A dwell made from a curve

A machine that must hold something still while something else happens to it has a standard answer, and the standard answer is a cam. Cut a profile with a flat stretch on it, run a follower over the flat, and the follower does not move while the cam turns.

That works and it is expensive. A cam has to be cut to a profile, it wears at a line of contact rather than in a bearing, it needs a spring or a groove to keep the follower on it, and its motion law has consequences in acceleration and jerk that must be designed rather than inherited.

A six-bar can do the same job out of nothing but pin joints, and the construction is a piece of opportunism worth admiring.

The observation #

A four-bar’s coupler point traces a sextic — a curve of degree six, generally with two branches, whose shape is a sensitive function of the four lengths and of where the point sits.

Parts of a sextic are very nearly circular.

That is not a theorem and it is not always true; it is an observation about a family of curves that happen to have stretches of nearly constant curvature. And it is enough, because if a stretch of the curve is nearly a circle of radius $r$ centred at $C$, then a rigid link of length $r$ from the coupler point to a pin at $C$ will hold that pin nearly still while the coupler point runs along the stretch.

Hang the output on that pin, and the output dwells.

Two ways to build the same idea #

There are two standard versions of this construction and it is worth knowing which one is here, because the pictures look similar and the mechanisms are not.

The arc on the coupler curve, which is what this essay builds. The tracing point runs along a stretch that is nearly a circle; a link of that radius holds its far end still; the output hangs on the far end. The dwell happens while the coupler point is on the arc.

The arc as a slot or a cam-like path, in which the link is replaced by a slotted member shaped to the curve. That is not a linkage any more — it has a higher pair, a sliding contact and a profile to cut — and it is really a cam with extra steps.

There is a third arrangement, sometimes called a dwell obtained from a circle-point, where the second dyad is chosen so that its own coupler has a stationary point rather than exploiting the first loop’s curve. It gives a briefer dwell and better proportions.

Only the first is built here. The others are named so that a reader comparing this figure against a textbook diagram can tell whether they are looking at the same thing, which is a real difficulty in this corner of the subject: the diagrams are nearly identical and the mechanisms are of different kinds.

Finding the arc #

Nothing about which stretch of the curve is most nearly circular can be read off the link lengths, so it is measured.

The coupler curve is traced at 360 crank angles. Every window of 94 consecutive samples — about a quarter of the curve — is fitted with a circle by the algebraic least-squares method, and the window whose fit is best is kept.

For the four-bar used here, with ground 3.2, crank 1, coupler 3.0, rocker 2.6 and the tracing point at $u = 0.55$, $v = 0.85$: the best window fits a circle of radius 3.006 with a worst radial deviation of $3.0 \times 10^{-3}$ — a relative error of just under a thousandth.

That thousandth is the whole quality of the mechanism. It is not zero, and no amount of design makes it zero: a sextic is not a circle, and the dwell is approximate for the same reason a straight-line linkage is approximately straight. Peaucellier’s linkage is exact and needs eight links; Watt’s is 9% out and needs four. This is the same trade in a different quantity.

The measure that had to be relative #

The first version of the search minimised the absolute radial deviation, and it returned a nonsense answer: a stretch of curve that was very nearly straight, fitted by a circle of radius nine hundred.

The arithmetic was correct. A nearly straight stretch is very well fitted by a very large circle, and its absolute deviation is small because the circle barely curves. The linkage built on it would have had a bar nine hundred units long on a mechanism whose links are of order three.

Minimising the deviation relative to the fitted radius removes the degeneracy, because a large circle now has to be proportionately as good as a small one. There is also a cap on the radius, which is a second guard doing the same job less elegantly.

This is a recurring shape and the site has recorded it before: an optimisation criterion that is correct arithmetic about a quantity nobody wanted. The spatial phase’s drift measurement reported 89° for a planar four-bar whose constraints certainly do not move, and the arithmetic was faultless — it was measuring the wrong subspace.

What the dwell is worth #

The output angle is measured through a full turn of the crank, unwrapped so that crossing the seam at $-\pi$ does not read as the fastest motion in the cycle, and the longest run over which it stays inside a stated band is recorded.

Inside a band of ±1°: the six-bar holds for 145.8° of crank. The four-bar it is built on holds for 41.9°.

The four-bar’s number is the one that makes the six-bar’s mean anything. Every rocker holds still to within a degree somewhere — at the ends of its swing, where it is momentarily stationary because it is turning round — so a dwell figure quoted without a baseline is quoting a property of rockers.

Three and a half times, at that band. assertTheDwellIsTheArc requires at least 1.5 times, which is deliberately loose: the assertion is that the six-bar does something categorically different, and pinning it to the measured 3.5 would make the check fail whenever the dimensions changed for any reason.

Why the six-bar wins by so much #

The ratio of 3.5 deserves an explanation rather than a bare number, because it is larger than the construction obviously earns.

A four-bar rocker holds still near its limit positions because its angular velocity passes through zero there. That is a point of zero velocity, and the interval over which the angle stays inside a band is however long it takes the velocity to grow past the point where it matters — which depends on how fast the velocity changes, that is, on the acceleration.

The six-bar’s output holds still for a completely different reason. Its pin sits at the centre of a circle the coupler point is running along, so its velocity is near zero not at an instant but throughout the stretch. The dwell interval is set by how long the coupler point stays on the arc, which is a matter of the arc’s angular extent rather than of anybody’s acceleration.

So the two mechanisms are producing small output motion by two unrelated mechanisms, and only one of them scales with the length of an interval. That is why the ratio is large and why widening the tolerance band helps the four-bar proportionally more — its interval grows as the square root of the band, roughly, while the six-bar’s grows only as the arc’s fit degrades.

Checking that prediction properly would mean measuring the interval against band width for both and comparing the exponents. It is not done here, and it is the obvious next measurement.

The dwell is a tolerance, not a stop #

Widening the band lengthens both intervals, and that is the honest character of the whole construction.

Nothing here stops. The output moves throughout the cycle. What it does is move less than a stated amount over a stated interval, and both of those are numbers a designer chooses rather than properties of the mechanism. A dwell quoted without its tolerance is not a measurement.

That is worth insisting on because “dwell mechanism” is usually said as though the mechanism has a dwell the way a gear has a tooth count. It does not. It has an output whose derivative is small over part of the cycle, and how small is small enough is a question about the machine it is going into.

The comparison with a cam is instructive here. A cam with a flat stretch has an exact dwell — the follower is on a circular arc of the cam and does not move at all, to the accuracy of the cut. The six-bar’s dwell is approximate. What the six-bar buys is that it is made of pins.

Building it needs the four-bar solved first #

There is a structural point about the construction that is easy to skip and is the reason this mechanism could not have been drawn.

The dwell six-bar cannot be specified until its own arc is known, and the arc is a property of a curve that only exists once the four-bar has been solved through a full turn. So dwellSixBar builds a four-bar, sweeps it 360 times, fits circles to every window of its coupler curve, picks the best, and then declares the two extra links — one of which has the fitted radius as its length and the other of which is grounded at a point derived from the fitted centre.

The mechanism’s dimensions are outputs of a computation over the mechanism it is built on. There is no way to draw this linkage and then check it; the drawing does not exist until the check has been run.

That is the site’s premise in its strongest available form. Everywhere else, “solved before drawn” means the positions come from a solve while the dimensions were chosen. Here two of the six link lengths are themselves solved quantities, and choosing them by eye would produce a mechanism that assembles and does not dwell.

The arm’s length is the whole thing #

The construction claims that the dwell comes from the arc, and that is exactly the kind of claim that needs a test it could fail. The test is to vary the arm and see whether the dwell follows.

The dwell is largest exactly at the fitted radius and falls away monotonically above it. Below it, the mechanism does not assemble at all — an arm shorter than the arc’s radius cannot reach from the coupler point to a pin the output can follow, so the linkage comes apart partway through the turn.

So the arc’s radius is not merely a good choice. It is the shortest arm that can be built, and the best one, and those are the same length.

That table replaced a weaker assertion. The first version required the dwell to more than halve when the arm was detuned by 12%, and it failed: 145.8° became 87.9°, a 40% loss rather than a 50% one. The assertion was asking the wrong question — sweeping the arm and requiring the maximum to sit at the arc radius is a stronger claim and a true one, and the weaker version was rejected by the measurement rather than adjusted to fit it.

What the fit residual buys and what it costs #

The arc fit has a residual of a thousandth of its radius, and it is worth tracing what that thousandth becomes downstream, because it is the only inexactness in the whole mechanism.

The coupler point wanders off the fitted circle by up to $3.0 \times 10^{-3}$ in units where the links are of order three. The arm is rigid and its length is the fitted radius, so that wander is taken up by the pin at the far end moving — and how far it moves depends on the geometry of the output linkage rather than on the arc directly.

Measured rather than propagated: the output stays inside one degree for the interval quoted, and one degree of a rocker of length 4.2 is about 0.07 in the same units. So the thousandth at the coupler point becomes something of order a hundredth at the output, which is an amplification of roughly twenty.

That amplification is not a defect and it is not avoidable. The output rocker exists to turn a small motion of its pin into a usable swing during the rest of the cycle, and a linkage that amplifies during three quarters of the turn amplifies during the other quarter too. A designer wanting a tighter dwell has to accept less output swing, and the two are traded through the same lever arm.

This is the same structure as the transmission angle argument one field over: a linkage’s ability to do one thing well and another thing badly is usually one ratio seen from two ends.

Where the design freedom is #

Three things can be chosen and only one of them is chosen here.

The four-bar. Its four lengths determine the coupler curve, and therefore which stretches of it are nearly circular and how nearly. A different four-bar gives a different arc, a different radius and a different dwell.

The coupler point. The same four-bar with a different tracing point traces a completely different sextic. This is the largest lever and the least explored: the site’s own coupler curve essay shows how sensitively the shape depends on it.

The output pivot. Where the output rocker is grounded, and how long it is, decide how much the output moves during the non-dwell part of the cycle — which is the other half of what the mechanism is for. A dwell that is followed by no useful motion is not a machine.

This site fixes all three and measures one mechanism. Searching over them for the longest dwell at a given tolerance is an approximate-synthesis problem of exactly the kind the previous rung describes, with an objective that is non-convex, disconnected and undefined wherever the linkage comes apart. It is named as a gap.

What can be said from the one mechanism is that the construction works, that the arc is what makes it work, and that the quality of the dwell is bounded by how nearly circular a sextic can be — which is a property of coupler curves and not of anybody’s cleverness.

That bound is the honest closing note. A designer who needs a better dwell than a thousandth of a radius cannot get it by being cleverer about this construction; the limit is in the curve. Getting past it means a different mechanism, and the different mechanism is a cam.

How near the near-circular stretch is #

The construction rests on a stretch of the coupler curve being nearly an arc, and “nearly” can be given a number that compares it against curves which are arcs exactly.

So the six-bar’s dwell sits between the exact ones and the crude ones, which is exactly what an approximate synthesis should do: it is the best arc available from a mechanism that contains no arcs.

The dwell being a tolerance rather than a stop is the sentence to carry, and it has a consequence for how such a mechanism is specified. A cam’s dwell is exact: the follower is on a concentric arc and does not move at all, so the specification is a range of input angle and nothing else. A curve dwell has two numbers — how long, and within what band — and neither means anything without the other. 146° inside one degree is a completely different mechanism from 146° inside five, and the same linkage delivers both figures depending on which band is asked for. So the honest specification is a curve of dwell length against tolerance band, and the two numbers usually quoted are one point on it. That also says how to compare a curve dwell against a cam: not by the dwell length, which the linkage can always win by loosening the band, but by asking what band the application actually needs and reading the length there. A mechanism that trades exactness for parts has to be specified with the exactness in the specification, or the comparison is between a number and a promise.