Open Belt Length⚠ unverified
Mechanical / Flexible Elements · Length of an open belt drive
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| D | D | m | 0.3 | Large pulley diameter |
| d | d | m | 0.15 | Small pulley diameter |
| C | C | m | 1.0 | Center distance |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| L | L | m | Belt length |
The science & history
Understanding the Parameters
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Center distance $C$ — dominates the length through the $2C$ straight-span term; the arcs and correction are smaller. The formula is most accurate when $C \gg D, d$ (widely-spaced pulleys), the common case.
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Pulley diameters $D$, $d$ — enter through the wrap arcs $(\pi/2)(D+d)$ and the small correction $(D-d)^2/4C$. The correction accounts for the belt spans being slightly angled (not parallel) when the pulleys differ in size.
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The $(D-d)^2$ term — note it uses the difference of diameters for an open belt (the crossed-belt formula uses the sum, Belt Length Crossed). It is a small term but distinguishes the two drive types.
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Design use — belts come in discrete standard lengths, so one usually computes $L$, rounds to the nearest stock belt, then back-solves for the exact $C$ (and provides a take-up adjustment for tensioning).
Derivation (Approaching a Proof)
The belt has four parts: two straight tangent spans and two pulley wrap arcs. For an open drive with center distance $C$ and pulley radii $R = D/2$, $r = d/2$, the belt leaves each pulley along a common tangent line inclined at a small angle $\alpha$ where $\sin\alpha = (R - r)/C$.
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Straight spans: each tangent has length $\sqrt{C^2 - (R-r)^2} = C\cos\alpha$; two of them give $2C\cos\alpha$.
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Wrap arcs: the larger pulley wraps $\pi + 2\alpha$, the smaller $\pi - 2\alpha$; their arc lengths sum to $R(\pi + 2\alpha) + r(\pi - 2\alpha) = \pi(R+r) + 2\alpha(R-r)$.
Adding and using the small-angle expansions $\cos\alpha \approx 1 - \alpha^2/2$ and $\alpha \approx (R-r)/C$, then converting radii to diameters, collapses to the standard approximation
$$L \approx 2C + \frac{\pi}{2}(D+d) + \frac{(D-d)^2}{4C}.$$
The last term is the second-order correction from the span angle $\alpha$.
Dimensional check. Every term is a length: $2C$, $(\pi/2)(D+d)$, and $(D-d)^2/4C = \text{m}^2/\text{m} = \text{m}$. So $L$ is in metres, as required.
History and Development
The open-belt length formula is a staple of machine-drive design going back to line-shaft days, appearing in every design handbook (Shigley, manufacturer catalogs). The companion crossed-belt formula (Belt Length Crossed) — used when the pulleys must turn in opposite directions — differs only in using $(D+d)$ in the wrap and correction terms, reflecting the larger belt-crossing angle.
Related Concepts: Belt Length Crossed, V Belt Contact Angle, Belt Power, Belt Tension Ratio, V-Belt Design
Notes: Open drive (same-direction pulleys). Uses $(D-d)^2$ (crossed uses $(D+d)^2$, Belt Length Crossed). Accurate for $C \gg D,d$. Compute $L$, round to a standard belt, back-solve $C$ with take-up. SI here; the crossed-belt card is in inches.