The smoothest coupler point is chosen in time

No drive holds the third pole found that a crank’s schedule cannot make a coupler point’s jerk nought over a stroke. A schedule is one function, the crank’s speed against its angle, and the two numbers that set a point’s jerk, $\kappa = \ddot\theta/\dot\theta^2$ and $\lambda = \dddot\theta/\dot\theta^3$, are tied by $\lambda = \kappa' + 2\kappa^2$. The numbers a point would need to be jerk-free miss that relation almost everywhere. The best schedule it could find took a fifth to a third off the jerk of the three points it tried.

That essay ended by turning the question round. The designer usually chooses the coupler point as well as the drive. Every point has a steady jerk and a best scheduled jerk, so a map of the coupler plane would show where scheduling helps most and which point a free designer should take. Is that point near Ball’s point, the point whose path is straightest at an instant, or somewhere the instant gives no hint of?

The first step was choosing what to measure, and the measure the earlier essay used turns out to choose a bad point.

Three ways to say smooth, two of them wrong #

Raw jerk is the first way, and it fails at once: a point near a fixed pivot hardly moves and so hardly jerks. The crank pin itself moves on a circle of radius one in these units. Smoothness has to be judged against travel, the jerk a point suffers for the distance it covers.

The second is the measure the schedule essay minimised, the root mean square over the stroke of the jerk per crank speed cubed. A coupler point’s jerk is $\dot\theta^3$ times a bracket, $z_3 + 3\kappa z_2 + \lambda z_1$, built from the point’s derivatives in crank angle and the two schedule numbers. Dividing out $\dot\theta^3$ leaves the bracket, a quantity that depends only on the crank’s angle and the schedule’s shape. That is convenient, and it hides something. A schedule that makes the crank crawl through part of the stroke and race through the rest changes $\dot\theta$ by a large factor. The jerk a mechanism feels in time is that factor cubed times the bracket, and the bracket measure does not see it.

The third is jerk in time. Fix the stroke’s duration, run the crank under the schedule, take the root-mean-square jerk over time, multiply by the duration cubed and divide by the point’s path length. The result has no units and does not change if the whole stroke is run faster or slower, so it judges a schedule’s shape and not its speed. A point on the crank pin turning steadily scores $\Theta^2$, the stroke’s angle squared, 4.39 for a stroke of 120°. That was confirmed to 1% as the measure’s first check.

The best schedule for each point under the third measure was found by Gauss–Newton on the schedule number $\kappa$ at 41 crank angles, with the crank’s speed $\omega = \exp\int\kappa\,d\theta$. The jerk formula it relies on was checked in time. The crank was run under a schedule, the coupler point placed from the solved four-bar at each instant, and its position differenced three times, which reproduces the formula’s jerk to 0.02%.

The map #

With a steady crank the scores run from 3.40 to 7.1 over the 625 points mapped, a square from −1 to 2 along the coupler and −1.5 to 1.5 across it. The smoothest steady point, (−1.00, 1.25), sits near the map’s corner. Points further out in that direction get worse again, and far out every point scores between 4.5 and 4.8, since a point far from the coupler moves mainly with the coupler’s own rotation. A steady crank leaves little to choose: the best point anywhere is only 23% smoother than the crank pin.

Scheduled for each point, the map changes character. The scores fall to between 1.26 and 5.12, and the plane breaks into valleys. The deepest lies just below the coupler’s line near A, and its floor, refined by a pattern search from the best grid point, is at (−0.02, −0.80). There the steady crank scores 4.56, and the best schedule 1.26, a cut of 72%. That is far beyond the fifth to a third the earlier essay found for its points, which were not in a valley.

A trench, not a pit #

The valley around the smoothest point has a shape a designer needs to know, because no coupler point is placed exactly. Of the 625 points mapped, 16 score below 1.5 when scheduled and 92 below 2.0, so good points are not rare. The deepest ones lie along a narrow trench parallel to the coupler’s line, about three quarters of a coupler length below it, running from $u = -0.75$ to $u = 0.25$ with scores between 1.34 and 1.54.

Along the trench the score hardly changes. Moving the smoothest point a tenth of a coupler length along the coupler’s line costs at most 3%, and two tenths at most 20%. Across it the score climbs fast: a tenth of a coupler length towards or away from the coupler’s line costs 38%, two tenths doubles it, and three tenths costs 130%. A designer who places the point within a few hundredths of the right distance from the coupler’s line can put it anywhere along a coupler length of the trench and give up little. The tolerance that matters is on the offset, not on the position along the coupler, and a drawing should dimension the tool point that way: its distance from the coupler’s line tightly, its position along it loosely.

A second, shallower valley lies near the map’s far corner, around $(1.4, 1.5)$ with scores near 1.46, on the way towards Ball’s point. Its existence is the first sign that the plane’s structure is not one basin around one best point. The section on Ball’s point below finds a third.

How much the choice of point is worth #

The question the earlier essay asked was whether choosing the point and then scheduling the crank does better than either alone, and by how much.

Scheduling alone, at a point chosen for other reasons, does what the earlier essay found. The crank pin is cut by 19%, from 4.39 to 3.56. Choosing the point alone, with a steady crank, takes the best score from the pin’s 4.39 to 3.40. Together, the smoothest scheduled point scores 1.26, 63% below the best steady point and 71% below the steady crank pin. Neither choice alone comes close. Put as factors, choosing the point with a steady crank is worth 1.29 against the pin, scheduling the pin is worth 1.23, and the product of the two would predict about 1.6. The combined choice is worth 3.5, more than twice the product, because the best point to schedule is not the best point to leave unscheduled. The two choices interact. Making them one at a time, first the point by its steady score and then the schedule, finds the smoothest steady point and schedules it to 2.08, two thirds worse than choosing them together. The schedule is worth most at points whose steady jerk is not especially low but whose jerk the schedule can reach, and those points are not the steady crank’s favourites: the smoothest scheduled point scores 4.56 when the crank turns steadily, worse than the pin.

The cheat in the other measure #

The second measure, jerk per crank speed cubed, has its own favourite. Over the same grid it is (0.63, −0.50), which that measure scores at 0.254 against 1.25 for a steady crank, an 80% cut that looks better than anything the time measure found.

The cut is bought with the crank’s speed. That schedule asks the crank to fall to a quarter of its starting speed across the stroke, a swing of four to one, and the bracket measure never charges for it. Run in time over a stroke of fixed duration, the same point and schedule score 2.59, twice the smoothest point’s 1.26. The time-measured schedule for the smoothest point asks the crank to fall to about half its starting speed, a swing of 1.92. Ball’s point’s schedule asks for an increase of 1.39.

In time the difference is plain. The smoothest point under its own schedule keeps its jerk low through the whole stroke, with a root mean square of 1.23 on this figure’s sampling against 4.53 steady. The other measure’s choice under its schedule peaks at 20.6, about four times anything the steady crank does to the smoothest point. Its root mean square stays at 2.20 only because the peak is brief. A designer who minimised jerk per crank speed cubed would have built a drive whose real jerk is dominated by one violent moment.

The algebra says exactly where the two measures part. In time, the quantity minimised is the integral over the stroke of $\omega^6$ times the bracket squared, against $dt = d\theta/\omega$, so each crank angle’s bracket is weighted by $\omega^5$, and then the stroke’s duration is fixed. The bracket measure weights every angle equally. When the schedule keeps $\omega$ within a few per cent of constant the two are the same up to a factor, and when it swings $\omega$ fourfold the weights differ by a factor of a thousand between the fastest and slowest parts of the stroke. A measure that cannot see a thousandfold weighting will happily choose a schedule that depends on it.

This is a correction to the measure rather than to the schedule essay’s conclusions. Its three points had best schedules that swung the crank by about one and a half, and over swings that small the bracket and the time measure rank schedules nearly alike. The difference appears only when the schedule is free to swing as far as it likes, which a map over the whole coupler plane allows.

Ball’s point is in another valley #

The straightest point there is found Ball’s point, the coupler point whose path has four-point contact with its tangent line at an instant, the straightest motion a four-bar offers there. At the stroke’s middle, 90° of crank, it lies at (1.55, 3.27) in the coupler’s frame, well off the map, more than four coupler lengths from the smoothest point.

It is not a bad point. Scheduled, it scores 1.70, with a gentle schedule that speeds the crank up by 39%. Walking from it towards the smoothest point, the scheduled score first falls to 1.50 two tenths of the way along, then climbs over a ridge of 4.15 at eight tenths, and drops into the deeper valley at the smoothest point, 1.26. A steady crank sees almost none of this structure; its scores along the same line stay between 4.6 and 5.1.

So the instant does give a hint, just not the best one. The circle of points going straight placed the instant’s straight-moving points on the inflection circle, and Ball’s point is the one of them whose straightness lasts longest. That is a property of one moment of the stroke, and the scheduled score integrates over all of it. Every point has a centre gave each coupler point its centre of curvature at an instant, and a point’s smoothness over a stroke is what happens to that centre, and to its rate of change, through the whole stroke. Ball’s point is straight at one instant, and a path that is straight at the stroke’s middle has small normal acceleration and small changes in it there, which helps a schedule. The smoothest point over the whole stroke owes nothing to the middle instant in particular. It sits below the coupler near the crank pin, where the coupler’s rotation and the crank’s own circle combine into a motion that one schedule can steady across the stroke rather than at one angle.

That matches what the third pole goes anywhere but the circle and the schedule essay found at the level of single angles. At one angle a schedule can make any point jerk-free. Over a stroke it has one function to spend, and the points where that function goes furthest are the ones whose needs at different angles are most nearly compatible. Nothing about the instant says which those are; the map has to be computed.

The ledger #

The ledger puts the five side by side. The smoothest scheduled point is the best by far, scheduled, and among the worst unscheduled, which is the one fact a designer could not guess. The smoothest steady point is a mediocre scheduled one. Ball’s point is good both ways. The crank pin is a fair reference and gains least from a schedule. The other measure’s favourite is good on paper and twice as rough as the best in time.

The circle a point stays on longest and where the curvature stands still chose coupler points by instantaneous properties of their paths, and those choices remain right for what they ask: a circle held, a curvature held, at an instant. A stroke’s smoothness is a different question, an integral over the stroke of a quantity a schedule can reshape. It has a different answer, found by computing it everywhere.

What the map does not settle #

The frame of reference. All positions are in the coupler’s own frame, A at the origin and B at one. The frame seen from the coupler found how the same motion looks from the coupler, and a point fixed in that frame is what a designer attaches a tool to. The map would look different drawn in the ground frame at any one crank angle, and the trench would be a curve rather than a line.

One stroke, one linkage. The map is for the standard four-bar over 30° to 150° of crank. Another stroke or another linkage has its own map, and nothing here says the smoothest point stays below the coupler near A.

The schedule’s own cost. A schedule that halves the crank’s speed across the stroke asks the drive for torque to decelerate the crank and whatever is on it. That torque, and the drive that could deliver it, are not modelled. The time measure charges for the schedule’s effect on the coupler point, not for the crank’s own inertia.

A full turn. The stroke is 120°, with free ends. A crank that turns continuously must return to its starting speed after a full turn, which forbids the steady deceleration the smoothest point’s schedule uses. The same map with a periodic schedule over a full turn is not computed and may look different.

Root-mean-square jerk. Smoothness is judged by the root mean square. A peak measure would judge the other measure’s favourite even more harshly, and might move the smoothest point.

Still open: the stroke a schedule can repeat #

A schedule over a stroke can end at any speed it likes. A machine running continuously cannot: its crank’s speed after a turn must be what it was before, so the schedule’s logarithm must return to where it started, $\int\kappa\,d\theta = 0$ over the turn.

The distinct argument would be the map redrawn under that constraint, with the schedule periodic over a full turn and smoothness judged over the whole turn or over a working stroke within it, the rest of the turn free to recover. The measurement would ask whether the smoothest point survives the constraint. It would also ask how much of its 72% a periodic schedule keeps, and whether the return stroke must then carry a jerk larger than the working stroke saved. If the working stroke keeps most of the gain and the return pays, a scheduled crank is a way to move jerk from the stroke that matters to the stroke that does not.