A free length is absorbed where the springs attach

The lamp needs three springs found the smallest springs that balance a desk lamp with a tilting head. The lamp has a lower arm on a base, an upper arm on the lower arm’s end, kept company by a parallelogram, and a head on the upper arm’s end. With springs of no free length it can be balanced exactly at every pose. The smallest such design uses three springs whose peak pulls add to 63.4 N. Its head spring meets the head at a threshold, 77% of the way to the head’s centre of mass, where the upper arm’s own spring has nothing left to do.

That essay stopped at a doubt. Beyond the threshold the upper arm’s spring comes back, and for ideal springs it is the head spring split in two, which buys nothing. But a real spring is not ideal. Springs of no free length are made by pre-stretching a coil and hiding its free length behind a pulley or a lever, and the hiding is never perfect. A four-spring design has a stiffness the three-spring design lacks, and a spare stiffness looks like exactly what a designer would want when the springs fall short of the ideal. The question is whether it is.

The answer, measured on the same lamp, is no, and the reason points somewhere else. A free length’s damage can be absorbed, but not by a spare spring. It is absorbed by a spring whose length hardly changes over the lamp’s range, and getting such springs means moving the attachment points. That works, and it is expensive.

Where a free length goes #

The lamp that stays where it is put took the problem apart for one arm. A spring of free length $\ell_0$ stretched to length $L$ stores $\tfrac12 k(L-\ell_0)^2$. That is the ideal spring’s $\tfrac12 kL^2$, less $k\ell_0 L$, plus a constant. The balance was built so that the $\tfrac12 kL^2$ terms cancel gravity at every pose. The term $-k\ell_0L$ is left over, and it changes as the spring’s length changes. On one arm, with the stiffness left at its ideal value, it leaves a torque that is exactly $\ell_0/s$ of the weight’s torque, where $s$ is the spring’s current length. The error is worst where the spring is shortest.

A designer is not obliged to leave the stiffness at its ideal value, though. Once the springs have free length, the natural thing is to choose the stiffnesses again, for the flattest total energy over the whole working range. That refit is a least-squares problem: the energy is linear in the stiffnesses, so making it as flat as possible is a small linear solve. The lamp’s worst unbalance is then the largest moment the refitted energy leaves on any joint, measured anywhere in the range. The range is the one the earlier essays used: the lower arm 60° either side of upright, the upper arm from 10° past upright to 150°, and the head from 60° to 200°.

With 2 cm of free length on every spring the refit brings the three springs to 96, 45 and 86 N/m: lower arm, head spring and wrist spring, against 87, 43 and 76 for ideal springs. Their peak pulls rise from 63.4 N in total to 64.8 N. And the lamp is no longer balanced. Its worst moment is 0.0931 N·m. Against the largest moment gravity puts on any joint, 3.40 N·m, that is 2.7%, which sounds small. The comparison that matters is with the head. The head’s own weight acts at 0.08 m from the wrist with a moment of 0.39 N·m, so an unbalance of 0.07 N·m at the wrist is a sixth of what holds the head down.

The unbalance is linear in the free length #

The first thing to measure is how the unbalance grows. Each free length from nought to 2.5 cm gets its own refit, and each gives a worst moment.

The answer is a straight line. At 0.5, 1, 1.5, 2 and 2.5 cm the worst moments are 0.0221, 0.0450, 0.0686, 0.0931 and 0.1184 N·m, close to 4.65 N·m for every metre of free length, and the model’s check holds the ratio steady to within ten per cent across the range. Nothing about the shape of the problem changes as the free length grows. It is one error, scaled.

That fixes how to read the rest of this essay. Every number at 2 cm can be halved for 1 cm or doubled for 4. The free length also has a ceiling of its own. The shortest any spring gets in the threshold design is the wrist spring at 7 cm, and no spring can be shorter than its own free length. A design with 2 cm of free length is therefore asking its shortest spring to sit, at its shortest, at three and a half times that length. That is an ordinary proportion for a coil hidden behind a pulley.

The straight line also says something about the cure. If the damage were dominated by one awkward pose or one spring near its shortest, it would grow faster than linearly as the free length approached that spring’s length, and it does not. The damage is spread across the range, and anything that cures it will have to work across the range too.

Two joints carry it everywhere #

Where the unbalance sits in the range says what kind of error it is.

Two of the three joints carry the same worst moment at every angle of the lower arm. The upper arm’s joint sees 0.093 N·m and the head’s 0.070 N·m, whatever the lower arm is doing. That is the parallelogram at work, the same one two springs stand in for a parallelogram weighed against extra springs. It keeps the elbow’s bracket upright, so the head spring, hung from that bracket, sees only the upper arm’s and the head’s angles. The lower arm drops out of their arithmetic entirely, and so it drops out of their error too.

The lower arm’s own joint behaves differently. Its leftover moment is nothing with the arm upright and rises to 0.073 N·m at either end of its 60° swing. The arm’s spring is symmetric about upright, so an upright arm feels no net moment from the spring’s error. At the ends of the swing it feels the most.

So the error is not a local defect at one bad pose. It is spread over the whole working range of two joints, and the parallelogram gives it the same structure it gives the balance. The upper arm and head form one problem and the lower arm another.

The refit makes up the average pull #

The refit does something definite to each spring, and seeing what it does explains what it cannot do.

Every spring gets stiffer, and in proportion to the free length. At 2 cm the lower arm’s spring is 10.7% stiffer than ideal, the head spring 5.4% and the wrist spring 12.5%. The reason is in the energy. A free length takes $k\ell_0$ of pull away from a spring at every length. Making the spring stiffer by a fraction $\varepsilon$ adds $\varepsilon kL$ of pull. If the spring’s length stayed near one value $\bar L$, those would cancel at $\varepsilon = \ell_0/\bar L$, so the refit should stiffen each spring by about its free length over its mean length.

It does, closely. The wrist spring’s mean length is about 15 cm, so 2 cm of free length predicts 13%, and the refit gives 12.5%. The lower arm’s spring averages about 20.5 cm, predicting 10%, against 10.7%. The head spring averages about 34.5 cm, predicting 5.8%, against 5.4%. The model checks all three to within fifteen per cent of the estimate. The springs are refitted together, so the match is not exact, but the mechanism is plainly this one.

What the refit cannot make up is the rest. Write the lost energy $-k\ell_0L$ near the mean length and the part a stiffness change can cancel is the part that goes like $L^2$. What remains, after the best $\varepsilon$, is of the order of $k\ell_0(L-\bar L)^2/\bar L$. The leftover is first order in the free length, which is the straight line above, and second order in how far the spring’s length swings about its mean. A spring that stays one length can lose any amount of pull to its free length, and a refit gives all of it back. A spring that swings from short to long cannot be repaired by any stiffness at all.

That is a sharper statement than the one-arm result. $\ell_0/s$ is the error at the ideal stiffness, and it makes the shortest pose look like the culprit. After a refit, the shortest pose matters only as one end of the swing.

The fourth spring finds nothing to do #

Now the question the earlier essay asked. At the threshold the upper arm’s spring was dropped because, for ideal springs, it had no stiffness to take. With free length it can be put back and given a stiffness of its own, refitted with the rest.

The refit takes it, barely. The fourth spring comes out at 1.1 N/m and a peak pull of 0.5 N, the other three adjust by a few per cent, and the worst moment goes from 0.0931 N·m to 0.0930. The least-squares refit flattens the energy, which is not quite the same as making the worst moment smallest. So the worst moment was also minimised directly, as a linear program: every moment at every sampled pose is linear in the stiffnesses, and the program asks for the stiffnesses, all of them non-negative, that make the largest of those moments least. That gives 0.0874 N·m for three springs and 0.0853 for four. The minimax answer beats least squares by about 6%, and the fourth spring buys 2.4% on top of it.

The argument about swings says why. The upper arm’s spring runs from the elbow bracket to a point 0.35 m along the upper arm, and over the range its length goes from 0.20 m to 0.49 m, a swing of ×2.38. That is no better than the springs it would be helping. A spare stiffness is only useful if its spring’s lost pull has a different shape from the error, so that adding a little of it cancels some of the error. A spring that swings as widely as the others adds an error of the same kind as the one it was meant to remove.

The plateau turns into a slope #

The other way to use the spare spring is to move the head spring beyond the threshold, where the four-spring design is not merely allowed but needed. For ideal springs that was a plateau. Every attachment beyond the threshold balances exactly, and with the head spring split in two the total pull did not get smaller anywhere along it. With free length it no longer has to be flat, and it might tilt in the designer’s favour.

It tilts the other way. At 1 cm of free length the worst moment rises from 0.0450 N·m at the threshold to 0.0479 at 0.15 m, and at 2 cm from 0.0930 to a little over 0.098, levelling off near the end of the head. No attachment beyond the threshold comes back down to the threshold’s value. The model checks that at seventeen attachments for each free length.

So the earlier essay’s rule survives real springs. With free length or without, the head spring belongs at the threshold. The split spring is not a reserve worth carrying. It was useless for ideal springs, and for real ones it is mildly harmful, because moving the head spring outward lengthens the head spring’s swing and brings back an upper arm spring that swings just as far.

Short swings are what the attachment points buy #

If no stiffness can repair a swinging spring, the remaining freedom is the attachment points themselves: where each spring is fixed and where it meets what it pulls. The search that found the three-spring design was rerun for real springs. It kept the same outline as before: brackets at most 0.15 m high, attachments along their arms, a wrist attachment at most 0.08 m past the wrist, and head attachments along the head’s 0.15 m. It started from the threshold design, used a pattern search on the worst moment of the least-squares refit, and refused any design that asks a spring to push.

It finds a design that leaves 0.0611 N·m at 2 cm of free length, against 0.0931. That is a third less, and the model’s check requires at least a quarter. The design moves almost every attachment point. The lower arm’s bracket drops from 0.15 m to 0.12 m. The head spring’s bracket drops from 0.15 m to 0.074 m, and its attachment on the head stays almost where it was, at 0.060 m. The upper arm’s spring keeps a stiffness too small to matter. The biggest change is the wrist spring. Its attachment on the upper arm moves from 0.08 m past the wrist to 0.01 m past it, and it still meets the head at 0.15 m.

Every one of those moves shortens a swing. The lower arm’s spring goes from a swing of ×1.73 to ×1.43. The head spring goes from ×3.84 to ×2.19. The wrist spring goes from ×3.28 to ×1.14: hung between two points a centimetre and fifteen centimetres from the wrist, it is a nearly constant 14 to 16 cm whatever the head does. The model checks that every working spring’s swing is smaller in the re-chosen design. The search was told nothing about swings. It was told only to minimise the moment, and it found the mechanism the energy argument predicts.

What a short swing costs #

The table above gives the price. The re-attached springs are 122, 0, 91 and 1,394 N/m, and their peak pulls add to 266 N, four times the 64.8 N of the threshold design and four times the ideal 63.4. Almost all of the increase is the wrist spring, which pulls 195 N at its longest.

That is not an accident of the search. It is what a short swing means for a spring that still has to balance something. A spring of no free length balances gravity through the change in $\tfrac12 kL^2$ across the range, and the change in $L^2$ between its shortest and longest is what the stiffness multiplies. A spring that stays nearly one length changes its $L^2$ very little, so it needs an enormous stiffness to change its energy by the same amount. Its pull at its longest is that enormous stiffness times its whole length, because a spring of no free length pulls in proportion to all of its length, not just its stretch.

So tolerating a free length and keeping springs small pull in opposite directions, and for the same geometric reason. A spring is small when its length swings widely, so that a modest stiffness changes its energy a lot. It is tolerant when its length hardly swings, so that a stiffness change can make up whatever pull its free length takes. The three-spring design sits at one end of that trade and the re-attached design at the other. A third of the unbalance costs a fourfold increase in pull, and that is a poor rate for a desk lamp.

Choosing the measure of unbalance #

Two measures of unbalance have appeared, and they deserve to be kept apart, because they answer different questions. Least squares flattens the total energy over the range, which treats a lamp that drifts slowly across a wide region the same as one that tips sharply in a small one. The linear program minimises the largest moment on any joint at any pose, a worst case with corners in it wherever the joint carrying the largest moment changes, as in a gap with corners in it, which is what decides whether joint friction can hold the lamp still. A lamp with 0.09 N·m of worst moment and 0.1 N·m of friction in every joint stays where it is put. One with 0.11 N·m creeps.

A calibration is a synthesis with more equations uses least squares because measurement errors average out. Here nothing averages out: a user feels the worst pose, not the average one, and the minimax measure is the honest one. It was solved only at the headline designs, though. As a linear program over every sampled moment it is large, and it needed rescaling and a working set of constraints, grown until no sampled moment exceeded the bound, before it solved reliably. The sweeps and the attachment point search use least squares, and at the headline designs the two measures agree on every comparison this essay makes: the fourth spring buys little under both, and the threshold design leaves about 6% more by least squares than the minimax optimum.

Exact costs more than close made a related point about straight-line mechanisms. An approximate linkage is not a step towards an exact one, and the two live on different branches. A spring balance with free length is in the same position. It is not an exact balance spoiled a little, and the exact design is not the right starting point for the approximate problem. The exact design’s attachment points were chosen for small springs, and small springs are exactly the ones a free length hurts most.

A practical reading #

For someone building a lamp, three things follow. First, the free length should be hidden as completely as the budget allows, because the unbalance is strictly proportional to whatever remains. A pulley offset of half a centimetre leaves a quarter of the moment that 2 cm would.

Second, the stiffnesses should be refitted for the real springs rather than taken from the ideal formulas. Raising each spring by about its free length over its mean length is the whole of the refit to a good approximation, and it is cheap: 1.4 N of extra pull in total at 2 cm.

Third, the spare spring should be left out. It cannot help, and the place it would need beyond the threshold makes things worse. If the remaining unbalance is still too much, the cure is joint friction or a better-hidden free length. Moving attachment points to shorten the swings works, but at a cost in pull that the springs and brackets of a desk lamp are unlikely to bear.

This connects back to a spring that must push is hung in the wrong place, where the threshold first appeared as a question of signs. It became the place for the smallest springs, and now it is also the most tolerant place for the head spring among the designs that keep the others’ attachment points. One point on the head turns out to answer three different questions.

What this does not settle #

One free length for every spring. Each spring here got the same free length. In a real lamp the pulley offsets differ, and a spring whose free length is hidden less well would want attachment points that give it a shorter swing more than its neighbours do. The proportionality holds spring by spring, so the measurement would extend directly.

The search found a good design, not the best. The pattern search on attachment points started from the threshold design and found one local least. A design with a smaller worst moment for the same total pull may exist elsewhere in the outline, and the trade between pull and unbalance was measured at two points only: the threshold design and the re-attached one.

Friction was named, not modelled. Whether a lamp with 0.09 N·m of worst moment stays put depends on its joints’ friction. That is what turns the unbalance into a lamp that holds or one that creeps, and it is a property of the bearings rather than of the springs.

Still open: the price of a balance in pull #

The two designs measured here are two points on a trade-off. One has small springs and leaves 0.0931 N·m, and one has springs four times larger and leaves 0.0611. Between them, and beyond them, lies a frontier: for every allowed total of peak pulls, the least worst moment any attachment points inside the outline can reach.

The distinct argument would trace that frontier. It would minimise the worst moment with the total peak pull capped, at a run of caps from 64.8 N upwards, and with the linear program rather than least squares. It would say how much balance the first few newtons of extra pull buy, whether the frontier is steep near the small design, which would make a slightly larger spring set a bargain, and which attachment point moves first as the cap rises. It would also say whether the wrist spring, which carried nearly all of the re-attached design’s cost, is the right place to spend pull at all.