| name | weibull-analysis |
| description | Weibull analysis — 2-parameter and 3-parameter Weibull, Weibull plotting, maximum likelihood estimation, B10/B50 life, confidence bounds, bathtub curve, reliability prediction. |
| metadata | {"priority":7,"promptSignals":{"phrases":["Weibull analysis","Weibull distribution","Weibull plot","beta Weibull","B10 life","reliability Weibull","failure distribution"],"minScore":3}} |
Weibull Analysis — Complete Skill
2-Parameter Weibull Distribution
CDF (cumulative distribution of failures):
F(t) = 1 - exp[-(t/η)^β]
Reliability:
R(t) = 1 - F(t) = exp[-(t/η)^β]
PDF:
f(t) = (β/η) × (t/η)^(β-1) × exp[-(t/η)^β]
Failure rate (hazard function):
h(t) = β/η × (t/η)^(β-1)
Parameters:
β (beta) = shape parameter (Weibull slope)
η (eta) = characteristic life (scale parameter; 63.2% of population has failed at t = η)
Shape Parameter β Interpretation
| β value | Failure rate | Failure mode |
|---|
| β < 1 | Decreasing | Infant mortality (manufacturing defects) |
| β = 1 | Constant | Random failures (exponential distribution) |
| 1 < β < 3 | Increasing (gentle) | Early wear out or random |
| β ≈ 3.4 | Increasing | Normal distribution approximation |
| β > 3 | Increasing (rapid) | Wear-out, fatigue (most mechanical failures) |
| β = 2 | Linear increase | Surface wear, linear degradation |
Typical β values:
Bearing fatigue (rolling contact): β ≈ 1.1–1.5
Electrical insulation: β ≈ 2–5
Structural fatigue: β ≈ 3–4
Gears: β ≈ 1.5–3
Weibull Probability Plot
Linearized form:
ln[ln(1/(1-F))] = β ln(t) - β ln(η)
Plot on Weibull paper: x-axis = ln(t), y-axis = ln[ln(1/(1-F))]
Straight line → 2-parameter Weibull fits; slope = β
Median Rank (plotting position for ordered failures):
F_i ≈ (i - 0.3) / (n + 0.4) [Bernard's approximation; n = sample size, i = rank]
Maximum Likelihood Estimation (MLE)
More accurate than linear regression for censored data (suspended tests)
Log-likelihood:
ℓ(β,η) = Σ_failures [ln(β/η) + (β-1)ln(t_i/η) - (t_i/η)^β] - Σ_suspensions [(t_j/η)^β]
Solved numerically (Newton-Raphson or optimization)
MLE automatically handles right-censored data (units still running)
3-Parameter Weibull
F(t) = 1 - exp[-((t - t₀)/η)^β]
t₀ = location parameter (failure-free period)
Useful when first failure cannot occur before some time t₀
(e.g., fatigue where cycles needed to initiate crack)
B Life Notation
B_x life: time at which x% of population has failed
B_10 = η × [-ln(1-0.10)]^(1/β) = η × (0.1054)^(1/β) (10% failure time)
B_50 = η × (ln 2)^(1/β) = η × (0.6931)^(1/β) (median life)
Bearing industry: L_10 = B_10 (10% of bearings failed; 90% reliability)
Confidence Bounds (Fisher Matrix)
95% confidence bounds on β and η (from inverse of Hessian of log-likelihood)
Bounds on B_x life:
Lower bound: B_x,lower = exp(ln(B_x) - z_α/2 × σ_ln B_x)
Upper bound: B_x,upper = exp(ln(B_x) + z_α/2 × σ_ln B_x)
z_0.975 = 1.96 for 95% two-sided confidence
Likelihood ratio confidence bounds: more accurate for small samples
Bathtub Curve (System Reliability)
Three phases:
- Infant mortality (β < 1): burn-in reduces field failures
- Useful life (β = 1): random failures; exponential distribution applies
- Wear-out (β > 1): scheduled replacement before η
Mixed Weibull (2-mode): sum of two Weibull CDFs
Used to model infant mortality + wear-out simultaneously
System Reliability
Series system: R_sys = ∏ R_i(t) (all must function)
Parallel system: R_sys = 1 - ∏ (1-R_i(t)) (any one suffices)
MTBF = ∫₀^∞ R(t)dt = η × Γ(1 + 1/β) [Gamma function of (1 + 1/β)]
Accelerated Life Testing (ALT)
Arrhenius acceleration (thermally activated):
t_use / t_test = exp[E_a/k × (1/T_use - 1/T_test)]
E_a = activation energy (0.3–1.2 eV typical); k = 8.617×10⁻⁵ eV/K
Power law (non-thermal):
t_use / t_test = (V_test/V_use)^n (for voltage stress, mechanical stress)
Assumption: same β at different stress levels; only η changes
Output
Provide: β (shape), η [hours or cycles], B_10 and B_50 [hours], failure mode characterization (infant/random/wear-out), 95% confidence bounds on B_10, sample size recommendation for desired confidence, ALT acceleration factor.
