E1C05 09/14/2010
14:36:27 Page 176
or we can write Equation 5.15 as
u y ¼ Ku E
ð
Þ
2 þ Eu K
ð
Þ
2
h
i 1=2
The operating point occurs at the nominal or the mean values of E ¼ 5:00 V and y ¼ 50:50 mm. With
E ¼ 5:00 V and K ¼ 10:10 mm/V and substituting for u E and u K , evaluate u y at its operating point:
u y
y¼50:5 ¼ 0:10
ð
Þ
2 þ 0:50
ð
Þ
2
h
i 1=2 ¼ 0:51 mm
Alternatively, we can use sequential perturbation. The operating point for the perturbation is again
y ¼ R o ¼ 50:5 mm. Using Equations 5.16 through 5.18 gives
i
x i
R
þ
i
R
À
i
dR
þ
i
dR
À
i
dR i
1
E
50.60
50.40
0.10
À0.10
0.10
2
K
51.00
50.00
0.50
À0.50
0.50
Then, using Equation 5.19,
u y
y¼50:5 ¼ 0:10
ð
Þ
2 þ 0:50
ð
Þ
2
h
i 1=2 ¼ 0:51 mm
The two methods give the identical result. We state the calculated displacement in the form of
Equation 5.11 as
y
0
¼ 50:50 Æ 0:51 mm ð95%Þ
Monte Carlo Method
A Monte Carlo simulation provides another effective way to estimate the propagation of the
uncertainties in the independent variables to the uncertainty in a result. As presented in Chapter 4,
the outcomes of a converged Monte Carlo simulation are the statistics of the predicted population in
a result R (i.e., x; s x , s x ) from which we calculate the random uncertainty in the result.
Generally, convergence is claimed when the computed standard deviation no longer changes by
1% to 5%. As one test for convergence, we define a numerical tolerance D, as being one-half of the
least significant digit of the estimated standard uncertainty (s x ),
D ¼
1
2
LSD
ð5:20Þ
For example, if the estimate of s x is 2 units, then D ¼ 0.5 units (i.e.,
1
2
= of the 10
0 digit); if s x is
0.2 units, then D ¼ 0.05 units. The Monte Carlo simulation is converged when 2s x < D (7).
5.7 ADVANCED-STAGE UNCERTAINTY ANALYSIS
In designing a measurement system, a pertinent question is, How would it affect the result if this
particular aspect of the technique or equipment were changed? In design-stage uncertainty analysis,
we only considered the errors due to a measurement system’s resolution and estimated instrument
176 Chapter 5 Uncertainty Analysis
14:36:27 Page 176
or we can write Equation 5.15 as
u y ¼ Ku E
ð
Þ
2 þ Eu K
ð
Þ
2
h
i 1=2
The operating point occurs at the nominal or the mean values of E ¼ 5:00 V and y ¼ 50:50 mm. With
E ¼ 5:00 V and K ¼ 10:10 mm/V and substituting for u E and u K , evaluate u y at its operating point:
u y
y¼50:5 ¼ 0:10
ð
Þ
2 þ 0:50
ð
Þ
2
h
i 1=2 ¼ 0:51 mm
Alternatively, we can use sequential perturbation. The operating point for the perturbation is again
y ¼ R o ¼ 50:5 mm. Using Equations 5.16 through 5.18 gives
i
x i
R
þ
i
R
À
i
dR
þ
i
dR
À
i
dR i
1
E
50.60
50.40
0.10
À0.10
0.10
2
K
51.00
50.00
0.50
À0.50
0.50
Then, using Equation 5.19,
u y
y¼50:5 ¼ 0:10
ð
Þ
2 þ 0:50
ð
Þ
2
h
i 1=2 ¼ 0:51 mm
The two methods give the identical result. We state the calculated displacement in the form of
Equation 5.11 as
y
0
¼ 50:50 Æ 0:51 mm ð95%Þ
Monte Carlo Method
A Monte Carlo simulation provides another effective way to estimate the propagation of the
uncertainties in the independent variables to the uncertainty in a result. As presented in Chapter 4,
the outcomes of a converged Monte Carlo simulation are the statistics of the predicted population in
a result R (i.e., x; s x , s x ) from which we calculate the random uncertainty in the result.
Generally, convergence is claimed when the computed standard deviation no longer changes by
1% to 5%. As one test for convergence, we define a numerical tolerance D, as being one-half of the
least significant digit of the estimated standard uncertainty (s x ),
D ¼
1
2
LSD
ð5:20Þ
For example, if the estimate of s x is 2 units, then D ¼ 0.5 units (i.e.,
1
2
= of the 10
0 digit); if s x is
0.2 units, then D ¼ 0.05 units. The Monte Carlo simulation is converged when 2s x < D (7).
5.7 ADVANCED-STAGE UNCERTAINTY ANALYSIS
In designing a measurement system, a pertinent question is, How would it affect the result if this
particular aspect of the technique or equipment were changed? In design-stage uncertainty analysis,
we only considered the errors due to a measurement system’s resolution and estimated instrument
176 Chapter 5 Uncertainty Analysis