How far the crank turns first

Turn the crank of a linkage one way, then the other. Somewhere in the reversal there is a moment when the crank is moving and the output is not, because the clearances in the joints are being taken up. That interval is the lost motion, and it is the quantity a designer is handed when they ask what the play in a mechanism costs.

It is not a number.

Where it comes from #

A clearance is a link of fixed length and free direction. While the load through a joint has one sign the pin rests against one side of the hole; when the load reverses the pin crosses to the other side, and during the crossing the joint transmits no motion at all.

So the output has a band of positions it can occupy for any given input, and the band’s width is what the previous essay computed. Lost motion is that band converted into input units: how far the crank has to turn to move the output across the whole width of its band.

The conversion is a division by the local velocity ratio. If the output moves dψ for an input dθ, then covering an output band of width w takes an input of w ÷ |dψ/dθ|.

That division is the entire structure of this essay. The numerator — the output band — is a smooth, well-behaved thing between 0.003 and 0.009 radians around the cycle. The denominator is the velocity ratio, which for a crank-rocker passes through zero twice per turn.

The two poles #

A crank-rocker’s output stops and reverses at two definite crank angles. The return stroke is quicker is entirely about where those are and how far apart; for the running linkage they are between 38° and 41°, and between 225° and 229°.

At exactly those positions dψ/dθ = 0, and the lost motion in input terms is infinite. That is not a numerical artefact and it is not a failure of the model. It is the correct answer, and it has a plain physical reading: at the instant the rocker is stationary, turning the crank does not move the rocker anyway, so there is no amount of crank rotation that takes up the play and gets the output going.

The consequence is that any plot of lost motion has two poles in it, and the height of the peak in the plot is decided by how near a sample landed to the pole.

The number in the table is a statement about the table #

This is worth demonstrating rather than asserting, because it is the sort of thing a reader is entitled to be suspicious of.

Computed at 48 crank positions, the largest lost motion found is 38.4°. Computed at 192 positions, it is 402.3° — more than a full turn of the crank with the output not moving, which is the pole announcing itself. Refining the sampling by a factor of four multiplied the peak by more than ten.

Meanwhile the plateau — the median over the cycle — is 2.80° at 48 samples and 2.80° at 192. It does not move at all.

That pair of behaviours is the signature of a pole beside a well-defined level, and it is why this site’s generator reports the plateau and the two limit positions rather than a maximum. A quantity that grows without bound as the grid is refined is not a property of the mechanism, and quoting it as one is quoting the grid.

The same discipline appears elsewhere here: the motion range of a spatial loop is reported with the step that produced it, for the same reason and with a different outcome — there the number does converge, and the step still has to be stated.

The plateau, and what it is proportional to #

Away from the two limits, the lost motion sits at about 2.8° with a minimum of 1.44°. It varies by a factor of two around the cycle, in the same way and for the same reason as the output band does.

It scales exactly with the clearance. At 0.001 instead of 0.01, the plateau is 0.280° — a tenth, to three figures. That linearity is worth having because it means a single computation serves every fit: the plateau in degrees is 280 times the clearance in link units, for this linkage, and a designer choosing between a close running fit and a free one can read the consequence straight off.

It is also the check that the computation is doing what it claims. A quantity that did not scale linearly with the clearance would mean the clearances were interacting with each other or with the geometry in some way the model does not contain, and that would need explaining rather than reporting.

Why this is the quantity that gets specified #

Lost motion is measured in input units rather than output units for a reason that is about instruments rather than about mechanisms.

An encoder, a handwheel, a stepper motor — the things that command a mechanism — are on the input. What a control system knows is where it has commanded the input to be. Lost motion is the answer to “how much of my command produced nothing”, and it is the term that goes into a positioning error budget.

That is also why the poles matter practically and not only mathematically. A positioning mechanism operated near one of its output’s limit positions has, in the most literal sense, no positioning authority there at all: the control can command as much input as it likes and the output will not move until the mechanism is driven back away from the limit. Machines are designed to avoid working there, and the reason is usually given as “the mechanical advantage is poor” — which is a different quantity that happens to be bad in the same place.

What it is not: a hysteresis loop of the mechanism #

The plot most people picture for lost motion is output against input, traced forward and then back, making a closed loop whose width is the play. It is a useful picture and it is worth saying what it does and does not contain, because it is easy to read a dynamic story into it.

The loop is entirely geometric. There is no energy dissipated going round it, because nothing in this model dissipates anything: the pin slides from one side of the hole to the other and no work is done against anything. A real joint has friction and the real loop does enclose energy, but that area is a friction quantity and not a clearance quantity, and the two are routinely conflated because they produce the same shape.

The geometric loop’s width also varies around the cycle, which the textbook picture never shows — the textbook draws a parallelogram, implying a constant offset, and the constant offset is the gear case rather than the linkage case.

So the honest version of the picture is a loop whose width breathes as it goes round and opens without limit at two points. That is harder to draw and it is what the curve in the hero figure is a section through.

Two clearances are not twice one clearance #

A question that comes up as soon as somebody tries to allocate: if one joint has play c and another has play c, is the lost motion the sum?

No, and the reason is the same one that governs tolerance stacking. Each joint’s clearance contributes an output displacement with its own sensitivity, and the four contributions combine — as a worst-case sum if all four clearances can point their worst way at once, and as something smaller if they cannot.

The computation here samples the clearance directions rather than assuming them, so what it reports is achieved rather than bounded. Measured at one position with 0.01 at each of four pins, the achieved output band is 0.0092 radians against a first-order bound of 0.0230 — the bound is loose by a factor of two and a half, because the four clearance directions are not independent once the loop has to close.

That is a real and useful gap. A mechanism rejected on a worst-case clearance stack may be perfectly capable, and the difference is a factor of two rather than a few per cent.

The quantity is not symmetric, and the asymmetry is usable #

One more property that falls out of the computation and is not in the textbook picture.

Lost motion at a position depends on the direction of the reversal only through the sign of the velocity ratio, so its magnitude is the same going either way at any given crank angle. But the two limit positions are not symmetrically placed: for this linkage they are at 38–41° and 225–229°, which are 187° apart on one side and 173° on the other.

That is the quick-return property showing up in a new quantity. The mechanism spends more of its cycle on one side of the limits than the other, so the average lost motion over the working stroke and over the return stroke are different — and a mechanism that does its useful work on one stroke and returns on the other can be arranged to do the useful part where the plateau is lower.

For the running linkage the minimum, 1.44°, sits around 300–310°, which is on the return side. A designer wanting accuracy on the working stroke would flip the crank direction, or re-proportion, and either way the decision needs the curve rather than a number.

Backlash is the same quantity with a different name #

In a gear train the same phenomenon has its own word, its own specification and a whole vocabulary of fixes, and it is worth noting that the kinematics is identical.

A gear pair’s teeth are thinner than the space they run in. Reversing the drive takes up that gap before the driven gear moves, and the amount of input rotation lost is the backlash divided by the pitch radius. That is the same division by a velocity ratio, with one difference that changes everything: a gear pair’s velocity ratio does not pass through zero. It is a constant, which is the whole point of an involute tooth.

So a gear train’s lost motion is a constant too, and it can honestly be quoted as a number. A linkage’s cannot. That is not a difference in the quality of the components; it is a difference between a mechanism whose ratio is constant by construction and one whose ratio is a curve through zero.

It also explains why gear backlash is a designed quantity with a table of values, and linkage lost motion is something people measure on the finished machine and are surprised by. Backlash is bought with a centre distance and can be specified in advance; linkage lost motion depends on the position, so there is no single value to specify.

Why a gear pair gets to have one number #

Worth drawing out, because the contrast is the sharpest statement of what a linkage is.

A gear pair’s backlash is fixed by the centre distance and the tooth thicknesses, and it converts to lost input rotation by dividing by a ratio that is constant by construction. Every part of that sentence is a design decision made once. The result is a number on a drawing, checkable with a dial indicator, and the same at every angular position of the pair.

A linkage has no such luxury and it is not a defect of linkages. The whole reason a four-bar exists is that its ratio varies — a quick-return mechanism is chosen precisely because the output goes one way faster than the other, and the time ratio is the specification. A mechanism whose ratio varies by design has a lost motion that varies with it, and asking for one number is asking the mechanism to be a gear pair.

So the two subjects diverge for a reason that is about what each mechanism is for. Where gears are used, backlash is specified. Where linkages are used, lost motion is a curve, and the design question is which part of the curve the machine works on.

The interference region on the left of the figure has no analogue in a linkage either, and it is worth noting as the place where the gear analogy finally breaks: a gear pair run too close does not merely have zero backlash, it cannot be assembled. A linkage’s clearance can go to zero and the mechanism still works — badly, hot, and briefly, but it works.

Measuring it, and why the measurement is harder than it sounds #

A note on how this quantity is obtained on a real machine, because the practical difficulty is the same one the computation had.

The usual test is to fix an indicator to the output, drive the input one way until the output moves, reverse, and count how far the input goes before the indicator moves again. That gives the lost motion at one position, which is a point on the curve above.

Done at a different position it gives a different answer. Done near a limit position it gives an enormous one, and — this is the part that catches people — the answer depends on how sensitive the indicator is, because near a pole what is being measured is how far the crank must turn to produce a displacement the indicator can see. A more sensitive indicator gives a smaller answer.

That is the physical version of the sampling artefact in the computation. Both are asking a question with an unbounded answer and getting a number back that describes the instrument.

So a measured lost-motion figure needs three things beside it to mean anything: the crank position, the indicator resolution, and the direction of the reversal. A number quoted without them is a number about somebody’s afternoon.

The fixes, and what each costs #

Three ways to reduce lost motion, all of them design decisions rather than manufacturing ones.

Reduce the clearance. The plateau is proportional to it, so halving the fit halves the lost motion everywhere. The cost is friction, heat and seizure risk, and there is a floor set by thermal expansion: a joint with no clearance when cold has none to spare when the machine warms up.

Preload the joint. Spring-load the mechanism so that the load through each pin never reverses. The pin then rests against the same side of every hole at every instant, the clearance stops contributing anything at all, and the lost motion goes to zero — not approximately, exactly, because the clearance link’s direction has stopped being free. This is what an anti-backlash gear does with a split gear and a spring, and what a cam follower’s return spring does. The cost is a permanent parasitic load and the wear that goes with it.

Move the operating region. Since the lost motion is a curve with two poles, a mechanism can be arranged so that the part of the cycle where accuracy matters is the part where the plateau is lowest. That is free, and it is invisible without the curve.

The third is the one this field exists to make available. The first two are known to anybody who has built a machine; the third requires knowing where in the cycle the quantity is small, and that is a computation rather than an intuition.

The lost motion being unbounded at the reversals is the finding, and its practical form is a rule about where a mechanism should be asked to reverse. Away from the extremes the lost motion is a few degrees and proportional to the clearance; at the two positions where the rocker turns back it is whatever the crank likes, because the output’s derivative with respect to the input is zero there and no amount of input produces any output. So a mechanism whose duty cycle happens to reverse near a rocker extreme has a lost motion with no useful bound, and one that reverses in the middle of the swing has one that is small and specifiable. That is a placement decision rather than a tolerance one, it costs nothing if it is made early, and it is invisible in any specification that quotes a single lost-motion figure. The number in the table is a statement about the table, as the essay says — and the position at which the mechanism actually reverses is the number that should have been quoted beside it.