故事 · 同一份寿命数据的三种问法
Story · Three Ways to Ask the Same Lifetime Data
可靠性工程师拿到一批轴承的失效记录,会同时问三个问题。质量经理问「到保修期为止,会坏掉百分之几」 —— 那是 F(t);
客户问「用三年还有多大把握不坏」 —— 那是 R(t);维修工程师问「这台机器现在最危险吗,要不要提前换」 —— 那是 h(t)。
三个问题指向同一个分布,却是完全不同的决策视角。威布尔的妙处在于:F、R、h 全由两个参数 β、η 唯一确定,
而 β 一个数就把「失效率是涨是跌」写死了 —— 这正是浴盆曲线背后的数学。
A reliability engineer holding a batch of bearing failure records asks three questions at once. The quality manager asks "by warranty expiry, what percent will have failed?" — that is F(t).
The customer asks "after three years of use, how confident can I be it's still working?" — that is R(t). The maintenance engineer asks "is this machine in the danger zone right now — should I swap it early?" — that is h(t).
Three questions, one distribution, three decision lenses. The beauty of Weibull is that F, R, and h are all locked down by just two parameters, β and η,
and β alone decides whether the failure rate is climbing, flat, or falling — which is exactly the math under the bathtub curve.
1 F(t) 与 R(t):坏掉的 + 活着的 = 100% F(t) and R(t): failed + alive = 100%
β=1.5 η=1.02 h(t) 失效率:β 跨过 1,走势就变天 h(t) hazard: cross β = 1 and the slope flips
随机期Randomβ<1 失效率一路下坡(早夭)、β=1 拉成水平(随机,退化成指数)、β>1 持续上扬(磨损)。把三段拼起来,就是经典的浴盆曲线。 β < 1 drops the hazard rate (infant mortality), β = 1 flattens it (random, collapsing to an exponential), β > 1 lifts it (wear-out). Splice the three segments and you get the classic bathtub curve.
3 现实里的 F / R / h F / R / h in the Real World
寿命分析:把现场失效数据拟合威布尔,得到 β、η,再用 F(t) 估「保修期内退货率」,直接换算售后成本。
Life data analysis: fit field failure data to Weibull, recover β and η, then use F(t) to estimate the in-warranty return rate — translated straight into service cost.
可靠度指标:R(t) 是合同里的硬承诺 ——「五年可靠度 ≥ 95%」就是要求 R(5)≥0.95,反推 β、η 的设计目标。
Reliability spec: R(t) is a contractual promise — "five-year reliability ≥ 95%" means R(5) ≥ 0.95, which back-solves into design targets for β and η.
失效率与换件:h(t) 上扬段陡升前是预防性维护的最佳窗口;h 平段则换件无益(坏全凭偶然)。
Hazard and replacement: just before h(t) ramps up steeply is the sweet spot for preventive maintenance; while h is flat, swapping parts gains nothing — failures are purely random.
浴盆曲线:β 的早夭→随机→磨损三态,正是浴盆三段的数学定义,把工程直觉变成可计算的曲线。
The bathtub curve: β's three regimes — infant mortality → random → wear-out — are the exact mathematical definition of the bathtub's three segments, turning engineering intuition into a curve you can compute.
一句话In One Line
F、R、h 不是三个独立公式,而是同一个威布尔的三种问法。它们由微积分锁死在一起:
R = 1 − F、h = f / R(密度除以幸存比例)。
所以你只要定下 β(失效率走势)和 η(时间尺度),三条曲线全部确定。
这就是为什么可靠性工程师拿到数据先估 β —— β 一旦定了,「该不该换、保修多久、还有多少活着」三个答案同时浮现。
F, R, and h are not three independent formulas — they are three questions asked of the same Weibull. Calculus welds them together:
R = 1 − F and h = f / R (density divided by the surviving fraction).
Pin down β (the shape of the hazard) and η (the time scale) and all three curves are fully determined.
That is why reliability engineers reach for β first — once β is known, "should we replace it, how long should we warrant it, how many are still alive" all answer themselves at once.
常见误用Common Mistakes
把 h(t) 当成 F(t)。h 是瞬时失效率(条件概率密度),F 是累积比例;h 可以超过 1,F 永远在 0~1 之间。
Confusing h(t) with F(t). h is an instantaneous hazard rate (a conditional density) and can exceed 1; F is a cumulative fraction and always lives in [0, 1].
磨损件(β>1)也按恒定失效率算。只有 β=1 才能用「恒定 λ」那套指数公式,否则会严重低估老化期的换件需求。
Treating a wear-out item (β > 1) as if it had a constant failure rate. The "constant λ" exponential math only holds at β = 1; using it on wear-out parts severely underestimates replacement demand late in life.
把 η 当平均寿命。η 是 F=63.2% 的特征寿命,均值要乘伽马因子 Γ(1+1/β),两者只有 β=1 时才相等。
Reading η as the mean life. η is the characteristic life at F = 63.2%. The true mean is η · Γ(1 + 1/β); the two coincide only when β = 1.