The right angle as a tolerance

Nine bars that ought to be rigid measured Dixon’s framework three ways and found something the count and the rank both miss. Nine bars joining each of three joints on one line to each of three on another leave no freedom by the count, wherever the joints are. The rank of the constraint Jacobian finds a freedom whenever the six joints lie on two crossing lines, at any angle between them. And a forced move — push one joint a distance $h$ along its line and find the placement that comes closest to keeping all nine lengths — separates those further still: at a right angle the push is taken exactly, and at every other angle it misses by an amount proportional to the tilt and to $h^2$.

So the right angle is a manufacturing condition on a framework that has no motion without it, and the miss is what a framework built off square cannot absorb. That essay ended by naming the other half: the miss has to go somewhere, and in a real framework it goes into the joints.

This is that measurement. The clearance a framework built off square needs, to move at all.

A clearance is a maximum, not a sum #

A pin joint with radial clearance ρ lets the two bodies it joins put their joint centres up to ρ apart. A bar has a joint at each end, so the distance its two joints have to span may be wrong by up to 2ρ without the bar minding, and a framework with such joints is assembled at a placement exactly when every one of its nine bars is inside its own allowance.

That “every” is the whole difficulty, and it is why this measurement needs its own apparatus rather than the one already in hand. The forced-move instrument minimises the sum of the nine squared length errors and returns a norm; a clearance is a statement about the largest of them. Those are different placements and different numbers, and the difference is not small.

So the question here is a feasibility question: given a tilt and a push, does a placement exist with all nine bars inside an allowance of $c$? The answer is found by bisection on $c$, with the feasibility test itself a descent on each bar’s excess over the allowance — nought while a bar is inside it, and the overshoot once it is not. A placement that satisfies every bar drives that whole vector to nought, and one that cannot leaves exactly the bars that do not fit.

There is a stronger reason to do it this way than tidiness. A four-bar’s clearance is a single scalar budget: the four bars form one loop, the four clearances enter it as a sum of four vectors, and only their total matters — so a four-bar cannot tell which of its bearings is loose. A Dixon framework has nine bars and no single loop, so there is no sum to take. Each bar is its own condition, and the framework’s tolerance is a statement about all nine at once. That contrast is the reason the two measurements read differently, and it is the reason this one is a bisection rather than a subtraction.

The law #

Measured over five tilts and a factor of forty in the push, the allowance follows one law and follows it cleanly.

Every line has slope 2.041. The allowance grows as the square of the push, which is what a framework with a first-order freedom and no motion has to do: the flex is available to first order, and what cannot be taken is the second-order part. That was already the shape of the miss the earlier essay measured, and it survives the change from a sum of squares to a maximum, which it did not have to.

The five lines are evenly spaced in the logarithm because the five tilts are, and that is the other half of the law.

The slope is 1.0055. Proportional to the tilt, not to some other power of it — which was the question worth asking, because nine joints are sharing the error and it would be reasonable for nine shares of a second-order miss to combine into something other than a first power. They do not. Dividing every measurement by its tilt and by the square of the push leaves one constant to within a factor of 1.016, so the whole law is

$$\text{allowance} \;\approx\; 8.5 \times 10^{-4} \;\times\; (\text{tilt in degrees}) \;\times\; (\text{push})^2$$

for this framework, whose bars are of order one and whose joints sit at −2, 1, 3 and −1.5, 1, 2.5 from the crossing point.

That is a gentle condition, and it is worth converting once. A framework a metre across, built a tenth of a degree off square, asked to move ten millimetres: the allowance is $8.5 \times 10^{-4} \times 0.1 \times 0.01^2$ of a unit, or about eight nanometres of bar length, which is four nanometres of radial clearance at each joint. A framework built ten degrees off square and asked to move a tenth of its size wants eight microns at each joint. Neither is a demanding fit, and the second is an ordinary one.

A fourth instrument #

Counting and measuring mobility set two instruments against each other — a count that knows the topology and nothing about the lengths, and a rank that knows the lengths and nothing about the topology. The earlier Dixon essay added a third, the forced move, which reads the second order and separates the right angle from every other. The allowance is a fourth, and it is worth saying what it sees that the third does not.

The push answers whether a framework takes a move, with a number that is nought or is not. The allowance answers how much room it needs to take one, and it is the first of the four that returns a quantity a drawing can carry. A count is dimensionless. A rank is dimensionless. A miss is a length but it is the length of a vector of nine residuals and no bar has it. An allowance is a length that belongs to a bar and to a bearing, and it is what a fitter is given.

That is also why it is the instrument that can be wrong in a new way. The first three can disagree with each other and the disagreement is the finding; this one can agree with all of them and still be the wrong number, because the placement it is computed at is a choice. The next section is about that choice.

Where the error goes #

The bisection returns a placement as well as a number, and what that placement does with the error is the part that separates this measurement from the one it replaces.

At the allowance, all nine bars are at the allowance, equal to nine significant figures. That is what minimising the largest of nine numbers does when every one of them can be traded against every other: there is no slack left anywhere, because any bar sitting below the maximum is a bar whose room could have been spent lowering the maximum.

The least-squares placement is a different object. It minimises the sum of the squares and leaves the nine spread from $4.35 \times 10^{-5}$ to $2.28 \times 10^{-4}$, a factor of five between the easiest bar and the hardest, and its worst bar asks for 1.508 times the clearance that is actually needed. Had the tolerance been read off the existing instrument, a designer would have specified half as much clearance again as the framework requires, and would have had no way of noticing.

That is the general hazard and it is worth stating without the framework in it. A fit that is right for the quantity it minimises is not right for a different quantity computed from the same residuals. A norm is a good measure of how badly a framework is being forced; it is not a tolerance, and turning it into one costs a factor that depends on how unevenly the error happens to be spread.

It is the same hazard the seventh contact meets from the other side. There a contact that adds no rank is the one that can fail to touch, and the question of which of several redundant conditions is the one that gives way is settled by geometry rather than by a count. Here nine conditions are redundant together — no one of the nine bars is the spare, and any eight of them imply the ninth — and the consequence for a tolerance is exactly that the error cannot be assigned to a culprit. It is shared, and the sharing is even, and both halves of that are consequences of the single dependency among the nine rows rather than of anything about clearance.

Two searches, one answer #

A descent on a clamped residual can stop in more than one place, so a single bisection is a single opinion. The check is to run the same bisection from a different starting point.

The second search starts where the perpendicular framework is when it has been pushed the same distance — a placement found by bisecting the closed form’s own parameter, since the distance between the first two joints is $\sqrt{1+t} + \sqrt{4+t}$ and increases with $t$. That is a genuinely different starting point, several times the allowance away from the framework as built, and it is arrived at without any reference to tilts or to bisections on clearance.

The two agree at every push, to the last digit either bisection carries. Neither is told the square law, and neither is told the other’s answer.

The crossing, and where the law stops #

The law holds over the whole range a framework would plausibly be asked to work in, and then it stops, and the way it stops is the answer to the second question the earlier essay left.

The perpendicular framework’s motion is a closed loop fifteen units long, and along it the joints pass the crossing point four times — once for each way of choosing which side of the crossing the two nearest joints are on. The first of those crossings happens when the first joint has been pushed 0.650. A framework built off square has no motion, only an allowance, and the question is whether an allowance sized for a small push carries it through.

It does not. Measured and fitted agree to a push of about 0.5 and then part company, and at the crossing the measured allowance is $3.05 \times 10^{-2}$ against the law’s $9.1 \times 10^{-5}$ — 336 times more. A framework whose bearings were sized on the square law does not arrive at the crossing with a little too little. It arrives more than two orders of magnitude short, and it arrives there having been given no warning by anything the law can say.

The failure is not a jam in the sense of a wall: the allowance rises steeply rather than to infinity, and past the crossing it goes on rising and the framework goes on being feasible at a large enough clearance. What the crossing costs is a change of régime. Below it the framework is a nearly-moving thing being asked for a small second-order concession; at it, the placement the framework has to reach is one where three bars of one joint lie along the other line, and reaching that with the lines not perpendicular is a different and much larger demand.

So a Dixon framework built off square is a mechanism with a bounded travel, and the bound is not set by the clearance running out gradually. It is set by the first crossing, and the travel available is whatever fraction of 0.650 the clearance buys under the square law — which for a framework a tenth of a degree off square with eight microns of play at each joint on a metre-wide frame is most of the way there, and for one a degree off is about a third of it.

That bound is a different kind of limit from the ones this field usually produces. The count says a framework moves or does not; the rank says the same thing with the geometry in it; neither has a way of saying how far. A framework built off square and given clearance has a definite travel, and the number that sets it is neither a count nor a rank but the distance to the first crossing divided among the terms of a quadratic law. It is a mechanism whose range is a manufacturing quantity.

What this does not settle #

The joints are perfect except for their size. A pin in a hole is a joint centre free within a disc, with no friction, no tilt, no wear and no load. Which side of its hole each pin sits against, in a framework being pushed, is a question about forces and the answer here is a question about reachability.

The clearance is the same at every joint. The law is measured with one allowance shared equally, which is what a drawing with one fit on it produces. A framework with one loose joint and eight tight ones is a different feasibility problem, and because the nine bars are nine separate conditions rather than one sum — the point the four-bar comparison turns on — there is no reason to expect the total to be what matters.

The travel is a push on one joint. Everything here measures how far $B_2$ can be moved along its line, which is one coordinate of a motion that moves all six joints. A framework asked to reach a stated placement rather than a stated push at one joint is being asked a related question with a different answer, and the two coincide only because the motion has one freedom and the push parametrises it.

One framework. Dixon’s placement with these six joints. The constant $8.5 \times 10^{-4}$ is a property of where the joints sit along their lines, and how it depends on that is not measured. The exponents are the general statement and the constant is not.

The bars still cross. In the plane several of the nine bars pass through one another as the framework moves, and a physical framework needs them in layers with pins long enough to reach. Nothing here asks whether the layering that permits the loop also permits the clearance, or whether a pin long enough to span the stack is stiff enough for the fit.

Only the first crossing. The loop has four, and only the one at a push of 0.650 is measured. Whether the other three cost the same, and whether the framework can be got round the whole loop by any clearance at all short of one that makes the bars meaningless, is not answered.

Still open: the clearance that makes the framework generic #

The measurement above holds the framework’s shape fixed and asks what clearance lets it move. The opposite question has the same ingredients and a different answer, and it is the one a designer would ask second.

Give every joint a clearance ρ and the six joints are no longer on two lines: each is somewhere within ρ of its nominal place, which is a placement in general position, and a generic placement of these nine bars has rank nine and no freedom at all. So clearance does two opposite things to this framework. It absorbs the misfit a tilt produces, which is what is measured here, and it also destroys the very coincidence — six joints on a conic — that gives the framework its rank deficiency in the first place.

Its distinct argument would be which of the two wins, and at what clearance. The first-order freedom survives a perturbation of size ρ only in the sense that the smallest singular value stays small rather than nought, and the question is whether the framework with clearance behaves as a mechanism with one freedom and a tolerance, or as a structure with a small compliance, and where the boundary between those two descriptions falls as a function of ρ and of the tilt. The instrument is already here: the smallest singular value along the loop, which the earlier essay measured at no less than 0.496 for the exact framework, taken over placements drawn from the clearance discs instead.