A centred slide has a best rod, and an offset beats it

A shaper cuts on the forward stroke of its ram and returns empty, so it wants two things of its drive. The return should be quick, a time ratio of 2 or so, and the cut should be even, the ram moving at nearly one speed through the middle of its stroke. A quick return that cuts evenly found that a drag link ahead of the ram’s own mechanism can supply both: its output shaft turns unevenly, and if its slow stretch is phased to fall where the second stage would otherwise run fastest, the two unevennesses partly cancel.

The first stage does not know what it drives then put that drag link ahead of three slider-cranks and found one exception to its rule. The centred slide, with no offset and no quick return of its own, reached a most even cut of 1.309 behind its best drag link, better than the symmetric crank-rocker’s 1.536. The reason was the short rod’s second harmonic: a rod of 2.72 cranks skews the ram’s stroke, and a drag link’s own lopsidedness can be phased against a skew. The essay ended on the rod’s length. A shorter rod skews more and leans more. Does the cut keep getting more even until the transmission angle stops it, or does it turn back first? And is a centred slide with the best rod better than an offset one, which buys its skew differently?

It turns back, at a rod of about two cranks, and the offset slide is better at every transmission angle a shaper would accept.

The search, repeated at every rod #

Everything here is the earlier essay’s search with one more dimension. The crank is 1 and the slide passes through its pivot. For a given rod, every drag link of the grid — 821 of them, ground 1, lengths in quarter steps — is put ahead of the slide at every angle between the two stages, in steps of 3°. Those designs whose combined time ratio reaches 2 are kept, and among them the one whose ram speed is most even is refined at 7,200 steps of the input. Evenness is the ratio of the fastest to the slowest ram speed across the central 80% of the working stroke, the stretch where a tool is actually cutting. The time ratio is the forward stroke’s duration over the return’s, the quantity the return stroke is quicker defined for a crank-rocker and the earlier essay carried to the slide in closed form. Every design keeps a transmission angle of at least 40° in both stages, the drag link’s and the slide’s.

The rod runs through fifteen lengths: 20, 10, 7, 5, 4, 3.2, 2.8, 2.5, 2.2, 2, 1.8, 1.6, 1.5, 1.4, and 1.305, where the rod’s worst lean, reached with the crank square to the slide, puts the transmission angle at exactly 40°. Plotted against crank ÷ rod, the size of the rod’s second harmonic, the long rods crowd at the left and the short ones spread to the right.

Two curves come out of it, and they go in opposite directions. Driven directly, without a drag link, the ram gets steadily less even as the rod shortens: 1.701 at twenty cranks, 2.276 at two, 2.752 at the 40° limit. That is the familiar cost of a short rod: the ram’s motion departs further from a pure sinusoid. Behind its best drag link the ram first gets more even as the rod shortens, from 1.435 at twenty cranks to 1.342 at 3.2 and 1.246 at 2. Then it turns: 1.250 at 1.8, 1.299 at 1.6, 1.401 at the limit. The rod the earlier essay used, 2.72, sits partway down the improving side at 1.309.

So the answer to the first question is no. There is a best rod, at about two cranks, with a worst transmission angle of 60°, and shortening the rod past it costs evenness before it costs transmission angle. The minimum is shallow — 1.246 at 2, 1.250 at 1.8, 1.266 at 2.2 — and with drag links in quarter steps its exact position is not resolved to better than a tenth of a crank. The turn is not shallow: the shortest rod is 12% worse than the best.

What the dragged ram does through its stroke #

The evenness is one number, and the stroke has a shape that says where it comes from.

Behind its drag link every one of these rams runs in two humps with a dip between. On the long rod the humps reach 1.28 and 1.29 of the stroke’s mean speed and the dip sags to 0.90, and the dip is what sets the evenness. The drag link can slow the crank in the middle of the stroke and speed it at both ends, but a nearly sinusoidal ram is fastest in the middle, so the best the drag link can do there is to overcorrect it into a saddle.

At the best rod of 2 the dip has come up to 1.01 of the mean while the humps stay at 1.26. The rod’s second harmonic makes the ram faster at one end of its stroke than at the other, and at this length it lifts the middle without raising either hump. At the shortest rod the dip is up to 1.10, but the late hump has grown to 1.39. The skew that was lifting the middle is now lifting one end as well, and the evenness is worse again.

The skew stops and the height does not #

The two direct-drive numbers behind this can be separated, and they behave differently.

The first is where the ram is fastest. On a very long rod the ram’s own speed peaks almost exactly halfway through the working stroke, at 0.516 of its duration on a rod of twenty cranks. As the rod shortens the peak moves forward, to 0.574 at four cranks and 0.624 at two. Then it stops: at the 40° limit it is at 0.647, having moved only 0.023 more over the whole range of rods shorter than two. The second is how tall the peak is against the stroke’s slowest part, and that grows the whole way, 1.70 to 2.28 to 2.75, with no sign of slowing at two cranks.

A drag link at a fixed combined ratio has a fixed amount of unevenness of its own to spend. Where its slow stretch falls can be chosen by the phase, and the best phase does follow the peak, moving from 261° on a long rod to 246° at two cranks. But the drag link cannot make its slow stretch deeper without changing the combined ratio. So a peak that moves is something the first stage can be phased against, and a peak that grows is something it has to absorb. Above two cranks the rod mostly moves the peak, and the drag link follows it into a better fit. Below two cranks the rod mostly makes the peak taller, and the fit gets worse.

The phase says the same thing from the other side. Below two cranks the best phase stops following the peak and drifts back, to 252° and 255° on the shortest rods. With the peak no longer moving, there is nothing further forward to phase against. The search has turned instead to spreading the drag link’s slow stretch across a taller peak that is also narrower, and it can only do that by accepting a worse fit on one side of it. The late hump in the stroke figure is that side.

Where the two factors part company #

The ram’s speed is a product, and the earlier essay showed that the product is exact: the crank’s angular speed, which the drag link sets, times the ram’s own travel per unit of crank angle, which the crank and rod set.

At both rods the phasing has done its job: the crank is slowest where the ram’s own lever is largest, and the two curves are close to mirror images across the mean. What differs is the size. At the best rod the lever’s arch reaches 1.47 of its mean and the drag link’s trough goes down to 0.58. At the shortest rod the arch reaches 1.62 and the trough only 0.53: the arch has grown by more than the trough has deepened. The shortest rod’s arch also comes down sooner at the end of the stroke, so the crank is already speeding up again while the lever is still high. That overlap is the late hump.

There is no drag link in the grid that keeps up, and the search has tried every one of them at every angle between the stages. A deeper trough is available only from a drag link whose combined ratio is higher than 2, which would change the problem.

The offset does it better #

The second question was whether a centred slide at its best rod beats an offset slide, which gets its quick return and its skew from the offset rather than from a short rod. A fair comparison needs both machines held to the same transmission angle, since a shaper designer limits the lean rather than the rod. So the search was run on slides at offsets of 0, 0.2, 0.4, 0.6 and 0.85 of a crank. Each had the rod that makes its worst transmission angle 45°, 55°, 65° or 75°, which for a slide is $(1 + e)/\cos\mu$ for an angle μ. That is twenty more searches of the whole grid.

Up to 65°, some offset slide beats the centred one at every angle, and by a margin the grid’s resolution cannot account for. At 45° the best offset slide gives 1.194 against the centred slide’s 1.382. At 55°, 1.190 against 1.266; at 65°, 1.181 against 1.289. The best design of the whole map is at an offset of 0.6, a rod of 3.79 and a worst transmission angle of 65°, where the slide on its own has a time ratio of 1.060 and the drag link brings it to 2 with an evenness of 1.181. That is 5% better than the centred slide’s best rod anywhere, and it keeps five more degrees of transmission angle.

The most offset slide shows the other half of the result. At an offset of 0.85 the evenness is 1.194, 1.192 and 1.195 at 45°, 55° and 65°, a spread of 0.2% across rods from 2.62 to 4.38. The centred slide spreads by 9.2% over the same angles. Once an offset is supplying the skew and a little quick return, the rod has almost nothing left to do for the cut, and a designer can choose it for the transmission angle alone.

At 75° the order reverses. The rods that give that angle are long, 4.6 to 7.2 cranks on the offset slides, and a long rod with an offset buys an own time ratio of only 1.01 to 1.02 and very little skew. The centred slide at 75° has the shorter rod, 3.86, and its second harmonic still helps: 1.363 against the offset slides’ 1.389 to 1.413. A designer who insists on a very upright rod gets a less even cut whichever way the slide is set. At that angle the offset has stopped paying for itself.

The map also puts the earlier essay’s three slides in context. Its offset slides were chosen to lean least for their own time ratios of 1.1 and 1.2, which happens to put them at offsets of 0.92 and 0.85 and worst angles of 59° and 47°. They reached 1.198 and 1.188, close to what the map’s most offset row gives at those angles, 1.192 to 1.194. The rod they were given was chosen for the lean, and the map says that was harmless: at that offset nothing better was available from the rod. Its centred slide was given the rod of the 1.2 machine, 2.72 cranks, which isolated the offset’s effect from the rod’s but left the centred slide 5% short of what it could do. Neither machine is an offset slide of the kinds whose crank cannot turn; every slide here has a turning crank and a ram that reaches both ends of its stroke once a turn.

What a designer can take from it #

If the slide is centred, the rod has a best length, and it is about two cranks. That is shorter than most shaper rods, and at a worst transmission angle of 60° it is not an extreme design. Shortening past it loses evenness before it loses transmission angle, so the 40° limit is not the binding constraint it looks like.

If the slide can be offset, offset it, and then choose the rod for the lean. Up to 65° the best offset slide beats the best centred one, and at an offset of 0.85 the rod makes no difference to the cut from 45° to 65°. The offset is doing two jobs at once, a little of the quick return and most of the skew. That is why the earlier essay’s offset slides at own ratios of 1.1 and 1.2 reached 1.198 and 1.188 without any tuning of their rods.

The first stage is chosen after the second, as before. Every one of these best designs uses a drag link of the same family: an input crank and coupler equal or nearly so, 2.5 to 5 on a ground of 1, and an output crank of 1.75 on every centred slide and 2 to 2.25 on the offset ones. What changes from design to design is mostly the phase, which a drag link ahead of a crank-rocker first found has two good values and which here moves by 15° as the rod moves the peak.

What the rod sweep does not cover #

One combined ratio. Every search here asks for a combined time ratio of 2. A higher ratio gives the drag link a deeper trough to spend, which is exactly what the shortest rods lacked. The best rod may move with the ratio, and the reversal at 75° may not survive a higher target.

The grid is the same grid. Drag links in quarter steps and phases in 3° steps resolve differences of about 1%. The best centred rod is somewhere between 1.8 and 2.2 cranks, and the offset slide’s flatness from 45° to 65° is inside that resolution. The differences between the centred and the best offset slides at equal transmission angle, 6% to 16%, are not.

Kinematics only. A shorter rod on a real shaper also carries larger side loads into the ram’s guides, since it leans further. The transmission angle stands in for that here and nothing measures the force.

Other ram drives. The Whitworth quick return, one of the slider-crank chain’s four inversions, drives its ram through a rocking slotted link and then a rod, and that final rod has a length too. Whether its best length behaves like the centred slide’s — a best setting, then a turn — or like the offset slide’s — hardly a knob at all — depends on whether the slotted link’s own motion supplies a skew, and it was not tried.

Still open: the combined ratio as a third knob #

The shortest rods lost because the drag link’s slow stretch could not deepen without raising the combined ratio above 2. A shaper that wants a quicker return, 2.5 or 3, gives the drag link more unevenness to spend. The distinct argument would be how the best rod and the offset’s advantage move with the ratio.

The measurement would repeat the centred sweep and the offset map at combined ratios of 2.5 and 3. The prediction the mechanism here makes is specific: a deeper trough can absorb a taller peak, so the centred slide’s best rod should shorten as the ratio rises, and the gap between it and the best offset slide should narrow. If both happen, the offset’s advantage belongs to low ratios and the rod is the knob at high ones. If the gap holds, the offset is simply the better way to skew a ram, whatever the drag link is asked to do.