A ratio of three costs forty degrees

A shaper cuts on the forward stroke of its ram and wastes the return, so its drive is built to make the return quick. The return stroke is quicker measured how quick a single crank-rocker can make it, and found the answer bounded by the transmission angle, the angle at which the coupler pushes the output: the more lopsided the motion, the more nearly the coupler pushes sideways instead of round. The drives the earlier essays then studied put two stages in series. A drag link ahead of a crank-rocker found that a double crank with a short ground turns its output unevenly, and a slider-crank behind it turns that unevenness into a quick return of the ram.

A centred ram cannot be timed past its drag link found the limits of that arrangement. A centred slide adds nothing to the quick return, so its ratio is the drag link’s alone; an offset slide adds a quick return of its own; and with the drag link held to a worst transmission angle of 40° and the slide to 45°, no train reached a ratio of 3. That essay drew which way the limits would have to give and left the price unmeasured: for each target ratio, the largest worst transmission angle a train can keep in both stages, how the burden is best shared between the stages, and whether two stages in series relax one stage’s bound by a product.

Very nearly by a product. A ratio of 3 costs forty degrees in both stages, and the drag link is the stage to let lean.

Three ceilings #

Each stage alone has a ceiling for each worst transmission angle μ it is made to keep.

The offset slider-crank alone, driven by a crank turning steadily, has a closed form. Its ram stops where crank and rod are in line, and with an offset $e$ the two dead centres are not opposite, the forward stroke taking $\pi + \delta$ of the crank and the return $\pi - \delta$. Its worst transmission angle is $90^\circ - \arcsin((1 + e)/b)$ for a crank of 1 and a rod of $b$, so keeping μ means a rod of at least $(1 + e)/\cos\mu$, and since a longer rod only makes the dead centres more nearly opposite, the best rod is that one. What is left is a search over the offset alone. The ceiling falls from 1.448 at 30° to 1.286 at 40° and 1.092 at 60°: an offset slide on its own makes a gentle quick return and nothing more.

The drag link alone, ahead of a centred slide, splits each half-turn of its output into a slow half and a quick one, and its ceiling is the most uneven split any double crank keeping μ can make. That has no closed form, and finding it is a search over the drag link’s three lengths with the ground held at one. It falls from 2.867 at 30° to 2.443 at 40° and 1.809 at 60°.

Both in series have five lengths free and a phase between the stages, every length searched with each stage keeping its own angle, starting from the two stages’ own optima. At 30° the train reaches 3.975, at 40° 3.006, at 50° 2.432, at 60° 1.957. A ratio of 3 is reached at 40.0° and no higher angle. Neither stage alone comes near 3 at any angle in the range: the drag link would have to lean below 30°, and the offset slide much further.

The ceiling at 40° was higher than reported #

The drag link’s own ceiling at 40° is the number the earlier essay rested its conclusion on, and it reported 2.365 with a caveat: a pattern search from one start finds a local best. The caveat was warranted.

Climbing from the best link of the quarter-step grid, with steps along every axis and every pair of axes so that the search can slide along the transmission-angle limit where it binds, the search stops at 2.364. Twelve more searches from seeded random starts over a wider box all end higher, along one long, nearly flat ridge: input crank and coupler of nearly the same length, from 2.8 to 5.5 times the ground, over an output crank of about 1.3. Along that ridge the split varies by under two per cent, and its best point is 2.443, at input crank 4.02, coupler 3.92 and output crank 1.34. The climb from the grid’s start stops at an input crank of 2.4, short of where the ridge begins. The earlier essay’s argument survives, since 2.443 is still well short of the 2.5 it showed a centred ram could not reach, and the ceiling it rests on is now the higher summit, found from thirteen starts.

At 35° and 30° the multi-start search finds 2.632 and 2.867, close to what the single search found there, 2.62 and 2.86; at 40° the difference is 0.08.

Two stages very nearly multiply #

The two-stage train beats each stage alone by a wide margin, and the question the earlier essay put was by how much: whether stages in series relax a single stage’s bound by a product, as their ceilings on the grid had seemed to.

Nearly. At every angle from 30° to 60° the train falls short of the product, never exceeds it, and the shortfall is a few per cent: 4.3% at 30°, 4.3% at 40°, 3.6% at 45°, 1.0% at 60°. It is largest where both stages are most uneven and smallest where each stage’s own ratio is near one, which is what a correction to a product ought to do.

One reason the product is not reached is visible in the stages’ own geometry. A product would need the drag link’s slow stretch to fall exactly across the slide’s slow stroke. The drag link’s most uneven split is a split of exactly half a turn of its output, while an offset slide’s slow stroke is more than half a turn of its crank, $\pi + \delta$, so the two unevennesses cannot be laid exactly on top of each other, and the best phase between the stages is a compromise between them. The shortfall shrinks with δ, as it should if that is the reason, but that is offered as a reading of the numbers rather than measured separately.

For a designer the reading is simple. Two stages each keeping μ reach about 0.96 to 0.99 of the product of what each reaches alone, so the frontier of a two-stage train can be sketched from the two single-stage frontiers, both of which are far easier to compute.

A ratio of 3 at forty degrees #

The train that reaches 3.006 at 40° has a drag link of input crank 4.02, coupler 3.92 and output crank 1.34 on a ground of 1, the drag link from the ridge above, and a slide offset 0.782 cranks with a rod of 2.326, the shortest rod 40° allows at that offset. Down the table the drag links lengthen as the angle rises, from input cranks of 2.3 at 30° to 7.3 at 60°, and the slides’ offsets grow while their rods lengthen faster, so that each slide’s own quick return falls even as its offset grows. In every train the transmission angle limit binds in both stages at once, each stage’s worst angle the limit to within a few thousandths of a degree.

So the answer to the earlier essay’s first question is a single number. A shaper drive of this kind with a ratio of 3 has a worst transmission angle of 40.0° in both its stages, which is the earlier limit on the drag link and five degrees below the earlier limit on the slide. The earlier finding, that 3 was out of reach with 40° and 45°, was out of reach by exactly the slide’s five degrees.

The drag link on the frontier is nearly a change point #

The drag links on the frontier all have the same shape, and the shape says why their limit is the transmission angle. Each is a double crank with the ground the shortest link, an output crank a little longer, and an input crank and coupler much longer and nearly equal to each other. A four-bar’s mobility is decided by Grashof’s sum: the shortest and longest links together against the other two. When the first sum is less, the shortest link turns fully; when the two are equal the linkage has a change point, a configuration where all four links lie on one line and the motion can go either of two ways.

The frontier’s drag links sit just on the turning side of that equality, and closer to it the lower the angle allowed. The other two links exceed the shortest and longest together by 0.106 of the ground at 30°, 0.158 at 35°, 0.239 at 40°, 0.470 at 50° and 0.838 at 60°. As a drag link approaches its change point its output crank and coupler come nearer to folding into one line once a turn, which is where the output turns slowest and where the coupler pushes the output most nearly along it; the transmission angle is the measure of how near, and the search pushes every drag link exactly as near as the angle allows. The first stage does not know what it drives found that the drag link’s motion is the same whatever is behind it, which is why its ceiling can be computed alone and why the product of the two stages is the right first guess for the train.

How to share the burden #

Both stages need not keep the same angle. A designer can give one stage more and the other less, and at a fixed ratio the two angles trade.

At a ratio of 3 the slide can keep 59.2° if the drag link is allowed 32°, 53.0° with 34°, 48.1° with 36°, 44.0° with 38°, and 38.5° with 41°. The curve is everywhere steeper than one degree for one: each degree given up by the drag link buys the slide about three at the left of the range and about one and a half at the right. So the cheaper stage to let lean is the drag link. A designer who must keep the slide’s guide loads low, since the slide’s transmission angle is what sets the side thrust on the ram, can buy that by letting the drag link lean, at three degrees of slide for one of drag link where the drag link already leans most.

The asymmetry follows from the ceilings themselves. The drag link’s ceiling changes by about 1.6% for each degree, the slide’s by about 1.2%; and the slide’s own ratio is close to one, so a small change in it is a large change in its angle. Holding the product fixed, the angle that can move most is the slide’s.

What the frontier cuts like #

A frontier is drawn on time alone, and a shaper is judged as much on how evenly its ram cuts as on how quickly it returns. A quick return that cuts evenly measured that evenness as the ram’s fastest speed over its slowest across the middle four fifths of its working stroke, and the earlier essay’s best centred trains cut with values from 1.12 to 1.47.

The ratio-3 train cuts for 270° of the input turn and returns in 90°, its return’s peak speed 3.1 times the cut’s, as a ratio of 3 requires. But its cut is uneven: the ram moves quickly at the start of the stroke, slows through the middle and speeds up again before the turn, and its fastest speed over its slowest across the middle four fifths of the cut is 3.25. Along the frontier the evenness gets worse as the angle falls, 1.27 at 60°, 1.94 at 50°, 3.25 at 40°, 6.12 at 30°. Every train on the frontier spends both stages’ unevenness on the timing of the strokes, and the cut inherits it.

That makes the frontier a limit and not a design. A designer asking for a ratio of 3 at 40° gets it from exactly one train, which cuts unevenly; asking for both a ratio of 3 and an even cut means giving up transmission angle beyond the frontier’s, by an amount the frontier does not say.

What a designer can take from it #

Sketch the two-stage frontier from the two single stages. The train reaches 96% to 99% of the product of the drag link’s and the slide’s separate ceilings at the same angle. For a target ratio, find the angle at which the product reaches it and take off about two degrees: the product reaches 3 near 42°, the train at 40°.

A ratio of 3 costs 40° in both stages. Below that angle in either, it is not available from a drag link and an offset slide. One chain, four mechanisms names the alternative the earlier essay raised, the Whitworth drive’s rocking slotted link, which reaches high ratios on its own and was not tried here.

Let the drag link lean, not the slide. At a fixed ratio each degree of the drag link buys the slide more than one, up to three. The slide’s angle sets the thrust on the ram’s guides, so this is the cheaper place to spend.

Do not expect an even cut on the frontier. The trains that reach the most ratio for an angle cut the most unevenly, 3.25 at a ratio of 3, where the trains of a centred slide has a best rod and the essay after it cut at 1.12 to 1.47 at ratios up to 2.2.

What the frontier does not settle #

Searches, not proofs. Every ceiling here is the best of pattern searches, the drag link’s from thirteen starts and the train’s from the two stages’ optima. The correction at 40° is a reminder that a search reports where it stopped. The train’s ceiling falling below the product at every angle, and the smoothness of the burden curve, are evidence that these searches are not stopping short, not a guarantee.

The slide’s rod at its limit. Every slide here has the shortest rod its angle allows, on the reasoning that a longer rod only lowers its quick return. That is true of the slide alone; behind a drag link a longer rod also changes the phase at which the stages combine best, and it was not searched.

Time and evenness, not force. As in the earlier essays the transmission angle stands for the loads, and no force on a guide or a pin is computed. Nor is the ram’s speed on the return, which at a ratio of 3 peaks at more than three times its cutting speed, and is what a shaper’s inertia has to absorb.

Still open: an even cut at a ratio of three #

The frontier’s train at a ratio of 3 cuts with a fastest-to-slowest speed of 3.25, more than twice the unevenness of the earlier essays’ best trains at a ratio of 2. A shaper that wants both a quick return and an even cut has to give up something, and the frontier only prices giving up time.

The distinct argument would be the second frontier: at a ratio of 3, the most even cut available at each worst transmission angle below 40°, found by a search that minimises the ram’s speed ratio across its cut while holding the combined ratio and both angles. If evenness can be bought back with a few degrees, the cost of a usable ratio-3 shaper is close to the 40° here. If it needs ten or fifteen, the frontier understates what a ratio of 3 costs, and the slotted link of the Whitworth drive, whose quick return comes with its own shape of cut, becomes the drive to compare against at the same angle.