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Engineering Example: Uncertainty Analysis in Mechanical Measurements

Introduction to the Engineering Problem

In mechanical engineering, precise measurements are critical for ensuring proper component functionality and system reliability. Let’s explore a practical example of uncertainty analysis in measuring a precision shaft diameter for an engine bearing.

The Engineering Context

A precision shaft needs to be measured for an engine bearing assembly:

The 8-Step Uncertainty Analysis Process

Step 1: Define Measurement Needs

Required Measurement:

Equipment Selection:

Measurement Conditions:

Step 2: Perform Measurements

Measurement Data:

ReadingValue (mm)ReadingValue (mm)
124.998625.001
225.001724.999
325.000825.000
424.999925.001
525.0001025.000

Recorded Conditions:

Step 3: Evaluate Uncertainties

Type A Uncertainty (Statistical Analysis)

  1. Calculate mean:

xˉ=∑xin=24.9999 mm\bar{x} = \frac{\sum x_i}{n} = 24.9999 \text{ mm}
(1)

  1. Calculate standard deviation:

s=∑(xi−xˉ)2n−1=0.001 mms = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} = 0.001 \text{ mm}
(2)

  1. Calculate standard uncertainty:

uA=sn=0.00110=0.00032 mmu_A = \frac{s}{\sqrt{n}} = \frac{0.001}{\sqrt{10}} = 0.00032 \text{ mm}
(3)

Type B Uncertainties (Other Sources)

  1. Calibration Uncertainty:

    • Given as ±0.002 mm (k=2)

    • Standard uncertainty: uB1=0.0022=0.001 mmu_{B1} = \frac{0.002}{2} = 0.001 \text{ mm}

  2. Resolution Uncertainty:

    • Resolution = 0.001 mm

    • Half-width = 0.0005 mm

    • Standard uncertainty: uB2=0.00053=0.00029 mmu_{B2} = \frac{0.0005}{\sqrt{3}} = 0.00029 \text{ mm}

  3. Temperature Effect:

    • Steel expansion coefficient: 11.5 × 10⁻⁶/°C

    • Temperature uncertainty: ±1°C

    • Length effect: 25 mm×11.5×10−6×1°C=0.00029 mm25 \text{ mm} × 11.5 × 10^{-6} × 1°C = 0.00029 \text{ mm}

    • Standard uncertainty: uB3=0.000293=0.00017 mmu_{B3} = \frac{0.00029}{\sqrt{3}} = 0.00017 \text{ mm}

Step 4: Check Independence

Independent Sources:

Potentially Correlated:

Step 5: Calculate Results

Base Result:

Measurement Conditions Verification:

Step 6: Find Combined Uncertainty

For independent sources, the combined standard uncertainty is calculated as:

uc=uA2+uB12+uB22+uB32u_c = \sqrt{u_A^2 + u_{B1}^2 + u_{B2}^2 + u_{B3}^2}
(4)

Substituting our values:

uc=(0.00032)2+(0.001)2+(0.00029)2+(0.00017)2u_c = \sqrt{(0.00032)^2 + (0.001)^2 + (0.00029)^2 + (0.00017)^2}
(5)
=0.0000001024+0.000001+0.0000000841+0.0000000289= \sqrt{0.0000001024 + 0.000001 + 0.0000000841 + 0.0000000289}
(6)
=0.0000012154=0.0011 mm= \sqrt{0.0000012154} = 0.0011 \text{ mm}
(7)

Step 7: Calculate Expanded Uncertainty

Using k = 2 for 95% confidence level:

U=k⋅uc=2×0.0011=0.0022 mmU = k \cdot u_c = 2 × 0.0011 = 0.0022 \text{ mm}
(8)

Comparison to Tolerance:

Step 8: Document Results

Final Result:

(24.9999±0.0022) mm (k=2,95% confidence level)(24.9999 ± 0.0022)\text{ mm }(k=2, 95\% \text{ confidence level})
(9)

Major Sources of Uncertainty:

  1. Calibration uncertainty (largest contributor)

  2. Measurement repeatability

  3. Resolution limits

  4. Temperature effects

Engineering Conclusions

Technical Decisions

Business Impact

Practice Questions

  1. Why is the expanded uncertainty (k=2) used rather than the standard uncertainty?

  2. How would you modify this analysis if the temperature variation was ±2°C instead of ±1°C?

  3. What would be the impact if the measurement uncertainty was larger than 1/4 of the tolerance?

  4. How would you handle correlated uncertainty sources if temperature effects couldn’t be treated as independent?

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