Belt Drive (Flat Belt / Capstan)

torque-power

Verified build 4 relations · 2 identities proven · 2 modeling steps · 6 parity samples

Before gears were cheap, factories ran on leather: one steam engine, one spinning lineshaft, and belts flapping down to every machine on the floor. Belts still drive your car’s alternator, your shop’s bandsaw, and every conveyor you’ve ever stood next to — and the same physics, run as a rope on a drum, is how a sailor holds a docking ship with one hand. The governing fact is an exponential:

$\frac{T_1 - T_c}{T_2 - T_c} = e^{\mu\theta} \qquad P = (T_1 - T_2)\,v \qquad T_c = m'v^2$

Three things to notice, one per formula:

Friction compounds. Each element of wrap multiplies the tension ratio by a little more than its neighbor did, because friction grows with the tension that friction has already built — the compound-interest condition, integrated into $e^{\mu\theta}$ . The numbers get silly fast: at $\mu = 0.3$ , one turn holds 6.6×, two turns 43×, four turns ~1900×. A capstan is a force amplifier with no moving parts, and the wrap angle knob in the widget is the whole machine.
Power is sold by the tension drop. The belt pulls with $T_1 - T_2$ , not with $T_1$ — the slack side rides along at $T_2$ ≥ its centrifugal floor, un-droppable. More wrap lets the drive convert more of its headroom ( $1 - e^{-\mu\theta}$ of it, 61 % at a half-turn for $\mu = 0.3$ , 98 % by two turns) — each added radian captures less than the one before, which is why an idler that adds wrap transforms a small drive and barely registers on one that’s already well wrapped.
Speed gives, then takes away. At fixed tension the power rises with $v$ — until the centrifugal slice $T_c = m'v^2$ , which presses the belt onto nothing and transmits nothing, eats the headroom. The peak sits at $v^* = \sqrt{T_1/3m'}$ (exactly one-third of the allowable tension doing centripetal duty), and past the ceiling $T_c \to T_1$ the widget refuses: the belt is spending all of itself staying on its own circle. Every number on this page is finite there — only the validity envelope knows the state is nonsense, which is the point.

The deliver configuration runs it backwards: name the power and the speed, and the widget reports the tension the belt must survive — then $v^*$ tells you whether you’re climbing the power curve or sliding down its far side.

Try it

Governing relations

T_c = m'\,v^2

Assumes: each belt element on the arc needs centripetal force m'v²·dφ, and the tension supplies it — a slice T_c of the tension does centripetal duty and never presses belt to pulley; belt speed well below the wave speed in the belt (no flutter dynamics)

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §17-2 "Flat- and Round-Belt Drives": the belting equation (F_1 − F_c)/(F_2 − F_c) = e^{fφ} (eq. 17-7), centrifugal tension F_c = (w/g)V² (lettered eq. (e), p. 877), transmitted power H = (F_1 − F_2)V.

\frac{T_1 - T_c}{T_2 - T_c} = e^{\mu\theta}

Assumes: belt on the verge of gross slip everywhere on the arc — this is the LIMIT a belt can exploit; a belt transmitting less sits inside it (Shigley writes the same equation as e^{fφ}); flat belt or rope on a drum — V-belts wedge and earn a larger effective μ; belt perfectly flexible, bending stiffness ignored · Valid while: Centrifugal tension has consumed the entire allowable tight-side tension — at this speed the belt is spending all of itself staying on its circle and can transmit nothing. This is the belt's hard speed ceiling; slow down, lighten the belt, or raise the allowable tension.

P = (T_1 - T_2)\,v

Assumes: steady transmission — the pulling force is the tension difference and it moves at belt speed

v^{*} = \sqrt{\frac{T_1}{3\,m'}} \quad (T_c = T_1/3)

Assumes: the stationary point of P(v) at fixed T_1, μ, θ — maximum power is transmitted when exactly one-third of the allowable tension is doing centripetal duty · Valid while: Past the power peak — above v* the centrifugal term steals tension faster than the speed adds power, so spinning this drive faster now delivers LESS power. You are on the wrong side of the hill.

Source: Khurmi, R. S., & Gupta, J. K., Theory of Machines, rev. ed., S. Chand — ch. 11 (Belt, Rope and Chain Drives), art. "Condition for the Transmission of Maximum Power": P is maximum when T_c = T/3, i.e. v* = √(T/3m).

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

T_{c} = m_{l} v^{2}

1. The modeling step, normal direction: an element of belt subtending dφ rides a circle at speed v, so it needs centripetal force m'v²·dφ pointing inward. The two tension ends of the element supply T·dφ of inward pull, and only the surplus N = (T − m'v²)·dφ actually presses the belt onto the pulley. A constant slice T_c = m'v² of the tension is therefore spoken for — it rides the belt and contributes nothing to grip. — element equilibrium: normal direction modeling step

T_{1} - T_{c} = \left(T_{2} - T_{c}\right) e^{\mu \theta_{w}}

2. The modeling step, tangential direction: on the verge of slip, each element adds friction dT = μ·dN = μ(T − m'v²)·dφ. The growth of tension is proportional to the tension excess itself — the compound-interest condition — and integrating from the slack side (T_2 at φ = 0) to the tight side (T_1 at φ = θ) gives exponential growth of the excess: T_1 − T_c = (T_2 − T_c)·e^{μθ}. Euler published this exponential in 1762 — it is the capstan equation a sailor exploits. (The build cannot integrate; the test pipeline solves the same ODE with a computer algebra system as the independent check.) — element equilibrium: tangential direction, integrated modeling step

P_{t} = v \left(1 - e^{- \mu \theta_{w}}\right) \left(T_{1} - T_{c}\right)

3. Power is the tension difference times the belt speed. Eliminating T_2 shows the structure: of the allowable tension T_1, the centrifugal slice T_c is lost off the top, and the wrap can only convert the fraction (1 − e^{−μθ}) of what remains into pull. More wrap asymptotically captures all of it — θ has diminishing returns — while T_c grows with v² without limit. — eliminate the slack side

3 m_{l} v_{star}^{2} = T_{1}

4. P(v) = (T_1 − m'v²)(1 − e^{−μθ})·v climbs linearly at low speed, but the centrifugal theft grows cubically; the peak sits where dP/dv = (T_1 − 3m'v²)(1 − e^{−μθ}) = 0, i.e. v* = √(T_1/3m') — maximum power when exactly one-third of the allowable tension is doing centripetal duty. Every belt drive has such a ceiling; high-speed flat belts are thin and light precisely to push v* up. (The calculus is re-done symbolically in the test pipeline.) — stationary point of P(v)

How it fails

The widget models a belt on the verge of slip, at steady speed, with honest tensions. Real belt drives die slower and stranger:

Slip is a regime, not an event. Below gross slip there is always creep: the belt stretches more on the tight side than the slack side, so it crawls relative to the pulley and the speed ratio is never quite the diameter ratio. Past the capstan limit, gross slip squeals, glazes the belt (burnishing it to a lower μ — a feedback loop), and cooks it with friction heat. A slipping belt fails by becoming a worse belt.
Fatigue at the pulleys. Every pass around a pulley bends the belt to that radius and back — a fully-reversed flexure cycle stacked on the tension cycle between slack and tight side. Small pulleys are the killer: belt life models are driven by the smallest sheave diameter in the drive, which is why minimum pulley diameters appear in every catalog.
Tension is a maintenance item. The capstan equation needs initial tension to exist — the belt only grips because it was stretched on. Belts creep, elongate, and bed into pulleys; the initial tension decays, the drive’s capacity decays with it, and one day the same load that ran for years starts to slip. Spring-loaded tensioners exist to make this failure impossible rather than periodic.
The wedge is a different bargain. V-belts multiply the effective friction by wedging into the groove (μ/sin of the half-angle) — more grip from the same μ, bought with extra flexure and a hotter belt. Timing belts opt out of friction entirely and inherit gear problems instead (tooth shear, ratcheting under shock).
Environment. Oil softens rubber and halves μ; ozone and UV crack it; heat ages it exponentially (industry rules of thumb put a halving of belt life at roughly every 10 °C of extra running temperature). Most real belt deaths are chemistry, not mechanics.
The flapping modes. A long free span is a tensioned string with its own natural frequencies; lineshaft-era engineers knew the sight of a span standing in resonance. Span flutter pumps tension cyclically, hammers the bearings, and can throw the belt — the dynamic failure the static model cannot see.

Planetary (Epicyclic) Gearset
Three coaxial members — sun, ring, planet carrier — share one gear mesh law. With two degrees of freedom, it has no single "ratio": fix a different member and the same hardware becomes a different transmission.
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Power Screw (Square Thread)
An inclined plane wrapped around a cylinder: torque in, brute linear force out. Friction is an honest knob here — it decides the torque, eats the efficiency, and (the good part) decides whether the load politely stays put or back-drives the screw when you let go.
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Spur Gear Pair (Lewis Bending)
Two meshing spur gears carry power through one tangential tooth force. The Lewis equation turns that force into a root-bending stress, and a cited form-factor table sets how much a tooth of N teeth can take. Same load, same module — yet the pinion, with fewer teeth, always works harder.
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Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)
The torque an axial plate clutch can pass depends on an assumption you cannot see: how the contact pressure is distributed across the friction annulus. A new, rigid clutch presses uniformly; a worn-in one wears until pressure ∝ 1/r, concentrating load at the inner edge. This page shows both torque predictions side by side — never picking a winner — with the worn-in model always giving the smaller (safe) number, and the r_i = r_o/√3 that squeezes the most torque from a given lining.
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Bolted Joint with Gasket (External Tensile Load)
A preloaded bolt clamping a gasketed joint, then pulled by an external tensile load. The bolt and the members act as two springs in parallel, so the external load does NOT all go to the bolt — it splits by stiffness. The build solves the coupled bolt/member force system exactly and refuses the moment the members go slack.
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Helical Compression Spring
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Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Composite Bar (Core + Sleeve)
- T_1 P
- T_2 P
- T_c P
Symmetric Two-Bar Truss
- T_1 P
- T_2 P
- T_c P
Cantilever Beam (End Load)
- T_1 P
- T_2 P
- T_c P
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
- T_1 P
- T_2 P
- T_c P
Simply Supported Beam (Center Load + UDL)
- T_1 P
- T_2 P
- T_c P
Transverse Shear in Beams (τ = VQ/Ib, Shear Flow, Fastener Spacing)
- T_1 V
- T_2 V
- T_c V
Shaft in Torsion (Solid, Circular)
- P_t P_w
Eccentric Column (Secant Formula)
- T_1 P
- T_2 P
- T_c P

+ 7 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §17-2 "Flat- and Round-Belt Drives": the belting equation (F_1 − F_c)/(F_2 − F_c) = e^{fφ} (eq. 17-7), centrifugal tension F_c = (w/g)V² (lettered eq. (e), p. 877), transmitted power H = (F_1 − F_2)V.
Khurmi, R. S., & Gupta, J. K., Theory of Machines, rev. ed., S. Chand — ch. 11 (Belt, Rope and Chain Drives), art. "Condition for the Transmission of Maximum Power": P is maximum when T_c = T/3, i.e. v* = √(T/3m).