KEVOS® MAKEOS · Gears
Enlarged Helical Pinion (20° Normal Pressure Angle) and Tight-Mesh Centre Distance
How much should I enlarge a helical pinion with few teeth to avoid undercut, what outside diameter does it take, and at what centre distance will it run in tight mesh with its mating gear?
Enlargement of full-depth helical pinions of 20° normal pressure angle by the handbook formulas: transverse diametral pitch Pt = Pn cos ψ, pitch diameter d = n ÷ Pt, transverse pressure angle tan φt = tan φn ÷ cos ψ, enlargement Kh = 2.1 − (n ÷ cos ψ)(sin φt − cos φt tan 5°) sin φt and outside diameter do = d + (2 + Kh) ÷ Pn; then the centre distance for tight mesh with the mating gear, inv φ = inv φt + (tnP + tnG − π) ÷ (n + N) and C′ = C cos φt ÷ cos φ with C = (n + N) ÷ (2 Pn cos ψ).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Teeth in pinion n | teeth | Number of teeth in the helical pinion. |
| Normal module mn (25.4 ÷ Pn) | mm | Normal module of the teeth; for a normal diametral pitch Pn it is 25.4 ÷ Pn. |
| Helix angle ψ | ° | Helix angle of the pinion. |
| Mating gear and tight mesh | ||
| Teeth in mating gear N | teeth | Number of teeth in the mating gear. |
| Pinion tooth thickness at 1 diametral pitch tnP (Table 9c) | Tooth thickness of the enlarged pinion at 1 diametral pitch, from the handbook’s Table 9c for its tooth number. | |
| Gear tooth thickness at 1 diametral pitch tnG | Tooth thickness of the mating gear at 1 diametral pitch; 1.5708 for a standard gear. | |
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: Teeth in pinion n
- What it is
- Number of teeth in the helical pinion to be checked for enlargement.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 1 teeth to 1000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 teeth as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Fewer teeth need more enlargement.
- Get it right
- Pinions with fewer than about 24 teeth usually need enlarging.
What this means: Normal module mn (25.4 ÷ Pn)
- What it is
- Normal module of the gear teeth, the size of tooth in the normal plane.
- 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 0.79375 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, lead and tooth dimension.
- Get it right
- For an inch normal diametral pitch Pn, enter 25.4 ÷ Pn.
What this means: Helix angle ψ
- What it is
- Helix angle of the pinion teeth, measured from the axis.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 60°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 18° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger helix angle needs less enlargement.
- Get it right
- The formulas are for 20° normal pressure angle teeth.
Mating gear and tight mesh
What this means: Teeth in mating gear N
- What it is
- Number of teeth in the standard mating gear.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 1 teeth to 10000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 24 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 standard and tight-mesh centre distances.
- Get it right
- Used only for the centre distances.
What this means: Pinion tooth thickness at 1 diametral pitch tnP…
- What it is
- Arc tooth thickness of the enlarged pinion at 1 diametral pitch.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.94703 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A thicker tooth pushes the gears further apart.
- Get it right
- Read it from the handbook’s Table 9c for the pinion’s tooth number.
What this means: Gear tooth thickness at 1 diametral pitch tnG
- What it is
- Arc tooth thickness of the mating gear at 1 diametral pitch.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5708 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A thicker tooth pushes the gears further apart.
- Get it right
- A standard gear has 1.5708 (half the circular pitch).
Method and formulas
- Pt = Pn cos ψ (transverse module mn ÷ cos ψ); d = n ÷ Pt; tan φt = tan 20° ÷ cos ψ.
- Kh = 2.1 − (n ÷ cos ψ)(sin φt − cos φt tan 5°) sin φt; do = d + (2 + Kh) ÷ Pn; no enlargement when Kh is not positive.
- C = (n + N) ÷ (2 Pn cos ψ); inv φ = inv φt + (tnP + tnG − π) ÷ (n + N) with inv x = tan x − x; C′ = C cos φt ÷ cos φ.
- Example: a 12-tooth pinion at 18° takes Kh = 0.8566 and an outside diameter of 12.283 mm; with a 24-tooth gear it meshes tight at 15.403 mm.
Limitations
- The formulas are for full-depth teeth of 20° normal pressure angle cut by hobs with sharp top corners; pinions cut by shaper cutters may need less enlargement.
- The handbook’s graph gives Kh directly and may differ slightly from the formula.
- An enlarged pinion needs a larger centre distance or a smaller mating gear.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the outside diameter of an enlarged fine-pitch helical pinion
- Find the centre distance for an enlarged pinion meshing with a standard gear
- Check whether a helical pinion needs enlarging at all
Who it is for
- Gear designers
- Machinists and gear cutters
- Students
What you get
- Transverse module and pitch diameter
- Transverse pressure angle
- Enlargement Kh and outside diameter of the pinion
- Standard centre distance
- Operating pressure angle and centre distance for tight mesh
Interpreting the result
- Fewer teeth or a smaller helix angle need more enlargement; pinions with about 24 or more teeth need none.
- The tight-mesh centre distance is the closest the gears can run; add backlash to it.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2206–2207, 2152. 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
Helical gears and axial thrust
What it covers: Convert between normal and transverse module, size a helical pair to a centre distance, and calculate the axial thrust and face overlap the helix creates.
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