Belt Efficiency⚠ unverified
Mechanical / Flexible Elements · Compute the approximate transmission efficiency of a belt drive
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F1 | F1 | N | 1.0 | Tight-side belt tension |
| F2 | F2 | N | 1.0 | Slack-side belt tension |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | efficiency | — | Approximate transmission efficiency, dimensionless. Returns 0.0 when ``F1`` is not greater than zero |
The science & history
Understanding the Parameters
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Tension ratio $F_1/F_2$ — the driver: $\eta = 1 - F_2/F_1$ increases as the ratio grows. At the slip limit $F_1/F_2 = e^{\mu\theta}$ (Belt Tension Ratio), so the maximum utilization is $\eta_{\max} = 1 - e^{-\mu\theta}$ — set by friction and wrap.
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What it really measures — the fraction of the peak belt tension converted to driving force. A high value means the belt is being used efficiently as a force transmitter (little "wasted" grip tension), which is why more wrap and higher friction (V-belts) improve it.
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What it does not measure — energy losses. A belt can have high tension utilization yet lose a few percent of power to creep (elastic stretching as tension varies around the pulley), flexing hysteresis, and bearing friction. Those set the true efficiency.
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Design use — as a guide to how hard the belt is working relative to its peak tension; combine with the actual $95$–$98\%$ power efficiency for energy calculations.
Derivation (Approaching a Proof)
The useful (driving) force is the difference between the two span tensions, $F_1 - F_2$. Normalising it by the tight-side tension $F_1$ gives the utilization fraction:
$$\eta = \frac{F_1 - F_2}{F_1} = 1 - \frac{F_2}{F_1}.$$
Substituting the slip-limit tension ratio $F_1/F_2 = e^{\mu\theta}$ gives the ceiling $\eta_{\max} = 1 - e^{-\mu\theta}$, showing utilization is bounded by friction and wrap — not by energy loss. True power efficiency instead compares output power to input power, $\eta_{\text{power}} = P_{\text{out}}/ P_{\text{in}}$, which requires the slip and hysteresis losses this formula omits.
Dimensional check. $\eta = 1 - F_2/F_1 = 1 - \text{N}/\text{N}$ = dimensionless — a ratio, as required.
History and Development
Belt drives are prized partly for their genuinely high power-transmission efficiency ($95$–$98\%$), better than many gear trains at comparable ratios. But that efficiency comes from low slip and flex losses, not from the tension-ratio quantity computed here. The distinction — between how much of the belt's tension is used and how much of the power survives — is a common source of confusion that this calculator's name invites; the page flags it so the two are not conflated.
Related Concepts: Belt Power, Belt Tension Ratio, Maximum Belt Power, Centrifugal Tension, Belt Stress, V-Belt Design
Notes: This is tension utilization $(F_1-F_2)/F_1$, not power efficiency (which is $95$–$98\%$, set by slip/creep/hysteresis/bearing losses). Ceiling $\eta_{\max} = 1 - e^{-\mu\theta}$ at the slip limit (Belt Tension Ratio).