A quick return that cuts evenly

A drag link ahead of a crank-rocker solved a shaper’s problem in the terms it was stated. A shaper’s ram should cut slowly and return quickly, and the classical specification is a swing and a time ratio — how many times faster the return is than the cut. A swing and a time ratio found that a crank-rocker with a 60° swing can keep a transmission angle of 40° only up to a ratio of 1.207, well short of the 2 a shaper wants. Putting a drag link in front — a four-bar whose two cranks both turn, but not at the same speed — lets the crank-rocker’s crank be driven unevenly, and with the right drag link at the right phase between the two stages the combined machine reaches 2 with both stages at 40°.

That essay chose its designs by the time ratio. It ended by naming the specification it had not used, and the one a machinist would ask about first: how evenly the working stroke is driven. A time ratio is a statement about how long each stroke takes. It says nothing about how the cut’s time is spent — whether the tool moves through the work at a steady speed or crawls, rushes and crawls again. A tool that cuts unevenly leaves a surface that varies with its speed, loads its edge hardest where it is fastest, and a quick-return mechanism that doubles its time ratio by making the cut lumpy has not obviously improved the shaper.

So the question is whether the best pair for the ratio is anywhere near the best pair for an even cut.

What evenness means for a rocker that has to stop #

A rocker cannot move at constant speed through its working stroke, because it must stop at each end to reverse. Every speed curve starts at nought and ends at nought whatever the mechanism, so a measure that looked at the whole stroke would report the reversal rather than the cut.

The measure used here reads the middle. Over the part of the working stroke in which the rocker covers the central 80% of its swing — from 6° to 54° of a 60° swing — it takes the rocker’s fastest angular speed and divides by its slowest. A value of one is a perfectly steady cut over that range. The ends, where the ram is approaching or leaving its reversal, are left out deliberately; a shaper’s tool enters and leaves the work in that region and the specification that matters is what happens in between.

The machine is the one the drag-link essay built: a drag link with ground 1 whose output crank carries the crank of a crank-rocker, turned ahead of it by a phase angle, the rocker angle found from two circles at each input angle on the same assembly the six-bar solver uses. The rocker’s speed is differenced on a grid of 1,200 to 7,200 input angles. Before anything else, the stroke read that way is checked against the machine it is supposed to be: at five phase angles on the drag link that gives the highest ratio, its time ratio matches the drag-link essay’s closed form to $6 \times 10^{-4}$ and its swing is 60° to the same relative precision, while a phase ten degrees off misses the ratio by 0.065.

The crank-rocker alone is not even #

The first thing the figure shows is that the baseline is not a steady cut either.

A crank-rocker with a 60° swing and a time ratio of 1.2 of its own, driven directly at constant speed, has a rocker speed that rises smoothly to a peak in the first half of its working stroke and falls away, roughly a half-wave. Its evenness is 1.91: the rocker is nearly twice as fast at its quickest point in the central 80% of its swing as at its slowest. The symmetric crank-rocker, with no quick return of its own, gives 1.58, which is close to what a pure sine wave gives over the same central 80% — the swing of a crank-rocker is nearly sinusoidal in its input.

So the drag link is not being asked to preserve an even cut. It is being asked not to make an uneven one much worse.

The design with the most ratio drives the cut in lumps #

The drag link that gives the highest time ratio on the grid — the same one on all three crank-rockers, with crank 2.5, coupler 2.25 and output crank 1.5 on a ground of 1 — reaches 2.71 on the quick crank-rocker at a phase of 282°. It does it by turning the crank-rocker’s crank slowly through the working stroke and fast through the return, and its own output speed varies by a factor of 5.93 over a turn.

That speed variation does not land evenly on the cut. The figure’s highest-ratio curve dips to two thirds of its mean in the first half of the stroke and rises to 1.8 times its mean in the second: an evenness of 2.77. The tool spends the first part of the cut crawling and the last part rushing.

On the other two crank-rockers the same drag link gives 2.94 and 3.58. Measured this way the most successful quick-return design on the grid is the least even cut in the table.

A different drag link at a different phase #

The search that found the highest ratio also records, for every design that reaches a ratio of 2, how evenly it cuts. On the quick crank-rocker 3,436 combinations of drag link and phase, in steps of 3°, reach 2 while keeping a transmission angle of 40° in both stages. The most even of them is a drag link with crank 5, coupler 4.5 and output crank 2.25 on the same ground of 1, at a phase of 300°. It reaches a ratio of 2.03, and its evenness is 1.21.

That is more even than the crank-rocker alone. The second stage has not traded evenness for quick return: it has bought both. Its own output speed varies by a factor of 2.67 over a turn — much gentler than the 5.93 of the highest-ratio link — and at that phase angle its slow stretch falls where the crank-rocker’s rocker would otherwise be fastest, so the drag link’s unevenness cancels a good part of the crank-rocker’s.

The blue curve in the first figure is flat through the middle of the stroke to within about ten per cent of its mean. Dragging the phase shows how narrow that is: ten degrees either way and the hump returns, on one side before the ratio falls below 2 and on the other after.

Two uneven factors that cancel #

Why a drag link should make the cut more even is worth taking apart, because the explanation is a product and the product can be drawn.

The rocker’s speed, per unit of input speed, is the chain rule applied to two stages. The crank-rocker has its own speed ratio — how many degrees its rocker turns per degree of its crank — which depends on where the crank is. The drag link decides how fast that crank is turning. The rocker speed is one times the other:

$$\frac{d\psi}{d\theta} = \frac{d\psi}{d\varphi}\bigg|_{\varphi = \Phi(\theta)} \cdot \frac{d\Phi}{d\theta}.$$

Over the central 80% of the swing the crank-rocker’s own ratio varies by a factor of 1.90 — it rises from the reversal, peaks, and falls towards the other — which is why the crank-rocker alone cuts at an evenness of 1.91. The drag link’s output speed over the same stretch varies by a factor of 1.88, but the other way round: it is fastest near the ends of the stroke and slowest in the middle. Their product varies by 1.21. Two factors each worth nearly two have multiplied to nearly one.

The product is checked, not assumed. The two factors are computed independently — one by differencing the crank-rocker’s rocker angle against its own crank angle, the other by differencing the drag link’s output against its input — and their product matches the rocker speed differenced directly on the whole machine to $4 \times 10^{-7}$.

That picture also explains why the highest-ratio drag link fails. Its output speed varies by 5.93 over a turn, far more than the crank-rocker’s own ratio could ever cancel, and at the phase that maximises the time ratio its slow stretch sits early in the working stroke rather than centred on it. The ratio wants the drag link slow for the whole working stroke and fast for the whole return; evenness wants it slow only in the middle of the working stroke. A drag link can be phased for one of those or for the other.

The two specifications peak at different phases #

For the drag link that gives the most even cut, both specifications can be drawn against the phase between the two stages.

The time ratio has two peaks, as the drag-link essay found for every drag link, and the higher reaches 2.042 at 310°. At that phase angle the evenness is 1.40. Among the phases where the ratio still reaches 2 — a window of 40° — the evenness is best at 302°, at 1.203, with the ratio 2.034.

So on one drag link the two specifications prefer phases eight degrees apart, and the choice between them is a trade of seven thousandths of time ratio against nearly a fifth of the cut’s speed variation. That is not a close decision. A designer who chose the phase by maximising the ratio would give up an even cut for a ratio no shaper would notice.

The evenness curve has breaks where the combined ratio passes through one, because there the slower stroke — which is what “working stroke” means here — changes from one direction to the other. Nothing physical happens at those phases; the measure is simply reading a different stroke.

How much evenness each extra step of quick return costs #

Holding the target fixed at 2 answers one question. Letting the target move answers the design question behind it: how even a cut can be had at each ratio a customer might specify.

The frontier is flat for a long way. The most even design reaching a ratio of 1.2 or 1.4 has an evenness of 1.14; from 1.6 to 2.0 it is 1.21; at 2.2 it is 1.25. Only past 2.2 does it climb steeply — 1.53 at 2.4 and 2.14 at 2.6 — and only between 2.4 and 2.6 does it cross the level of the crank-rocker driven directly.

Read as design advice: up to a ratio of about 2.4, a well-chosen drag link improves the cut and the return together, and beyond it every further step of quick return is paid for with an uneven cut. A shaper specified at 2 is inside the region where a second stage is free.

That is the reverse of the intuition the time ratio suggests. A quick return is usually described as an asymmetry — a swing and a time ratio leave a whole family of crank-rockers with the same one — and asymmetries in a linkage usually come with distortion. Here the drag link’s asymmetry is a second, adjustable distortion, and a second distortion can cancel a first.

Three crank-rockers, scored #

The same search on all three crank-rockers of the drag-link essay — time ratios of 1, 1.1 and 1.2 of their own, all with a 60° swing — gives the same pattern with different margins.

The highest-ratio drag link is the same one on all three, with an output speed ratio of 5.93, and on all three it gives the least even cut in the table: 3.58, 2.94 and 2.77.

The most even drag link is different on each, and always gentler — speed ratios of 3.94, 2.90 and 2.67. On the symmetric crank-rocker it only just beats the crank-rocker alone, 1.54 against 1.58, because a symmetric crank-rocker needs the whole of its time ratio of 2 from the drag link and there is little freedom left for evenness. On the two crank-rockers with a quick return of their own it gives 1.21, far better than either alone.

Every design keeps both stages at 40° or better. The most even pairs are not buying evenness with transmission angle, and since a transmission angle has no size, that holds for a shaper of any scale.

The practical reading is a division of labour. A crank-rocker with some quick return of its own lets the drag link spend its unevenness on the cut rather than on the ratio; a symmetric one makes the drag link do everything and leaves it no room. That is the same composition the drag-link essay described for the ratio — the two asymmetries add when they are phased together — applied to a second specification.

How much of the stroke is judged #

Reading the evenness over the central 80% of the swing is a choice, and a choice that could decide the answer. Near the reversals — the two places a crank-rocker’s output stops — every rocker slows to nought, so reading closer to the ends makes every design look worse; if it made them look worse at different rates, the ranking could flip.

It does not flip. Read over the central half of the swing, the crank-rocker alone, the highest-ratio pair and the most even pair give 1.24, 2.77 and 1.20. Over 95% they give 3.89, 2.77 and 2.19. The most even pair is the most even at every share.

Two details are worth reading off the figure. The highest-ratio pair’s evenness does not change at all between 50% and 95%, because its slowest and fastest points both sit well inside its stroke, so widening the window adds nothing new to either extreme. And the crank-rocker alone is more even than that pair over the middle of its swing and less even once the window reaches past about 90%, where its own speed is already falling towards the reversal — so a specification that cared about the whole stroke up to its last few degrees would rank the direct drive last.

What this does not settle #

The cut is measured as a rocker angle. A shaper’s ram is driven from the rocker by a further link, and a ram’s speed is the rocker speed times a lever arm that changes through the stroke. An evenness measured on the ram would move the numbers, and a rocker optimised for an even angle is not automatically a ram optimised for an even speed. The ram’s own linkage is the next stage of the same composition.

The frontier was read at 80%. The ranking of the three designs holds at every share, but the frontier’s knee near a ratio of 2.4 was located at one share only, and a wider window would move it towards lower ratios.

The grid is coarse. Drag links in quarter steps of length and phases in steps of 3° are enough to show that the two specifications pull apart and to find designs well inside the free region; they are not enough to locate the frontier’s knee to better than a tenth of a ratio.

Dynamics are absent. An uneven cut is a kinematic fact; the force on the tool at each point also depends on the cutting load, the ram’s inertia and the drive’s compliance.

Still open: the same stage in front of a Whitworth quick return #

The drag link’s success here comes from placing its slow stretch where the crank-rocker would otherwise be fast. The slider-crank chain’s inversions include the Whitworth quick-return, which gets its asymmetry from a single stage and drives a ram almost directly, and an offset slider-crank has a quick return of its own with no rocker at all.

Its distinct argument would be the same two specifications — a ratio and an evenness, each measured on the output that actually cuts — for a drag link ahead of those mechanisms. Two things would come out of it. Whether the cancellation found here is a property of putting any gently uneven first stage ahead of any output that is roughly sinusoidal, which would make it a rule about quick-return trains in general; and, set against the six-bar chains that buy a quick return in one loop, which way of reaching a ratio of 2 gives the most even cut for the fewest links.