E1C05 09/14/2010
14:36:26 Page 175
5. Evaluate the approximation of the uncertainty contribution from each variable,
dR i ¼
dR
þ
i À dR
À
i
2
% u i u i
ð5:18Þ
Then, the uncertainty in the result is
u R ¼
X L
i¼1
dR i
ð Þ
2
"
# 1=2
ð5:19Þ
Equations 5.15 and 5.19 provide two methods for estimating the propagation of uncertainty to a
result. In most cases, each equation yields nearly the identical result and the choice of method is left
to the user. The method can also be used to estimate just the sensitivity index of Equation 5.14 (2). In
this case, steps 2 and 3 would apply a small deviation value, typically 1% of the nominal value of the
variable, used in place of the actual uncertainty to estimate the derivative (5).
We point out that sometimes either method may calculate unreasonable estimates of u R . When
this happens the cause can be traced to a sensitivity index that changes rapidly with small changes in
the independent variable x i coupled with a large value of the uncertainty u x i . This occurs when the
operating point is close to an minima or maxima inflection in the functional relationship. In these
situations, the engineer should examine the cause and extent of the variation in sensitivity and use a
more accurate approximation for the sensitivity, including using the higher order terms in the Taylor
series of Equation 5.7.
In subsequent sections, we develop methods to estimate the uncertainty values from available
information.
Example 5.3
For a displacement transducer having the calibration curve, y ¼ KE, estimate the uncertainty in
displacement y for E ¼ 5:00 V, if K ¼ 10:10 mm/V with u K ¼ Æ0:10 mm/V and u E ¼ Æ0:01 V at
95% confidence.
KNOWN y ¼ KE
E ¼ 5:00 V
u E ¼ 0:01 V
K ¼ 10:10 mm/V u K ¼ 0:10 mm/V
FIND u y
SOLUTION Based on Equations 5.12 and 5.13, respectively,
y ¼ f E; K
À
Á
and u y ¼ f u E ; u K
ð
Þ
From Equation 5.15, the uncertainty in the displacement at y ¼ KE is
u y ¼ u E u E
ð
Þ
2 þ u K u K
ð
Þ
2
h
i 1=2
where the sensitivity indices are evaluated from Equation 5.14 as
u E ¼
qy
qE
¼ K and u K ¼
qy
qK
¼ E
5.6 Uncertainty Analysis: Error Propagation 175
14:36:26 Page 175
5. Evaluate the approximation of the uncertainty contribution from each variable,
dR i ¼
dR
þ
i À dR
À
i
2
% u i u i
ð5:18Þ
Then, the uncertainty in the result is
u R ¼
X L
i¼1
dR i
ð Þ
2
"
# 1=2
ð5:19Þ
Equations 5.15 and 5.19 provide two methods for estimating the propagation of uncertainty to a
result. In most cases, each equation yields nearly the identical result and the choice of method is left
to the user. The method can also be used to estimate just the sensitivity index of Equation 5.14 (2). In
this case, steps 2 and 3 would apply a small deviation value, typically 1% of the nominal value of the
variable, used in place of the actual uncertainty to estimate the derivative (5).
We point out that sometimes either method may calculate unreasonable estimates of u R . When
this happens the cause can be traced to a sensitivity index that changes rapidly with small changes in
the independent variable x i coupled with a large value of the uncertainty u x i . This occurs when the
operating point is close to an minima or maxima inflection in the functional relationship. In these
situations, the engineer should examine the cause and extent of the variation in sensitivity and use a
more accurate approximation for the sensitivity, including using the higher order terms in the Taylor
series of Equation 5.7.
In subsequent sections, we develop methods to estimate the uncertainty values from available
information.
Example 5.3
For a displacement transducer having the calibration curve, y ¼ KE, estimate the uncertainty in
displacement y for E ¼ 5:00 V, if K ¼ 10:10 mm/V with u K ¼ Æ0:10 mm/V and u E ¼ Æ0:01 V at
95% confidence.
KNOWN y ¼ KE
E ¼ 5:00 V
u E ¼ 0:01 V
K ¼ 10:10 mm/V u K ¼ 0:10 mm/V
FIND u y
SOLUTION Based on Equations 5.12 and 5.13, respectively,
y ¼ f E; K
À
Á
and u y ¼ f u E ; u K
ð
Þ
From Equation 5.15, the uncertainty in the displacement at y ¼ KE is
u y ¼ u E u E
ð
Þ
2 þ u K u K
ð
Þ
2
h
i 1=2
where the sensitivity indices are evaluated from Equation 5.14 as
u E ¼
qy
qE
¼ K and u K ¼
qy
qK
¼ E
5.6 Uncertainty Analysis: Error Propagation 175