A constant pull holds the band for less

The arm belongs to the poses left the rocking slide with one spring and one price. The slide meets five prescribed poses of a moving body only with play in its guide, and only if its block bears on alternate walls, + − + − +, which is a matter of which wall the load picks. A single spring of no free length can put it there, hung from a fixed attachment point to a point on an arm below the body, never from the body itself. To keep those walls for every working load from nought to the design load, that essay found, the spring would need a pull of about a hundred times the load.

It ended on a guess about the cure. A spring with free length pulls with a constant term along its line as well as with its stiffness. A constant force at a body point is what the earlier certificates had ruled out at the body’s own points, but not at the arm’s, so the constant term might buy back much of the hundredfold pull.

The measurement below says no, for a reason that runs deeper than the certificates. It also finds a cheaper spring of a different kind, and it finds that the hundred was never the right number to start from.

A spring is a tension along a line #

Start with what a spring does to the slide. Hung from an attachment point $G$ to a body point $X$, it pulls the body towards $G$ along the line joining them. Whatever its law, it acts as a single force of some tension $T$ along that line. At pose $i$ the body point is at $P_i$ and the crank pin at $A_i$, and the spring’s moment about the crank pin is the tension times the signed distance from the pin to the line:

$$M_i = T_i\,\delta_i, \qquad \delta_i = \frac{(P_i - A_i)\times(G - P_i)}{|G - P_i|}.$$

The block bears on the wall that moment sends it to. So everything about which wall the spring chooses is in the sign of $\delta_i$, because a spring that only pulls never has a negative tension. The law of the spring enters only through how large $T_i$ is at each pose.

A spring of no free length has $T_i = k|G - P_i|$, a tension in proportion to its length. A spring of free length $\ell_0$ has $T_i = k(|G - P_i| - \ell_0)$, the same tension less a constant $k\ell_0$. A coil wound with initial tension is the opposite case, the same tension plus a constant. All three are one family, $T_i = k|G - P_i| + \tau$, with the constant $\tau$ negative for a free length and positive for an initial tension. The question the earlier essay asked is whether letting $\tau$ be nonzero lowers the price.

With the attachment point held, the tension at every pose is linear in $k$ and $\tau$, and so is every moment and every force component. So finding the lightest spring at one attachment point is a small linear program, as it was for springs of no free length, with one more variable. The attachment point then has to be searched for, because $\tau$ multiplies a direction that depends on the attachment point in no linear way.

Five lines cut out the attachment points #

Before any price is computed, the sign condition above has a consequence of its own. The sign of $\delta_i$ is the sign of $(P_i - A_i)\times(G - P_i)$, and that is linear in the attachment point $G$. At each pose the attachment points that give the wall needed form a half-plane, bounded by the line through that pose’s crank pin and that pose’s image of the body point. Five poses give five half-planes, and the attachment points from which any pulling spring can choose all five walls are their intersection, a convex polygon.

For the lightest arm point, $(-0.60, -1.20)$ in the body’s frame, the polygon is a sliver about 0.31 units wide and 0.52 tall, with its corners at $(3.27, 2.46)$, $(2.96, 2.14)$, $(2.96, 1.94)$ and $(3.11, 2.03)$. Only four of the five lines bound it. The first pose’s line passes well clear of this region, and every attachment point near the others puts the first pose’s wall right with room to spare, a lever of 0.23 against a few hundredths at the other poses.

Nothing in that construction mentions stiffness, free length or load. That is the reason the earlier essays’ wedge of arm points never moved. A body point can carry a spring that chooses the walls exactly when its attachment polygon is not empty. If it is not empty, a spring attached inside it can be scaled up until it overpowers any fixed load. Run over the same grid of the body’s frame, 0.15 apart and six units square, the polygon is not empty at exactly the 155 points the arm belongs to the poses found by solving its program at each one, and at no point of the body.

So the certificates were right, but they were a special case of a simpler fact. Every spring that pulls, of any law, is a tension along a line, and the line decides the walls. A free length changes the tension and leaves the line alone, so it cannot open the body, and nothing else that pulls along a line can either. The body is shut out by geometry: none of its points has five images whose lines from any single attachment point weave across the five crank pins in the right order.

The hundred was a scaling, not a price #

The earlier essay’s hundred came from a different calculation from the one the question needs. It designed a spring against no load, with a margin of 0.25 at every pose, and then asked how far a working load could rise before a wall flipped. That design keeps its walls up to 0.0915 of the unit working load, and pull and band scale together, so a spring scaled up until its band reached the full load would pull about 101.6 times the load.

But a spring meant to hold a band should be designed for the band. Since each pose’s moment is linear in the working load, the walls hold for every load between nought and the design load exactly when they hold at both ends. The program can simply be asked for that. It keeps the margin of 0.25 with no load, as before, and requires the walls only to hold at the design load, which is how the earlier essay measured the edge of a band, so the two prices compare.

Designed for the band directly, a spring of no free length holds the band from nought to the design load with a pull of 51.96, about half the scaled figure. The two agree only where they should. For a band of a tenth of the load the scaled design needs 10.2 and the direct one 9.5, because a design for no load is very nearly a design for almost no load. By a quarter of the load they have parted, 25.4 against 16.3, and from there the direct design’s price grows at about 47.6 units of pull for each further unit of band against the scaled design’s 101.6.

The difference is where the attachment point sits. The no-load design’s attachment point was chosen to clear every pose by the margin at no load, and scaling it up keeps that attachment point. A design for the band can move its attachment point towards wherever the working load does its damage, and it does. The arm belongs to the poses found that the band is set by the third pose, where the working load most opposes the wall needed, with a moment about the crank pin of 2.73. The band’s design moves the attachment point so that the spring’s line has a longer lever about the third pose’s pin.

So the earlier essay’s rule, a hundred units of pull for every unit of band on a unit arm, overstated the price of a wide band by about a factor of two. It is still right that a band costs pull in proportion to its width, but the rate is about 48, not 102. The question the earlier essay actually asked was whether a free length would cut the price. That question now applies to 52, not to 100.

Offered a free length, the design takes none #

At the lightest arm point the program was run three ways for the band. In the first the spring has no free length. In the second it has a true free length, a constant term no greater than nought. In the third it has any constant term at all.

With a free length allowed, the design takes none. Its price for the band is 51.96, exactly the price without one, with the same attachment point at $(3.09, 2.04)$ and the same stiffness, 14.04. The same is true with no working load at all: 9.30 either way. A free length is available to the design, and making it anything but nought only makes the spring dearer.

With any constant term allowed, the design takes the other sign and drops the stiffness entirely. The lightest spring for the band is a pull of 44.63 that does not change with length, from an attachment point at $(3.10, 2.05)$, almost the same place. Its largest force component, the measure of pull used throughout these essays, is 44.61, fourteen per cent less than the 51.96 of the spring of no free length. For no working load the same thing happens, 8.25 against 9.30.

That answers the question in an unexpected direction. A spring with free length does not lower the band’s price, but a spring with the opposite of a free length does. Taken all the way, it becomes a spring whose pull does not depend on its length at all. Such springs exist: a coiled strip spring, a weight on a cord over a pulley at the attachment point, or a gas spring at the flat part of its curve. Each delivers close to a fixed tension over its working stroke.

The design was tested by running it through the whole of the crank’s travel, not only at the five poses the program saw. The mechanism was solved at four hundred steps, with the constant pull acting along the line from the attachment point, and the wall read off the sign of the moment about the crank pin at each step. At no working load, at half the design load and just inside the band’s top, the block takes + − + − + with exactly four changes of wall, and the spring never pushes. The band’s top itself is its edge: there the third pose’s moment is exactly nought, which is what holding the walls with no margin means.

Three poses need the same tension #

Why a constant pull, and why never a free length, is easiest to see pose by pose. At a given attachment point, each pose needs a tension of at least the margin, less whatever help the load gives, divided by the lever $\delta_i$. A spring of a given law then gives some tension at each pose, and its pull is set by the largest.

At both designs’ attachment points, three poses bind: the second, third and fourth. The first needs almost nothing, 1.1 units, because its lever is long, 0.23. The fifth needs about 24. The middle three need much more, because the attachment point sits at the polygon’s corner where the second and fourth poses’ lines cross. Their levers there are only a few thousandths, 0.004 and 0.006 for the spring of no free length.

That corner is not an accident of the search. The third pose is where the working load most opposes its wall, and its lever is what the band’s price depends on most. At the polygon’s centre the third pose’s lever is 0.030 and it needs 91.9 units of tension. At the corner its lever is 0.062 and it needs 43.9. Moving towards the corner lengthens the third pose’s lever and shortens the second’s and fourth’s. The attachment point stops where the three of them need tensions the spring can supply with the least waste.

What the spring can supply is where the two designs differ. A spring of no free length has a tension in proportion to its length, and from the attachment point its lengths at the second, third and fourth poses are 4.11, 3.12 and 2.93. Its attachment point is chosen so that the three poses’ needs fall in that ratio: 57.7, 43.9 and 41.0. Those are exactly the tensions it gives there, 57.7, 43.9 and 41.1, so three poses sit on their walls with nothing to spare. But the spring must also give the first pose a tension of 69.0, because that is where it is longest, and the first pose needed 1.1.

A constant pull gives the same tension everywhere. Its attachment point is chosen so that the three binding poses need the same tension: 44.6, 44.6 and 44.6. It gives exactly that at every pose, wasting nothing on the second pose for being the longer of the three and nothing on the first for being longest of all. That is the whole saving. A spring that follows its length is forced to pull hardest at the pose that needs it least, the first, because that is where it is longest. It must also pay more than the others at the second pose of the binding three, for the same reason.

The same reading says why a free length can only hurt. A free length makes the tension rise with length faster than in proportion, since a spring of free length $\ell_0$ gives $k(L - \ell_0)$. The binding poses’ lengths differ by more than a third, so a steeper profile forces the attachment point to a worse compromise between them. A free length helps only when the poses that need the most tension are also the ones where the spring is longest. At these five poses that is never the case, because the longest pose is the one the walls hardly care about.

Across the wedge, the same answer #

One arm point could be a coincidence of where its poses fall. So the comparison was repeated at every other grid point of the wedge within two units of the body, eighteen points in all, from 0.32 to 1.92 below the body’s outline.

At all eighteen, a true free length lowered the price by nothing, and a constant term lowered it by between 6.7 and 29.1 per cent, every time with the stiffness at nought. The saving is largest near the body, where the wedge narrows and the arm points are expensive anyway: 29% at 0.32 below the outline, where the band costs 606 units without a constant pull and 430 with one. It is smallest at the cheap points far down the arm, 6.7% at 1.92 below, where the band costs 34.6 and 32.3.

The pattern fits the reading by poses. Far from the body the spring is long and its length changes little across the five poses, so a tension in proportion to length is already nearly constant, and there is little for a constant pull to fix. Near the body the spring is short, its length swings by a larger share, and the wasted tension at the long poses is a larger part of the price.

A designer looking for a cheap spring is therefore pushed two ways at once. The cheapest arm points are far below the body, which the arm belongs to the poses already found, and a long arm is awkward to build. A constant-force spring lowers the price everywhere and most where the arm is short. On an arm a unit long the band from nought to the design load costs about 44.6 units with a constant pull; on one two units long, about 32.

What the spring’s law is for #

This connects to two other places where these essays balanced loads with springs. The lamp that stays where it is put and two springs stand in for a parallelogram needed springs of no free length, because cancelling gravity exactly at every pose needs an energy that is a quadratic in position, and only a spring whose tension is in proportion to its length supplies one. A free length is absorbed where the springs attach then found that a free length is damage to be repaired, and that repairing it means short swings in length.

The rocking slide’s spring has a different job. It must leave a moment of the right sign at each of five poses, with room to spare, and nothing about the size of that moment needs to follow any law. So the spring of no free length, which was the only spring that could balance the lamp, is merely one choice here, and not the cheapest. The best spring for a sign problem is the one whose tension is flattest, because the needs at the binding poses are made equal by choosing the attachment point. Whatever the spring adds to one pose beyond the others is waste.

Read back into the synthesis that set the problem, the result belongs with what a synthesis assumes it knows. The five poses decide where any spring may hang, through the polygon, and that decision is fixed before any spring is chosen. Five is already one more pose than a slide can meet exactly, as how many points may be prescribed counted, and the play that pays for the fifth is what the spring exists to manage. The spring’s law decides only the price of hanging it there. A designer who wants to change where the spring can go has to change the poses, and one who wants to change what it costs has to change the spring.

The limits of the comparison #

The pull is a largest force component. These essays have measured a spring’s pull as the largest component of its force along either axis of the frame, so that the numbers here compare with the earlier ones. For the constant pull that is 44.61 against a tension of 44.63, and the order of every comparison is the same if tension is used instead.

A constant pull is an idealisation too. A strip spring or a weight on a cord holds its tension only over its working stroke, and the spring here must run from its shortest length at the fourth pose, 2.93, to its longest at the first, 4.92. That is a stroke of two units at a fixed tension. A gas spring would add friction, and a weight would add inertia, neither of which the measurement includes.

Eighteen points, one band. The comparison across the wedge used every other grid point within two units of the body and one band, from nought to the design load. The reason given for it, that the longest pose needs the least, is a reading of this slide’s poses, not a theorem about all of them.

Still open: how close to the corner a real attachment point can go #

The lightest attachment points for the band sit 0.014 units from the edge of a polygon a third of a unit wide, at the corner where the second and fourth poses’ lines cross. There those poses’ levers are a few thousandths, and a spring whose line passes that close to a crank pin is choosing its wall by a hair. Moving the constant pull’s attachment point by up to 0.02 in any direction left the walls right at half the design load. But nothing here says what an attachment point that can only be placed to within some tolerance costs.

The distinct argument would be the price of a tolerance on the attachment point. It would require the walls to hold for every attachment point in a disc of radius $\varepsilon$ about the nominal one, a family of constraints that can be written for each pose as the polygon shrunk inwards by $\varepsilon$. It would measure how fast the band’s price rises as $\varepsilon$ grows from nought to a tenth of a unit. The answer would show whether the corner the lightest designs crowd into is a usable design or only a limit. If the price rises steeply, a practical spring sits well inside the polygon and pays for it. If it rises gently, the constant pull’s saving survives a realistic bracket, and so does the halving of the hundred.