reliability-engineering

name	reliability-engineering
description	Reliability engineering — MTBF, Weibull analysis, fault tree analysis (FTA), reliability block diagrams (RBD), FMECA, spare parts planning, RAM analysis, reliability growth (Duane).
metadata	{"priority":7,"promptSignals":{"phrases":["reliability","MTBF","Weibull","fault tree","FTA","reliability block","RBD","FMECA","availability","RAM"],"minScore":3}}

Reliability Engineering — Complete Skill

Basic Reliability Metrics

Reliability R(t)

R(t) = P(T > t) = probability of surviving beyond time t F(t) = 1 - R(t) = cumulative failure probability (unreliability) f(t) = dF/dt = failure probability density function

Hazard Rate (Failure Rate) λ(t)

λ(t) = f(t)/R(t) = instantaneous failure rate [failures/hr] Bathtub curve: high early (infant mortality) → constant (useful life) → increasing (wear-out)

Constant failure rate (exponential): R(t) = e^(-λt) MTTF = 1/λ (mean time to first failure) MTBF = 1/λ (mean time between failures, for repairable)

Relationship: R(t) = e^(-t/MTBF) At t = MTBF: R = e^(-1) = 0.368 (only 36.8% survive to MTBF!) At t = 0.1×MTBF: R = e^(-0.1) = 0.905 (90.5% survive to 10% of MTBF)

Weibull Distribution

Two-Parameter Weibull

f(t) = (β/θ)(t/θ)^(β-1) exp(-(t/θ)^β) R(t) = exp(-(t/θ)^β) λ(t) = (β/θ)(t/θ)^(β-1)

β = shape parameter (Weibull slope) θ = scale parameter (characteristic life; R(θ)=0.368 always) MTTF = θ × Γ(1+1/β)

β interpretation: β < 1: decreasing failure rate (infant mortality) β = 1: constant failure rate (exponential — random failures) β = 2: linearly increasing rate (linear aging — bearings, gears) β = 3.5: approximates normal distribution (symmetric wear-out) β > 1: wear-out (fatigue typically β=1.5-3)

Three-Parameter Weibull

R(t) = exp(-((t-γ)/θ)^β) where γ = location parameter (failure-free period) Use when: no failures in early life (γ = minimum life guarantee)

Weibull Analysis (Parameter Estimation)

Median rank (plotting position): F_i = (i-0.3)/(n+0.4) for i-th ranked failure Probability paper: plot ln(-ln(R)) vs. ln(t) → straight line with slope β MLE (Maximum Likelihood): more accurate for small samples; software: Reliasoft Weibull++, JMP

B10 life: time at which 10% of population has failed: t_B10 = θ × (ln(10/9))^(1/β) B10 for ball bearings: typically 10^6 revolutions at rated load (ISO 281 definition)

System Reliability

Series System (all must work)

R_sys = Π R_i = R₁ × R₂ × ... × Rₙ System R < any component R Weakest link dominates

Parallel System (redundancy, any one survives)

R_sys = 1 - Π(1-R_i) = 1 - (1-R₁)(1-R₂)...(1-Rₙ) For n identical: R_sys = 1 - (1-R)ⁿ Active redundancy: both running (standby: one idle until needed)

k-out-of-n system (k of n must work): R_sys = Σ_{i=k}^{n} C(n,i) × R^i × (1-R)^(n-i) Example: 2-out-of-3: R = C(3,2)R²(1-R) + C(3,3)R³ = 3R²-2R³

Reliability Block Diagram (RBD)

Draw blocks in series/parallel per functional architecture Series block: R = R_A × R_B Parallel block (redundant): R = 1-(1-R_A)(1-R_B) Mixed: simplify block by block

Software: Reliasoft BlockSim, Isograph Reliability Workbench, ITEM Toolkit

Fault Tree Analysis (FTA)

Gate Types

AND gate: all inputs must occur for output (lower probability) OR gate: any input causes output (higher probability) NOT gate: event occurs when input does NOT occur Vote gate: k of n inputs required

Minimal Cut Sets

Smallest combination of failures causing top event Boolean algebra to reduce tree

Probability of top event (independent basic events): OR gate: P_top = 1 - Π(1 - P_i) ≈ ΣP_i (for small P_i) AND gate: P_top = ΠP_i

Quantitative FTA: assign failure rates → compute probability of top event per unit time Qualitative FTA: find minimal cut sets → identify critical failure paths

Common Cause Failure (CCF): single event causing multiple failures simultaneously Beta factor method: λ_CCF = β × λ_independent (typical β = 0.01-0.1)

FMECA (Failure Mode, Effects & Criticality Analysis)

FMEA vs. FMECA

FMEA: qualitative (RPN = S×O×D) — see sensitivity-analysis skill FMECA adds: criticality Cr = ΣP_failure_mode × α_mode × β_effect × t α_mode = failure mode ratio (fraction of failures in this mode) β_effect = conditional probability of loss given failure mode occurs

Criticality matrix: severity category (I-IV) vs. probability (A-E) Category I (catastrophic) + Probability A (frequent): unacceptable — eliminate Category IV (negligible) + Probability E (improbable): acceptable

Standards: MIL-STD-1629A, SAE J1739, IEC 60812

Availability

Inherent Availability (no logistics delays)

A_i = MTBF / (MTBF + MTTR) MTTR = mean time to repair [hr]

Operational Availability

A_o = UPTIME / (UPTIME + DOWNTIME) = MTBM / (MTBM + MDT) MTBM = mean time between maintenance (all maintenance) MDT = mean down time (includes logistics, admin, active repair)

Target: A_o > 0.95 (industrial), > 0.99 (critical power), > 0.9999 (telecom "four nines")

Maintainability

MTTR distribution: lognormal typically P(repair in time t) = Φ(ln(t/M)/σ) where M = median repair time Maintainability design: accessibility, diagnostics (BIT/BITE), modular replacement

Spare Parts Planning

Economic Order Quantity (EOQ)

Q* = √(2DS/H) (optimal order quantity) D = annual demand [units/yr], S = order cost [$], H = holding cost [$/unit/yr] Reorder point: ROP = d × L (demand rate × lead time)

Sparing Level (Repairable Systems — METRIC model)

For each base and depot: Optimal stock based on: pipeline (parts in repair/transit) + safety stock P(backorder) = Σ_{n>S} (n-S) × P(n parts in pipeline | Poisson demand)

Practical shortcut: stock enough for 95% availability Required spares N: N such that P(X ≤ N) ≥ 0.95 where X ~ Poisson(λ × MTTR)

Reliability Growth (Duane Model)

As design matures, failure rate decreases with test experience λ(T) = λ₀ × T^(-α) [T = accumulated test time, α = growth slope 0.3-0.6] Plot: log(cumulative failures) vs. log(T) → straight line slope = 1-α

AMSAA model (MIL-HDBK-189C): statistical basis for Duane Used to: project final reliability from test data, plan test program length

Output

Provide: R(t) at design life, MTBF [hr], Weibull β and θ, B10 life [hr], system R via RBD, top event probability from FTA, A_o availability, spare parts quantity recommendation.