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3.1: Component Uncertainty

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    121537
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    1. Errors are properties of all measurements
      1. \(\hat{x}_{\mathrm{gage}} = x_{\mathrm{actual}} + \epsilon\), where \(\epsilon\) arises from contributing error sources
      2. Results should be presented as value ± uncertainty range
    2. Categories of measurement error
      1. Systematic error – constant offset to the true value
      2. Random errors – variations about the measurand value
      3. Comparison of types:
        Systematic Random
        Not determined statistically Scatter of data measurements allows statistics
        Estimated via comparison methods Affected by repetition and resolution
        1. Calibration
        2. Comparison to other references
        3. Engineering experience
    3. Measurand uncertainty analysis (Chapter 10)
      1. Identifies the ± range for individual sensor and resulting measurand
      2. Like magnitude of a rank \( \mathbb{R} \) vector, magnitude determined by root-sum-square: \[ u_{r} = \pm \sqrt{u_{0}^{2} + u_{1}^{2} + u_{2}^{2} + \ldots + u_{J}^{2}} \]
        1. Since it's a summation → all units must match (e.g., cannot add \( \pm 0.03 \) cm and 3% of measurement)
        2. Assumes unit sensitivity where none are more important than others
        3. Built from two types: zero-order and instrument uncertainty
      3. Zero-order uncertainty
        1. Based on resolution (physical divisions of transducer stage)
        2. Variations create random scatter above/below scale value
        3. Equation: \( u_{0} = \pm 0.5 \times \text{resolution scale} \)
        4. Explains importance of significant figures in science
      4. Instrument errors
        1. Determined through calibration methods or manufacturer specs
        2. Types:
          1. Hysteresis
          2. Linearity
          3. Sensitivity
          4. Repeatability
          5. Zero-shift
          6. Stability
          7. Thermal drift
        3. Equation: \[ u_{I} = \sqrt{u_{H}^{2} + u_{S}^{2} + \ldots} \]
      5. Total resulting uncertainty: \[ u_{r} = \pm\sqrt{u_{0}^{2} + u_{I}^{2}} \]
      6. Example: Speedometer graduated in 5-mph increments and instrument accuracy of ±4% at 60 mph
      7. Important distinctions (when reading words):
        1. % of the measurement → multiply result by percentage
        2. % FSO (full scale output) → use full sensor range

    3.1: Component Uncertainty is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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