3.1: Component Uncertainty
- Page ID
- 121537
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Errors are properties of all measurements
- \(\hat{x}_{\mathrm{gage}} = x_{\mathrm{actual}} + \epsilon\), where \(\epsilon\) arises from contributing error sources
- Results should be presented as value ± uncertainty range
- Categories of measurement error
- Systematic error – constant offset to the true value
- Random errors – variations about the measurand value
- 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
- Measurand uncertainty analysis (Chapter 10)
- Identifies the ± range for individual sensor and resulting measurand
- Like magnitude of a rank \( \mathbb{R} \) vector, magnitude determined by root-sum-square:
\[ u_{r} = \pm \sqrt{u_{0}^{2} + u_{1}^{2} + u_{2}^{2} + \ldots + u_{J}^{2}} \]
- Since it's a summation → all units must match (e.g., cannot add \( \pm 0.03 \) cm and 3% of measurement)
- Assumes unit sensitivity where none are more important than others
- Built from two types: zero-order and instrument uncertainty
- Zero-order uncertainty
- Based on resolution (physical divisions of transducer stage)
- Variations create random scatter above/below scale value
- Equation: \( u_{0} = \pm 0.5 \times \text{resolution scale} \)
- Explains importance of significant figures in science
- Instrument errors
- Determined through calibration methods or manufacturer specs
- Types:
- Hysteresis
- Linearity
- Sensitivity
- Repeatability
- Zero-shift
- Stability
- Thermal drift
- Equation: \[ u_{I} = \sqrt{u_{H}^{2} + u_{S}^{2} + \ldots} \]
- Total resulting uncertainty: \[ u_{r} = \pm\sqrt{u_{0}^{2} + u_{I}^{2}} \]
- Example: Speedometer graduated in 5-mph increments and instrument accuracy of ±4% at 60 mph
- Important distinctions (when reading words):
- % of the measurement → multiply result by percentage
- % FSO (full scale output) → use full sensor range