When Will a Bearing Fail? The Weibull Model

Parts don't all fail at once, and not at a steady rate either. Some fail young from manufacturing defects, some fail at random, and most mechanical parts wear out. One shape number, β, captures which story you're in, and the scale η says when: by time η, 63.2% of parts have failed.

How it works

Weibull reliability and failure rate

In one sentence

A Weibull lifetime has $F(t) = 1 - e^{-(t/\eta)^\beta}$ for $t \ge 0$ , reliability $R = 1 - F$ , hazard $h(t) = \frac{f(t)}{R(t)} = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}$ and mean $\eta\,\Gamma(1 + 1/\beta)$ . The case $\beta = 1$ is the exponential distribution with rate $1/\eta$ .

General formula

{\htmlData{sym=0}{R(t)}} = e^{-(t/{\htmlData{sym=3}{\eta}})^{\htmlData{sym=2}{{\beta}}}}, \qquad {\htmlData{sym=1}{h(t)}} = \frac{\htmlData{sym=2}{{\beta}}}{\htmlData{sym=3}{{\eta}}}\left(\frac{t}{\htmlData{sym=3}{{\eta}}}\right)^{{\htmlData{sym=2}{\beta}} - 1}, \qquad {\htmlData{sym=5}{\text{MTTF}}} = {\htmlData{sym=3}{\eta}}\,{\htmlData{sym=4}{\Gamma}}\!\left(1 + \frac{1}{\htmlData{sym=2}{{\beta}}}\right)

The hazard asks: of the parts still running, what fraction fails in the next hour? Its trend over time tells you whether age makes failure more likely, less likely, or makes no difference.

What the symbols mean

$R(t)$
Reliability
The share of parts still working at time t.
$h(t)$
Failure rate
Failures per hour among the parts still working at time t.
$\beta$
Shape
Below 1: infant mortality; 1: random failures; above 1: wear-out.
$\eta$
Scale
The characteristic life: by then 63.2% have failed.
$\Gamma$
Gamma function
Extends the factorial to fractions: Γ(n + 1) = n!.
$\text{MTTF}$
Mean time to failure
The average life of a part.

The reliability $R(t)$ is the share of parts still working at time $t$ ; the Weibull model takes $R(t) = e^{-(t/\eta)^\beta}$ .
The failure rate (hazard) is the density divided by the survivors, $h(t) = f(t)/R(t)$ , which works out to $\frac{\beta}{\eta}(t/\eta)^{\beta-1}$ : falling when $\beta < 1$ , flat when $\beta = 1$ (the exponential model), rising when $\beta > 1$ .
Setting $R = 0.9$ gives the B10 life, $\eta(-\ln 0.9)^{1/\beta}$ , when 10% have failed; at $t = \eta$ , $R = e^{-1}$ , so 63.2% have failed whatever $\beta$ is.
The mean life is $\int_0^\infty R(t)\,dt = \eta\,\Gamma(1 + 1/\beta)$ .

Learn the math

The Weibull DistributionEng Stats · Lesson 6, Probability Models for Engineering Reliability and Hazard FunctionsEng Stats · Lesson 10, Reliability Engineering