When the index law becomes a choice

A Geneva wheel and a cam indexer are sold for the same job, and it is the job of every mechanism that waits. An input shaft turns steadily; an output shaft turns one step and stops, holds, turns one more step. What differs is who decides how the step is taken.

A cam indexer’s motion law is drawn by its designer. The profile is cut to whatever displacement the machine wants — cycloidal, modified sine, a polynomial with its end derivatives fixed — and the law that costs least sets out what each of those buys. A Geneva’s law is not drawn at all. A pin on a crank enters a radial slot along the slot, which is the one requirement the Geneva’s arithmetic starts from, and from that requirement everything else follows: the centre distance, the angle the crank turns while indexing, and the entire displacement curve. The slot count is the only dial.

So the usual comparison — a Geneva has one fixed law, a cam can have a good one — is framed as a choice against no choice. That is not a fair fight, because a cam that is free to pick its law is also free to pick how long it takes to index, and a longer index is gentler whatever the law. The fair comparison fixes everything the Geneva fixes. Give the cam the Geneva’s index angle and the Geneva’s step, and let only the law differ.

The Geneva’s law, written down #

With $n$ slots and $\lambda = \sin(\pi/n)$, the pin’s crank radius is $\lambda$ times the centre distance, and when the crank has turned an angle $\alpha$ from the line of centres the wheel has turned

$$\tan\varphi = \frac{\lambda \sin\alpha}{1 - \lambda\cos\alpha},$$

for $\alpha$ between $-(\pi/2 - \pi/n)$ and $+(\pi/2 - \pi/n)$. Over that interval $\varphi$ runs from $-\pi/n$ to $+\pi/n$, one step of $2\pi/n$. Differentiating twice, with $D = 1 + \lambda^2 - 2\lambda\cos\alpha$,

$$\varphi' = \frac{\lambda(\cos\alpha - \lambda)}{D}, \qquad \varphi'' = -\frac{\lambda(1-\lambda^2)\sin\alpha}{D^2}.$$

These are derivatives with respect to the crank angle, so multiplied by the crank’s angular speed squared $\varphi''$ is the wheel’s angular acceleration in real units. Nothing about the size of the machine appears.

The closed form is not taken on trust. The Geneva the cam field already builds is constructed differently — the pin’s position is computed, and the wheel angle is whatever puts a slot through it, by an arctangent of coordinates — and at four slot counts and some sixty crank angles the two agree to $3 \times 10^{-13}$, with the second derivative checked against a second difference of the constructed angle. The same comparison made against a wheel with one slot more misses by at least 0.007 radians, so the check could tell a wrong Geneva from the right one.

Putting them in one unit #

A cam law is a function $s(u)$ that rises from 0 to 1 as $u$ goes from 0 to 1. Cut to a step $h$ over an input angle $T$, its acceleration peaks at $C_a h / T^2$, where $C_a$ is the law’s acceleration coefficient — a pure number, and the number cam tables quote. A lift is a size and a law is a shape explains why a law is portable in exactly this form: the coefficient is the shape, and $h/T^2$ is the size.

The Geneva’s step is $h = 2\pi/n$ and its index angle is $T = \pi - 2\pi/n$, so dividing its acceleration by $h/T^2$ puts it into the same unit as every cam law, and its peak becomes a coefficient too. The first figure draws the four curves that way. The cam laws’ coefficients are measured by differencing their displacement functions, not typed in, and they come out where cam tables put them: cycloidal $6.2832$, modified sine $5.5280$, simple harmonic $4.9348$, constant acceleration $4.0000$.

The Geneva’s coefficient depends on $n$. At six slots it is 5.653.

That is below cycloidal. A six-slot Geneva, the commonest there is, has a lower peak acceleration than a cycloidal cam indexer given the same 120° of input for the same 60° of output — by ten per cent. It is above the modified sine by two per cent and above simple harmonic by fourteen.

Where each law stops winning #

Treating the slot count as a real number turns each comparison into a curve, and each curve crosses one at a definite place.

The crossings, found by bisection on the ratio:

• cycloidal at 5.19 slots,
• modified sine at 6.23 slots,
• simple harmonic at 8.06 slots,
• constant acceleration never.

So for wheels of three, four or five slots every cam law beats the Geneva on peak acceleration, and by a great deal at three — a three-slot Geneva peaks at 2.6 times a cycloidal cam’s. From six slots a cycloidal cam has the higher peak. From seven a modified sine does. From nine even simple harmonic motion does.

The one law that is never crossed is the one that cannot be. Among all laws that rise through a step in a given time from rest to rest, constant acceleration has the least possible peak, four — a Geneva’s law is one of those laws, so its coefficient can approach four and never go under it, and the lowest it reaches is about 4.46, near eighteen slots.

The bound of four is short enough to prove in a paragraph, and it is worth having because it is the floor under every curve in the figure. Suppose a rise of $h$ takes an input angle $T$, starts and ends at rest, and never accelerates harder than $A$. Starting from rest, the velocity at input $t$ can be no more than $At$; ending at rest, it can be no more than $A(T - t)$. The largest displacement those two ceilings allow is the area under the lower of them, a triangle of base $T$ and height $AT/2$, which is $AT^2/4$. The rise must fit under it, so $h \le AT^2/4$ and $A \ge 4h/T^2$. Constant acceleration is the law that runs along both ceilings and meets the bound; every other law, the Geneva’s included, sits above it.

The crossings also come in the order the laws’ smoothness would predict. Cycloidal buys a finite jerk everywhere by spending the most peak acceleration, so it is the first to be overtaken; simple harmonic spends the least of the three and holds out longest. A Geneva sits among them because, measured by peak acceleration alone, it is a moderately economical law — and that is the half of the story the usual comparison leaves out.

The step no slot count removes #

Peak acceleration is not the whole of an index. The other thing a motion law decides is how it meets the dwell on either side.

A cycloidal or modified-sine cam reaches the end of its rise with nought velocity and nought acceleration, so the follower goes from moving to held with no step in anything below the jerk. That is why those laws exist. Simple harmonic motion and constant acceleration reach the dwell with nought velocity and a finite acceleration, so their acceleration steps at each end and their jerk is momentarily infinite.

Tangential entry gives the Geneva nought velocity at entry and exit. It does not give it nought acceleration. Setting $\alpha = -(\pi/2 - \pi/n)$ in the closed form, $\cos\alpha = \lambda$ and $\sin\alpha = -\sqrt{1-\lambda^2}$, and the second derivative collapses to

$$\varphi''_{\text{entry}} = \frac{\lambda}{\sqrt{1-\lambda^2}} = \tan\frac{\pi}{n}.$$

At six slots the wheel is at rest and accelerating at 0.577 of the crank speed squared the instant the pin touches the slot. The closed form holds at eight slot counts from three to a hundred to $10^{-15}$, and the smallest value in that range is 0.031 — not nought at any slot count a wheel can have.

As a share of the peak, the step is 0.18 at four slots, 0.43 at six and 0.77 at twelve, and it keeps climbing. So the slot counts at which a Geneva wins on peak acceleration are exactly the ones at which its step on entry is largest relative to that peak. The wheel does not trade one defect for another along the way — it trades a high peak with a small step for a lower peak that is nearly all step.

The value has a construction as well as a derivation, and the construction says why it cannot be nought. At the instant of entry the pin is moving along the slot — that is what tangential entry means — so its velocity turns the wheel not at all. Its acceleration is the centripetal acceleration of a point on a turning crank, $a\omega^2$ toward the crank’s centre, and the crank radius is square to the pin’s velocity, so that acceleration is square to the slot. All of it acts across the slot. The pin is at a distance $C\cos(\pi/n)$ from the wheel’s centre, where $C$ is the centre distance, so the wheel’s angular acceleration is $a\omega^2 / (C\cos(\pi/n))$, and with $a = C\sin(\pi/n)$ that is $\omega^2\tan(\pi/n)$. Choosing the entry direction removed the velocity across the slot; nothing about the entry direction can remove the crank’s centripetal acceleration, and the wheel is handed all of it.

This is the law a Geneva actually is. The mechanism that waits grouped the Geneva with the machines that start from rest, and it does; what it starts from is rest with an acceleration already on.

A Geneva of many slots is simple harmonic motion #

The limit is clean enough to state as a result.

When $n$ is large, $\lambda$ is small, $D$ is close to one, and $\tan\varphi \approx \lambda\sin\alpha$. The input interval approaches the whole half-turn from $-\pi/2$ to $\pi/2$, over which $\lambda\sin\alpha$ rises through a step of $2\lambda$ along a sine — which is simple harmonic motion, the law whose displacement is half a cosine wave. So the Geneva’s coefficient must tend to simple harmonic’s $\pi^2/2$ and its entry step must become its whole peak, since simple harmonic motion’s acceleration is largest at its ends.

Measured at two thousand slots, the coefficient is within 0.2% of $\pi^2/2$, the step is 0.99999 of the peak, and the displacement curve matches simple harmonic motion to $10^{-4}$ of the step. At eight slots — where the peak accelerations of the Geneva and simple harmonic motion happen to agree to 0.2% — the two displacement curves still differ by 3.4% of the step, so equal peaks are not the same law.

That is why the simple harmonic crossing is the last one and why the curves in the second figure turn back up after sixteen slots: every ratio is heading for the value it will have when the Geneva has become simple harmonic motion, which is one for that law and 0.79, 0.89 and 1.23 for the other three.

Four wheels, scored #

A buyer choosing an indexer reads more than one column, and the columns do not agree.

Four things stand out in the table.

The dwell is not free. A Geneva of $n$ slots dwells for $(n+2)/2n$ of the input turn, which is 75% at four slots and falls toward a half as the slots multiply. A cam indexer may be given any dwell; the Geneva’s is part of the price of its slot count.

The velocity coefficient at six slots is exactly two. At the middle of the index the pin is on the line of centres, its distance from the wheel’s centre is the centre distance less the crank, and at six slots the crank is half the centre distance — so the wheel turns at exactly the crank’s speed, $\lambda/(1-\lambda) = 1$. Expressed as a coefficient that is $2$, the same as cycloidal and constant acceleration. Peak velocity matters to an indexer because it sets the kinetic energy the output has to shed before it stops, and on that column a six-slot Geneva ties the cycloidal cam exactly.

On peak acceleration the ranking flips inside the table. At four slots both cams beat the Geneva comfortably; at eight and twelve the Geneva beats both.

On the step, the Geneva loses at every slot count, and by more as the slot count rises.

No row wins every column, which is the honest outcome. It also explains the division of practice that grew up without this table: small Genevas in light, slow machines where the peak does not matter and the lock is valued; cams in fast indexing tables where the step in acceleration, and the vibration it excites in whatever the table carries, is what limits speed.

What the cam gives up to match #

The comparison so far held the index angle fixed, because that is what makes it fair. A designer choosing a cam is not obliged to hold it fixed, and the question they would ask is the reverse one: to be no worse than the Geneva on peak acceleration, how much dwell does the cam have to give back?

For a six-slot wheel: a cycloidal cam matches the Geneva’s peak acceleration of 1.350 crank speeds squared at a dwell of 64.9% of the turn, against the Geneva’s 66.7%. So a cycloidal cam that accepts 1.8 points less dwell is as gentle at its peak as the Geneva and has no step at all. A modified-sine cam matches the peak at 67.0% — a little more dwell than the Geneva — so at six slots that law is better on every column in the scorecard at once.

That is the most practical sentence here: at six slots a modified-sine cam indexer beats a Geneva outright, and a cycloidal one beats it for the price of two per cent of dwell. At eight slots and above the Geneva’s peak is genuinely lower than either cam’s at equal dwell, and a designer who wants the lower peak without the step has to accept a shorter dwell to get it.

What this does not decide #

It is kinematics, not dynamics. Every number here is an acceleration of the output for a steadily turning input. A real indexer’s input slows under the load of accelerating the output, a real follower deflects, and the step in acceleration excites whatever the frame’s lowest mode is. The comparison says what each law asks of the machine; how much of it the machine delivers is a question about stiffness and inertia.

It says nothing about the lock. A Geneva holds its wheel through the dwell with a locking disc and an arc concentric with the pivot; a cam indexer holds its output with the dwell portion of its profile, preloaded against backlash. Which lock is stiffer, and which wears into play, is a separate question and often the deciding one.

It compares with four laws. Polynomial laws with prescribed end derivatives, modified trapezoidal motion and laws optimised for a particular load exist and would each have their own crossing. The four here were chosen because they bracket the family: constant acceleration is the lower bound, cycloidal the smooth benchmark, and simple harmonic the law a Geneva becomes.

Several pins change the arithmetic. Two pins and no dwell at all put more than one pin on the crank, which halves the time per index and so changes the index angle against which a cam is compared. The law of each engagement is the one-pin law, so the entry step is still tan(π/n); the peak ratio has to be recomputed at the shorter angle.

Still open: the locking disc as the real limit on the pin count #

With one pin the comparison above is complete. With several, the arithmetic of pins and slots decides which combinations index at all, and it says nothing about the other half of the mechanism: the locking disc, which must be cut away wherever a pin is engaging and must still hold the wheel through the dwell between pins.

Its distinct argument would be the disc’s geometry as a second inequality — the angular width each cut-away needs so that the wheel’s locking arc can pass, set against the $360°/p$ available between pins — and whether that inequality bites before the pin arithmetic does. Two things would come out of it. The largest pin count a buildable Geneva can carry on a given wheel, which may be smaller than the count the slots allow and would then be the number a designer needs. And, joined to the comparison here, whether a multi-pin Geneva can ever beat a cam on peak acceleration at a slot count where a one-pin Geneva cannot — since more pins shorten the index angle, the Geneva’s coefficient is unchanged while the angle it is measured over shrinks, and it is not yet known whether the locking disc lets the pin count rise far enough for that to matter.