Nine bars that ought to be rigid

The mechanism Grübler says cannot move is the standard example of a count that is wrong. Three equal cranks hang from a frame and carry one coupler; the count of links and joints says the chain is a structure, and it swings freely. The explanation is equally standard, and it is easy to point at. The third crank repeats the first. Whatever the first crank’s condition allows, the third’s allows too, so one of the three conditions adds nothing, and a count that treats every condition as independent comes out one short.

That explanation has a shape worth noticing: there is a culprit. Some particular bar is the redundant one, and taking it away leaves the same mechanism. This essay is about a mechanism that moves when its count says it cannot and has no culprit at all.

In 1899 the mathematician A. C. Dixon described a framework of nine bars that ought to be rigid by every count and is not, in a particular placement. What follows measures it three ways and finds something the count and the rank both miss: its redundancy belongs to none of its bars and to all nine together.

Nine bars, six joints, and a count of nought #

The framework has two sets of three joints. Call them $B_1$, $B_2$ and $B_3$, and $W_1$, $W_2$ and $W_3$. Every B is joined to every W by a bar, and no two joints of the same set are joined, so there are nine bars. As a graph it is the one mathematicians call $K_{3,3}$, the pattern of three houses each connected to three utilities.

Grübler’s count sees nine binary links and six joints, each joint shared by three bars and therefore worth two pins. With one bar taken as the frame that is 3 × 8 − 2 × 12 = 0. Counted in coordinates instead, six joints in a plane have twelve, the plane’s own rigid motions take three, and nine bars take nine, which again leaves nought. Either way the count says a structure.

For almost every placement the count is right. With the joints put down anywhere in general position, the nine rows of the constraint Jacobian, each the derivative of one bar’s length with respect to the twelve joint coordinates, have rank nine. Nothing is left over, and the framework cannot change shape without some bar changing length.

Two lines at right angles #

Dixon’s placement puts $B_1$, $B_2$ and $B_3$ on a horizontal line, at −2, 1 and 3 from a chosen point, and $W_1$, $W_2$ and $W_3$ on the vertical line through that point, at −1.5, 1 and 2.5. The bar from a joint at distance x on one line to a joint at distance y on the other then has length $\sqrt{x^2 + y^2}$, which is what makes the placement special. Every bar’s squared length is a sum of one number belonging to its B and one belonging to its W.

That sum can be held fixed while both numbers change. Add the same amount t to every B’s squared distance and subtract it from every W’s, so that each B sits at $\sqrt{x^2 + t}$ and each W at $\sqrt{y^2 - t}$, and every one of the nine squared lengths is unchanged, because each gains t from one end and loses t from the other. That family of placements is a motion, and the right angle is doing all the work in it. With the two lines at any other angle α, the squared length of the bar from x to y is $x^2 + y^2 - 2xy\cos\alpha$, and the last term ties the two ends together: shifting the B’s squared distances and the W’s in opposite directions no longer cancels, because the cross term changes by an amount that depends on both. Only when cos α is nought does every bar’s length separate into a part owned by each end. That is also a prediction about what tilting the lines should do, and it is tested below.

The first figure draws three members of the family: $B_2$ at 0.632 and $W_2$ at 1.265, the placement as built with both at 1, and $B_2$ at 1.265 and $W_2$ at 0.632. The nine bar lengths are the same in all three to 4.4 × 10⁻¹⁶.

So the count is wrong about this placement. The rank of the Jacobian is not.

The table starts with a placement near Dixon’s but with every joint moved a few tenths off its line. Its rank is nine, it has no freedom, and when $B_2$ is forced 0.01 along its line the best placement that can be found still misses keeping all nine lengths by 4.2 × 10⁻⁴. That is a rigid framework being forced, and the miss is what forcing a rigid framework costs.

At Dixon’s placement the rank is eight, one row of the nine depends on the others, and the framework has one freedom by rank. Forced 0.01, it misses by nothing at all, to rounding. That agrees with the formula.

The last two rows are the reason the push column exists. Tilt the vertical line to 80° or to 60° from the horizontal, keeping every joint at the same distance from the crossing point, and the rank is still eight. By rank, each of these placements has a freedom exactly as Dixon’s does. Pushed, they miss by 2.3 × 10⁻⁶ and 9.4 × 10⁻⁶. A rank measures the first order and a push measures the motion, and here the two disagree. That disagreement is taken up below. First, the placement where they agree.

The motion, followed without the formula #

The formula gives the motion, and a formula is a claim. The second route to it uses nothing but the nine bar lengths. Starting from the placement as built, the framework is moved a small step along the one direction its bars leave free, pulled back onto exact lengths by Newton’s method, and moved again. Nothing in this knows about lines, square roots or the parameter t. It is told only that nine distances must stay fixed, and it is not told which joints to move.

Each placement it reaches is then compared with Dixon’s formula by all fifteen joint-to-joint distances, which fix a placement up to a rigid motion. Every placement along the way matches a member of the formula to 4.2 × 10⁻¹⁴. Both routes find one motion.

The formula runs out and the motion does not #

The formula has a range. Subtracting t from every W’s squared distance works only while every one stays positive, and $W_2$, at 1, is the nearest to the crossing point; at t = 1 it arrives there. In the other direction $B_2$ arrives at t = −1. So the formula describes a motion between two placements in which a joint sits exactly on the crossing point, and says nothing beyond them.

The framework has no such limit. Followed on its bar lengths, it reaches the placement with $W_2$ at the crossing and passes straight through it.

Past the crossing, the placements are again members of Dixon’s formula, with $W_2$’s distance now counted on the other side of the crossing point. The parameter turns round and runs back down to −1, where $B_2$ reaches the crossing; past that the formula resumes with $B_2$ on its other side as well; and so on. The motion passes the crossing point four times, once for each of the four ways of choosing sides for $B_2$ and $W_2$, and after a distance of 15.00 in joint coordinates it is back at the placement it started from. It is a closed loop.

Nothing about the rank marks the crossings. The smallest singular value the rank keeps is never below 0.496 anywhere on the loop, including at the four crossings, so the framework’s one freedom is never a close decision. The formula’s range ends where a square root reaches nought, and that is a fact about the formula, not about the framework.

The middle placement looks alarming, and a reader of the essay on a parallelogram at its flat position might expect a change point there. $W_2$’s three bars lie along one line, over $B_1$, $B_2$ and $B_3$, which is the kind of alignment at which a four-bar has to choose between two motions. Here there is no choice. The framework has one freedom on either side of that placement and one freedom at it, and the measured motion runs through without branching. A flat arrangement makes a change point only where it also loses rank, and this one does not.

No bar is the redundant one #

Back to the question the introduction raised. A framework with nine conditions and one freedom has one condition too many. In the three-crank chain the extra one can be named. Which is it here?

Removing bars answers that. Take away one bar, recompute the rank of the other eight, and follow the motion the eight allow; take away two, and recompute again.

Every single bar can go. With any one of the nine removed, the other eight have rank eight and one freedom, and the motion they allow is the nine-bar motion exactly. Followed over a travel of 0.29 on their own lengths, the missing bar’s length would have changed by no more than 7.1 × 10⁻¹⁵ in any of the nine cases. No pair can go. With any two of the nine removed, and there are thirty-six pairs, the remaining seven have two freedoms.

That is a precise statement of what the framework’s redundancy is. It is a single dependency among the nine rows of the constraint Jacobian, and every one of the nine takes part in it, so any eight rows are independent and any one row is implied by the other eight. The count missed a redundancy; there is no redundant bar. A repeated condition, like the third crank, is the special case in which a dependency happens to involve only a few rows that visibly copy each other. Dixon’s framework is the general case, and it cannot be diagnosed by looking for a duplicate.

It also changes what “the constraint that can be removed” means for anybody building one. The three-crank chain has a bar that can be left out and one that cannot. Here any bar is the spare, which is also why a Dixon framework carries a manufacturing condition on all nine at once rather than on one, as a seventh contact does for a part held by six. What the condition is, is the next question.

Tilt the lines #

The dependency among the nine rows exists at the placements with the lines at 80° and 60° too; that is what their rank of eight says. What those placements lack is the motion, and the push is the instrument that shows it.

A rank is a statement about derivatives at one placement: there is a direction in which every bar’s length is unchanged to first order. Whether a real motion starts in that direction is a question about the second order and beyond, and a freedom to first order can exist with no motion at all. The push asks it directly. Force $B_2$ a distance h along its line, hold the plane’s rigid motions, and find the placement that comes closest to keeping all nine lengths. If the framework moves, the miss is nought for every h. If it has a first-order freedom and no motion, the lengths can be kept to first order and not to second, so the miss grows as $h^2$.

That is what the measurement finds, over two decades of push. At 85° the miss rises from 1.1 × 10⁻⁸ at h = 0.001 to 1.2 × 10⁻⁴ at h = 0.1, a slope of 2.015; at 80° the slope is 2.016, and at 60° it is 2.023. At 90° it is never above 8.0 × 10⁻¹⁶.

Why the tilted placements have a first-order freedom at all is a classical result, quoted here rather than derived: nine bars in the $K_{3,3}$ pattern lose rank whenever their six joints lie on one conic section, and two crossing lines are a conic, at any angle. Every placement with the joints on two crossing lines therefore has rank eight. Dixon’s observation was that at a right angle, and only there, the first-order freedom extends to a motion.

“Only there” is itself a claim a measurement can test, and it could have been otherwise. The moving placements might have formed a narrow band of angles around 90°, or the miss might have fallen away much faster than the tilt.

Near perpendicular the miss is proportional to the tilt: within two degrees of 90° the log-log slope is 1.004, at 2.15 × 10⁻⁷ for each degree when $B_2$ is forced 0.01. At 89.75° it is 5.39 × 10⁻⁸, and at exactly 90° it is rounding. There is no band of moving frameworks round the right angle; there is the right angle, and a miss that starts growing the moment the lines leave it.

That is the shape the cross term predicted. The term that spoils the motion carries a factor cos α, and cos α at 90° less a small tilt ε is sin ε, which is ε to first order. So near perpendicular the obstruction should be linear in the tilt, and the measured slope of 1.004 says it is. That is a statement about small tilts and nothing more. Further out the miss grows faster than the tilt, from 2.15 × 10⁻⁷ a degree near 90° to 3.15 × 10⁻⁷ a degree at 60°, where it is 9.44 × 10⁻⁶: eight and a half times its value at 85° for six times the tilt. The cross term is not the only thing that changes as the lines close up, and nothing here separates the others.

The same reading explains why the tilted placements keep their rank of eight. The cross term changes a bar’s length only at second order in a small displacement along the first-order freedom, so the first derivatives, which are all a rank can read, are exactly those of a placement on two crossing lines at any angle. The rank belongs to the space of configurations only near one point; the push sees the space a finite distance away.

That proportionality makes the right angle the framework’s manufacturing condition, and a gentle one. The miss grows as the tilt times the square of the push. A framework built a degree off square and pushed through a small move misses by the product of a small number and the square of another. Whether clearance in nine joints can absorb that is a question about how the miss divides among them, which this essay does not settle.

What the count, the rank and the push each see #

Three instruments have been pointed at one object, and each saw something different.

The count reads bars and joints. It says nought for every placement of nine bars on six joints, because it cannot see where anything is.

The rank reads the first derivatives at one placement. It separates a generic placement, rigid at rank nine, from every placement with the joints on two crossing lines, which have rank eight and a freedom to first order. It cannot tell Dixon’s right angle from 89°.

The push reads the second order. It separates the right angle, where every forced move is taken, from every other angle, where a move of h misses by something proportional to $h^2$.

The three-crank chain was a disagreement between the first instrument and the second, and it is sometimes taken as the whole story of paradoxical mechanisms: the count is wrong and the rank is right. Dixon’s framework has that disagreement too, and then a second one, between the rank and the motion, with the rank on the wrong side of it. One freedom and four hundred links could lean on the rank because every surplus equation there was added deliberately and the motion was known. Here the rank is the instrument that needs checking.

What the nine bars leave out #

The bars cross. In the plane, several of the nine bars pass through one another as the framework moves. A physical framework needs the bars in layers, with pins long enough to reach, and whether a layering exists that lets the whole loop be traversed without collisions is not examined.

Clearance and elasticity. Every joint is a perfect pin and every bar exactly its length. What a real framework does near a tilted placement, where a small miss must be absorbed somewhere, depends on play and stiffness.

Other flexible placements. Dixon described more than one way to make these nine bars move. Only the placement on two perpendicular lines is measured here, and the claim that it is the only moving one among placements on two crossing lines rests on the push at the angles tried, not on a proof.

Forces. The dependency among the nine rows has a counterpart in the forces the bars can carry with nothing applied from outside. That is a statement about loads, and nothing here computes one.

What comes next: the right angle as a tolerance #

The push measured a miss proportional to the tilt and to the square of the move. That is half of a tolerance. The other half is how the miss is shared among nine joints with clearance, and it has a definite form: for a framework built with its lines a small angle off perpendicular, the smallest radial clearance, the same at every joint, that lets it pass through a stated travel.

Its distinct argument would be whether that clearance grows as the tilt, as the miss does, or as a different power because nine joints share it; and whether a framework whose clearance is sized for one travel can traverse the whole loop, crossings included, or jams at the first placement where a joint passes the crossing point.