A link may be bent

For twenty-five fields a link here was a distance between two points. This one gave it a body, and the body it gave it was a straight bar — which is a modelling choice, has been treated as though it were a fact, and is the reason several of this field’s results are gloomier than they need to be.

A link is a rigid body carrying two pins at a fixed distance. The distance is kinematics; the metal between the pins is not.

The claim, checked the only way it can be #

The claim is that bending is kinematically free, and the check is not an argument but a measurement.

The mechanism is solved first, from its four lengths and its driving angle, and the bodies are hung on the solved joints afterwards. So bending a body cannot move a joint — but “cannot by construction” is exactly the kind of statement that has gone wrong here before, where a check that restated its own construction announced a result to the last bit and tested nothing.

So the two machines are solved independently, sixty configurations each, and every joint of one is compared against the corresponding joint of the other. The worst displacement over the whole drive is 2.4 × 10⁻¹⁴, which is the solver’s own residual and not a small number.

And the other half, which is the half that could actually fail: the worst clearance must move. It does — from 0.1042 to −0.3192 on the same machine at an offset of 0.2 — because a parameter that changed nothing measurable would be a parameter drawing nothing.

What the shape actually is, and what it is not #

The bent bar drawn here has its centreline running from one pin, through a point offset perpendicular from the chord’s midpoint, to the other — a dog-leg rather than a curve, thickened to the same width as the straight bar and finished with the same bosses. h is that offset as a fraction of the link’s length, so an offset of 0.12 on a coupler three and a half units long moves its middle 0.42 of a unit sideways.

Two details of the construction are worth recording because both were wrong first.

The bar is boss-wide throughout, not waisted. A straight bar in these figures runs its sides at the boss radius rather than at the half-width, so it is uniformly 1.35 widths thick with cut corners. The first bent bar mitered its inner corner at the half-width instead, which made it narrower in the middle than the bar it was being compared against — and the four-bar with a post then read 0.1275 of clearance at zero bend against the straight bar’s 0.1042. A difference of 0.023 that has nothing to do with bending would have been attributed to bending in every sentence below. A bend of nothing must be the straight bar, and it is now required to the last bit.

The inner corner is a miter and it runs away. The offset lines on the inside of a bend meet at a point whose distance from the centreline goes as 1/cos of half the turn, so a sharp enough bend produces an outline that crosses itself. The bound is on the offset rather than on the resulting polygon, because a message about the bend is more use than a message about edges three and seven.

What a bend buys #

Three things are visible and each is worth a sentence.

Bending away from an obstacle buys clearance, and then stops. The four-bar with a post reads 0.1042 straight, rises to 0.1884 at an offset of −0.12, and goes no higher at −0.18, −0.24 or −0.30. The gain is 0.084 of a unit — about half a link width — and after that the critical contact has moved to a feature the bend cannot reach.

Bending toward it costs at about the rate it is applied. The same machine reads −0.0033 at +0.06 and −0.2723 at +0.18, so the wrong sign of bend is expensive and roughly linear. That asymmetry is the practical content: a bend is a cheap fix applied in the right direction and a good way to ruin a machine applied in the wrong one.

And it can decide whether the machine exists. The crank passing a stud is 0.0100 inside the stud with a straight coupler. Bend the coupler by −0.12 and it clears by 0.0018; at −0.30 it clears by 0.0138. The straight machine cannot be built and the bent one can, on identical kinematics and with the same two pins.

What a bend cannot buy #

The negative result is sharper than the positive one and it is the reason this essay is worth writing.

Every four-bar that turns all the way round sweeps a link straight over one of its own ground pivots. That is Grashof’s inequality arriving as a statement about bodies: the same condition that makes a crank rotate fully forces one of its links to pass over a pivot in plan. And a ground pivot has to be carried by a pedestal standing up from the base plate, which is in every layer at once — so the plane-assignment trick that solves the links’ quarrel with each other cannot solve this one.

The natural hope is that a bend can. It cannot, and the measurement is unusually flat.

A bend moves the metal between the pins, and the pins do not move. The contact here is the coupler’s own boss passing over a pedestal that stands at a pin — and the boss is at the pin, because a boss is the material that surrounds the pin. There is no offset of the middle of the link that moves its ends.

That is why the number is not merely large but constant. A bend that bought a little would show a slope; a bend that bought nothing at all shows a flat line, and a flat line to five decimal places is a statement about which part of the link the obstruction is against.

A machine that was refused, and is not #

The stud case deserves the numbers laid out, because it is the first time in this field that a design change has moved a machine across the line rather than along it.

A crank passing a stud a tenth of a unit across, with links 0.16 wide, reads −0.0100: the coupler is a hundredth of a unit inside the stud, at 2.103 radians, for a small part of every turn. That is a machine that does not run.

The remedies the field already had are all expensive. Make the links thinner — the gap is affine in the width, so a width of 0.145 would clear it, at the cost of every link in the machine. Move the stud — it is somebody else’s part. Put the coupler in another plane — the stud is bolted to the frame and is in every plane.

The bend costs none of those. At an offset of −0.12 the same machine reads +0.0018; at −0.18, +0.0054; at −0.24, +0.0123; at −0.30, +0.0138. The coupler is the same length, the crank is the same length, the drive is the same drive, and the coupler curve every other field has drawn for this machine is the same curve to the last bit.

The margins are small — a hundredth of a unit is not a comfortable clearance — and that is honest rather than disappointing: the stud was placed to be a near thing, and what the measurement shows is the sign changing rather than a generous escape.

The three places an obstruction can be #

Putting the two results together gives a small classification, and it is the useful outcome here.

Against the middle of a link. A bend works, and works well — a tenth of a link length of clearance for nothing. The post and the stud are both of this kind.

Against a link’s own boss. A bend does nothing. The pedestal is of this kind, and so is any obstruction near a pin.

Against another link’s middle. A bend on either of them works, and bending both works twice — the two offsets add, because the gap between two features is affine in the positions of both.

A designer meeting an interference therefore has a question to ask before choosing a remedy, and it is a question this field can answer directly: where on the link is the witness pair? The closest-approach segment has two endpoints and one of them is on the link in question. If it is near a pin, bend nothing and change something else. If it is in the middle, bend.

That is a better rule than the one the field has been running on, which was that a straight link either fits or does not.

What this changes about earlier results #

Three of this field’s findings are worth revisiting in this light, and the honest summary is that two of them survive and one is weakened.

Two bars that have to cross survives. The four-bar drawn throughout has its coupler inside its frame by a full link width for the whole of a turn — not at one configuration but throughout — and a bend of a tenth of a length cannot clear a width for a whole turn. The conclusion that the machine needs more than one plane is unaffected.

The width limit survives, and now has a reason. The limit is set by the pedestal contact, which is exactly the contact a bend cannot reach, so the widest link a four-bar will take is a property of the link’s ends rather than of its shape.

The plane count is weakened, and the amount is not measured here. A plane assignment is computed from the conflict graph, and the conflict graph is computed from straight bars. Some of its edges are conflicts against a link’s middle, and every one of those is an edge a bend might remove — which would lower the chromatic number and cost a plane less. Whether it does on any of this field’s machines is a question this essay does not answer, and it is the obvious next one.

The cost, which is not in this field #

A bend is free kinematically and it is not free. Three costs are worth naming, and all three are outside what is computed here.

A bent link is longer than a straight one. Its centreline runs $2\sqrt{\tfrac14 + h^2}$ rather than 1, so an offset of 0.12 costs 2.8 per cent more material and an offset of 0.24 costs 10.8 per cent. That is a manufacturing quantity and it is computable from the offset alone.

A bent link carries a bending moment where a straight one carries an axial force. A two-force member loaded along its own line is in pure tension or compression; move the material off that line and the same load produces a moment proportional to the offset. That is the reason connecting rods are straight wherever they are allowed to be, and it needs a modulus and a load, which the structural field owns and this one does not.

And a bend has a hand. The five bodies in the first figure are five different parts, so a machine with a bent coupler has a left-handed and a right-handed version and they are not interchangeable. A straight bar is its own mirror image and that is worth something in a stores.

None of the three changes the geometry above. All three are why the straight bar stayed the default long after it stopped being the only option, and stating them is the difference between “a bend is free” and “a bend is free in this field’s currency”.

Why straight bars were the right place to start #

It is worth saying why this was not an oversight, because the reasoning still holds for the other fields.

A straight bar is the canonical body for a link: it is the one determined by the link’s own kinematic data and nothing else. Every other body needs a design decision that no loop equation supplies. So a field that draws links as straight bars is a field reporting what it knows rather than a field guessing, and every number it produces is a statement about the worst body a link could reasonably have.

Which makes most of this field’s results conservative in a useful direction: the clearances are lower bounds, the width limits are lower bounds, and the plane counts are upper bounds. A machine this field declares buildable is buildable. A machine it refuses may still be buildable with a shape it was not told about.

The one number that direction spoils is the one it is now natural to want, which is how much room a well-shaped machine needs — and that is not a measurement this field can make without being told what shapes are allowed.

The bend that a designer would actually reach for #

The offset above is a single dog-leg at the middle, and it is worth saying what a real part does instead, because the difference is instructive rather than merely cosmetic.

A connecting rod avoiding a crankcase, a rocker clearing a shaft, a coupler passing a boss: all of them are local bends. The metal leaves the chord where it has to and comes back, and the departure is a bulge rather than a slant. A single mid-span offset is the crudest member of that family and it costs the most material for the clearance it buys, because it moves every part of the link including the parts that had nothing to avoid.

That points at the quantity a designer is really trading, and it is not the offset. It is how much of the link has to move, and by how much. A bend that clears an obstacle occupying a tenth of the link’s length need only displace that tenth; the mid-span offset displaces all of it, so it buys the same clearance at ten times the material and ten times the bending moment.

This field can say something firm about the geometry of that and nothing about the cost. Where the obstruction is, along the link, is exactly what the witness pair reports: the closest-approach segment has an endpoint on the link, and that endpoint’s position along the chord is where the metal has to move. A design rule follows without any optimisation — bend where the witness is — and it is a rule the straight-bar model could not state, because a straight bar’s witness is always somewhere and never anywhere useful.

What this essay leaves is therefore a parameterisation question rather than a geometry one. The affine reasoning says clearances are linear in offsets; the witness says where to apply one; and how many offsets a link should carry is a question about parts rather than about mechanisms.

Still open: bending is one degree of freedom out of many #

A bend of a stated offset is a single number, and it was chosen because it is the smallest parameterisation that makes the point. A real link’s shape is not one number.

Two directions are open and neither is measured here. The first is where to bend: the offset above is applied at the middle, and a bend applied nearer one pin gives a different profile of clearance along the link. The second is how many bends: two offsets in opposite directions give an S, which is what a connecting rod avoiding two obstacles actually looks like, and the affine reasoning suggests they compose.

There is also a question about the shape optimum worth being careful of. Given a set of obstacles and a drive, the widest possible link body is the complement of the swept union of everything it must avoid, expressed back in the link’s own coordinates — a well-posed problem with a computable answer and no design judgement in it at all. That object would be the largest link rather than a well-shaped one, and it would not be a bar; naming it is worth more here than computing it, because it says what the design variable actually is.