Put the motor where the rocking bar is longest

A drive that goes round is an angle found how to carry Dixon’s framework all the way round its loop. The framework has its black joints on one line and its white joints on a perpendicular one, with a bar from every black to every white. Its motion is a single loop through four crossings, where one of the two joints nearest the lines’ crossing point passes through it. With those two joints, $B_2$ and $W_2$, both starting 1 from the crossing point, they sit at $\sqrt2\cos\psi$ and $\sqrt2\sin\psi$ round the loop. They are the two ends of a trammel bar. Any drive that is a single number read off the joints has a top and a bottom on that circle and folds there. What goes round without folding is an angle, and of the framework’s eighteen joint angles exactly four turn a whole revolution: the four with the bar $B_2W_2$ for one side. A motor at the steadiest of them, at $W_2$ from $W_2B_2$ to $W_2B_3$, varies its speed by 2.62 to 1.

That essay ended with a framework whose two nearest joints start at different distances, $B_2$ at 1 and $W_2$ at 1.5. It expected their ends to trace an ellipse rather than a circle, and one of the four angles perhaps to come close to stopping where the ellipse was flattest. It asked for a family of such frameworks and a search for the ratio at which no angle could be driven at a useful speed ratio.

The expected ellipse is not there, and the reason gives the whole answer in closed form.

The loop is a circle at every ratio #

In Dixon’s construction each black joint moves from $x_i$ to $\sqrt{x_i^2 + t}$ along its line and each white from $y_j$ to $\sqrt{y_j^2 - t}$ along its own. Every bar keeps its length because its squared length is $x_i^2 + y_j^2$ whatever $t$ is. The bar from $B_2$ to $W_2$ is one of those bars. So $B_2^2 + W_2^2 = x_2^2 + y_2^2$ at every point of the motion, and with $B_2$ starting at 1 and $W_2$ at $r$ the two lie on a circle of radius $\sqrt{1 + r^2}$. Writing $t = \tfrac12\big((r^2 - 1) + (r^2 + 1)\cos 2\psi\big)$ puts them at $\sqrt{1 + r^2}\cos\psi$ and $\sqrt{1 + r^2}\sin\psi$, and the bar’s direction is ψ itself.

The expected ellipse came from imagining each joint running between its own starting distance and nought. Neither does. As $W_2$ crosses the black line, $B_2$ reaches out to the bar’s whole length, $\sqrt{1 + r^2}$, well past its start, and the same holds the other way round. A bar with its ends on two perpendicular lines is the elliptic trammel at any length, and its ends trace the circle of its own length in the lines’ coordinates. Points along the bar trace ellipses; its ends never do.

That was checked on two families of frameworks round 2,880 points of each loop. In the first the whole white line is scaled, $W_2$ starting at $r$ from 1 to 3 and the other whites in proportion. In the second only $W_2$ moves, from 1 towards $W_1$’s 1.5, with every other joint where Dixon put it. On both, $B_2$ and $W_2$ stay on their circle to $10^{-12}$, every bar keeps its length to $10^{-12}$, and the trammel bar’s direction follows ψ to $10^{-6}$, the limit set by evaluating the loop’s formula a hair from its crossings. The four angles that include the bar each turn exactly once round the loop on every framework of both families.

Each angle is one radian, plus or minus a bar that rocks #

Each of the four turning angles has the trammel bar for one side, and the bar turns at exactly one radian per unit of ψ. Its other side is a bar from $B_2$ or $W_2$ to a joint further out, and that bar only rocks. So the angle’s speed is one, plus or minus the rocking bar’s rate, and the angle stops only if the rocking bar ever turns as fast as the trammel bar does.

The rocking bar turns fastest where its moving end passes through the crossing point. Take the bar from $W_2$ to $B_k$. When $W_2$ crosses the black line, $W_2$ is moving along its line at the trammel bar’s full rate, $\sqrt{1 + r^2}$ per unit of ψ, while $B_k$ is momentarily still. The bar then turns at $W_2$’s speed over its own length, which at that instant is just $B_k$’s distance from the crossing point, $\sqrt{x_k^2 + r^2}$. So the largest rocking rate is

$$m = \frac{\sqrt{x_2^2 + y_2^2}}{\sqrt{y_2^2 + x_k^2}},$$

the trammel bar’s length over the rocking bar’s length at the crossing, and the same with the roles exchanged for a bar from $B_2$. The angle’s speed runs from $1 - m$ to $1 + m$, and its speed ratio is $(1 + m)/(1 - m)$. At the two crossings of the other kind the rocking bar is momentarily still, and the angle turns at exactly one.

For every framework of both families, and every one of the four angles, the least and greatest speeds measured round the loop by differences match $1 \mp m$ to a thousandth, and they fall at the crossings. On Dixon’s own framework the formula gives back the earlier essay’s numbers: for the motor at $W_2$ with $B_3$, $m = \sqrt2/\sqrt{10} = 1/\sqrt5$ and a speed ratio of 2.62.

Dixon’s own framework, read off the formula #

The formula also reproduces the whole of the earlier census of Dixon’s framework, not just its motor. With $B_2$ and $W_2$ both at 1 the trammel bar is $\sqrt2$ long. The four rocking bars at their crossings are $\sqrt{1 + 1.5^2}$, $\sqrt{1 + 2.5^2}$, $\sqrt{1 + 2^2}$ and $\sqrt{1 + 3^2}$ long, for $B_2$ to $W_1$ and $W_3$ and $W_2$ to $B_1$ and $B_3$. So $m$ is 0.784, 0.525, 0.632 and 0.447, and the four speed ratios are 8.28, 3.21, 4.44 and 2.62. The earlier essay measured the four angles’ speeds between 0.22 and 1.78 in size, never crossing nought. Those are the extremes of the first angle, $1 - 0.784$ and $1 + 0.784$, the least steady of the four. Its motor at 2.62 was the steadiest, and now it is clear why. $B_3$ is the joint furthest from the crossing point on either line, so its bar to $W_2$ is the longest of the four.

That is also why the earlier essay could call the four turning angles “a motor’s” without qualification. Every one of them is usable on Dixon’s framework. The worst varies its speed eightfold, which a geared motor or a flywheel could absorb, but it never stops. The fourteen angles that rock are a different kind of object. Their speeds pass through nought twice a circuit whatever the proportions, because they come back to where they started. No formula of this kind applies to them, and no motor can drive them round.

What an uneven speed costs a motor #

A speed ratio of 2.62 means a motor running at a steady speed carries the framework round at a speed that varies by that factor. Seen from the framework, the motor’s angle runs unevenly against the loop’s own parameter. Seen from the motor, the framework speeds up and slows down twice a circuit, at the two crossings of the motor’s own joint. A machine driven this way behaves like a crank driving a linkage near a dead centre. Locked on purpose found toggles that exploit exactly the slow stretch for mechanical advantage, and a framework driven by its motor gains force where it moves slowly, in proportion.

So the speed ratio is a measure of how uneven the drive’s force and speed are, not of whether it works. A designer who wants an even drive puts the motor where $m$ is least. One who wants the most force at a particular point of the loop might put it where $m$ is larger and time that point to the motor’s own crossing. Either way the choice is now a reading of four bar lengths, where it was a search round the loop.

No turning angle ever stops #

The formula settles the earlier question without a search. Counting and measuring mobility found that the count calls such frameworks rigid and the measurement finds them moving; here the measurement says more, that they move evenly enough to be driven. An angle stops only if its $m$ reaches one, which needs the rocking bar’s far joint to be no further from the crossing point than the nearest joint on its line. But $B_2$ and $W_2$ are by definition the nearest joints on their lines; that is what makes them the ends of the loop’s trammel bar. So every $m$ is under one, and every one of the four angles turns round the whole loop at a speed bounded away from nought.

An angle comes close to stopping only as another joint comes nearly as close to the crossing point as the nearest one. The second family shows it happening. As $W_2$ moves from 1 out towards $W_1$’s 1.5, the angle at $B_2$ between its bars to $W_2$ and to $W_1$ has $m$ approaching one, and its speed ratio climbs from 8.28 to 42.8 at 1.4 and 433 at 1.49. At 1.5, $W_1$ and $W_2$ would reach the crossing point at the same moment, and the loop as constructed here, with $W_2$ alone changing sign there, would no longer describe the framework’s motion.

The motor goes where the rocking bar is longest #

A motor wants the angle with the smallest speed ratio, which is the one with the smallest $m$: the one whose rocking bar is longest at its crossing compared with the trammel bar. That is a rule a builder can apply by looking at the framework. Of the four candidate bars, $B_2$ to $W_1$, $B_2$ to $W_3$, $W_2$ to $B_1$ and $W_2$ to $B_3$, take the longest when its own end is at the crossing point, and put the motor on the angle it makes with the trammel bar.

On Dixon’s own framework the longest is $W_2$ to $B_3$, which is why the earlier essay found its motor there. Scale the white line and the answer moves. Its joints get further out, so the bars from $B_2$ to the whites grow almost in proportion to $r$, while the bars from $W_2$ to the blacks grow only as the trammel bar does and their ratio to it approaches one. The angle at $B_2$ to $W_3$ becomes steadier than the angle at $W_2$ to $B_3$ when $6.25r^2 + 1 = r^2 + 9$, at $r = 1.234$. At that crossover both have a speed ratio of 2.92. Beyond it the angle at $B_2$ to $W_3$ keeps improving, to 2.56 at $r = 2$ and 2.44 at $r = 3$. The angle at $W_2$ to $B_3$ worsens to 4.27 and 6.85. Across the whole family the best speed ratio a motor can have stays between 2.3 and 3.

Moving $W_2$ alone towards $W_1$ shows the other side of the rule. The angle at $B_2$ to $W_1$, whose rocking bar becomes as short as the trammel bar, slows without limit. The other three hardly notice. The steadiest, at $W_2$ to $B_3$, goes from 2.62 to 3.31 at 1.49, because the trammel bar is a little longer than before. So the earlier essay’s worry has a precise answer. There is no ratio at which a framework stops being one a motor can drive, short of the ratio at which its nearest joints stop being uniquely nearest. Up to that point some angle always runs at a speed ratio near three.

Why the angles do not care about the crossings’ kind #

The earlier essay found its motor’s slowest and fastest points both at $W_2$’s crossings, and its two other crossings at exactly one. With the formula that is no coincidence. An angle at $W_2$ has its rocking bar running from $W_2$ to a black joint. That bar turns fastest when $W_2$ is moving fastest along its line, at $W_2$’s crossings, and is momentarily still when $W_2$ stops, at $B_2$’s crossings. An angle at $B_2$ does the same with the roles exchanged. So each angle sees only one kind of crossing, its own joint’s, and at the other kind it runs exactly at the trammel bar’s rate.

A push turns back at the crossing found the push on $B_2$ folding at $W_2$’s crossing and the side drive at $B_2$’s, each drive blind to one kind of crossing. The angles have the same asymmetry in a milder form: they do not fold at their own crossings, but that is where their speed varies most. A builder choosing between an angle at $B_2$ and one at $W_2$ is choosing which pair of crossings the drive will be most uneven at.

What a builder can take from it #

The loop is a trammel at any proportions. For any Dixon framework of this kind, the two nearest joints go round a circle whose radius is the length of the bar between them, and that bar turns once per circuit. Nothing about the proportions changes the drive’s topology, only its evenness.

Read the motor’s place off the bars. Each of the four turning angles varies its speed by $(1 + m)/(1 - m)$, with $m$ the trammel bar’s length over its other side’s length at the crossing. The motor goes on the angle whose other side is longest. No search round the loop is needed.

Keep a second joint away from the crossing point. An angle slows towards stopping only when the joint at the far end of its rocking bar is nearly as close to the crossing point as the nearest joint on its line. A framework built with two joints nearly equally close has one angle that is nearly useless as a drive, and three that are still good.

What the circle does not settle #

The price round the loop. The price is one self-stress and the earlier drive essay priced the play a framework as built needs to be driven round. That price was computed only for Dixon’s own proportions. Whether it grows as the nearest joints move apart, and whether the motor’s best place for evenness is also its best place for play, is not measured here.

Frameworks whose lines are not perpendicular. Every framework here has its two lines at a right angle, which is what makes the trammel’s ends a circle in the lines’ coordinates. Nine bars that ought to be rigid found Dixon’s motions with the lines at other angles too. There a bar of fixed length between the two lines is a tilted ellipse in the joints’ coordinates, and the rates here would need to be redone.

Joint clearance. The clearance that makes the framework generic found that a framework as built moves only through its joints’ play. A turning angle’s speed near a crossing depends on where in its play the framework sits, and the formula here is for the exact framework.

Torque. A speed ratio is kinematic. The torque a motor needs at each point of the loop depends on the load the framework drives and how the bars carry it, and a slow point of the angle is a point of mechanical advantage, not only of unevenness.

Still open: the motor and the play together #

The motor’s best place depends on which rocking bar is longest, and the play a framework as built needs to go round depends on how its errors load the self-stress at each point of the loop. The two are set by different things and could point to different joints.

The distinct argument would be both measured on the same frameworks. For the scaled family at a few ratios, build each framework with its joints out of place, drive it round with a motor at each of the four turning angles in turn, and find the least play each needs to get round. The measurement would ask whether the angle with the steadiest speed is also the one needing least play, or whether a motor placed for evenness asks more of the bars than one placed elsewhere. If they agree, the rocking-bar rule is the whole design rule. If they disagree, a builder chooses between a drive that runs evenly and one that tolerates the framework’s errors, and the numbers would say what each costs.