Design a transmission that carries torque between two sprockets with a roller chain. Adjust the tooth counts, pitch, speed and centre distance to see the speed ratio, chain velocity, tension, link count and polygon effect (speed variation) update in real time, and build a positive, quiet chain drive.
Parameters
Small sprocket teeth z₁
Tooth count of the driving sprocket
Large sprocket teeth z₂
Tooth count of the driven sprocket
Chain pitch p
mm
Distance between adjacent roller centres (standard size)
Small sprocket speed n₁
rpm
Centre distance C
mm
Distance between the two shaft centres. Typically 30-50× the small pitch diameter
Transmitted power P
kW
Results
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Speed ratio i
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Output speed n₂ (rpm)
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Chain speed V (m/s)
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Chain tension T (N)
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Chain link count
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Polygon effect (speed var. %)
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Roller chain drive — rotation animation
Torque is carried from the small sprocket (left) to the large sprocket (right) by the roller chain. The polygonal wrap and the travelling chain links are shown.
Pitch circle diameter d and chain velocity V. p: chain pitch, d₁: small sprocket pitch diameter (in metres). Chain tension is T = power/V.
$$\frac{\Delta V}{V}=1-\cos\!\frac{\pi}{z_1}$$
Chordal speed variation from the polygon effect. As the small sprocket tooth count z₁ rises the variation shrinks and the drive runs smoother.
What is the Roller Chain Drive Simulator?
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A "roller chain" — is it the same thing as a bicycle chain? How does it actually transmit power?
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Yes, a bicycle chain is the textbook example. A roller chain runs over toothed wheels called "sprockets". Each tooth positively engages a roller of the chain one by one, so it cannot slip the way a belt does. And unlike meshing two gears directly, the two shafts can sit far apart. So it transmits power "farther than gears" and "more positively than belts" — a handy element right in between.
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If it doesn't slip, it sounds unbeatable. Doesn't it have a weak point?
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Good question. The biggest weakness is the "polygon effect". You might picture the chain wrapping the sprocket as a smooth circle, but because the links are rigid straight segments it actually wraps as a regular polygon. So even when the sprocket turns at constant speed, the linear chain speed keeps speeding up and slowing down. Drop the "small sprocket teeth z₁" on the left down to 9 — the polygon-effect figure on the right jumps right up.
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You're right, lowering z₁ pushes the speed variation up near 6%. Is that really so bad?
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A pulsating speed means the chain accelerates and decelerates with a little "jolt" every cycle. That is a source of vibration and noise, and every time a link seats onto the sprocket it adds an impact that speeds up wear of the pins and bushings. That is why "at least 17 teeth on the small sprocket" is the rule of thumb on the shop floor. Look at the formula: the speed variation is (1−cos(π/z₁)), and it drops away nicely as z₁ rises. That is the curve in the "polygon effect vs teeth" chart below.
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How do you decide the chain length? You can't just cut it anywhere, right?
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A chain can only be joined in whole-pitch (whole-link) steps. So you use L = 2C/p + (z₁+z₂)/2 + ((z₂−z₁)/2π)²·p/C to get the number of pitches, then round it up to an even link count. Why even? Because an odd count needs a special "offset link", which is weaker. So in practice you tweak the centre distance a little so the chain comes out at exactly an even number of links.
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And how do you get the tension in the chain?
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Basically "power = force × speed". Divide the transmitted power P by the chain speed V and you get the working tension that pulls the chain, T = P/V. For the same power, the faster the chain moves the lower the tension. So designing the chain on the high-speed, low-torque side lowers the tension and makes life easier. The catch is that going faster brings the polygon-effect vibration into play, so balancing speed against tooth count is where the skill of chain design lies.
Frequently Asked Questions
The speed ratio is i = z2/z1, the ratio of the small to the large sprocket tooth counts. The output speed is n2 = n1·z1/z2. Chain velocity is found from the pitch circle diameter as V = π·d1·n1/60 (with d1 converted to metres). The pitch diameter of a sprocket follows from the tooth count z and pitch p as d = p/sin(π/z). This tool computes all of these and also the working chain tension T = power/V.
A chain wraps a sprocket not as a circle but as a regular polygon, so even when the sprocket turns at constant speed the linear chain speed pulsates periodically. This is the polygon effect (chordal action). The speed variation is (1−cos(π/z1))×100[%] and grows sharply as the small sprocket tooth count z1 falls. The pulsation causes chain vibration, noise, tension fluctuation and shorter life, so a small sprocket with at least 17 teeth is the usual recommendation.
The chain length in pitches is L = 2C/p + (z1+z2)/2 + ((z2−z1)/2π)²·p/C, where C is the centre distance and p the pitch. The first term is the two straight runs, the second the wrap around both sprockets and the third a correction for the difference in sprocket diameters. A chain can only be joined in whole-pitch steps, and to avoid offset links the calculated value is rounded up to the next even link count.
Few teeth make the polygon-effect speed variation rise sharply. For example z1=9 gives a speed variation of about 6%, while z1=17 keeps it near 1.7%. A large variation makes the chain move in jerks with vibration and noise, and the impact as each link seats on the sprocket grows, accelerating pin and bushing wear. Few teeth also increase the chain articulation angle and so the flexing fatigue. For these reasons a small sprocket should have at least 17 teeth, and 21 or more for high-speed drives.
Real-World Applications
Bicycle and motorcycle drives: Power from a bicycle pedal to the rear wheel, and from a motorcycle engine to the rear wheel, is carried almost entirely by roller chains. The gear ratio is set by the tooth-count ratio of the chainring (the large sprocket) and the rear-wheel sprocket, and the derailleur switches between combinations. Chain drives are chosen because they transmit human-scale or small-engine power positively, lightly and over a long distance.
Industrial machinery and conveying equipment: Conveyor drives, agricultural machinery, machine-tool spindle drives and hoists — many rotating machines that demand positive torque transfer use chain drives. They are adopted because they hold an exact speed ratio without slipping like a belt, withstand harsh environments (dust, oil, temperature swings) and allow a relatively free choice of centre distance.
Escalators and assembly lines: Escalator step chains and the overhead conveyors of car factories carry large loads positively at low speed. On the low-speed, high-torque side the chain speed V is small and the tension T large, so double-pitch chains or larger-pitch chains are chosen to spread the tension.
Preliminary machine-design study: Before a detailed strength calculation or pulling a chain-maker's selection table, a quick estimate like this tool fixes the rough figures for speed ratio, chain speed, tension and link count. Problems such as "chain speed too high → polygon vibration" or "tension too large → insufficient life" can be spotted at the layout stage, so the sprocket teeth and centre distance can be revised early.
Common Misconceptions and Pitfalls
The most common misconception is that "a smaller sprocket tooth count is better because it is compact". Fewer teeth do make the sprocket smaller, but the polygon-effect speed variation rises steeply. As the formula (1−cos(π/z₁)) shows, the speed variation differs by more than three times between z₁=9 and z₁=17. The variation leads to chain vibration, noise and impact loading, accelerating pin and bushing wear and shortening life. Aim for at least 17 teeth on the small sprocket, and 21 or more for high-speed, high-load drives.
Next, "the chain length can be set exactly to the calculated value". A chain can only be joined in whole-pitch steps, and an odd link count requires a weaker offset link. So the calculated number of pitches must be rounded up to an even link count. Conversely, adjusting the centre distance C by a few millimetres lets you tune the chain to exactly an even number of links. In real designs the centre distance is usually made "adjustable", or a tensioner (idler) is added, so the slack side does not sag. Use the theoretical length from this tool as a starting-point guide.
Finally, "only the chain tension T = power/V matters". The T this tool computes is the working tension needed to transmit power. A real chain also carries centrifugal tension (significant at high speed), the catenary tension from the chain's own weight, and impact tension during start-up and load changes. At high rotational speed or long centre distance the centrifugal and weight effects become non-negligible, and the safety factor against the allowable tension must be checked separately. This tool is for quickly grasping the basic layout quantities — speed ratio, speed, working tension, link count and polygon effect; the final chain selection should be made from the maker's rated-power tables and a service factor.
How to Use
Enter pinion sprocket tooth count (z1, typically 13–25 teeth) and driven sprocket tooth count (z2, up to 120 teeth) to establish speed ratio i = z2/z1
Select chain pitch (9.525 mm for #60, 12.7 mm for #80, 15.875 mm for #100) matching your torque and speed requirements
Set input shaft speed n1 (50–3000 rpm) and run the simulator to calculate output speed n₂, chain velocity V, chain tension T based on power transmission, and polygon effect deviation caused by sprocket tooth engagement
Worked Example
A conveyor drive transmits 5.5 kW at n1 = 1200 rpm using a #60 roller chain (pitch = 9.525 mm). Pinion z1 = 17 teeth, driven z2 = 68 teeth. Speed ratio i = 68/17 = 4.0. Output speed n₂ = 1200/4 = 300 rpm. Chain speed V = (1200 × 17 × 9.525)/(60 × 1000) = 3.24 m/s. At 5.5 kW and 3.24 m/s, chain tension T ≈ 1697 N. Required chain length = 2 × 200 mm center distance + (68+17)/2 = approximately 65 links. Polygon effect = (360/(2×17))² × sin²(360/34) ≈ 1.8% speed variation on the pinion shaft.
Practical Notes
Maintain center distance between 30–50 times the chain pitch; excessive distances increase slack and polygon effect; use 200–300 mm for industrial #60/#80 drives
Polygon effect becomes critical at high speeds (>2000 pinion rpm) with small sprockets; limit z1 ≥ 13 teeth to keep variation below 3% and reduce noise
For heavy shock loads (cranes, presses), upsize chain one grade and reduce speed ratio; a #100 chain at 1.5 m/s outlasts #80 at 3.5 m/s
Monitor actual chain tension using calipers on installed chain; deviation >5 N/mm² from calculated T indicates wear, misalignment, or lubrication failure