BIZOMAT MAKEOS · Gears
Gear Backlash at Operating Temperature
How much backlash will this gear pair have when the housing and gears warm up, and could it jam?
Change in circular backlash of a spur gear pair when the housing and the gears expand differently: the change in centre distance of the housing bores, the growth of the gears’ pitch radii, the net change in operating centre distance and the resulting change in backlash (2 Δa tan φ), giving the hot backlash and a jamming check.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module | mm | Module. |
| Pinion teeth | Pinion teeth. | |
| Gear teeth | Gear teeth. | |
| Pressure angle | ° | Pressure angle. |
| Cold circular backlash | mm | Circular backlash at assembly temperature. |
| Housing expansion coefficient | µm/m·K | Expansion coefficient of the housing (23 for aluminium, 11 for cast iron). |
| Housing temperature rise | K | Rise in housing temperature from assembly to running. |
| Gear expansion coefficient | µm/m·K | Expansion coefficient of the gears (11.5 for steel; plastics 50–100 or more). |
| Gear temperature rise | K | Rise in gear temperature (gears usually run hotter than the housing). |
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
- What it is
- Gear module, pitch diameter ÷ teeth, used with the tooth numbers to find the centre distance.
- 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 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Pinion teeth
- What it is
- Number of teeth on the pinion, the smaller gear of the pair.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 500. Use a whole number. 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.
What this means: Gear teeth
- What it is
- Number of teeth on the mating gear, the larger wheel of the pair.
- Units and format
- Enter a plain number with no unit. The form accepts 6 to 2000. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Pressure angle
- What it is
- Pressure angle of the gear teeth, from the gear drawing or tooth standard.
- 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
- The backlash change is proportional to the tangent of this angle.
What this means: Cold circular backlash
- What it is
- Circular backlash measured or specified at assembly temperature, along the pitch circle.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 5 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.08 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Circular backlash, not normal backlash measured across the tooth flanks.
What this means: Housing expansion coefficient
- What it is
- Linear expansion coefficient of the housing material; as the housing warms, the bearing bores move apart.
- Units and format
- Enter a number in µm/m·K: micrometres per metre per kelvin. The form accepts 0.000001 µm/m·K to 200 µm/m·K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 23 µm/m·K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Housing temperature rise
- What it is
- Temperature rise of the housing from assembly to running; a negative value is a fall.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts -100 K to 300 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Gear expansion coefficient
- What it is
- Linear expansion coefficient of the gear material; as the gears warm, their pitch radii grow.
- Units and format
- Enter a number in µm/m·K: micrometres per metre per kelvin. The form accepts 0.000001 µm/m·K to 200 µm/m·K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 11.5 µm/m·K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Gear temperature rise
- What it is
- Temperature rise of the gears themselves from assembly to running; negative for cooling.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts -100 K to 300 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 70 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Gears often run warmer than the housing; measure both if you can.
Method and formulas
- Standard centre distance a = m (z₁ + z₂) ÷ 2.
- Housing bores move apart by a α_h ΔT_h; the gears’ pitch radii together grow by a α_g ΔT_g.
- Net operating centre distance change Δa = a (α_h ΔT_h − α_g ΔT_g).
- Backlash change Δj = 2 Δa tan φ (handbook relation between centre distance and backlash); hot backlash = cold + Δj.
- Example: 20/60 teeth, module 2, go from 0.08 mm to 0.10009 mm of backlash when warm.
Limitations
- Uniform temperatures; tooth thickness growth is included through the pitch radius approach.
- Bearing clearances and shaft deflection also change running backlash.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Set cold backlash for a gearbox with an aluminium housing
- Check plastic or bronze gears for jamming when warm
- Explain noise or rattle changes as a gearbox warms
Who it is for
- Gear designers
- Design engineers
- Maintenance engineers
What you get
- Centre distance change from the housing
- Pitch radius growth of the gears
- Hot backlash and jamming check
Interpreting the result
- An aluminium housing with steel gears gains backlash when warm; plastic gears in a steel housing lose it and can jam.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2163–2167, 375–378. Topic reference; no source tables or recommended settings are reproduced.
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