Problem statement
The length of the calibrated part with a nominal length of 50mm is determined by comparison with another sample, in advance of known length. Comparing the two samples, we get the length difference, which is expressed by the formula:
(1) d = l(1 + αθ) - l_{s}(1 + α_{s}θ_{s})
- l - the length of the calibrated part at a temperature of 20 °C
- l_{s} - the length of the sample at a temperature of 20 °C, indicated in the relevant documentation
- α, α_{s} - coefficients of thermal expansion of the part and sample
- θ, θ_{s} - the standard deviation of the measurement at a temperature of 20 °C
Additional information to the task condition
Based on 25 independent measurements using the instrument used, an experimental standard deviation of 13 nm was determined. The value of the effective number of degrees of freedom is 18. Calibration certificate of a measuring instrument for comparing two lengths indicates that based on six measurements, the uncertainty due to random errors is 0.01 µm s the confidence level is 95%, the uncertainty in case of systematic errors is 0.02 microns at the level of three RMS deviations (25% confidence). The coefficient of thermal expansion of the sample: α_{s} = 11.5 × 10^{-6} °C^{-1}, the uncertainty is set square distribution with boundaries ± 2 × 10^{-6} °C^{-1}. The temperature in the measuring chamber (19.9 ± 0.5) °C. Temperature difference of part and sample is in the range ±0.05 °C.
Measurement model
(2) l = [l_{s}(1 + α_{s}θ_{s}) + d] / (1 + αθ) = l_{s} + d + l_{s}(α_{s}θ_{s} - αθ) + ...
If we use the sample temperature and the part temperature in the measurement model, then it will be necessary take into account the correlation between these values. To avoid complications, we introduce the difference temperatures of the part and sample: δθ = θ - θ_{s} and we introduce the assumption that θ and δθ do not correlate with each other. The same is true for the difference in expansion coefficients δα = α - α_{s}
Assume that δθ and δα are equal to zero, but the corresponding uncertainties are not equal to zero.
(3) l = f(l_{s},d,α_{s},θ,δα,δθ) = l_{s} + d - l_{s}[δα·θ + α_{s}·δθ]
It follows from equation (3) that a statistical estimate of the length of the part, l, can be obtained from a simple equation l_{s} + d_{μ}, where l_{s} is the length of the sample indicated in the documentation when 20°C, d_{μ} - statistical estimate of the mean obtained by calculating the arithmetic mean of five (n=5) independent measurements.
Total uncertainty
According to the formula of total certainty, the uncertainty of formulas of the measurement model (3):
(4) u^{2}_{c}(l) = c^{2}_{s}·u^{2}_{c}(l_{s}) + c^{2}_{d}·u^{2}_{c}(l_{d}) + c^{2}_{αs}·u^{2}_{c}(α_{s}) + c^{2}_{θ}·u^{2}_{c}(θ) + c^{2}_{δα}·u^{2}_{c}(δα) + c^{2}_{δθ}·u^{2}_{c}(δθ)
where
c_{s} = ∂f/∂l_{s} = 1 - (δα·θ + α_{s}·δθ) = 1
c_{d} = ∂f/∂d = 1
c_{αs} = ∂f/∂α_{s} = -l_{s}δθ = 0
c_{θ} = ∂f/∂θ = -l_{s}δα = 0
c_{δα} = ∂f/∂δα = -l_{s}θ
c_{δθ} = ∂f/∂δθ = -l_{s}α
From where follows:
(5) u^{2}_{c}(l) = u^{2}(l_{s}) + u^{2}(d) + l^{2}_{s}θ^{2}u^{2}(δα) + l^{2}_{s}α^{2}_{s}u^{2}(δθ)
Sample uncertainty u(l_{s})
The documentation for the sample indicates an extended uncertainty U = 0.075 microns with an overlap coefficient k=3. Hence:
u(l_{s}) = (0.075 microns) / 3 = 25 nm
Uncertainty of the length difference u(d)
The experimental standard deviation of comparing the difference of lengths l and l_{s} is based on the result is 25 independent measurements and it is equal to 13 nm. In this example, we have made 5 measurements, from where the standard uncertainty of the average value of the measurement data is:
u(d_{μ}) = s(d_{μ}) = (13 nm)/√5 = 5.8 nm
The uncertainty of the random error of the measuring instrument, according to the Student's distribution with the stpenu freedom ν = 6 - 1 = 5, with an overlap coefficient k = t_{95}(5) = 2.57:
u(d_{1}) = (0.01microns) / 2.57 = 3.9 nm
Uncertainty of the systematic error of the measuring instrument:
u(_{d2}) = (0.02 microns) / 3 = 6.7 nm
Total uncertainty:
u^{2}(d) = u^{2}(d_{μ}) + u^{2}(d_{1}) + u^{2}(d_{2}) = 93 nm^{2}
u(d) = 9.7 nm
Uncertainty of the coefficient of thermal expansion u(α_{s})
u(α_{s}) = (2 × 10^{-6} °C^{-1}) / √3 = 1,2 × 10 ^{-6} °C^{-1}
Uncertainty of thermal expansion of the sampleu(θ)
The maximum temperature deviation during the measurement, Δ = 0.5°C. Suppose that the temperature changes within the given limits according to the cyclic law, according to the sine wave, then:
σ_{t} = √[∫(t-t_{μ})^{2}sin(t)dt] = √2
u(Δ) = (0,5 °C) / √2 = 0,35 °C
The uncertainty of the average temperature in the measuring chamber follows from the standard deviation of the average value:
θ_{μ} = 19,9°C - 20°C = -0,1 °C
u(θ_{μ}) = ±(-0,1 °C) = 0,2 °C.
The standard deviation θ can be taken as the standard deviation of the mean θ_{μ}, hence the uncertainty:
u^{2}(θ) = u^{2}(θ_{μ}) + u^{2}(Δ) = 0,165 °C^{2}
u(θ) = 0,41 °C
Uncertainty of the difference in thermal expansion coefficientsu(δα)
The statistical expectation δα is 1 × 10^{-6} °C^{-1} with equal probability, that the value of δα will be within the specified limits, hence the standard uncertainty:
u(δα) = (1 × 10^{-6} °C^{-1}) / √3 = 0,58 × 10 ^{-6} °C^{-1}
Uncertainty of temperature differenceu(δθ)
u(δθ) = (0,05 °C) / √3 = 0,029 °C
Standard total uncertaintyu_{c}(l)
The total uncertainty is calculated by the formula (5):
u_{c}^{2}(l) = (25 nm)^{2} + (9.7 nm)^{2} + (0.05 m)^{2}(-0,1 °C)^{2}(0,58 × 10^{-6} °C^{-1})^{2} + (0.05 m)^{2}(11,5 × 10^{-6}°C^{-1})^{2}(0.029°C)^{2} = 1002 nm^{2}
u_{c}(l) = 32 nm
The calculations show that the main contribution to the uncertainty is the uncertainty of the sample
If the measurement model Y = f(X_{1}, X_{2}, ..., X_{n}) has a nonlinearity on in the measured area, it is necessary to include higher-order uncertainties, then:
u_{c}^{2}(l) = (25 nm)^{2} + (9.7 nm)^{2} + (0.05 m)^{2}(-0,1 °C)^{2}(0,58 × 10^{-6} °C^{-1})^{2} + (0.05 m)^{2}(11,5 × 10^{-6}°C^{-1})^{2}(0.029 °C)^{2} + (0.05 m)^{2}(0,58 × 10^{-6}°C^{-1})^{2}(0.41 °C)^{2} + (0.05 m)^{2}(1,2 × 10^{-6}°C^{-1})^{2}(0,029 °C)^{2} + 1140 nm^{2}
u_{c}(l) = 34 nm
Measurement result
From the sample certificate we have l_{s} = 50,000623 mm at 20°C. The average value of the length difference as a result of five independent measurements is 215 nm. The length of the part l =l_{s} + d_{μ} at 20 °C is 50,000838 mm.
Calculation of relative uncertainty
Relative uncertainty is calculated as the ratio of uncertainty to the measurement result. Measurement result l = 50,000838 mm, with total uncertainty u_{c} = 32 nm. Then the relative total uncertainty is u_{c} /l = 6,4 × 10^{-7}
Calculation of effective degrees of freedom
The calculation of effective degrees of freedom is performed according to the Welch–Satterthwaite equation:
ν_{eff} = u_{c}^{4}(y) / Σ^{n}_{i=1}u_{i}^{4}(y)/ν_{i}
If u(x_{i}) is an uncertainty of type B, then, as a rule, the number of degrees of freedom tends to to infinity, otherwise the number of degrees of freedom for n dimensions is calculated by the following algorithm: if the statistical estimate the average is calculated using the arithmetic mean formula, then ν = n - 1 if the statistical estimate is determined by by the least squares method using m independent factors, then ν = n - m.
The number of degrees of freedom of the sample is set by the calibration certificate, ν_{eff}(l_{s}) = 18.
The value d_{μ} was obtained based on five measurements, but the uncertainty value was
obtained based on 25 measurements, the number of degrees of freedom d_{μ}: ν(d_{μ})
= 25 - 1 = 24. The number of degrees of freedom d_{1}:
v_{eff}(l) = (32 nm)^{4}/[(25 nm)^{4}/18 + (9.7 nm)^{4}/25.6 + (2.9 nm)^{4}/50 + (16.6 nm)^{4}/2] = 16,7
u_{i}(x) | Source of uncertainty | Value of uncertainty | c_{i}≡ ∂f/∂x_{i} | u_{i}(l) ≡ |c_{i}|u (x_{i}) (nm) | Number of degrees of freedom |
---|---|---|---|---|---|
u(ls) | Sample calibration | 25 nm | 1 | 25 | 18 |
u(d) | Measuring the difference between the length of the part and the sample | 9.7 nm | 1 | 9.7 | 25.6 |
u(d_{μ}) | Independent measurements | 5,8 nm | 24 | ||
u(d_{1}) | Random errors | 3.9 nm | 5 | ||
u(d_{2}) | Systematic errors | 6,7 nm | 8 | ||
u(α_{s}) | Thermal expansion coefficient of the sample | 1,2× 10^{-6} °C^{-1} | 0 | 0 | |
u(θ) | Measuring chamber temperature | 0.41°C^{-1} | 0 | 0 | |
u(θ_{μ}) | The average temperature of the measuring chamber | 0.2 °C | |||
u(Δ) | Cyclic temperature change in the measuring chamber | 0.35 °C | |||
u(δα) | Difference of expansion coefficients | 0.58 × 10^{-6} °C^{-1} | l_{s}θ | 2,9 | 50 |
u(δθ) | Difference of expansion coefficients | 0.029°C | -l_{s}α_{s} | 16,6 | 2 |
u_{c}^{2}(l) = Σu_{i}^{2}(l) = 1002 nm^{2} u_{c}(l) = 32 nm ν_{eff}(l) = 16 |
|||||
Table 1. Total uncertainty and its components |
Extended uncertainty
Suppose that it is necessary to obtain an extended uncertainty U_{99} =k_{99} u_{c}(l) with a confidence interval of approximately 99%. The effective value of the degrees of freedom of the standard total uncertainty, equal to 16.7 we round it up exclusively in the smaller direction. The value of the Student's distribution, according to the table, is t_{99}(16) = 2.92, from where U_{99} = t_{99}(16)u_{c}(l) = 2.92 × (32 nm) = 93 nm.
l = (50,000 838 ± 0.000 093) mm, where the error is calculated as the extended uncertainty U =ku_{c}. u_{c} is an extended uncertainty with a coverage coefficient k = 2.92, which was determined from Student distributions for 16 degrees of freedom and a confidence interval of 99%. Corresponding extended relative uncertainty U/l = 1,9 × 10^{-6}