monte-carlo-tolerance

name	monte-carlo-tolerance
description	Monte Carlo tolerance analysis — statistical vs. worst-case tolerancing, process capability (Cp/Cpk), GD&T tolerance stacks, simulation setup (N iterations, distributions), probability of assembly failure, RSS method, ANOVA-based sensitivity, Six Sigma tolerance design, DFSS, ISO 286 fit selection.
metadata	{"priority":7,"promptSignals":{"phrases":["Monte Carlo tolerance","tolerance analysis","tolerance stack","Cp Cpk tolerance","statistical tolerancing","RSS tolerance"],"minScore":3}}

Monte Carlo Tolerance Analysis — Complete Skill

Tolerancing Methods Comparison

Worst-Case (WC) Tolerancing

Method: all parts at extreme tolerance limits simultaneously → compute assembly gap Assembly gap: G_WC = G_nominal - Σ |t_i| [sum of all component tolerances subtracted from nominal gap] Conservative: extremely conservative; assumes all parts simultaneously at worst tolerance Problem: probability of all parts at worst limits simultaneously ≈ 0 for N > 5 components → parts over-toleranced → cost excessive

RSS (Root Sum of Squares) Method

Assumption: component dimensions normally distributed; independent Assembly variation (1σ): σ_G = √(Σ σ_i²) [σ_i = standard deviation of i-th component = t_i / 3 for ±3σ tolerance; t_i = half-tolerance]

Assembly tolerance (±3σ): T_G_RSS = 3 × √(Σ (t_i/3)²) = √(Σ t_i²) [RSS assembly tolerance]

Example: 5 components, each t_i = 0.05 mm WC: T_G = 5 × 0.05 = 0.25 mm RSS: T_G = √(5 × 0.05²) = √(0.0125) = 0.112 mm (55% of WC)

RSS validity: normal distributions; Cp = 1 (centered processes)

Monte Carlo Simulation

Setup

Variable distributions per component: Normal (Gaussian): N(μ, σ) where σ = t_i/3 (for ±3σ process); use for most machined dimensions Uniform: U(nominal - t, nominal + t); for crude estimate or non-normal processes Log-normal: for positive-only dimensions with right-skew; some casting dimensions Truncated normal: if process has 100% inspection or hard physical limits

Key parameters: Mean (μ): nominal dimension (0 shift assumed for Cp process; shift for Cpk) Standard deviation (σ) = tolerance / (3 × Cp) = tolerance / 3 for Cp = 1

Number of iterations: N = 10,000: adequate for P_fail > 0.1%; error ≈ 1/√N ≈ 1% relative error N = 100,000: for P_fail > 0.01%; 0.3% error N = 1,000,000: for P_fail > 0.0001%; 0.1% error (rare events → very long runs) Rule: N × P_fail > 100 for reliable estimate (at least 100 failure events in simulation)

Simulation Execution

Algorithm:

1. For iteration j = 1 to N: a. Sample each component x_i ~ distribution(μ_i, σ_i) b. Compute assembly gap G_j = f(x₁_j, x₂_j, ..., x_n_j) c. Record G_j
2. From {G_j}: compute mean G̅, std_dev σ_G
3. Count failures: N_fail = count(G_j < G_min or G_j > G_max)
4. P_fail = N_fail / N

Python implementation:

import numpy as np

N = 100000
# Component tolerances (half-tolerances), nominal values
components = {'A': (10.0, 0.05), 'B': (5.0, 0.03), 'C': (15.0, 0.04)}

# Sample all components N times
samples = {name: np.random.normal(nom, t/3, N) for name, (nom, t) in components.items()}

# Compute assembly gap (example: G = A + B - C)
G = samples['A'] + samples['B'] - samples['C']

# Assembly requirement: G must be between G_min and G_max
G_min, G_max = -0.05, 0.10
P_fail = np.mean((G < G_min) | (G > G_max))
sigma_G = np.std(G)
print(f"P_fail = {P_fail:.4f} = {P_fail*1e6:.0f} ppm")
print(f"Assembly sigma = {sigma_G:.4f} mm")

Process Capability Integration

Cp (process capability index): Cp = T / (6σ) = (USL - LSL) / (6σ) [T = tolerance; USL, LSL = spec limits; Cp = 1 → ±3σ; Cp = 1.33 → ±4σ]

Cpk (centered process capability): Cpk = min((USL - μ) / 3σ, (μ - LSL) / 3σ) [accounts for process shift; Cpk ≤ Cp]

Six Sigma quality target: AIAG: Cpk ≥ 1.33 for production (minimum); Cpk ≥ 1.67 for safety-critical Six Sigma: Cpk = 2.0 → 6σ process → 3.4 ppm defect rate (with 1.5σ long-term shift)

Monte Carlo σ adjustment for process shift: μ_i = nominal + shift (use shift from historical Cpk data) σ_i = t_i / (3 × Cp) [adjust σ for Cp ≠ 1]

Example: Cp = 1.33 → σ = t / (3 × 1.33) = t / 4 (tighter distribution than Cp = 1) Reassemble G distribution → lower P_fail

GD&T Tolerance Stack Analysis

Linear Stack

For assembly gap G = Σ a_i × x_i: Sensitivity factor a_i: +1 (adds to gap), -1 (reduces gap) G_nominal = Σ a_i × x_i_nominal σ_G = √(Σ (a_i × σ_i)²)

GD&T contributions to stack: Position tolerance: contributes t_position / (√3 × 3) to σ for uniform distribution within circular zone Perpendicularity: angular error → linear equivalent at length L: δ = L × tan(angle) Flatness: adds directly to stack as planar variation

ANOVA Sensitivity Analysis

Which tolerances dominate? Contribution of each factor: η_i = (a_i × σ_i)² / σ_G² × 100% [percent contribution to total variance] Sum Σ η_i = 100%

Example analysis: 5 components: η = [30%, 25%, 20%, 15%, 10%] Dominant factors (30% + 25%) = 55% of variance → tighten these two for most impact

Decision: tighten vs. loosen: High η_i → dominant; tighten tolerance → significant assembly variation improvement Low η_i → minor contributor; safe to loosen tolerance → cost saving

Geometric Nonlinear Stacks

For complex geometries (e.g., angular chains): G = f(x₁, x₂, ..., x_n) [nonlinear function] Linearized sensitivity: ∂G/∂x_i (partial derivatives computed numerically or analytically) σ_G = √(Σ (∂G/∂x_i × σ_i)²) [first-order propagation; add second-order terms for high curvature]

Monte Carlo handles nonlinear exactly: no linearization needed; direct simulation → preferred for complex 3D stacks

ISO 286 Fit Selection

Clearance fits: H7/g6: sliding fit (precision mechanism); 0–50 μm clearance H7/f7: running fit (bearings, spindles); 20–100 μm clearance

Interference fits (press fits): H7/p6: light interference; press with thermal assist H7/s6: medium interference; permanent assembly

Transition fits: H7/k6: locating fit; may have clearance or interference (centered distribution) H7/n6: snug fit; slight interference likely

ISO 286 fundamental deviation and IT grade: Fundamental deviation: position of tolerance zone relative to nominal IT grade (International Tolerance grade): 01, 0, 1, 2, ..., 18 (finer to coarser) IT6 = 0.016 mm for 50 mm diameter; IT7 = 0.025 mm; IT8 = 0.039 mm

Monte Carlo for fit analysis: Sample shaft: x_shaft ~ N(μ_shaft = 50 + deviation, σ_shaft = IT_grade/6) Sample bore: x_bore ~ N(μ_bore = 50 + deviation, σ_bore = IT_grade/6) Clearance C = x_bore - x_shaft; find P(C < 0) = P_interference; P(C > C_max) = P_excessive_clearance

DFSS (Design for Six Sigma) Tolerance Design

Transfer Function Development

Step 1: identify Critical-to-Quality (CTQ) output y Step 2: identify key inputs x_i (component dimensions, material properties) Step 3: develop transfer function y = f(x_i) from physics or DOE Step 4: compute sensitivities ∂y/∂x_i Step 5: allocate tolerances inversely proportional to sensitivity (equal contribution rule)

Equal contribution rule: σ_xi = σ_y_target / (n^0.5 × |∂y/∂x_i|) [n = number of contributors; distributes variance equally] Alternative: proportional to manufacturing cost sensitivity

Taguchi Tolerance Design

Signal-to-noise ratio: S/N = -10 × log₁₀(σ²) [higher S/N → more robust to variation] L9 or L18 orthogonal array: vary component nominal values → identify optimal nominal → robust to manufacturing variation

Loss function (quadratic): L = k × (y - target)² [k = loss constant; economic loss from deviation] Expected loss: E[L] = k × (σ_y² + bias²) [minimize by reducing σ and bias] Tolerance = t → loss per defect × defect rate = break-even

Tools

Commercial: 3DCS (Dimensional Control Systems, Dassault): full 3D Monte Carlo; GD&T; mechanical assembly VisVSA (Dimensional Control Systems): similar; automotive industry standard CETOL 6σ (Siemens Tecnomatix): NX integrated; ANOVA; sensitivity charts Sigmetrix CETOL: popular in automotive and aerospace

Open source / custom: Python NumPy/SciPy: flexible; custom distributions; fast for N = 10⁶ MATLAB Statistics Toolbox: distributions, regression, Monte Carlo built-in

Standards

Standard	Scope
ISO 286-1/-2	Limits and fits for holes and shafts
ISO 5459	Datums and datum systems (GD&T)
ASME Y14.5-2018	Dimensioning and tolerancing (GD&T)
AIAG PPAP	Production part approval (Cpk ≥ 1.67 requirement)
ISO 22514-1	Statistical methods in process management (Cpk)
ISO 10303-210 (STEP)	CAD representation including tolerances

Output

Provide: assembly gap requirement (G_min and G_max [mm]), number of contributing components N, each component (nominal [mm], tolerance t_i [mm], Cp and Cpk from supplier data), distribution type (normal/uniform), assembly σ_G from RSS [mm], Monte Carlo results (P_fail [ppm], confidence interval), sensitivity analysis (η_i [%] for each component; top contributors), recommended tolerance tightening (which components; expected P_fail reduction), ISO 286 fit (if bearing/shaft: H7/g6 or equivalent; clearance distribution statistics), DFSS transfer function (if developed: y = f(x); sensitivities ∂y/∂x_i), and applicable standard (ASME Y14.5, ISO 286, ISO 22514, AIAG PPAP Cpk ≥ 1.67).