tolerance-stack-analysis

name	tolerance-stack-analysis
description	Tolerance stack-up analysis — worst-case, RSS, DRSS, GD&T loop diagrams, clearance/interference calculation, Monte Carlo, ASME Y14.5, design for manufacture.
metadata	{"priority":7,"promptSignals":{"phrases":["tolerance stack","tolerance analysis","tolerance stackup","GD&T stack","worst case tolerance","RSS tolerance"],"minScore":3}}

Tolerance Stack-Up Analysis — Complete Skill

Purpose

Determine if assembly will function over entire tolerance range:

• Will clearance be maintained (e.g., gap ≥ 0)?
• Will interference be within limits?
• What is probability of conforming assembly?

Loop Diagram (Assembly Equation)

1. Identify the gap/clearance to analyze (the "closed loop")
2. Create vector loop through all contributing dimensions
3. Assign each dimension a direction (+/-)
4. Total: Gap = ΣD_i × ±sign_i

Sign convention: choose consistent direction (e.g., right = +) Each dimension that increases the gap gets +; each that decreases gets –

Worst-Case Analysis

Total tolerance accumulation: T_gap,WC = Σ |t_i| (arithmetic sum of all contributing tolerances)

Gap limits: Gap_min = Gap_nominal - T_gap,WC Gap_max = Gap_nominal + T_gap,WC

When to use: safety-critical assemblies; small batch; 100% compliance required Disadvantage: very conservative for many tolerances; forces tight component tolerances

RSS (Root Sum Square) — Statistical

Assumes normal distribution of each dimension (all within ±3σ; Cpk ≥ 1.0) Assembly tolerance at 3σ: T_gap,RSS = √(Σ t_i²) [3σ result; 99.73% compliance]

Or more precisely: T_gap = z_p × √(Σ (t_i/z_i)²) where z_i = process sigma level

Gap probability: μ_gap = Gap_nominal (expected value) σ_gap = T_gap,RSS / 3 (if each t_i = ±3σ_i) P(Gap ≥ 0) from normal CDF

When to use: mass production; moderate risk; parts from well-controlled process Advantage: allows wider component tolerances for same assembly confidence

DRSS (Dynamic RSS) — Six Sigma

Account for mean shifts (1.5σ shift in DMAIC convention): Modified RSS: T_gap,DRSS = √(Σ (t_i × f_i)²) f_i = 1.5 for dynamic conditions; 1.0 for well-controlled process

Or using Motorola 6σ: use Cpk = 1.5 to account for drift

DRSS result: assembly compliance at 99.99966% (6σ level)

GD&T Contributions to Stack

Flatness, straightness: contribute to tilt → worst at max material condition Perpendicularity/angularity: contribute rotation → proportional to distance True position (MMC bonus): extra tolerance when at MMC → reduce actual stack if at LMC

MMC concept in stacks: Bonus tolerance = departure from MMC Actual Mating Envelope (AME) → use for functional gages

Sensitivity Analysis

When tolerance i changes by Δt_i, gap changes by: ΔGap = Δt_i × s_i s_i = sensitivity = ±1 for linear; can be partial derivative for angular/geometric

Percent contribution: PC_i = (t_i²/Σt_j²) × 100% (RSS contribution) Focus optimization on top contributors (Pareto)

Monte Carlo Tolerance Analysis

1. Assign distribution to each dimension (normal, uniform, lognormal)
2. Sample n = 10,000–100,000 assemblies
3. Calculate gap for each assembly
4. Estimate P(gap < 0) = defect rate

When to use: non-normal distributions; nonlinear relationships; complex mechanisms Tools: MATLAB, Python (scipy.stats), 3DCS, Sigmetrix, VSA

Key Formulas Summary

Method	Formula	Use case
Worst case	T = Σt_i	Safety-critical, small n
RSS (3σ)	T = √Σt_i²	Production, moderate risk
RSS (6σ)	T = √Σ(1.5t_i)²	6σ target, with drift
Monte Carlo	Numerical	Complex geometry

Practical Rules

• Start with RSS; verify with worst-case for safety margins
• If worst-case too tight → loosen manufacturing tolerances or change design
• Target Cpk ≥ 1.33 on mating features (4σ process)
• Identify the top 2–3 contributors (usually 80% of variance) for optimization
• Include datum shift, setup variation, fixture repeatability

ASME Y14.5 Implications

Regardless of Feature Size (RFS): no bonus tolerance; tightest stack Maximum Material Condition (MMC): bonus tolerance reduces effective stack at LMC Virtual Condition (VC): worst-case mating boundary = MMC + position tolerance

Output

Provide: loop diagram, nominal gap [mm], worst-case T_gap [mm], Gap_min and Gap_max (WC), RSS T_gap [mm], P(conforming) [%], top tolerance contributors (Pareto), recommended tolerance changes for compliance.