The lamp needs three springs

A spring that must push is hung in the wrong place settled which sign each spring takes on 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, and a head on the upper arm’s end. Two springs stand in for a parallelogram balanced it exactly, at every pose, with one parallelogram and four springs of no free length. The essay on signs found that the upper arm’s spring pulls exactly when the head spring meets the head beyond a threshold, $G_3L_2/G_2$, where $G_2$ and $G_3$ are the gravity coefficients of the upper arm and head and $L_2$ is the upper arm’s length. It ended with a rule: hang every spring that reaches the head at or beyond the head’s centre of mass, and every spring pulls on every lamp.

A rule about signs says nothing about size, and a builder buys springs by size. A spring of no free length pulls with its stiffness times its length, so over the lamp’s working range its largest pull is its stiffness times its longest stretch. That peak pull is what sets the coil, the pulley it hides behind and the bracket it hangs from. The question left open was the obvious one. Among the designs that balance the lamp with springs that all pull, inside the lamp’s own outline, which has the smallest springs? Does the best design sit on the threshold or beyond it, and can a design with no parallelogram and six springs ever do better?

The answer is that the smallest design uses three springs, not four. Its head spring sits exactly on the threshold, short of the centre of mass the rule named. The reason is simple enough to carry to any exact spring balance. Beyond the threshold, the upper arm’s spring and the head spring together are one spring split in two, and splitting a spring cannot make it smaller.

What a spring costs #

Every design here balances the lamp exactly, so they differ only in their springs. The lamp’s weight puts the same moment on each joint whatever the springs are, so it costs nothing to compare. What differs is how hard each spring has to pull to supply that moment, and that depends on where its ends are.

The lamp in the figure is the one the earlier essays used. Its springs were placed to make the balance easy to show, not to make them small. The lower arm’s spring runs from a bracket 0.07 m above the base pivot to a point 0.1 m along the lower arm. The upper arm’s spring runs from the elbow bracket, held upright by the parallelogram, to 0.09 m along the upper arm. A head spring runs from the elbow bracket to the head, and a wrist spring runs from near the upper arm’s end to the head. Their peak pulls add to 162.3 N, and the wrist spring’s 61.7 N is the largest.

Two measures of size are worth separating. The total of the peak pulls is what the springs cost together: the coil wire, the pulleys and the brackets they load. The largest single pull is what the stoutest bracket must carry, and on a desk lamp it is often what decides whether a design fits. The two need not agree, and here they do not.

The search is over eight attachment points, two for each spring, kept inside an outline. Brackets are at most 0.15 m high. Arm attachment points lie along their arms, an attachment past the wrist reaches at most 0.08 m past the wrist, and attachments on the head lie along the head’s 0.15 m. For each set of attachment points the stiffnesses come from closed forms the earlier essays derived. Each spring’s longest stretch over the working range is then measured on a grid of poses, so the totals in this essay are measured over the lamp’s real range, not estimated from a formula. A design that asks for a negative stiffness is excluded. A pattern search from seven starts, one of them the lamp as built, finds the least of each measure.

The lower arm’s spring is fixed by its bracket #

The springs are nearly independent, which makes the search easier to read than its eight dimensions suggest. The lower arm’s spring depends on its own two attachment points only. Its stiffness is $G_1/(a_1b_1)$, where $G_1$ is the gravity coefficient of everything the lower arm carries, $a_1$ is the base bracket’s height and $b_1$ is how far along the arm the spring meets it.

The lower arm swings 60° either side of upright. Its spring is longest at either end of that swing, where the arm leans 60° away from the upright bracket, and its length there is $\sqrt{a_1^2 + b_1^2 - a_1b_1}$. The spring’s peak pull is $G_1\sqrt{a_1^2 + b_1^2 - a_1b_1}/(a_1b_1)$. For a fixed bracket that pull is least where $b_1 = 2a_1$, and the least is $(\sqrt{3}/2)\,G_1/a_1$.

The closed form has one consequence that the search confirms on its own. The attachment on the arm should be twice the bracket’s height, and the bracket should be as tall as the outline allows. With a 0.15 m bracket the spring meets the arm at 0.3 m and pulls 22.7 N at its peak, against 49.8 N as built. The search found exactly that attachment point from every start, and the model’s check holds it to the closed form to one part in a million.

Nothing else in the lamp affects this spring. A taller base bracket is the only way to make it smaller, so this is the first place a designer who is short of room should look. It is also the spring that will turn out to set the largest pull in the best designs. Its 22.7 N is a floor that no arrangement of the other three springs can lower.

The upper arm’s spring is the one that disappears #

The head spring and the upper arm’s spring are coupled through a single attachment point, $b_3$, the distance along the head at which the head spring meets it. The head spring’s stiffness is $G_3/(a_3b_3)$, so it falls as $b_3$ grows. The upper arm’s spring’s stiffness is $(G_2 - G_3L_2/b_3)/(a_2b_2)$, so it rises as $b_3$ grows. That is the threshold the earlier essay found: below it the upper arm’s spring would have to push, and at it the spring has no stiffness at all.

The search for the least total puts $b_3$ exactly on the threshold, 0.0615 m from the wrist on a head whose centre of mass is at 0.08 m. The upper arm’s spring then has no stiffness and is not needed. The model’s check removes it and balances the lamp again with the other three alone, by a separate route: the stiffnesses that make the lamp’s total energy flattest over its range. The energy stays flat to better than a billionth of the head’s own moment.

The three springs left are each at most half the size of the largest spring as built. The total falls from 162.3 N to 63.4, a factor of 2.6. Most of that comes from using the outline: taller brackets and longer reaches lower every stiffness, as the lower arm’s spring showed. The rest comes from the threshold, and that part needs explaining. A rule about signs would have put the head spring further out, and it is not obvious why stopping short should help.

The head spring now does two jobs. It balances the head about the wrist, as before. Because it leaves the elbow bracket, which the parallelogram holds upright, it also pulls on the upper arm through the wrist joint, and at the threshold that pull is exactly what the upper arm needs. The upper arm’s own spring has nothing left to do.

The wrist spring’s stiffness does not depend on $b_3$. The closed form gives it as the head spring’s stiffness times $L_2b_3$, divided by its own attachment points, and the $b_3$ cancels. Its 17.5 N is set by its attachment points at the outline’s limits, a reach 0.08 m past the wrist and 0.15 m along the head.

Beyond the threshold a spring is split in two #

What happens if the head spring is moved out from the threshold, towards the centre of mass the rule named? The head spring’s stiffness falls and its peak pull falls with it, from 23.3 N to 11.1 at the end of the head. The upper arm’s spring comes back, from nothing to 12.2 N. The two changes very nearly cancel.

With the upper arm’s spring anchored at the wrist, the total stays at 63.44 N to within 0.002 N across the whole head. That flatness is not a coincidence, and it explains the whole result. At the least total, both springs leave the elbow bracket at the same height, 0.15 m, the outline’s limit. The upper arm’s spring reaches the wrist, which is also a point on the head’s line. So there are two springs of no free length running from one point to two points on one line.

Two such springs from a common point add exactly to one spring. Each pulls with its stiffness times the vector from the common point to its other end. Their sum is the combined stiffness times the vector to a weighted point between the two ends. Put the closed forms in and the arithmetic is short. The combined stiffness is $G_2/(aL_2)$, whatever $b_3$ is. The weighted point lies $k_3b_3/(k_2 + k_3)$ along the head, and that is $G_3L_2/G_2$: the threshold.

So beyond the threshold, the two springs together are the threshold spring itself, split in two. The model’s check confirms the identity to twelve figures at three head attachments. Splitting a spring cannot reduce its peak pull. The two halves pull in slightly different directions, so the sum of their peak pulls is at least the peak pull of the single spring they replace. Here the angle between them is small and the excess is two thousandths of a newton. That is why the curve is flat and not strictly rising: the extra spring buys nothing and costs a spring.

When the two springs do not share an attachment point, they no longer add to one spring. Attachment point the upper arm’s spring 0.2 m along the upper arm instead of at the wrist, and every step beyond the threshold costs force, ending 2.7 N worse at the head’s end. Choose different bracket heights for the two and the same happens. Only at the threshold does none of this matter. There the upper arm’s spring has no stiffness, so where it would have been anchored makes no difference, and the total is 63.44 N whatever its attachment points.

The earlier rule, hang the head spring at the head’s centre of mass, is therefore safe but not small. At the centre of mass, with the shared attachment point, the total is the same 63.44 N but uses four springs. With the upper arm’s spring anchored 0.2 m up its arm, it is 64.5 N. The rule was written to make every stiffness positive, and it does. The threshold is the stronger statement: the least total is reached where one stiffness is exactly zero, which is the boundary the sign rule kept away from.

The largest pull is set by the base #

The least largest pull leads to a different design. The lower arm’s spring cannot go below 22.7 N with a 0.15 m bracket, so no design’s largest pull can either. The question is whether the other springs can all be brought under it, and they can.

At the threshold the head spring pulls 23.2 N, just over the floor. Moving it out brings it down, but brings back the upper arm’s spring. Any head attachment that leaves both under 22.7 N gives the same largest pull. The search stopped at 0.137 m, but the least is a plateau, not a point. The wrist spring’s attachment points also moved, trading its pull against the others under the same ceiling.

This design costs 73.1 N in total, 9.6 more than the three-spring design, to save 0.6 N on the largest spring. The three-spring design is the better buy. A spring balanced against the load tested a spring’s placement by a linear program and found that whether a spring can do a job at all is decided by where it is hung, before its stiffness is chosen. The same is true here of how large it must be. Its largest spring is 23.2 N, only 2.6% above the floor the base bracket sets, and it does without a spring and a bracket. The two measures agree on what matters: the base bracket sets the largest pull, and the threshold sets how many springs are needed.

Every lamp’s threshold is its best place #

The lamp here has a head heavier than its upper arm, 0.5 kg against 0.3. Is the threshold best only for this ratio? A light head moves the threshold close to the wrist, and a heavy one moves it out towards the centre of mass without reaching it. If the threshold’s advantage depended on where it fell, it might fail at one extreme.

It does not fail. For heads from a tenth of the upper arm’s mass to ten times it, a scan of 300 head attachments from the threshold outwards finds the least total of the two coupled springs exactly at the threshold. The threshold moves from 0.013 m to 0.076 m, a factor of almost six, and the best place moves with it. The split-spring identity does not use the masses, so this is what it predicts. The scan finds it at all five ratios.

Six springs are never smaller #

The other way to balance the lamp has no parallelogram and six springs. Three run from the base to each of the lower arm, upper arm and head; one from the lower arm to the head; one across the elbow; one across the wrist. Without a parallelogram there is no upright bracket at the elbow, so springs that act on the head have to start from the base or the lower arm. More springs share the weight, and it is tempting to think each can then be smaller.

They are not. The same search over the six-spring design’s twelve attachment points finds a least total of 213.0 N and a least largest pull of 87.7 N. Its lower arm spring is smaller than the parallelogram design’s, because other springs carry part of the lower arm’s load. But the springs that reach the head from the base are long, and long springs of no free length pull hard. The wrist spring alone is 87.7 N, larger than the parallelogram design’s three springs together.

The no-parallelogram design also pins its wrist spring against the outline. Its least total and its least largest both stop at 87.7 N, with the wrist attachment points at their limits. A larger outline would help it. Inside the same outline it is three times the parallelogram design on total and almost four times on its largest spring. A parallelogram carries an angle is the reason: it gives the head spring an upright bracket close to the head, so the spring is short. No arrangement of springs from the base can match that.

The rule, sharpened #

The lamp that stays where it is put showed that springs of no free length can balance a linkage exactly. The essay on signs gave a rule that keeps every spring pulling. This one turns that rule into a design for the smallest springs, in three steps.

Make every bracket as tall as the outline allows, and meet each arm at twice the bracket’s height. The lower arm’s spring then reaches $(\sqrt{3}/2)\,G_1/a_1$, and that sets the largest pull of any design that keeps it.

Hang the head spring at the threshold, not at the centre of mass. There the head spring balances the upper arm as well as the head, and the upper arm’s spring is left out. Beyond the threshold, a second spring is only the first split in two.

Prefer the parallelogram. It lets the head spring start close to the head, which keeps it short.

The second step generalises past lamps. In any exact balance with springs of no free length, two springs that share an attachment point and end on one rigid line add to a single spring. A design that uses both where one would do has added a spring and gained no strength. The search found the three-spring design without being told about the identity, and the identity explains why it was found. Exact costs more than close found nothing between approximate straight-line linkages and exact ones: adding bars to an approximation does not walk it towards exactness. Here the balance is exact from the start, and adding a spring to an exact balance buys nothing at all.

What this does not settle #

Springs with free length. Every design here is exact only for ideal springs. A real spring has a free length, and a length error is undone by its own size measured how far that spoils a single arm. The split-spring identity uses the fact that springs of no free length add as vectors, and springs with free length do not. A real three-spring lamp may lose more balance than a four-spring one, and that is not measured here.

The cost of a spring. The total of peak pulls is a stand-in for cost. A real price has a fixed part for each spring, its pulley and its bracket, and that favours the three-spring design more than the totals show. It also has a part that grows with stretch, which this measure ignores. A spring is also not the only way to balance a weight. The horsehead is a drum meets a pumping unit whose crank carries counterweights instead, and leaves their sizing as open as this essay leaves a spring’s price.

Friction. Why a hinge works treats joints with friction. A lamp balanced to within its friction stays put, which would let a designer trade a little balance for smaller springs. A spring sized with its friction in mind is a different optimisation.

The outline. The outline here is a stated box, and every best design pushed some attachment point against its edge. A different outline gives different numbers, though the threshold and the split-spring identity would be unchanged.

Still open: a balance that tolerates a free length #

The three-spring design is the smallest for ideal springs, but it is also the design with no spare spring. When a real spring’s free length spoils the balance, the four-spring design beyond the threshold has a spring whose stiffness can be adjusted to absorb part of the error, and the three-spring design does not.

The distinct argument would put a stated free length on every spring, as a pulley’s offset does, and measure the worst unbalance over the working range for each design on the plateau beyond the threshold. The measurement would say whether a split spring, useless for ideal springs, is worth its cost once springs have free length. It would also say whether the best place for the head spring moves off the threshold when the springs are real, and in which direction.