union college mechanical engineering mer301 -measurement system and uncertainty analysis lecture...

Union CollegeMechanical Engineering

MER301 -Measurement System and Uncertainty Analysis LECTURE 16-17: Measurement System Analysis and

Uncertainty Analysis-Homework

MER301: Engineering ReliabilityLecture 16

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Measurement System Analysis Homework Problem #0

What is the definition of measurement error? What is measurement system accuracy? What is measurement system precision? What is the relationship between observed values of Y, actual values of Y, and

measurement error? What is the relationship between the variation of observed values of Y, actual

values of Y and measurement error? What are the three elements that contribute to measurement system variation? What is the definition of gage R&R? What does it mean? What is the relationship between standard deviation and uncertainty? What does

relative uncertainty mean? What is propagation of errors? How is it used to estimate the variance of Y? What is the dimensionless or relative variance of


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cba xxxY 321



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For a displacement transducer having a calibration curve R = kE2, estimate both the uncertainty and the relative uncertainty in displacement R at E = 5.00v.

From manufacturer’s information sheets, it is known that k = 10 mm/v2 with = 0.10 mm/v2 and that = 0.20v

kuEu




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A voltmeter is to be used to measure the output from a pressure transducer that outputs an electrical signal. The nominal expected pressure read by the transducer w ill be about 3 psi. Estimate the instrument uncertainty and the relative uncertainty based on the follow ing information:

Voltmeter:Resolution: 10 mV Repeatability w ithin 0.001% of reading

Linearity negligibleTransducer:

Range: +/- 5 psiSensitivity: 1V/psiInput power: 10 VDC +/- 1%Output: +/- 5VLinearity: w ithin 2.5 mV/psi over rangeRepeatability w ithin 2.0 mV/psi over rangeResolution negligible


A voltmeter is to be used to measure the output from a pressure transducer that outputs an electrical signal. The nominal expected pressure read by the transducer will be about 3 psi. a) Estimate the instrument standard deviation, uncertainty and the relative uncertainty

based on the following information:

b) These instruments were installed in a complete measurement system and a well controlled short term test was run where a standard reference pressure of 3psi was repeatedly read. The results from that test gave an observed standard deviation of 5.68mv Calculate the repeatability standard deviation

c) Long term operation of the complete measurement system was audited and the resulting standard deviation was found to be 7.56mv. What is the reproducibility contribution in long term use?


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mvSTtmeasuremen 68.5, ityrepeatibil

ilityreproducib



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Uncertainty in Volume of a Cylinder Homework Problem #3

Obtain expressions for the dimensionless variance and the relative uncertainty U in the volume of a cylinder in terms of variation in its radius and height. The volume of a cylinder in terms of radius and height and observed r and h values are given by

a) Estimate the value of V, and find the uncertainty (use as the definition of uncertainty) in the estimate

b) Which variable- r or h- has the greatest effect on the uncertainty in V? c) A recent set of ten volume measurements gave a sample mean and standard

deviation of 0.802 and 0.035. Test whether these values are consistent with the results of part a)? What are your conclusions? Use

025.0,0.1

005.0,5.0

, 2

hh

rr

hrhrVV

05.0

VV /4



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System Comparison Using Uncertainty Analysis- Homework Problem #4

Two methods of measuring the heat transfer q from a computer board are being considered. One uses a convection heat transfer technique (air flow from a fan) and the second uses a conduction heat transfer method (heat sink). The following are observed values in tests

Method 1: where h is the heat transfer coefficient and is the temperature difference between the board and the convective cooling medium

Method 2: where is the thickness of a heat sink which conducts heat away from the board, the temperature difference across the heat sink is and k is the thermal conductivity of the heat sink material

Given , for both methods, h = 100BTU/ft2 hrF

and k is constant , which method provides the least uncertainty in the

observed heat transfer from the board? If is 134 BTU/ft2 hrF, what are the measurement standard deviations and the Gage R&R’s for each method?

Thq 1

XTkq /2

FT 055.5

FT 10

FT 1

X

FhrftBTUkininXhrFBTU xh /15,025.0,1,/10 21 qq

actualq )(



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Uncertainty in the Modified Wobbe Index of a Gas Fuel- Homework Problem #5

The Modifed Wobbe Index for a gas fuel is defined as above. Write the equations for the dimensionless variance, and the standard deviation of MWI in terms of the fuel Lower Heating Value(LHV), Temperature, and Molecular Weight. Assume that the molecular weight of air is a constant.

Measurements of LHV, MWfuel ,and Tfuel for the fuel delivered to a European power plant were made daily over a yearlong period. For each of these quantities the ratio of the resulting standard deviation to the mean is given in the table. The mean MWI for the actual gas was 50 At the same time each measurement of the actual fuel was made, a measurement of a standard reference fuel (MWIref =50)was made and these results are also shown above.

For MWI, calculate both the observed and measurement system dimensionless variances and the standard deviations. What is the actual dimensionless variance and standard deviation of the gas? Which element (LHV, MWgt, Tfuel) is the biggest source of measurement system error? What is the Gage R&R for this measurement system? If the Gage R&R must be reduced to no more than 30%, what is the required (dimensionless) measurement system standard deviation?

fuelair

fuel TMW

MW

LHVMWI

DATA SETS

-----ratio of StDevX /mean value X for----

LHV MWgtf uel Tf uel

observed 0.02 0.02 0.02ref gas 0.00707 0.01 0.01



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Linear Variable(induction) Displacement Transducers (LVDT’s) are often used to measure movement of a part. The output of such transducers is a voltage that is proportional to the position of a armature that is connected to the part and moves within a three segment transformer core. Two of the segments-A and B- are wired such that their voltages have opposite sign and the difference between them is proportional to the position of the armature. The following information is available for a particular model of transducer where M is the instrument “constant” and VA and VB are the segment voltages.

Calculate the instrument uncertainty and sigma. The transducer manufacturer claims the linearity and repeatability are less than 1.0% and 0.5% of full scale, respectively. Resolution is negligible. Are your results consistent with that claim?

)(

)(

BA

BA

VV

VVMD

Mean Uncertainty,u'M 50mm 0.3mmVA 8v 0.01v

VB 2v 0.01v



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Measurement System Analysis Homework Problem #6(con’t)

The actual positions of eighteen parts are given in the table to the right. The population position is intended to be 30mm with a standard deviation of 0.6mm. What is your estimate of the actual population mean position and actual population standard deviation based on these eighteen measurements?

Using these results and those on the previous page, calculate the observed variance and standard deviation that you would expect when using the LVDT to measure the positions of these parts. What is the GRR of these results?

X30.09130.00530.20930.42229.68630.48730.88629.78229.12130.87629.77430.34530.47630.20429.37530.25829.46829.541


Measurement System Analysis Homework Problem #6 (con’t)

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Repeatability and Non-Linearity

union college mechanical engineering mer301 -measurement system and uncertainty analysis lecture...

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