Source: https://guthlabs.ai/docs/instruments/gear-ratio-train-calculator/mechanism/

instrument mechanism / verified-with-binding-errata

# Gear Ratio & Train Calculator mechanism

The inspectable method record for Gear Ratio & Train Calculator, kept separate from implementation and qualification.

Status: **VERIFIED_WITH_ERRATA**Lifecycle: **verified-with-binding-errata**

## Model scope

The core ideal magnitude formula is standard, but the specification is not sufficiently explicit to verify implementation unambiguously. Define separate symbols for speed (n) and tooth count (Z), explicitly define each stage ratio as n_out/n_in, add n_out = n_in × GR and T_out = efficiency × T_in/GR, and state the valid gear geometry and unit constraints.

- The stated ideal relation GR = N_out/N_in = T_in/T_out is correct only when GR is explicitly defined as the output-to-input speed ratio and losses are zero; the symbol N is ambiguous because it commonly denotes both speed and tooth count.

- For meshing gears, the tooth-count and pitch-diameter relations are missing: n_out/n_in = Z_in/Z_out = d_in/d_out for external gears. The calculator must state that diameters are pitch diameters and use consistent units.

- The compound-train rule is valid only when each stage ratio is defined consistently as output speed divided by input speed, with the compound gears on each intermediate shaft rotating at the same speed.

- If an efficiency factor is entered, the required torque formula is T_out = efficiency × T_in/GR; the specification provides no formula for applying that factor.

- Gear direction is not addressed: each external mesh reverses rotational direction, so output speed should be signed or direction should be reported separately.

## Units and conventions

- Speeds are in the same unit, such as rpm; the ratio is dimensionless.

- Input and output torques use the same torque unit, such as N·m.

- Tooth counts are dimensionless integers and must be positive.

- Gear diameters must be pitch diameters, not outside or root diameters, and must be converted to the same length unit before taking their ratio.

- The test vectors assume external spur gears, ideal efficiency of 100%, no slip, and negligible losses unless otherwise stated.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "train_type": "simple",
    "input_gear_teeth": 20,
    "output_gear_teeth": 40,
    "input_speed_rpm": 1800,
    "input_torque_Nm": 10,
    "efficiency": 1
  },
  "expected_outputs": {
    "overall_ratio": 0.5,
    "output_speed_rpm": 900,
    "output_torque_Nm": 20
  },
  "working": "GR = Z_in/Z_out = 20/40 = 0.5. Output speed = GR × input speed = 0.5 × 1800 = 900 rpm. Ideal output torque = input torque/GR = 10/0.5 = 20 N·m."
}
````

## Failure boundary

- Division by zero occurs if input speed is used as the denominator while calculating a ratio and input speed is zero.

- Division by zero occurs if output tooth count, output pitch diameter, overall ratio, or efficiency-adjusted denominator is zero.

- Zero or negative tooth counts, diameters, ratios, and efficiencies outside 0 < efficiency <= 1 must be rejected.

- A gear mesh requires compatible module or diametral pitch; arbitrary tooth counts or diameters do not guarantee physically meshing gears.

- Compound gears must be constrained so the paired gears on each intermediate shaft share the same angular speed.

- Pitch-diameter units must be normalized before division; mixing inches and millimetres without conversion produces an incorrect ratio.

- Ideal torque conservation is not physically honest for real gears with friction, windage, bearing losses, and mesh losses.

- External gear direction reverses at every mesh; reporting only a positive speed omits output rotation direction.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/gear-ratio-train-calculator/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/gear-ratio-train-calculator/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/gear-ratio-train-calculator/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-tranche4/tranche4-results.json`