| 14. 9606| 36. 3| 0. 354| 8. 9916| 1. 382| 35. 1028| 14. 986| 36. 3| 0. 350| 8. 89| 1. 384| 35. 1536| 14. 986| 36. 3| Table A. 3. Time and Period Measured from Optical Sensor and Stopwatch Total Measured Time (s)| Period (s)| cycle| 3 cycles| 25 cycles| Oscilloscope| 1 cycle| 3 cycles| 25 cycles| 0. 88| 2. 66| 22. 28| 0. 8920| 0. 88| 0. 89| 0. 89| 0. 87| 2. 66| 22. 35| 0. 8924| 0. 87| 0. 89| 0. 89| 0. 54| 2. 25| 23. 09| 0. 8930| 0. 54| 0. 75| 0. 92| 0. 91| 2. 69| 22. 22| 0. 8900| 0. 91| 0. 90| 0. 89| 0. 91| 2. 78| 22. 63| 0. 8910| 0. 91| 0. 93| 0. 91| 0. 91| 2. 62| 22. 31| 0. 8920| 0. 91| 0. 87| 0. 89| 0. 87| 2. 66| 22. 31| 0. 8920| 0. 87| 0. 89| 0. 89| 0. 9| 2. 06| 21. 62| 0. 8930| 0. 90| 0. 69| 0. 86| 0. 88| 2. 69| 22. 31| 0. 8900| 0. 88| 0. 90| 0. 89| 0. 75| 2. 53| 22. 12| 0. 8900| 0. 75| 0. 84| 0. 88| 0. 72| 2. 75| 22| 0. 900| 0. 72| 0. 92| 0. 88| 0. 81| 2. 85| 22. 41| 0. 8920| 0. 81| 0. 95| 0. 90| 0. 89| 3. 03| 28. 78| 0. 8910| 0. 89| 1. 01| 1. 15| 0. 6| 2. 41| 22. 1| 0. 8960| 0. 60| 0. 80| 0. 88| 0. 9| 2. 66| 21. 38| 0. 8902| 0. 90| 0. 89| 0. 86| 0. 69| 2. 5| 22. 1| 0. 8881| 0. 69| 0. 83| 0. 88| 0. 81| 2. 66| 22. 16| 0. 8881| 0. 81| 0. 89| 0. 89| 0. 81| 2. 66| 22. 16| 0. 8881| 0. 81| 0. 89| 0. 89| Table A. 4. Absolute Error and Regular Error Using a Stopwatch 1 cycle| Measured Value| Absolute Error| Relative Error| 0. 88| 0. 07| 8. 12%| 0. 87| 0. 06| 6. 89%| 0. 54| -0. 27| -33. 65%| 0. 91| 0. 10| 11. 81%| 0. 91| 0. 0| 11. 81%| 0. 91| 0. 10| 11. 81%| 0. 87| 0. 06| 6. 89%| 0. 90| 0. 09| 10. 58%| 0. 88| 0. 07| 8. 12%| 0. 75| -0. 06| -7. 85%| 0. 72| -0. 09| -11. 54%| 0. 81| 0. 00| -0. 48%| 0. 89| 0. 08| 9. 35%| 0. 60| -0. 21| -26. 28%| 0. 90| 0. 09| 10. 58%| 0. 69| -0. 12| -15. 22%| 0. 81| 0. 00| -0. 48%| 0. 81| 0. 00| -0. 48%| 3 cycle| 25 cycle| Measured Value| Absolute Error| Relative Error| Measured Value| Absolute Error| Relative Error| 0. 89| 0. 01| 1. 61%| 0. 89| -0. 01| -1. 30%| 0. 89| 0. 01| 1. 61%| 0. 89| -0. 01| -0. 99%| 0. 75| -0. 12| -14. 05%| 0. 92| 0. 02| 2. 29%| 0. 90| 0. 02| 2. 76%| 0. 89| -0. 01| -1. 7%| 0. 93| 0. 05| 6. 20%| 0. 91| 0. 00| 0. 25%| 0. 87| 0. 00| 0. 08%| 0. 89| -0. 01| -1. 17%| 0. 89| 0. 01| 1. 61%| 0. 89| -0. 01| -1. 17%| 0. 69| -0. 19| -21. 31%| 0. 86| -0. 04| -4. 23%| 0. 90| 0. 02| 2. 76%| 0. 89| -0. 01| -1. 17%| 0. 84| -0. 03| -3. 35%| 0. 88| -0. 02| -2. 01%| 0. 92| 0. 04| 5. 05%| 0. 88| -0. 02| -2. 54%| 0. 95| 0. 08| 8. 87%| 0. 90| -0. 01| -0. 73%| 1. 01| 0. 14| 15. 75%| 1. 15| 0. 25| 27. 49%| 0. 80| -0. 07| -7. 94%| 0. 88| -0. 02| -2. 10%| 0. 89| 0. 01| 1. 61%| 0. 86| -0. 05| -5. 29%| 0. 83| -0. 04| -4. 50%| 0. 88| -0. 02| -2. 10%| 0. 89| 0. 01| 1. 61%| 0. 89| -0. 02| -1. 83%|