Timing Belt Tooth Load⚠ unverified
Mechanical / Flexible Elements · Compute the load carried by each tooth in a timing belt
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| P | P | N | 1.0 | Total belt load (force) |
| teeth_in_mesh | teethinmesh | — | 1.0 | Number of teeth simultaneously engaged in mesh, dimensionless |
| pitch | pitch | in | 1.0 | Belt tooth pitch |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Ftooth | N | Load carried per tooth, in newtons (N) |
The science & history
Understanding the Parameters
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Total belt force $F$ — the tangential (effective driving) force at the pulley, $F = (F_1 - F_2)$, equal to torque ÷ pitch radius. This is what the meshed teeth collectively transmit.
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Teeth in mesh $n_{mesh}$ — how many teeth are engaged at once, roughly the wrap fraction times the tooth count ($n_{mesh} \approx N\phi/2\pi$). More teeth in mesh means each carries less, so designers keep the small pulley's tooth count (and wrap) high enough that $n_{mesh} \ge 6$ is commonly recommended.
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Per-tooth load $F_{tooth}$ — compared against the tooth shear/allowable to prevent tooth shear, and it drives the belt's fatigue and "ratcheting" (tooth-jump) limits. It is the timing-belt analog of a gear-tooth load.
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Equal-sharing assumption — the formula assumes the load divides equally. In reality the first tooth in mesh carries the most (like a bolt group or threaded joint), so the true peak tooth load is higher; the equal-share value is a design idealisation.
Derivation (Approaching a Proof)
The pulley transmits torque through the teeth currently engaged with the belt. If the total tangential force at the pitch line is $F$ and it is shared equally among the $n_{mesh}$ teeth in contact, each tooth carries
$$F_{tooth} = \frac{F}{n_{mesh}}.$$
The number in mesh follows from the wrap: with $N$ teeth around the pulley and a wrap angle $\phi$ (V Belt Contact Angle), $n_{mesh} \approx N\phi/(2\pi)$. The equal-sharing assumption parallels bolted- joint and gear analysis; the leading tooth actually takes a larger share, so the design allowable includes margin for that non-uniformity.
Dimensional check. $F_{tooth} = \dfrac{F}{n_{mesh}} = \dfrac{\text{N}}{(\text{–})} = \text{N}$ — a force, as required ($n_{mesh}$ dimensionless).
History and Development
Tooth-load sharing is central to synchronous-belt design: the belt's advantage over a friction belt (positive, slip-free drive) depends on the teeth not shearing or jumping under load. Keeping enough teeth in mesh — through pulley size, tooth count, and wrap — is the standard rule (e.g. $\ge 6$ teeth engaged), and the per-tooth load here is what that rule protects. It complements the pitch geometry (Timing Belt Pitch Diameter) to define a synchronous drive.
Related Concepts: Timing Belt Pitch Diameter, Belt Power, V Belt Contact Angle, Belt Tension Ratio, V-Belt Design
Notes: Input $P$ is a force (effective tension $F_1-F_2$), not power; the pitch input is unused.
$n_{mesh} \approx N\phi/2\pi$ — keep $\ge 6$ to avoid tooth shear/jump. Equal-sharing is idealised (lead tooth
carries more).