Taking up the play

Lost motion is proportional to the clearance, so the obvious way to reduce it is to reduce the clearance. That works, and it runs into a wall: a joint with no play when cold has none to spare when the machine warms up, and a bearing that seizes is worse than one that rattles.

There is a second way, and it does not reduce the clearance at all.

Magnitude and direction are different properties #

A clearance link has a length and a direction. The length is what a fit table specifies. The direction is what nothing specifies, and it is where the trouble is.

Because the direction is free, it changes whenever the load through that joint reverses — and it therefore changes at a different crank angle for each of the four pins, several times a cycle. The error it causes is consequently not repeatable: the same mechanism given the same input twice can be in measurably different places, depending on what it did last.

Preload removes the freedom rather than the length. Load the mechanism against a spring so that the force through every pin keeps one sign at every position, and each pin rests against the same side of its hole all the way round. The clearance is still there, still 0.01, still as large as it ever was — and it has become a fixed offset.

A fixed offset is a dimensional error. It is exactly the object the first half of this field is about: repeatable, computable, and correctable by adjusting something else.

Why the answer is exactly zero #

Lost motion is the width of the output’s band divided by the local velocity ratio. Fixing the directions collapses the band from an interval to a point, so the width is zero, so the lost motion is zero — including at the two positions where the velocity ratio passes through zero and the unpreloaded quantity was unbounded.

That is worth pausing on, because those poles were the alarming part of the earlier essay. A preloaded mechanism has no lost motion at its limit positions, not because the division has been made safe but because the numerator is zero. Nothing is being divided.

This is the one result in this field that is exact rather than measured, and it is exact because it is a statement about the dimension of a set rather than about its size. Take a free direction away from a mechanism and the set of positions it can occupy for a given input stops being a region and becomes a point, whatever the magnitudes involved.

Where the technique came from #

Preload is not a modern idea and it is worth knowing what it displaced, because the history explains the vocabulary.

Before it, the answer to play was fitting: make the parts, assemble, and then scrape, lap or shim until the mechanism moved sweetly with no perceptible slop. That produces one excellent machine and no interchangeable parts, which is the arrangement the tolerance itself replaced.

Preload is what makes interchangeability compatible with precision. The parts are made to ordinary tolerances with ordinary fits, and a spring takes up whatever play they happen to have — so the accuracy comes from the assembly rather than from the parts, and a worn joint is compensated automatically as it wears.

That last property is the one that gets undersold. A preloaded mechanism does not merely start accurate; it stays accurate as its clearances grow, because the spring keeps taking them up. The 0.343° residual grows as the clearance does, but the 2.80° of lost motion never comes back at all — which for a machine expected to run for years is worth considerably more than the initial factor of eight.

The vocabulary follows the history: “taking up the play” is what a fitter did with a scraper, and it is what the spring is doing now.

What preload does to the tolerance band #

A question this essay has not answered and should: preload removes the clearance’s contribution to the error. It does nothing at all about the lengths.

The band from ±0.01 on the four lengths is 0.73° at its widest and it is unaffected by any spring. So a preloaded mechanism’s total error is the length band plus the preload offset — 0.73° plus 0.34° at these numbers — and the second is now the smaller of the two.

That reorders the whole design problem. Unpreloaded, the 2.80° of lost motion dominates everything and no amount of tolerance work is worth doing until it is dealt with. Preloaded, the length tolerances are the dominant term and the allocation becomes the thing to spend money on.

Which is a general lesson about ordering: fix the term that dominates, then re-rank, because the ranking changes when the largest contribution leaves. Doing the tolerance allocation first on a mechanism with 2.80° of play would have been careful work on the wrong quantity.

What is left instead #

Not nothing, and the honest accounting is the reason this essay exists rather than a sentence in the previous one.

With every pin held hard over in one direction, the output sits at an offset from where the nominal mechanism would put it. That offset is not the same at every crank angle: it runs from −0.330° to +0.013° around the cycle, because the same fixed clearance vectors project differently onto the output as the geometry turns.

Split that into two parts:

• Its average, −0.136°, is a constant. It comes out in a calibration — zero the encoder with the mechanism assembled and preloaded, and the constant is gone.
• Its variation, 0.343° peak to peak, does not. It is a position-dependent error, and it is what survives.

So the trade is 2.80° of unrepeatable lost motion against 0.343° of repeatable position error, of which some is removed by calibration. A factor of about eight, and it is a factor rather than an elimination.

What it costs, and why that is not computed here #

A preload is a force that is always there.

It loads every bearing all the time, including when the mechanism is doing nothing. That means friction — a torque the drive must overcome at every position, on top of whatever the mechanism is for — and it means wear, at a rate set by a contact pressure that exists whether or not the machine is working.

None of that is kinematic and none of it is computed on this site. What can be said is the geometry: the lost motion goes to zero, the offset is this, and the calibratable part of it is that. What cannot be said is whether the spring that achieves it makes the mechanism too stiff to drive, or wears its bearings out in a year.

That is the boundary drawn deliberately elsewhere in this field, and preload is the place it bites hardest, because the technique’s whole cost is on the far side of it.

Why the offset varies at all #

The residual error deserves an explanation rather than a measurement, because “a fixed clearance gives a varying offset” sounds like a contradiction.

The clearance vectors are fixed in magnitude and direction once the load is one-signed. What is not fixed is how they project onto the output. A clearance at the crank pin, pointing along the crank, adds to the crank length — and the output’s sensitivity to the crank length varies through the cycle and changes sign twice. Multiply a constant by a varying sensitivity and the product varies.

So the preloaded mechanism’s error curve has exactly the shape of the sensitivity curves that the tolerance half of this field computed, which is the strongest possible confirmation that preload has converted a clearance problem into a dimensional one. It is not merely like a length error. It is a length error, and it is described by the same four numbers.

That also says where the residual is smallest: at the crank positions where the sensitivities are smallest, which are the positions where the tolerance band is narrowest. A preloaded mechanism arranged to do its precise work there is more accurate again, for free, and the arrangement is visible only once the curve has been drawn.

The calibration is worth doing and is not free either #

The constant that comes out in a calibration — 0.136° here — is removable in principle. In practice it is removable per mechanism, not per design, because it depends on the actual clearances in the actual joints, and those vary from one assembly to the next and grow with wear.

So a calibration is a step in the assembly process: build it, preload it, run it to a reference position, and zero. That is cheap for one instrument and expensive for a production run, and it has to be repeated after any service that disturbs a joint.

What it buys is real, though. Without it the preloaded mechanism has a 0.343° error band about the nominal; with it, the band is about the calibrated mean and its width is unchanged. The calibration removes the offset, not the variation — and stating that clearly is the difference between a technique that works and one that disappoints.

How much preload, and the one number this site can supply #

“Enough to keep the load one-signed” is the specification, and most of it is a force calculation this site cannot do. One part of it is geometric, and it is the part that decides whether the technique is available at all.

The load through a joint reverses when the direction of the force it transmits crosses the joint. Whether that happens, and where in the cycle, is decided by the mechanism’s geometry together with whatever it is driving — so a preload has to exceed the largest reversing component over the whole cycle, and the position at which that maximum occurs is a geometric question with a geometric answer.

What follows is a design rule that is available before any force is known: the preload has to be sized against the worst position, and the worst position can be computed. A designer who sizes a spring against the average is under-preloaded for part of every cycle, and a mechanism that is preloaded for nine tenths of its cycle has all the cost of preload and, for that last tenth, all the lost motion it was meant to remove.

That is a failure worth naming because it looks like success on a test bench. Run the mechanism slowly through its range, measure the backlash, find none — and then meet it at speed, or under load, at the one position where the spring is not winning.

The same idea, four ways #

Preload is not one trick. It is a family, and recognising the family is more useful than the four members.

A spring-loaded follower. A cam follower held against its cam by a spring never leaves it, so the contact never reverses and the pressure angle never changes sign. This is the oldest version and the one nobody calls preload.

A split anti-backlash gear. Two halves on one hub with a spring between them, each bearing on one flank of the mating tooth. Backlash goes to zero for exactly the reason above: the play has had its direction taken away.

An over-centre or gravity bias. A mechanism arranged so its own weight or a toggle keeps the load one-signed. Free, and it only works over the range where the bias dominates whatever else is acting.

A flexure. The limiting case, and worth a paragraph because it is the one that removes the magnitude too.

The flexure, and where it stops being kinematics #

Replace the pin with a thin elastic strip and the clearance is not held to one side — it is gone. There is no pin and no hole and nothing to take up. A flexure pivot has exactly zero play, perfect repeatability, and no wear, because nothing slides against anything.

In exchange it has a very limited range of motion, and it exerts a restoring force that grows with deflection.

Both halves of that exchange are outside what this site computes, and it is worth being precise about which is which. The range limit is elastic — how far the strip can bend before it yields — and there is no geometric answer. The restoring force is elastic too. And the subtler one: a flexure’s instantaneous centre moves as it deflects, so it is not a revolute pair at all but a compliant mechanism whose “pivot” wanders, and where it wanders to depends on the deflected shape, which depends on the load distribution.

So a flexure is the limiting case of this essay’s idea and it is not a mechanism this site can position. That is not a gap in the field so much as the field’s boundary appearing exactly where the technique becomes most effective — which is a pattern worth noticing rather than apologising for.

The measurement, and what it assumed #

The number in the title — a factor of eight — comes from holding every clearance in one direction and re-solving around the cycle, so it is worth saying which direction and why it does not matter much.

The figure uses direction zero, meaning every pin displaced along the positive x-axis of the frame. That is not a physically motivated choice: the real directions are set by the load path, and each pin’s is different.

The reason it is defensible is that the lost motion result — exactly zero — does not depend on the directions at all, only on their being fixed. Any choice gives zero. What the choice affects is the residual offset, and the numbers here are one sample of that.

A more careful treatment would take the directions from the load path at each position, which is a force calculation, and it would land somewhere in the range this one samples. Since the conclusion is a factor of eight rather than a factor of 8.17, the sampling is adequate for the claim being made — and stating that it is a sample rather than the worst case is the difference between a measurement and a number.

Where it is the wrong answer #

Two situations, both common.

A mechanism whose load must reverse cannot be preloaded away from it — an engine’s connecting rod takes tension and compression by construction, and no spring changes that. What is available there is to make the reversal happen where it costs least, which needs the lost-motion curve rather than a spring.

And a mechanism that is already heavily loaded in one direction is preloaded for free and nobody had to do anything. A press, a clamp, a hoist: the working load holds every pin hard over, the play never reverses under load, and the lost motion is the value that shows up on the return stroke when the load comes off — which is why so many machines are accurate at work and sloppy when idle.

The lost motion that is not a tolerance #

One mechanism on this site has lost motion that no accuracy improves, and it is worth setting beside the clearance-driven kind measured here.

The two add on the same mechanism — the input takes up the clearance first and then travels to the next root — and their sizes are not comparable: the four-bar here loses about 2.7° of crank to a hundredth of a unit of joint clearance, against 15° of design lost motion on a coarse ratchet.

The distinction the rung turns on is worth stating in the most general form it takes, because it applies well beyond preloaded joints. A clearance has a magnitude and a direction, and only one of them is a tolerance. The magnitude is set by manufacture and is what a drawing controls; the direction is set by whatever last pushed the part, and it is not controlled by anything. So an error whose direction varies is unrepeatable, and an error whose direction is fixed is a bias — larger, perhaps, but constant, and therefore measurable and removable by calibration. Preload does not shrink the error; it fixes its direction, and everything the technique buys follows from that one change. Which says where else to look for the same move: any mechanism whose error is unrepeatable because something is free to sit at either end of a gap can be improved by removing the choice rather than the gap, and the improvement is a factor rather than a fraction.