The flattest dwell is not the longest

A dwell made from a curve built a six-bar whose output holds nearly still for a large part of every turn. It searched a four-bar’s coupler curve for the stretch most nearly a circular arc, hung a link of that arc’s radius from the tracing point, and let the link’s far end stand almost still while the tracing point ran along the arc. The machine is Stephenson’s chain, with the dwell link and an output link as its second dyad.

A symmetric curve from a lopsided machine found a place where no search is needed. When a four-bar’s rocker, coupler and the arm from the rocker pin to the tracing point are all one length, the coupler curve is its own mirror image, and where it crosses its mirror line its curvature is stationary. A link pinned at the centre of curvature there holds its far end still to fourth order, by the same amount before the crossing as after. That essay located the place, measured the curve there, and did not build the machine.

This essay builds it. On the way it finds a dwell of sixth order that no search would have looked for, and then finds that sixth order is not what a dwell mechanism wants.

A circle the curve leaves the same way on both sides #

The four-bar is the symmetric one from the earlier essay: ground 3, crank 1, and coupler, rocker and arm all 2.5. The tracing point sits on a circle of radius 2.5 about the rocker pin, and where on that circle is one angle γ, the angle at the rocker pin between the coupler and the arm. Every γ gives a symmetric curve, and every one crosses its mirror line at crank angle 0° and at 180°.

Take the circle that best fits the curve where it crosses the line at 0°, the osculating circle, with its centre on the mirror line. Measure how far the tracing point is from that circle a crank angle θ before the crossing and the same θ after. The mirror carries one position onto the other and carries the circle onto itself, so the two distances are equal exactly. As a function of θ the distance is even: its first, third and fifth derivatives at the crossing are all nought, not approximately but by symmetry.

The second derivative is nought too, by the choice of circle, and a crossing of the mirror line is a point where the curvature stands still, which removes the third. So the first term left is the fourth, and a dwell link of the osculating radius has its far end move at fourth order in the crank angle. That is what the earlier essay measured.

A vertex of a curve that is not symmetric is also fourth order, but it is not even. The circle a point stays on longest found the points of a general motion whose curvature is stationary and also stops accelerating, the Burmester points, and measured them holding their circle to fifth order. A plain vertex has a fifth-order term, and it is odd: the dwell comes in and goes out differently.

One angle for the fourth-order term #

The fourth-order coefficient depends on γ, and a symmetric curve offers something a general one does not. Remove that coefficient, and the next term is not the fifth, which symmetry has already removed. It is the sixth.

It does vanish, at γ = 120.8024°, where the osculating radius is 5.7691. The coefficient is positive below that angle and negative above it, and the angle is found by bisection on its sign, with the coefficient read from two spans so that the sixth-order term, which a single reading would mix in, drops out. Read from one span, the zero lands a thousandth and a half of a degree away: small, and exactly the size of the digits being printed.

In the language of the curvature essay, the tracing point at that angle is a Burmester point, since its curvature is stationary and has stopped accelerating. What that essay found only where two Burmester points merge — a circle held to sixth order — is available here from one angle, because the symmetry pays for the fifth order in advance. It is the same kind of move as Ball’s point, where one extra condition buys one extra order of straightness, with symmetry supplying a condition for nothing.

The other crossing of the mirror line, at crank angle 180°, has no such angle in this family. Its fourth-order coefficient changes sign only where the curve’s curvature does, between 100° and 105°, where the osculating radius runs off to infinity and there is no circle to hold.

Sixth order, measured #

A coefficient that vanishes says the fourth-order term is gone. It does not say that the next is sixth, and a claim about orders deserves a slope.

At 120.8024° the distance grows as the 5.97 power of the crank angle over spans from 2° to 16°, and its odd part never exceeds 4.4 × 10⁻¹⁶. Two degrees below, the power is 3.92, fourth order with the sixth-order term still showing at the widest spans. On the unsymmetric curve, with the same construction at its nearest vertex, the even part grows as the 3.99 power and the odd part is there, growing as the 4.96.

So the symmetric vertex at this one angle holds its circle two orders longer than an ordinary vertex and one order longer than a Burmester point of a general motion, and it is found by setting a single dimension.

The six-bar on it #

The machine needs two more links. The dwell link runs from the tracing point to a pin that starts at the centre of curvature. An output link from a third ground pivot holds that pin, set square to the dwell link at the vertex, so the pin’s small motion is along the output’s direction of travel.

The output link’s length scales the output angle, and every dwell below is measured as a share of the output’s own swing, so the length is a drawing choice for the dwell. At output links of 2, 3, 5 and 8 the dwell inside each tolerance agrees to within two tenths of a degree, the size of the step at which the turn is sampled. It is not a drawing choice for the machine. With an output link of 3 the output swings 7.43° over a turn; the six-bar the earlier essay searched for swings 14.0° on an output link of 5.17. To give the symmetric machine the same swing, its output link has to come down to about 1.6, and a short output link means a large force on the dwell link for a given output torque.

With every position computed from circles rather than by iteration, the output can be followed to rounding. Near the vertex it changes at sixth order in the crank angle, as the curve does.

Whether the curve’s symmetry survives into the output is a sharper question than it looks. The output is a function of the tracing point’s position, and the tracing point’s positions before and after the vertex are mirror images. But the second loop is not symmetric about the mirror line. Its ground pivot is off to one side, because the only way to put it on the line would put the output link along the dwell link at the vertex, which is a toggle. So nothing forces the output’s approach and departure to match exactly.

They match to rounding out to 8° of crank. At 16° the odd part is 9.4 × 10⁻¹³ against an even part of 4.0 × 10⁻⁶, more than six decades apart. On the six-bar built on the unsymmetric vertex the odd part grows as the fifth power and at 16° is 1.4 × 10⁻⁵, against an even part of 3.3 × 10⁻⁵: its dwell visibly comes in differently from how it goes out. The symmetry does not survive exactly, and it survives far past the angles a dwell uses.

The flattest is not the longest #

A dwell mechanism is judged by how long its output stays inside a stated tolerance, not by the order of its error at one instant, and the two are different questions. So each six-bar is measured by the longest interval of crank angle over which its output stays inside a band that is a stated share of its swing: 1%, 0.3% or 0.1%.

Measured in absolute degrees instead, the comparison would mislead. A band of one degree is 13.5% of the sixth-order machine’s swing and 7.1% of the searched machine’s, so the first holds it for 160.2° of crank and the second for 146.2°, and a symmetric machine detuned to 116.1°, whose swing is the smallest of all, holds it for 190.2°. Those numbers rank the machines by how little they move, which is not the question. A share of the swing asks how still the output is compared with how far it goes, and that answer does not change when the output link is lengthened or shortened.

At the sixth-order angle the answers are 90.5°, 71.9° and 58.7°. Turning the angle away from 120.8024° makes the contact fourth order again, and it makes the dwell longer.

Inside 1% of the swing the longest dwell is 122.9°, at 116.1°. Inside 0.3% it is 98.3°, at 117.6°. Inside 0.1% it is 80.1°, at 118.6°. At every tolerance the best angle lies below the sixth-order one, and it moves towards it as the tolerance narrows.

What the detuned machines do differently shows in their outputs.

The sixth-order output leaves its dwell value slowly and does not come back. The detuned outputs leave it, turn, pass back through it and only then leave for good. Moving γ below 120.8024° gives the error a small fourth-order term of the opposite sign to the sixth-order one, and near the vertex the two partly cancel: the output moves off at fourth order, the sixth-order term overtakes and turns it back, and it crosses its starting value again before the sixth order carries it away. The whole excursion fits inside a band that the sixth-order output had left long before.

The measured excursions say how closely each detuned machine is fitted to its band. At 118.6° the output rises 0.092% of its swing above its value at the middle of the dwell and comes back, inside a band of 0.1%; at 116.1° it rises 0.94% inside a band of 1%. Each spends nine tenths of its band on the excursion and the rest on the final departure, which is what the longest interval inside a band has to do: use all of it.

That explains the jumps in the curves above, too. The dwell at a given tolerance grows steadily as γ moves down until the output’s return swing first fits inside the band, and then it jumps, because the interval inside the band suddenly includes the whole excursion. Just past the jump is the longest dwell for that tolerance. A narrower band admits only a smaller excursion, which needs a smaller fourth-order term, which is an angle closer to 120.8024°: the three best angles sit 4.7°, 3.2° and 2.2° below it, falling by a factor of about 2.1 for each factor of ten in the tolerance.

This is the same trade the straight-line problem made with Watt’s linkage, and that exact costs more than close measured in general: a curve that matches its target to the highest order at one point is not the curve that stays closest to it over an interval. The highest order is the right answer only in the limit of no tolerance at all.

Against the searched dwell #

The six-bar of the earlier essay was found by fitting circles to windows of a general coupler curve and keeping the window whose fit had the smallest relative error. It has no symmetry and no special order.

Measured the same way, as a share of its own swing, it stays inside 1% for 100.0° of crank, inside 0.3% for 67.2° and inside 0.1% for 24.4°. The symmetric six-bar tuned for each tolerance stays inside for 122.9°, 98.3° and 80.1°: longer at every one, and more than three times longer at the tightest. Even untuned, at the sixth-order angle, it beats the searched machine at the two tighter tolerances.

Why the searched machine falls off so sharply is visible in its output about the middle of its tightest dwell. It is not a stationary point. Taken apart the same way as the symmetric output, its odd part grows at first order in the crank angle and its even part at second: 2 × 10⁻⁶ and 1.8 × 10⁻⁶ at one degree, 2.1 × 10⁻⁵ and 4.0 × 10⁻⁴ at sixteen. The output is still moving at the middle of its dwell, only slowly, and it is curving as well. A band wide enough to contain a slow drift for a hundred degrees is not wide enough to contain it for more than twenty-four once the band is ten times narrower.

The reason is the search’s objective. A least-squares circle through a window of the curve spreads its error across the window, which is a reasonable thing to do for a 1% dwell and a poor one for a 0.1% dwell. There the error that matters is the part near the middle, and a fitted window has no reason to make that part vanish to any order at all. The symmetric construction makes the middle of the error exactly even by geometry and lets one angle set its size.

One family, and no claim of the best #

One family. Every number belongs to four-bars with ground 3, crank 1 and the other three lengths 2.5. Other lengths meeting the symmetry condition have their own sixth-order angle, or none, and nothing here surveys them.

The output link’s placement. The output link is set square to the dwell link at the vertex, and its length and side are shown not to change the dwell. Other placements are not compared, and the short output link a large swing needs would matter to a real machine whatever the kinematics say.

Optimality. The best angle at each tolerance is the best on a scan with steps of 0.1°, not a proven optimum, and the explanation of why it lies below the sixth-order angle is a reading of the measured curves.

Clearance and wear. The dwell is a kinematic quantity of a rigid machine. A band of 0.1% of a 14° swing is 0.014°, and a clearance behaves like a short extra link that can move the output by more than that on its own. How much of any of these dwells survives real pins is a separate question and a pressing one.

What comes next: the tolerance as the design variable #

The three best angles sit 4.7°, 3.2° and 2.2° below the sixth-order angle for tolerances of 1%, 0.3% and 0.1%, which is a factor of about 2.1 per factor of ten in the tolerance, and 2.15 is ten to the power one third. The reading of the output curves above predicts exactly that power. If the error is a fourth-order term proportional to how far γ is from 120.8024° plus a fixed sixth-order term, the excursion that just fits a band of width ε has a half-width growing as ε to the power one sixth, and the fourth-order term needed to produce it grows as ε to the power one third. The dwell at the sixth-order angle itself should grow as the one sixth power too, and its measured values, 58.7° and 90.5° a factor of ten apart, grow by 1.54 against the 1.47 that power gives.

Its distinct argument would be that law, tested rather than read off three points: the best angle and the longest dwell measured at tolerances across four or five decades, the gap to the sixth-order angle fitted against the tolerance, and the dwell length fitted too, to see whether both follow the powers the two-term picture predicts — and at what tolerance the terms it leaves out take over.