A V holds the wheel to first order

A Geneva drive turns its wheel with a pin in a slot and holds it still between turns with a disc on the driver. The disc decides the pin count found that the second job, not the first, limits the mechanism. When the wheel is held by the classical concave arc cut into its rim, the disc is as large as the crank and must be cut away over 236° of a six-slot driver’s turn, and one pin is all it can carry. A flat holds the wheel to second order replaced the arc with a flat chord, shrank the disc to the crank times $\sin(\pi/n)$ and got the second pin back. But the flat holds only to second order. A running clearance $c$ lets the wheel rock by $\sqrt{2c/a}$ rather than $c/(a\sin\psi)$, which on a six-slot wheel at a close fit is sixty-seven times as much.

That essay stated the result as a trade: a lock is either first order or small, and never both. It ended by doubting that the trade was a law. A face that moves off the disc when the wheel turns is what first order needs, and the arc is one way to get one. A shallow V — two short flats angled towards each other between each pair of slots — might be another, and such a V need not reach deep into the wheel.

This essay cuts that V and measures it. It is both.

How large a disc a V allows #

A V lock is two faces meeting at an apex on the lock’s centre line, each tilted by an angle $\gamma$ off the chord a flat would have used. The notch deepens towards its middle. The driver’s disc sits in it, touching both faces. Since the disc is centred on the driver’s axis at a distance $a$ from the wheel’s centre, and each face is tilted $\gamma$, the apex has to sit at $h = a - r_d/\cos\gamma$ from the wheel’s centre for the disc of radius $r_d$ to reach both faces.

The limit on the disc is the same as the flat’s: the faces must not open past the slot mouths on either side, or the V would cut the slots the pin runs in. Setting the face through the slot mouth, at the rim and at $\beta = \pi/n$ from the centre line, gives the largest disc a V allows:

$$r_d \le \text{crank}\cdot\sin(\beta + \gamma).$$

At $\gamma = 0$ this is the flat’s cap, the crank times $\sin\beta$. At $\gamma = \pi/2 - \beta$ it is the crank itself, the arc’s cap. The V is a family that runs from one classical lock to the size of the other, and the disc grows steadily along it. The cap is checked by construction: a V at its largest disc, drawn at four angles, meets the rim at the slot mouths exactly, to $10^{-9}$ of the crank.

A larger disc must be cut away over a wider arc of the driver, and the width decides the pins. Swept through the index in the driver’s frame, a six-slot wheel’s V needs 127° cut away at nought — the flat’s figure — and more as the faces steepen: 152° at 10°, 188° at 30°, 221° at 50°, against the arc’s 236°.

The hold is the normal’s lever #

Why the flat holds only at second order, and what the V changes, is one geometric fact.

The disc touches a face along its own radius, so the contact normal always passes through the driver’s axis. What the face’s shape changes is the angle that normal makes with the line of centres. The wheel turns about its own centre, and a face resists that turning with the distance of its contact normal from that centre — the normal’s lever arm about the wheel’s pivot. For a face tilted $\gamma$, that lever is $a\sin\gamma$.

The flat’s face is square to the line of centres, so its normal runs along that line, straight through the wheel’s centre. Its lever is nought. A torque on the wheel has no first-order component pressing the flat into the disc, and the only thing that eventually closes the gap is the curvature of the flat’s path, which is the square-root law. The arc’s normal meets the rim at the arc’s end, 59.8° from the line of centres on a six-slot wheel, so its lever is $a\sin 59.8°$. A V at 14° has a lever of 14.5 on a wheel whose centres are 60 apart; at 30°, a lever of 30.

This is the same rule a contact turns and slides at once found for a stop on a four-bar. A contact holds a motion in proportion to the distance of its normal from the motion’s pole, and a normal through the pole holds nothing at first order. For a wheel on its own pin the pole is the pin, and the flat is the stop aimed at it.

How far a V lets the wheel rock #

The lever gives the first-order part of the rock. The whole of it is short to derive. Turn the wheel through $\delta$ about its centre, and a face whose normal is at $\gamma$ from the line of centres moves towards the disc by $a[\cos\gamma - \cos(\gamma + \delta)]$. The disc runs with a clearance $c$, so the wheel can turn until that equals $c$:

$$\delta_V = \arccos\!\left(\cos\gamma - \frac{c}{a}\right) - \gamma.$$

At $\gamma = 0$ this is the flat’s $\arccos(1 - c/a)$. For small clearances it is $c/(a\sin\gamma)$, linear. For large ones it approaches the flat’s $\sqrt{2c/a}$. The two regimes cross where the two asymptotes meet, at a clearance of $2a\sin^2\gamma$.

The law is checked against a measurement that uses none of it. The wheel’s material — rim, slots and notches — is turned about its centre, every point of the disc’s boundary facing it is tested for contact, and the first turn that touches is found by bisection. Across four decades of clearance and five angles the measured rock and the exact law agree to $3 \times 10^{-4}$. Below the crossover the measured slope on log axes is 0.997; well above it, on a V of half a degree, it is 0.54 and falling towards a half. Every V with $\gamma$ above nought is a first-order lock at small clearance: at 2° below a clearance of 0.146, at 5° below 0.912, at 10° below 3.6, on a crank of 30.

A flat is the limit of the family, and the family shows why the flat was a poor place to stop. The flat’s law is not a law of flat faces. It is what $\arccos(\cos\gamma - c/a) - \gamma$ becomes at the one angle that puts the contact normal through the wheel’s pivot. Tilt the face by a single degree and every clearance below 0.037 is held to first order.

Where the flat essay’s argument slipped #

The trade was stated with a reason, and the reason is worth reading again now that it has been measured. A lock that holds to first order, it said, needs its face to move off the disc as soon as the wheel turns, which means curving round the driver’s axis, and a face that curves round the driver’s axis reaches deep into the wheel.

The first clause is right. The second is where the argument went wrong. It treated “moves off the disc at first order” and “curves round the driver’s axis” as one condition, because the only first-order lock on the table was the arc. They are different conditions. Moving off the disc at first order needs the face’s normal at the contact to miss the wheel’s centre, and a straight face tilted by any angle does that. Curving round the driver’s axis is one way to make the normal miss — the arc’s normal misses by the most any lock can manage, $a\sin 59.8°$ — but it is the most expensive way, because the arc must reach round the disc until the rim cuts it off.

The angle that holds the lock met the same distinction on an escapement, from the other direction. There a locking face cut concentric with the pallet arbor has no draw at all: the tooth’s push aims at the pivot, so the face neither pulls the pallet in nor pushes it out. Tilting the face gives it draw, and the draw is the lever the tooth’s push has about the arbor. A Geneva’s flat is the concentric face of that story with the roles swapped. The disc pushes along a normal through the wheel’s pivot, and a face tilted off that normal is the only thing that turns the push into a hold. The arc that is concentric with the pivot is the concentric case on the driver’s side: it lets the driver turn freely because its normal passes through the driver’s axis, and it holds the wheel because the same normal misses the wheel’s.

The second pin, and the angle it allows #

The pin count is decided the way the disc decides the pin count decided it. Each index costs the disc the whole angular width of its cut-away, and every index needs some locking between cut-aways, twenty degrees here. So two pins need a cut-away of at most 160°. Measured at every angle, the cut-away rises from the flat’s value as the V steepens and crosses 160° once. The largest V that keeps a second pin is 34.1° at four slots, 20.8° at five, 14.1° at six, 8.6° at eight, 6.5° at ten and 5.6° at twelve.

The limit falls with the slot count for two reasons. More slots mean a smaller $\beta$, so a flat’s disc starts smaller and its cut-away starts wider relative to the index. And the V’s disc, $\text{crank}\cdot\sin(\beta + \gamma)$, grows faster in $\gamma$ when $\beta$ is small, because the sine is steepest near nought.

Found by scanning upward from the flat, these limits are the honest ones. Scanning the whole range would find something else. As $\gamma$ approaches $\pi/2 - \beta$ the faces run parallel to the slots, and the apex falls towards the wheel’s centre; the notch hollows out the whole sector between two slots and leaves them as thin walls. A wheel reduced to that little material needs almost nothing cut away, and a search over every angle reported two pins at six slots with the V at 60°. The wheel it described had no rim left to lock on.

Both at once #

The trade the flat essay stated can now be priced at every slot count. At a clearance of 0.1 on a crank of 30 — a clearance of a third of a per cent of the crank, a free-running fit — the flat lets the wheel rock 3.94° at four slots and 2.38° at twelve. The arc, carrying one pin, allows 0.19° and 0.05°. The largest two-pin V allows 0.24° at four slots, 0.39° at six and 0.49° at twelve, between five and sixteen times less than the flat.

At four slots the V is nearly the arc. It rocks 0.240° against the arc’s 0.192°, with a disc of 26.5 against the arc’s 27, and it carries two pins where the arc carries one. At six slots it rocks three and a half times as much as the arc and eight and a half times less than the flat. At twelve it rocks nearly ten times as much as the arc, because its angle has fallen to 5.6° and its lever with it — but the clearance of 0.1 is still far below its crossover of 2.2, so it is still holding at first order, and still five times tighter than the flat.

So the flat essay’s trade was between two particular shapes, as it suspected. What a first-order lock needs is a contact normal that misses the wheel’s centre, and a V supplies that at any angle. How much it misses by is what a steeper V buys, and the disc it costs is $\text{crank}\cdot\sin(\beta + \gamma)$, which the second pin can afford up to the angles above. The arc buys the largest lever of all and pays for it with the whole driver.

How shallow a V can be #

The V’s crossover says how much angle a given fit needs. A V holds to first order at any clearance below $2a\sin^2\gamma$, so a designer who knows the running clearance can read off the least angle that keeps the lock first order at that clearance. On a six-slot wheel with a crank of 30 — centres 60 apart — the arc essay’s close fit of 0.02 needs a V of at least 0.75°, whose crossover is 0.021. A V of one degree has a crossover of 0.037 and rocks 0.79° at that fit; two degrees, 0.49°; five, 0.21°; and the six-slot limit of 14.1°, 0.078°.

So the pin budget leaves a great deal of room. Almost any V a second pin allows is far past the least angle a close fit needs, and the choice of angle is not about staying first order at all. It is a choice of lever. Past the crossover, the rock is $c/(a\sin\gamma)$, so it falls in proportion to $\sin\gamma$, and the steepest V the pins allow is the tightest lock available with two of them. The design rule is short: take the largest two-pin angle for the slot count, and the lock is first order at every clearance below its crossover, which at every slot count from four to twelve is more than two units on a crank of thirty.

That also says where the flat sits in the family: it is the one angle whose crossover is nought. Every other V is first order below some clearance, and the flat is first order below none. A flat cut a fraction of a degree out of square — as any real flat is — is a V with a crossover of a few thousandths, and at clearances below that it holds to first order after all, with a lever too small to be worth anything. It moves to first order and not at all met the same knife-edge from the mechanism side: a special position that the smallest error in any direction removes.

What this means at the projector #

Stopping thirty times a second described the application that made the Geneva famous. The wheel carries the sprocket, the sprocket carries the film, and the frame’s position at the aperture is the wheel’s angle times the sprocket’s radius at the instant the shutter opens. The flat essay found that a flat lock lets a six-slot wheel rock by a degree and a half at a close fit, which moves every frame visibly. A second pin would halve the time spent pulling down each frame, and two pins and no dwell at all counted what that costs in dwell. The lock’s rock is lost motion of the kind backlash is an allowance describes: an amount a designer chooses to spend, here spent at the aperture.

The V takes the question out of the trade. A six-slot wheel with a 14° V carries two pins and, at the arc’s close fit of 0.02, rocks by 0.078° against the arc’s 0.022° and the flat’s 1.48°. It is not as tight as the arc, but it is a lock in the sense an indexer needs: its lost motion shrinks in proportion to the clearance, so a better fit buys a proportionally tighter hold, which the flat’s never does.

What the V does not settle #

Load, not only clearance. The rock is the free play before contact, measured with no torque on the wheel. Under a torque, the contact force on a V face has a component along the face, and whether friction holds the disc in the V or lets the wheel wedge round depends on the friction angle against $\gamma$. A V steeper than the friction angle is not self-locking and pushes the disc; a shallower one grips it. The geometry does not decide which the design wants.

Wear. A V’s hold is its lever, and its lever is set by the angle of its faces. Wear at the contact line changes that angle, and how quickly a worn V drifts back towards a flat’s second-order hold is not measured.

Entry and exit. The lock has to release the wheel as the pin enters a slot and take it again as the pin leaves. The cut-away is sized for the pin’s passage, but whether a V’s faces are met by the disc’s edge at a steep angle at the start of locking, and knock, is not examined.

The notch in the rim. A V takes material from the wheel between the slots, and a steep one leaves the slot walls thin. The scan stops at the second pin’s limit, well short of the hollowed wheel, but the wall thickness at each limit is not computed.

Still open: a lock that uses its pin’s slot #

Every lock so far — arc, flat, V — holds the wheel with material between the slots, and every one pays for its disc in cut-away. The pin itself is already in a slot at the end of each index, and a slot is a pair of faces whose normals miss the wheel’s centre by the pin’s own lever.

The distinct argument would be a lock that keeps the pin engaged: a driver whose pin dwells at the bottom of the slot — as a pin on a curved path can — and holds the wheel through the slot’s walls rather than through a disc. There is then no disc to cut away and no cut-away to limit the pins, and the measurement would be how far such a wheel rocks, which is the slot’s own clearance divided by the pin’s radius from the wheel’s centre, against how much dwell a pin path that pauses at the slot’s end can give. Whether that dwell is long enough for a real indexer, and at what cost in the path’s acceleration, is the question.