Machine Kinematics
solved, then drawn
Links with a width

The crank that cannot turn all the way

Grashof's inequality says which four-bars turn fully. Ask instead how wide their links may be with a bearing at each ground pivot, and the same inequality answers the opposite question: every four-bar that turns all the way round sweeps a link straight over one of its own pivots.

Assumes A gap with corners in it.

15 min read 8 figures The crank is the argumentTwo routes to the same number

Grashof’s condition is the oldest classification in the subject and this site has already run it twice: predicted from four lengths and then measured by sweeping, with the prediction and the measurement required to agree.

It says nothing about width, because width is not one of its four numbers. This essay asks the width question of the same linkages and gets an answer that is not independent of Grashof’s at all — it is the same condition, pointing the other way.

crank rocker: the closest pair at one positionThe site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: **0.1494** here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.gap 0.1494positioned by solving, not by drawing
Fig. 1 What layering does answer: the links’ quarrel with each other, resolved by putting them in different planes.

The obstacle that layers cannot fix

Two essays ago the answer to a pair of links being in the same place was to build them in different planes. That works for links, and it does not work for the one thing in a planar linkage that is in every plane at once.

A ground pivot has to be carried, and the links pinned to it are two links sharing a joint. Something stands up from the base plate and holds the pin, and whatever that something is — a pedestal, a boss on a bracket, a bearing housing — it occupies its own footprint through the whole stack. A moving link that passes over that pivot in plan passes through the pedestal in fact, whatever plane the link is in.

So the pedestal is the one obstacle a planar linkage cannot design around by choosing heights, and it turns out to be what actually limits how wide a four-bar’s links can be.

The instant a coupler crosses its own driving pivot. A crank rocker at the driving angle where the sign of (B − A) × (O₂ − A) changes: the coupler's centreline is passing through the pivot the crank turns about. The seven grey positions either side are 0.09 radians apart and show the line sweeping across. A sampled distance can only ever report a small number here — this sweep's smallest is 2.4e-3 at 720 samples — and a sign change reports the fact: the closest approach is zero, so a pedestal of any size at that pivot leaves no width for the coupler at all. Real machines answer it by hanging the crank on a stub shaft with a bearing only on one side, which is a decision about the third dimension and not about the four lengths.
Fig. 2 And what it does not answer: a coupler passing over the pivot its own crank turns about, which no choice of plane avoids.

The model, and what it assumes

A pedestal is a square of side 2·1.6·w centred on each ground pivot, where w is the half-width every link is cut at. It grows with the stock, and the reason is mechanical rather than tidy: a wider link needs a bigger pin, a bigger pin needs a bigger bearing, and a bigger bearing needs a bigger pedestal. Holding the pedestal fixed while the links grew would be a different question with a larger answer and would say nothing about proportion.

The links pinned to a pivot are not tested against its pedestal. Their bosses pass over it by construction — that is what a bearing is — and testing them would report every four-bar as jammed at every width. That exemption was missing in the first version of this measurement, and every entry in the table read zero: a table of small numbers that looks exactly like a result, because a coupler really does hit a pedestal at some width.

Everything else is tested: the coupler against both pedestals, the crank against the far one, the rocker against the near one, and the links against each other.

The width search, and why bisection is sound here. The worst gap over a whole drive, against the width every link is cut at, for a non-Grashof rocker whose links keep clear of both pivots. The curve is decreasing — every body only grows with the width, and a growing body cannot open a gap — so the crossing is unique and bisection converges to it rather than to whichever root a search happened to find first. The answer is 0.3966, at which point the machine is exactly touching somewhere on its travel, and the search cost 13 full sweeps to get there.
Fig. 3 The search on a machine that admits a width: the worst gap over the whole drive, falling as every body grows.

What the search finds

Bisect on w, sweeping at each step, with every frame marched rather than jumped to. The gap is a decreasing function of the width — every body only grows, and a growing body cannot open a gap — so the crossing is unique and bisection converges to it rather than to whichever root it met first.

For the site’s own crank rocker the search returns 0.0094, which is to say: essentially nothing. A link a hundredth of a unit wide, on a mechanism five units across. For the short-crank rocker and the drag link it returns zero.

For the two non-Grashof rockers it returns 0.396 and 0.199 — real widths, a fifth and a tenth of the shortest link, on machines that will not turn all the way round.

The pattern is not subtle and it is not what Grashof’s condition was for.

The gap is a function, and it has corners. The smallest gap over every tested pair, at each of 360 solved positions of the crank. Two things are visible that a check at the ends could not report. The minimum, -0.0100, is in the middle of the travel and not at either end. And the curve has 4 corners, each one a change in which pair is closest — the colours — so the function is piecewise smooth rather than smooth, and its minimum is not where a derivative vanishes. Refining off the sample grid by golden section moves the answer by 6.1e-5, which is what a corner rather than a smooth minimum looks like.
Fig. 4 The same sampling failure in its own essay’s machine: a narrow dip that a coarse sweep steps over.

Why the answer is zero, and how that is known

A bisected width of 0.0094 is a small positive number and a reader is entitled to ask whether it is real. It is not, and the way that is established is the second route.

Take the sign of (B − A) × (G − A), where A and B are the ends of a link and G is a ground pivot. It says which side of the link’s line the pivot is on. A change of that sign, with the pivot’s perpendicular foot on the segment at both samples and the two samples adjacent, is the link sweeping across the pivot — at which the closest approach is exactly zero and no positive width whatever is admissible.

Every one of the four crank rockers has one such crossing, and it is the coupler over the driving pivot. The drag link has two: the crank over the far pivot and the rocker over the near one. Neither of the two rockers has any, and their closest approaches are 1.43 and 0.60 units — enormous, not marginal.

So the positive widths in the crank-rocker rows are artefacts of sampling. The sweep steps over the instant of the crossing, finds a small positive gap at the samples either side, and the bisection converges to a width at which those samples touch. It is the sampling failure of two essays back appearing in a completely different measurement, found by the machinery that essay built — and the certificate refuses those widths, as it must.

Why a crossing must happen

The measurement is a sweep and the reason is a sentence.

The crank pin A goes all the way round the driving pivot O₂ — that is what turns fully means — so the direction from A to O₂ takes every value in a full turn. The coupler’s direction from A to B varies continuously and, on a crank rocker, does not itself take every value: B rocks. A continuously varying direction and a direction sweeping through every value must coincide somewhere, and where they coincide the pivot is on the coupler’s line.

That is not yet a crossing of the segment — the pivot has to be between A and B as well — and that is what the measurement establishes rather than the argument. But it explains why the phenomenon is universal among the linkages that turn rather than a coincidence of these dimensions: it is forced by the crank going all the way round, and it does not happen on a rocker for the same reason.

The crossing test, and the two ways it was wrong first

A sign change is a discrete quantity and therefore robust where a distance is not, which is why the crossing count is the number this result rests on. Getting it right took two corrections, both of which produced numbers that contradicted the distances in the same table.

The side has to be taken from the segment, not the line. The sign of (B − A) × (G − A) is about the infinite line through the link, and a line can sweep over a pivot that the link’s material never approaches. The triple rocker was reported as crossing three pivots while its closest approach was 1.43 units. The repair is to record the side only while the pivot’s perpendicular foot lies strictly between the two joints.

And the two samples have to be adjacent. With the side recorded only over the segment, a pivot can leave the far end of a link on one side and return past the near end on the other, having never been swept over at all. That was still two crossings on the same triple rocker. A crossing is a sign change between consecutive samples of a dense sweep, and with that the rockers report none.

Both errors were caught by the same thing: a row of a table saying crossed beside a column saying 1.43 units away. Two quantities in one row contradicting each other is the cheapest error detector this site has, and it only works because both are printed.

The gap is a function, and it has corners. The smallest gap over every tested pair, at each of 360 solved positions of the crank. Two things are visible that a check at the ends could not report. The minimum, 0.1041, is in the middle of the travel and not at either end. And the curve has 4 corners, each one a change in which pair is closest — the colours — so the function is piecewise smooth rather than smooth, and its minimum is not where a derivative vanishes. Refining off the sample grid by golden section moves the answer by 1.4e-5, which is what a corner rather than a smooth minimum looks like.
Fig. 5 What the width search minimises at each step: the worst gap over a whole drive.

The two instruments, and what they agree about

Grashof’s inequality is a statement about four numbers: s + l ≤ p + q, decided in arithmetic, with no motion in it. The crossing count is a statement about a sweep: sign changes of a cross product, decided by driving the mechanism. They have no step in common.

They agree exactly, on all seven rows. Every four-bar the inequality says turns fully, crosses a pivot. Every one it says does not, does not.

That is a genuinely new statement about the oldest classification in the subject, and it is worth being careful about what it claims. It is measured on seven linkages, not proved for all of them, and the argument above says why the mechanism is there rather than deriving the result. What can be said with confidence is the negative form: Grashof’s condition is not silent about buildability, as this site has treated it for twenty-three fields. It is the same condition and the sign of its usefulness reverses.

drag link: the closest pair at one positionA double crank — ground 1, crank 2, coupler 1.6, rocker 2.2 — where both turn fully. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: **-0.4320** here, between crank · rocker. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.gap -0.4320positioned by solving, not by drawing
Fig. 6 The drag link, whose two long links each sweep across the other’s bearing — two crossings rather than one.

The crossings are not all the same crossing, and the pattern is worth reading.

On all four crank rockers it is the coupler over the driving pivot O₂. The crank pin circles O₂ and the coupler hangs off the crank pin, so the coupler’s line sweeps across the pivot the crank is turning about — the machine’s own input bearing, passed over by the link two joints away from it.

On the drag link it is the crank over the far pivot O₄, and the rocker over the near pivot O₂. A drag link is the inversion in which both the input and the output turn fully, its frame is the shortest link, and both long links sweep right across the other’s bearing. Two crossings rather than one, which is the arithmetic version of what the drawing already looks like.

There is a small asymmetry worth noticing in the crank-rocker rows: the crank does not cross the far pivot, although it turns fully, because O₄ is four units away and the crank is one unit long — it never reaches. Crossing needs the pivot to be within the link’s own reach as well as on its line, and on a crank rocker only the coupler is long enough to span the frame.

What real machines do instead

Everybody has seen a four-bar with a crank that turns — an engine’s slider crank is one. The result above is not that they do not exist; it is that they are not built with a bearing on both sides of both ground pivots.

The standard answer is an overhung crank: the crank pin is carried on a stub shaft with its bearing on one side only, so there is no pedestal on the side the coupler sweeps over. An engine’s crankshaft is the same idea at a larger scale — the connecting rod passes the main bearing’s plane and the main bearings are elsewhere along the shaft.

That is a decision about the third dimension, and it is exactly the kind of decision this field exists to make visible. It is not free: an overhung pin is a cantilever, and everything about how it is supported is harder, which is a force argument and is outside this site. What is inside this site is the geometric fact that forces the choice.

Watt's linkage: the closest pair at one positionThe straight-line linkage at the proportions the curves field draws it at. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: **-0.4320** here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.gap -0.4320positioned by solving, not by drawing
Fig. 7 A linkage that rocks rather than turns, and can therefore be made of much more material.

What the rockers buy

The two non-Grashof rockers take widths of 0.198 and 0.099 of their shortest link, and a link a fifth as wide as it is long is a chunky, stiff part.

That is the trade in its plainest form. A mechanism that turns all the way round has to be built with a bearing missing somewhere; a mechanism that rocks can be built with bearings everywhere and made out of much more material. That is a trade the applied field’s machines all make one way or the other. Neither is better. It is a real design axis that has been invisible on this site because a link had no width, and it runs in the opposite direction to the one Grashof’s classification is usually read in — where turning fully is the desirable property and rocking is the compromise.

crank rocker: the closest pair at one positionThe site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: **0.5772** here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.gap 0.5772positioned by solving, not by drawing
Fig. 8 The machine at the position the crossing happens nowhere near: everything about it is comfortable.

What the closest approach predicts

For the machines with no crossing there is a third number, and it is a check on the first.

A link of half-width w touches a square pedestal of half-side 1.6w when its centreline comes within about w + 1.6w of the pivot, so the widest admissible width is roughly d/2.6 with d the closest approach — and between d/2.6 and d/3.26, since a square pedestal is reached sooner across its corner than across its face.

For the triple rocker, d = 1.426, which predicts a width between 0.437 and 0.549 against a bisected 0.396. For the compact rocker, d = 0.600, predicting between 0.184 and 0.231 against a bisected 0.199.

The bisected answers are inside the band in one case and just below it in the other, and the reason for the shortfall is stated rather than fitted away: the prediction assumes the pedestal is the binding constraint, and at those widths the links have started to interfere with each other as well. That is the closed-form route doing exactly what a second route should — agreeing where its assumptions hold and disagreeing informatively where they do not.

A width is a design variable this site has never had

It is worth stepping back from the four-bars to notice what kind of quantity has just been introduced.

Every design variable on this site until now has been a length, an angle or a count — four lengths, a tooth count, a pressure angle: four lengths for a four-bar, a tooth count, a number of stages, a pressure angle. All of them enter the constraint equations, all of them change the motion, and the whole of synthesis is about choosing them so that the motion is what was asked for.

A width enters nothing. Change it and every position, velocity, ratio, curve and rank is exactly what it was. It is the first parameter on this site that is invisible to every instrument the site had, and it is decided entirely by questions asked after the solve: does the machine assemble, how deep is the stack, how much material is there.

That makes the width search a different kind of design step from anything in the synthesis field. There, choosing a parameter trades one aspect of the motion against another. Here, the motion is fixed and settled before the question is asked, and what is being traded is stiffness and material against buildability — with the mechanism watching, unaffected.

What is being assumed about the pedestal

Three things, and all three are choices a designer would make differently.

Its size scales with the stock. A pedestal 1.6 times the link half-width is a proportion, not a law, and a machine with generously sized bearings has a larger one.

It is square. A round pedestal — which is what a bearing housing usually is — is reached at d/2.6 uniformly rather than sooner at the corners, which is why the prediction is quoted as a band.

And there are exactly two of them. A four-bar has two ground pivots and both are assumed to be carried from below. A machine with one pivot on a pedestal and the other overhung is the arrangement the previous section describes, and its answer is a different search with a different table.

None of that changes the crossings, which are what the result actually rests on. A crossing means the link passes over the pivot’s centre, so it holds for a pedestal of any shape and any size, including one of zero size — which is the honest statement of the finding: at a pivot that is crossed, no pedestal of any size fits.

The shape of the whole thing

Six of the seven rows can be read as one sentence. Ask a four-bar to turn all the way round and it will sweep a link over one of its own bearings; ask it to be made of material with bearings at both ends and it will only rock.

The measurement did not set out to find that. It set out to answer how wide may these links be, expecting a table of widths, and what came back was a table with zeros in it and a reason. That is the shape of most of what this field has produced: a question about material, asked of mechanisms this site has been drawing for twenty-three fields, coming back with an answer about the mechanisms rather than about the material.

Where this goes next

Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept. Grashof, predicted and then swept Add the shortest link to the longest. If the total does not exceed the other two, some link can turn a full revolution. It is a sentence about four numbers, it was published in 1883, and it is the kind of claim this site refuses to print without measuring — so every linkage here is also asked for all 360 positions and required to agree. Peaucellier's cell: the closest pair at one position. Eight links, ten pins and an exact straight line — the site's densest planar loop. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1983 here, between long arm A · crank. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour. A gap is a number A collision test that answers yes or no cannot say by how much, and therefore cannot say what would fix it. The quantity this field is built on is one signed number: positive is a gap, negative is how far the parts would have to be moved to stop overlapping. The bound that says the sweep missed nothing. Every corner of every body is an affine function of its link's two joints, so its speed is the same combination of the joint velocities the mechanism already solves for — and the largest corner speed anywhere on this drive is V = 1.255 per radian. A distance between point sets is 1-Lipschitz in those points, so from each sample the gap can fall no faster than V: the fine lines are those cones. Where two cones cross is the least the gap can be between the samples, and at 60 samples that is 0.0383 — positive, so nothing was missed. The same bound refuses the twelve-sample sweep of the stud machine, where the bound is -0.281. A sweep that missed nothing A swept clearance check looks at finitely many positions of a machine that has infinitely many, and cannot report what it did not look at. Here is a twelve-sample sweep declaring a machine clear by 0.007 while it is 0.010 inside a stud — and the bound that refuses to certify it.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 14 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

BearingBearing pedestalCrank-rockerGrashof's conditionInterferenceLink bodyPivot crossingSigned clearance