1 Quantifying & Propagation of Uncertainty Module 2 Lecture THREE (4-4) 3/14/05
Dec 13, 2015
2
What have you learned so far?
Determine Random Uncertainty in the Measurement of the Measurand Using Single measurement Using ONE sample Using M samples
Determine Overall Random uncertainty caused by Elemental Errors
Determine Total Uncertainty caused by Bias and Random uncertainties
Determine Total Uncertainty caused by more than ONE variable
5
Determination of Total (Systematic and Random ) Uncertainties
21
1 1
22
k
i
m
jjix PBW
Total Systematic and Random Uncertainty Wx (RSS)
Bi’s are the systematic uncertainties caused by k elemental error sources and Pi’s are the random uncertainties caused by m elemental error sources
7
Mathematical Approach for Determining Uncertainties
o It allows us to study the impact of uncertainties caused by MORE THAN ONE INDEPENDENT variable on the TOTAL uncertainty on the DEPENDENT variable
o The mathematical mechanism to do this is “Partial Derivative”.
8
Calculate Total uncertainty
21
1
2
n
ii
i
dxx
RdR
dR is the Total uncertainty in the measurement of R (result-dependent variable) caused by Elemental uncertainties dxi in the variables xi (independent variables)
Using the RSS method
9
Partial Derivative - Notation
The “total” change in the area is represented by the derivative dA as
dWW
AdL
L
AdA
Total Change in
area
Partial Change in area due to d W
Partial Change in area due to d L
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In Terms of Uncertainty in Measurements
dWW
AdL
L
AdA
Total Uncertain
ty
Sensitivity of A with respect to
L
Uncertainty in L
Uncertainty in W
Sensitivity of A with respect to W
11
TOTAL Uncertainty of Dependent Variable in Terms of Random and Bias Uncertainties of Independent Variables
dWW
AdL
L
AdA
Uncertainty in L
Uncertainty in W
21
1 1
22
L Lk
i
m
jLLL PBWdL
21
1 1
22
W Wk
i
m
jWWW PBWdW
12
Determine Total Uncertainty
Using the RSS Formula
?)(30.28
0.4603.050
LdWdLdA2
1
21
22
22
units
W
Assume the dimensions of the rectangular (LxW= 60x50) and uncertainties in the measurements of L and W are + 0.4 mm and + 0.3 mm, respectively.
Meaning?
We are 95% Confident that True Value of Area = 3000.00 + 28.30 m2
Assuming that THERE is No Bias Errors
Other wise We are 95% Confident that
Mean Value of Area Measurements (population) = 3000.00 + 28.30 m2
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How to perform PD?
Transform the Multi-variable function to ONE variable function replace all variables with constants, except
the ONE variable that is differentiated Perform ordinary differentiation Replace back the constants with the
equivalent variables
14
Example: Area A = L x W
11
1 i.e ,constant assume
calculateTo
cLdL
dcL
L
cWW
LWLL
A
W
c
11
WLWLL
A
isThat
15
Example: Area A = L x W
WcdW
dWc
W
cLL
LWWW
A
22
2 i.e ,constant assume
calculateTo
L
c
12
LLWWW
A
isThat
22
13
213
113
13
13
1
A
L
A
cc
Acc
AdA
dcc
dA
d
A
cc
dA
d
A
cc
A
A
L
AA
R
Another Differentiation
Rule?
22
Example: Resistance
Determine the Total uncertainty in measuring the
resistance R, for the nominal values of L = 2m,
A= 1 mm2 , and resistivity = 0.025x10-6 Ω.m. The
uncertainties in the measurement of L, A, and
resistivity are + 0.01m, + 0.1mm2, and +
0.001x10-6 Ω.m, respectively?
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Example: Electrical Resistance
21
222
dAA
RdL
L
Rd
RdR
21
22
22
22
2
dA
A
LdL
Ad
A
LdR
The RSS formula
21
22
22
22
2
dA
A
LdL
Ad
A
LdR
21
6
2
26
62
2
6
626
2
6101.0
101
210025.001.0
101
10025.010001.0
101
2
dR
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Determine Total Fractional Uncertainty Using Fractional Uncertainties of Variables
If the dependent variable R is a product of the measured variables, i.e
Nn
cba xxxCxR ....321Then, the fractional uncertainty in R is directly related to the fractional uncertainty of the variables
21
22
2
2
2
1
1 ....
n
n
x
dxN
x
dxb
x
dxa
R
dR
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The Two Forms are Equivalent
21
22
2
2
2
1
1 ....
n
n
x
dxN
x
dxb
x
dxa
R
dR
Which is equivalent to
21
22
22
2
11
.......dR
nn
dxx
Rdx
x
Rdx
x
R
2
22
22
mm30.28
50
3.0
60
4.0
5060
30,405060For
RSS theUsing
21
21
dA
dA
mm. dW mm. , dL, WL
W
dW
L
dL
A
dA