KEVOS® MAKEOS · Gears
Enlarged Spur Pinion Mesh (Long and Short Addendum or Enlarged Centre Distance, Tight-Mesh Centre Distance)
When I enlarge a spur pinion to avoid undercut, what tooth thicknesses and outside diameters do the pinion and its mating gear take, at what centre distance will they mesh tight, and how few teeth can the reduced gear have?
Enlarged spur pinions by the handbook formulas, in the standard centre-distance (long and short addendum) system or the enlarged centre-distance system: outside diameters, arc tooth thickness at the standard pitch circle t = p ÷ 2 + e tan φ for the enlarged pinion and p ÷ 2 − e tan φ for a reduced gear, the centre distance for no backlash from inv φ1 = inv φ + (P (t + T) − π) ÷ (n + N) and C1 = C cos φ ÷ cos φ1, the smallest gear teeth without undercut N = 2 a csc² φ, and the smallest teeth of standard proportions a hob cuts without undercut, Nmin = 2 P csc² φ [aH − rt (1 − sin φ)].
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module m (25.4 ÷ P) | mm | Module of both gears; for a diametral pitch P it is 25.4 ÷ P. |
| Pressure angle φ | ° | Pressure angle of the gears. |
| Teeth in pinion n | teeth | Number of teeth in the enlarged pinion. |
| Teeth in gear N | teeth | Number of teeth in the mating gear. |
| Enlargement of pinion outside diameter e | mm | Amount the pinion outside diameter is increased over standard, from the handbook’s enlarged pinion tables or your design. |
| Reduction of gear outside diameter (0 = standard gear) | mm | Amount the gear outside diameter is reduced under standard; equal to the pinion enlargement keeps the standard centre distance. |
| Hob for the undercut check | ||
| Addendum of the hob aH | mm | Addendum of the hob or rack cutter, from its pitch line to its tip. |
| Tip radius of the hob rt | mm | Radius at the tip corners of the hob or rack cutter. |
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: Module m (25.4 ÷ P)
- What it is
- Module of the enlarged pinion and its mating gear, the pitch diameter per tooth in millimetres.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.254 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales every diameter, tooth thickness and centre distance.
- Get it right
- For an inch diametral pitch P, enter 25.4 ÷ P.
What this means: Pressure angle φ
- What it is
- Pressure angle of the gears at the standard pitch circle.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 10° to 35°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the base circles and every involute angle.
- Get it right
- 20° is the usual standard; 14½° and 25° are also used.
What this means: Teeth in pinion n
- What it is
- Number of teeth in the enlarged pinion of the pair.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 3 teeth to 10000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 teeth as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the pinion diameters and the centre distance.
- Get it right
- The handbook enlarges coarse-pitch pinions of 10 to 17 teeth at 20°.
What this means: Teeth in gear N
- What it is
- Number of teeth in the gear that meshes with the enlarged pinion.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 3 teeth to 10000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 teeth as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the gear diameters and the centre distance.
- Get it right
- A reduced gear needs enough teeth to avoid its own undercut.
What this means: Enlargement of pinion outside diameter e
- What it is
- Amount the pinion outside diameter is made larger than standard.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.21089 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Thickens the pinion teeth by e tan φ.
- Get it right
- Take it from the handbook’s enlarged pinion tables for the tooth number, or your design.
What this means: Reduction of gear outside diameter (0 = standar…
- What it is
- Amount the gear outside diameter is made smaller than standard.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Thins the gear teeth by e tan φ and raises its undercut limit.
- Get it right
- Equal to the pinion enlargement keeps the standard centre distance; 0 for a standard gear.
Hob for the undercut check
What this means: Addendum of the hob aH
- What it is
- Addendum of the hob or rack cutter, from its pitch line to its tip.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 300 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3175 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer addendum raises the smallest tooth number without undercut.
- Get it right
- From the hob maker; it equals the dedendum the hob cuts on a standard gear.
What this means: Tip radius of the hob rt
- What it is
- Radius at the tip corners of the hob or rack cutter teeth.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0762 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger radius lowers the smallest tooth number without undercut.
- Get it right
- From the hob maker; 0 for a sharp-cornered hob.
Method and formulas
- Do (pinion) = n m + 2 m + e; Do (gear) = N m + 2 m − e (gear).
- t = π m ÷ 2 + e tan φ (pinion); T = π m ÷ 2 − e (gear) tan φ.
- inv φ1 = inv φ + (t + T − π m) ÷ ((n + N) m), with inv x = tan x − x; C = (n + N) m ÷ 2; C1 = C cos φ ÷ cos φ1.
- Smallest gear teeth N = 2 a csc² φ with a = 1 + (gear reduction × P ÷ 2), the enlarged pinion’s addendum at 1 diametral pitch when the gear is reduced by the same amount.
- Nmin = 2 csc² φ [aH − rt (1 − sin φ)] ÷ m for teeth of standard proportions.
- Example: the enlarged 10-tooth pinion has a tooth 0.47574 mm thick and meshes tight with the 30-tooth gear at 5.1787 mm, 0.098658 mm over standard.
Limitations
- The tight-mesh centre distance has no backlash; add the backlash you need.
- The smallest-teeth formulas are for hob or rack cutters; shaper cutters undercut less.
- Take the enlargement for a given tooth number from the handbook’s tables or the gear standard you follow.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the tooth thicknesses and outside diameters for an enlarged pinion and reduced gear
- Find the tight-mesh centre distance when only the pinion is enlarged
- Check the reduced gear has enough teeth to avoid undercut
Who it is for
- Gear designers
- Machinists and gear cutters
- Students
What you get
- Outside diameters of pinion and gear
- Arc tooth thicknesses at the standard pitch circle
- Operating pressure angle and centre distance for no backlash
- Smallest gear teeth without undercut
- Smallest pinion teeth of standard proportions without undercut
Interpreting the result
- When the gear is reduced by the pinion’s enlargement the centre distance and pressure angle stay standard.
- With a standard gear the centre distance grows by about half the enlargement.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2142, 2154–2155. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Useful alongside · KEVOS® original book
Undercut and profile shift
What it covers: See why pinions with few teeth are undercut, calculate the minimum profile shift that prevents it, and find the new centre distance when shifts are added.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.