Six things a balancing spring is not

The lamp that stays where it is put began a run of essays about springs that balance a mechanism. A desk lamp whose arms and head stay wherever they are left is not held there by stiff joints. It is balanced: its springs store exactly the energy its weight gives up, at every pose, so there is no moment for the joints to resist. The essays that followed sized the springs, found where they must be hung, gave them free length, and priced the balance in pull. A second run, on the rocking slide, used a spring for a different purpose: to put a block on the right wall of its guide at five poses.

Springs that balance things have gathered a body of confident statements, and six are collected here. Each is what a careful designer might conclude from one part of the problem, and each has been measured in those essays and found false, incomplete or misapplied. Like six things a hold is not, the six fail in a common way, which is the last section’s subject.

One: more springs share the load #

A lamp with three joints might be balanced by one spring for each, or by more. The intuition for more is the one behind a shelf with more brackets: if several springs share the weight, each carries less, and the springs can be smaller. A lamp with six springs and no parallelogram should be a lamp of small springs.

The lamp needs three springs searched every attachment point inside the same outline for both designs. The one-parallelogram design reaches a least total of 63.4 N with three springs, the largest of them 23.2 N. The six-spring design reaches 213.0 N at its best, and its largest spring alone pulls 87.7 N, more than the three springs of the other design together.

The six springs do not share a weight, because a balancing spring does not hold up a weight. It cancels a term in the lamp’s energy. With no parallelogram, the springs that cross the elbow and the wrist each bring in terms in the angles between links. Two springs stand in for a parallelogram found that each such term must be taken away again by another spring. Most of the six springs are cancelling each other. The parallelogram carries the vertical out to the elbow, so the springs see the angles the weight sees, and nothing has to be cancelled twice.

The arithmetic is worth seeing once. A weight on an arm puts into the lamp’s energy a term in the cosine of that arm’s angle from the vertical, and nothing else. A spring of no free length between two links puts in a term in the cosine of the angle between those links. If the links are the ground and the arm, that is the weight’s own term, and one spring cancels it. If the links are two arms, it is a term in their relative angle, which no weight produces, and it must be cancelled by yet another spring across the same joint. Without a parallelogram, every spring that reaches past the lower arm brings such a term with it. The six-spring design is three springs doing the balancing and three undoing what the first three introduced.

Two: a spring that must push wants a compression spring #

Solve the balance equations for a lamp with a heavy head and a light upper arm, and the upper arm’s spring sometimes comes out with a negative stiffness. The natural reading is that this lamp needs a compression spring at that place, one that pushes instead of pulling, and that a design with only extension springs cannot balance it.

A spring that must push is hung in the wrong place found that the sign belongs to one attachment point, not to the lamp. The upper arm’s spring pulls exactly when the head spring meets the head beyond a threshold. That threshold is always short of the head’s centre of mass: 77% of the way on this lamp, and no more than 99% even for a head fifty times the arm’s mass. Moving the head spring’s attachment from 0.03 m to 0.08 m along the head takes the upper arm’s spring from −434 N/m to +95 N/m, with the balance still exact. A negative stiffness is a message about where a spring is hung. Hang the head spring at its centre of mass and every spring pulls on every lamp.

The reason the sign moves with the attachment is that the head spring does two jobs. Hung from the elbow’s upright bracket, it balances the head, and it also brings a term into the upper arm’s balance. The closer to the wrist it meets the head, the stiffer it must be to balance the head, and the more it brings to the upper arm. Hung close to the wrist it brings more than the upper arm’s own balance can absorb, and the upper arm’s spring would have to take the excess away, which is a push. Hung further out it is softer and brings less, and the upper arm’s spring makes up the rest, which is a pull. The threshold is where the head spring brings exactly what the upper arm needs. A compression spring at the upper arm would work on paper. But it is a sign that the head spring is hung too close to the wrist, and the cheaper repair is to move it.

The threshold turned out to matter for more than signs. It is also where the springs are smallest: the least total of peak pulls in One is reached with the head spring hung exactly there, where the upper arm’s spring has no work left and can be dropped.

Three: a stiffer spring balances better #

A real spring is not a spring of no free length, and a lamp built with real springs is left slightly unbalanced. The natural repair is to give it stronger springs. More pull, the reasoning goes, should hold the lamp more firmly against whatever the free length disturbs.

The first pull goes to the spring the lamp did without measured exactly that exchange. With 2 cm of free length on every spring and the smallest design’s attachment points held, the least worst moment any stiffnesses can leave is 0.0874 N·m at 62 N of total peak pull and 0.0854 from 62.2 N upwards. It stays there however much pull is allowed. Stiffer springs pull harder at every pose, but they leave the shape of the error as it was.

The reason is in what a free length does. It takes the same pull, its stiffness times the free length, from a spring at every length, and that loss has a shape across the lamp’s range: it tilts the lamp most where the spring’s length departs most from its average. A stiffer spring with the same free length loses more pull in exactly the same shape, and the refitted stiffnesses rearrange the loss without changing its shape. The minimax program at fixed attachment points therefore has a floor, 0.0854 N·m here, set by the geometry alone.

Let the attachment points move as the pull rises and the same pull buys a great deal: 0.0620 N·m at 100 N, 0.0521 at 227 N, the most any attachment points inside the outline can reach. Every bit of that improvement came from moving the springs, none from stiffening them in place. And it came at a sharply falling rate: the first ten newtons above the smallest design bought forty times as much balance per newton as the last thirty. The first thing they bought was a spring the smallest design had dropped, the upper arm’s, hung low on its bracket.

Four: a refitted stiffness repairs a free length #

The more careful version of Three is not to stiffen the springs indiscriminately but to refit them: choose each spring’s stiffness again, for the real spring, so that the lamp’s energy is as flat as it can be. If a free length takes pull away, a refit should give it back.

A free length is absorbed where the springs attach found that a refit gives back exactly the part of the lost pull that does not change with the pose. A free length $\ell_0$ takes $k\ell_0$ of pull from a spring at every length. The refit stiffens each spring by about its free length over its mean length, 10.7%, 5.4% and 12.5% for the three springs at 2 cm, and that cancels the average loss. What it cannot cancel grows with how far the spring’s length swings across the range. A spring that stays one length can lose any amount of pull and have it all refitted; one that swings from short to long cannot be repaired by any stiffness.

So the repair is in the attachment points, which set the swings. Re-chosen for real springs, they cut the wrist spring’s swing from ×3.28 to ×1.14 and leave 0.0611 N·m instead of 0.0931. The refit is still worth doing, and it is cheap: 1.4 N of extra pull at 2 cm. It just does not do what the statement says.

The measure of the damage makes the point sharply. Write the lost energy near a spring’s mean length and the part a stiffness change can cancel is the part that goes like the square of the length; what is left over is of the order of the free length times the square of the swing, divided by the mean length. The leftover is first order in the free length, which is why the unbalance grew in exact proportion to it, 4.65 N·m for every metre at the smallest design. It is second order in the swing, which is why the attachment points, which set the swings, are the repair.

Five: least squares finds the balance #

The obvious way to fit the stiffnesses is to make the lamp’s total energy as flat as possible over its range, by least squares. It is a small linear solve, it uses every pose, and a flatter energy sounds like a better balance.

Least squares flattens the energy by area, and a user of the lamp meets the worst pose of the worst joint. Fitted that way the smallest design leaves its joints unequal: 0.093 N·m on the upper arm, 0.070 on the head, and on the lower arm 0.073 at the ends of its swing and nothing in the middle. Minimising the worst moment directly, as a linear program, leaves 0.0874 with the same three springs, and at the re-chosen attachment points the difference is larger still. There least squares leaves 0.0611 N·m and the minimax refit 0.0528, at about the same pull.

The frontier essay found the signature of the right measure. At every design on the frontier, the minimax answer leaves all three joints with exactly the same worst moment, to the fourth decimal. A fit that leaves one joint with less than the others had slack it could have traded to the worst one. A calibration is a synthesis with more equations uses least squares for good reason, since measurement errors average out. Nothing averages out in a balance.

The lower arm’s curve shows least squares at work. Its leftover moment is nothing with the arm upright and largest at the two ends of its swing, because the lower arm’s spring is symmetric about upright and its error is too. Least squares, weighing every pose equally, is content to leave the ends large if the middle is small. The two joints beyond the parallelogram carry the same leftover at every angle of the lower arm, 0.093 and 0.070 N·m, because the parallelogram keeps the lower arm out of their arithmetic. Neither of those features is wrong in itself. What is wrong is that a lamp user, who meets the worst pose, would have been better served by a fit that traded some of the middle for the ends.

Six: the best spring is one of no free length #

The lamp essays turn on springs of no free length, which are made by pre-stretching a coil and hiding its free length behind a pulley. They are the only springs whose energy is exactly a quadratic in position, and so the only ones that can balance a weight at every pose. It is tempting to conclude that they are the best balancing spring, full stop.

That is true where the job is to cancel a weight. A constant pull holds the band for less found it false where the job is different. The rocking slide’s spring must leave a moment of the right sign at each of five poses, with room to spare, so that its block rides on alternate walls of its guide. For that job the spring’s tension at each pose is all that matters, and a spring of no free length is forced to pull hardest where it is longest, which there is the pose that needs it least. A true free length, which steepens the same profile, lowered the price of the band at none of eighteen arm points. A pull that does not vary with length lowered it at all eighteen, by 7 to 29 per cent: at the lightest arm point, 44.6 units of pull against 52.0.

The polygon the rocking slide’s essay found makes the point from the other side. Any spring that pulls along a line, whatever its law, can choose the slide’s walls only from attachment points inside a polygon cut out by five half-planes, one a pose. So the law of the spring decides nothing about where it may hang, and a spring of no free length has no special standing there. It is one tension profile among many, and at the lightest attachment point the profile that wastes least is the flat one.

So the ideal spring is ideal for one task. A designer choosing a spring should first ask whether the mechanism needs an energy cancelled, which wants a spring of no free length, or a set of signs kept, which wants the flattest tension available.

The way they fail #

All six statements treat the spring as the thing to be sized: more of them, stiffer, compression instead of extension, refitted, fitted in the flattest way, made ideal. And all six are settled somewhere else.

Five of the six are settled by where the spring is hung. The parallelogram decides how many springs a balance needs. A threshold along the head decides whether a spring pulls. Attachment points, not stiffness, decide how much a free length costs, and they decide it through the swing, which is a fact about geometry. On the rocking slide the attachment points any pulling spring may use are cut out by five half-planes, one a pose, before any spring is chosen. The sixth is settled by how the error is judged, worst case or average, and that is a decision about the user, not the spring.

The common error is the same one six things a hold is not found in holds: reading the strength of an element where the geometry decides. A balancing spring is a force along a line between two points, and the two points do most of the work. The spring’s size follows once they are chosen. Choosing the size first and the points afterwards is how a lamp ends up with 162 N of springs where 63 would do, as the lamp of the lamp needs three springs was first built.

The order the six imply #

Read together, the six give a designer an order of decisions, and it is the reverse of the order the statements assume.

First, the structure. Decide whether the springs can see the angles the weight sees, a parallelogram on a lamp or the right links on any other mechanism. That decides how many springs the balance needs, and the wrong structure multiplies them.

Second, where each spring hangs. Thresholds, attachment points and swings settle whether every spring pulls, how small the springs can be and how much a free length costs. On a sign problem like the rocking slide, the attachment polygon settles where a spring may go at all.

Third, how the error is judged. If a user meets the worst pose, fit by the worst moment, not the flattest energy.

Fourth, the spring’s law. A weight to cancel wants a spring of no free length, with its free length hidden as well as the budget allows. A sign to keep wants the flattest tension available.

Last, the size. Stiffnesses follow from the four decisions above, and they are the cheapest thing to change.

Still open: six things a redundant arm is not #

The serial essays have now gathered a set of statements about seven-joint arms that are confidently made and measured otherwise. A seventh joint is said to get an arm round singularities; on straight moves singularities ended three moves in a hundred. It is said to get an arm round its stops, and it cannot move the elbow, whose stop it never gets round. It is said to extend the reach, and 94 of 95 moves that left the reach ended where they did without it. Its self-motion is said to be a solution to the inverse problem with one more answer, when it is a curve of them.

The distinct argument would collect six such statements and answer each with a measurement already made in the serial field. As here, it would look for the common way they fail. The candidate is that each reads the extra joint as an extra capability of the tool, when the tool’s reach, its singular directions and the elbow’s angle on a line are all fixed by the six task coordinates, and the seventh joint can rearrange only what those leave free.