V Belt Speed⚠ unverified
Mechanical / Flexible Elements · Compute belt linear speed from pulley diameter and rotational speed
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| d | d | in | 1.0 | Pulley pitch diameter |
| rpm | rpm | — | 1.0 | Pulley rotational speed |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | v | ft/min | Belt linear speed, in feet per minute (ft/min) |
The science & history
Understanding the Parameters
-
Pitch diameter $d$ — the effective diameter at which the belt's tension line sits (for a V-belt, inside the groove). Belt speed scales with it, so a larger drive pulley gives a faster belt at the same rpm.
-
Rotational speed $n$ — in rev/min; speed scales linearly with it.
-
The $\pi$ and the $12$ — $\pi d$ is the circumference (distance per revolution, in inches); multiplying by $n$ rev/min gives inches/min; dividing by $12$ converts to ft/min, the customary belt-speed unit.
-
Why speed matters — three ways: it multiplies tension into power (Belt Power); it drives centrifugal tension as $v^2$ (Centrifugal Tension), which caps power (Maximum Belt Power); and it affects belt fatigue (flexes per minute over the pulleys). Hence the recommended-speed window.
Derivation (Approaching a Proof)
The belt moves with the pulley rim (no slip). In one revolution a point on the pitch circle travels one circumference, $\pi d$. At $n$ revolutions per minute the distance per minute is
$$v = \pi d\,n \quad(\text{length per minute}).$$
With $d$ in inches, this is in/min; dividing by $12$ in/ft gives feet per minute:
$$v = \frac{\pi d\,n}{12} \quad[\text{ft/min}].$$
The $12$ is purely the inch-to-foot conversion; the physics is just circumference × rotation rate. (Chain velocity Chain Velocity uses the identical structure with pitch × teeth in place of $\pi d$.)
Dimensional check. $\dfrac{\pi\,d[\text{in}]\,n[\text{1/min}]}{12\,[\text{in/ft}]} = \dfrac{\text{in}}{\text{min}}\cdot\dfrac{\text{ft}}{\text{in}} = \text{ft/min}$ — a speed, as required.
History and Development
Belt speed in feet per minute is the traditional unit of US power-transmission practice, and the $\pi d\,n/12$ form appears throughout V-belt and pulley catalogs. The recommended speed band (peaking around $4000$–$5000$ ft/min for standard V-belts) reflects the trade-off this page's related concepts formalise: higher speed transmits more power until centrifugal tension and flex fatigue take over (Maximum Belt Power).
Related Concepts: Belt Power, Centrifugal Tension, Maximum Belt Power, Chain Velocity, Timing Belt Pitch Diameter, V-Belt Design
Notes: Imperial — $d$ in inches, $n$ in rpm, $v$ in ft/min; the $12$ is inch→foot. SI form is simply $v = \pi d n$. $\pi d$ = circumference (distance per rev). Same structure as Chain Velocity. Speed sets power, centrifugal tension, and flex fatigue.