The return stroke is quicker

Drive a crank-rocker at constant speed and watch the rocker. It swings out, stops, swings back, stops, and repeats — and the two halves of that cycle take different amounts of time.

That is not a defect. It is the property the whole machine is built around, in a shaper, a slotting machine, a drilling head, or anything else that does work in one direction and returns idle in the other.

The number that describes the asymmetry is the time ratio, and it is one of the few quantities in this subject that a designer is handed as a specification rather than deriving. It also has two routes to it — a closed form from the geometry and a measurement from a sweep — which makes it a good test of whether the geometry has been understood, and one of the two routes was wrong the first time it was written here.

Where the output stops #

The rocker is at an extreme of its travel exactly when the crank and coupler are in line.

The reason is short. The rocker pin B sits at a fixed distance from O₄, so its position is determined by the angle of the rocker; and it also sits at the end of the crank-plus-coupler chain from O₂. The distance from O₂ to B is what the crank and coupler between them can reach, and that distance is greatest when the two are stretched out in line (a + b) and least when the crank has folded back over the coupler (b − a).

At those two configurations B is as far from O₂ as it can get and as near as it can get, and since B is confined to a circle about O₄, being at an extreme distance from O₂ puts it at an extreme angle from O₄. There is nowhere further to go.

Nothing about the rocker’s length enters that argument, which is the useful part: the condition is entirely about the crank and coupler, and the rocker only determines where on its circle the extremes fall.

This is worth contrasting with the mechanism’s other special configurations. Grashof’s condition is a statement about all four lengths and decides whether the crank rotates at all; the limit positions are a statement about two of them and decide where the output stops. A linkage can be reproportioned to move its limits without changing its Grashof class, and that is most of what quick-return design consists of.

The angles, in closed form #

Both limit configurations give a triangle with all three sides known — O₂ to O₄ is the ground g, O₂ to B is (a + b) or (b − a), and O₄ to B is the rocker c. The cosine rule gives the crank angle directly, with no iteration:

$$\cos\psi = \frac{g^2 + d^2 - c^2}{2\,g\,d}$$

with d the reach, and the two crank angles are ψ for the extended limit and π + ψ for the folded one.

That “π plus” was wrong first, and the error is worth recording because it does not cancel. At the folded limit the crank has doubled back inside the coupler, so B lies at distance b − a from O₂ in the direction opposite to the crank — which puts the crank angle a half turn away from the direction of O₂B, on the same side as the extended limit. Writing π − ψ instead reflects the configuration into the other assembly branch, and the two limits then belong to configurations the mechanism cannot move between. The predicted time ratio came out 2.97 against a measured 1.09, and the measurement was right: the two limit cranks are 188° apart, not 91°.

Each assembly branch has its own pair — {+ψₑ, π + ψ_f} on one and {−ψₑ, π − ψ_f} on the other — and they are mirror images. So the time ratio is the same for both, which is worth knowing because it means the ratio is a property of the four lengths and not of how the linkage was put together.

The time ratio #

The crank turns at constant speed. Between the two limit positions it covers some angle one way and 360° minus that angle the other. The rocker covers the same swing both times. So the two strokes take times in the ratio of those two crank angles, and that ratio is the time ratio:

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

where δ is how far the two limit cranks are from being exactly opposite. For the figure’s linkage δ is 7.7°, and Q is 1.089 — the working stroke takes 9% longer than the return.

The measurement is worth describing because it is genuinely independent of the prediction. It drives the linkage through 7,200 crank angles, solving each, and records the rocker’s angle at every one. The extremes of that record are the limit positions, and the crank angles at which they occurred are read off. Nothing in that path knows about triangles or the cosine rule; it is a search for a maximum and a minimum.

The three rows exist because a symmetric linkage returns exactly 1 by both routes. A check run only on that case would pass with the prediction deleted, which is the same kind of empty test that a seeded solver agreeing with the formula that seeded it produces.

What the swing depends on #

Alongside the time ratio, the other number a designer specifies is the rocker swing — how far the output travels — and it is worth noting that the two are set by different things and pull against each other.

The swing follows from the same two triangles. At the extended limit the rocker’s angle from the ground line is the triangle O₂O₄B with O₂B = a + b; at the folded limit it is the same triangle with O₂B = b − a. The difference between those two angles is the swing, and it depends strongly on the crank length: a longer crank makes the two reaches more different, so the swing grows.

But a longer crank also brings the two limit cranks nearer to being opposite, so δ falls and the time ratio approaches 1. That is the trade in its simplest form. More swing, less quick return, from the same lengths, and a plain crank-rocker cannot have both.

There is a second constraint pulling the same way. Grashof’s condition requires the crank to be short enough for the linkage to be a crank-rocker at all, so the crank length is bounded above before either of the design numbers is considered. Push it and the mechanism ceases to have a rotating input, at which point the time ratio stops meaning anything because there is no constant-speed revolution to measure it against.

Getting a bigger ratio #

A time ratio of 1.089 is not much of a quick return, and the natural question is how to get more.

δ is the departure of the two limit cranks from being opposite, and it grows as the mechanism becomes less symmetric. Making the ground link long relative to the crank and coupler pushes it up; so does an offset. But a plain crank-rocker is limited — the geometry that gives a large δ also gives a small rocker swing and a poor transmission angle — and the classical machines do not use one.

They use an inversion. The Whitworth quick-return mechanism and the crank-and-slotted-lever are both slider-crank chains with a different link grounded, and the reason they exist is precisely this: grounding a different link changes the relationship between the input’s constant rotation and the output’s travel, and the inverted forms reach time ratios of two or three where the direct one struggles past 1.2.

That is the practical answer to why kinematic inversion is worth a chapter rather than a remark. It is not a curiosity about chains; it is how a designer gets a number the un-inverted mechanism cannot reach.

Measuring rather than deriving #

The sweep in the audit figure does something slightly unusual and it is worth saying why.

It could have found the limit positions by driving to the two predicted crank angles and confirming that the rocker’s derivative is zero there. That would be a cheaper check and it would be circular: the predicted angles came from the collinearity condition, and confirming that the derivative vanishes where the condition says it should is a statement about calculus rather than about the mechanism.

Instead it searches. Seven thousand two hundred crank angles, each solved, each recorded, and the extremes taken from the record. That finds the limit positions without knowing they exist, and it would find them in the wrong place if the closed form were wrong — which is exactly what happened when the closed form was wrong, and is how the sign error above was caught.

The cost is resolution. A sweep at 7,200 steps locates a maximum to about a twentieth of a degree, and the crank angles it reports are therefore good to about that. The predicted and measured ratios agree to a few parts in ten thousand, which is consistent with that resolution and is quoted alongside it in the figure rather than presented as if the agreement were exact.

Limit positions are not toggles #

There is a confusion here that this site has met from the other direction, and the two essays should be read together.

At a limit position the crank and coupler are in line. At a toggle the coupler and rocker are in line. Both are configurations where two links are collinear, both are places where something extreme happens, and they are completely different situations.

At a limit position the output has stopped. The mechanism passes through smoothly, driven from the crank, and nothing is wrong: the rocker reverses and the crank carries on. This is a normal part of the cycle and it happens twice per revolution.

At a toggle the input cannot be driven. The transmission angle is zero, the crank has no leverage on the rocker at all, and a mechanism driven from the rocker at that instant will not move — which is what makes a toggle useful for clamping and dangerous for anything else.

A crank-rocker has limit positions and no toggles, because Grashof’s condition guaranteeing the crank rotates fully is precisely the condition that the coupler and rocker never line up. A double-rocker has toggles and its “limit positions” are toggles. So the two phenomena tend not to occur in the same mechanism, and which one a given collinearity is depends on which two links are in line — a distinction that a single frame of a drawing does not make and that the site’s figures print explicitly for that reason.

The machines this was for #

The quick return is one of the clearest cases on this site of a mechanism existing because somebody had a specific problem, and the problem is worth stating because it explains the whole shape of the design.

A shaper cuts metal with a single-point tool on a reciprocating ram. The tool cuts on the forward stroke and does nothing on the return. Cutting speed is limited by the tool and the material — go faster and the edge burns — so the forward stroke cannot be hurried. The return stroke is limited by nothing except the machine’s own inertia.

So every second spent on the return is wasted, and the whole of the machine’s productivity is the ratio of useful time to total time. A time ratio of 2 means the return takes half as long as the cut, and the machine does 50% more work per hour than one with a ratio of 1. That is not a refinement; it is the difference between a machine worth buying and one that is not.

The same argument applies to slotting machines, to some drilling heads, and to any press whose working stroke is force-limited and whose return is not. What all of them have in common is an asymmetry in the job, and a mechanism whose asymmetry matches it.

What they also have in common is that the asymmetry is produced by geometry alone. There is no clutch, no gearing change, no control system — the input shaft turns at one constant speed and the output is fast one way and slow the other because of where the pivots are. That is the kind of thing linkages are unreasonably good at, and it is worth putting beside the mechanisms whose whole purpose is to produce an intermittent motion from the same constant input.

What the ratio leaves out #

The time ratio is a single number summarising a whole cycle, and like every single number summarising a varying quantity it hides something.

It says how long each stroke takes. It says nothing about the speed profile within a stroke, and the two strokes do not merely differ in duration — they differ in shape. The rocker’s angular velocity through the working stroke is not the return stroke’s reversed, and for a machine whose cutting force depends on speed that matters as much as the total time.

It also says nothing about where in the stroke the mechanism is strong. The transmission angle varies through the cycle independently of the time ratio, and a mechanism with a good time ratio may deliver its worst leverage in the middle of the working stroke, which is the wrong place for it.

There is a third omission and it is the one a designer feels first. The time ratio is computed from a mechanism turning at constant speed, and a real machine’s input shaft is driven through a belt or a gearbox from a motor with a speed-torque curve. The output’s varying resistance feeds back to the input, so the crank does not turn at constant speed under load, and the actual times differ from the kinematic prediction. That is dynamics and this site does not compute it; the point is only that the ratio is a geometric quantity and the machine is not.

So the time ratio is a specification number rather than an analysis. It is what a designer is given — “the return must be at least twice as fast” — and satisfying it is necessary and a long way from sufficient. The rest of the cycle is what the swept figures on this site are for.

Where the number comes from in a real machine #

One more thing the ratio does not contain, and it is the reason a shaper’s specification lists more than one number.

The time ratio is computed from the linkage alone. A shaper’s ram is driven through the linkage from a gearbox, and the gearbox’s output shaft turns at a speed the operator selects for the material being cut. So the ratio fixes the proportion of the cycle spent cutting and the gearbox fixes the rate, and the two are independent: a machine with a ratio of 2 running slowly and one with a ratio of 1.2 running fast can have the same cutting speed and very different productivity.

That separation is what makes the ratio a design parameter rather than an operating one. It is built into the geometry, it cannot be adjusted, and it is the same at every speed — which is exactly the property a geometric quantity has and a dynamic one does not.

It also explains why the quick-return linkage sits where it does in the machine. It is between the gearbox and the ram rather than between the motor and the gearbox, because its job is to shape the cycle and the gearbox’s job is to set its rate, and putting them the other way round would make the ratio depend on the speed selected.

The time ratio being a ratio of crank angles is the whole of the mechanism’s usefulness, and it is worth saying why that makes it robust in a way most linkage properties are not. The crank turns at a constant rate, so time is proportional to crank angle, and the ratio of the two strokes’ durations is the ratio of two angles read off the geometry. No speed enters, no mass, no load — the ratio is the same at any driving rate whatever, and it is exactly what the geometry says. That is unusual: most quantities a machine is sold on vary with how hard it is being run. Here the quick-return ratio is a pure number, fixed at design time, unaffected by the motor, and measurable on a drawing. Which is why the property survived into machines that have nothing else in common with a shaper — it is one of very few kinematic quantities that a change of duty cannot touch.