Five degrees buy an even cut at a ratio of three

A ratio of three costs forty degrees priced a shaper’s quick return in transmission angle. The drive is a drag link, a double crank whose output turns unevenly, ahead of an offset slider-crank that moves the ram. Asked for a return three times as fast as the cut, the two stages together can give it only if neither keeps more than 40.0° of worst transmission angle. At 40° exactly one train does it, and its ram cuts badly. Through the middle four fifths of the working stroke its fastest speed is 3.25 times its slowest, more than twice the unevenness of the earlier essays’ best trains at a ratio of 2.

That essay’s frontier was drawn on time alone, and it ended by naming the second one. At a ratio of 3, what is the most even cut each worst transmission angle below 40° allows? The answer decides what a usable ratio-3 shaper costs. If a few degrees buy the evenness back, the 40° frontier is close to the real price. If ten or fifteen are needed, it understates the price badly, and the Whitworth drive’s slotted link, which makes its quick return a different way, becomes the comparison to draw.

A few degrees buy it back. Five of them buy nearly all of it.

What is searched, and what is held #

The drive is the earlier essay’s. A drag link of ground 1 has an input crank, a coupler and an output crank. The output crank drives an offset slider-crank of crank 1, with the slide offset from the crank’s centre and a rod to the ram. Between them is a phase, the angle at which the drag link’s output is fixed to the slide’s crank. The ratio is the input angle spent on the cut over the angle spent on the return, read off the ram’s two dead centres in closed form.

The evenness is a quick return that cuts evenly’s measure: the ram’s fastest speed over its slowest while it covers the middle four fifths of its working stroke, with the input turning steadily. The ends of the stroke are left out because the ram has to stop there to reverse, whatever the design. One is a ram that cuts at constant speed.

The search holds three things. The ratio must be at least 3. The drag link must remain a double crank whose worst transmission angle is at least μ. And the slide’s worst transmission angle must also be at least μ, the same angle in both stages, as on the earlier frontier. Six numbers are free: the drag link’s three lengths, the slide’s offset, the slide’s rod and the phase.

The rod is the one freedom the earlier frontier did not use. There every slide had the shortest rod its angle allows, because a longer rod only lowers the slide’s own quick return, and the time frontier wanted all the ratio it could get. Here the ratio is fixed at 3, so a longer rod is no longer obviously wasted. It moves the ram’s peak speed in the stroke, which a centred slide has a best rod found is something a drag link can phase against.

Each angle’s search is a pattern search over the six numbers, minimising the evenness. It starts from the time frontier’s own train at that angle, which below 40° reaches more than 3 and so has ratio to trade. It is then repeated with each angle seeded from its neighbours’ answers, keeping the better.

Below forty, the price falls fast #

At 40° there is almost nothing to trade. The time frontier’s train reaches 3.007, barely above 3, and the most even train that still reaches 3 cuts at 2.97 against its 3.25. Holding the ratio at 40° leaves the cut uneven.

One degree lower, the most even train cuts at 2.73. At 38° it is 2.53, at 37° 1.90, at 36° 1.64 and at 35° 1.42. Below 35° the curve flattens: 1.32 at 34°, 1.26 at 32°, 1.22 at 30°. Most of the improvement happens in the three degrees from 38° to 35°. After that, each further degree buys little.

The shaded band is the comparison the earlier essays make possible. Their best trains at a ratio of 2 cut at 1.12 to 1.47, depending on the slide. By 35° a ratio-3 shaper cuts as evenly as those did, and by 30° it is within a tenth of the best of them. So an even cut at a ratio of 3 costs about five degrees below the 40° the ratio alone costs. Five, not ten or fifteen.

The dashed curve shows why the earlier frontier could not see this. It is the train with the largest ratio at each angle, and its evenness gets worse as the angle falls: 4.35 at 35°, 6.12 at 30°. Below 40° the frontier spends every degree on ratio it does not need, 3.43 at 35° and 3.98 at 30°. Each extra unit of ratio is paid for in unevenness. Held at a ratio of 3, the same degree goes to the cut instead.

The cut straightens #

The speed profile above shows what the evenness numbers mean for the ram. At 40° the most even train still slows through the middle of the stroke. Its speed rises to about 1.6 times its mean near the start, falls to about 0.6 in the middle, and rises again before the turn. That is the sag the earlier essay drew for its 40° train, a little reduced.

At 35° the same curve flattens to a gentle wave. The ram reaches 1.33 times its mean speed early in the cut, dips to 0.93 at about two fifths of the stroke, and rises to 1.32 near the end. At 30° the wave is smaller still. The trains all keep the two peaks and the dip between them, because the slide’s own motion has that shape and the drag link can only reshape it. What the drag link does at the lower angles is fill the dip.

Through a whole turn of the input the two trains look alike at first glance. Both cut for about three quarters of the turn and return in the last quarter, as a ratio of 3 requires. The difference is in the long slope. The 40° train’s cut bends, quick at the start, slow through the middle, quick again near the end. The 35° train’s cut is close to a straight line. On a shaper, a straight cut means the tool meets the work at nearly one speed across the surface it is planing. That is the reason for wanting evenness at all: a surface finish that does not vary along the stroke, and a tool that is not worked hardest at one point.

The drag link leaves the ridge #

The trains that buy the evenness are not the frontier’s trains slightly adjusted. They are a different kind of drag link.

The earlier essay found every drag link on the time frontier on one long ridge. Each had an input crank and coupler of nearly the same length, several times the ground, over a short output crank of about 1.3. Such a drag link is close to a change point, where all four links can lie on one line. That is where its output turns slowest and where it splits a half-turn of output most unevenly. That shape buys ratio, and it is the shape the even trains keep from 40° down to 38°: input crank 4.02 and coupler 3.92 at 40°, 3.79 and 3.69 at 39°, 3.19 and 3.07 at 38°, each over an output crank of 1.34 or 1.35.

At 37° the drag link has changed. Its input crank and coupler are 2.72 and 2.64 and its output crank 1.45. By 35° they are 2.31, 2.24 and 1.62, and by 30° 2.24, 2.37 and 1.99. The output crank grows towards the length of the other two, and the drag link moves away from its change point. Eight kinds of four-bar gives the measure: the two middle links exceed the shortest and longest together by more and more. The drag link becomes a gentler double crank, one whose output speed varies less sharply.

That is where the knee in the evenness comes from. A drag link near its change point produces a brief, sharp slow-down of its output once a turn. That is excellent for splitting a turn unevenly, which is what the ratio needs, and bad for the cut, because the slow-down lands somewhere in the working stroke. Above 38° the searches find no other drag link that reaches a ratio of 3 within the angle, so the even trains keep the sharp one. Below 37° a compact drag link reaches 3 as well. Its unevenness is spread over a wider arc of the turn, and the phase can lay it across the slide’s own uneven stroke so that the two largely cancel. The first stage does not know what it drives is why that cancellation can be designed: the drag link’s motion is the same whatever is behind it, so its unevenness and the slide’s can be laid over each other by the phase alone.

Every rod at its limit #

The rod was the freedom the search was given that the time frontier had not used, and it was never taken. At every angle the most even train’s rod is the shortest the angle allows for its offset, to within the search’s resolution. The offsets vary, from 0.36 cranks at 35° to 1.06 at 30° and 0.66 at 40°, with rods of 1.66, 2.39 and 2.17 cranks to match.

A longer rod lowers the slide’s own quick return, which the train must then make up in the drag link. A ratio of 3 leaves no ratio to spare at any of these angles. The slide’s rod is pressed against its limit by the ratio, as on the time frontier. What changes with the angle is how the remaining freedom, the drag link’s shape and the phase, is spent.

The phases sit between 250° and 267°, close together, and they place the drag link’s slowest output across the middle of the slide’s working stroke. At 30° the drag link keeps 31.7° of transmission angle, more than it is asked for. There the slide’s angle is what binds, and the drag link has room the cut does not need.

A keying error costs evenness, not ratio #

The phase between the stages is set when the drag link’s output is keyed to the slide’s crank, and a key, a spline or a clamped hub sets it only to within a degree or two. The search put every train’s phase exactly where the evenness is least, so the natural question is how much a small error costs.

It costs evenness and almost no ratio. Moving the 35° train’s phase by two degrees either way leaves its ratio between 2.995 and 3.004 and its evenness between 1.47 and 1.50, against 1.42 at the search’s value. Five degrees take the evenness to about 1.55 one way and 1.63 the other, and ten degrees to 1.70 and 1.90, with the ratio still within a twentieth of 3. The ratio barely moves because it is decided by where the ram’s dead centres fall in the input’s turn, and a small phase change slides both dead centres almost equally. The evenness is decided by where the drag link’s slow stretch falls in the cut, and that moves with the phase degree for degree.

The evenness is least at a corner, not at a smooth bottom: a degree to one side costs 0.02, a degree to the other 0.04. That is what a fastest-over-slowest measure does. Like the clearance in a gap with corners in it, it is a maximum taken over the stroke, and its best value sits where two of its points, here the two speed peaks, are pulled level. So the tolerance is lopsided, and a designer who cannot key the stages exactly should bias the key towards the side that costs less.

The same holds at 30° and 40°, with the size of the effect set by how even the train was to begin with. The 30° train stays below 1.36 within four degrees. The 40° train, already uneven at 2.97, rises past 3.2 within four degrees on its costly side.

What a designer can take from it #

An even cut at a ratio of 3 costs about 35° in both stages. At 35° the most even train cuts at 1.42, as evenly as the best ratio-2 trains of the earlier essays. Asking for both a ratio of 3 and that evenness costs five degrees more than the ratio alone.

At 40° the ratio is all there is. The one train that reaches 3 there can be evened only from 3.25 to 2.97. A designer who must keep 40° in both stages and wants a ratio of 3 gets an uneven cut, and none of the search’s freedoms changes that.

Do not start from the train with the largest ratio. Below 40° that train spends its spare angle on ratio and gets steadily less even, 4.35 at 35° against 1.42 for a train holding the ratio at 3. The even trains have different drag links, compact rather than long and nearly folded, and searching near the ratio frontier finds them only by leaving it.

Key the stages to within a couple of degrees, biased towards the cheaper side. A two-degree error costs the 35° train about a twentieth of its evenness and none of its ratio.

Keep the slide’s rod short. None of the even trains lengthens its rod past the angle’s limit. The ratio needs every bit of the slide’s own quick return.

The Whitworth comparison is not forced. A ratio of three costs forty degrees said the Whitworth drive’s rocking slotted link, one of the inversions one chain, four mechanisms names, becomes the comparison if an even cut needed ten or fifteen degrees. It needs about five. The drag link and slide remain a competitive way to build a ratio-3 shaper with an even cut, at an angle that the transmission angle puts at the edge of comfortable practice rather than past it.

Where the second frontier stops #

Local searches, carefully seeded. Each angle’s answer is the best of pattern searches from the time frontier’s train and from the neighbouring angles’ even trains. The evenness rises with the angle at every step of the table, which is some evidence the searches are not stopping short. But a drag link family the seeds never reached could do better, particularly between 37° and 39°, where the knee is.

One angle in both stages. The earlier essay found that at a fixed ratio the drag link is the cheaper stage to let lean, at one and a half to three degrees of slide for each degree of drag link. The even frontier here holds the two angles equal. A designer who gives the drag link less and the slide more would very likely find a still more even cut at the same slide angle, and that trade was not measured.

Evenness is a kinematic measure. It reads the ram’s speed with the input turning steadily. A real shaper’s input slows under the cutting load and speeds up on the return, and the cutting force varies with the ram’s speed. Neither is modelled, any more than in the earlier essays.

Still open: whether the slide should lean instead #

The time frontier found that the drag link is the cheaper stage to let lean: at a ratio of 3, each degree the drag link gives up buys the slide between one and a half and three. The even frontier here kept both stages at the same angle. Yet the knee it found is about the drag link’s shape, which the drag link’s angle decides. It leaves the long ridge when that angle drops below about 38°.

The distinct argument would be the even frontier with the angles split. At a ratio of 3, with the slide held at 40° or 45°, it would find the most even cut for each angle allowed to the drag link alone. If the drag link can drop to 35° and buy the whole of the evenness while the slide keeps 45°, the guide loads that the slide’s angle sets stay low and the even cut is nearly free. If evenness needs both stages to lean, the five degrees here are the real price, and a designer cannot move them onto the stage that matters less.