MEEN 260 - Exam 1 Review

Systematic Error (Bias)

Error with the same magnitude and sign (always positive or always negative); caused by instrument bias, parallax, procedure, etc.

Blunder (Mistake)

Catastrophic error that invalidates the result; caused by human error or mechanical failure; results should be rejected.

Offset Error

A fixed/constant error each time the measurement is made, independent of input magnitude; can be positive or negative.

Proportional Error

An error equal to a fixed fraction of the input/measurand; increases as the input magnitude increases (fixed relative error).

Systematic Uncertainty (Us)

Uncertainty estimate due to systematic effects, commonly computed from manufacturer accuracy specifications (often offset plus percent-of-reading).

Precision Uncertainty (Up)

Uncertainty estimated from the scatter of repeated measurements under fixed conditions; random errors appear as normally distributed scatter.

Error (E)

Error is the difference between the measured value (X-bar) and the true value (mu): E equals X-bar minus mu.

True Value (mu)

The actual or exact value of the measurand (unknown).

Measured Value (measurement result)

The assumed value of the measurand; usually the sample mean (x-bar).

Random Error

Error with random magnitude and sign (plus/minus); caused by vibrations, temperature fluctuations, observer movement, etc.

Precision Uncertainty Formula (mean of n)

Up equals t-sub-(nu, alpha over 2) times (s-sub-x divided by the square root of n).

Total Uncertainty (uncorrelated)

Combine independent uncertainties by root-sum-square: U-total equals the square root of (U1 squared plus U2 squared plus ...).

Propagation of Uncertainty (Kline-McClintock)

For w equals f of (x, y, z): Uw equals the square root of ((partial f partial x times Ux) squared plus (partial f partial y times Uy) squared plus (partial f partial z times Uz) squared).

Sensitivity Coefficient

The partial-derivative term (e.g., partial f partial x) that weights how an input uncertainty contributes to the output uncertainty.

Rounding with Uncertainty (NIST GLP-9)

Do not round intermediate calculations; round expanded uncertainty to two significant digits, then round the reported value to the same decimal place as the uncertainty.

Modeling

Process of selecting the general form of a predictive equation: y equals f of x.

Curve Fitting

Process of selecting coefficients of a predictive equation to match a data set (example: y equals a x plus b).

Regression

Mathematical optimization process used to select best coefficient values (e.g., by least squares).

Residual

Difference between measured response and predicted response: R equals y-sub-i minus y-hat-sub-i.

Prediction Error (e-sub-i)

e-sub-i equals y-sub-i minus y-hat-sub-i.

Least Squares Objective

Choose parameters to minimize S-squared equals the sum from i equals 1 to N of (e-sub-i squared).

SSE (Explained Squared Variation)

SSE equals the sum from i equals 1 to N of (y-hat-sub-i minus y-bar) squared.

SSR (Squared Prediction Error / Residuals)

SSR equals the sum from i equals 1 to N of (y-hat-sub-i minus y-sub-i) squared.

SST (Total Squared Variation)

SST equals the sum from i equals 1 to N of (y-sub-i minus y-bar) squared.

Correlation Coefficient (r-squared)

r-squared equals Se-squared divided by ST-squared.

Adjusted Correlation Metric (r-prime)

r-prime equals (one minus r-squared).

Coefficient of Determination (R-squared)

R-squared equals one minus (Sr-squared divided by ST-squared).

Nonlinear Linearization (exponential example)

If y equals c1 times e to the (c2 x), then take ln of both sides: ln(y-sub-i) equals ln(c1) plus c2 times x-sub-i.

Transducer

A device that converts energy (or a signal) from one form to another.

Sensor

A class of transducer that converts a physical phenomenon into a signal/information.

Accuracy (Sensor)

How close the sensor's measured value is to the true/actual value; usually the primary source of systematic uncertainty.

Resolution (Sensor)

The smallest change in input that produces an observable change in output; used to compute zero-order uncertainty U0.

Range/Span (Sensor)

Range: the upper and lower input limits; span: the numerical difference between those limits.

Sensitivity (Sensor)

Change in output per change in input: S equals partial(output) partial(input), approximately delta-y over delta-x.

Linearity Error (%)

Linearity error percent equals (absolute value of maximum deviation divided by full-scale output) times 100 percent.

Hysteresis

Difference in output for the same input depending on whether the input is increasing or decreasing; often expressed as percent full scale.

Time Constant (tau)

Time to reach 63.2% of final value after a step input (or decay to 36.8% for decreasing).

Response Time

Time to reach about 99.3% of final value, typically about five times tau.

Normal Stress (sigma)

sigma equals P divided by A.

Normal Strain (epsilon)

epsilon equals delta-L divided by L.

Hooke's Law

sigma equals E times epsilon.

Poisson Relation (transverse strain)

epsilon-trans equals negative nu times epsilon.

Strain Gage (function)

Electromechanical sensing element that transduces strain to resistance (resistance changes with applied strain).

Gage Factor (K)

K equals (delta-R over R) divided by epsilon.

Strain Gage Output Function

R of epsilon equals R-zero times (one plus K epsilon).

Wheatstone Bridge Balance Condition

Bridge is balanced when R1 over R2 equals R3 over R4 (equivalently R1 over R3 equals R2 over R4); then Vo equals zero.

Wheatstone Bridge Deflection Voltage

Vo equals V-ex times (R3 divided by (R3 plus R4) minus R1 divided by (R1 plus R2)).

Quarter-Bridge Shortcut (approx.)

Vo equals V-ex times (K epsilon over 4) under the stated approximation conditions.

Bridge Factor (BF)

Bridge factor depends on configuration: quarter bridge BF equals 1, half bridge BF equals 2, full bridge BF equals 4.

Sampling Frequency

fs equals one divided by Ts.

N-bit Quantization Resolution

Delta equals (V-max minus V-min) divided by (two to the N).

ADC Uncertainty (quantization)

ADC uncertainty equals one-half times the resolution (one-half times Delta).

Digital Resolution (DAQ)

Smallest change in the input analog voltage that can be represented unambiguously by a unique digital code.

Digital Range (levels)

Digital range equals two to the (number of bits).

Digital Assignment of Input (counts)

Counts equals round (or truncate) of: digital range times (input minus min input) divided by input range.

ADC Key Concept

The ADC does not simply convert the analog value to binary; it converts the input to a number of binary steps across the expected input range.