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MER301 -Measurement System
and Uncertainty Analysis
LECTURE 16-17:
Measurement System Analysis and
Uncertainty Analysis-Homework
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
1
Measurement System Analysis
Homework Problem #0
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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 Y  x a  x b  x c
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Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
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Measurement System Analysis
Homework Problem #1
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 u k = 0.10 mm/v2 and that
u E = 0.20v
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
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Measurement
System
Analysis
Measurement
System
Analysis
Homework
Problem
Homework
Problem#2
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A voltmeter Aisvoltmeter
to be used
the output
fromfrom
a pressure
transducer
is toto
bemeasure
used to measure
the output
a pressure
transducer that
that
outputs an electrical
signal.
The
nominal
expected
pressure
read
by
the
will
outputs an electrical signal. The nominal expected pressure read by
thetransducer
transducer will
be about 3 psi.
be about 3 psi. Estimate the instrument uncertainty and the relative uncertainty based
on the the
following
information:
a) Estimate
instrument
standard deviation, uncertainty and the relative
uncertainty based on the following information:
Voltmeter:
Resolution:
Repeatability
Linearity
Transducer:
Range:
Sensitivity:
Input power:
Output:
Linearity:
Repeatability
Resolution
10 mV
within 0.001% of reading
negligible
1V/psi
+/- 5 psi
10 VDC +/- 1%
+/- 5V
within 2.5 mV/psi over range
within 2.0 mV/psi over range
negligible
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
measurement , ST  5.68mv
deviation repeatibility
c) Long term operation of the complete measurement system was audited and the
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resulting
standard deviation wasMER301:
found
to be
Mechanical Engineering
Lecture
16 7.56mv. What is the reproducibility
contribution
reproducibility in long term use?
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Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
4
Uncertainty in Volume of a Cylinder
Homework Problem #3
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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
V  V r , h   r 2 h
 r  0.5,  r  0.005
 h  1.0,  h  0.025
a) Estimate the value of V, and find the uncertainty (use 4   / V as the
V
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
  0.05
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
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System Comparison Using Uncertainty
Analysis- Homework Problem #4
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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: q  hT where h is the heat transfer coefficient and T  10F
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is the temperature
difference between the board and the convective
cooling medium
Method 2: q2  k  T / X where  X is the thickness of a heat sink
which conducts heat away from the board, the temperature difference
across the heat sink is T  5.550 F and k is the thermal conductivity
of the heat sink material
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Given
1
2 ,  T  1F for both methods, h = 100BTU/ft hrF
q q
 h  10BTU / hrF , X  1in,  x  0.025in, k  15BTU / ft  hr  F
and k is constant , which method provides the least uncertainty in
the observed heat transfer from the board? If ( q ) actual is 134 BTU/ft2 hrF,
what are the measurement standard deviations and the Gage R&R’s
for each method?
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
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Uncertainty in the Modified Wobbe Index of a
Gas Fuel- Homework Problem #5
MWI 
MW fuel
MWair
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DATA SETS
-----ratio of StDevX /mean value X for----
LHV
 T fuel
observed
ref gas
LHV
0.02
0.00707
MWgt f uel
0.02
0.01
Tf uel
0.02
0.01
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?
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Mechanical Engineering
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Lecture 16
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Measurement System Analysis
Homework Problem #6
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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 Bare 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.
(V  VB )
DM A
(V A  VB )
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M
VA
Mean
50mm
8v
Uncertainty,u'
0.3mm
0.01v
VB
2v
0.01v
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?
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
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Measurement System Analysis
Homework Problem #6(con’t)
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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?
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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?
Union College
Mechanical Engineering
MER301: Engineering Reliability
Lecture 16
X
30.091
30.005
30.209
30.422
29.686
30.487
30.886
29.782
29.121
30.876
29.774
30.345
30.476
30.204
29.375
30.258
29.468
29.541
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Measurement System Analysis Homework Problem #6 (con’t)
Repeatability and Non-Linearity
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