Calculation
vmax = ωr; vmin = vmax·cos(180°/z); δ = (vmax−vmin)/vmax
The chain does not run on a true circle but on a z-sided polygon, so instantaneous speed oscillates periodically around the mean value used elsewhere.
δ = 1 − cos(180°/z)
Chain polygon (chordal) effect and speed variation
The chain does not run on a true circle but on a z-sided polygon, so instantaneous speed oscillates periodically around the mean value used elsewhere.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Non-uniformity δ — Relative speed swing between the fastest and slowest instant within one pitch cycle.
- Tooth count z — Tooth count of the sprocket considered, usually the smaller one.
- Rotational speed n — Rotational speed of the sprocket considered.
- Chain pitch p — Distance between adjacent pin centers of the roller chain.
- Maximum instantaneous speed vmax — Chain speed at the instant of largest effective radius.
- Minimum instantaneous speed vmin — Chain speed at the instant of smallest effective radius, at the pitch transition.
Example
17 teeth at 1,000 rpm and a 19.05 mm pitch give vmax ≈ 5.428 m/s, vmin ≈ 5.336 m/s and δ ≈ 1.7%.
Assumptions and limits
Ideally rigid chain links and constant drive speed; chain elongation, bearing clearance and the feedback of polygon action onto the driven sprocket are not included.
Understand Chain polygon (chordal) effect and speed variation
This calculator finds the instantaneous chain speed variation caused by sprocket polygon (chordal) action, from tooth count, rotational speed and pitch.
What does this quantity describe?
A chain does not run on a true circle but follows a z-sided polygon whose vertices are the sprocket teeth. At constant rotational speed, instantaneous chain speed therefore oscillates periodically between a maximum vmax = ωr, when a chain pin sits at the greatest distance from the axis, and a minimum vmin = vmax·cos(180°/z) at the transition between two pitches, where r is the pitch-circle radius.
Picture rolling a hexagon instead of a circle: a point on the hexagon's edge does not move uniformly under constant rotation but slightly speeds up and slows down within each edge. The more corners the polygon has, the closer it approaches a true circle and the smaller this variation becomes.
Formula and variables
vmax = ωr; vmin = vmax·cos(180°/z); δ = (vmax−vmin)/vmax
vmax = [p/(2·sin(180°/z))]·ωvmin = vmax·cos(180°/z)δ = 1 − cos(180°/z)
| Symbol / input | Meaning |
|---|---|
| Non-uniformity δ | Relative speed swing between the fastest and slowest instant within one pitch cycle. |
| Tooth count z | Tooth count of the sprocket considered, usually the smaller one. |
| Rotational speed n | Rotational speed of the sprocket considered. |
| Chain pitch p | Distance between adjacent pin centers of the roller chain. |
| Maximum instantaneous speed vmax | Chain speed at the instant of largest effective radius. |
| Minimum instantaneous speed vmin | Chain speed at the instant of smallest effective radius, at the pitch transition. |
Choose the inputs correctly
Tooth count z, together with chain pitch p and rotational speed n, sets the absolute speeds vmax and vmin; the non-uniformity δ, by contrast, depends only on tooth count z.
How to use the calculator
Choose δ as the target to find the relative speed variation for a given tooth count, or z as the target to find the minimum tooth count for an allowable variation. For absolute speeds, additionally enter pitch and rotational speed.
Worked example
At 17 teeth, a 19.05 mm pitch and 1,000 rpm, vmax ≈ 5.428 m/s and vmin ≈ 5.336 m/s, giving a non-uniformity of δ ≈ 1.7%.
Understand the result and units
A δ of 1.7% at 17 teeth is considered uncritical; at only 9 teeth, δ instead rises to about 6% and can cause noticeable vibration and noise. This is why at least 17 teeth are recommended in practice for the smaller sprocket.
Tooth count is dimensionless, rotational speed is given in rpm, chain pitch in mm, speeds in m/s, and the non-uniformity δ dimensionless or as a percentage.
Why at least 17 teeth are recommended
As tooth count falls, the angle 180°/z that each chain link sweeps through while engaging and disengaging grows, and with it both the speed variation δ and the relative articulation angle at the joints. At z = 9, δ is already about 6%; at z = 17 it is only about 1.7%; at z = 20 about 1.2%. Because a larger variation goes hand in hand with stronger vibration, more noise and accelerated joint wear, a minimum tooth count of 17 is recommended in practice for the smaller, usually driving, sprocket wherever space allows.
Typical applications
The calculation is used to judge how smoothly a chain drive runs, to select a sufficient minimum tooth count for the smaller sprocket, and to estimate potential vibration excitation at low tooth counts or high speeds.
Assumptions, limits and common mistakes
The model considers only the kinematic geometry of a single, ideally rigid sprocket at constant drive speed. Elastic chain elongation, bearing clearance, the feedback effect on a second sprocket with a different tooth count, and dynamic resonance effects are not captured.
Common mistake: A common mistake is confusing δ with the slip or efficiency loss of a chain drive — polygon action is a purely kinematic, geometric effect without frictional slip or wear loss in the strict sense, even though it can contribute to higher wear over time. It is also easy to overlook that δ is independent of speed and pitch and depends only on tooth count.
Frequently asked questions
What is polygon (chordal) action?
As the chain engages and disengages the sprocket, it follows a z-sided polygon rather than a true circular path, so instantaneous chain speed oscillates periodically even at constant rotational speed.
Why does δ depend only on tooth count, not speed?
Because δ is defined as the ratio (vmax−vmin)/vmax, and the speed factor ω appears equally in both vmax and vmin, canceling out when the ratio is formed.
How does polygon action affect noise and wear?
A larger variation δ means stronger periodic acceleration of the chain mass, which can excite vibration, generate noise and accelerate joint wear over the service life.
What is the minimum recommended tooth count for the smaller sprocket?
At least 17 teeth is recommended in practice, since δ has already fallen to an uncritical level of about 1.7% by that point.
Is polygon action the same as chain slip?
No. It is a purely geometric, kinematic effect of the positive tooth engagement and not a friction loss like the slip seen in friction-wheel drives or flat belts.