One input at one end

The lazy tong is in this field for two reasons, and the first is that it is the case where counting works.

Sixteen bodies, twenty-two pins, and Grübler’s $3(n-1) - 2j$ gives four: three rigid motions of the whole assembly and one internal freedom. The rank of the constraint Jacobian gives four, with no dependency among the constraints at all. That holds at one unit and at thirty-two.

A field about mechanisms whose counts are wrong needs a control, and this is it. When the deployable ring’s count comes out at nought and the Miura sheet’s at minus ninety-nine, it will not be because the assembly is large.

What makes it easy #

Each unit is two bars pinned at their middles; consecutive units share their end pins. Adding a unit adds two bodies and three pins, so it adds $3 \times 2 - 2 \times 3 = 0$ to the count and one to the number of independent loops.

Each of those loops is a genuine quadrilateral and its closure says two independent things. There is no arrangement, no chosen angle, nothing that has to be got right to a hundredth of a degree: the mechanism works for any bar length and any pin position, and it works for the same reason at every size. That is exactly what the deployable ring is not, and the two together are the field’s two ends.

The graph, and what it does not know #

Reading the tong as a graph first is worth the paragraph, because it is the case where the graph tells the truth.

Ten bodies and thirteen pins on a five-unit tong is a circuit rank of four: four independent loops, four closure equations to find, and a mobility of $(n-1) - 2L$ in the form the loop arithmetic takes. That is nine less eight, which is one, plus the three rigid motions if nothing is grounded.

The graph does not know where any pin is. A tong drawn with the pins a third of the way along the bars rather than at their middles has the same graph and the same count, and it is a different mechanism: the span is no longer $2nL\cos\theta$, the connection lines are no longer parallel, and the assembly curves. What the graph gets right here it gets right by luck rather than by knowing anything, and the only way to find that out is to take the rank.

The reach, and why it is exact #

Take the opening angle $\theta$ to be the half-angle of the scissor. One unit spans $2L\cos\theta$ and stands $2L\sin\theta$ high, and the span of $n$ of them is

$$2nL\cos\theta.$$

Measured against the mechanism’s own solved positions over eight sizes and sixty openings, the worst departure is $10^{-13}$ — which is to say the formula is not an approximation.

That exactness is what the mechanism is bought for. A hand moving a few centimetres at the near end moves the far end by $n$ times as much, with no gearing, no ratio that varies through the stroke and no ratio that is not a number — the multiplication really is a multiplication.

It also survives being differentiated. The rate at which the span grows with the opening is $-2nL\sin\theta$: $n$ times one unit’s rate, at every opening, so the velocity ratio is the same $n$ as the displacement ratio. Very few mechanisms on this site have that property; a four-bar certainly does not.

The lines it grows along #

The tong’s other structural fact is the one that connects it to the ring, and it is a measurement rather than an observation.

The two pins joining one unit to the next lie on a line. Draw that line for every junction of the tong and extend it.

All parallel, to $10^{-15}$ radians, at every opening. That is why the tong grows in a straight line — and why no number of ordinary scissors ever makes a ring, since a ring needs its connection lines to be radii and radii are not parallel.

Bending each bar at its pivot turns those lines by the amount of the bend, which is the whole of the deployable ring. The two mechanisms differ in one number and in nothing else: the same graph, the same pins, the same count, and a wholly different assembly.

Two openings, and what stays true #

The mechanism has one freedom, so one number decides every position in it, and it is worth seeing what does and does not depend on that number.

At an opening of 0.22 radians a five-unit tong on unit bars spans 9.76 and stands 0.44 high; at 1.46 radians it spans 1.09 and stands 1.98. That is a range of nearly nine to one in reach, from one parameter, with no part of the mechanism doing anything but rotating about its own pins.

Through all of it the span is $n$ times one unit’s — that is the exactness above — and the connection lines stay parallel, which is the structural fact. Neither of those is a small-angle result or an approximation near the middle of the range; both hold at both ends and everywhere between.

What does change through the range is the rate: $-2nL\sin\theta$ runs from almost nothing at the closed end to nearly $2nL$ at the open one. So the mechanism is a constant ratio in displacement and a wildly varying one in the relation between the opening angle and the span, which is a distinction worth keeping straight and is the same one a slider-crank’s stroke and its velocity make.

Where one unit’s error goes #

Now the second reason the tong is here, and it is the one that generalises past it.

A mechanism that multiplies its input by $n$ multiplies everything by $n$. Cut one unit’s bars so that its opening angle is a hundredth of a radian away from the drawing and the span is out by that unit’s share: 0.0117 in units of $L$, at an opening of 0.62 radians, and it is 0.0117 whether the tong has two units or thirty-two.

Cut every unit that way — which is what a machine setting, a worn tool or a mistaken drawing produces — and the errors add.

The total is $n$ times one unit’s, exactly: 0.0234 at two units, 0.0936 at eight, 0.374 at thirty-two. The ratio between the two columns is 2.000000, 4.000000, 8.000000, 16.000000, 32.000000 — the unit count, to six figures, because the mechanism is linear in the openings and the errors go through the same multiplication the motion does.

The third column is what would happen if the errors were independent and equally likely either way: the accumulation would go as the square root of the count instead, giving 0.0165, 0.0331, 0.0468 and 0.0662. At thirty-two units the two columns differ by a factor of $\sqrt{32} = 5.66$.

Which column applies is not a question about the mechanism. It is a question about how the parts were made, and it is exactly the worst-case-against-statistical question this site’s practice field measures on a stack of four lengths — arriving here with thirty-two terms instead of four and with a strong reason to expect the terms to be correlated rather than independent.

That reason is the repetition itself. Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution, and the honest model for a repeated unit is the first column rather than the third.

The compensation nobody gets #

There is a hope worth killing here, because it is natural and wrong.

A tong is symmetric, so it is tempting to think that an error in one direction somewhere will be cancelled by an error in the other direction somewhere else. It will, if the errors are independent — that is precisely what the square-root column is. But the mechanism has no mechanism for cancelling anything: the span is a plain sum of the units’ contributions, and nothing in the assembly feeds one unit’s error back to another.

Compare a Peaucellier linkage, where the straightness is an algebraic identity and holds for a whole family of link lengths. There the exactness survives some errors because the identity does. Here the exactness is a sum, and a sum of thirty-two wrong numbers is wrong.

What a multiplier does to a specification #

The error arithmetic above has a practical shape worth drawing out, because it is what a repeated unit does to a drawing’s tolerances.

A designer specifies a tolerance on the finished span. On a single-unit mechanism that translates straight into a tolerance on the parts. On a tong of $n$ units it does not: the span’s tolerance has to be divided by $n$ before it reaches the part, because the part’s error is multiplied by $n$ on the way back. A tong of thirty-two units built to a span tolerance of one millimetre needs its units held to a thirty-second of a millimetre, and the same tong of two units needs a half.

That is a specification that gets harder as the mechanism gets bigger, and it gets harder linearly, which is the worst of the two available behaviours. Had the errors been independent it would get harder as the square root, and a tong of thirty-two would need its units held to a fifth of a millimetre rather than a thirty-second.

The site’s practice field has a stack-up whose worst case and statistical estimate diverge for the same reason, and the same warning applies with more force here: the assumption of independence is doing all the work, and on a repeated unit it is usually false.

There is one further wrinkle that belongs to networks and not to stacks. A tong’s error shows up as a wrong length, which is an inaccuracy. A ring’s or a sheet’s shows up in a closure that has to come back to where it started, which is not an inaccuracy at all — it is an assembly that does not go together. That difference is the subject of the tolerance essay this field adds, and it is why a mechanism with dependencies among its constraints is harder to build than one without, quite apart from being harder to count.

What the tong cannot do #

Three limits, and the first is why deployable structures are not usually tongs.

It grows in one dimension. A tong reaches; it does not open an aperture, cover an area or close a ring. Everything about that follows from the parallel connection lines, and no amount of tong makes it otherwise.

Its mechanical advantage goes the wrong way at the ends. The span’s rate of change with the opening is $-2nL\sin\theta$, which vanishes as $\theta \to 0$: fully extended, a tong moves a long way at the far end for very little at the near one, which is another way of saying that holding it extended is hard. That is a force statement and is not developed here; the geometric half of it is the sine, and the sine is on the page.

And it has no redundancy at all, which sounds like a virtue and is a plain fact with two sides. Nothing about a tong depends on a dimension being exact, so it is easy to make; and nothing about a tong is held in place by anything but its pins, so it has exactly as much stiffness as its bars do.

The unit, and the thing it is a unit of #

One last observation, and it is the one that puts the tong in a field about networks rather than in the linkages field.

Nothing above needs the tong to have any particular number of units. Every statement is either about one unit — its span, its rate, its error — or about the count of them, and there is no length at which the mechanism becomes a different one. That is unusual. A four-bar’s behaviour is a function of four lengths and changes completely when one of them changes; a tong’s behaviour is a function of one length, one angle and an integer.

So the mechanism to analyse is the unit, and the assembly is arithmetic. That is the best case a network can offer, and it is available exactly because the constraint matrix has full rank at every size: with no dependencies, nothing about the assembly is more than the sum of its units, and the sum can be written down.

The rest of this field is what happens when that is not true. A deployable ring’s behaviour is not the sum of its pairs’ — its pairs cannot even be assembled into a ring unless one angle is right. A Miura sheet’s is not the sum of its vertices’ — each vertex folds and the sheet may not. In both cases the assembly is the object and the unit is a component of it, which is the reverse of the relationship the tong has, and it is the reason the ledger’s other five rows need a rank taken and this one does not.

Gain and error amplification are one derivative #

The tong multiplies its input by the number of units, and it multiplies every error by the same number, and those two sentences are not two properties. They are one derivative read twice, and saying so makes the trade unavoidable rather than unfortunate.

The span is $2nL\cos\theta$, so its rate against the opening angle is $-2nL\sin\theta$. That is the mechanism’s gain: how much output a given input produces, and it is proportional to $n$, which is what the mechanism is bought for. It is also the coefficient by which an error in the input arrives at the output — because an error is a small change, and a small change is what a derivative multiplies.

So an actuator positioned to a hundredth of a radian drives a thirty-two-unit tong to $2 \times 32 \times L \sin\theta \times 0.01$ of span error, which is thirty-two times what it would do to one unit. The input’s resolution is divided by the gain in exactly the way the input’s travel is multiplied by it.

There is no arrangement in which one of those happens and the other does not. A mechanism whose output moves $n$ times as far as its input, for any reason, delivers $n$ times its input’s error to the output; a mechanism that filtered the error while keeping the gain would be one whose derivative had two values. That is why the trade cannot be designed around, only priced.

The same identity sorts the mechanisms this site has collected under one heading. A compound epicyclic’s reduction is its own sensitivity, squared, for the same reason. A tackle’s velocity ratio is a derivative of a length and its shortfall follows the same derivative. A difference mechanism’s gain is its conditioning. All four are the statement that a gain is a derivative and a derivative amplifies whatever is fed to it.

Which gives the tong’s design rule its final form, covering both of the essay’s error arguments at once. Manufacturing errors accumulate as $n$ because there are $n$ of them; the input’s error is multiplied by $n$ because the gain is $n$; and both are consequences of the same multiplication the mechanism exists to perform. A repeated-unit mechanism is a multiplier, and a multiplier has no way to be selective about what it multiplies.

What it is for, in this field #

Two things, and they are worth separating.

As a mechanism, the tong is the demonstration that repetition on its own produces no dependencies. Thirty-two copies of a unit, ninety-four pins, thirty-one independent loops, and a constraint matrix of full rank. Whatever makes a network’s count wrong, it is not being large.

As an argument, it is where the field’s second theme starts. A network is one unit repeated, so whatever is true of the unit is true $n$ times over — its motion, its exactness, and its error. The ring and the sheet spend that repetition on making constraints coincide; the tong spends it on reach. In both cases the interesting quantity is per-unit and the assembly is a multiplier.

That multiplication is what the next essay on tolerance is about, at the point where the multiplied thing is not a length but a closure that has to come back to where it started — which is where an accumulated error stops making a mechanism inaccurate and starts making it impossible to assemble.