A swing and a time ratio

The return stroke is quicker went from lengths to numbers. Given the four bars of a crank-rocker, it found the two limit positions where the output stops, the crank angles between them, and so the rocker’s swing and the time ratio between its two strokes.

A designer is handed those numbers the other way round. A shaper’s specification says the ram must travel so far and must come back at least so much faster than it cut, and those translate into a swing ψ for the rocker and a time ratio Q. The lengths are what has to be found. That reversal is the whole difference between analysis and synthesis, and for this problem it has an answer that is exact, is a construction rather than a search, and turns out not to be one answer.

Two numbers that do not fix four #

Count what is being asked for. A crank-rocker has four lengths. One of them can be set to 1 at no cost, because the swing and the time ratio are both ratios of angles, and multiplying every length by the same factor changes neither: they are properties of a linkage’s shape and not its size. So there are three shape numbers — the ground, the crank and the coupler, each measured in rocker lengths — and two demands.

Three unknowns and two equations leave one freedom. The specification does not pick out a linkage; it picks out a curve of them, and every point of that curve delivers the swing and the ratio exactly. Something else has to choose.

The classical way to see the curve is a construction, and it is worth doing carefully because every member it produces can then be checked against the closed forms that analysed the forward problem.

The construction #

Put the rocker’s pivot O₄ at the origin and draw its circle. The rocker stops at two limit positions, so draw two points on the circle, B₁ and B₂, separated by the swing. For a 60° swing on a rocker of length 1 they are exactly one unit apart.

At both limits the crank and coupler are in line. At one limit they are stretched out, so the crank pivot O₂, the crank pin and B all lie on one line and the distance from O₂ to B is the crank plus the coupler. At the other they are folded back, and the distance is the coupler less the crank, with the crank pointing away from B. Between those two positions the crank turns through 180° plus some angle δ one way and 180° minus δ the other, and the time ratio is the ratio of the two:

$$Q = \frac{180^\circ + \delta}{180^\circ - \delta}, \qquad \delta = 180^\circ\,\frac{Q - 1}{Q + 1}$$

Now look at the crank pivot. The line from O₂ through the stretched limit points at one of the B’s; the line through the folded limit points directly away from the crank at that moment, which puts it at the other B. The angle between the two lines, measured at O₂, is δ. For a time ratio of 1.2 that is 16.36°.

So the crank pivot must be a point from which the chord B₁B₂ is seen at an angle of 16.36°. The points that see a fixed chord at a fixed angle are an old object: arcs of a circle through the chord’s two ends, one on each side of it. That is the whole construction. Choose any point on those arcs as O₂; the distances to B₁ and B₂ are then the coupler less the crank and the coupler plus the crank, which gives both lengths by halving their difference and their sum; and the ground is the distance from O₂ to O₄.

The figure at the head of this essay draws it for a 60° swing and a time ratio of 1.2, with the two limit positions, the arcs, and one member of the family drawn at both of its limits.

Not every point, but every member exactly #

A point on the arcs gives four lengths, and those lengths need not make a crank-rocker. Some points give a linkage whose crank is not the shortest bar, some a double crank, and some pivots on the wrong stretch of the circle reproduce the swing with the roles of the two limits exchanged. So each candidate is kept only if Grashof’s condition says its crank turns fully and the closed forms for the limit positions return the demanded swing and ratio. Of the points sampled round the two circles for this specification, 828 survive, and they are the thick stretches of the arcs in the construction.

Each survivor was checked by closed forms, which is one route. The second route shares nothing with the construction: build the linkage, drive the crank through 7,200 positions with the solver, and read the swing and the time ratio off the rocker’s recorded angles.

The member drawn is the one the rest of this essay is about, with ground 1.2223, crank 0.4783 and coupler 1.1298 in rocker lengths. Its limit cranks are at 38.4° and 234.8°, so the crank spends 196.4° on the working stroke and 163.6° on the return, and the ratio of those is 1.2000. The sweep, which knows nothing about arcs or chords, finds 1.2005 — agreement to one step of a sweep at a twentieth of a degree.

What varies along the family #

Every member delivers the swing and the ratio. What differs between them is everything else, and the quantity that matters first is the transmission angle: the angle at which the coupler meets the rocker, whose sine is the share of the coupler’s force that turns the rocker rather than loading its bearing.

For a crank-rocker the worst transmission angle has a closed form, and it is worth seeing why. The transmission angle is decided by the distance from the crank pin to the rocker’s pivot, through the cosine rule in the triangle of coupler, rocker and that distance, and it grows steadily with the distance. The distance is least when the crank points straight at O₄ and greatest when it points straight away. So the two extremes of the transmission angle occur with the crank lying along the ground line, and the worse of them is the worst in the whole cycle.

Across the family that worst angle runs from nearly nothing to 40.32°. The members near the ends of each stretch are linkages whose crank has almost become as long as some other bar, where the chain nearly lies flat and the coupler and rocker come close to a straight line. The best member sits on the circle whose centre is below the chord. The other circle’s best is a little over 29°, so half of the construction can be discarded at once for this specification: nothing on it is a machine anyone would choose.

A family is also a freedom, and a designer can spend it on something other than the transmission angle. A frame with no room for a long ground link, or a crank that has to clear a housing, pushes the choice away from the best member along its arc, and the figure says what that costs: every step away gives up transmission angle, slowly near the best member and steeply towards the ends of the stretch. Nothing has to be re-derived to make that trade, because every point of the arc already delivers the swing and the ratio exactly; the arc is the design space and the transmission angle is the price attached to each point of it.

The best member is the linkage already drawn and swept. Its transmission angle through one turn is worth seeing directly rather than as a single number, because the single number is the whole reason it was chosen.

It runs from 40.3° to 105.8°. The low point is reached once a revolution, when the crank lies along the ground line pointing towards the rocker’s pivot, which is where the closed form said it would be; the sweep reads it off solved positions and gets the same figure to the precision printed.

Only just 40° #

That best value, 40.32°, is worth a pause, because 40° is the usual rule and a 60° swing at a time ratio of 1.2 lands a third of a degree above it. No member of the family does better, and every member either side does worse. The specification sits exactly on the edge of what the design rule allows.

That is the source of a remark made twice in these essays: that a plain crank-rocker struggles to get past a time ratio of 1.2, in the essay on limit positions and again in the essay on inversions, each time as the reason shapers use a different mechanism. For a 60° swing, the remark is exactly right. Whether it is right for any other swing is a question the construction can now answer, and before answering it there is a case without a quick return at all that needs its own treatment, because there the family behaves differently.

No quick return, and a bound nobody reaches #

At a time ratio of 1, δ is zero, and a chord seen at zero angle is seen from anywhere on the line through its two ends. The arcs have straightened into that line: the crank pivot sits on the continuation of the chord B₁B₂, beyond one end or the other, and the two choices are mirror images.

Moving the pivot out along the line improves the worst transmission angle at every step, and there is no best member: the improvement goes on for ever and slows down as it goes. The best of the members sampled, thirty rocker lengths out, has a worst angle of 59.987° for a 60° swing, and the value it is approaching is 60° — exactly 90° less half the swing.

The limit is visible in the geometry. As the pivot recedes, the crank tends to half the chord, while the ground and coupler both grow without bound and become nearly parallel. The coupler then arrives at the rocker pin almost exactly along the chord’s direction at both extremes, and a rocker pointing at the limits of a swing ψ makes an angle of 90° − ψ/2 with that chord. The best symmetric crank-rocker is a linkage with an infinitely long coupler, which is to say that no crank-rocker attains the bound, and the bound is still the number that matters: it says that without a quick return, the quality of the best possible linkage is set by the swing and nothing else.

That has a hard consequence. A swing of 100° gives a bound of exactly 40°, which no member reaches, so a crank-rocker with a swing of 100° or more cannot keep the 40° rule even with no quick return at all. That is not a limitation of this construction. Every crank-rocker with that swing and that ratio is a point on this line, and the line has been searched out to where the improvement is smaller than any design could use.

A quicker return always costs transmission #

Raise the ratio and δ grows, the arcs tighten round the chord, and the family shrinks.

At a time ratio of 1.5 the chord is seen at 36.00°, 667 of the sampled pivots survive, and the best worst angle any of them reaches is 29.76°, with ground 0.941, crank 0.444 and coupler 0.865. The best member has moved in towards the rocker and its crank has shortened; the machine has become more compact and considerably worse at transmitting force.

There is a reason to expect this and not merely to observe it. The quick return comes from the two limit cranks failing to be opposite, and that asymmetry is produced by bringing the crank pivot off the line of the chord, towards the rocker. A crank pivot close to the rocker makes a linkage whose coupler meets the rocker steeply at one end of the cycle. The same geometry that makes one stroke quicker makes the transmission worse, and the construction turns that trade from a remark into a curve.

How quick, for each swing #

Doing the same thing for several swings and a range of ratios produces the answer to the question the 40° remark raised: for each swing, the ratio at which even the best member of the family falls below the rule.

Every curve falls as the ratio rises — checked at twenty-one ratios for each swing, and not once does a quicker return improve the best available transmission. Each curve starts at 90° − ψ/2 and crosses 40° at a definite ratio. A 30° swing keeps the rule up to a time ratio of 1.322; a 45° swing up to 1.286; a 60° swing up to 1.207; a 75° swing up to 1.106; a 90° swing only up to 1.025. Beyond 100° there is no crossing, because the curve starts below the line.

So the remark that a plain crank-rocker struggles past 1.2 is precisely true of a 60° swing, generous to anything larger, and unfair to anything smaller. It is still true in the sense that matters for the machine it was said about. A shaper wants a time ratio near 2, and at a ratio of 2 the best transmission angle available to any of these swings is below 20°. A crank-rocker is not a shaper’s mechanism at any swing, and the reason is now a measured curve rather than an impression.

What the construction leaves out #

The rest of the motion. Swing, ratio and worst transmission angle are three numbers about a cycle. The speed profile of each stroke, where in the stroke the transmission is poor and whether the cutting stroke is uniform enough to cut well all vary along the family and were not asked about.

The rule itself. Forty degrees is a convention with a reason, not a threshold at which anything fails. A lightly loaded mechanism can run lower, and below about 30° friction can make a joint stick, which is a different sense of jamming whose onset depends on a coefficient no drawing contains. The ratio limits above move with the rule, and the curves are drawn so that a different threshold can be read off them.

The other assembly. Every member was analysed on one assembly branch. The mirror-image assembly has the same swing, ratio and transmission angles, so nothing changes, but a machine built on the other branch is still a different machine with its stroke on the other side of the ground line.

Everything physical. Pin sizes, the space the links sweep, the drive to the crank and the load on the ram are all outside this analysis, and some of them will rule out members the transmission angle prefers.

What comes next: two stages instead of one #

When a crank-rocker cannot deliver a ratio, the classical answer is not a better crank-rocker but a different chain, and there are two ways to get one. The first is an inversion — the Whitworth quick-return and the slotted lever, which ground a different link of the slider-crank chain. The second is to put a stage in front: a drag link, whose two cranks both turn fully but at a varying ratio of speeds, driving the crank of a crank-rocker, so that the crank-rocker’s crank no longer turns at constant speed.

The second is the one worth measuring. A drag link spends more time on one half of its output turn than the other, and that asymmetry can be phased against the crank-rocker’s own, so the two stages’ time ratios combine. The distinct question is how: whether a six-bar built that way can reach a ratio of 2 with both stages above 40°, what the combined ratio is as a function of the phase between the stages, and whether the best combination is ever one in which neither stage alone would have been acceptable.