Four kinds of slider-crank

Eight kinds of four-bar sorted every four-bar by the signs of three sums of its lengths, found eight regions of length space each moving in its own way, and ended on the case it had not done. The slider-crank is usually introduced as a different mechanism, a four-bar with one pin replaced by a slide. It is also the limit of a four-bar, and a limit can be taken on a classification as well as on a machine.

This essay takes it. The eight regions become four, and the four that remain are not a new classification bolted on for sliders. They are four of the eight, carried unchanged to the limit, and the other four can be watched disappearing into them.

The limit, written out #

Keep the eight-kinds names: g the ground, a the input crank, b the coupler, c the output, and

$$T_1 = g + c - a - b,\qquad T_2 = g + b - a - c,\qquad T_3 = b + c - a - g.$$

Now lengthen the ground and the output bar together, holding their difference fixed at e, so that c = g − e. The output’s pivot runs away along the ground line, and the output pin moves on a circle whose radius grows with it but which always passes through the point a distance e from the crank’s pivot. Near the crank that circle straightens. In the limit the output pin runs on a straight line square to the ground line, a distance e from the crank’s pivot: the slide of an offset slider-crank, with the coupler as its connecting rod and the offset e.

Substituting c = g − e into the three sums gives

$$T_1 = 2g - a - b - e,\qquad T_2 = b - a + e,\qquad T_3 = b - a - e.$$

The first grows without bound and is eventually positive. The other two do not contain g at all. So every four-bar on the line c = g − e, however it starts, has the same second and third signs all the way to the limit, and its first sign is negative until $g^* = (a + b + e)/2$ and positive after it. The slider-crank keeps two signs out of three, and its first is always +. Two signs cut four regions, and they are the four of the eight whose first sign is +.

What the four regions predict #

The eight-kinds table already says what each of those four regions does, and the prediction can be written in slider-crank terms before anything is measured.

The crank at angle θ puts its pin a horizontal distance |e − a cos θ| from the slide, and the rod can close that gap exactly when it is no longer than b. A full turn needs that for every θ, and the largest gap is a + |e|, so the crank turns fully exactly when b ≥ a + |e|, which is $T_2 \ge 0$ and $T_3 \ge 0$ together. That is the textbook condition, a rod longer than the crank plus the offset, and in these terms it is simply the region + + +: the crank-rocker.

When the rod is shorter than the crank less the offset, both remaining sums are negative, and the crank can reach neither the direction pointing at the slide nor the one pointing away. It swings in two separate arcs, one either side of the ground line: the + − − region, the double rocker. Between those two cases lies a band of width 2|e| in which one sum is positive and the other negative, and there the crank swings through one of the two directions along the ground line. With a positive offset it swings through the direction pointing at the slide, which is the inner position of the 0–π triple rocker; with a negative offset it swings through the outer one, the π–π triple rocker’s.

The slider has its own vocabulary in the limit. A four-bar’s output could turn, or swing through its inner position, pointing back along the ground line at the crank’s pivot, or through its outer position, pointing away. A slider cannot turn, and the outer position has gone to infinity with the output pivot. What is left of the inner position is the place where the slide crosses the ground line. The slider reaches it exactly when |e| ≥ |a − b|, which is the band of the two triple rockers, and otherwise it runs back and forth on one side of the line only.

Its reach has a closed form, and it is an old one. The slider can be anywhere its pin is between |a − b| and a + b from the crank’s pivot, so its travel runs from $\sqrt{(b-a)^2 - e^2}$ to $\sqrt{(a+b)^2 - e^2}$. The difference is the offset slider-crank’s stroke, which the essay on the slider-crank gave as a formula; here it appears as the ends of the slider’s range, and in the two triple-rocker regions the first root has no real value, the range runs through the ground line, and that formula stops describing a stroke at all.

Slices at an offset #

With the offset fixed, the walls are the lines b = a + e and b = a − e in the plane of crank and rod lengths, and the slider-crank assembles above the line a + b = |e|. So each region’s share of a slice is a convex polygon and can be cut exactly.

Each slice holds three of the four kinds, never four. At a positive offset the band between the walls is the 0–π rocker, and the π–π rocker cannot appear, because it would need b − a + e negative and b − a − e positive at once, which a positive e forbids. At a negative offset the roles swap. Between them the two slices show all four, which mirrors the eight-kinds slices, where holding the ground and output apart showed six of eight.

At no offset the two walls are one line, the band has no width, and only the crank-rocker and the double rocker are left. A centred slider-crank has two kinds, and the two triple-rocker kinds exist only because of the offset. The offset is not a small correction to a centred mechanism: it opens a band of rod lengths, 2|e| wide, in which the crank swings rather than turns and the slider runs right through the ground line.

On the slices drawn, from nought to six in both lengths at an offset of 1.2, the band takes 12.24 of the slice against 11.52 each for the crank-rocker and the double rocker, which are equal because swapping the crank’s length with the rod’s carries one into the other. Every unit of offset widens the band and narrows both Grashof regions.

For a designer that band is the price of the offset. The standard slider-crank drawn in these essays has a crank of 1 and a rod of 3.5, and it can be offset by anything up to 2.5 and still turn fully; at 2.5 it sits on the wall b = a + e, and past it the crank stops short of the direction pointing away from the slide. An offset is usually introduced to make the return stroke quicker than the working one, and the band is where that stops being available, because a crank that no longer turns no longer has a return stroke. And since multiplying the crank, the rod and the offset by the same factor changes no sign, the band is a statement about the ratios of the three, exactly as Grashof’s condition is a statement about shape and not about size.

Four thousand slider-cranks #

The predictions are measured the way the eight kinds were, and the way Grashof’s condition was first swept before it was believed. Draw a crank and a rod uniformly between 0.2 and 5 and an offset between −5 and 5, discard the draws that cannot assemble, and redraw any that land within two per cent of a wall or of the edge of assembly, where a sampled test cannot resolve the range. Then sort each by its two signs and measure it by asking, at 360 crank angles and along the slide, whether the chain closes.

The census holds 727 crank-rockers, 764 double rockers, 1,230 0–π rockers and 1,279 π–π rockers, and every one of the 4,000 moves as its region predicts: the crank’s kind, whether the slider crosses the ground line, and the number of circuits. The four regions with a negative first sum hold none at all. 168 draws were redrawn for sitting near a wall.

The textbook condition is checked on the same draw. A rod longer than the crank plus the offset decides a full turn on all 4,000. A rod longer than the crank alone, the condition for a centred slider-crank, misjudges 1,275 of them. Every one of those misjudgements has to be of the same kind, because the two conditions differ only where the rod is longer than the crank but not longer than the crank plus the offset: they are slider-cranks in the triple-rocker band with a rod that looks long enough, and a crank that cannot get past the direction pointing away from the slide.

A kind is also a count of circuits, and the census checked that too. Eight kinds of four-bar found two circuits in every Grashof region and one in every triple rocker, and tied the difference to the change points where the circuits meet; the serial field had already found that an assembly branch is a connected component of the configuration space. The slider shows the circuits without a configuration curve. In the crank-rocker and the double rocker its range is two separate stretches of the slide, one either side of the ground line, and nothing can carry it from one to the other without taking the mechanism apart. In the two triple rockers the range is a single stretch running through the ground line, and the two assemblies are one circuit. All 4,000 have the number of circuits their region predicts.

The proportions are a fact about the draw, not about slider-cranks. An offset drawn uniformly up to five is usually larger than the difference between a crank and a rod drawn up to five, so the draw lands in the triple-rocker band more than half the time. A designer choosing an offset of a few per cent of the rod would fill the regions the other way, and would not move a wall.

Two routes on each kind #

The census rests on the closure test, which compares a gap with a rod length and knows nothing about signs or limits. The second route is a general linkage solver: the crank driven round, the slide a constraint on a line, Newton’s method at every angle, and every failure retried from both sides of the crank pin before it is accepted.

On a slider-crank from each region the two routes refuse exactly the same crank angles: none of 1,440 decisions differ. The crank-rocker turns fully by both, the double rocker reaches 22% of a turn by both, and the two triple rockers 65% each by both, through opposite directions.

The agreement is about where the slide is and not merely about how long the parts are, and the way to show that is to put the slide on the wrong side of the crank’s pivot in the closure test and leave the solver alone. The crank-rocker still agrees on every angle: its crank turns fully whichever side the slide is on, because b ≥ a + |e| does not care about the sign of e. The double rocker disagrees on 84 of its 360 angles, because its two swinging arcs are placed unevenly about the ground line by the offset and mirroring moves them. The two triple rockers disagree on 254 each. For them the offset’s sign is the whole of the kind: a 0–π rocker’s crank swings through the direction pointing at the slide, and with the slide moved to the other side it swings through the opposite direction and becomes a π–π rocker. The sign of e that the census sorted by is the one thing the two regions do not share.

Where the other four went #

The limit is not only a place the classification arrives at. It is a path, and the path can be walked with a four-bar from each region.

Take the double crank used for the eight-kinds figures: ground 1, input 3.2, coupler 3.6, output 3.4, so e = −2.4. Its first sum is negative until $g^* = (3.2 + 3.6 - 2.4)/2 = 2.2$. Lengthen the ground and output together. At g = 2.125 it is still a double crank whose input turns fully. At g = 2.206 it has crossed the wall and is a π–π rocker, its input swinging through its outer position over 98% of a turn. By g = 10 that share is 67%, by 40 it is 63.5%, and at 400 it is 62.36%, exactly the slider-crank’s.

The rocker-crank goes the same way into the other band. With e = 3 its wall is at g* = 4.75; at 4.686 its input swings between the ground-line directions over 40% of a turn, at 4.832 it is a 0–π rocker swinging through its inner position over 46%, and at 400 it reaches 55.14% against the slider-crank’s 55.42%. The π–0 rocker, with e = 0.5, becomes a crank-rocker at g* = 3.55 and turns fully from then on. The 0–0 rocker, with e = 0.1, becomes a double rocker at g* = 3.65 and settles on 59.17%.

So the four vanishing kinds do not vanish into nothing. Each becomes the survivor that shares its second and third signs, at a ground length that can be written down before the walk is run, and every walk crosses exactly one wall on its way.

The survivors’ own representatives cross the same kind of wall from the other side. The crank-rocker used for the eight-kinds figures has e = 1, and a little below its g* of 2.75, at g = 2.708, the same chain with a shorter ground is a π–0 rocker; above it, it is a crank-rocker again and stays one. The 0–π rocker’s line passes through the rocker-crank, the π–π rocker’s through the double crank, and the double rocker’s through the 0–0 rocker. Every line c = g − e passes through exactly two of the eight kinds, one with $T_1$ negative and one with it positive, and the eight fold onto four, two to one.

Why these four #

Which four survive can be read off the eight-kinds table without doing the limit at all, and the reason is the slider.

The four regions with a negative first sum are the double crank, the rocker-crank, the π–0 rocker and the 0–0 rocker. In the first two the output turns fully. In the last two it swings through its outer position, pointing away from the input’s pivot along the ground line. Those are precisely the two things a slider cannot do: it has no full turn, and the position pointing away from the crank along the ground line has gone to infinity with the output’s pivot.

The four survivors are the regions whose output either swings without reaching the ground line, the crank-rocker and the double rocker, or swings through its inner position, the two triple rockers. Those are the two things a slider can do: run on one side of the ground line, or run through it.

That is also why the first sum is the one that goes. $T_1$ is zero where a four-bar can lie flat with its input pointing at the output’s pivot and its output pointing away. As the output’s pivot runs off to infinity, that flat arrangement needs an ever longer coupler, and past $g^*$ no coupler of the given length can make it. The change point it marked has left the mechanism, and with it every kind of motion that needed the output to pass through it.

A square slide, classified #

The slide is square to the ground line. That is what the limit c = g − e produces. A slider-crank whose slide is inclined is a different limit, taken along a different line in length space, and whether its kinds are the same four is not examined here.

The census does not sample the walls. Slider-cranks within two per cent of b = a ± e, or of the edge of assembly, were redrawn. What happens exactly on a wall — a flat arrangement, a change point, two circuits touching — is argued from the four-bar case and shown by the walks, not counted.

Classification only. The regions say what the crank and the slider can reach. They say nothing about the time ratio, the transmission angle or the stroke inside a region, all of which vary continuously there and are where a design is actually decided. The crank angles at which the slider stops, which set the time ratio of any quick return, are not computed here.

What comes next: the chain’s other inversions #

The slider-crank chain’s other inversions. Grounding the rod instead of the frame gives the oscillating-cylinder engine, and grounding the crank gives the Whitworth quick-return. In the four-bar each inversion permutes the three sums and flips some of their signs; in the limit only two sums are left, and whether each inversion’s kinds are the same four, re-labelled, or a different selection from the eight is a question with a definite answer.

An inclined slide. Let the output’s pivot run off in a direction that is not along the ground line and the slide comes out at an angle. The limit then holds a different combination of lengths fixed, and the question is whether the two surviving sums are still b − a ± e with e measured square to the slide, or whether the angle enters them.

The time ratio across the band. An offset slider-crank is a quick-return mechanism, and its ratio grows with the offset. The band of width 2|e| in which the crank stops turning is where that growth must end. How large a time ratio a crank-rocker slider can reach before its rod falls into the 0–π band, as a function of the offset, would be the slider-crank’s version of the limit that essay found for the four-bar.