Chain Velocity⚠ unverified
Mechanical / Flexible Elements · Compute chain linear velocity from sprocket geometry and speed
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| pitch | pitch | in | 1.0 | Cha |
| teeth | teeth | — | 1.0 | Number of teeth on the driving sprocket, dimensionless |
| rpm | rpm | — | 1.0 | Sprocket rotational speed |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | v | ft/min | Chain linear velocity, in feet per minute (ft/min) |
The science & history
Understanding the Parameters
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Chain pitch $p$ — the distance between adjacent roller centers; it is the defining size of a chain (e.g. #40 chain is 1/2-inch pitch). The chain advances one pitch per tooth engaged.
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Tooth count $N$ — teeth on the driving sprocket. Since $pN$ is the pitch circumference, each revolution feeds $N$ pitches of chain. Small sprockets (few teeth) also worsen the chordal action — a cyclic speed variation because the chain wraps as a polygon, not a circle.
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The $12$ factor — inches-to-feet conversion, so the answer is in ft/min (US chain-drive convention).
- Why velocity matters — it multiplies the working tension into power, sets the required lubrication regime (drip/bath/spray as speed rises), and bounds the smoothness (chordal variation grows at low tooth counts and high speed). Sprockets are usually kept to $\ge 17$ teeth for smooth high-speed running.
Derivation (Approaching a Proof)
The chain wraps the sprocket as a regular polygon with $N$ sides of length $p$ (the pitch). One full sprocket revolution therefore pays out the full polygon perimeter:
$$\text{distance per revolution} = N\,p.$$
At $n$ revolutions per minute, the distance per minute is $N p\,n$ (in inches if $p$ is in inches). Converting to feet (÷ 12):
$$v = \frac{p\,N\,n}{12} \quad[\text{ft/min}].$$
This is the same circumference × rotation-rate logic as belt speed (V Belt Speed), with the polygon perimeter $Np$ playing the role of $\pi d$. (The true instantaneous speed oscillates by the chordal action, $\pm(1 - \cos(\pi/N))$; this gives the mean.)
Dimensional check. $\dfrac{p[\text{in}]\,N\,n[\text{1/min}]}{12[\text{in/ft}]} = \dfrac{\text{in}}{\text{min}} \cdot\dfrac{\text{ft}}{\text{in}} = \text{ft/min}$ — a speed, as required.
History and Development
Roller-chain drives (Renold, 1880s) transmit power positively and efficiently over longer center distances than gears and at higher loads than friction belts — bicycles, motorcycles, conveyors, and industrial drives. The pitch-based velocity relation is the foundation of chain kinematics; the polygon (chordal) effect it hints at is the reason chains run rougher than belts at low tooth counts, driving the "$\ge 17$-tooth" guidance for smooth high-speed drives.
Related Concepts: V Belt Speed, Chain Sag Allowance, Chain Wear Elongation, Timing Belt Pitch Diameter, Belt Power
Notes: Imperial — $p$ in inches, $n$ in rpm, $v$ in ft/min (the $12$ is inch→foot); SI form $v = pNn$. Distance per rev $= Np$ (polygon perimeter). Chordal action worsens at low $N$/high speed — use $\ge 17$ teeth. Sets power, lubrication regime, smoothness.