The arm belongs to the poses

The slide five poses almost have found the best slide for five prescribed poses of a moving body, and that it misses each by 0.0140 on alternate sides. Read as play in its guide, that miss lets the slide reach all five poses if the block bears on alternate walls, + − + − +. The load picks the wall found that the block bears on whichever wall the total moment about the crank pin sends it to. The body’s weight gives + − − + +, which play cannot use, and a working load along the guide gives + − − − −.

A spring balanced against the load then added a spring of no free length from the frame to a point of the body. Whether any attachment point and stiffness give + − + − + is a linear program for each body point. No point of the body would do. A certificate, a set of weights on the five poses, showed that every spring hung from a body point has moments that cancel under those weights. Points on an arm below the body would do, and the lightest pulled at most 11.7 times the load and kept the walls only for working loads within nine per cent of the design.

That essay’s question for the next step was whether the body’s own weight, which does not change, would do better. The weight’s moments about the crank pin differ from the working load’s, and run against them the program might move the reachable region onto the body. If it did, a spring hung straight from the body could carry it through five poses on its play. If it did not, the arm would be a fact about the five poses rather than about the working load, and the certificates would say why.

It did not, and the reason is cleaner than the question.

Four loads, one region #

The program is the earlier essay’s with the load as a parameter. A spring from frame point $G$ to body point $X$ with stiffness $k$ pulls with $k(G - X)$. Its moment about the crank pin at pose $i$ is linear in $w = kG$ and in $k$. Each pose’s total moment, spring plus load, must lie on the side the walls need by a margin, and the least largest pull that achieves it is the optimum. The weight is a unit force downward at the centre of the body’s outline. The working load is a unit force along the guide opposing the block’s motion.

Run over a grid of the body’s frame, 1,681 points 0.15 apart over six units square, under four loads: none, the weight, the working load, and both together.

The four regions are the same 155 points: the wedge below the body the earlier essay drew, and not one point of the body. The lightest spring within a unit of the outline is at the same body point under every load, (−0.60, −1.20), 0.92 below the body’s lower edge, with the same stiffness, 2.513. What changes is the attachment point. It sits at (3.04, 2.10) with no load, (2.93, 1.97) for the weight, (3.41, 1.16) for the working load and (3.29, 1.03) for both. The pull changes too, 9.30, 8.97, 10.10 and 9.80 times the unit load. The weight helps slightly and the working load hinders.

What the attachment point does #

A load is a constant force on the body, and a spring with its stiffness fixed and its attachment point moved adds a constant force too, at the spring’s own body point. So the program has a natural way to answer a change of load: leave the stiffness where it is and slide the attachment point until the spring’s extra constant force does what the load undid. That is what it chooses. The stiffness is 2.513 under all four loads, and the attachment point moves by up to 1.10 between them, furthest between no load and the weight and working load together, the two most different loads.

The match is not exact, because the load acts at a different body point from the spring. The weight acts at the body’s centre and the working load at the block, while the spring acts on its arm, and the price of the difference is the change in pull, from 8.97 to 10.10. A designer who builds the spring for one load and then finds the load different has an adjustment to make, and it is the attachment point’s position, not the spring. That is how such mechanisms are tuned in practice, with a threaded attachment point, and the program says it is the right knob: one setting per load, the band around it fixed by the spring’s pull.

Why no load can move it #

The region does not move because it never depended on the load. The springs hung from one body point that give the right walls form a cone in $(w, k)$, since each pose’s condition is linear in them and $k$ must not be negative. Where that cone contains a spring that gives + − + − + with no load at all, scaling the spring up makes each pose’s margin as large as wanted, and any fixed load is simply overpowered. Where it contains no such spring, Farkas’s lemma supplies the certificate: weights on the five poses under which every spring’s moments cancel in $w$ and do not help in $k$. A load could then rescue the point only if its own moments, weighted the same way, came out positive.

For the body’s points they do not, under either load. For a spring from the body’s centre, the certificate against the weight puts weights 0.08, 0.30, 0.41, 0.21 and 0 on the poses, and the weight’s weighted moments come to nought. Against the working load the weights are 0, 0.20, 0.40, 0.30 and 0.10, the earlier essay’s, and the load’s weighted moments also come to nought. A sum of five conditions, each required to be positive, cannot be nought, so neither load makes the centre usable.

That the weight’s weighted sum is exactly nought at the centre has a short reason. The weight is a constant force at the body’s centre, and a spring with no stiffness is exactly that, a constant force at its body point. The certificate’s first condition says every constant force at the centre cancels under its weights, the weight included. So for a spring at the centre the weight is not even a different load. It is one of the springs the certificate already rules out.

The arm, then, is a property of the five poses. A body point can carry the spring only if the line from some fixed attachment point to that point’s five images crosses the crank pin’s five images on the right sides in turn, four changes of side. The body’s own points move too much like the crank pin for that line to weave across it. The test is about where the points go, not about what is loading them.

The nine per cent was a margin #

The second finding is about the band. The earlier essay found its spring keeping the walls only for working loads between 0.903 and 1.091 of the design load, and read that as a limitation of balancing a load with a spring. It is a property of the margin the program was given.

The program asks each pose’s total moment to clear its wall by a margin, 0.25 throughout the earlier essay, and finds the lightest spring that does. Sweep that margin and both the pull and the band follow it. Designed against no load, a margin of 0.05 gives a pull of 1.9 and keeps the walls for working loads from nought to 0.018 of the unit; a margin of 2 gives a pull of 74.4 and keeps them up to 0.732. Pull and band are both exactly proportional to the margin, so the band is the pull divided by about 102. Designed against the working load, the band is centred on the design load and grows the same way: 0.981 to 1.018 at a pull of 2.8, 0.903 to 1.091 at the earlier essay’s 10.1, 0.227 to 1.732 at 74.3.

The band itself turns out not to depend on the spring at all. At a margin of 0.25 it is 0.0915 of the unit working load wide on its open side for every point of the wedge, the lightest and the heaviest alike. At the pose where the working load most opposes the walls needed, pose three, the load’s moment about the crank pin is 2.73. A spring that clears the walls there by the margin and no more can absorb an extra working load of the margin over 2.73 before that pose’s wall flips, and the optimum spring always clears that pose by exactly the margin. So the band is the margin divided by the working load’s worst opposing moment, a property of the poses and the load.

What a band costs is the point’s own price. Every spring at a point scales with its margin, so each point has a pull per unit of margin, and with the band fixed by the margin the pull per unit of band follows: about 102 at the lightest point within a unit of the body, 83 within one and a half units, 74 within two, 63 within two and a half and 56 within three. Near the wedge’s tip, half a unit below the body, it runs from 212 to 1,520. So a spring does not balance a load to within nine per cent and no further. It holds the walls for any band of working loads whose width it pays for, at a rate its arm decides. A spring meant to keep the walls for every working load from nought to the design load needs a pull of about a hundred times that load on an arm a unit long, and about fifty-six times on an arm three units long. The earlier essay’s spring, at ten, was never asked for either.

That changes the design question. The earlier essay offered two bad choices: a spring tuned to one load, or no spring. The real choice is continuous. A designer states the range of working loads the slide must reach its poses under, and the spring’s pull follows at a hundredfold. The attachment point is then set to centre the band on the loads expected, which is the only thing the load being designed against actually does.

A spring for the weight #

The question asked was about the weight, and it has its answer. Designed against the weight alone, the lightest spring is at the same arm point, with the same stiffness and an attachment point at (2.93, 1.97). It pulls at most 8.97 times the body’s weight. With the weight on and no working load it keeps + − + − + through the travel, and it does so for as long as the body weighs what it weighs, with no band to fall out of.

What it does not do is tolerate a working load. With the weight on, the weight’s design keeps the walls only for working loads from nought to 0.091 of the unit, the same band of 0.09 that every design at this margin has. Designed against both the weight and the working load, the spring keeps the walls from 0.903 to 1.091 of the working load with the weight on, and none at all without it. Every design in the ledger keeps a band about 0.09 wide on either side of what it was designed for, because they share the margin that sets it. The weight is just one more load the band is centred on.

Two springs stand in for a parallelogram and the lamp that stays where it is put balanced a body’s weight exactly at every pose, with zero-free-length springs on parallelograms. That is a different task: a spring that cancels a load exactly cannot also choose which wall the block presses. Here the spring must leave a moment of the right sign at each pose, not nought, and the band is the room between the two.

What the certificates are telling a designer #

Four poses are the most a slide can meet exactly, as the slide a fourth pose leaves found, and five the most a pin can, as how many points may be prescribed counted, so the compromise slide at five is already an approximation and its play is what pays for the fifth pose. What a synthesis assumes it knows found that a synthesis is handed its demand in a frame and takes the frame as known. Here the demand is the five poses, and the certificates are the poses’ verdict on every point of the body. A certificate’s weights pick out which poses fight each other. For the body’s centre, under either load, they fall mostly on the middle poses, two, three and four. The middle three poses together forbid any spring at the centre, and the first pose is not involved.

That says where a designer could not relax the demand. A certificate that puts no weight on a pose still proves its point when that pose is dropped from the demand, since the remaining conditions already contradict each other. The weight’s certificate puts nothing on pose five and the working load’s nothing on pose one. So a spring from the body’s centre would stay out of reach against the weight with pose five dropped, and against the working load with pose one dropped. The arm comes from the middle three poses, which every certificate weighs most heavily.

What the program does not settle #

Springs with free length. Every spring here has no free length, which keeps the program linear. A real spring adds a constant tension along its line; whether that helps is the open question below.

Margin and play together. The margin is in units of moment and is independent of the slide’s play. A real block in a guide with play needs a moment of a definite size to cross its gap and seat firmly, set by friction and stiffness the model does not have. The band-to-pull rate here is for the margin as given.

One wall, one pose. The band’s width is set by pose three, where the working load most opposes the walls. A demand that changed which pose is critical, a different load or a different slide, would change the band’s rate; the measurement here is for this slide and these two loads.

Still open: a spring with free length #

Every spring here has no free length, which keeps the program linear and makes its pull a hundred times the load for a band from nought to the design load on a short arm. A spring with free length pulls with a constant term along its line as well as with its stiffness, and a constant force at a body point is exactly what a zero-stiffness spring is. The certificates rule out constant forces at the body’s own points, but not at the arm’s.

The distinct argument would be whether that constant term, set against the working load, lowers the price of a wide band. The program would gain one variable per spring, the free length’s tension, and stop being linear in the attachment point, so it would be solved as a sequence of linear programs or a small nonlinear one. The measurement would find, at the lightest arm point, the least pull for a band from nought to the design load with and without free length. If the free length takes a large share of the hundredfold pull, the zero-free-length springs that make the program clean are also what make the band expensive. If it does not, the band’s price belongs to the arm’s geometry and nothing about the spring will lower it.