A centred ram cannot be timed past its drag link

A shaper cuts on the forward stroke of its ram and comes back empty, so its drive has two jobs: a quick return, measured by the time ratio of the forward stroke to the return, and an even cut, measured by how nearly constant the ram’s speed is 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 do both, its uneven output phased against the ram’s own unevenness. A centred slide has a best rod then held that two-stage drive to a combined time ratio of 2 and swept the slider-crank’s rod. The centred slide’s most even cut improved as the rod shortened, to 1.246 at a rod of two cranks, and then got worse again. An offset slide beat it at every worst transmission angle up to 65°.

That essay explained the turn with the drag link’s budget. At a fixed combined ratio the drag link has a fixed amount of unevenness to spend, and the shortest rods made the ram’s own speed peak taller than the drag link’s slow stretch could absorb. So it predicted that a quicker return, which gives the drag link more unevenness, would let a shorter rod pay off, moving the best rod down and closing the gap to the offset slide. It proposed asking for 2.5 and 3.

The first half of the prediction comes true. The second runs the other way. And neither 2.5 nor 3 can be asked of a centred slide at all.

Half a turn apart at every rod #

A slider-crank’s ram stops at its two dead centres, where the crank and the rod lie along one line. For an offset slide those two crank angles are not opposite each other, and the difference is the slide’s own quick return. For a centred slide they are exactly opposite: the ram is furthest out with the crank pointing along the slide and furthest in with it pointing back. That is true of every rod, long or short, because the slide passes through the crank’s pivot and the line of crank and rod is the slide’s own line at both ends.

So behind a drag link the forward stroke and the return of a centred ram are the two halves of a turn of the drag link’s output, and the time each takes is the input angle the drag link spends turning its output through that half. The combined time ratio is the drag link’s split of a half-turn. The rod does not enter it, and neither does anything else about the slide. The rod shapes the ram’s speed within each stroke, which is what made it a knob for evenness, and it has no say in how long either stroke lasts.

The figure above shows the split on the drag link that makes it most uneven. Its ground is 1, its input crank 4.019, its coupler 3.920 and its output crank 1.339, and its worst transmission angle is 40.0°, the limit every drag link here is held to. At the best phase between the two stages half a turn of output takes 104.5° of input and the other half 255.5°, a time ratio of 2.443. Dragging the phase moves the two dead centres along the curve, and the split falls away on either side of that phase.

The ceiling #

The largest split over the whole grid of drag links the earlier essays searched — ground 1, the other three bars from 1.25 to 5 in quarter steps, double cranks that keep 40° — is 2.304, found with the closed form for the dead centres at every phase angle in steps of a degree. That number is the centred slide’s ceiling on the grid, and it is the same at every rod tried, to the sixth decimal place. Off the grid, a pattern search over the drag link’s three lengths holding the 40° limit, climbing from the best grid link, stops at 2.365. Twelve more climbs from random starts end along a long, nearly flat ridge — an input crank and a coupler of nearly the same length over a short output crank — whose best point is 2.443, and everywhere the limit binds, each search running until the worst transmission angle is exactly 40°. Thirteen climbs are not a proof that nothing higher exists, but 2.5 is still 0.057 above the best of them, which settles the question.

An offset slide breaks through the ceiling, because its own dead centres are already unequally spaced. The slide brings a quick return of its own and the drag link multiplies it, very nearly. A slide offset 0.6 cranks with the rod that gives 45° has an own time ratio of 1.219, and the largest combined ratio on the grid behind it is 2.750, against the product 2.304 × 1.219 = 2.809. At 75° the same offset needs a rod of 6.18 cranks, its own ratio falls to 1.021 and its ceiling to 2.346. Across the map the ceilings are 0.1% to 2.1% under the product of the two stages’ separate ratios, closest where the slide contributes least and furthest at 45°, where it contributes most. So the quick return a two-stage drive can reach is set by how much the slide supplies on its own, and the slide supplies it only with a short rod and a large offset, which is to say a low transmission angle.

That puts a hard edge on the earlier essay’s proposal. A ratio of 2.5 is reached only by offset slides at 45° and 55°. None of the four offsets reaches it at 65° or 75°, whose ceilings run from 2.33 to 2.43. A ratio of 3 is reached by nothing on the map: the highest ceiling, an offset of 0.85 at 45°, is 2.760. A shaper wanting a return three times as quick as its cut, with this kind of drive, has to accept a worse transmission angle than 45° in the slide or 40° in the drag link.

What holds the drag link back #

A drag link’s split of half a turn is a statement about how unevenly it turns. To give a ratio of 2.443 its output must cover half a turn in 104.5° of input and the other half in 255.5°, so on average it runs more than twice as fast through one half as through the other, and at its extremes the ceiling link’s output turns 7.49 times as fast as at its slowest. A drag link ahead of a crank-rocker found that unevenness comes from a short ground between two long cranks, and that it is limited by what such a linkage does to its coupler: the more lopsided the output, the more nearly the coupler and the output crank fold onto one line at the slow point, and the transmission angle is the measure of how nearly.

So the centred ram’s ceiling is the drag link’s own transmission angle in another form. Holding the drag link to 35° instead of 40° lets the same search reach 2.63; holding it to 30° reaches 2.87. Even at 30°, a lean at which the coupler pushes the output crank more sideways than round, a centred ram cannot be timed to a ratio of 3. The rod, which decides how the ram moves inside each stroke, has no lever on any of this: the slider-crank reaches its dead centres where crank and rod are in line, and for a slide through the crank’s pivot that happens at the same two crank angles whatever the rod’s length.

Below the ceiling the rod does what was predicted #

The question about the rod can still be asked where the centred slide can reach, and the answer is the one the earlier essay expected.

At a ratio of 1.6 the most even train runs between 1.118 and 1.215 across rods from ten cranks to the 40° limit of 1.305, and at 1.8 between 1.171 and 1.233. The best rod is about five cranks at both. The rod is barely a knob there: the differences among the long rods are around 1%, the same as the grid’s resolution in drag links and phases, and only the shortest rods are clearly worse. At 2 the rod has the best setting the earlier essay found, 1.246 at two cranks. At 2.2 every shortening helps, from 2.004 at ten cranks to 1.466 at the 40° limit, and the best rod is the shortest the transmission angle allows.

So the best rod does shorten as the ratio rises, from about five cranks to two to the limit, and for the reason the earlier essay gave. The drag links that reach 2.2 are the most lopsided on the grid, with the deepest slow stretches, and a deep enough slow stretch can absorb the tall speed peak a short rod gives the ram. Near the ceiling that is the only way left to even the cut, because the drag links that could be phased more gently against a long rod’s lower peak no longer reach the ratio.

What the prediction did not foresee is how fast the cut gets worse. The best centred train is 1.118 at 1.6, 1.171 at 1.8, 1.246 at 2 and 1.466 at 2.2. A tenth of ratio near the ceiling costs more than the whole climb from 1.6 to 2.

The offset’s lead grows instead of closing #

Compared at equal worst transmission angle, from 45° to 65°, the best centred train is 1.134 at a ratio of 1.6 and the best offset train 1.122. At 1.8 they are 1.173 and 1.161. At those ratios the offset buys about 1%, inside the grid’s resolution, and the two kinds of slide are equally good. At 2 they are 1.266 and 1.181, the earlier essay’s 7%. At 2.2 they are 1.552 and 1.208, and the centred slide is 28.5% less even. At 2.5 the offset slide still reaches, at 1.386 with an offset of 0.6 and 45°, and the centred one does not.

The gap grows from nothing to everything. Taken angle by angle at 2.2 it is 1.552 against 1.208 at 45°, 1.669 against 1.288 at 55° and 1.773 against 1.430 at 65°. Only at 75°, where every slide’s rod is long and an offset buys almost no quick return, does the centred slide stay level, 1.914 against 1.943, as it did at 2. The offset’s advantage belongs to the high ratios, not the low ones, the reverse of what the earlier essay’s mechanism suggested.

Choice is what the offset buys #

The reason is in how many trains are left to choose from.

At a ratio of 1.6, 10,412 trains reach it behind the centred slide, and 16,091 and 26,257 behind the two offset slides. At 2 the counts are 1,006, 1,986 and 3,961. At 2.2 they are 256, 608 and 1,465. At 2.5 the centred count is nought, and the offset slides keep 39 and 213. The centred slide’s count falls fastest because every part of its ratio has to come from the drag link, and only the most lopsided drag links supply it. An offset slide brings some of its own, so drag links of more moderate lopsidedness reach the same total, and the search has many more phases and shapes to choose an even cut from.

A rough rule follows, and it holds to a few per cent where it can be tested. An offset slide asked for a ratio $Q$ cuts about as evenly as a centred slide asked for $Q$ divided by the offset slide’s own ratio. The slide offset 0.6 at 45°, own ratio 1.219, reaches 2.5 at 1.386; the centred slide at 45°, asked for 2.5 ÷ 1.219 = 2.05, reaches 2 at 1.382. The same offset slide reaches 2.2 at 1.208, and the centred slide at 45° reaches 1.8, close to 2.2 ÷ 1.219, at 1.173. The drag link is doing the same job in each pair, and the offset has taken the rest of the ratio off its hands. That is the earlier finding that the first stage does not know what it drives seen from the other side: the drag link’s contribution depends on how much ratio it is asked for, not on which slide asks.

What the two rams look like at 2.2 #

Both rams run in two humps with a dip between them, the shape the earlier essay found at a ratio of 2. Behind its best train, the centred ram at 2.2 rises to 1.40 of its mean in each hump and dips to 0.95 between them. The offset ram’s humps reach 1.23 and its dip stays above the mean, at 1.02. The centred slide’s drag link is one of the 256 that reach the ratio at all, and its slow stretch, deep enough to reach 2.2, is narrower than the centred ram’s stroke. The crank is fast again at both ends of the stroke while the ram’s own lever is still high, and the humps are the overlap. The offset slide’s drag link has room to be gentler, and its humps are lower.

What a designer can take from it #

A centred ram’s quick return is the drag link’s alone. Nothing about the slide’s rod changes it, and the drag link’s split of half a turn has a ceiling set by its own transmission angle: 2.44 or so at 40°. A centred slide is the wrong second stage for any ratio near that.

Below about 1.8 the choice of slide hardly matters. Centred and offset slides cut equally evenly there, within the grid’s resolution, and a designer can choose the slide for the transmission angle, the stroke or the guide load.

Above 2 the offset is the knob. It is the only slider-crank that reaches 2.5, and every offset slide here is of the kind whose crank turns fully, so nothing is given up in the motion to get there. It cuts 28% more evenly at 2.2, and it does so by taking part of the quick return off the drag link. The rod on a centred slide is a knob for evenness at moderate ratios and a lever of last resort near the ceiling, where the best rod is the shortest the transmission angle allows.

What the ratio sweep does not cover #

The same grid. Drag links in quarter steps and phase angles in 3° steps resolve differences of about 1%, so the 1% advantage of the offset at 1.6 and 1.8 is not a finding, and the best rod at those ratios is not resolved better than “long”. The 7% and 28% at 2 and 2.2 are well outside it.

Two transmission-angle limits held fixed. Every drag link keeps 40° and every slide 45° or more. A ratio of 3 is out of reach under those limits, not under all limits. How far they must be relaxed to reach it is a different measurement.

The ceiling’s continuous value is a search, not a proof. 2.443 is the best of thirteen climbs, one from the best grid link, which stops short at 2.365, and twelve from random starts; nothing here proves no drag link keeping 40° splits a half-turn more unevenly.

Kinematics only. As in the earlier essays, the transmission angle stands for the guide loads and the rod’s lean, and no force is computed.

Other second stages. Every second stage here is a slider-crank. The Whitworth drive, one of the slider-crank chain’s four inversions, gets its quick return from a rocking slotted link rather than an offset, and it reaches high ratios on its own. Behind a drag link it would be the natural second stage for a ratio of 3, and whether it also cuts evenly there was not tried.

Still open: what a ratio of 3 costs in transmission angle #

A ratio of 3 is out of reach with 40° in the drag link and 45° in the slide, and the ceiling map shows which way the limits would have to give: the slide’s own ratio rises as its rod shortens and its offset grows, and the drag link’s split rises as its own transmission angle falls. The transmission angle is the price of both.

The distinct argument would be that frontier drawn. For each target ratio up to 3, find the largest worst transmission angle that some two-stage train can keep in both stages at once, and ask how the burden is best shared: whether it is cheaper to let the drag link lean or the slide, and whether the most even cut at a ratio of 3 lies on that frontier or well inside it. The return stroke is quicker found a single crank-rocker’s quick return bounded by its own transmission angle; the measurement would say whether two stages in series relax that bound by a product, as their ceilings nearly do here, or by less.