ISO 376 Calibration Uncertainty C. Ferrero For instruments classified for interpolation, the calibration uncertainty is the uncertainty associated with using the interpolation equation to calculate a single value of applied force from a measured deflection. It is not the uncertainty associated with the mean deflection (or force) value, as the subsequent use of the device (in ISO 7500-1) demands that each individual value be within certain limits. For instruments classified for specific forces only, the calibration uncertainty is the uncertainty associated with the range of deflections obtained at a specific calibration force, expressed in units of force. This device s uncertainty during its subsequent use will also be subject to further contributions, such as drift of its sensitivity with time, the effect of temperature on its sensitivity, the effect of different end-loading conditions, and the effect of different time-loading profiles. The sensitivity of the device to these two final effects can be calculated from the results of the preliminary checks (e.g. a bearing pad test and a creep test), and the drift of its sensitivity with time can be calculated from its calibration history. Its change of sensitivity with temperature can either be directly measured or the manufacturer s specification figures can be used. DEFLECTION Firstly, I think we need to define deflection if labs can t calculate the same deflection values from the same raw data, there s not much hope for an uncertainty annex! At the moment, ISO 376 (6.4.4) defines deflection as the difference between a reading under force and a reading without force, but doesn t specify which reading without force to use. I can see three possibilities: 1. The zero reading before the run starts 2. The average of the initial and final zero readings 3. An interpolated value, assuming a linear change between the initial and final zero readings throughout the calibration run At NPL, we use option 3, as this is how it is defined in ASTM E 74 - we occasionally issue ISO 376 and ASTM E 74 certificates based on the same raw data, and it would look strange if the two gave different deflection values for what was the same calibration. CALIBRATION UNCERTAINTY At each calibration force, a combined uncertainty value (u c, in force units) is calculated from the readings obtained during the calibration. These combined uncertainties are plotted against force, and a worst case fit covering all values calculated. This equation is used to calculate a combined uncertainty for each force these uncertainties are then multiplied by the coverage factor k = 2 to give an expanded uncertainty value, U.
Calculation of expanded uncertainty, U, and combined uncertainty, u c U = k and u c 3 2 u c = ui + i= 1 B 2 where: u 1 = standard uncertainty associated with calibration force, expressed in units of force u 2 = standard uncertainty associated with reproducibility of calibration results, expressed in units of force u 3 = standard uncertainty associated with resolution of indicator, expressed in units of force B = for interpolation: maximum deviation of deflection (incremental or decremental) from interpolated value, expressed in units of force for specific forces: difference between incremental and decremental deflections, expressed in units of force Note: B is divided by 2 in the equation so that, when the expanded uncertainty is calculated (using k = 2), its effect is of the correct magnitude. The effect of B is added linearly, rather than in quadrature as, at each specific calibration force, it is the result of a systematic difference between the measured deflection and the value obtained from the interpolation equation. This is due to the lack of fit of the equation and/or to the hysteresis of the instrument. For specific forces, it is a systematic error resulting from the worst case scenario of using the incremental deflection for decremental forces but can be ignored in the subsequent use of the instrument if the actual decremental deflection is used for decremental forces and the maximum force applied was the maximum calibration force. Alternatively, the magnitude of B can be halved if the certified deflection is taken as the mean of the incremental and decremental deflections. Calculation of calibration force uncertainty, u 1 u 1 is simply the certified standard uncertainty associated with the forces generated by the calibration machine, expressed in units of force. Calculation of reproducibility uncertainty, u 2 u 2 is the standard deviation associated with the population of incremental deflections obtained during the calibration, expressed in units of force. Calculation of resolution uncertainty, u 3 Each deflection value is calculated from two readings (the reading with an applied force minus the reading at zero force). Because of this, the resolution of the indicator needs to be included twice, either as two rectangular distributions, each with a standard uncertainty of r 2 3, or as a single triangular distribution with a standard uncertainty of r 6, where r is the resolution, expressed in units of force.
Calculation of maximum deflection deviation, B The following table gives the criteria for the calculation of B: Incremental only Incremental and decremental Interpolation B = difference between mean deflection and value calculated from interpolation equation, expressed in units of force B = the greater of: a) difference between mean incremental deflection and value calculated from interpolation equation b) difference between mean decremental deflection and value calculated from interpolation equation expressed in units of force Specific forces B = 0 B (incremental) = 0 and B (decremental) = difference between incremental and decremental deflections, expressed in units of force (if certified deflection is incremental value) Alternatively, B (incremental) = B (decremental) = 0.5 x difference between incremental and decremental deflections, expressed in units of force (if certified deflection is mean of incremental and decremental values)
Example: Force X 1 X 2 X 3, X 4 X 5, X 6 X r Fit Fit Error kn µvv -1 µvv -1 µvv -1 µvv -1 µvv -1 µvv -1 µvv -1 100 220,48 220,49 220,41 220,32 220,40 220,37 0,03 200 440,61 440,62 440,58 440,45 440,55 440,57-0,02 300 660,60 660,59 660,48 660,46 660,51 660,53-0,02 400 880,33 880,32 880,24 880,18 880,25 880,27-0,02 500 1 099,86 1 099,86 1 099,79 1 099,70 1 099,78 1 099,78 0,00 600 1 319,17 1 319,16 1 319,13 1 318,90 1 319,07 1 319,06 0,00 700 1 538,26 1 538,24 1 538,17 1 537,96 1 538,13 1 538,12 0,01 800 1 757,04 1 757,03 1 757,03 1 756,83 1 756,97 1 756,95 0,02 900 1 975,67 1 975,66 1 975,63 1 975,40 1 975,57 1 975,55 0,01 1 000 2 194,05 2 194,03 2 193,94 2 193,72 2 193,90 2 193,93-0,03 900 1 975,53 1 975,31 1 975,47 1 975,55-0,08 800 1 756,94 1 756,72 1 756,87 1 756,95-0,08 700 1 538,07 1 537,88 1 538,04 1 538,12-0,08 600 1 319,05 1 318,90 1 319,03 1 319,06-0,03 500 1 099,82 1 099,68 1 099,79 1 099,78 0,01 400 880,33 880,20 880,31 880,27 0,04 300 660,61 660,48 660,58 660,53 0,05 200 440,67 440,52 440,62 440,57 0,06 100 220,42 220,33 220,41 220,37 0,04 Note 1: For decremental forces, X r is calculated as the incremental value, corrected for the average hysteresis observed in the final two series. Note 2: The fit values come from a second order fit to the incremental deflections, given by the equation: X a = 1,135 10 5 Uncertainty contributions: F 2 + 2,205 32 F 0,044 Force u 1 u 2 u 3 B u c kn % kn µvv -1 kn µvv -1 kn µvv -1 kn kn 100 0,005 0,005 0,080 0,036 0,004 0,002 0,04 0,017 0,045 200 0,005 0,010 0,085 0,039 0,004 0,002 0,06 0,026 0,053 300 0,005 0,015 0,076 0,034 0,004 0,002 0,05 0,024 0,050 400 0,005 0,020 0,075 0,034 0,004 0,002 0,04 0,018 0,049 500 0,005 0,025 0,080 0,036 0,004 0,002 0,01 0,006 0,047 600 0,005 0,030 0,146 0,066 0,004 0,002-0,03-0,015 0,080 700 0,005 0,035 0,154 0,070 0,004 0,002-0,08-0,035 0,096 800 0,005 0,040 0,118 0,054 0,004 0,002-0,08-0,037 0,086 900 0,005 0,045 0,146 0,066 0,004 0,002-0,08-0,036 0,098 1 000 0,005 0,050 0,168 0,077 0,004 0,002-0,03-0,012 0,097
0,14 0,12 0,10 u c / kn 0,08 0,06 0,04 0,02 0,00 0 200 400 600 800 1 000 Force / kn The equation of the line encompassing all values is: u c = 0,000 083 F + 0,038 The expanded calibration uncertainty is therefore given by the equation: U = 0,000 166 F + 0,076 For incremental forces only, the uncertainty is reduced (due to the incremental deflections being closer to the fitted curve than the decremental ones), as shown below: 0,14 0,12 0,10 u c / kn 0,08 0,06 0,04 0,02 0,00 0 200 400 600 800 1 000 Force / kn The expanded calibration uncertainty for incremental forces only is given by: U = 0,000 122 F + 0,078
These uncertainties can be calculated at each calibration force and also, if required, expressed as percentage values: Force U (inc & dec) U (inc only) kn kn % kn % 100 0,093 0,093 0,090 0,090 200 0,109 0,055 0,102 0,051 300 0,126 0,042 0,115 0,038 400 0,142 0,036 0,127 0,032 500 0,159 0,032 0,139 0,028 600 0,176 0,029 0,151 0,025 700 0,192 0,027 0,163 0,023 800 0,209 0,026 0,176 0,022 900 0,225 0,025 0,188 0,021 1 000 0,242 0,024 0,200 0,020 Note: In its actual calibration, the instrument received a Class 00 classification down to 200 kn, and a Class 0,5 classification down to 100 kn, failing to satisfy only the Class 00 repeatability with rotation limit (achieving a value of 0,070 %). The uncertainty values obtained therefore agree well with EA 10/04 which gives maximum uncertainties of 0,06 % for Class 00 and 0,12 % for Class 0,5.
UNCERTAINTY DURING THE TRANSDUCER S SUBSEQUENT USE The uncertainty associated with the force calculated from the deflection obtained from the transducer subsequent to its calibration will comprise of contributions from a number of sources: 1. Calibration uncertainty 2. Drift in sensitivity since calibration 3. Effect of being used at a different temperature 4. Effect of being used with different end-loading conditions 5. Effect of being used with a different time-loading profile 6. Effect of linear approximations to interpolation equation 7. If applicable, effect of replacement indicator It can be assumed that none of these effects are correlated, so their standard uncertainties can be summed in quadrature to calculate a combined standard uncertainty at each force (expressed in units of force), assuming that any known errors have been corrected for. For example, if the temperature sensitivity of the transducer is known, and so is the temperature difference (between calibration and subsequent use), either a correction should be made to the calculated force or the effect should be added to the uncertainty arithmetically, rather than in quadrature. 1. Calibration uncertainty This is simply the value of u c calculated in the previous section. 2. Drift in sensitivity since calibration This component may be estimated from the history of the transducer s sensitivity changes between previous calibrations. The exact uncertainty distribution (and possibly even an estimated error correction) will depend on the individual transducer, but a rectangular distribution with an expanded uncertainty of ± the largest previous change is suggested. If such information is not available, an estimate should be made based on the performance history of similar devices. 3. Temperature effect The temperature effect on zero output can be ignored, as the calculation of deflection makes it insignificant, but the effect of temperature on sensitivity, or span, needs to be allowed for. If the actual temperature sensitivity of the load cell is known, a correction should be made to the calculated force. If, as is more likely to be the case, the only information is the manufacturer s specification, an uncertainty component based on this figure and the difference in temperature between the load cell s calibration and its subsequent use shall be employed. It is recommended that a rectangular distribution be used. 4. End-loading effect The bearing pad test specified in Annex B.2 of ISO 376 gives an indication of the sensitivity of a compression load cell to end-loading effects. Clause 6.1.2 also calls for a preliminary test to ensure that the loading attachments for tension transducers allow axial force application. The results of these tests, together with information as to the conditions in which the transducers will subsequently be used, should enable realistic uncertainty contributions for these effects to be determined.
5. Time-loading profile The load cell calibration method (as defined in ISO 376) and its subsequent use (as defined in ISO 7500-1) specify different time-loading profiles (a wait of 30 s before taking a reading in ISO 376, whereas ISO 7500-1 allows calibration with a slowly increasing force). If the load cell is sensitive to time-loading effects, these different protocols would lead to errors in the calculated force, so an uncertainty contribution is needed. It is proposed that ISO 376 be amended to include a once in a lifetime creep / creep recovery test which will determine the change in deflection between the value obtained immediately after the maximum calibration force is applied / removed and the value obtained 30 s later. The changes will be expressed as a percentage of maximum deflection and must fall within given limits to meet particular classifications. In addition, the values can also be used as inputs to the uncertainty budget for the subsequent use of the device. 6. Effect of approximations to equation Some indicators allow points from the calibration curve to be input, so that the display is in units of force, but carry out linear interpolation between these points, rather than use the actual equation. If this is the case, the effect of this approximation to the curve should be investigated and, if significant, an uncertainty contribution should be included. For the majority of cases, it is expected that this effect will be insignificant. 7. Effect of replacement indicator Both indicators must have been calibrated against electrical standards. Ideally, the difference in the indicator sensitivities across the range shall be calculated and used to make corrections to the forces calculated from the interpolation equation. In this case, the additional uncertainty contributions are those due to the indicator calibrations, including any components due to drift since calibration. In the case where the indicator calibrations simply give an error limit within which they fall, the additional uncertainty contributions will include these two error values and any associated drift components.