Tolerance Stack-Up Analysis Skill

Purpose

The Tolerance Stack-Up Analysis skill provides capabilities for dimensional tolerance analysis and stack-up calculations, enabling verification of assembly fits and functional requirements through systematic tolerance chain analysis.

Capabilities

Worst-case tolerance analysis
Statistical (RSS) tolerance analysis
Monte Carlo tolerance simulation
GD&T-based stack-up analysis
Assembly feasibility verification
Tolerance allocation optimization
CETOL/3DCS integration
Stack-up report generation

Usage Guidelines

Tolerance Analysis Methods

Method Comparison

Method	Approach	Application	Result
Worst-case	All tolerances at limit	Safety critical	Maximum variation
RSS	Statistical combination	High volume production	Probable variation
Monte Carlo	Random sampling	Complex assemblies	Distribution
6-Sigma	Process capability	Quality control	Defect rate

Worst-Case Analysis

Linear Stack-Up

Gap = Nominal gap +/- sum of all tolerances

For a simple assembly:
Gap_min = Nominal - sum(all positive contributors)
Gap_max = Nominal + sum(all negative contributors)

Or using sensitivity:
Gap = sum(ai * xi)
Tolerance = sum(|ai| * ti)

Where:
ai = sensitivity coefficient (+1 or -1)
xi = nominal dimension
ti = tolerance on dimension i

Direction Convention

Define positive direction:
- Dimensions adding to gap: positive (+1)
- Dimensions subtracting from gap: negative (-1)

Example (shaft in hole):
Gap = Hole_dia - Shaft_dia
Hole: +1 (increases gap)
Shaft: -1 (decreases gap)

Statistical Analysis

Root Sum Square (RSS)

Statistical tolerance (RSS):
T_rss = sqrt(sum(ti^2))

For unequal distributions (weighted):
T_rss = sqrt(sum((ai * ti)^2))

Assumes:
- Normal distribution
- Independent variables
- Process centered at nominal

Process Capability

Cp = (USL - LSL) / (6 * sigma)
Cpk = min((USL - mean)/(3*sigma), (mean - LSL)/(3*sigma))

For 6-sigma quality:
Cpk >= 2.0
PPM defective < 3.4

For tolerance analysis:
sigma = T / (3 * k)

Where k depends on desired Cpk:
k = 3 for Cpk = 1.0
k = 4 for Cpk = 1.33
k = 6 for Cpk = 2.0

Monte Carlo Simulation

Simulation Process

1. Define distribution for each dimension
   - Normal: mean, sigma
   - Uniform: min, max
   - Skewed: appropriate parameters

2. Generate random samples (N = 10,000+)
3. Calculate assembly result for each sample
4. Analyze output distribution
5. Determine percent out-of-spec

Distribution Selection

Scenario	Distribution	Parameters
Machined feature	Normal	Nominal, T/6 (Cpk=2)
Purchased part	Normal/Uniform	Per vendor data
Press fit	Truncated normal	Limits at tolerance
Unknown process	Uniform	Min, max

GD&T in Stack-Ups

Including GD&T

Position tolerance contribution:
Dia_positional / 2 = linear contribution (per direction)

For MMC position:
Contribution = (Position_tol + Bonus_tol) / 2

Bonus tolerance:
Bonus = |Actual_size - MMC_size|

Datum Reference Frame

Stack-up must follow datum precedence:
1. Establish primary datum (constrains normal)
2. Establish secondary datum (constrains one rotation)
3. Establish tertiary datum (constrains remaining DOF)

Feature control frame specifies:
|Position|0.5 MMC|A|B|C|

Analysis Process

Stack-Up Procedure

Define the Problem
- What gap/clearance is being analyzed?
- What is the acceptance criterion?
- What components are involved?
Create the Loop Diagram
- Start at one surface
- Follow chain to other surface
- Identify all contributors
- Assign directions
Gather Data
- Nominal dimensions
- Tolerances (bilateral, unilateral)
- Process capabilities
- Distribution data
Perform Calculation
- Calculate nominal gap
- Calculate worst-case variation
- Calculate statistical variation
- Compare to requirement
Document Results
- Stack-up spreadsheet
- Loop diagram
- Conclusions and recommendations

Tolerance Allocation

Optimization Strategies

If tolerance too tight:
1. Increase gap nominal (if possible)
2. Tighten critical dimension tolerances
3. Add adjustment or shim
4. Change assembly method
5. Accept higher defect rate

If tolerance too loose:
1. Relax non-critical tolerances
2. Reduce manufacturing cost

Cost-Tolerance Relationship

Approximate relationship:
Cost ~ 1 / Tolerance^n

Where n ~ 1.5 to 2 for machining

Tighten tolerances on:
- Lower cost features
- Higher sensitivity contributors

Process Integration

ME-004: GD&T Specification and Drawing Creation

Input Schema

{
  "analysis_name": "string",
  "requirement": {
    "type": "gap|clearance|interference|alignment",
    "nominal": "number",
    "min": "number",
    "max": "number"
  },
  "contributors": [
    {
      "name": "string",
      "nominal": "number",
      "tolerance": "number (bilateral half)",
      "direction": "+1|-1",
      "distribution": "normal|uniform",
      "cpk": "number (if normal)"
    }
  ],
  "method": "worst_case|rss|monte_carlo|all"
}

Output Schema

{
  "analysis_summary": {
    "requirement": {
      "min": "number",
      "max": "number"
    },
    "nominal_result": "number"
  },
  "worst_case": {
    "min_result": "number",
    "max_result": "number",
    "pass_fail": "pass|fail",
    "margin": "number"
  },
  "statistical": {
    "mean": "number",
    "sigma": "number",
    "min_3sigma": "number",
    "max_3sigma": "number",
    "percent_out_of_spec": "number",
    "cpk": "number"
  },
  "monte_carlo": {
    "mean": "number",
    "sigma": "number",
    "min_observed": "number",
    "max_observed": "number",
    "percent_out_of_spec": "number",
    "histogram": "data reference"
  },
  "sensitivity_ranking": [
    {
      "contributor": "string",
      "sensitivity": "number",
      "percent_contribution": "number"
    }
  ],
  "recommendations": "array"
}

Best Practices

Define acceptance criterion before analysis
Include all contributors in the chain
Verify dimensions from actual drawings
Use realistic process capabilities
Document assumptions and simplifications
Perform sensitivity analysis on tight results

Integration Points

Connects with GD&T Drawing for tolerance inputs
Feeds into DFM Review for manufacturing feasibility
Supports FAI Inspection for verification
Integrates with Design Review for approval

tolerance-stackup