A cam on a chain steps at every tooth

Every motion law a cam designer uses is written in cam angle. A cycloidal rise over 120° is a displacement as a function of how far the cam has turned, and on a cam turning at constant speed it delivers the law’s acceleration in time, scaled by the speed squared. A cam behind a bent joint drove the camshaft through a single Hooke’s joint, whose output turns unevenly, and found the follower’s acceleration picking up a second term: the law’s velocity times the shaft’s angular acceleration, nought on a steady shaft and invisible to a designer who assumes one. The joint’s unevenness has a period of half a turn, long compared with a rise, so the rise could be clocked into the joint’s slow stretch and most of the harm avoided.

That essay ended on a drive whose unevenness is fast. The chain is a polygon found that a roller chain leaves its sprocket at a radius that varies once for every tooth, so a sprocket driven by a chain at steady speed turns unevenly with a period of one tooth. Many ripples fall inside every rise, and no clocking can put a rise between them. The question was which law survives best, and whether more teeth — a smaller ripple, but faster — help or hurt.

They help, and the reason overturns the expectation that prompted the question. But the more important finding is about the shape of the ripple, not its size.

The shaft’s motion, from the pin geometry #

A sprocket’s pins sit on a circle, and the chain leaves along a tangent. Between one pin seating and the next the seated pin swings through an angle ψ from $-\pi/n$ to $+\pi/n$, and the chain leaves at that pin’s radius times $\cos\psi$. With the chain moving at a steady speed, the sprocket must turn faster when the effective radius is small:

$$\dot\theta = \frac{k}{\cos\psi},\qquad k = \frac{\sin(\pi/n)}{\pi/n},$$

with $k$ chosen so that the mean speed is one. Differentiating along the motion gives the shaft’s angular acceleration,

$$\ddot\theta = \dot\theta^{2}\tan\psi.$$

On an eleven-tooth sprocket the speed ripples by 4.2% once per tooth, rising as the pin swings towards its seating and dropping back as the next pin takes over. The angular acceleration does something the speed hides. It rises steadily through each tooth to $\tan(\pi/n)$, 0.294 on eleven teeth, and then — as the next pin seats and ψ jumps from $+\pi/n$ to $-\pi/n$ — it jumps to $-0.294$. The shaft’s speed is continuous at a seating, because $\cos\psi$ is the same at both ends, but its acceleration is not. The ripple is a sawtooth, not a wave.

A step in every law #

The follower’s acceleration in time is the same two terms the bent joint gave:

$$a = s''(\theta)\,\dot\theta^{2} + s'(\theta)\,\ddot\theta.$$

The first term is continuous, since the speed is. The second jumps at every seating by $2\,s'(\theta)\tan(\pi/n)\,\dot\theta^{2}$, wherever the rise is moving. So every law acquires a step in acceleration at every tooth, however smooth it is on a steady shaft, and a step in acceleration is an infinite jerk.

That matters most to the law chosen for its smoothness. A cycloidal rise is used instead of simple harmonic or constant acceleration precisely because its acceleration is continuous everywhere, beginning and ending at nought: the law that costs least is not the smoothest found what that continuity buys and what it costs in peak acceleration. On an eleven-tooth chain drive the cycloidal rise has about four seatings inside a 120° rise, and at the worst of them its acceleration steps by 0.41 of its steady peak. The continuity it was chosen for is gone, and nothing about the cam can restore it. Only the drive can.

The two routes agree on this. The closed forms above give the acceleration directly. Composing the law’s displacement with the shaft’s angle as a function of time — found from the chain’s own geometry by integrating $\cos\psi$ — and differencing twice in time, sharing nothing with the closed forms but the displacement, agrees with them to $8 \times 10^{-8}$ of the unit lift everywhere except in a narrow band either side of each seating, where a difference straddling the step cannot resolve it.

More teeth help, as one over their number #

The bent joint’s lesson was that the second term’s size is the law’s velocity times the shaft’s angular acceleration. The lead that asked for this measurement reasoned that a fluctuation of fixed size at a period of one tooth has an angular acceleration growing as the square of the tooth count, and asked whether the product rises or falls.

The fluctuation is not of fixed size. The speed ripple is $1 - \cos(\pi/n)$, which shrinks as $1/n^2$, and its rate is proportional to $n$. Their product, the shaft’s angular acceleration, is at most $\tan(\pi/n)$ and falls as $1/n$: halving the ripple’s period quarters its size, and the net effect is to halve the acceleration.

Measured at the worst clocking of 24 against the teeth, a constant-acceleration rise’s peak exceeds the steady shaft’s by 38.2% on eleven teeth and 8.7% on forty, falling with slope −1.15 on log axes. A cycloidal rise’s excess falls from 16.5% to 3.1%, slope −1.30. A simple harmonic rise’s falls from 7.7% to 0.5%, slope −2.05. Every law is helped by more teeth, and none is hurt.

Why the harmonic law is hurt least #

The three slopes differ, and the difference is where each law’s peak sits. The second term is the law’s velocity times the shaft’s angular acceleration, so it can only raise the peak where the law is moving.

A simple harmonic rise has its largest acceleration at its two ends, where its velocity is nought. The second term cannot reach it there at all, and the only thing that raises the peak is the first term’s squared speed, $1/\cos^2\psi$, whose excess shrinks as $1/n^2$. A constant-acceleration rise has the same acceleration throughout each half, including its middle, where its velocity is largest, and there the second term adds in full: slope −1. A cycloidal rise peaks a quarter of the way through, moving at half its top speed, and pays a mixture: its slope lies between.

So on a chain drive the ranking of the three laws by peak acceleration is the reverse of their ranking by smoothness. The simple harmonic law, dismissed for its step at the ends of every rise, is the one whose peak the chain least disturbs; the cycloidal law, chosen for having no steps, acquires them at every tooth.

The step itself, in closed form #

The largest step in a rise is at the seating nearest the law’s fastest point, and as a share of the law’s steady peak it is close to $2\,s'_{\max}\tan(\pi/n)/s''_{\max}$. For the simple harmonic and cycloidal laws that share is the same, because both have the ratio of peak velocity to peak acceleration equal to the rise’s span over π: 0.392 on eleven teeth for a 120° rise, against 0.414 measured. Constant acceleration’s is π/2 times as large, 0.615 against 0.649 measured. The measured steps sit a few per cent above the closed form, because at a seating the shaft is turning a few per cent faster than its mean — $k/\cos(\pi/n)$, 1.035 on eleven teeth — and that speed enters squared; the seating nearest the law’s fastest point is never exactly at it, which takes a little of that back.

The step grows with the rise’s span, since a longer rise moves faster in cam angle for the same lift: at eleven teeth the cycloidal law’s worst peak is 10.9% above the steady one over a 60° rise and 22.3% over 180°. A cam with a short, sharp rise is hurt less by its chain than one with a long, gentle one, which is the opposite of how a designer usually thinks about harshness.

Clocking helps less #

Behind a bent joint, clocking decided almost everything: a rise placed in the joint’s slow stretch could even come out below its steady peak. On a chain, about four teeth pass inside a 120° rise on eleven teeth, and clocking can only decide which ripple meets the law’s peak. The cycloidal law’s peak runs from 1.014 to 1.165 of the steady one as the rise’s start moves through a tooth, constant acceleration’s from 1.292 to 1.382, simple harmonic’s from 1.009 to 1.077. The best clocking recovers most of the peak for the smoother laws, but it cannot remove a single step: those come with every seating, wherever the rise starts.

Two polygons: the timing drive #

Everything so far had the chain moving at a steady speed, as if the driving sprocket were a perfect pulley. In an engine’s timing drive, or any drive that turns a camshaft from another shaft, the driving sprocket is a polygon too, and usually a smaller one turning faster: a crankshaft sprocket of 21 teeth driving a camshaft sprocket of 42 at half its speed.

The same geometry handles both ends without a differential equation. While a driving pin swings through $\psi_1$ the chain travels $R_1\sin\psi_1$; the same travel on the camshaft’s side is $R_2\sin\psi_2$; so the camshaft’s angle follows from the crank’s directly, and its mean speed comes out at exactly half the crank’s, as the tooth counts require, though the construction is never told them.

The camshaft’s acceleration now steps at the seatings of both sprockets, and the driving sprocket’s steps are the larger. On the 21/42 drive the camshaft’s own seatings step its acceleration by 0.149 of its mean speed squared — the $2\tan(\pi/42)$ of the section above, to 1% — while the crankshaft’s seatings step it by 0.598. The driver’s polygon shakes the chain’s own speed, and a sprocket with half the teeth turning twice as fast does it, in the camshaft’s terms, by $2\sin(\pi/n_1)(n_2/n_1)$ to within half a per cent: four times as much. At 17 driving 34 it is 0.738 against 0.183. Only when the two sprockets are equal, 25 and 25, are the two kinds of step the same size.

The strand’s phase — how much of a pitch the taut strand holds, set by the chain’s length — decides where the two kinds of seating fall relative to each other, and on a two-to-one drive with twice the teeth they fall at the same rate. At most phases they fall apart, and the camshaft’s own step is 0.134 to 0.152. At one phase they coincide, and since the two steps have opposite signs they merge into a single step of 0.451, smaller than the crank’s alone. A timing chain’s length is chosen for other reasons, but it is also choosing this.

So a timing drive’s camshaft is shaken mostly by the crank’s sprocket, not its own, and the lever that matters is the driving sprocket’s tooth count. Doubling the camshaft sprocket’s teeth halves one kind of step and leaves the larger kind alone. The joint that is not constant velocity found a Hooke’s joint’s unevenness set by its angle alone; a chain drive’s is set by its smaller sprocket.

How large a step is, in a machine #

The shares above are dimensionless, and a lift is a size and a law is a shape says how to put units back: acceleration scales with the lift and with the square of the camshaft’s speed. Take a cycloidal rise of 10 mm over 120° of cam angle on a camshaft turning at 3,000 revolutions a minute, 314 radians a second. On a steady shaft its peak acceleration is 1,414 m/s², and its acceleration is continuous.

Driven from an eleven-tooth sprocket by a chain at steady speed, the same rise’s acceleration steps by up to 553 m/s² at the seatings inside it, and those steps arrive about 550 times a second. Driven through a 21-tooth crank sprocket on a two-to-one drive, the crank’s seatings alone step it by up to 564 m/s² at 2,100 seatings a second, whatever the camshaft sprocket’s tooth count. Those are not small corrections to a smooth law. They are a second, stepped signal of about four-tenths the law’s own size, at a frequency set by the teeth rather than by the cam.

That is why a cam designer’s choice of law, made on paper with a steady shaft, is only half the design of what the follower feels. Prescribing motion framed a cam as a way to write any displacement a follower should have; on a chain-driven shaft the writing is done in cam angle, and the reading in time, and the chain is what translates between them.

What a designer can do with it #

The measurement gives three levers, in order of how much they buy.

More teeth reduce both the peak excess and the steps as one over the tooth count, and for the simple harmonic law the excess as its square. A camshaft sprocket of forty teeth instead of eleven cuts every step to a quarter.

A shorter rise reduces them in proportion to the span.

A different law changes which of the two effects dominates. If the peak matters — a valve train limited by its spring’s ability to hold the follower — the simple harmonic law is least disturbed. If the jerk matters — a mechanism excited by steps — no law helps, because every law steps.

What does not help is the one a designer might reach for first: a smoother law. The cycloidal law’s advantage is a property of the cam on a steady shaft, and a chain drive’s steady shaft is a fiction at the scale of one tooth.

What the chain does not settle #

The chain’s own stretch. The model takes the chain as inextensible, moving at a perfectly steady speed where it leaves the driving sprocket, as the chain is a polygon did. A real chain stretches under the cam’s torque, which smooths the step into a fast transient whose shape depends on the chain’s stiffness and the camshaft’s inertia.

The strand’s own phase, beyond one drive. The chain is a polygon found the fraction of a pitch in the taut strand deciding how the two ripples combine. On the 21/42 drive it moves the camshaft’s own step between 0.134 and 0.152 and, at one phase, merges it with the crank’s; other ratios, where the seatings do not fall at one rate, are not measured.

Dynamics. The follower’s acceleration is the kinematic demand, and a lift is a size and a law is a shape scales it to any lift and speed. Whether the follower leaves the cam at a step depends on its spring and mass, which are not modelled. A step in demanded acceleration is exactly the event a follower spring is least able to follow, and in a valve train it reaches the valve through a rocker whose own ratio is not constant either.

Still open: a sprocket cut to cancel its own polygon #

The ripple comes from the pins sitting on a circle while the chain leaves along a tangent. A sprocket whose pitch line is not a circle — slightly non-round between teeth — could in principle keep the chain’s effective radius constant, and some timing drives use profiled sprockets to reduce chordal action.

The distinct argument would be that sprocket designed and measured: the non-circular pitch curve on which the effective radius stays constant as each pin swings through its tooth, and whether it exists, since the pin must still seat at the tooth’s centre. A ratio that is a function of the angle found that a demanded ratio fixes a pair of non-circular gears. Here the demanded ratio is constant and the given is a polygon, and the measurement would be how much of the step such a profile removes, and whether the pin’s seating forces it to leave a residual step no profile can cancel.