Tolerance Stackup

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

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

1. Define the Problem

   • What gap/clearance is being analyzed?
   • What is the acceptance criterion?
   • What components are involved?

2. Create the Loop Diagram

   • Start at one surface
   • Follow chain to other surface
   • Identify all contributors
   • Assign directions

3. Gather Data

   • Nominal dimensions
   • Tolerances (bilateral, unilateral)
   • Process capabilities
   • Distribution data

4. Perform Calculation

   • Calculate nominal gap
   • Calculate worst-case variation
   • Calculate statistical variation
   • Compare to requirement

5. 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

1. Define acceptance criterion before analysis
2. Include all contributors in the chain
3. Verify dimensions from actual drawings
4. Use realistic process capabilities
5. Document assumptions and simplifications
6. 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