Centrifugal Tension⚠ unverified
Mechanical / Flexible Elements · Compute the centrifugal tension in a moving belt
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| m | m | kg/m | 1.0 | Belt mass per unit length |
| v | v | m/s | 1.0 | Belt linear speed |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Tc | N | Centrifugal tension, in newtons (N) |
The science & history
Understanding the Parameters
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Mass per unit length $m$ — a linear density (kg/m), not a total mass. Heavier belts build more centrifugal tension at a given speed, which is why high-speed drives favour light, strong belts.
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Belt speed $v$ — centrifugal tension grows with the square of speed, so it is negligible at low speed but dominant at high speed. This quadratic growth is what ultimately caps belt power.
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Adds to both spans, drives nothing — $T_c$ appears equally in the tight and slack sides, so it raises the total tension (and belt stress) without increasing the difference $F_1 - F_2$ that does the work. It is pure overhead.
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Effect on grip — because it adds equally, it also reduces the pulley normal force available for friction; the effective tension ratio at the slip limit becomes $(F_1 - T_c)/(F_2 - T_c) = e^{\mu\theta}$, shrinking the usable driving force at high speed.
Derivation (Approaching a Proof)
Take a small belt element of length $ds = r\,d\theta$ on a pulley of radius $r$, subtending angle $d\theta$. Its mass is $dm = m\,ds = m r\,d\theta$, and it moves in a circle at speed $v$, so it requires a centripetal force directed inward:
$$dF_c = \frac{dm\,v^2}{r} = \frac{(m r\,d\theta)\,v^2}{r} = m v^2\,d\theta.$$
This inward force is supplied by the radial components of the belt tension $T_c$ on the two ends of the element, which sum to $T_c\,d\theta$ (the same geometry as the capstan derivation). Setting the supplied force equal to the required centripetal force:
$$T_c\,d\theta = m v^2\,d\theta \;\Longrightarrow\; T_c = m v^2.$$
The pulley radius cancels — centrifugal tension depends only on the belt's linear density and speed, not on pulley size.
Dimensional check. $T_c = m v^2 = (\text{kg}/\text{m})\cdot(\text{m}/\text{s})^2 = \text{kg}\cdot\text{m}/\text{s}^2 = \text{N}$ — a force, as required.
History and Development
Centrifugal tension became important as belt drives ran ever faster to transmit more power (power scales with speed, Belt Power). The insight that it adds to tension without adding to drive — and its quadratic growth — explains the existence of an optimum belt speed: maximum power occurs when the centrifugal tension equals one-third of the maximum allowable tension ($T_c = F_{\max}/3$, i.e. $v = \sqrt{F_{\max}/3m}$), a classic result in Shigley and belt-drive design (Maximum Belt Power).
Related Concepts: Maximum Belt Power, Belt Power, Belt Tension Ratio, Belt Stress, V Belt Speed, V-Belt Design
Notes: $m$ is mass per unit length (kg/m). Grows as $v^2$ — negligible at low speed, dominant at high. Adds equally to both spans → raises stress, drives nothing; shrinks usable tension at high speed. Optimum power at $T_c = F_{\max}/3$ (Maximum Belt Power).