A drag link ahead of a crank-rocker

A swing and a time ratio took the classical crank-rocker design problem — a swing and a time ratio for the return — and found a one-parameter family of linkages that deliver both exactly, chosen between by the transmission angle. It also found the limit. For a 60° swing, the best member keeps the usual 40° only up to a time ratio of 1.207. At a ratio of 2, which is what a shaper wants, no crank-rocker with any swing keeps even 20°.

Its closing section named the classical answer and the question the answer raises. Put a stage in front: a drag link, whose two cranks both turn fully but at a varying ratio of speeds, driving the crank-rocker’s crank, so that the crank-rocker’s crank no longer turns at constant speed. Whether a six-bar built that way can reach a ratio of 2 with both stages above 40°, and how the combined ratio depends on the phase between the stages, was left to measure.

It can, comfortably, and the phase turns out to matter more than either stage’s shape.

A crank that does not turn at constant speed #

A four-bar whose ground is its shortest bar has two cranks that both turn fully. Grashof’s condition calls it a double crank; a machine designer calls it a drag link. Its output makes exactly one turn for each turn of its input, and not at the same speed: the output lags through part of the turn and catches up through the rest.

How uneven it is can be measured by one number, the ratio of the output’s fastest speed to its slowest, both as multiples of the input’s.

A drag link with ground 1, crank 2.5, coupler 2.25 and output crank 1.5 turns its output between 0.32 and 1.91 times the input’s speed — slowest at an input angle of 111°, fastest at 339° — a ratio of 5.93, while its transmission angle never falls below 41.41°. A gentler one, with ground 1, crank 4.5, coupler 5 and output 2.75, runs between 0.62 and 1.48 times the input’s speed, a ratio of 2.40. On a grid of drag links with ground 1 and the other three bars from 1.25 to 5 in quarter steps, 821 turn both cranks and keep 40°, and their speed ratios run from 1.56 to 5.93.

The speed is found three ways that share no step: from a table of the output angle solved in closed form at 7,200 input angles, from a general linkage solver driven round with the output angle read at every position, and from the velocity ratio of the two cranks worked out from the link angles alone. All three give the same slowest and fastest speeds to four decimal places, at the same input angles, on every drag link compared. The speed ratio a number in this essay rests on is therefore a property of the linkage and not of the table that records it.

It is worth seeing why a drag link can be this uneven without transmitting badly, because the two sound as if they should go together. The output crank’s speed, as a multiple of the input’s, is the input crank’s length times the sine of the angle between the input crank and the coupler, divided by the output crank’s length times the sine of the transmission angle at the output. The sine on top is never more than one, so the output can never run faster than a/(c·sin μ) times the input, and a transmission angle held above 40° caps that. For the most uneven drag link the cap is 2.52, against a fastest speed of 1.91. The unevenness is not coming from a near-dead position, where the sine at the bottom would shrink. It comes from cranks of unequal length, 2.5 driving 1.5, whose pins travel at different speeds round circles of different sizes while the coupler between them stays well away from lying along either. That is why the search can ask for both at once. Every drag link here sits in the double-crank region of the eight kinds of four-bar, where both cranks turn, and inside that region the transmission angle and the speed ratio are two different things to choose.

The time ratio, composed #

A crank-rocker driven at constant speed reaches its two limit positions at two crank angles, and its time ratio is the larger of the two crank turns between them over the smaller. The essay on limit positions gave those angles in closed form. Driven through a drag link, the crank-rocker’s crank still has to reach the same two angles, but the input now reaches them at whatever input angles the drag link maps them to.

Fix the crank-rocker’s crank to the drag link’s output shaft, turned ahead of the output crank by a phase α. The crank-rocker’s crank then points at the drag link’s output angle plus α. If Φ is the drag link’s output angle as a function of its input angle, the input reaches the crank-rocker’s two limits at

$$\theta_e = \Phi^{-1}(\lambda_e - \alpha), \qquad \theta_f = \Phi^{-1}(\lambda_f - \alpha),$$

with λe and λf the limit crank angles in the crank-rocker’s own frame. The whole machine’s time ratio is the larger input interval between those two over the smaller. Every piece of it is a closed form except $\Phi^{-1}$, which is read off a monotone table of the drag link’s output angle at 7,200 input angles.

The formula also says what the drag link can do and what it cannot. It cannot change the swing, which is the crank-rocker’s alone, or either stage’s transmission angles. What it moves is where in the input’s turn the crank-rocker’s two strokes begin, and it can only move them by spending time: a stroke placed where the drag link’s output is slow takes more of the input’s turn, and one placed where it is fast takes less.

Two routes to one ratio #

The second route shares nothing with the composition. The six-bar is built in a general linkage solver: the drag link’s four pins, the crank-rocker’s crank pin attached rigidly to the drag link’s output link at the phase angle, its coupler and rocker, all started from their closed-form positions and driven round through 3,600 input angles, with the time ratio read off the rocker’s recorded extremes and both transmission angles off every solved position.

At eight phases round the circle the two routes agree to within 0.001: 2.1071 and 2.1061 at nought, 1.8029 and 1.8037 at 90°, 2.6504 and 2.6511 at 270°. The swing is 60.00° in every sweep, the least transmission angles are 41.41° and 40.32° in every sweep, exactly the two stages’ closed forms, and no input angle fails to solve.

The agreement depends on building the right machine, and two wrong machines are close enough to the right one to be built by mistake. A crank-rocker has a mirror-image assembly, with its rocker pin on the other side of the line from its crank pin to its rocker’s pivot. That assembly has the same swing and the same transmission angles, and at a phase of 45° its time ratio is the true machine’s at 315°. The phase itself has a direction, and measured the wrong way round it gives the same mirror: 2.5608 at 45° against the closed form’s 1.3791. Both routes here are pinned to the assembly the closed-form limit positions describe, and the phase is measured anticlockwise from the drag link’s output crank to the crank-rocker’s crank, so the eight rows of the table are one machine at eight settings.

The phase decides almost everything #

With the two stages fixed, the phase alone sweeps the whole machine’s ratio over a wide range.

With the crank-rocker whose own ratio is 1.2, the most uneven drag link gives a ratio as high as 2.7108 at a phase of 284.8° and as low as 1.003 at 213.5°. Nothing else has changed. A two-stage machine assembled with its cranks keyed at the wrong angle is not slightly worse than intended: at the worst phase the drag link cancels the crank-rocker’s own quick return almost exactly, and the machine works and returns at the same speed.

Near the best phase the ratio is forgiving. Keyed 5° either side of 284.8° it is 2.7046 or 2.7054, and 10° either side 2.6847 or 2.6902, so an error of a few degrees in keying costs less than a hundredth. Over 128.5° of the full circle of phases the machine still returns at least twice as fast as it works. The steep parts of the curve are elsewhere, round the phases where the drag link’s slow stretch passes from under one stroke to under the other.

The same drag link ahead of the middle crank-rocker, whose own ratio is 1.1, gives two peaks of 2.507 at 276.5° and 2.119 at 102°. So the gap between the peaks grows with the crank-rocker’s own quick return: nothing for the symmetric one, 0.388 for the one at 1.1, 0.747 for the one at 1.2. That gap is the crank-rocker’s asymmetry counted twice, once added to the drag link’s at the higher peak and once taken away at the lower.

The lowest point of each curve says the same thing from the other side. With the crank-rocker at 1 the worst phase gives 1.001, at 1.1 it gives 1.002, and at 1.2 it gives 1.003. Whatever quick return the crank-rocker brings, there is a phase at which the drag link’s unevenness cancels it almost exactly. The drag link’s speed variation is larger than any of these crank-rockers’ asymmetry, so it can always be set to undo it — which is the same fact that lets it be set to double it.

The curve has two peaks, and their heights explain what the phase is doing. For the symmetric crank-rocker, whose own ratio is 1, the peaks are equal, 2.3038 at 65° and at 245°, half a turn apart. That crank-rocker’s two strokes are the same length of crank turn, so it makes no difference which of them the drag link’s slow stretch is placed under; either way one stroke is lengthened and the other shortened by the same amount. For the crank-rocker at 1.2 the peaks are 2.7108 and 1.9644. Its working stroke is already the longer, and placing the drag link’s slow stretch under that stroke lengthens it further, while placing it under the return stroke first has to undo the crank-rocker’s own quick return before it can make one of its own.

A ratio of 2, with both stages keeping 40° #

Searching every drag link on the grid, each at its best phase, with each of the three crank-rockers gives the answer the time-ratio essay asked for.

The symmetric crank-rocker, least transmission angle 59.99°, reaches 2.3038. The one whose own ratio is 1.1 and whose least angle is 46.25° reaches 2.5074. The one at 1.2, least angle 40.32°, reaches 2.7108. All three best pairs use the same drag link, the most uneven on the grid. A ratio of 2 is not a corner of the search: 30 of the 821 drag links reach it with the symmetric crank-rocker, 72 with the one at 1.1 and 147 with the one at 1.2, and the gentlest of those needs a speed ratio of only 2.40.

A single crank-rocker with the same swing keeps at best 19.47° at a ratio of 2, and 10.39° at 2.71. The drag link has no time ratio of its own at all, since both its cranks turn. So the combination does something that neither stage could do alone: each keeps the transmission rule, and together they deliver a ratio at which any one crank-rocker has lost half of its transmission angle.

The ordering of the three is the other finding. A crank-rocker with more quick return of its own gives a quicker two-stage machine, 2.30, then 2.51, then 2.71. The two asymmetries add when they are phased together. That makes the best pair for a 60° swing the one that spends both margins at once: the crank-rocker at exactly the edge of what keeps 40° by itself, and the most uneven drag link that keeps 40° by itself.

How uneven the first stage has to be #

Spread across the grid, the relation between a drag link’s unevenness and the ratio it gives is strong but not exact: the correlation between the two is 0.966. Two drag links with the same speed ratio can give different ratios, because what matters is not only how slow the output gets but over how much of its turn it stays slow, and whether that stretch is long enough to sit under a whole working stroke. Every drag link on the grid helps, at its best phase: the least of them still takes the crank-rocker from 1.2 to 1.557. And the scatter has a floor: no drag link on the grid with a speed ratio below 2.40 reaches a ratio of 2 with this crank-rocker.

For a designer that turns the specification round. A shaper that needs a ratio of 2 from a 60° swing needs a first stage whose output speed varies by at least a factor of about two and a half, keyed at the right phase, and it can then keep both stages at 40° with room to spare in the drag link. The gentlest such pair has little room in the phase, though: 5° either side of its best phase it falls from 2.0003 to 1.997, just short.

A Watt six-bar, and a grid that covers every size #

Counted as a chain, the machine has six links — the frame, the drag link’s input crank and coupler, the rigid pair of cranks on the middle pivot, the crank-rocker’s coupler and its rocker — and seven pins. Grübler’s count gives 3 × 5 − 2 × 7 = 1, a single freedom, which is the input crank. Two of the six links carry three joints: the frame, with its three pivots, and the pair of cranks, with the middle pivot and the two crank pins. Those two are joined to each other directly at the middle pivot, and that is what the essay on the two six-bar chains uses to tell them apart: this is a Watt six-bar.

That essay’s Watt six-bar hung its second loop from a rocker, so the second loop could only ever be driven through a swing. Here the second loop hangs from a crank that turns fully, which is what lets the first loop’s unevenness pass into the second loop’s timing without taking anything from its swing. The swing is still exactly 60° in every sweep, because the crank-rocker’s crank still turns through every angle; only the input angles at which it does so have moved.

The search’s grid fixes the drag link’s ground at 1, and nothing is lost by that. The time ratio is a ratio of two crank turns, the swing is an angle, and both transmission angles are angles, so multiplying every length of either stage by the same factor changes none of them, as a transmission angle has no size found for a single four-bar. The drag link’s size relative to the crank-rocker’s does not enter either: the stages share only a shaft, and the angle that shaft turns through is the same whatever the cranks on it measure. So the 821 drag links stand for every drag link with those proportions, at any scale, ahead of any crank-rocker with those proportions.

A grid, two angles, and the swing and ratio only #

The grid is a grid. The drag links searched have ground 1 and the other three bars in quarter steps up to 5. The best pair is best on that grid, not over all drag links, and a finer or wider search could find a quicker return that still keeps 40°.

Two transmission angles, not every one. The rule is applied to the drag link at its output crank and to the crank-rocker at its rocker. A six-bar has other places force is transmitted, including the shaft that carries both cranks, and none of them is examined.

Only the swing and the ratio. A quick return bought with an uneven first stage is bought with accelerations: the working stroke is no longer driven at anything like constant speed, and whether it cuts evenly, and what the drag link’s speed changes cost in inertia, are not computed.

Tolerances other than the phase. The phase’s effect near the best setting is measured above. How the ratio responds to errors in the bar lengths, or to play in the shaft that carries both cranks, is not.

What comes next: how evenly the working stroke is driven #

How evenly the working stroke is driven. A shaper wants its cutting stroke at nearly constant speed and its return as fast as possible. The two-stage machine’s rocker speed through the working stroke can be read off the same composition, and the question is whether the best pair for the ratio is anywhere near the best pair for an even cut, or whether the two specifications pull the phase in different directions.

The same stage in front of a slider-crank. An offset slider-crank has its own quick return and its own band of offsets beyond which its crank stops turning. Put a drag link ahead of it and the combined ratio composes the same way, while the offset, the slider-crank’s only source of quick return, can stay small enough to keep the crank well away from that band.

One chain instead of two. The Whitworth quick-return is an inversion of the slider-crank chain that gets its asymmetry from a single stage. Measured by the same pair of rules, a transmission angle and a ratio, it can be set beside this six-bar and the six-bar chains to say which way of buying a quick return costs least.