KEVOS® MAKEOS · Gears
Spur Gears Cut for a Given Centre Distance and Ratio (Operating Pitch, Tooth Thickness, Root and Outside Diameters)
When no standard pitch fits the centre distance, how do I cut a pair of spur gears with a standard hob to run at that centre distance: what operating pitch and pressure angle result, and what tooth thicknesses, root diameters and outside diameters do I cut?
Spur gears of standard diametral pitch P and pressure angle φ redesigned to run at a specified centre distance C1 by the handbook procedure: operating pitch P1 = (NP + NG) ÷ 2 C1, operating pressure angle φ1 = arccos (P1 cos φ ÷ P), operating pitch and base diameters, basic tooth thickness 1.5708 ÷ P1 at the operating pitch circle, tooth thicknesses at the standard pitch circle t = (N ÷ P)(1.5708 ÷ N + inv φ1 − inv φ), dedendums b1 = bc − (t − 1.5708 ÷ P) ÷ 2 tan φ for a hob of addendum bc, root diameters N ÷ P − 2 b1 and outside diameters 2 C1 − (mating root diameter) − 2 (bc − 1 ÷ P).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module of the hob m (25.4 ÷ P) | mm | Module of the hob or cutter; for a diametral pitch P it is 25.4 ÷ P. |
| Pressure angle of the hob φ | ° | Pressure angle of the hob or cutter. |
| Teeth in pinion NP | teeth | Number of teeth in the pinion. |
| Teeth in gear NG | teeth | Number of teeth in the gear. |
| Centre distance required C1 | mm | Centre distance the gears must run at. |
| Addendum of the hob bc | mm | Addendum of the hob or cutter, measured from its pitch line to its tip. |
| Operating pressure angle range | ||
| Smallest operating pressure angle you accept | ° | Smallest operating pressure angle you accept; the handbook suggests about 18°. |
| Largest operating pressure angle you accept | ° | Largest operating pressure angle you accept; the handbook suggests about 25°. |
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 of the hob m (25.4 ÷ P)
- What it is
- Module of the hob or cutter used to cut both gears, 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 3.175 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 of the hob φ
- What it is
- Pressure angle of the hob or cutter 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 NP
- What it is
- Number of teeth in the pinion of the pair.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 5 teeth to 10000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 28 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 ratio and the pinion tooth thickness.
- Get it right
- Choose the teeth for the ratio you need first.
What this means: Teeth in gear NG
- What it is
- Number of teeth in the gear of the pair.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 5 teeth to 10000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 88 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 ratio and the gear tooth thickness.
- Get it right
- Gear teeth ÷ pinion teeth gives the ratio.
What this means: Centre distance required C1
- What it is
- Centre distance the gear pair must run at in the housing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 190.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the operating pitch and pressure angle.
- Get it right
- Take it from the housing bores, measured between shaft centres.
What this means: Addendum of the hob bc
- What it is
- Addendum of the hob or 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 4.2926 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the dedendums, root diameters and tip clearance.
- Get it right
- From the hob maker; it equals the dedendum the hob cuts on a standard gear.
Operating pressure angle range
What this means: Smallest operating pressure angle you accept
- What it is
- Smallest operating pressure angle you accept for the pair.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 1° to 45°. 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
- Flags a pressure angle that is too low.
- Get it right
- The handbook suggests about 18°.
What this means: Largest operating pressure angle you accept
- What it is
- Largest operating pressure angle you accept for the pair.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 1° to 45°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Flags a pressure angle that is too high.
- Get it right
- The handbook suggests about 25°.
Method and formulas
- Operating module m1 = 2 C1 ÷ (NP + NG) (P1 = (NP + NG) ÷ 2 C1); cos φ1 = m cos φ ÷ m1.
- DP1 = NP m1, DG1 = NG m1; base diameters × cos φ1; t1 = π m1 ÷ 2.
- tP = NP m (π ÷ 2 NP + inv φ1 − inv φ) and the same for the gear, with inv x = tan x − x.
- bP1 = bc − (tP − π m ÷ 2) ÷ (2 tan φ); root diameter DPR1 = NP m − 2 bP1; the same for the gear.
- DPO = 2 C1 − DGR1 − 2 (bc − m); DGO = 2 C1 − DPR1 − 2 (bc − m).
- Example: 28 and 88 teeth at 190.5 mm run at 24.719°; cut the pinion to 97.189 mm and the gear to 295.03 mm outside diameter.
Limitations
- The tooth thicknesses give zero backlash at C1; thin the teeth for the backlash you need.
- A negative dedendum means the root lies outside the standard pitch circle; check the teeth are not pointed.
- Check the contact ratio and the undercut of each gear separately.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Design a spur gear pair for a centre distance no standard pitch fits
- Find the tooth thicknesses and blank diameters to cut with a standard hob
- Check the operating pressure angle stays in a sensible range
Who it is for
- Gear designers
- Machinists and gear cutters
- Students
What you get
- Operating module (pitch) and pressure angle
- Operating pitch and base diameters
- Basic tooth thickness at the operating pitch circle
- Tooth thicknesses at the standard pitch circle
- Dedendums, root diameters and outside diameters to cut
Interpreting the result
- The tooth thickness is shared so each gear has the basic thickness at its operating pitch circle.
- Choose the hob pitch and pressure angle so the operating pressure angle stays within your range.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2139–2140. 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
Centre distance and contact ratio
What it covers: Calculate the standard centre distance and the transverse contact ratio, and see how a centre distance error changes the pressure angle and overlap.
Also in Undercut and profile shift
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.