Calibration of a gauge block of nominal length 50 mm

The calibration of the grade 0 gauge block (ISO 3650) of 50 mm nominal length is carried out by comparison using a comparator and a calibrated gauge block of the same nominal length and the same material as reference standard. The difference in central length is determined in vertical position of the two gauge blocks using two length indicators contacting the upper and lower measuring faces.

The actual length of the gauge block to be calibrated is related to the actual length of the reference standard by the equation

lX' = lS' + δl

with δl being the measured length difference. lX blocks under the measurement conditions, in particular at a temperature which on account of the uncertainty in the measurement of laboratory temperature may not be identical with the reference temperature for length measurements.

Model Equation:

lX = lS + δlD + δl + δlC - L * (αav * δt + δα * Δtav + uat) - δlV

List of Quantities:

Quantity Unit Definition
lX mm length of the gauge block to be calibrated
lS mm length of the reference gauge block at the reference temperature of t0=20 °C according to its calibration certificate
δlD mm Change of the length of the reference gauge block since its last calibration due to drift
δl mm observed difference in length between the unknown and the reference gauge block
δlC mm correction for non-linearity and offset of the comparator
L mm nominal length of the gauge blocks under consideration
αav K-1 average of the thermal expansion coefficients of the unknown and the reference gauge block
δt K difference in temperature between the unknown the reference gauge block
δα K-1 difference in the thermal expansion coefficients between the unknown and the reference gauge block
Δtav K deviation of the average temperature of the unknown and the standard gauge block from the reference temperature
uat   coorection for second order terms of (δα * Δtav)
δlV mm correction for non-central contacting of the measuring faces of the unknown gauge block

lS: Type B normal distribution
Value: 50.000020 mm
Expanded Uncertainty: 30·10-6 mm
Coverage Factor: 2.0
REFERENCE STANDARD: The length of the reference gauge block together with the associated expanded uncertainty of measurement is given in the calibration certificate of a set of gauge blocks as 50,000 02 mm ±30 nm (coverage factor k=2).

δlD: Type B triangular distribution
Value: 0 mm
Halfwidth of Limits: 30·10-6 mm
DRIFT OF THE STANDARD: The temporal drift of the length of the reference gauge block is estimated from previous calibrations to be zero with limits ±30 nm. General experience with gauge blocks of this type suggest that the zero drift is the most probable value and that a triangular probability distribution can be assumed.

δl: Type A
Method of observation: Direct
Number of observation: 5
No. Observation
1 -100·10-6 mm
2 -90·10-6 mm
3 -85·10-6 mm
4 -95·10-6 mm
5 -100·10-6 mm

Arithmetic Mean: -94.00·10-6 mm
Pooled Standard Deviation: 12·10-6 mm
Standard Deviation of the Mean: 4.75·10-6 mm
Pooled Degrees of Freedom: 9

MEASUREMENTS: A pooled estimate of standard deviation of 15 nm derived from previous measurements is used for the evaluation of the repeatability. The following observations are made for the difference between the unknown gauge block and the reference standard, the comparator being reset using the reference standard before each reading.

δlC: Type B rectangular distribution
Value: 0 mm
Halfwidth of Limits: 32·10-6 mm
COMPARATOR: The comparator has been verified to meet the specifications stated in EAL-G21. From this it can be ascertained that for length differences D up to ±10 µm corrections to the indicated length difference are within the limits ±(30 nm + 0,02 * abs(D)). Taking into account the tolerances of the grade 0 gauge block to be calibrated and the grade K reference gauge block the maximum length difference will be within ±1 µm leading to the limits of ±32 nm for non-linearity and offset corrections of the comparator used.

L: Constant
Value: 50 mm
Nominal length of the gauge block to be calibrated.

αav: Type B rectangular distribution
Value: 11.5·10-6 K-1
Halfwidth of Limits: 1·10-6 K-1
TEMPERATURE CORRECTION: Based on the calibration certificate of the reference gauge block and the manufacturer’s data for the gauge block to be calibrated the linear thermal expansion coefficient of the steel gauge blocks is assumed to be within the interval (11,5±1,0)×10-6 °C-1.

δt: Type B rectangular distribution
Value: 0 K
Halfwidth of Limits: 0.05 K
TEMPERATURE CORRECTION: Before calibration care is taken to ensure that the gauge blocks assume ambient temperature of the measuring room. The remaining difference in temperature between the standard and the gauge block to be calibrated is estimated to be within ±0,05 K.

δα: Type B triangular distribution
Value: 0.0 K-1
Halfwidth of Limits: 2·10-6 K-1
TEMPERATURE CORRECTION: Combining the two rectangular distributions the difference in linear thermal expansion coefficient is triangularly distributed within the limits ±2E-6 °C-1.

Δtav: Type B rectangular distribution
Value: 0 K
Halfwidth of Limits: 0.5 K
TEMPERATURE CORRECTION: The deviation of the mean temperature of measurement from the reference temperature t0 = 20 °C is estimated to be within ±0,5 °C.

uat: Type B normal distribution
Value: 0
Expanded Uncertainty: 0.236·10-6
Coverage Factor: 1.0
TEMPERATURE CORRECTION: The best estimates of the difference in linear expansion coefficients (δα) and the deviations of the mean temperature from the reference temperature (Δtav) are zero. Therefore second order terms have to be taken into account in the evaluation of their uncertainty contribution resulting in the product of standard uncertainties associated with the factors of the product term (δα × Δtav) in the model equation. The final standard uncertainty is u(δα × Δtav) = 0,236·10-6.

δlV: Type B rectangular distribution
Value: 0 mm
Halfwidth of Limits: 6.7·10-6 mm
VARATION IN LENGTH: For gauge blocks of grade 0 the variation in length determined from measurements at the centre and the four corners has to be within ±0,12 mm (ISO 3650). Assuming that this variation occurs on the measuring faces along the short edge of length 9 mm and that the central length is measured inside a circle of radius 0,5 mm the correction due to central misalignment of the contacting point is estimated to be within ±6,7 nm.

Uncertainty Budgets:

lX: length of the gauge block to be calibrated
Quantity Value Standard
Uncertainty
Distribution Sensitivity
Coefficient
Uncertainty
Contribution
Index
lS 50.0000200 mm 15.0·10-6 mm normal 1.0 15·10-6 mm 19.3 %
δlD 0.0 mm 12.2·10-6 mm triangular 1.0 12·10-6 mm 12.8 %
δl -94.00·10-6 mm 4.75·10-6 mm normal 1.0 4.7·10-6 mm 1.9 %
δlC 0.0 mm 18.5·10-6 mm rectangular 1.0 18·10-6 mm 29.2 %
L 50.0 mm          
αav 11.500·10-6 K-1 577·10-9 K-1 rectangular 0.0 0.0 mm 0.0 %
δt 0.0 K 0.0289 K rectangular -580·10-6 -17·10-6 mm 23.6 %
δα 0.0 K-1 816·10-9 K-1 triangular 0.0 0.0 mm 0.0 %
Δtav 0.0 K 0.289 K rectangular 0.0 0.0 mm 0.0 %
uat 0.0 236·10-9 normal -50 -12·10-6 mm 11.9 %
δlV 0.0 mm 3.87·10-6 mm rectangular -1.0 -3.9·10-6 mm 1.3 %
lX 49.9999260 mm 34.2·10-6 mm

Results:

Quantity Value Expanded
Uncertainty
Coverage
factor
Coverage
lX 49.999926 mm 68·10-6 mm 2.00 95% (t-table 95.45%)

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