Gear Ratio Design (`gear-ratio-design`)

Gear Ratio Design

Tooth count optimization for target ratio

Standards catalog

Validation: indicative · Method band: formula

Open calculator

Indicative method: Indicative closed-form or numerical model

Assumptions

Linear elastic material behavior unless noted otherwise.
User is responsible for load combinations and load factors per the selected design code.
Design standard (US/EU/ISO) sets unit defaults and screening check labels — not a full code worksheet.

Limitations

Professional screening / indicative workspace — does not replace a licensed PE or official code compliance review.
Where specialized evaluators are not implemented, checks map solver outputs to catalog templates for orientation only.

Engineering checks

Check	IND	US	EU	ISO
Ratio error	implemented	—	—	—

Purpose

Search integer tooth-count combinations to achieve a target speed ratio within specified tolerance. Optimizes for compactness, balanced wear, or minimum total teeth subject to interference and manufacturing constraints.

Physics & theory

Gear ratio for external spur gears is $i = N_{g} / N_{p} = ω_{p} / ω_{g}$ . Only integer tooth counts are manufacturable, so exact ratios are approximated: $i_{actual} = N_{g} / N_{p} \approx i_{target}$ . Error $ϵ = ∣ i_{actual} - i_{target} ∣/ i_{target}$ must fall within tolerance for synchronized drives.

Minimum tooth counts avoid undercut in standard involute profiles (typically $N \geq 17$ for 20° pressure angle without profile shift). Hunting tooth combinations (where common factors of $N_{p}$ and $N_{g}$ exceed 1) distribute wear unevenly — coprime tooth counts are preferred for long life.

Governing equations

i = \frac{N _{g}}{N _{p}}, ϵ = \frac{∣ N _{g} / N _{p} - i _{target} ∣}{i _{target}}

N_{p} + N_{g} \to min subject to g cd (N_{p}, N_{g}) = 1

Numerical method

Exhaustive or bounded integer search over tooth count ranges. Filters candidates by minimum teeth, interference, and ratio error. Ranks solutions by total teeth, center distance, or hunting tooth preference.

Inputs

Parameter	Description
`targetRatio`	Desired $i$
`tolerance`	Maximum ratio error
`minTeeth`, `maxTeeth`	Search bounds
`module`	For center distance estimate
Preferences	Min total teeth, coprime requirement

Outputs

Ranked tooth-count pairs, actual ratio, ratio error, center distance, hunting tooth flag.

Design codes & checks

Indicative: Ratio error screening

Assumptions & limitations

External spur pair; internal or compound trains not searched.
No profile shift or helical overlap considered.
Center distance assumes standard involute with zero backlash.
Does not verify bending/contact capacity — use Gear Design module.

References

Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 13.
AGMA 917-B97. Design Manual for Parallel Shaft Fine-Pitch Gearing.
ISO 21771:2007. Cylindrical involute gears and gear pairs — Concepts.
Buckingham, E. Analytical Mechanics of Gears. Dover.
PhyCalcPro verification benchmarks in src/data/verification/ where available for this module.
Beer, F. P., et al. Mechanics of Materials, 8th ed. McGraw-Hill — foundational stress and deformation theory.

Maintainer note: Integer tooth ratio search.

Gear Ratio Design

Assumptions

Limitations

Engineering checks

Gear Ratio Design (gear-ratio-design)

Gear Ratio Design (`gear-ratio-design`)