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ArticlePublished 11 Jul 2026Updated 21 Jul 20268 min readBy Kevin Jogin
KEVOS® Knowledge Library · Engineering → Mechanical Engineering

Engineering / Mechanical Engineering

Transmission Chains

A chain is a gear mesh unrolled into a strap: positive drive, exact average ratio, brutal load capacity — wrapped around a geometric confession. A chain cannot lie on a circle; it lies on chords, and everything odd about chain drives follows from that polygon.

  • Reading time · 8 min
  • 7 sections
  • Chordal action worked
  • A bearing per joint
a circle built of chords rise & fall: r − r·cos(180/z) each joint: a tiny plain bearing pulse per tooth: 1 − cos(180/z) → 4.05% at z = 11 seventeen teeth calm it to 1.70% — the smooth-running floor
Doc №KL-ENG-MECH-194
SectionEngineering → Mechanical Engineering
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DrawnKEVOS®
Date2026-07-11

§1Between belt and gear

The chain fills the gap the last page opened: positive drive like a gear, long centres like a belt — with the debts of both, and a geometry all its own.

Where the belt gripped and could slip, the chain engages: rollers seat in sprocket teeth, the average ratio is exactly the tooth-count ratio, and no tension pre-load is needed to make it so — a chain drive loads its bearings only with the working pull, sparing them the belt page’s standing tribute. Where the gear demanded rigid, precise centres, the chain spans whatever distance the strand reaches, forgives centre-distance error entirely (the slack side simply hangs differently), and delivers gear-class torque through a component sold by the metre and joined with a spring clip. The bills are equally definite. Positive engagement returns the timing belt’s clause: no slip means no fuse. Steel on steel returns the noise the belt hushed. Every joint is a loaded, articulating metal contact — §2 will call it exactly what it is — so the chain inherits the lubrication page as a survival manual. And underneath everything sits the hero’s confession: a chain of rigid links wraps a sprocket as a polygon, not a circle, and §3 computes what that costs. The chain, in short, is the honest middleweight: nearly a gear, nearly a belt, and priced accordingly in maintenance.

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§2Anatomy of the roller chain

Three concentric parts per joint — pin, bushing, roller — and the middle pair is the part that matters: every link of a roller chain is a small plain bearing, working for a living.

The construction alternates two link types: inner links, whose plates carry press-fitted bushings with free-turning rollers around them, and outer links, whose plates carry press-fitted pins passing through the neighbouring bushings. Trace the duties and the design explains itself. When the chain articulates onto and off a sprocket, the motion happens between pin and bushing — a loaded, oscillating journal exactly per the plain-bearings page, boundary-lubricated at every engagement, and the site of essentially all chain wear (§5’s subject). The roller’s whole job is the sprocket’s protection: as a tooth picks up the chain, the roller rolls into the seat instead of sliding, so tooth-face wear is traded down to a rolling touch — remove the rollers (as cheap bushed conveyor chain does) and sprockets pay the difference. And the plates carry the tension in pure tab-tension, their figure-eight waisting a fatigue shape, their press fits the joints’ structure. One sentence carries the maintenance philosophy of the whole page: a chain is dozens of tiny plain bearings hinged in series — so oil is not corrosion protection or a courtesy, it is the working film of every one of those bearings, and §6 delivers it accordingly.

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§3Chordal action

The chain lies on chords of the pitch circle, so the effective winding radius rises and falls once per tooth — and the pulse it puts into chain speed is pure trigonometry.

Speed pulse per tooth: variation = 1 − cos(180/z)
Sprocket teeth zVariationReading
114.05%rough — slow drives only
132.91%still lumpy
171.70%the classic smooth-running floor
211.12%quiet
250.79%near-circular
The mechanism is the hero’s picture: the chain’s span leaves the sprocket from a chord, whose distance from centre swings between the full pitch radius r (roller at the tangent) and r·cos(180/z) (mid-chord), so at constant sprocket rpm the chain is alternately winched fast and slow — a speed and tension pulse at tooth-passing frequency, felt as vibration, whip in the spans, and the characteristic chain thrum. The cure is in the left column: the pulse falls as the square of the tooth count’s reciprocal, roughly, so seventeen teeth is the traditional floor for anything running at speed, elevens live on hand-cranked and crawl duty, and the polygon is one reason chain, for all its strength, hands genuinely high-speed refinement back to the belt and the gear.
1.00 z = 11 — dips 4.05% z = 17 — dips 1.70% 0 1 2 3 rotation (tooth pitches) chain speed (fraction of mean)
Fig. 1. Chain speed against sprocket rotation for three tooth passes: the eleven-tooth sprocket scallops 4.05% deep at tooth-passing frequency, while seventeen teeth calm the same drive to 1.70% — the table’s numbers, drawn as the vibration the shafts actually feel.
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§4Speed and the pitch series

Chain speed is a counting argument — teeth per revolution times pitch — and the standard pitch series turns capacity into a stocking decision.

v = z p N  — teeth × pitch × rev/s: the sprocket pays out one pitch of chain per tooth passed
Example 1 — a mid-size drive’s tape measure

A 17-tooth sprocket in the everyday 12.7 mm pitch (the size sold as 08B in the ISO series and ANSI 40 in the American — half-inch pitch under both flags) at 900 rpm pays out v = 17 × 0.0127 × 15 = 3.24 m/s of chain. The formula’s shape carries the selection logic. Pitch is the capacity dial — plates, pins and bearing areas all scale with it — but §3 taxes big pitch on small sprockets, and inertia taxes it at speed, so the refined answer to more power is often not a bigger pitch but more strands: duplex and triplex chains run the smooth small pitch and share the pull across parallel rows, the standard move on compact, faster drives. Ratio arithmetic stays gear-simple (tooth counts, with the small sprocket setting both the ratio’s reach and §3’s roughness), and one civilised convention rounds out the drive: an odd-and-even pairing of sprocket teeth against an even chain-link count walks each roller around the tooth population rather than marrying roller to tooth — the hunting-tooth idea from the gear pages, spreading wear at the price of nothing.

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§5Wear is pitch growth

Chains do not stretch — their joints wear, every pin-and-bushing gains a few hundredths of clearance, and the sum reads as a chain that has grown measurably longer.

The word “stretch” misleads usefully corrected: the plates never yield; the pin–bushing bearings of §2 wear, each articulation transferring a whisper of steel to the oil, and since pitch is measured pin-centre to pin-centre, joint clearance is pitch growth. That makes chain condition uniquely measurable for a wear part: pull a span taut and measure over a counted number of pitches against the nominal — a vernier over a dozen links, or the go/no-go wear gauge that does the arithmetic in one bite. The service bands are standard trade wisdom: retire around 1.5–2% elongation on ordinary drives (tighter where the chain times something), because by 3% the rollers no longer land in the seats — each sits progressively higher and further up its tooth flank until the chain rides over and jumps, the endgame of every neglected drive. And the flank-riding explains the page’s hardest-learned rule: a worn chain has spent its miles re-cutting the sprocket teeth into hooks that match its grown pitch, so a new chain meshed with hooked sprockets is destroyed in a fraction of its life while the old sprockets feel “fine”. Chain and sprockets are a wearing set; on anything that matters they are measured together and replaced together.

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§6Chain-side practice

Four disciplines keep a chain drive honest — slack, line, oil and inspection — and each is five minutes standing between the drive and §5’s endgame.

Slack: a chain needs some — a taut chain loads every joint and bearing continuously and amplifies §3’s pulse into tension spikes — and the working setting is a modest total sag in the slack span, a few per cent of the span’s length, arranged on the bottom run where layout allows so gravity helps the catenary rather than fighting it; adjusters or an idler on the slack side (never the tight side) hold the setting as §5’s growth arrives. Line: sprockets must be coplanar and shafts parallel — a straightedge across both faces, exactly as at the sheaves — because offset running works the rollers against the tooth sides and grinds link plates against sprocket cheeks, wear no lubricant addresses. Oil: §2’s verdict delivered — the film must reach the pin–bushing interface, so oil is applied to the slack span’s link edges where joints are open, by can and brush at the bottom, drip feed above that, oil-bath dip and pumped stream as speed and power rise; grease on a dirty chain, famously, builds a grinding paste that out-wears neglect. Inspection: the §5 measurement on a calendar, plus eyes for hooked teeth, stiff links and rusty joints. None of it is skilled work; all of it is the difference between a drive that runs for years and the one that announces itself by climbing its sprocket.

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§7Quick reference

The working core of the page on one card rack.

Character

positive · long centres

no pre-tension on bearings

Anatomy

pin · bushing · roller

a plain bearing per joint

Chordal

1 − cos(180/z)

4.05% @11 → 1.70% @17

Numbers

v = zpN = 3.24 m/s

08B / ANSI 40 = 12.7 mm

Wear

measure pitch growth

retire ~2% · 3% jumps · replace as a set

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