Stopping thirty times a second

Some machines need motion that stops. A film projector must hold each frame still while it is lit and then advance it in the dark; an indexing table must present each station and stay there while work happens.

Continuous rotation in, a step and a pause out — which is a motion no four-bar produces, because a linkage’s output is continuous wherever its input is. The Geneva mechanism is the classical answer.

The one requirement #

Everything about the geometry follows from a single condition: the pin must enter and leave the slot along the slot.

If it entered at an angle it would strike the slot wall, and the driven wheel would go from rest to a finite speed instantaneously — an infinite acceleration, and a hammer blow every index.

Entering along the slot means the crank is perpendicular to the slot at entry. With n slots the slot spacing is $2\pi/n$, and the perpendicularity condition fixes the centre distance:

$$c = \frac{r_{\text{crank}}}{\sin(\pi/n)}$$

For six slots and a crank of 30, that is exactly 60. Nothing about it is free: choose the slot count and the crank radius and the centre distance is determined.

Measured, the driven wheel’s angular velocity at entry is 1.5 × 10⁻³ against a peak of 1.000 — zero to the accuracy of the sampling, which is what tangential entry buys.

What it does not fix #

The velocity starts and ends at zero. The acceleration does not.

Measured on the six-slot wheel, the peak angular acceleration is 1.35 times the driver’s angular velocity squared, and it occurs shortly after entry. That is finite, which is the improvement over a pin hitting a stop, and it is large.

This is the mechanism that ran cinema projectors for a century, advancing 24 frames a second with a pull-down time of a few milliseconds. The acceleration is why sprocket holes tear, why film is the most mechanically abused medium in common use, and why the intermittent movement was the part of a projector that wore out.

Why the pin must enter along the slot #

The condition is easy to state and worth deriving, because it is the only design freedom the mechanism has and everything else follows from it.

While the pin is engaged, the driven wheel’s angular position is whatever puts the slot through the pin. If the pin arrived at an angle to the slot, the wheel would already have to be turning at the instant of contact — the slot’s direction and the pin’s velocity would not be parallel, so the component across the slot would have to be taken up by the wheel rotating.

But the wheel is stationary an instant earlier, held by the locking disc. So the angular velocity would have to jump from zero to a finite value in no time, which is an infinite angular acceleration and, in a real mechanism, a blow limited only by the stiffness of the parts.

Entering along the slot removes the discontinuity: at the moment of contact the pin’s velocity is entirely along the slot, so the component across it is zero and the wheel starts from rest smoothly. Geometrically that means the crank is perpendicular to the slot at entry, which is a right angle in a triangle whose other two sides are the crank radius and the centre distance — and that fixes the centre distance completely.

Measured on a six-slot wheel, the driven angular velocity at entry is 1.5 × 10⁻³ against a peak of 1.000. That is zero to the accuracy of the sampling, and it is what the geometry was chosen to buy.

The locking disc #

Between indexes the wheel must not move, and nothing described so far stops it.

Real Geneva mechanisms carry a locking disc: a circular segment on the driver running against a matching concave face on the wheel, holding it while the pin is disengaged. The entry slot is cut through the disc so that the pin takes over as the disc releases and vice versa, and the handover is the part that has to be made accurately.

That is a second mechanism sharing the driver’s shaft, and it is why a real Geneva drive looks more complicated than its kinematics. This site draws the kinematic part only, which is honest and incomplete in a specific way: the locking action is what makes the mechanism usable, and it is geometry rather than dynamics, so it could be drawn and is not.

Fewer slots, worse #

The slot count sets how much of the driver’s turn is spent indexing: one turn in n. A four-slot wheel indexes 90° in a quarter turn; a twelve-slot wheel indexes 30° in a twelfth.

Fewer slots means a larger index in a shorter time, so the accelerations rise steeply — and unlike a cam, where the acceleration is a design choice, there is no dial to turn. Three slots is the practical minimum and is rarely used; six or eight is common.

There is also a geometric floor. With three slots the centre distance is $r/\sin 60° = 1.155r$, which puts the wheel very close to the driver, and the wheel radius $c\cos(\pi/n)$ becomes small relative to the crank — the mechanism runs out of room before it runs out of theory.

The locking arc #

Between indexes the wheel must not move, and nothing in the description so far stops it.

Real Geneva mechanisms add a locking disc: a circular segment on the driver that runs against a matching concave face on the wheel, holding it while the pin is disengaged. The pin’s entry slot is cut through the locking disc so the two hand over cleanly.

That is a second mechanism sharing the same shaft, and it is why a Geneva drive looks more complicated than its kinematics. This site draws the kinematic part only.

Cam or Geneva #

The choice is a real one and it turns on the same trade as everywhere in this field.

A Geneva is simple, cheap, self-locking, and its motion law is fixed by the geometry — the law the slots give is the only law available. Its acceleration is what it is.

A cam can produce any law, including cycloidal, which arrives at its dwell with essentially no acceleration. It needs an accurately cut profile, a follower held in contact, and it does not lock itself.

Where the indexing is fast and the load small, the Geneva’s fixed law is acceptable and its simplicity wins. Where accelerations matter, the cam’s freedom is worth the manufacturing.

What the acceleration means in practice #

The peak angular acceleration on a six-slot wheel is 1.35 times the driver’s angular velocity squared. That number is dimensionless because both are angular, and it scales exactly as one would expect: double the driver’s speed and the acceleration quadruples.

For a projector running at 24 frames a second with a pull-down occupying a sixth of each cycle, the film is accelerated and decelerated 24 times a second, and the force on the sprocket holes is that acceleration times the mass of the film being moved. This is why sprocket holes tear, why the intermittent movement was the part of a projector that wore, and why the mechanism was made as small and light as the frame size permitted.

It is also why fewer slots is worse in a way that compounds. A four-slot wheel indexes 90° in a quarter of the driver’s turn, so the same rotation is completed in less time and covers more angle — and the acceleration rises with both. Three slots is the practical minimum and is rarely used.

Where it sits among the alternatives #

Intermittent motion has several standard answers and the choice between them is a clean example of the trade this whole field turns on.

A Geneva is simple, cheap, self-locking, and has a law it cannot change.

A cam with a dwell can produce any law including one with finite jerk, at the cost of an accurately cut surface and a follower that must be held in contact and kept below a pressure-angle limit.

A ratchet and pawl indexes by an amount set by the tooth pitch, is the cheapest of all, and has an impact at every step — the thing tangential entry exists to avoid.

A stepper motor removes the mechanism entirely and moves the problem into control, which is the modern answer wherever electricity is available and the loads are small.

The Geneva survives where the load is large, the speed is fixed, and the mechanism must not need adjustment — which is a narrower niche than it once had and a real one still.

The intermittent-motion family #

The Geneva mechanism is one answer to a question with several, and the alternatives sort themselves by what they are willing to pay.

The ratchet and pawl indexes by an amount set by the tooth pitch, is cheap, and has no control over the motion between positions at all — the pawl drops and the wheel arrives when it arrives. Acceptable when the indexed part is light and the rate is low.

The star wheel and cam replaces the Geneva’s slot with a cam profile, which buys the freedom to choose the acceleration programme rather than accepting the one the geometry gives. A properly designed indexing cam has finite jerk and can run faster than a Geneva of the same size, and it costs a cam.

The parallel indexer (Ferguson drive) uses a cylindrical cam with a rib engaging rollers on a turret, and is the arrangement most high-speed machinery actually uses. The engagement is continuous, the motion law is whatever the rib is cut to, and the cost is a precision cylindrical cam and a turret of rollers.

The Geneva sits between: no cam to cut, an acceleration profile fixed entirely by the slot count and the geometry, and a jerk discontinuity at entry and exit that no amount of care removes. It is the cheapest mechanism that gets the tangential-entry condition right, and that condition is the whole reason it works at all.

Reading them together, the pattern is the one that runs through the choice of a motion law: the more of the motion the designer specifies, the more the mechanism costs to make, and the higher it can be run.

What the acceleration figure is measuring #

The motion of a Geneva wheel is not a designed programme — it is whatever follows from a pin travelling on a circle and a slot that must accommodate it. That makes the acceleration curve a consequence rather than a choice, and its features are worth naming.

Angular velocity starts at zero, rises to a maximum at the moment the pin passes closest to the wheel centre, and returns to zero. Angular acceleration starts at a finite non-zero value, passes through zero at the velocity peak, and ends at the negative of the starting value.

The finite non-zero acceleration at entry is the mechanism’s defining compromise. The velocity is continuous — that is what the tangential entry buys — and the acceleration is not, so the jerk is infinite at entry and exit. In a real machine that appears as an audible click and a transient in the drive torque, and it is the reason a Geneva has a speed above which it cannot be run quietly.

Adding slots reduces it: with more slots the wheel turns less per index, the pin’s path is shallower relative to the slot, and the entry acceleration falls. That is why three-slot Genevas are rare outside demonstrations and why six is the common choice — the geometry works at three and the accelerations are unpleasant.

The figure computes this from the mechanism rather than from a formula for it, which means the entry condition being exactly tangential is something the curve demonstrates rather than something the caption claims. A velocity that starts at a non-zero value would be visible immediately, and it is what a Geneva with the pin circle wrongly proportioned produces — a failure the assertion is there to catch.

Why this mechanism, on a site about solving #

The Geneva is the simplest mechanism here whose entire behaviour follows from one geometric requirement, and that makes it the clearest demonstration of what the site’s method is for.

The requirement is that the driving pin enters the slot along the slot. Everything else is a consequence: the pin circle radius, the wheel radius, the slot length, the angular travel per index, the locking arc, and the acceleration profile the wheel experiences. A designer choosing the number of slots has chosen all of them.

That structure means the figures can be checked against the requirement rather than against an expected picture. The wheel’s angular velocity must be zero at entry and at exit — not small, zero — and a Geneva with the pin circle wrongly proportioned produces a velocity that starts at a finite value, which is visible in the plot and caught by the assertion. There is no way to draw the mechanism such that the check passes and the geometry is wrong.

Compare that with what a drawing of a Geneva conveys. It shows a wheel, a slot, a pin and a locking disc, and it looks correct for any proportions at all. Whether the mechanism actually starts and stops smoothly is invisible in the drawing and is the only thing the mechanism is for. That is the same gap a drawn four-bar has, on a mechanism where the consequence is easier to state.

The jerk discontinuity at entry is the honest limit, and it is a limit of the mechanism rather than of the analysis: no proportioning removes it, because the acceleration is finite at entry by geometry and zero before. Fixing it requires a different mechanism — an indexing cam, where the profile is specified rather than derived — and that is the trade the whole intermittent-motion family is organised around.

The other way of not going at constant speed #

A Geneva mechanism produces motion that stops. There is a milder version of the same idea that produces motion which merely goes at two different speeds, and it is worth putting alongside because the two solve adjacent problems with very different machinery.

A crank-rocker’s output stops at two definite places — where the crank and coupler line up — and the crank angles between those two limits are unequal. So the output covers the same swing twice in different times, and the ratio of the two is what a shaper’s productivity is made of: cut on the slow stroke, return on the fast one.

The difference between that and a Geneva is what the mechanism does at the ends of the travel. A quick-return linkage reverses smoothly, with the output’s velocity passing through zero and out the other side; there is no dwell, and the tool is only momentarily stationary. A Geneva actually stops and stays stopped, because the driving pin has left the slot entirely and the wheel is locked.

Which one a machine needs depends on whether something has to happen while the output is still, and the price is exactly the acceleration this essay is about. A quick return costs nothing in acceleration beyond what any four-bar has. A dwell costs a finite, large acceleration at every entry, and that is why film sprocket holes tear and shaper rams do not.

Against the other ways of stopping #

The Geneva was later joined on this site by a whole field of mechanisms that do the same job, and the comparison is worth having in one place.

What the tangential entry buys is visible in the last column. A mutilated gear — a gear with most of its teeth removed — indexes the same way and engages at the full pitch-line velocity, so its output’s angular velocity has a step and its acceleration is not a number at all.