Gear Power Capacity⚠ unverified
Mechanical / Gears · Compute transmitted power from tangential load and rotational speed
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| Wt | Wt | N | 1.0 | Transmitted tangential load |
| d | d | m | 1.0 | Pitch diameter |
| n | n | — | 1.0 | Rotational speed |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | P | kW | Transmitted power, in kilowatts (kW) |
The science & history
Understanding the Parameters
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Tangential load $W_t$ — the tooth force that does the work. The gear's strength (from Lewis bending and Hertzian contact) sets the maximum allowable $W_t$; feeding that into this formula gives the gear's power rating.
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Pitch diameter $d$ — sets both the torque arm ($d/2$) and, with speed, the pitch-line velocity. A bigger gear transmits more power for the same tooth load — but tooth load is itself limited by stress, so there is a design optimum.
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Rotational speed $n$ — in rev/min. Power is directly proportional to speed: the same tooth load at twice the speed transmits twice the power. This is why high-speed gearboxes pack a lot of power into small teeth (and why the dynamic velocity factor becomes important).
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The constants — $2\pi/60$ converts rev/min to rad/s (giving angular speed $\omega$), and $/1000$ converts watts to kilowatts.
Registry note: $n$ is labelled dimensionless but must be rev/min for the $2\pi/60$ factor to be correct. Noted in Known Issues.
Derivation (Approaching a Proof)
Power transmitted by a rotating shaft is torque times angular velocity, $P = T\omega$. Express each factor in gear variables:
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Torque from the tangential tooth load acting at the pitch radius: $$T = W_t\,\frac{d}{2}.$$
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Angular velocity from the rotational speed $n$ (rev/min) converted to rad/s: $$\omega = \frac{2\pi n}{60}.$$
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Power: $$P = T\omega = W_t\,\frac{d}{2}\,\frac{2\pi n}{60},$$ and dividing by 1000 expresses it in kilowatts.
Equivalently, group the geometry and speed into the pitch-line velocity $v = \omega\,(d/2) = \pi d n/60$ (the linear speed at which the teeth mesh), and the formula reduces to the intuitive
$$P = W_t\, v,$$
power = (tooth force) × (mesh velocity) — the same $P = Fv$ that governs any moving force (Power From Force Velocity). The pitch-line velocity is also what drives the dynamic (velocity) factor in the stress equations, tying rating and strength together.
Dimensional check. $W_t\,(d/2)\,\omega = \text{N}\cdot\text{m}\cdot\text{s}^{-1} = \text{W}$; dividing by 1000 gives kW. ✓
History and Development
Power = torque × speed is elementary rotational mechanics, but its gear form — via tangential load and pitch-line velocity — is the bridge between AGMA/Lewis stress limits and catalogue power ratings in Shigley and every gear-drive selection guide. Pitch-line velocity is a headline gear parameter because it sets both the transmitted power and the dynamic loading regime.
Related Concepts: Gear Bending Stress Lewis, Gear Contact Stress, Power From Force Velocity, Motor Torque, Gear Scoring Index
Notes: Power = torque × ω = tooth load × pitch-line velocity. $n$ in rev/min (see note). The allowable $W_t$ from bending/contact stress converts to a power rating here.