BIZOMAT MAKEOS · Gears
Gear Train Tooth Number Planner
What tooth numbers give this reduction ratio in one or two stages, and how close can I get?
Search for spur or helical gear tooth numbers that produce a target reduction in one or two stages within limits on tooth count and stage ratio: the best combination by ratio error, preferring hunting-tooth pairs (no common factor) and the smallest total teeth, with the actual ratio, error, output speed and a table of the five best trains.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Target reduction ratio (input ÷ output speed) | Required overall reduction (for a step-up drive enter the reciprocal and reverse the train). | |
| Input speed | rpm | Input shaft speed. |
| Stages | choice | Number of gear meshes in series. |
| Minimum teeth | Smallest pinion allowed (17 or more avoids undercut at 20°). | |
| Maximum teeth | Largest gear allowed. | |
| Maximum ratio per stage | Largest ratio allowed in one mesh (about 6–8 for spur gears). | |
| Prefer hunting-tooth pairs | choice | Prefer pairs whose tooth numbers share no common factor, so every tooth meets every other tooth (more even wear). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Target reduction ratio (input ÷ output speed)
- What it is
- How many times slower the output shaft must turn than the input shaft. The search looks for tooth numbers whose driven ÷ driver product comes closest to it.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 1000. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 23.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The tool searches reductions only; for a step-up drive enter the reciprocal ratio and run the train the other way round.
What this means: Input speed
- What it is
- Speed of the shaft that drives the first pinion. It does not change which teeth are chosen; it is used to report the output speed of each train.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1450 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Output speed is input speed ÷ actual ratio, so it rises in direct proportion.
What this means: Stages
- What it is
- How many gear meshes in series the train may have: one pinion and gear, or two pairs with an intermediate (compound) shaft between them.
- Units and format
- Choose one of 2 options from the list: One stage; Two stages.
- Example
- The form opens with “Two stages” selected as a worked example. Choose the option that matches your case.
- Get it right
- With one stage the target ratio must not exceed the maximum ratio per stage, otherwise the tool asks for two stages.
What this means: Minimum teeth
- What it is
- The fewest teeth any gear in the train may have, usually set by the smallest pinion that avoids undercut or still fits on its shaft.
- Units and format
- Enter a plain number with no unit. The form accepts 8 to 200. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 17 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raising it narrows the search, which can increase the ratio error or leave no solution.
- Get it right
- The undercut limit depends on pressure angle and profile shift; check it for your tooth form rather than reusing another design’s figure.
What this means: Maximum teeth
- What it is
- The most teeth any gear in the train may have, usually limited by the space available, the gear blank size or cost.
- Units and format
- Enter a plain number with no unit. The form accepts 10 to 300. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger limit widens the search and often finds a closer ratio, at the cost of bigger gears.
What this means: Maximum ratio per stage
- What it is
- The largest reduction a single mesh may take, driven teeth ÷ driver teeth. High single-mesh ratios mean a small pinion working against a large gear.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 6 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- This limits one pair, not the whole train; two stages can together reach up to the square of this value.
What this means: Prefer hunting-tooth pairs
- What it is
- Whether pairs whose tooth numbers share no common factor are ranked ahead of others with the same ratio error, so each tooth meets every tooth of its mate in turn.
- Units and format
- Choose one of 2 options from the list: Yes; No.
- Example
- The form opens with “Yes” selected as a worked example. Choose the option that matches your case.
- Get it right
- This only breaks ties in ratio error; it never accepts a less accurate ratio to get hunting-tooth pairs.
Method and formulas
- One stage: for each pinion z₁, gear z₂ = round(R z₁) within the limits.
- Two stages: for each first pair z₁/z₂ (ratio r₁ ≤ limit), the second pair z₃/z₄ = round(R ÷ r₁ × z₃).
- Ranked by ratio error, then hunting-tooth pairs (greatest common divisor 1), then fewest total teeth.
- Actual ratio = Π z_driven ÷ Π z_driver; output speed = input ÷ ratio.
- Example: a 23.5:1 reduction in 2 stage(s) is 17/94 × 20/85 (error 0%).
Limitations
- Ratio only; centre distances, strength and space need their own checks.
- Two stages searched exhaustively within the limits; three or more stages are not covered.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Pick tooth counts for a custom reduction gearbox
- Match an odd ratio for a feed or timing drive
- Avoid repeating tooth contact with hunting-tooth pairs
Who it is for
- Gear designers
- Machine designers
- Toolmakers
What you get
- Best tooth numbers
- Actual ratio and error
- Output speed and the five best trains
Interpreting the result
- If the error is too large, widen the tooth range or add a stage.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2131–2160, 2043–2047. Topic reference; no source tables or recommended settings are reproduced.
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