Torque Wrench Calibration Presentation
Content
Calibration of a Torque Wrench as per ISO6789
by Eddie Tarnow
NLA Test & Measurement Workshop 20 September 2011
© NMISA 2011
Calibration Setup 350,0 N•m
Clockwise Rotation Unit Under Test Torque Wrench
Reference Standard Torque Transducer & Readout Unit
Force Applied
Top View of Setup
© NMISA 2011
Calibration Scenario
• The Unit Under Test Torque Wrench is a Type II class A tool (adjustable click type) and has a full scale of 350 N•m. • It has a setting dial resolution of 2 N•m • We are to calibrate it according to ISO 6789 which requires a calibration point at full scale (100 % of range) viz. at 350 N•m and the estimation of the measurement uncertainty at this point
© NMISA 2011
GUM Steps
Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result • • • •
© NMISA 2011
Measurement Model
TUUT = TSTD Ind + CorrSTD
© NMISA 2011
Identifying the sources of uncertainty
TSTD
SCal SRes SCF
TUUT
URes
URep
TUUT
© NMISA 2011
Quantifying the sources of uncertainty
• SCAL • Calibration Results Table from the Calibration Certificate
APPLIED TORQUE (N•m) 0,0 99,9 199,9 299,8 399,8 499,7 599,7 699,6 MEAN UUT CALCULATED TORQUE (N•m) 0,0 100,0 200,2 300,5 400,8 501,1 601,5 701,8 UNCERTAINTY OF MEASUREMENT (± N•m) 0,1 0,3 1,0 1,0 1,0 1,0 1,0 1,0
© NMISA 2011
Quantifying the sources of uncertainty (2)
• SCAL
• This is the uncertainty due to the accuracy of the Reference Standard Torque Transducer, which is not perfect • Corrections must first be applied, or the uncertainty increased, to take the error into account (largest error on values either side of the calibration point was +1,0 N•m) • The Reference Standard Torque Transducer used has a full scale of 700 N•m and was calibrated in 100 N•m steps (See calibration certificate) • Therefore we will have to use the polynomial equation to determine the actual torque generated by the UUT at 350 N•m since it is a measurement point in between 300 N•m and 400 N•m. • Since we have to interpolate a value we will use the largest reported uncertainty from the calibration certificate for the values on either side of the calibration point which is ± 1 N•m.
© NMISA 2011
Quantifying the sources of uncertainty (3)
• SCAL • Since we will be using the polynomial to interpolate a value at 350 N•m, we DO NOT need to correct for the + 1 N•m error at 399,8 N•m. • Therefore total uncertainty for the “accuracy” of the Reference Standard Torque Transducer is ±1 N•m • This is treated as normal at 95,45% Level of Confidence • The divisor is the coverage factor k which for 95,45% LOC is 2 • The degrees of freedom are always ∞ or 100 % Reliable due to the source of traceability being accredited and reputable.
© NMISA 2011
Quantifying the sources of uncertainty (4)
• SRES • This is due to the resolution of the Reference Standard Torque Transducer Readout Unit • We must first determine the “effective resolution” • The least significant digit displayed is 0,1 N•m • Resolution is always treated as a Rectangular distribution source of uncertainty and this is the range. • The semi-range is therefore (0,1 N•m/2)=0,05 N•m • The divisor is √3 • The degrees of freedom are always ∞ or 100 % Reliability
© NMISA 2011
Quantifying the sources of uncertainty (5)
• SCF • Polynomial Equation Coefficients Table from the Calibration Certificate
POLYNOMIAL EQUATION POLYNOMIAL COEFFICIENTS a b c d Standard Error of the polynomial curve fit for a Level of Confidence of 68,27% and 4 degrees of freedom NORMAL FUNCTION 2,71846 x10 9,99825 x10 -7,89039 x10 5,16535 x10
-2 -1 -6 -9
y=a+bx+cx +dx
2
3
INVERSE FUNCTION -2,70765 x10 1,00017 7,95708 x10 -5,22137 x10
-6 -9 -2
± 0,045 N•m
± 0,045 N•m
© NMISA 2011
Quantifying the sources of uncertainty (6)
• SCF • This is the additional uncertainty which results from the interpolation calculation to determine the torque generated by the UUT at a point in between the calibration points of the Reference Standard Torque Transducer • It is obtained directly from the calibration certificate as the “Standard Error of the polynomial curve Fit” value = ± 0,045 N•m • This is treated as a normal distribution at a 68,27% Level of Confidence • The divisor is the coverage factor k which for 68,27% LOC is 1 • The degrees of freedom are also obtained directly from the calibration certificate = 4
© NMISA 2011
Quantifying the sources of uncertainty (7)
• URES • This is due to the resolution of the UUT Torque Wrench scale. (How it influences the setting of the wrench to a specified torque) • Typically this would be the smallest graduation on the UUT setting dial which for this UUT is 2 N•m • This is the range of the rectangular distribution • Therefore the semi-range is (2 N•m/2)=1 N•m • The divisor for Rectangular Distributed uncertainty contributors is √3 • The degrees of freedom for resolution is always ∞ or 100 % Reliability
© NMISA 2011
Quantifying the sources of uncertainty (8)
• UREP • This results from the variability in the measurement results obtained after repeating the measurement 5 times. • It can be dealt with either as “repeatability” or as “reproducibility” • “Repeatability” – all conditions remained the same during the repeated measurements • “Reproducibility” – any one or more of the conditions changed during the repeated measurements • Different approaches can be used to repeat the measurement • Take 5 measurements at one setting sequentially • Take 5 sets of measurements from zero to full scale
© NMISA 2011
Quantifying the sources of uncertainty (9)
Repeatability
20 % Meas 1 20 % Meas 2 20 % Meas 3 20 % Meas 4 20 % Meas 5 20 % Mean
60 % Meas 1
60 % Meas 2
60 % Meas 3
60 % Meas 4
60 % Meas 5
60 % Mean
100 % Meas 1
100 % Meas 2
100 % Meas 3
100 % Meas 4
100 % Meas 5
100 % Mean
© NMISA 2011
Quantifying the sources of uncertainty (9)
Reproducibility
20 % Meas 1 20 % Meas 2 20 % Meas 3 20 % Meas 4 20 % Meas 5 20 % Mean
60 % Meas 1
60 % Meas 2
60 % Meas 3
60 % Meas 4
60 % Meas 5
60 % Mean
100 % Meas 1
100 % Meas 2
100 % Meas 3
100 % Meas 4
100 % Meas 5
100 % Mean
© NMISA 2011
Quantifying the sources of uncertainty (9)
• UREP • Treating as “Repeatability” (as per ISO 6789) • We use the ESDM • ESDM = ESD/SQRT (n) = 0,98/sqrt (5) = 0,436348 N•m • Treating as “Reproducibility” (preferred option but contrary to ISO 6789) • We use the ESD • ESD = 0,98 N•m
© NMISA 2011
GUM Steps
Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result • • • •
© NMISA 2011
Categorize as type A or type B
• • • • • SCAL - type B, not statistically determined SRES - type B, not statistically determined SCF - type A, statistically determined (standard deviation) URES - type B, not statistically determined UREP - type A, statistically determined (standard deviation)
© NMISA 2011
GUM Steps
• • • • Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result
© NMISA 2011
Uncertainty Budget
Torque Wrench Calibration Uncertainty Budget.xls
© NMISA 2011
GUM Steps
Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result • • • •
© NMISA 2011
Reporting the result
• The final result is calculated using the “Normal Function” polynomial coefficients • This is because we want to know the true torque applied to the Reference Standard Torque Transducer when it reads the mean measured value of 350,7 N•m • The calculated interpolated value was 349,938089 N•m • The calculated measurement uncertainty was ± 1,794160789 N•m • Rounding the uncertainty to two significant digits gives ± 1,8 N•m • Rounding the interpolated value to the same number of digits gives 349,9 N•m • The measurement result is then reported as: 349,9 N•m ± 1,8 N•m at a Level of Confidence of 95,45%
© NMISA 2011
Graphical Representation of results
© NMISA 2011
Conclusions
• Both methods in this case prove that the UUT is well within the allowable ± 4% of Maximum (± 14 N•m) • Using the ESDM (In accordance with ISO 6789) results in the smallest uncertainty (unrealistic??) • Using the ESD (contrary to ISO 6789) results in the largest uncertainty (realistic??) • Always use the polynomial for calibrations using the laboratory Reference Standard Torque Transducer • This will correct for any error on the Reference Standard eliminating the need to apply corrections • This will solve the problem of the “Applied Torque” not being exactly at the nominal values
© NMISA 2011
by Eddie Tarnow
NLA Test & Measurement Workshop 20 September 2011
© NMISA 2011
Calibration Setup 350,0 N•m
Clockwise Rotation Unit Under Test Torque Wrench
Reference Standard Torque Transducer & Readout Unit
Force Applied
Top View of Setup
© NMISA 2011
Calibration Scenario
• The Unit Under Test Torque Wrench is a Type II class A tool (adjustable click type) and has a full scale of 350 N•m. • It has a setting dial resolution of 2 N•m • We are to calibrate it according to ISO 6789 which requires a calibration point at full scale (100 % of range) viz. at 350 N•m and the estimation of the measurement uncertainty at this point
© NMISA 2011
GUM Steps
Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result • • • •
© NMISA 2011
Measurement Model
TUUT = TSTD Ind + CorrSTD
© NMISA 2011
Identifying the sources of uncertainty
TSTD
SCal SRes SCF
TUUT
URes
URep
TUUT
© NMISA 2011
Quantifying the sources of uncertainty
• SCAL • Calibration Results Table from the Calibration Certificate
APPLIED TORQUE (N•m) 0,0 99,9 199,9 299,8 399,8 499,7 599,7 699,6 MEAN UUT CALCULATED TORQUE (N•m) 0,0 100,0 200,2 300,5 400,8 501,1 601,5 701,8 UNCERTAINTY OF MEASUREMENT (± N•m) 0,1 0,3 1,0 1,0 1,0 1,0 1,0 1,0
© NMISA 2011
Quantifying the sources of uncertainty (2)
• SCAL
• This is the uncertainty due to the accuracy of the Reference Standard Torque Transducer, which is not perfect • Corrections must first be applied, or the uncertainty increased, to take the error into account (largest error on values either side of the calibration point was +1,0 N•m) • The Reference Standard Torque Transducer used has a full scale of 700 N•m and was calibrated in 100 N•m steps (See calibration certificate) • Therefore we will have to use the polynomial equation to determine the actual torque generated by the UUT at 350 N•m since it is a measurement point in between 300 N•m and 400 N•m. • Since we have to interpolate a value we will use the largest reported uncertainty from the calibration certificate for the values on either side of the calibration point which is ± 1 N•m.
© NMISA 2011
Quantifying the sources of uncertainty (3)
• SCAL • Since we will be using the polynomial to interpolate a value at 350 N•m, we DO NOT need to correct for the + 1 N•m error at 399,8 N•m. • Therefore total uncertainty for the “accuracy” of the Reference Standard Torque Transducer is ±1 N•m • This is treated as normal at 95,45% Level of Confidence • The divisor is the coverage factor k which for 95,45% LOC is 2 • The degrees of freedom are always ∞ or 100 % Reliable due to the source of traceability being accredited and reputable.
© NMISA 2011
Quantifying the sources of uncertainty (4)
• SRES • This is due to the resolution of the Reference Standard Torque Transducer Readout Unit • We must first determine the “effective resolution” • The least significant digit displayed is 0,1 N•m • Resolution is always treated as a Rectangular distribution source of uncertainty and this is the range. • The semi-range is therefore (0,1 N•m/2)=0,05 N•m • The divisor is √3 • The degrees of freedom are always ∞ or 100 % Reliability
© NMISA 2011
Quantifying the sources of uncertainty (5)
• SCF • Polynomial Equation Coefficients Table from the Calibration Certificate
POLYNOMIAL EQUATION POLYNOMIAL COEFFICIENTS a b c d Standard Error of the polynomial curve fit for a Level of Confidence of 68,27% and 4 degrees of freedom NORMAL FUNCTION 2,71846 x10 9,99825 x10 -7,89039 x10 5,16535 x10
-2 -1 -6 -9
y=a+bx+cx +dx
2
3
INVERSE FUNCTION -2,70765 x10 1,00017 7,95708 x10 -5,22137 x10
-6 -9 -2
± 0,045 N•m
± 0,045 N•m
© NMISA 2011
Quantifying the sources of uncertainty (6)
• SCF • This is the additional uncertainty which results from the interpolation calculation to determine the torque generated by the UUT at a point in between the calibration points of the Reference Standard Torque Transducer • It is obtained directly from the calibration certificate as the “Standard Error of the polynomial curve Fit” value = ± 0,045 N•m • This is treated as a normal distribution at a 68,27% Level of Confidence • The divisor is the coverage factor k which for 68,27% LOC is 1 • The degrees of freedom are also obtained directly from the calibration certificate = 4
© NMISA 2011
Quantifying the sources of uncertainty (7)
• URES • This is due to the resolution of the UUT Torque Wrench scale. (How it influences the setting of the wrench to a specified torque) • Typically this would be the smallest graduation on the UUT setting dial which for this UUT is 2 N•m • This is the range of the rectangular distribution • Therefore the semi-range is (2 N•m/2)=1 N•m • The divisor for Rectangular Distributed uncertainty contributors is √3 • The degrees of freedom for resolution is always ∞ or 100 % Reliability
© NMISA 2011
Quantifying the sources of uncertainty (8)
• UREP • This results from the variability in the measurement results obtained after repeating the measurement 5 times. • It can be dealt with either as “repeatability” or as “reproducibility” • “Repeatability” – all conditions remained the same during the repeated measurements • “Reproducibility” – any one or more of the conditions changed during the repeated measurements • Different approaches can be used to repeat the measurement • Take 5 measurements at one setting sequentially • Take 5 sets of measurements from zero to full scale
© NMISA 2011
Quantifying the sources of uncertainty (9)
Repeatability
20 % Meas 1 20 % Meas 2 20 % Meas 3 20 % Meas 4 20 % Meas 5 20 % Mean
60 % Meas 1
60 % Meas 2
60 % Meas 3
60 % Meas 4
60 % Meas 5
60 % Mean
100 % Meas 1
100 % Meas 2
100 % Meas 3
100 % Meas 4
100 % Meas 5
100 % Mean
© NMISA 2011
Quantifying the sources of uncertainty (9)
Reproducibility
20 % Meas 1 20 % Meas 2 20 % Meas 3 20 % Meas 4 20 % Meas 5 20 % Mean
60 % Meas 1
60 % Meas 2
60 % Meas 3
60 % Meas 4
60 % Meas 5
60 % Mean
100 % Meas 1
100 % Meas 2
100 % Meas 3
100 % Meas 4
100 % Meas 5
100 % Mean
© NMISA 2011
Quantifying the sources of uncertainty (9)
• UREP • Treating as “Repeatability” (as per ISO 6789) • We use the ESDM • ESDM = ESD/SQRT (n) = 0,98/sqrt (5) = 0,436348 N•m • Treating as “Reproducibility” (preferred option but contrary to ISO 6789) • We use the ESD • ESD = 0,98 N•m
© NMISA 2011
GUM Steps
Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result • • • •
© NMISA 2011
Categorize as type A or type B
• • • • • SCAL - type B, not statistically determined SRES - type B, not statistically determined SCF - type A, statistically determined (standard deviation) URES - type B, not statistically determined UREP - type A, statistically determined (standard deviation)
© NMISA 2011
GUM Steps
• • • • Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result
© NMISA 2011
Uncertainty Budget
Torque Wrench Calibration Uncertainty Budget.xls
© NMISA 2011
GUM Steps
Model the measurement Identify and quantify the sources of uncertainty Categorize as type A or type B Manipulate appropriately to obtain • Standard uncertainties, u(xi) • Sensitivity coefficients, ci • Uncertainty contributor, u(yi) • Combine to obtain combined standard uncertainty, uc(y) • Expand to obtain an expanded uncertainty, U, at an appropriate level of confidence • Report the result • • • •
© NMISA 2011
Reporting the result
• The final result is calculated using the “Normal Function” polynomial coefficients • This is because we want to know the true torque applied to the Reference Standard Torque Transducer when it reads the mean measured value of 350,7 N•m • The calculated interpolated value was 349,938089 N•m • The calculated measurement uncertainty was ± 1,794160789 N•m • Rounding the uncertainty to two significant digits gives ± 1,8 N•m • Rounding the interpolated value to the same number of digits gives 349,9 N•m • The measurement result is then reported as: 349,9 N•m ± 1,8 N•m at a Level of Confidence of 95,45%
© NMISA 2011
Graphical Representation of results
© NMISA 2011
Conclusions
• Both methods in this case prove that the UUT is well within the allowable ± 4% of Maximum (± 14 N•m) • Using the ESDM (In accordance with ISO 6789) results in the smallest uncertainty (unrealistic??) • Using the ESD (contrary to ISO 6789) results in the largest uncertainty (realistic??) • Always use the polynomial for calibrations using the laboratory Reference Standard Torque Transducer • This will correct for any error on the Reference Standard eliminating the need to apply corrections • This will solve the problem of the “Applied Torque” not being exactly at the nominal values
© NMISA 2011
