Machine Kinematics
solved, then drawn
Motion that stops

The arc that is concentric with the pivot

A rotation carries a curve into itself exactly when the curve is an arc of a circle about the centre of rotation. Every exact dwell on this site is that one sentence applied — a cam's dwell, a Geneva's locking disc, a deadbeat escapement's locking face — and the six-bar dwell that is merely very good is what happens when the curve is nearly one.

Assumes The mechanism that waits and A dwell made from a curve.

16 min read 7 figures Exact and approximateSolved, not drawnTwo routes to the same number

Three mechanisms in three different fields of this site hold something still, and all three do it with the same piece of geometry. It is worth stating that piece of geometry on its own, because once it is stated the three mechanisms stop looking like three ideas.

A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point.

That is the whole of it. If a surface must not move when the part carrying it turns, the surface has to be an arc about the axis it turns about, and nothing else will do — not a good approximation to one, not a flat that is tangent to one, not a stretch of some other curve that happens to be nearly circular. Those all move. How much they move is the subject of the rest of this essay.

The curves that do not move. A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point, and that one sentence is every exact dwell on this site. Each bar is how far a curve moves when it is turned two degrees about the axis its mechanism turns about, as a fraction of its own radius. The two arcs about their own centres — a cam's dwell and a deadbeat's locking face — sit at the sampling floor, which is the sagitta of the polyline they are measured as and not a property of the geometry; the number is quoted with the floor beside it because an agreement quoted without its resolution is a mistake this site has already made once. Everything else is orders of magnitude above it, including the near-circular stretch of a coupler curve that a six-bar builds its approximate dwell out of.
Fig. 1 Five curves, rotated two degrees about the axis their mechanism turns about, with the largest distance each one moves from its own former self. Two of them do not move; three do, by between thirty and ten thousand times the resolution the measurement was made at.

The three mechanisms

A cam’s dwell. The profile of a cam is the follower’s motion law drawn in polar coordinates: radius equals base circle plus displacement. A dwell is a stretch of the programme on which the displacement is constant, so the profile there is a stretch of a circle of constant radius about the cam’s own axis. The follower does not move while the cam turns through it, and it does not move exactly.

A Geneva’s locking disc. Between indexes the driven wheel has to be held, and it is held by a convex segment on the driver bearing against a concave face on the wheel. The segment is an arc about the driver’s shaft, so turning the driver slides the segment along the face and moves the face nowhere. The dwell that follows is not the pin being absent from the slot — the pin is absent for the whole dwell and would leave the wheel free — it is the arc.

A deadbeat escapement’s locking face. A tooth resting on a pallet is held while the pendulum finishes its swing, and the pallet is moving the whole time. If the face the tooth rests on is an arc about the pallet’s arbor, the pallet’s rotation carries that arc into itself, the tooth’s resting place does not move, and the escape wheel does not move. This is the case where the argument is least obvious and most valuable, because the part carrying the surface is in motion throughout.

The two faces of one palletAn escape wheel of 30 teeth and a pair of pallets spanning 4 and a half tooth pitches. The heavier line at each pallet is the **locking face**, here an arc about the arbor; the lighter one is the **impulse face** the tooth slides along once it is let go. The wheel is drawn where the contact puts it, not where it looks well: at this pallet angle the tooth in play sits on the lock face and the wheel is exactly at its nominal position. Dragging the pallet through its whole engagement moves the wheel by nothing at all — this face is an arc about the arbor and rotation carries it into itself. Of the 6.0° the wheel turns each beat, 65.3% is drop and does nothing.pallet 2.00° · wheel 0.0000° · lockthe wheel is placed by the contact, not by the drawing
Fig. 2 The third of the three, close up and at true scale. The heavier line is the locking face: an arc of the pallet arm’s own radius, centred on the pallet arbor. The pallet swings on through its supplementary arc and the tooth stays exactly where it is.
The two faces of one palletAn escape wheel of 30 teeth and a pair of pallets spanning 4 and a half tooth pitches. The heavier line at each pallet is the **locking face**, here a flat; the lighter one is the **impulse face** the tooth slides along once it is let go. The wheel is drawn where the contact puts it, not where it looks well: at this pallet angle the tooth in play sits on the lock face and the wheel is -0.0128° from it. Dragging the pallet through its whole engagement moves the wheel by 0.238° of recoil. Of the 6.0° the wheel turns each beat, 65.5% is drop and does nothing.pallet 2.00° · wheel -0.0072° · lockthe wheel is placed by the contact, not by the drawing
Fig. 3 The same pallet cut flat rather than on an arc about its own pivot. The tooth now rides up the face as the pallet swings, so the wheel is driven back through the whole of the lock — which is the failure the concentric arc exists to remove.

What “exactly” is worth, measured

The site’s rule is that an agreement is quoted with the resolution that measured it, and this measurement has a resolution worth naming, because it comes from the curves being polylines rather than from anything about the geometry.

A rotated point lands a sagitta away from a sampled chord however perfect the arc is — the sagitta being ℓ2/8r\ell^2/8r for a chord of length ℓ\ell on a circle of radius rr — so a deviation at or below that number is a curve that did not move, and one far above it is a curve that did. Both are reported.

curve moves by (÷ its radius) sampling floor ratio
deadbeat locking face 1.92 × 10⁻⁷ 3.20 × 10⁻⁷ 0.60
cam dwell, base circle 3.35 × 10⁻¹⁶ 6.61 × 10⁻⁶ 0.00
coupler curve, best stretch 2.14 × 10⁻⁴ 7.02 × 10⁻⁶ 30
the same face, cut flat 3.38 × 10⁻³ 3.17 × 10⁻⁷ 10 700
cam rise, same cam 2.51 × 10⁻² 6.61 × 10⁻⁶ 3 800

The cam’s dwell is the cleanest of the five and by a distance: its points are generated at a constant radius, so rotating them is arithmetic that returns the same numbers, and the answer is 3×10−163 \times 10^{-16} — one bit. The escapement’s face is generated the same way but measured against a polyline that has been rotated rather than regenerated, which is why it sits at the floor rather than under it.

The near miss, which is a mechanism in its own right

The third row of the table is the interesting one, because it is not a mistake. It is the six-bar dwell this site built three phases ago.

A four-bar’s coupler point traces a sextic, and stretches of a sextic are very nearly circular. Find the most nearly circular stretch, hang a link of that stretch’s radius on the coupler point, and the far end of the link stands almost still while the point runs along it. The linkage dwells without a cam, out of nothing but pin joints, and it does it well: 146° of crank inside a one-degree band.

Measured on this essay’s question the same construction gives 2.1×10−42.1 \times 10^{-4} of its own radius — thirty times its sampling floor, so the stretch genuinely moves, and a hundred and sixty times less than the flat face in the row below it. That is exactly the right shape of answer. The coupler stretch is not an arc and does not pretend to be; what it is, is the best arc available from a mechanism that has no arcs in it, and the number says how good the best is.

Why a flat is not good enough

The fourth row is the one that matters for the rest of this field, and it is the row a workshop is most likely to produce, because a flat surface is what a straight tool cuts.

Take the deadbeat’s locking face and replace the arc by the straight line tangent to it at the corner. The two agree to first order — the tangent is, after all, the tangent — and they disagree by 3.4×10−33.4 \times 10^{-3} of the pallet arm under a two-degree rotation, which is ten thousand times the resolution of the measurement.

The consequence is measured in the next essay but one and is worth previewing here in one sentence, because it is the whole reason this piece of geometry is worth an essay: a locking face cut flat is deadbeat at exactly one point of itself — the corner, which is the one point the tooth never rests on.

How far back the wheel is pushed. Recoil against the part of the pendulum's swing that happens after the tooth has landed. The arc cut concentric with the pallet arbor is the flat line on zero, and it is zero as a matter of arithmetic rather than of smallness: rotation about the arbor carries that arc into itself, so the tooth's resting place does not move and every sampled value is the same double. Every other face rises. The flat face with no draw at all is the interesting one — it starts at zero and curves, because it agrees with the arc to first order and not to second, which is precisely the amplitude sensitivity a deadbeat exists to remove.
Fig. 4 The same fact as the escape wheel sees it. The flat line on zero is the arc; the curve rising from zero is the same face cut flat, whose error is second order and therefore grows as the square of the pendulum’s swing. Every other line is a face deliberately tilted, for reasons the essay after next is about.
One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.
Fig. 5 And the deliberate version of the same thing: a face cut past concentric on purpose, so the wheel pulls the pallet in rather than pushing it out. The recoil is now a design quantity with a sign, which is a different mechanism from a face that is merely wrong.

Why nothing else can do it

The statement is short enough to argue in a paragraph, and worth arguing because the “only if” half is the part that does the work.

A rotation about OO preserves distance from OO. So if a curve is carried into itself by such a rotation, every point of the curve has an image on the curve at the same radius. Take a connected curve and a rotation by any angle ε\varepsilon smaller than the curve’s own angular extent: the point at one end is carried to a point of the curve at the same radius, and that point is carried to another, and the chain of images sweeps the curve’s whole extent at one radius. A curve of constant radius about OO is an arc of a circle about OO.

The “if” half is immediate and the “only if” half is what rules out every clever alternative. There is no non-circular curve with the property, no family of them, and no way to buy it approximately that does not have a residue. A designer who needs an exact dwell has one curve available and the only remaining choices are where its centre is and how long it is.

That also explains the shape of the table above. The three curves that move are not three degrees of the same failure — they are three different curves, none of which has the symmetry, and their residues have nothing to do with each other. The flat is second-order wrong because it is tangent; the coupler stretch is wrong by whatever a sextic’s curvature does over that stretch; the cam’s rise is wrong by the whole of the motion law. Compared as fractions they run from 2×10−42 \times 10^{-4} to 3×10−23 \times 10^{-2}, and the only thing they have in common is being above the floor.

Where the disc is cut away

The Geneva’s locking arc is the one of the three that has to be interrupted, and the interruption is where the mechanism is usually got wrong.

The pin and the locking segment share a shaft, so the segment’s arc has to have a gap in it wide enough for the wheel to turn through its index — and the two have to hand over cleanly at both ends. Cut the gap too narrow and the wheel is still locked when the pin starts to drive it, which is a mechanism that jams. Cut it too wide and the wheel is free for a moment before the pin arrives and after it leaves, which on a machine that is being driven at speed is a wheel that arrives at its next stop having already moved.

This site draws the kinematic Geneva and not the locking disc, and the reason is worth being explicit about: the disc is a second mechanism sharing the driver’s shaft, its dimensions do not follow from the slot count the way the rest of the Geneva’s do, and what decides the width of the gap is how far the wheel overruns — which is a question about inertia. The arc is geometry and belongs here. The gap is not.

Two routes to the same zero

The escapement row of the table is measured by rotating a sampled curve and finding the largest distance back to the original. There is a second and completely different way to reach the same conclusion, and the two agreeing is the reason to believe either.

Rotate the pallet by φ\varphi and ask the contact solve where the tooth is. The locking face is an arc of radius ρ\rho about a centre CC; the tooth tip runs on the wheel’s tip circle of radius RR about OO; the contact is the intersection of two circles and has a closed form. When the face is concentric, CC is the arbor itself, and a rotation about the arbor leaves the arbor where it is — so CC does not move, both circles are unchanged, the intersection is unchanged, and the wheel angle comes back bit for bit identical at every pallet angle sampled.

That is a stronger statement than the table’s, and it is the one the recoil essay rests on: not a small number but the same double. It arrives at it through the mechanism’s own solve rather than through a measurement on a curve, and it is the same fact in a different language.

The count says nothing about any of this

It is worth putting the dwell through the mobility count, because the count is the site’s first instrument and it comes back entirely wrong-footed.

A Geneva during its dwell is three links — frame, driver, wheel — with a revolute at each shaft and one higher pair where the locking arc bears. Grübler gives

M=3(3−1)−2(2)−1=1M = 3(3-1) - 2(2) - 1 = 1

one degree of freedom, which is the same answer it gives while the pin is in the slot and the wheel is turning. Nothing has been constrained away and nothing is about to be. What is zero during the dwell is not a freedom but the derivative of the transmission function, and it is zero over an interval rather than at a point.

That distinction separates this field from everything the site has done with singularities. A toggle position is a point at which an output’s velocity vanishes and a Jacobian drops rank; a dwell is an interval on which it vanishes and no rank drops anywhere. The first is a property of a configuration. The second is a property of a surface, and the surface has to be an arc.

What each kind of joint takes away. Grübler's formula is M = 3(n − 1) − 2j₁ − j₂, and the 2 and the 1 in it are not conventions. A lower pair — a pin or a slide — holds two bodies together over a surface and leaves one relative freedom, so it costs 2. A higher pair — a cam against a follower, a wheel on a rail — touches at a point, the contact travels along both surfaces, and it costs 1. Five chains, each built and each measured from the rank of its constraint Jacobian, which has never heard of the formula. The last row is the one worth having: count that cam contact as a pin, as is very easily done, and the formula returns 0 where the mechanism has 1. The Jacobian does not move.
Fig. 6 The pair table, with the cam-and-roller chain lit. A contact along a line removes one freedom of the three a planar body has, which is what makes a Geneva’s locking arc a legal joint and what makes the count above come out at one. Nothing in the table distinguishes an arc concentric with the driver from an arc concentric with anything else.

Where else the same curve turns up

Two more places, both of them older than this field, and both worth naming because they make the device look less like a trick.

The first is the base circle of a cam, which is not merely where the profile starts: it is the radius the profile falls back to at every dwell, and the reason a cam with two dwells at different heights needs two arcs rather than one. The second is a gear’s own base circle, where the role is different — the involute is generated from it rather than being an arc of it — but the geometric reason it is singled out is the same: it is the one circle about the gear’s axis that the construction has to be anchored to.

And the general form, which is what this essay is really about: when a mechanism has to produce nothing over an interval, it needs a curve with a symmetry, and the only curves with a rotational symmetry about a given point are the circles about it. Everything else — the coupler stretch, the flat, the fitted arc, the profile that is nearly right — is an approximation to a symmetry rather than a symmetry, and approximations to symmetries have residues that can be measured.

Which they were, above, and the residues run over nine orders of magnitude.

The same theorem for the other motions

The statement is about rotations, and the reason it is worth having in that form is that a dwell in a rotating mechanism is what this field is made of. It is a special case, and naming the general one places the whole argument.

A one-parameter family of rigid motions carries a curve into itself exactly when the curve is an orbit of that family. For rotation about a point, the orbits are circles about that point, which is the theorem above. For translation along a direction, the orbits are straight lines in that direction. For a screw, they are helices about its axis.

So the three motions give three curves, and each of them is the shape a dwell face must have in a mechanism whose input moves that way. A cam driven by a rotating shaft holds its follower on an arc. A linear cam — a plate cam driven by a slide rather than a shaft, which is what an indexing bar or a shaped guide is — holds its follower on a straight face, and a straight face is exactly what such mechanisms are drawn with. A helical cam on a rotating and advancing shaft would hold on a helical face.

That generalisation is worth having because it explains the shape of the answer rather than merely stating it. There is nothing special about circles; what is special is that a circle is what a rotation leaves alone, and the mechanism’s own motion decides which curve that is. Asking what shape gives an exact dwell is asking what are this motion’s orbits, and the second question has an answer for every motion.

It is also the pair field’s classification one dimension down. There the question is which surfaces slide on themselves under a continuous family of displacements, and the answer is the six lower pairs. Here it is which curves do, under a one-parameter family, and the answer is the orbit of that family. The same argument, applied to a curve rather than a surface, and producing a shorter list because a one-parameter family has one orbit through each point.

Which gives the field’s three exact dwells a common ancestry with something quite far from them. A locking face, a cam’s base circle, a Geneva’s locking arc, and the cylindrical surface of a plain bearing are all the same construction: a shape chosen so that the motion leaves it where it is. In the bearing’s case that is what makes it a joint; here it is what makes it a dwell; and the difference is only whether the moving body is the one being held or the one doing the holding.

What this does not settle

The theorem says which curve holds something still. It says nothing about whether the mechanism can get onto that curve and off it again, and that is where the next three essays go.

A cam’s dwell has to be joined to the rise on either side of it, and a join that is merely continuous in position produces a step in the follower’s velocity — which is why the join is designed rather than drawn and why a cycloidal law exists at all. A Geneva’s locking disc has to be cut away where the pin enters, and the handover between the arc and the slot is the part of the mechanism that gets made wrong. And an escapement’s locking face has to be left, by a pallet that is moving, at a moment decided by a corner — and everything the field argues about from here on is about that corner.

Arriving at a dwell. The rise ends at 120° and the follower then stands still, so its acceleration must be zero from there on. Simple harmonic motion arrives at 6.854e-3 per degree² and drops to nothing instantly — an impulsive jerk, which in a real train is a shock the whole mechanism feels. Cycloidal motion arrives at 2.056e-4, 33 times smaller, because its acceleration is a sine that reaches zero exactly where the dwell begins. That is the only reason to prefer it, and it is enough. Both numbers are read off the curves drawn here, and the site's standing check on the pair — which reaches them by its own route — puts harmonic at 6.85e-3 and cycloidal at 2.06e-4.
Fig. 7 The join, on a cam. An arc holds the follower still and a law brings it away again; whether the two meet with a step in the acceleration is decided by which law, and it is the one design decision the arc does not make for itself.

Where this goes next

A dwell is a measurement, not a stop. The Stephenson six-bar's output against a full turn of the crank, with the four-bar it is built on for comparison. Inside a band of ±1° the six-bar's output holds still for 145.8° of crank and the four-bar's for 41.9°. The dwell comes from a stretch of the coupler curve that fits a circle of radius 3.006 to within 2.96e-3, and the arm from the coupler point is that radius. Nothing here stops; it moves less than the band. A dwell made from a curve Parts of a coupler curve are very nearly circular arcs. Put a link of the arc's own radius on the coupler point and its far end stands almost still while the point runs along it, so the output dwells — 146° of crank inside a one-degree band, against 42° for the four-bar it is built on. A dwell linkage does not stop. It moves less than the tolerance, and how much less is a number. A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding. Prescribing motion A linkage gives the motion its geometry allows. A cam gives the motion it was asked for, which sounds like an improvement and is a trade — the displacement becomes free and the derivatives stop being. How far back the wheel is pushed. Recoil against the part of the pendulum's swing that happens after the tooth has landed. The arc cut concentric with the pallet arbor is the flat line on zero, and it is zero as a matter of arithmetic rather than of smallness: rotation about the arbor carries that arc into itself, so the tooth's resting place does not move and every sampled value is the same double. Every other face rises. The flat face with no draw at all is the interesting one — it starts at zero and curves, because it agrees with the arc to first order and not to second, which is precisely the amplitude sensitivity a deadbeat exists to remove. The wheel that goes backwards While a pendulum finishes its swing the escape wheel is doing something, and what it does is decided entirely by the shape of the face the tooth is resting on. An arc about the pallet arbor sends it nowhere — not nearly nowhere, the same double at every sample. A flat cut tangent to that arc is dead at exactly one point of itself, and it is the one point the tooth never rests on.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Base circleCamCircular arcConcentric arcCoupler curveDeadbeatDwellEscapementthe Geneva mechanismIntermittent motionLocking faceTransmission function