A spring balanced against the load

The slide five poses almost have found the best slide for five prescribed poses of a moving body. No slide reaches all five, and the best misses each by 0.0140 on alternate sides. Read as play, that is a block with 0.0140 of clearance either side of its guide, able to reach every pose if it bears on the right wall at each: + − + − +.

The load picks the wall found that nothing about a frictionless slot decides the wall. The block bears on whichever wall the total moment about the crank pin sends it to, and changes walls only where that moment changes sign. The body’s own weight gives + − − + +, a pattern no play can satisfy. A working load along the guide gives + − − − −, which needs nineteen times the play. The essay ended with the obvious remedy. A load can be designed. A spring from the frame to the body adds a moment of its own, and if it could be placed so that the total changes sign between every pair of poses, the compromise’s play would reach all five.

This essay places it. The spring can be found, and the search is exact. But the search’s answer says something the question did not expect about where the spring can hang and what it is really doing.

A spring is a linear program #

A spring of no free length from a point $G$ on the frame to a body point $X$, with stiffness $k$, pulls the body with $k(G - X)$. Its moment about the crank pin $A$ at a pose is

$$k\,(X - A)\times(G - A) = (X - A)\times w - k\,(X - A)\times A, \qquad w = kG.$$

For a chosen body point, whose five positions and the five crank pins are fixed by the poses, that is linear in the three numbers $(w, k)$. Asking whether some attachment point and stiffness make the total moment, load plus spring, take the needed sign at every pose by some margin is a set of five linear inequalities in three unknowns. That is a linear program. The least force such a spring needs, measured as the largest component of its pull over the five poses, is a linear objective on the same program. Two springs at two body points are the same program with six unknowns.

So the question stops being a search over five continuous numbers with a pattern to hit. For each candidate body point the program either returns the lightest spring that gives the walls or proves that none exists. The working load is the one the earlier essay used, a unit force on the block opposing its motion along the guide. The margin is a quarter of a unit of moment, so the walls do not depend on a knife-edge.

No point of the body will do #

The program was run for every point of a grid 0.15 apart over a square six units on a side in the body’s own frame, 1,681 points in all.

Of the 98 grid points on the body, none can carry the spring. Of the rest, 155 can, and they form a wedge opening down and to the left, starting 0.17 below the body’s lower edge. At the wedge’s tip the lightest spring pulls 1,710 times the load. At the grid’s far corner, three units out, it pulls 5.8.

The result for the body was checked by a route that shares nothing with the program. Sixty thousand random body points on the body, attachment points anywhere in a region fifteen units across and stiffnesses over five decades, seeded, were each tested directly by computing the five moments and their signs. None gave the pattern.

The program also says why. When it finds no spring for a point, the same algebra supplies weights on the five poses that prove there is none. For the body’s centre they are 0, 0.20, 0.40, 0.30 and 0.10. Multiply each pose’s condition by its weight and add, and the terms from any spring hung from that point, whatever its attachment point or stiffness, add to nothing or less, because the weighted sum of its arms from the crank pins is zero. The load’s terms add to nothing too. So the five conditions, each of which must be positive, have a non-negative combination that is not. The slide five poses almost have found the obstruction to a slide as weights on the poses. The obstruction to a spring is the same kind of object, and for points on the body it always exists. For the point on the arm the program finds no such weights.

The weights differ from point to point on the body, and they share one feature. At the body’s lower left corner they are 0, 0.12, 0.34, 0.37 and 0.17, and at its upper right 0, 0.32, 0.49, 0.20 and 0. The first pose never carries any weight. The obstruction lives in poses two to five, where the working load presses the block the same way throughout and the needed walls alternate three times. A spring on the body has to produce that alternation against a load that does not help, and the weights show it cannot. At the first pose the load already presses the needed wall, so that pose asks nothing of the spring.

Checking the program against the mechanism #

A linear program is a model of the question, and it is only as good as its algebra. The spring it returns was therefore put back into the mechanism and run. The crank was turned through its travel from the first pose to the fifth in four hundred steps. At each step the body was placed by the crank and the rigid compromise slide, and the moments of the working load and of the spring about the crank pin were added. That total was divided by the rod’s component along the guide to give the force across the slot. The crank is the one of Burmester’s four that the earlier essay found driving the poses in order, and with it the force’s sign at the five pose angles is + − + − + for the lightest single spring and for the lightest pair.

The run also counts how often the block crosses its slot on the way, and the answer is four for both, exactly the four the compromise needs and no more. A spring that gave the right walls at the five poses and flickered between them in the stretches between would be a worse mechanism than the pattern suggests, since every crossing is a flight across the play. The time a crossover takes measured such flights for a cam’s roller in its groove, and the same physics applies here. These springs cross only where they must.

With the walls right, the play does the rest. The earlier essay’s law for the best worst miss with play $e$ and walls $s$ is $|F - e\sum\lambda_i s_i| / \sum|\lambda_i|$. With the compromise’s own walls it falls to nought at $e = 0.0140$, so a block with that much play either side reaches all five poses. That is what the slide five poses almost have read into its miss, now with a force that makes the reading true.

What the spring is really doing #

The natural picture of a spring chosen for the walls is a spring that chooses them, pressing the block one way at some poses and the other way at others. The found spring does not do that.

The lightest spring within a unit of the body’s outline hangs from a point 0.92 below its lower edge, is fixed at (3.41, 1.16) in the frame, and pulls at most 11.7 times the load. On its own, its moment about the crank pin keeps one sign through the whole travel. It would press the block onto one wall throughout, which is exactly the failure the earlier essay found for a working load. The working load on its own changes sign once. Together they change sign four times, in the right places. At the first pose load and spring agree. At the third and fifth the spring outweighs the load, and at the second and fourth the load outweighs the spring.

So the spring does not choose the walls. It balances the load, so that the load’s moment and its own trade places four times. Run through the mechanism, with the block in its slot and the force across the slot read along the crank’s travel, the pair puts the block on + − + − + at the five poses and crosses the slot exactly four times.

A spring chosen for one load #

A balance has a consequence the picture of a choosing spring would miss. If the load changes, the balance moves.

The walls are all right only for a working load between 0.903 and 1.091 times the one the spring was chosen against. With no load the spring presses one wall throughout. At twice the load the block is on the right wall at three poses of five. A slide chosen for five poses and a spring chosen for its walls make a mechanism that reaches its poses at one load, within nine per cent either way. A machine whose working load varies, a press that meets harder and softer stock or a transfer arm carrying parts of different weights, would lose the walls as soon as the load left the band, and the block would miss the poses by the amounts the earlier essay measured for the load’s own walls. The lamp that stays where it is put balanced a weight with springs exactly, at every position, because the weight never changes. A working load does change, and this balance holds at five positions for one size of it.

That band is a design fact rather than a failing of this particular spring. Any spring whose own moment keeps one sign can produce alternating walls only by trading places with the load, and trading places is a comparison of sizes. A spring whose moment alternated on its own would not need the load. It would press the block onto alternate walls unloaded, and the load’s moment would then only have to stay below it, which is a one-sided limit rather than a band. The lightest springs the program finds are not of that kind, because using the load’s own moment is what lets them be light. The band’s width, though, is set by the margin each pose was asked to clear, 0.25 throughout here, and not by the mechanism: the arm belongs to the poses found it widening in proportion to the spring’s pull, about one unit of working load for every hundred units of pull on this arm.

A longer arm buys a lighter spring #

Within the wedge, the least pull falls as the body point moves out.

Within half a unit of the outline the lightest spring pulls 23.9 times the load, within one unit 11.7, within two 8.0 and within three 5.8. The attachment point hardly moves as the arm lengthens. It stays near (3.41, 1.16) in the frame, up and to the right of the crank’s centre, while the body point walks down and to the left. The spring’s line therefore runs across the whole mechanism, from low on the body’s left to high on the frame’s right, and the lengthening arm changes how far its line passes from the crank pins rather than which way it points. A designer choosing between a short arm and a light spring is choosing a point on this curve. The curve says that the first half unit of arm is worth a factor of seventy in spring force and the next two and a half units a factor of four.

What an arm off the body costs is the obvious. The body now carries a lever, the lever sweeps a region the body did not, and whatever is being moved through the five poses has to tolerate it. A clearance is a link treated play as a link of its own. Here the arm is a real link welded to the body, added so that the play can be used.

What the spring costs the crank #

A spring that pulls eleven times the working load is not free. The crank now drives the body against the spring as well as against the load, and a spring of no free length stores energy in proportion to the square of its length. Over the travel the spring’s length changes as the body point moves relative to the attachment point. The work it takes or gives back is then a real part of the crank’s torque, and it may be much larger than the work done on the load. The slider-crank is the mechanism whose crank torque has been worked out most often, and with a spring on it that working would need redoing. That is not measured here. It is the price a designer pays, alongside the arm, for a slide that reaches its poses by play.

Two springs on the body #

The body itself cannot carry one spring. The same program answers whether it can carry two.

Of the 11,781 pairs, 94 can. Every one of them includes a point within a grid step of the body’s lower edge. The lightest hangs its springs from (−0.60, −0.28) and (0.45, −0.28) on that edge, fixed at (2.17, −0.35) and (1.86, −0.31) in the frame, and needs pulls of up to 76 times the load. That is six and a half times what one spring on a unit-long arm needs, and it keeps the walls right for loads between 0.915 and 1.091 of its design load, almost the same band. Two springs on the body’s edge and one on an arm below it are two answers to the same balance. The arm is the cheaper.

What the program does not settle #

Springs with free length. Every spring here has no free length, so its pull is proportional to its length and its moment is linear in the attachment point. A spring with a free length adds a term that is not, and the problem stops being a linear program. Two springs stand in for a parallelogram used zero-free-length springs for the same reason; a real spring approaches one only with a long pre-stretched coil.

Friction. The block’s wall is decided here by moments alone, with a frictionless slot. With friction the wall changes where the moment passes a threshold rather than nought, and the band of loads narrows by it.

The play itself. The spring gives the walls the compromise needs, and the earlier essays showed that with those walls a play of 0.0140 reaches all five poses. The play is not modelled as the block actually rocking across its slot. The walls are read at the rigid compromise, which is exact only as the play goes to nought.

Clearance for the spring itself. The attachment points and arms are placed by the program with no regard for what else occupies the space. The single spring’s line runs across the mechanism from low left to high right, and whether it clears the crank, the rod and the guide at every point of the travel is not checked.

The body’s own weight. Only the working load was balanced. A spring for the weight, which is fixed, would not have the narrow band, and the program would find it the same way.

Still open: a spring for the body’s weight #

The band of loads is the cost of balancing a load that changes. The body’s own weight does not change, and the load picks the wall found it pressing + − − + +, which makes play worse.

The distinct argument would be the same program run against the weight instead of the working load. Its answer would be a spring that gives + − + − + for as long as the body weighs what it weighs, with no band to fall out of. The measurement would map the body points from which such a spring exists and find whether the weight’s different moments move the wedge onto the body itself. That would be the only case in which a spring hung from the body, with no arm, reaches five poses by play. If the wedge stays off the body, the arm is not a feature of the working load but of the five poses, and the certificate weights would say why.