Thermal expansion and backlash calculation
A technopolymer gear, mounted with care, that seizes after a few hours of operation. The cause, almost always, is the same: the gap between the teeth wasn't calculated taking into account how the material expands with temperature. Backlash isn't an assembly imperfection. It's a design parameter, and it must be calculated before assembling.
What backlash is and why it's needed
Flank backlash, or backlash, is the space separating the non-meshing flanks of two mating teeth when the opposite flanks are in contact. It isn't a machining error or a defect: it's a deliberate distance, calculated at the design stage.
Without backlash, the system becomes statically indeterminate: the teeth lock kinematically as soon as the real conditions (temperature, tolerances, deformation) deviate from the theoretical value. The result is damage to the transmission.
The thermal expansion formula
The dimensional change of a component due to temperature follows this relation:
Where $L_0$ is the initial length, α is the material's thermal expansion coefficient (expressed in 10⁻⁶/K) and ΔT is the temperature change in degrees Celsius.
In technopolymers, the coefficient α is about ten times higher than in steel. A POM-C gear 100 mm in diameter that heats up by 50 °C expands by 0.55 mm: more than half a millimeter. Without a clearance adequate at the time of assembly, this expansion brings the tooth flanks into continuous contact on both sides, the seizing condition.
The thermal expansion coefficient of POM-C is about 110 × 10⁻⁶/K, against the 10-12 × 10⁻⁶/K of steel. For PA6 the value is similar to POM-C; for PK it settles around 85 × 10⁻⁶/K. Always verify the figure on the supplier's datasheet. The values change with the formulations and the presence of fiber reinforcement.
Table of recommended minimum clearances
The following values are minimum clearances for standard industrial applications with thermal swings up to 50-60 °C. For more severe conditions, increase the clearance proportionally.
| Module m (mm) | Recommended minimum clearance (mm) | Notes |
|---|---|---|
| 0.5 – 1.5 | +0.03 / +0.10 | Typical small transmissions |
| > 1.5 | +0.08 / +0.30 | Standard machines and power transmissions |
How to set the clearance
In practice, you set the clearance by slightly increasing the center distance relative to the theoretical value, which is the sum of the pitch radii of the two paired gears. For standard gears with a 20° pressure angle, a radial increase of the center distance of 0.1 mm generates an increase of the flank backlash of about 0.07 mm.
What happens without adequate clearance
A gear mounted without sufficient clearance is destined for rapid failure. When the temperature rises and the material expands, the flanks come into continuous contact. Local friction increases, the temperature rises further, the material expands even more, the friction rises again. It's a positive-feedback mechanism, the so-called thermal runaway, which ends in a few minutes with plastic deformation and collapse of the material.
No technopolymer application should be designed with zero clearance. If positioning precision is critical and you can't tolerate even a small backlash, the technopolymer gear isn't the suitable solution. Evaluate a metal profile or a transmission with an elastic element that absorbs the expansion.
If your application involves thermal swings above 60 °C or materials with off-standard expansion coefficients, write to the technical office. We calculate the correct clearance for your real operating conditions.