Source: https://guthlabs.ai/docs/instruments/gear-tooth-bending-stress-lewis-equation/mechanism/

instrument mechanism / verified-clean

# Gear Tooth Bending Stress (Lewis Equation) mechanism

The inspectable method record for Gear Tooth Bending Stress (Lewis Equation), kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

The arithmetic and conditional formula are valid for a simplified Lewis bending check using N, mm, and a dimensionless 20-degree full-depth Lewis form factor. Review is required because the spec leaves the Y-versus-y convention, tooth-system table convention, interpolation policy, and allowable-stress comparison undefined. The honest label should explicitly state that this is not an AGMA rating calculation and that gear and pinion members require separate checks.

- The stated formula is correct for the simplified Lewis equation only when Y is the dimensionless Lewis form factor y used in the selected table: sigma = Ft/(b*m*Y). Some references instead define a Lewis geometry factor Y = pi*y or use an AGMA geometry factor, which would make the numerical result differ by a factor of pi.

- The Lewis form factor depends on tooth count, pressure angle, tooth-depth standard, and whether the pinion or gear is being checked. The spec does not identify these table conventions.

- The stated output includes comparison to allowable, but no allowable bending stress input or comparison rule is specified, so that output cannot be computed.

- The formula is a basic static bending estimate and does not include AGMA dynamic, overload, size, load-distribution, rim-thickness, reliability, or stress-concentration factors.

## Units and conventions

- Ft is tangential tooth load in newtons at the pitch circle.

- b and m are entered in the same length unit; with b and m in millimetres, sigma is N/mm^2, numerically equal to MPa.

- Y is dimensionless and must be the Lewis form factor from the same tooth-system convention as the calculation; the table values below use 20-degree full-depth involute teeth.

- b, m, and Y must be positive; Ft is assumed nonnegative for a magnitude calculation.

- These examples check one gear tooth member only; the pinion and gear should both be checked using their respective tooth counts and form factors.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "Ft_N": 2000,
    "face_width_b_mm": 40,
    "module_m_mm": 2.5,
    "tooth_count": 20,
    "Lewis_factor_Y": 0.322
  },
  "expected_outputs": {
    "sigma_bending_MPa": 62.1118
  },
  "reference_data": {
    "Lewis_form_factor_20deg_full_depth_involute": {
      "tooth_count_20": 0.322
    },
    "source_convention": "Widely tabulated standard value; 20-degree full-depth involute Lewis factors, common Shigley machine-design textbook convention"
  },
  "working": "Denominator = b*m*Y = 40 mm * 2.5 mm * 0.322 = 32.2 mm^2. sigma = Ft/denominator = 2000 N / 32.2 mm^2 = 62.1118 N/mm^2 = 62.1118 MPa."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The zero-face-width input produces an undefined result and must be rejected. The specification also does not define the Lewis-factor table, interpolation behavior, or whether tangential force is signed or a positive magnitude, leaving lookup and convention hazards unresolved.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/gear-tooth-bending-stress-lewis-equation/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/gear-tooth-bending-stress-lewis-equation/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/gear-tooth-bending-stress-lewis-equation/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche7/tranche7-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`