Belt_(mechanical) - Wikiparency

# Belt (mechanical) A **belt** is a loop of flexible material that runs over two or more [[Pulley|pulleys]] and carries [[Power_(physics)|power]] from a driving shaft to a driven one. Because the belt's speed is common to every pulley it touches, the shafts' speeds stand in inverse proportion to the pulley diameters, and a drive's ratio is set by choosing two diameters. The belt itself is the cheapest way yet found to move power across a gap: it needs no lubrication, tolerates misalignment and shaft movement that would destroy a [[Gear|gear]] pair, damps shock instead of transmitting it, and can be replaced in minutes. What a friction belt can carry is limited not by its strength alone but by its grip. A belt pulls harder on the side entering the driving pulley than on the side leaving it, and the difference between those two tensions, multiplied by the belt's speed, is the power delivered. Friction caps that difference: around a wrap angle `θ` at a [[Friction|coefficient of friction]] `μ` the tensions can differ at most by the factor `e^(μθ)`, the same exponential law that lets a rope turned a few times round a post hold a ship. Everything a belt drive designer does — moving the shafts apart, crowning or grooving the pulleys, adding an idler, raising the initial tension — is an attempt to buy wrap angle or effective friction. At high speed a second effect intervenes: the belt's own mass, swung round the pulleys, adds a [[Centrifugal_force|centrifugal]] tension that uses up the allowance the belt was rated for and carries no power at all. The framework microsim *Belt drives: the wrap angle sets what friction can carry* makes that competition visible. Moving the pulleys apart or shrinking the small one changes the wrap angle on the small pulley; the tension ratio the friction can hold follows as `e^(μθ)`; the tight side sits at the belt's maximum allowable tension while the slack side falls to what the ratio permits; and the difference times the belt speed is the power — until the speed is raised far enough that centrifugal tension eats the margin and the drive can carry nothing at all. ## History Belts entered engineering with the first factories, because a factory's problem was distribution rather than generation. One [[Water_wheel|water wheel]] or engine turned one shaft, and a [[Line_shaft|line shaft]] running the length of the building carried that rotation to every machine, with a flat leather belt dropping from the shaft to each one. The belt was the switch as well as the coupling: shifting it from a fast pulley to a loose idler stopped one machine without stopping the mill, and the [[Industrial_Revolution|Industrial Revolution]]'s [[Textile_manufacturing|textile]] mills ran on shafting and belting for more than a century. Long distances were covered with rope drives, several ropes running in grooved sheaves, which carried power hundreds of metres from a waterfall to a [[Factory|factory]] in the decades before electrical distribution made the question moot. Two inventions changed the picture. The rubber V-belt, conventionally credited to John Gates of the Gates Rubber Company around 1917, wedged itself into a grooved pulley and multiplied the available friction, so that a short drive between closely spaced shafts could carry real power for the first time.[^gates] Then the individual [[Electric_motor|electric motor]] removed the need for distribution altogether: each machine got its own motor, the line shafts came down, and the belt survived as a short drive between a motor and the thing it turned. The toothed belt, which meshes rather than grips, later took over work that had needed a chain — most visibly the [[Timing_belt_(camshaft)|timing belt]] that drives a [[Camshaft|camshaft]] at half crank speed, quietly and without oil. ## Power transmission For a drive of two pulleys of diameters `d1` and `d2`, the belt speed is common to both, so the [[Rotational_frequency|shaft speeds]] stand as `n2/n1 = d1/d2`.[^b109belt] The microsim's default drive — a 100 mm pulley at 1,450 rpm, a 250 mm pulley, shafts 500 mm apart — therefore turns its driven shaft at 580 rpm, with a belt speed `v = π d1 n1 = 7.59 m/s`.[^engsim-belt] The wrap angle on the smaller pulley, which is the one that slips first and is therefore the angle the whole drive is judged by, follows from the geometry of an open belt as `θ = π − 2 asin((d2 − d1)/2C)`, here 2.8404 rad or 162.7 degrees.[^manual3] The belt's length is `2C cos s + π(d1 + d2)/2 + (d2 − d1)s` with `s = asin((d2 − d1)/2C)`, which comes to 1,561 mm — the length that has to be ordered.[^engsim-belt] Power is the tension difference times the belt speed, `P = (T1 − T2) v`. With the tight side at a 1,000 N allowance and a friction-limited slack side of 433 N, the default drive carries 567 N × 7.59 m/s ≈ 4.30 kW, which is 28.4 N·m of [[Torque|torque]] at the motor shaft and 70.9 N·m at the driven one.[^engsim-belt] Nothing in that calculation depends on the pulleys' material or the belt's width except through `μ` and the allowable tension, which is why belt catalogues are organised by exactly those two quantities. The losses are small but not zero — a well-tensioned belt drive is an efficient machine element — and they are of two kinds. A belt stretches on the tight side and relaxes on the slack side, so material creeps backwards along each pulley's arc of contact as it passes from one tension to the other, and the driven pulley turns slightly slower than the diameter ratio predicts, by a fraction of a percent, even when nothing slips.[^shigley17] That elastic creep is unavoidable and distinct from gross slip, which is a failure of the drive rather than a property of it. The rest goes into bending the belt round the pulleys — hysteresis in the rubber, worse on small diameters — and into windage. Against those losses a belt drive offers quiet running, tolerance of misalignment, no lubrication, cheap replacement and, in the friction types, an overload release: a jammed machine slips its belt instead of shearing a key. The costs are an inexact ratio, a tension that must be maintained, and a shaft load the [[Bearing_(mechanical)|bearings]] carry whether or not power is being transmitted. *Try: slide the centre distance C down from 500 mm and watch the wrap angle on the small pulley close, the ratio `e^(μθ)` fall with it and the power bar shrink although nothing about the belt has changed; then raise the speed until the centrifugal bar `Tc` climbs into the tight-side allowance and the power collapses.* ### Flat belts The flat belt is the original and the simplest: a plain strap of leather, [[Natural_rubber|rubberised]] fabric or polymer running on cylindrical pulleys, which are given a slight crown so the belt self-centres on the high point. It bends easily, so it runs on small pulleys without much hysteresis loss and tolerates very high speeds; it can be crossed to reverse the driven shaft's direction, or twisted a quarter turn to drive a shaft at right angles, tricks no other drive can manage. Its weakness is grip: with nothing but plain friction on a plain pulley, the tension ratio available at a 180-degree wrap is modest, so a flat-belt drive needs large pulleys, long wraps and high initial tension for the power it carries. Round belts — small circular cross-sections running in V-grooves — do the same job for light drives such as [[Sewing_machine|sewing machines]] and appliance motors; coiled spring belts, wound from wire and joined into a loop, serve the same light duties where a belt must be fitted without dismantling the shafts; and film belts, thin and wide, work where a drive must be very light and very fast, as in tape transports. Wide flat belts scaled up become the [[Conveyor_belt|conveyor belt]] and the rolling road of a chassis dynamometer, in which the belt itself, rather than the shaft it drives, is the point of the machine. ### V belts A V-belt is a flat belt's answer to its own weakness. The belt's cross-section is a trapezoid that wedges into a matching groove, and the wedge multiplies the normal force: for a groove of included angle `φ`, the friction available becomes `μ/sin(φ/2)` instead of `μ`. With the usual 38-degree groove that factor is `1/sin 19° = 3.07`, so an [[Elastomer|elastomer]] belt at `μ = 0.3` behaves as though its coefficient were about 0.92, and the tension ratio the microsim's 162.7-degree wrap can hold rises from 2.35 to roughly 13.6. The microsim computes the plain-belt law `e^(μθ)`; the wedge case is the same arithmetic with `μ` replaced by the effective value, which is why the sim's friction slider is the right control to think with even for a grooved drive.[^shigley17] The practical consequences are large: V-belt drives work at short centre distances and small wrap angles where a flat belt would slip, several belts can be run side by side in a multi-groove [[Pulley|sheave]] to multiply the power, and the ribbed or [[Serpentine_belt|serpentine belt]] — many small V-ribs on one thin back — is flexible enough to be bent backwards round idlers, which is how a single automotive belt reaches every accessory on the front of an engine. ### Timing belts A timing belt has teeth moulded into its inner face that mesh with a toothed pulley, so the drive is positive: the ratio is exact, set by tooth counts rather than diameters, and no tension is needed to generate grip. Only enough tension to keep the teeth engaged is required, so shaft and [[Bearing_(mechanical)|bearing]] loads fall sharply compared with a friction drive of the same power. This is the drive of choice where phase matters — a [[Camshaft|camshaft]] that must stay in step with a [[Crankshaft|crankshaft]], a printer carriage, a machine-tool axis — and its failure mode is not slip but tooth jump, which loses timing all at once rather than gradually. The friction law above does not apply to it, and neither does the microsim's model: the positive-drive case is carried by the See-also variant *Roller chain: an exact ratio, and the chordal ripple of a polygon*, which draws a 17-tooth and a 38-tooth sprocket as pitch polygons on 12.7 mm pitch, an exact ratio of 2.235, a chain speed of 1.08 m/s at 300 rpm, and the chordal speed ripple `1 − cos(π/17)` of 1.7 percent that a polygon inflicts and a smooth belt does not.[^engsim-chain] ## Standards for use A belt drive is not calculated from first principles in practice; it is selected. The designer multiplies the driven machine's nominal power by a service factor for the duty — smooth motor and steady load at one end, reciprocating compressor and shock loading at the other — and takes the result to a catalogue whose tables give the power one belt of each cross-section will carry at a given small-pulley diameter and speed, with corrections for the wrap angle and the belt's length.[^shigley17] Published standards make that possible across manufacturers: they fix the cross-sectional dimensions, the groove angles of the sheaves, and the datum or pitch length system by which a belt is ordered, so that belts and pulleys from different makers interchange and a catalogue rating means the same thing everywhere. The same standards govern what is checked afterwards. Belt [[Wear|wear]] appears as glazed sidewalls, cracking of the backing, or loss of section depth where the belt has been running down into its groove; a worn belt in a new sheave, or a new belt in a worn one, sits at the wrong depth and loses the wedge that the section was rated for. Free spans [[Vibration|vibrate]]: each span behaves like a string under tension, and a [[Resonance|resonance]] excited by a shaft order or a pulley imbalance will flutter a belt until it fails. That same string behaviour is the basis of the ordinary field measurement of tension, since the span's fundamental frequency is `f = (1/2L)·√(T/m')`. The default drive's free span is 494 mm long and its mean tension 717 N, so with a belt of 0.2 kg/m the span should sound at about 61 Hz[^engsim-belt] — which is what a sonic tension meter listens for when a technician plucks the belt, and one place where [[Standing_wave|standing waves]] are a maintenance tool. Belt dressing, the sticky compound sold to stop a belt squealing, is deprecated in most modern guidance: it treats a symptom of low tension or a worn groove, adds a contaminant, and does nothing the correct tension would not do better. ### Belt friction The exponential law behind every friction drive comes from an element of belt subtending an angle `dθ` on the pulley. The normal force on that element is `dN = T dθ`, the friction it can supply is `μ dN`, and equilibrium along the belt gives `dT = μT dθ`, whose integral around the contact is `T1/T2 ≤ e^(μθ)`. It is a remarkable result for how little it contains: no pulley diameter, no belt width, no material property except `μ`, and no tension level — only the wrap angle and the friction coefficient.[^shigley17] It is conventionally named for [[Leonhard_Euler|Leonhard Euler]] and [[Johann_Albert_Eytelwein|Johann Albert Eytelwein]], who treated the problem of a rope on a drum in the eighteenth and early nineteenth centuries.[^euler] The inequality matters as much as the formula: the ratio is a ceiling, not a working value. A lightly loaded belt runs well below it with only part of its arc slipping, and a drive that reaches the ceiling is a drive that is about to squeal and glaze. At the microsim's defaults — `μ = 0.3` from the friction table the sim prints as its reference, and the 2.8404 rad wrap computed above — the ceiling is `e^(0.8521) = 2.345`.[^b077fric][^engsim-belt] That is the whole difference between a drive that carries 4.3 kW and one that does not. The exponent is what makes the law so strong when angle is cheap: the See-also variant *The capstan equation: each turn of rope multiplies what a hand can hold* winds a rope twice round a post, for `θ = 4π`, and the same coefficient gives `e^(0.3 × 4π) = 43.4`, so a 100 N pull by hand holds 4,338 N at the other end.[^engsim-capstan] Sailors at a [[Capstan_(nautical)|capstan]], climbers and riggers have used that arithmetic for centuries, and it is the same equation the belt lives by. *Try: raise μ from 0.2 to 0.4 and watch the point on the `e^(μθ)` curve climb steeply while the wrap angle stays where it was; then shrink the small pulley and see the wrap angle and the ratio fall together, the two ways a friction drive can lose its grip.* ### Belt tension A friction belt must be installed with an initial [[Tension_(physics)|tension]], because the two sides' tensions rise and fall about it: to a good approximation `T1 + T2 ≈ 2·Ti` for a belt of fixed length on fixed centres. Too little and the belt slips; too much and it overloads the belt, the shafts and the bearings, which is the most common way a belt drive is wrecked by a well-meaning fitter. The bearing load is roughly the sum of the two spans' pulls — about 1.43 kN for the default drive's 4.3 kW — and it is carried whenever the drive turns, loaded or not.[^engsim-belt] Speed adds a third tension. A belt of mass `m'` per unit length running at speed `v` needs an extra tension `Tc = m' v²` just to be forced round the pulley's curve, and that centrifugal tension is present on both sides: it raises the tension the belt must survive without adding anything to the difference that carries power. In the default drive it is trivial — 11.5 N out of a 1,000 N allowance, one percent.[^engsim-belt] Drive a 300 mm pulley at 3,000 rpm instead, and the belt runs at 47.1 m/s, where `Tc = 444 N`; with a 100 N allowance the sim reports zero transmissible power, because the belt has spent its entire rating on holding itself together.[^engsim-belt] Between those extremes lies an optimum: substituting the friction limit into `P = (T1 − T2)v` and maximising gives the classical result that a belt carries the most power when centrifugal tension takes exactly one third of the allowable tension, at `v* = √(Tmax/3m')` — 41 m/s for the default belt, where the same drive would carry about 15.6 kW instead of 4.3.[^shigley17] Real drives stay well below it, limited by belt materials, sheave balance and span [[Vibration|vibration]] rather than by the arithmetic. The belt's mass per unit length in the microsim, 0.2 kg/m, is a typical V-belt figure and is labelled ILLUSTRATIVE on the sim's own sheet and in its sources note: it is not a manufacturer's datum, and any drive near the centrifugal limit must be recomputed with the real belt's mass.[^engsim-belt] ## V-belt profiles V-belts are sold in a small number of standardised cross-sections, and the section is the first choice in a selection: it sets the power one belt can carry, the smallest [[Pulley|sheave]] it will bend round without excessive hysteresis, and the groove it needs. The classical inch series runs A, B, C, D and E in order of increasing size, and the narrow or wedge series — 3V, 5V and 8V — carries comparable power in a deeper, narrower section that grips better and runs on smaller sheaves, with metric equivalents designated in the same spirit.[^shigley17] Groove angles cluster between 34 and 38 degrees, with the smaller angles used on the smaller sheaves, where a belt bent round a tight radius spreads and would otherwise ride up out of its groove; the wedge factor `1/sin(φ/2)` of the section above follows directly from this angle, so the groove is as much a part of the drive's rating as the belt. Within a section, a belt is designated by its length in the datum or pitch system, which measures the length at the belt's neutral axis rather than at its inside or outside face, so that a belt's designation can be compared with a sheave's pitch diameter without correction. Variants of the basic profile exist for particular problems: cogged or notched belts, with slots moulded across the inner face, bend round small sheaves with less hysteresis heating and less [[Fatigue_(material)|fatigue]] in the backing; banded belts join several V-sections under a common back so that they cannot jump out of their grooves under shock; and the multi-rib serpentine profile, many shallow ribs on a thin flexible back, trades some wedge depth for the ability to bend both ways around a train of idlers. Where several plain V-belts share a drive they should be replaced as a matched set, since one longer belt in the set carries less than its share and the rest are overloaded. ## See also - [[Belt_friction]] - [[Capstan_equation]] - [[Roller_chain]] - [[Timing_belt_(camshaft)]] - [[Pulley]] - [[Conveyor_belt]] - [[Belt-driven_bicycle]] - [[Belt-drive_turntable]] - [[Chain_drive]] - [[Gear]] (section 9) - [[Cam_(mechanism)]] (section 11) - [[Friction]] (section 14) ## References [^b109belt]: Jensen (Portal Book 109). *Introduction to Mechanical Design and Manufacturing*, pp. 228–229, belt and chain drives: the speed ratio `d2/d1` of a belt pair and the exact tooth ratio of a chain. These are the forms the framework's `design.belt.ratio` implements. [^b077fric]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. Chapter 6 "Applications of Newton's Laws," §6.2 "Friction," Table 6.1 of coefficients, pp. 277–287. https://openstax.org/details/books/university-physics-volume-1 (Portal Book 077). Two rows of Table 6.1 are printed on the microsim's notes sheet as the reference for its `μ` control. [^shigley17]: Shigley's *Mechanical Engineering Design*, chapter 17, "Flexible Mechanical Elements" — the friction (capstan) limit on the tension ratio, centrifugal tension, initial tension, elastic creep, the classical A–E and narrow 3V/5V/8V V-belt sections and the selection procedure by service factor and catalogue rating. Cited as a named textbook form; edition and page numbers are not pinned for this article, and the run's build report records the sim's numbers as agreeing with these forms. [^engsim-belt]: Engineering portal microsim spec `specs/sims/Belt_(mechanical).json` and its build report (job E-E, `reports_engrun_2026-09-18.md`, September 18, 2026). Wrap angles and belt length by `design.belt.wrap`, ratio by `design.belt.ratio`, belt speed by `design.belt.speed`, the friction limit by `design.belt.capstanRatio(μ, θ)` evaluated at the smaller wrap, tensions and capacity by `design.belt.maxPower(Tmax, μ, θ, v, m')`. Hand check at the defaults (d1 100 mm, d2 250 mm, C 500 mm, μ 0.3, 1,450 rpm, Tmax 1,000 N): θ = 2.8404 rad = 162.7°, `e^(μθ)` = 2.345, v = 7.59 m/s, Tc = 11.5 N, T2 = 433 N, P = 4.30 kW, belt length 1,561 mm; and at d1 300 mm and 3,000 rpm, Tc = 444 N exceeds a 100 N allowance so the readout falls to zero power. ILLUSTRATIVE, as labelled on the sim's picture sheet and in its sources note: the belt mass per unit length of 0.2 kg/m (a typical V-belt value). The drawing is to scale but shrunk to fit, and a centre distance that would overlap the pulleys is raised to the touching distance plus 10 mm, with a note on the sheet. The report records no library gap for this sim. [^engsim-capstan]: See-also variant `Capstan_equation` of the same spec (job E-E, September 18, 2026): a rope wound two turns on a post, `θ = 4π`, `e^(0.3 × 4π) = 43.4`, so a 100 N hand tension holds 4,338 N; the ratio is drawn as a straight line on a logarithmic chart, clamped at 10⁶ with a caption. [^engsim-chain]: See-also variant `Roller_chain` of the same spec (job E-E, September 18, 2026): 17-tooth and 38-tooth sprockets on ISO 08B 12.7 mm pitch drawn as pitch polygons, exact ratio 2.235, chain speed 1.08 m/s at 300 rpm, and the chordal speed ripple `1 − cos(π/17)` = 1.7 percent. [^manual3]: Wikitube MICROSIM_GUIDE sub-manual `12_design_mfg`, §3 (belts and chains): the `design.belt` namespace — `wrap`, `ratio`, `speed`, `capstanRatio` and `maxPower` — and the convention that the friction limit is evaluated at the smaller of the two wrap angles. [^gates]: The rubber V-belt is conventionally credited to John Gates of the Gates Rubber Company, about 1917. Attribution and date as recalled from company and industry histories; a company history page or the original patent record would settle both, and neither was re-checked for this article. [^euler]: The exponential rope-friction law is conventionally named for Leonhard Euler and Johann Albert Eytelwein, who treated the rope-on-drum problem in the eighteenth and early nineteenth centuries. The attribution is standard in histories of mechanics; the precise priority and the original memoirs are not pinned here.  **Microsim — three.js (Wikitube framework):** *Belt (mechanical)* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/engineering/Belt_(mechanical).html" data-title="Belt (mechanical)"></div> *Built from `MICROSIM_GUIDE/specs/sims/Belt_(mechanical).json`; part of the [[PORTAL_Engineering|Engineering portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Belt_(mechanical)) : [Wikitube](https://en.wikitube.io/wiki/Belt_(mechanical)) - skeleton pinned to revision 1241115019 (2026-09-18).