weibull-analysis

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:

1. Infant mortality (β < 1): burn-in reduces field failures
2. Useful life (β = 1): random failures; exponential distribution applies
3. 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.