The first pull goes to the spring the lamp did without

A free length is absorbed where the springs attach gave every spring of the smallest balanced desk lamp 2 cm of free length and measured what that costs. The lamp has a lower arm on a base, an upper arm kept company by a parallelogram, and a tilting head, balanced by springs. The free length left the lamp with 0.093 N·m of unbalance at its worst pose. No stiffness could repair it, and the spare fourth spring that ideal springs had no use for recovered almost none of it. Moving the attachment points could absorb a third of it, by giving each spring a shorter swing in length. The design that did so needed 266 N of peak pull against the smallest design’s 65.

That essay had two points on a trade-off and nothing in between. It asked for the curve: for every allowed total of peak pulls, the least worst moment any attachment points inside the lamp’s outline can reach, with the moment minimised directly rather than by least squares. It also asked which attachment point moves first, whether the first few newtons are a bargain, and whether the wrist spring, which carried nearly all of the expensive design’s pull, is the right place to spend it.

All of those now have answers, and the last one is no.

The measure, and the cap #

The lamp, its range and its outline are the earlier essays’. The lower arm swings 60° either side of upright, the upper arm from 10° past upright to 150°, and the head from 60° to 200°. The brackets are at most 0.15 m high, and each attachment point lies along the link that carries it. There are four springs: one for the lower arm, one for the upper arm on the parallelogram’s bracket, one from that bracket to the head, and one across the wrist. Every spring has 2 cm of free length.

For any choice of attachment points, the moment each joint feels at each pose is linear in the four stiffnesses. So is each spring’s peak pull, its stiffness times its longest stretch over the range. That makes “the least worst moment with the total peak pull at most $P$” a linear program in the stiffnesses and the bound. It is the minimax program of the earlier essay with one more row, so that without the cap it is that program exactly. The worst is taken over every joint at 512 poses spread through the range.

The attachment points enter that program in no linear way, so each cap gets a search over the eight of them. The caps are walked upwards from the smallest the lamp can meet. Each search starts from the attachment points found for the cap below, which traces one connected family of designs and shows the order in which the attachment points move.

The program has five unknowns and some three thousand rows, one for each sign of each joint’s moment at each pose. Solved directly, a tableau that deep sometimes failed to close, and two solutions of the same design disagreed. Solved through its dual, which has five rows, it is quick and stable. The stiffnesses are then recovered from the dual’s binding moments, priced over every row, and required to leave exactly the dual’s value. They do at every point of the frontier, to a millionth.

The first newtons buy most of it #

The frontier falls steeply and then flattens. At 62 N, the least the springs can pull and still balance the lamp at all, the worst moment is 0.0856 N·m. Ten newtons more bring it to 0.0735, and 80 N to 0.0687. After that it eases down, to 0.0620 at 100 N, 0.0575 at 140 and 0.0529 at 200. Past 227 N nothing more can be bought: the best attachment points inside the outline leave 0.0521 N·m, and a design allowed more pull does not use it.

So the whole available improvement is from 0.0856 to 0.0521, and half of it, down to 0.0687, costs 18 N on top of the smallest design’s 62. The other half costs the next 147. Measured as moment bought per newton, the first ten newtons buy 1.2 thousandths of a newton-metre each. The last thirty buy three hundredths of a thousandth, about forty times less.

The dashed line is the comparison that matters most. Hold the smallest design’s attachment points where they are and let the stiffnesses alone take more pull. Then the least worst moment drops from 0.0874 at 62 N to 0.0854 at 62.2 N, and stays there however much pull is allowed. All of the improvement comes from moving the attachment points. Stiffer springs at the same places pull harder at every pose, but they cannot change the shape of the error, which is what the free length damaged.

The earlier designs sit off the frontier, and where they sit says something about the measure. The threshold design refitted by least squares, 64.8 N and 0.0931, is well above the curve. Minimising the worst moment directly at the same attachment points gives 62.2 N and 0.0854, almost on it. The re-chosen design found by least squares, 266 N and 0.0611, is far to the right of it. The frontier reaches 0.0611 at about 106 N, two-fifths of the pull. But the same attachment points with their stiffnesses chosen by minimax give 0.0528 at 256 N, close to the curve’s end. The earlier search found good places to attach the springs. What it did not find was the right stiffnesses, because it judged them by the flattest energy rather than by the worst moment.

Every joint is left with the same worst #

One feature of every design on the frontier is easy to miss and explains a good deal of it. The least-squares refit of the smallest design left the joints unequal: 0.093 N·m on the upper arm, 0.070 on the head, 0.073 on the lower arm at the ends of its swing. That is the shape two springs stand in for a parallelogram led one to expect, since the parallelogram makes the lower arm one problem and the upper arm and head another. On the frontier, at every cap from 62 N to 227 N, all three joints are left with exactly the same worst moment, to the fourth decimal: 0.0856 on each at 62 N, 0.0620 on each at 100 N, 0.0521 on each at the end.

That is the mark of a minimax answer. A design that left one joint with less than the others would have slack it could trade, a little more pull on the spring that helps the worst joint at the expense of the joint with room to spare. The optimum is where no such trade remains, and the worst is shared. It has the same structure as the worst case in a gap with corners in it, a maximum taken over several things at once, whose least sits at a corner where several of them are equal rather than where any one is flat. Exact costs more than close found Chebyshev’s straight-line linkage working on the same principle, its error swinging equally either side of the line.

It also says why least squares was the wrong judge for the earlier design. Flattening the total energy spreads the error by area, not by height, and lets one joint carry more of the worst than it needs to. A user of the lamp meets the worst pose of the worst joint, not the average. Least squares is right for a calibration, where many measurements each carry an error that averages out, and wrong for a balance, where nothing averages out.

The lower arm shares the worst too, and it does so without more pull. Its spring takes 22 N throughout, but its attachment moves out along the arm, from 33.7 to 37.3 cm, and its stiffness falls from 82 to 74 N/m. Its error follows the others down by reshaping rather than strengthening, which is the dashed line’s lesson in miniature.

The spare spring comes back first #

Where the pull goes as the cap rises answers the earlier essay’s last question.

The lower arm’s spring takes 22 N at the smallest design and 22.6 N at the largest. It is alone on its joint, the parallelogram keeps the upper arm and head out of its arithmetic, and the free length’s damage to it is small. Every newton the cap adds goes to the other three springs.

The first newtons go to the upper arm’s spring. In the smallest design it pulls nothing at all. That is the threshold the lamp’s design is built on: a spring that must push is hung in the wrong place found that with the head spring attached 77% of the way to the head’s centre of mass, the upper arm’s spring has no work left, and the lamp needs three springs built the smallest design around dropping it. At 64 N it pulls 2.1 N, at 72 N 14.1, at 100 N 28.7, and at 140 N 49.0, more than any other spring. Meanwhile the spring from the elbow bracket to the head gives pull back, from 23.3 N to 13.2 by 75 N.

The wrist spring grows slowly at first, from 16.4 N to 21.7 N by 75 N and 34.9 N at 100. It takes the largest share only from 140 N on, reaching 148.6 N at the frontier’s end. By then the upper arm’s spring has given some back, down to 27.8 N.

So the wrist spring is the right place to spend pull, but only late, and only once the cheap balance is already bought. The design the earlier essay found at 266 N was spending at the far end of the frontier, where each newton buys little. A designer who can afford 80 or 100 N has a much better use for it, and it is the spring the threshold design did without.

The head spring moves past the threshold #

The spare spring cannot simply be switched back on at the smallest design’s attachment points. The essay on free length tried exactly that and found it bought 2.4%. What changes along the frontier is where everything hangs.

Two attachment points move first and together. The upper arm’s spring drops down its bracket, from 15 cm at the smallest design to 11.1 cm at 64 N, 6.1 cm at 80 N and 4.0 cm at 100 N. The head spring’s attachment on the head moves out, from 5.7 cm to 10.8 cm by 75 N, past the threshold at 6.15 cm. At the smallest design it sits just inside the threshold, where the upper arm’s spring would have to push.

Those two moves are one decision. With the head spring attached beyond the threshold, the upper arm needs a spring of its own again. For ideal springs that spring was the head spring split in two and bought nothing. That is the plateau the lamp needs three springs found beyond the threshold, where every attachment balances exactly and the split spring costs as much as the spring it replaces. The free-length essay found that, with the other attachment points held, moving out along the plateau made things worse. Here the other points are free. The upper arm’s spring can be hung low on its bracket, where its length barely changes, and that changes the trade.

The other two attachment points move later. The wrist spring’s attachment on the upper arm comes in from 43 cm, eight centimetres past the wrist, to 36 cm, one past it. The head spring’s attachment on the elbow bracket stays at the bracket’s top until 90 N and then comes down to 8.8 cm. The remaining four attachment points sit on the outline’s edge throughout, or move by less than four centimetres.

Pull goes where it shortens a swing #

The free-length essay found why moving attachment points helps. A spring with free length loses the same pull, its stiffness times the free length, at every pose. A refitted stiffness makes up that loss on average, and what it cannot make up grows with how far the spring’s length swings across the range. A spring that stays one length can absorb any free length; one that swings from short to long cannot be repaired.

Along the frontier that mechanism decides where every newton goes. The upper arm’s spring, hung low on its bracket, has a swing of 2.38 at the smallest design and 1.24 at 100 N. By 170 N it is 1.11, a spring whose length changes by a tenth over the whole range of the lamp. The wrist spring’s swing falls from 3.28 to 1.48 at 100 N and 1.14 at the end. The springs that take pull are the springs whose swings shrink as they take it.

The head spring is the exception, and the exception confirms the reading. As its attachment moves out past the threshold its swing grows, from 3.70 to 6.20 at 75 N. That is the widest any spring swings anywhere on the frontier. And it is exactly the spring that gives pull back. The design lets the worst-swinging spring do less and moves its work to springs that hardly swing. Only when the cap is large enough to stiffen the wrist spring a great deal does the head spring take pull again, and its swing comes back down to 2.59.

The lower arm’s spring needs no such care: its swing is 1.5 from start to finish, already short, and it takes the same pull all along.

The price of a short swing is the one the free-length essay identified. A spring of no free length balances through the change in its stored energy, which goes with the change in its length squared. One whose length barely changes needs a large stiffness to store the same change, and it pulls in proportion to all of its length, not just its stretch. The upper arm’s spring at 140 N has a stiffness of 141 N/m, and the wrist spring at 227 N has 1,062 N/m. A short swing is bought with stiffness, and stiffness at a long spring is pull.

Six designs on the frontier #

The ledger gives a designer the choice in a form that can be built. At 72 N, ten newtons above the smallest, four springs of 78, 35, 27 and 112 N/m leave 0.0735 N·m, fourteen per cent less than the smallest design. At 100 N they leave 0.0620. At 140 N, 0.0575, with the upper arm’s spring the largest. Past that, the wrist spring stiffens from 354 N/m towards a thousand, and the last thirty newtons of pull buy less than a thousandth of a newton-metre.

Whether any of that matters depends on the bearings. The lamp that stays where it is put relied on joint friction to hold a lamp whose balance is not quite exact. A lamp with 0.06 N·m of worst moment holds where the joints offer more than that, and the frontier says how much pull each step down in the friction that is needed costs. The cheapest step is the first: the drop from 0.086 to 0.069 costs 18 N, a little less than the pull of the lower arm’s spring alone.

This also changes the practical reading the free-length essay gave. It advised leaving the spare spring out, and with the attachment points of the smallest design that is right. But the smallest design is one point on the frontier, and the spare spring is out only there. A lamp designed for real springs from the start, with a modest pull budget, wants the upper arm’s spring back, hung low on its bracket. It also wants the head spring moved out past the threshold, which ideal springs made pointless.

What the frontier rests on #

One free length, one outline. Every spring has 2 cm of free length, and the attachment points are confined to the earlier essays’ outline. The free-length essay found the unbalance proportional to the free length at fixed attachment points, but the frontier’s designs move, and whether the whole frontier scales with the free length was not measured.

A connected family, not a proof of the best. Each cap’s design is the end of a local search started from the cap below. A search started afresh from the smallest design’s attachment points at 100 N ended within a twentieth of a per cent of the frontier, 0.06197 against 0.06195. At the large caps a second search, started from the earlier essay’s re-chosen design, was raced against the first, and the first won. A design better than the frontier somewhere else in the outline is not ruled out.

Peak pull is a proxy for size. The cap is on the sum of the springs’ peak pulls, as in the earlier essays. A real spring’s size also depends on its stroke, and the springs that take pull here are those whose length barely changes, so their strokes are short. A cap on stored energy rather than pull would favour them even more, and was not run.

Still open: whether a bracket is worth more than a spring #

Along the frontier, two attachment points press against the edge of the outline from the first newton to the last. The upper arm’s spring wants to hang lower on its bracket than the 2 cm the outline allows by 170 N, and the head spring’s bracket starts at its highest, 15 cm. The outline was chosen for a desk lamp’s proportions, not derived, and the frontier’s cheapest moves are exactly the ones it stops.

The distinct argument would be the frontier’s sensitivity to the outline itself. It would repeat the capped search with the brackets allowed to 0.20 and 0.25 m and the lowest attachment on the bracket allowed to nought. It would ask how far the whole curve moves, and in particular whether its steep first part gets steeper. If a few centimetres of bracket buy as much balance as fifty newtons of pull, the lamp’s proportions are the cheaper place to spend. That would turn the balance problem into a question about the shape of the lamp rather than the size of its springs.