Linearizing Engineering Models

Linearizing is simply rewriting a curved law so it becomes a straight line, then fitting that line with the regression you already know. Logs are the tool because they turn multiplication into addition and powers into multiplication: a power law $T = CV^{-k}$ becomes $\ln T = \ln C - k\ln V$, and an exponential law becomes a line in $1/T$. The only new step is translating back: the slope is the exponent (or the activation energy), and the intercept, un-logged, is the constant. It's a change of glasses, not a new method.

The scene. A machinist cuts steel at 150 m/min and the carbide insert lasts about 50 minutes; at 300 m/min it lasts 3. Doubling the speed cut tool life by a factor of 16. That's no line. It's Taylor's tool-life equation, $VT^n = C$, a power law, and a plot of the raw data curves hard while a plot of $\ln T$ against $\ln V$ is straight.

Why logs straighten these curves. Logs turn products into sums and powers into multiples:

• Power law $y = ax^b$: $\ln y = \ln a + b\ln x$, a line on log–log axes with slope $b$. Taylor's $VT^n = C$ gives $\ln T = \frac1n\ln C - \frac1n\ln V$.
• Exponential $y = ae^{bx}$: $\ln y = \ln a + bx$, a line on semi-log axes. Newton's law of cooling, $T - T_a = (T_0 - T_a)e^{-kt}$, the solution of a first-order linear ODE, gives $\ln(T - T_a) = \ln(T_0 - T_a) - kt$.
• Arrhenius $k = Ae^{-E_a/(RT)}$: $\ln k = \ln A - (E_a/R)(1/T)$, a line in $1/T$ with slope $-E_a/R$.

You've met one of these already: the Weibull plot of $\ln(-\ln(1 - F))$ against $\ln t$ is a line with slope $\beta$.

Fit, then transform back. Fit $\ln y$ on $x$ (or on $\ln x$) by least squares exactly as for a calibration curve, then undo the logs: $a = e^{b_0}$; for Taylor, $n = -1/b_1$ and $C = e^{-b_0/b_1}$; for Arrhenius, $E_a = -R\,b_1$ when the predictor is $1/T$.

The error model changes too. Least squares on logs assumes constant scatter on the log scale, which means multiplicative error on the original scale: tool lives that vary about ±10% whether they're 3 minutes or 50. That's usually how such data behave, and it's why the fan-shaped residuals of the diagnostics lesson often disappear after a log. If the error really is additive, a log fit overweights the small values, so check the residual plot on the log scale.

Median, not mean. If $\ln y$ is normal with mean $\mu$ and SD $\sigma$, then $e^{\mu}$ is the median of $y$, and the mean is $e^{\mu + \sigma^2/2}$, a little larger (the lognormal mean). So $e^{\hat\mu}$ predicts the typical life, with half failing sooner and half later. With $s = 0.04$ the difference is under 0.1%; with $s = 0.5$ it's 13%.

Arrhenius and accelerated testing. Parts run hot fail sooner, by the ratio of the rates. That ratio is the acceleration factor:

A life measured at the stress temperature, multiplied by $AF$, estimates the life in service, which feeds straight into the MTTF of the reliability lessons.

A common mistake. Temperatures usually come in °C, so it's tempting to plug them straight into $1/T$. The Arrhenius law needs kelvin; with °C the line is wrong and so is the activation energy. Also watch for mixing $\log_{10}$ and $\ln$ (the intercepts differ, so convert consistently), quoting $e^{\hat\mu}$ as a mean, and extrapolating an Arrhenius line across a change of failure mechanism.

Machining handbooks list Taylor constants fitted on log–log axes, and shops use them to set cutting speeds for planned tool changes. Electronics makers run accelerated life tests at high temperature and convert the results to field life with Arrhenius acceleration factors. Chemical engineers estimate activation energies from Arrhenius plots, and battery engineers fit calendar aging the same way. Fatigue (S–N) curves, pump and fan laws and creep rates are all power laws fitted after a log.