KEVOS® MAKEOS · Gears
Angular Backlash of a Gear Pair (and Backlash from a Centre Distance Change)
With a given backlash between two gears, how far can each gear rock back and forth while the other is held, and how much does a change in centre distance change the backlash or the tooth thickness I measure on a variable-centre fixture?
Angular backlash of a pair of gears from the backlash on the pitch circles by the handbook formulas: the angular movement of the larger gear with the smaller held, 6875 B ÷ D minutes, and of the smaller gear with the larger held, 6875 B ÷ d minutes, with the angular play also in degrees and arc-seconds; and the approximate relation used when tooth thickness is checked on a variable-centre-distance fixture, change in tooth thickness (or backlash) = 2 × change in centre distance × tan(transverse pressure angle), so a centre distance change can be turned into a backlash change.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Backlash on the pitch circle | mm | Backlash between the gears measured on the pitch circles. |
| Pitch diameter of the larger gear | mm | Pitch diameter of the larger gear. |
| Pitch diameter of the smaller gear | mm | Pitch diameter of the smaller gear. |
| Centre distance change | ||
| Increase in centre distance (negative for a decrease) | mm | Change in the centre distance from the one at which the backlash was measured; positive when the gears move apart. |
| Transverse pressure angle | ° | Transverse pressure angle of the gears, 20° for most modern spur gears. |
| Accepted play | ||
| Most angular play accepted at the smaller gear | min | Largest angular play you accept at the smaller gear (often the motor or input side). |
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: Backlash on the pitch circle
- What it is
- Backlash between the two gears, measured on the pitch circles.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The angular play grows in direct proportion.
- Get it right
- Measure at the tightest point of mesh with one gear held and an indicator on the other.
What this means: Pitch diameter of the larger gear
- What it is
- Pitch diameter of the larger gear of the pair.
- 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 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The larger gear rocks through a smaller angle.
- Get it right
- Module times teeth, or teeth divided by diametral pitch.
What this means: Pitch diameter of the smaller gear
- What it is
- Pitch diameter of the smaller gear of the pair.
- 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 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The smaller gear rocks through the larger angle.
- Get it right
- Module times teeth, or teeth divided by diametral pitch.
Centre distance change
What this means: Increase in centre distance (negative for a dec…
- What it is
- Change in centre distance from the one at which the backlash was measured.
- Units and format
- Enter a number in mm: millimetres. The form accepts -100 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.02 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moving apart adds backlash; moving together removes it.
- Get it right
- Positive when the gears move apart, negative when they move together.
What this means: Transverse pressure angle
- What it is
- Transverse pressure angle of the gear teeth.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 5° to 45°. 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
- A larger pressure angle changes the backlash more for a centre distance change.
- Get it right
- 20° for most modern spur gears, 14.5° for older ones.
Accepted play
What this means: Most angular play accepted at the smaller gear
- What it is
- Largest angular play you accept at the smaller gear.
- Units and format
- Enter a number in min: minutes. The form accepts 0.000001 min to 10000 min. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 min as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the play check.
- Get it right
- Set it from the positioning accuracy the drive must hold at that shaft.
Method and formulas
- θ_D = 6875 B ÷ D minutes (larger gear rocked, smaller held); θ_d = 6875 B ÷ d minutes (smaller gear rocked, larger held).
- 6875 = 2 × 180 × 60 ÷ π (rounded): the backlash divided by the pitch radius, in minutes.
- Change in backlash ≈ 2 × change in centre distance × tan(pressure angle), the handbook’s approximation for tooth thickness change on a variable-centre fixture.
- Example: 0.1 mm backlash gives 3.4375′ of play at the 200 mm gear and 13.75′ at the 50 mm pinion; moving the centres apart by 0.02 mm adds about 0.014559 mm.
Limitations
- The backlash is the one on the pitch circle; backlash measured normal to the teeth is smaller by the cosine of the pressure angle (and helix angle for helical gears).
- The centre distance relation is approximate and applies to small changes.
- The backlash a gear set needs is set by its accuracy, temperature range and duty; enter the backlash you have or intend.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Convert a gear backlash into the angular play at each shaft
- Find how much backlash a centre distance adjustment adds or removes
- Interpret a variable-centre-distance gear check
Who it is for
- Gear designers
- Machine designers
- Inspection technicians
What you get
- Angular backlash of the larger gear
- Angular backlash of the smaller gear
- Backlash change from the centre distance change
- Backlash and angular play after the change
Interpreting the result
- The smaller gear always rocks through the larger angle; on a reduction drive the motor side sees the play multiplied by the ratio.
- Adjusting the centre distance is a common way to set backlash at assembly in spur and helical gearing.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2168–2169. 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
Backlash and tooth thickness
What it covers: Specify backlash through tooth thickness and centre distance tolerances, and check that temperature and errors cannot close the clearance.
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.
Open the Angular Backlash of a Gear Pair (and Backlash from a Centre Distance Change) calculator