A crank that must come round picks another point

The smoothest coupler point is chosen in time mapped the coupler plane of the standard four-bar for the point a scheduled crank keeps smoothest over the stroke from 30° to 150°. The schedule is the crank’s speed as a function of its angle. Smoothness was measured as the jerk a mechanism feels in time, root mean square over a stroke of fixed duration, per unit of the point’s own travel. The best point was (−0.02, −0.80) in the coupler’s frame, with the crank pin at the origin and the rocker pin at (1, 0). Its schedule cut its jerk by 72%.

That schedule was free at both ends of the stroke. It could end the stroke at any speed it liked, faster or slower than it began. A machine that runs continuously cannot do that. After a whole turn the crank must be turning at the speed it started at, or the next turn begins differently and the speed drifts. The stroke the schedule can repeat is a different object, and the essay ended by asking whether the smoothest point survives that constraint, how much of its 72% it keeps, and whether the return stroke then pays.

The answer has two halves. The point keeps its gain only if the return pays heavily. A different point, chosen for the whole turn, keeps most of the gain without the return paying at all.

The constraint, stated exactly #

The crank’s speed $\omega$ against its angle $\theta$ is the schedule. Its logarithm, $\ell = \ln\omega$, has derivative $\kappa = \omega'/\omega$, the number the jerk depends on alongside $\kappa'$. No drive holds the third pole wrote the jerk of a coupler point per crank speed cubed as $z''' + 3\kappa z'' + (\kappa' + 2\kappa^2) z'$, with primes taken in the crank angle.

Repeating every turn means $\ell(\theta + 2\pi) = \ell(\theta)$, so $\int \kappa\, d\theta = 0$ round the turn. The cleanest way to impose that is to write the logarithm of the speed as a Fourier series in the crank angle, $\ell = \sum a_n \cos n\theta + b_n \sin n\theta$, which is periodic by construction. Then $\kappa$ and $\kappa'$ are exact derivatives of the series. Sixteen harmonics are used, and the point’s jerk is evaluated every degree round the turn.

The series matters for a reason found while measuring. A schedule sampled point by point, every three degrees, gave the optimiser freedom at the scale of the samples, and it took it. One schedule tripled the crank’s speed for a single sample, racing through a feature the samples could not see between them. A series of sixteen harmonics has no such freedom. Whether sixteen are enough is checked by comparing twelve and twenty-four, which agree where the schedule is smooth.

The second constraint is the stroke’s share of the cycle. A schedule that slows the crank through the working stroke gives the stroke more time, and since jerk falls as the cube of the time, that alone lowers the stroke’s jerk. Left free, the best schedule for the stroke gives it 49% of the cycle instead of a third. A machine with a fixed cycle and a working stroke of fixed share cannot buy smoothness that way, so the stroke is held to a third of the cycle, a steady crank’s share. Every comparison below is at equal cycle time and equal share.

Repeating every turn, the stroke keeps its gain only if the return pays #

The return, from 150° round to 390°, is scored the same way as the stroke, by its own jerk per its own travel over its own duration. The schedule then minimises the stroke’s score squared plus a weight $\mu$ times the return’s. A small weight leaves the return free to recover however it must. A weight of one counts it as much as the stroke.

With the return nearly free the stroke’s jerk falls to 0.324 of a steady crank’s. That is 68% off, close to the open stroke’s 72%. The return then carries 5.50 times a steady crank’s jerk. Count the return at a thousandth of the stroke and the stroke keeps 44% off while the return carries 2.77 times steady. As the weight grows the two scores trade along a curve, the stroke rising and the return falling at every step, which the model checks.

The curve never passes below and left of the steady crank. At a weight of 0.1 the stroke is at 1.008 of steady and the return at 0.991; at 0.03 they are 0.946 and 1.033. No weight from $10^{-4}$ to 1 leaves both strokes even 2% below a steady crank, and the model checks that too. At this point a steady crank is already on the frontier, so a schedule can move jerk between the strokes but cannot remove it.

Left free to give the stroke more of the cycle, the schedule recovers the open stroke’s optimum. The stroke’s score falls to 1.254 against the open stroke’s 1.263, found by a different optimiser on a different grid, which is the first route’s check. But it gets there by taking 49% of the cycle for the stroke, and its return carries over twice a steady crank’s jerk even with that extra time.

The jerk goes where the crank must hurry #

The time profile shows where the return’s jerk comes from.

Through the working stroke both schedules hold the jerk below the steady crank’s. After it the crank has to make up time. The stroke-first schedule slows the crank to about two-thirds of its mean speed a third of the way into the return, then accelerates it to 1.85 times its mean just before the next stroke begins, as the speed curve in the opening figure shows. The point’s jerk follows the crank’s own acceleration: it is quiet through most of the return and then rises to over 4,000 in these units near the end of the cycle, more than twenty times the steady crank’s level there.

That is the price in its plainest form. The stroke’s quiet is bought by a burst on the return, and the burst lands just before the stroke begins again, which is where a real machine would least want it.

The open stroke’s point is not the turn’s #

The stroke’s best point was chosen with both ends of its stroke free. There is no reason the same point should be best when the stroke has to hand the crank back at its starting speed, and it is not.

A grid step from the open stroke’s point, at (−0.15, −0.80), the stroke’s score under the same periodic schedule is 2.265 against 2.571, 12% lower, with a slightly lower return too. The open stroke’s point sits in what the earlier essay called a trench, a long shallow valley of nearly equal points, and the periodic constraint tilts the trench.

None of the six points near the open stroke’s choice changes the picture much, though. Under the same schedule their working strokes run from 2.27 to 2.97 and their returns from 61.6 to 68.5, all of them paying more than two and a half times a steady crank’s return for a stroke at half to two-thirds of a steady crank’s. The tilt of the trench moves the best point a little; it does not make that neighbourhood a good place to schedule a continuously running crank. A point whose return can share in the schedule’s benefit has to be looked for elsewhere in the plane.

A search for the turn’s own point started from a grid every half coupler length over the earlier map’s range. It then refined by pattern search, at a weight of a thousandth, for the least working-stroke score. It ended at (1.44, 1.89), far from the open stroke’s choice, at the edge of the earlier map. The model confirms it as a minimum among its four neighbours at the search’s last step, a sixty-fourth of a coupler length.

At that point a steady crank gives a working stroke of 5.38 and a return of 30.9, both a little worse than at the open stroke’s point. Under the periodic schedule the stroke falls to 1.401 and the return to 25.4, 74% off the one and 18% off the other, together. The crank’s speed swings only 1.66 to one, and its curve in the next figure is a single smooth rise and fall: slow at the start of the stroke, fastest just after it, slow again before the next. Twelve harmonics give 1.409, sixteen 1.401 and twenty-four 1.398, so the schedule is resolved.

Drawn over a whole turn, the three schedules put the crank’s fastest and slowest moments in different places. At the open stroke’s point with the return nearly free, the crank is slowest at 221°, deep in the return, and fastest at 10°, just before the stroke begins, a swing of 2.83 to one. It enters the stroke at speed and spends the stroke braking. Weighting the return a thousandth narrows the swing to 1.76 without moving those two moments much: slowest at 229°, fastest at 13°. The turn’s own point is slowest at 341°, just before the stroke, and fastest at 169°, just after it, at 1.66 to one. Its crank does not brake through the stroke; it gathers speed through it, and the return is where it eases off.

At the open stroke’s point the same comparison does not settle. Twelve harmonics give 2.725, sixteen 2.571 and twenty-four 2.441, each added harmonic sharpening the schedule and lowering the score further. A schedule that keeps improving as it is allowed to change faster is asking the crank for abrupt changes of speed. A real drive delivers that only approximately, while the turn’s point needs nothing of the kind.

Why the turn’s point can have both #

The scores say what differs between the two points, though not everything about why. At the open stroke’s point the steady crank already gives a return of 23.7, and every schedule that eases the stroke worsens the return: the frontier has no part below and left of steady. The speed the stroke gives back must be taken up on the return, and at this point there is nothing on the return that taking it up can improve.

At the turn’s point the steady crank gives a worse return, 30.9, and a worse stroke, 5.38. Both have room to improve, and one schedule improves both. Its shape is a single rise and fall of crank speed: slow at the start of the stroke, fastest just after the stroke ends, slow again before the next. Whatever this point’s path asks of the crank, it asks for one slow-fast-slow cycle per turn, where the open stroke’s point asks for a sharp slowing in the stroke and a hard acceleration late in the return, which no gentle schedule can give.

The smoothest coupler point is chosen in time found that Ball’s point, the instant’s straightest from the straightest point there is, was a poor guide to the stroke’s smoothest point. Here the stroke’s smoothest point is a poor guide to the turn’s. Each widening of the question moves the answer, because each brings in parts of the motion the narrower question did not see.

The ledger #

Read across, the table gives a designer of a continuously running machine three findings. The point a single stroke prefers is not worth scheduling for continuous running: either the return pays several times over or the gain largely vanishes. A point chosen for the turn can be scheduled for both strokes at once, with a crank speed that varies gently. And the turn’s point gives up only a little on the stroke alone: its open-stroke score is 1.395 against the open optimum’s 1.263, about a tenth more, for a return that is better instead of five times worse.

A schedule that closes is a gear that closes #

The constraint here has a twin elsewhere in these essays. A ratio that is a function of the angle cut a pair of non-circular gears to deliver a prescribed speed law, and found that the demand fixes both pitch curves completely, leaving only the question of whether they close after a turn. A crank schedule delivered by such gears has to close in exactly the same sense, its speed back where it started after a turn, and $\int\kappa\,d\theta = 0$ is that closing condition written in the schedule’s own terms.

A drive that goes round is an angle made the related point about a constraint that goes all the way round: a periodic input is a different object from an interval of one. The same is true of a schedule. One defined over a stroke is a function on an interval. One defined over a turn is a function on a circle, and the circle’s single condition is what moved the best point three coupler lengths away.

The third pole goes anywhere but the circle showed that a point’s jerk depends on two numbers of the drive, and no drive holds the third pole found that those two are tied by one relation, $\lambda = \kappa' + 2\kappa^2$, so a schedule cannot zero the jerk. Periodicity adds a second tie, $\int\kappa = 0$. Neither forbids smoothness; each narrows where it can be found.

What a cam would do instead #

A designer who wants a smooth coupler point over a working stroke has an older option than a scheduled crank: drive the output from a cam and prescribe its motion outright. Prescribing motion described that trade. A cam gives the displacement it is asked for, and the price is that the derivatives are no longer free: they follow from the displacement the designer chose. The law that costs least measured how much a law’s jerk costs against its peak acceleration.

A cam must also come round. Its rise and return are one closed profile, and the return has to undo whatever the rise did. The cam designer meets the same constraint this essay found, in a different form: a gentle return makes room on the rise and a quick one takes it away. A scheduled crank on a four-bar inherits that discipline as soon as it runs continuously.

What a scheduled four-bar has that a cam does not is a choice of point. A cam’s follower is where the cam’s profile puts it. A coupler point is chosen from a whole plane, and the plane holds points whose stroke and return ask for compatible schedules. The turn’s point is one such, and the search that found it was not exhaustive, so there may be better.

What the turn does not settle #

The search was local. The turn’s point came from a coarse grid and a pattern search, and it sits at the edge of the range searched, so a better point may lie beyond it. Its neighbours at a sixty-fourth of a coupler length are all within a part in a thousand of it, so it lies in a valley as flat as the open stroke’s trench.

The weighting. The turn’s point was found with the return weighted a thousandth. A designer who counts the return more heavily would find a different point, and the frontier at the turn’s point was not traced.

The drive. A schedule is a speed law, and something has to deliver it: non-circular gears, a servo motor, or a second linkage driving the first. The turn’s point’s gentle schedule is the easier to deliver. How closely each drive delivers it is a separate question.

Still open: a schedule a pair of gears can cut #

The turn’s point needs a crank speed that rises and falls once per turn by a ratio of 1.66. That is the kind of law a pair of non-circular gears delivers exactly, and the gear essays found that a law fixes both pitch curves.

The distinct argument would cut that pair. It would take the turn’s point’s schedule as the law, derive the two pitch curves, and check that they close and stay convex. It would then ask what the pair’s pressure angle and curvature do over a turn, whether the gentler schedule makes a better-behaved pair than the open stroke’s point’s sharper one, and how the coupler point’s jerk changes when the gears carry a small tooth error. If the pair is reasonable, a continuously running four-bar can be smoothed by one gear pair chosen together with its coupler point. If it is not, the schedule is a servo’s to deliver.