sensitivity-analysis

name	sensitivity-analysis
description	Engineering sensitivity analysis — parameter sensitivity, Monte Carlo simulation, DOE (factorial design, RSM), FMEA, tolerance stack analysis, Taguchi methods.
metadata	{"priority":5,"promptSignals":{"phrases":["sensitivity analysis","Monte Carlo","design of experiments","DOE","FMEA","Taguchi","response surface","parametric study"],"minScore":4}}

Sensitivity Analysis — Complete Skill

Partial Derivative Sensitivity

For output Y = f(x₁, x₂, ..., xₙ): Local sensitivity: S_i = ∂Y/∂x_i × x_i/Y [normalized, dimensionless]

For failure criterion σ = Mc/I: S_M = ∂σ/∂M × M/σ = 1 (linear → 1:1) S_c = ∂σ/∂c × c/σ = 1 S_I = ∂σ/∂I × I/σ = -1 (inverse)

Importance ranking: |S_i| gives relative influence of each parameter High |S_i|: tighten tolerance on x_i; reduce manufacturing variation

Worst-Case (WC) Tolerance Analysis

Y_max = f(x_1+δ₁, x_2+δ₂, ...) for monotonic f Y_min = f(x_1-δ₁, x_2-δ₂, ...) for monotonic f δY = Σ|∂Y/∂x_i| × δx_i [linear WC]

Used when: 100% part acceptance required (safety-critical, small batch) Conservative: assumes all worst-case dimensions simultaneously

RSS (Root Sum Square) / Statistical Tolerance

δY_RSS = √[Σ(∂Y/∂x_i)² × δx_i²]

Interpretation: δY_RSS = ±1σ of output variation if inputs are ±1σ (Gaussian) For ±3σ inputs giving δx_i = 3σ_i: δY_RSS represents ±3σ output variation

Probability of Y within ±δY_RSS: ~99.73% (3σ both ways) When to use: large production runs (statistical approach valid)

Monte Carlo Simulation

1. Define input distributions: x_i ~ Normal(μ_i, σ_i) or uniform, log-normal
2. Sample N random inputs (N = 1,000-100,000)
3. Evaluate Y = f(x₁, x₂, ...) for each sample
4. Histogram of Y → estimate P(Y < Y_limit) = probability of failure

Advantages: handles nonlinearities, non-Gaussian distributions, any function When: analytical WC/RSS not feasible; need probability of failure P_f

Required N for accuracy: P_f ≈ 10⁻³: N ≥ 10,000 P_f ≈ 10⁻⁶: N ≥ 10⁸ (use importance sampling instead)

Design of Experiments (DOE)

Full Factorial 2^k Design

k factors, each at 2 levels (+1 high, -1 low) Experiments: 2^k total (exhaustive) Identifies all main effects and interactions

Main effect: A = (sum of Y at A=+1 minus sum at A=-1) / (2^(k-1)) Interaction: AB = mean(Y at A=+,B=+) + mean(A=-,B=-) - mean(A=+,B=-) - mean(A=-,B=+)

2^(k-p) Fractional Factorial

Alias structure: some effects confounded (aliased) Resolution III (min): main effects free of other mains but aliased with 2-factor interactions Resolution IV: main effects free of 2-factor interactions Resolution V: 2-factor interactions free of each other

Common design: 2^(7-4) = 8 runs for 7 factors (Plackett-Burman screening)

Response Surface Methodology (RSM)

Center Composite Design (CCD): adds axial and center points to 2^k Fits 2nd-order model: Y = β₀ + Σβᵢxᵢ + Σβᵢᵢxᵢ² + Σβᵢⱼxᵢxⱼ Optimum: ∂Y/∂xᵢ = 0 → find stationary point

Box-Behnken: 3-level design, no corners; good for constrained parameter space

FMEA (Failure Mode and Effects Analysis)

For each component → list all failure modes → for each failure mode:

Risk Priority Number: RPN = S × O × D

• S = Severity [1-10]: 1=no effect, 10=safety hazard
• O = Occurrence [1-10]: 1=almost never, 10=almost certain
• D = Detection [1-10]: 1=always detected, 10=not detectable

Action priority: RPN > 100 (or custom threshold) → corrective action required Focus on high RPN items first; then re-evaluate after mitigation

DFMEA: Design FMEA — product design phase PFMEA: Process FMEA — manufacturing process Automotive: AIAG-VDA FMEA 1st edition (2019) replaced AIAG 4th edition

Taguchi Method

Goal: minimize variance without expensive tolerance tightening Signal-to-Noise (S/N) ratio: Smaller-is-better: S/N = -10 log(Σy²/n) Larger-is-better: S/N = -10 log(Σ(1/y²)/n) Nominal-is-best: S/N = -10 log(σ²) or 10 log(μ²/σ²)

Orthogonal arrays: L9 (3^4), L18 (2^1 × 3^7), L27 — reduced experiments Identifies control factors (adjustable, cheap) vs. noise factors (environment, wear)

Robustness: choose factor levels maximizing S/N → design insensitive to noise

Reliability (MTTF, B10, Weibull)

Weibull distribution for time-to-failure: f(t) = (β/θ)(t/θ)^(β-1) × exp(-(t/θ)^β) β = shape parameter (β<1: infant mortality, β=1: exponential, β>1: wear-out) θ = scale parameter (characteristic life = 63.2% failed)

MTTF = θ × Γ(1 + 1/β) [mean time to failure] B10 life: time at which 10% failed = θ × (-ln(0.9))^(1/β)

Reliability: R(t) = exp(-(t/θ)^β) [fraction surviving at time t]

Failure Probability from Normal Distribution

P_f = Φ(-β_reliability) where β_reliability = (μ_R - μ_S) / √(σ_R² + σ_S²) μ_R = mean resistance, μ_S = mean load effect σ_R, σ_S = standard deviations

β_reliability required: structural = 3.5 (P_f ≈ 2×10⁻⁴), nuclear = 4.5, medical device = 5.0

Output

Provide: ranked sensitivity S_i table, δY_WC and δY_RSS, P_failure from Monte Carlo [%], RPN table for FMEA, DOE design matrix, S/N ratios for Taguchi, Weibull β and θ.