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Last Rev.: 11 JUN 08

Kline-McClintock Uncertainty : MIME 3470

KLINE-MCCLINTOCK1 METHOD

Page 1

and the uncertainty is calculated as:

(EXPERIMENTAL UNCERTAINTY) ~~~~~~~~~~~~~~

The first thing the student must be aware of is that this procedure does not calculate an error; instead, it states what the possible error —the uncertainty—is in a final result based on experimental measurements or a tolerance in fabrication. So what causes the uncertainty? Basically, it is due to the coarseness of measuring tools. A simple six-inch scale graduated in increments of 0.1 inches can only be read accurately to the nearest 0.1 of an inch. The uncertainty is taken as one half of a graduation or 0.05 inches. An alternate dimensional uncertainty is a tolerance as used in drawing of machine parts. If a dimension is 8mm 0.1mm, then the 0.1mm is the uncertainty in the dimension. The Kline-McClintock Method determines the uncertainty of a calculation given certain measurements and the tolerances on those measurements. To exemplify how this method works, assume one is to determine the volume of a cube containing a cylindrical hole (see the figure below). D

dV V dL

L

2

2

3 2

2

2

L

0.2

2D L 4

2

3 5 2

2

2

2

184.5870 5.5516

2

1/ 2

D

1/ 2

2

67.9314 0.2 23.5619 0.1 2

0.1

1/ 2

2

2 1/ 2

13.7891in . 3

This value is not a percent!

D V L3 2

L2 L1

2

D V L3 L 2 If one measures D and L with measurements

(A.1)

L 5"0.2"

0.2 100 4.000% 5

D 3"0.1"

0.1 100 3.333% 3

what is the anticipated possible error (either positive or negative) when the volume is calculated using Equation A.1? In general, if n measurements xn are being made, each with a measurement tolerance of n, and a function F is calculated using the measured values, then the uncertainty or tolerance in the calculation can be determined as: 2 dF dF 1 2 F dx dx1 2 For the proposed example,

2

L 2

5

125 35.3429 89.6571in3 . Thus, one could have a percent error as large as

To make this a simple explanation of the procedure, assume that L = L1 = L2= L3. The volume, ideally, would then be:

2

2

1/ 2

2

.

2

1 0.2 1 0.1

x2 D 1

2

2

2

3 5 3 2

L

D

1/ 2

2

The ideal volume calculation is

L3

D F V L3 2 x1 L

2

dV dD

D 3L2 2

3 5

2

Kline, S. J., and F. A. McClintock. “Describing Uncertainties in Single-Sample Experiments.” Mechanical Engineering, Vol. 75, No. 1, January 1953: 3-8.

13.7891 100 15.38% 89.6571 even though the errors in individual measurements are less than 4%.

Last Rev.: 11 JUN 08

Kline-McClintock Uncertainty : MIME 3470

Page 2

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Kline-McClintock Uncertainty : MIME 3470

KLINE-MCCLINTOCK1 METHOD

Page 1

and the uncertainty is calculated as:

(EXPERIMENTAL UNCERTAINTY) ~~~~~~~~~~~~~~

The first thing the student must be aware of is that this procedure does not calculate an error; instead, it states what the possible error —the uncertainty—is in a final result based on experimental measurements or a tolerance in fabrication. So what causes the uncertainty? Basically, it is due to the coarseness of measuring tools. A simple six-inch scale graduated in increments of 0.1 inches can only be read accurately to the nearest 0.1 of an inch. The uncertainty is taken as one half of a graduation or 0.05 inches. An alternate dimensional uncertainty is a tolerance as used in drawing of machine parts. If a dimension is 8mm 0.1mm, then the 0.1mm is the uncertainty in the dimension. The Kline-McClintock Method determines the uncertainty of a calculation given certain measurements and the tolerances on those measurements. To exemplify how this method works, assume one is to determine the volume of a cube containing a cylindrical hole (see the figure below). D

dV V dL

L

2

2

3 2

2

2

L

0.2

2D L 4

2

3 5 2

2

2

2

184.5870 5.5516

2

1/ 2

D

1/ 2

2

67.9314 0.2 23.5619 0.1 2

0.1

1/ 2

2

2 1/ 2

13.7891in . 3

This value is not a percent!

D V L3 2

L2 L1

2

D V L3 L 2 If one measures D and L with measurements

(A.1)

L 5"0.2"

0.2 100 4.000% 5

D 3"0.1"

0.1 100 3.333% 3

what is the anticipated possible error (either positive or negative) when the volume is calculated using Equation A.1? In general, if n measurements xn are being made, each with a measurement tolerance of n, and a function F is calculated using the measured values, then the uncertainty or tolerance in the calculation can be determined as: 2 dF dF 1 2 F dx dx1 2 For the proposed example,

2

L 2

5

125 35.3429 89.6571in3 . Thus, one could have a percent error as large as

To make this a simple explanation of the procedure, assume that L = L1 = L2= L3. The volume, ideally, would then be:

2

2

1/ 2

2

.

2

1 0.2 1 0.1

x2 D 1

2

2

2

3 5 3 2

L

D

1/ 2

2

The ideal volume calculation is

L3

D F V L3 2 x1 L

2

dV dD

D 3L2 2

3 5

2

Kline, S. J., and F. A. McClintock. “Describing Uncertainties in Single-Sample Experiments.” Mechanical Engineering, Vol. 75, No. 1, January 1953: 3-8.

13.7891 100 15.38% 89.6571 even though the errors in individual measurements are less than 4%.

Last Rev.: 11 JUN 08

Kline-McClintock Uncertainty : MIME 3470

Page 2