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MECH215-Chapter05-SolutionstoSampleProblems

# MECH215-Chapter05-SolutionstoSampleProblems - PROBLEM 5.5...

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PROBLEM 5.5 KNOWN: Micrometer Resolution: 0.001 inch (0.025 mm) FIND: u d at 95% confidence ASSUMPTIONS: Instrument error is negligible compared with error due to resolution. SOLUTION The interpolation error due to instrument resolution of an analog instrument is approximated by half its least increment: u o = 0.0005 inch or 0.0125 mm It is not unusual for this type of instrument to have mostly negligible instrument errors. However, its zero set point can only be controlled to within the precision of the resolution, so we set u c ~ u o . Then, the design-stage uncertainty becomes 22 1 / 2 () do c uu u + = ± 0.0007 inch (95%) = ± 0.0180 mm (95%)

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PROBLEM 5.7 KNOWN: Speedometer Resolution: 5 mph (8kph) Accuracy: within ± 4% reading FIND: u d at 60 mph (90 kph) SOLUTION The design-stage uncertainty is 22 1 / 2 () do c uu u + where u o = ± 2.5 mph (4 kph) u c = ± 4% of reading = ± 2.4 mph at 60 mph = ± 3.6 kph at 90 kph This yields 1 / 2 (2.5 2.4 ) d u + = ± 3.5 mph (95%) = ± 5.4 kph (95%) COMMENT In the United States, automobile speedometers and their connected odometers have acceptability tolerance limits set by the government at u = ± 4% of the reading. During the tight fuel availability times in the western hemisphere of the 1970's, some automakers were known to misrepresent (legally) automobile performance by holding a tighter precision (lower random error) on their units while purposely building a 2 to 4% systematic error into their automobile odometers. This made the vehicle slightly more attractive to a fuel cost conscious consumer. Unless the consumer actually made a careful test of the speedometer and odometer performance, usually by some comparison means such as a road sign check, they never would have reason to disbelieve their car's indicated, but inaccurate, performance. Caveat emperor!
PROBLEM 5.8 KNOWN: Temperature sensor Error limit: ± 0.5 o C Readout Device Resolution: 0.1 o C Accuracy: 0.6 o C FIND: u d SOLUTION The design-stage uncertainty is for the combined system is 22 1 / 2 [( ) ( ) ] dd R d S uuu + where (u d ) R is the design-stage uncertainty of the readout device and (u d ) s is that of the sensor. In either case, the individual design-stage uncertainty is found from 1 / 2 () do c uu u + Sensor u o = 0 (i.e. no output stage in the sensor; readout is separate) u c = ± 0.5 o C 1 / 2 (0 0.5 ) d S u =+ = 0.5 o C Readout Device u o = 0.05 o C and u c = 0.6 o C So that 1 / 2 (0.05 0.8 ) d R u = 0.8 o C Then, the design-stage uncertainty for this combined system becomes 1 / 2 (0.5 0.8 ) d u + 0.8 o C (95%) COMMENT Although the error limit on the sensor was given in this problem, similar information is available through various publications – an example of estimating error by using published values. For example, there are readily available ASTM and ASME standards (e.g.,

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MECH215-Chapter05-SolutionstoSampleProblems - PROBLEM 5.5...

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