Backlash is an allowance

A gear pair’s teeth are thinner than the spaces they run in. Reverse the drive and the driven gear does not move until the gap is taken up.

That gap is backlash, and almost everything said about it in ordinary speech is wrong. It is not wear. It is not slop. It is not evidence of a badly made pair. It is a quantity on a drawing, chosen deliberately, and a pair that has none of it cannot be run at all.

What it is measured from #

The measurement here computes no backlash formula at all.

Two gears mesh on a pair of operating pitch circles which roll on each other without slipping. Because they roll without slipping, rw₁/z₁ = rw₂/z₂, so one circular pitch serves both and the two teeth are directly comparable: they have to fill that pitch between them. Backlash is what is left over.

$$j = \frac{2\pi r_{w1}}{z_1} - (s_1 + s_2)$$

where s₁ and s₂ are the tooth thicknesses at the operating radii, computed from the involute geometry the flanks were generated from. No involute equation enters — only the thickness of the tooth that was drawn.

That is the whole of it, and it is worth noticing what it does not contain. Nothing about how the gears were cut, nothing about the cutter, nothing empirical. Given the two tooth forms and the distance between the shafts, the backlash follows.

Zero is not the bottom of a range #

The figure’s left half is the part worth sitting with.

At the centre distance where the two teeth exactly fill the pitch, backlash is zero. Bring the shafts any closer and the arithmetic gives a negative number, and a negative backlash is not a tight mesh. It is an interference: the tooth needs more room than exists, and the pair will not go together.

So the zero is not one end of a tolerance band with a comfortable region below it. It is a wall. A design that specifies zero backlash is specifying a centre distance that has to be exactly right — at every temperature, with every bearing at every point of its own clearance, on every pair that comes off the machine.

Which is to say it is specifying the impossible, for exactly the reason an overconstrained linkage is unbuildable: a constraint that must hold exactly, in a world where nothing holds exactly. And it is rescued in exactly the same way — by an allowance, chosen large enough to swallow the errors that will occur.

What the allowance has to swallow #

Four things, and none of them is manufacturing sloppiness.

Centre-distance error. The two shafts are located by bearings in a housing, and the distance between them carries the housing’s tolerance. It is the dominant term and it is the one the figure plots.

Thermal expansion. A steel housing and steel gears grow together, but not equally if the housing is aluminium and the gears are steel — which is the usual combination. A 100 mm centre distance in an aluminium case, 60 °C above the gears, moves by about a tenth of a millimetre. That is more than the whole backlash allowance on a small pair.

Runout. No gear is exactly concentric with its bore. The centre distance therefore varies once per revolution, and the mesh must have room for the tightest moment of the tightest pair.

Tooth-form error and swelling. Cutting tolerances on the flank itself, and — on plastic or filled gears — moisture absorption, which is why nylon gears are cut thinner than steel ones for the same duty.

Backlash is the sum of what those need. It is specified as a tooth-thickness allowance on the drawing, usually by thinning both gears rather than by opening the centres, because the centre distance is set by the housing and the tooth thickness is set by how deep the cutter goes.

The textbook formula, and where it drifts #

The expression everybody uses is

$$j = 2\,\Delta a\,\tan\alpha_w$$

— open the centres by Δa and the backlash grows twice that times the tangent of the operating pressure angle. It is a linearisation, and its error is measurable.

The formula holds α_w fixed. It does not stay fixed: opening the centres raises the operating pressure angle, because the base circles cannot change and the line of action must still be tangent to both. So the true relationship curves gently upward and the linearisation runs below it, by half a per cent at a centre-distance error of one part in a thousand and by four and a half per cent at one part in a hundred.

One part in a hundred on a centre distance is not a tolerance anybody would accept. So the honest verdict on the formula is that it is right everywhere it is used — and that saying so requires having computed the thing it approximates, which is the only reason this site can say it rather than repeat it.

What it costs at the input #

A gear train’s backlash converts to lost input rotation by dividing by the pitch radius: the driven gear stands still while the driver turns through the arc that closes the gap.

For the pair above at Δa = 0.06 — a sixth of a per cent on a 30 mm centre, an ordinary machining tolerance — the backlash is 0.044 and the pinion’s pitch radius is 10, so the input turns a quarter of a degree before the output moves.

That number is the whole reason backlash gets attention in positioning machinery, and it is worth putting beside the same quantity in a linkage, where it is 2.8° and is not a number at all but a curve with two poles.

The difference is not that gears are made better. It is that a gear pair’s velocity ratio does not pass through zero. It is a constant, by construction, because that is what an involute tooth is for — so dividing by it gives a constant, and backlash can honestly be quoted as one figure and checked with a dial indicator.

A linkage’s ratio varies by design, so its lost motion varies with it and there is no single figure to specify. Two mechanisms, the same geometry of play, and one of them gets a number because of a property it was chosen for.

Where the allowance is not enough #

The four contributions the allowance has to cover are not equally well behaved, and one of them defeats a fixed allowance entirely.

Centre-distance error, tooth-form error and cutting tolerance are all fixed once made. A pair assembled with them has whatever backlash it has, and a single number covers the population.

Runout is not fixed. A gear eccentric on its bore brings its teeth closer to the mating gear once per revolution and further away half a turn later, so the backlash breathes: the pair is tight at one angular position and loose at the opposite one. An allowance chosen for the average leaves the tight position with none.

That is why gear inspection reports a composite error rather than a single thickness — the pair is rolled through a full mesh cycle against a master gear and the variation is recorded, not just the mean. And it is why the specification has to be written against the tightest moment rather than the typical one: the failure mode at zero backlash is not inaccuracy, it is interference, and interference does not average.

The kinematic parallel is exact. A linkage’s tolerance band is widest at one part of its cycle and the specification has to be met there rather than on average, for the same reason: the thing that goes wrong at the worst position is a different kind of failure from the thing that goes wrong at the typical one.

In a train, it accumulates #

A single pair’s backlash is small. A train’s is the sum, referred to the output — and the referral is what makes it painful.

Each mesh contributes its own backlash, divided by the pitch radius at which it acts, and multiplied by the ratio between that shaft and the output. In a reduction train the first mesh is the one that matters least, because its error is divided down by everything after it; the last mesh, nearest the output, contributes its full amount.

That is the opposite of the intuition that the fast-running input stage is the critical one, and it is the same shape of result as the tolerance allocation on a linkage: where the error matters is not where the load is, and the ranking is arithmetic rather than judgement.

Two routes, and what the second one is for #

The mesh geometry in this essay is computed twice, as everything on this site is, and the two routes are worth separating because one of them is the one that generalises.

The closed form is the involute function: for the teeth to fill the pitch exactly, inv α_w = inv α + 2(x₁ + x₂) tan α / (z₁ + z₂), and the centre distance follows from the base circles. It is exact and it answers one question — where is zero backlash?

The measurement bisects on the centre distance until the two tooth thicknesses fill the circular pitch, using only thicknesses computed from the drawn flanks. It answers the same question and agrees to a part in 10¹².

But it also answers a question the closed form does not: what is the backlash at any other centre distance. That is the whole of the curve in the hero figure, and the involute equation has nothing to say about it — it is an equation for the zero-backlash condition, not a function of the separation.

So the two routes here are not merely a check on each other. The slow one is strictly more general, and the fast one is the special case it agrees with. That is worth noticing because it is the reverse of the usual arrangement on this site, where the closed form is general and the solve is the demonstration.

Taking it out #

Three ways, and each buys something with something.

Cut the teeth full and set the centres exactly. The purist’s answer and rarely available, because it needs the housing tolerance, the thermal range and the runout all to be smaller than the allowance being dispensed with.

Split the gear and spring it apart. An anti-backlash gear is two halves on one hub with a spring between them, each half bearing on one flank of the mating tooth. Backlash goes to zero because the play has had its direction taken away — which is precisely what a preload does to a linkage’s clearance, and it costs the same thing: a permanent load, and the friction and wear that follow.

Shift the profiles. Profile shift moves the operating pressure angle and the centre distance together, so a pair can be made to run at a non-standard centre with a chosen backlash. This is the version that costs nothing extra to manufacture, because the shift is a setting on the machine that was cutting the gear anyway.

The number that is actually specified #

A last piece of practice, because the quantity a drawing carries is not the one plotted above.

The figure plots backlash against centre distance, which is how it is measured on an assembled pair: run the shafts at a known separation, hold one gear, and rock the other. What a drawing specifies is a tooth-thickness allowance — each gear cut a stated amount thinner than nominal — because the centre distance is fixed by the housing and cannot be used as an adjustment.

The two are related by the same geometry: thinning both teeth by t in total gives the same backlash as opening the centres by t / (2 tan α_w). At 20° that divisor is 0.728, so a tooth-thinning allowance and the centre-distance error it tolerates are almost the same number, which is a coincidence of the standard pressure angle and a convenient one.

Thinning is preferred for a reason worth stating. Opening the centres also changes the contact ratio — the pair runs further out on the flanks, where less of the line of action lies between the tip circles — and a contact ratio that falls below one means the teeth let go before the next pair picks up. Thinning the teeth barely touches it. So the two ways of buying the same backlash are not equivalent, and the one that is free of side effects is the one everybody uses.

What a gear pair cannot borrow from a linkage #

Worth naming the asymmetry, since this field keeps insisting the two subjects are the same.

A linkage’s play can be removed by preloading it — hold the load one-signed and every pin sits against the same side of its hole. A gear pair driven in one direction is preloaded in exactly that sense, and its backlash costs nothing while it runs.

The difference is that a gear pair is normally expected to reverse. That is what a positioning drive does, and it is the case backlash is specified for. A linkage under a working load frequently never reverses at all — a press, a clamp, a hoist — so the same geometry produces a problem in one subject and not in the other, purely because of what the mechanisms are asked to do.

Which is why gear backlash has a table of standard values and linkage lost motion does not.

The last mesh is nearly all of it #

The accumulation in a train is described above as dominated by the final stage, and the arithmetic behind that is short enough to do and gives a bound worth carrying.

Each mesh’s backlash arrives at the output divided by whatever reduction comes after it. A stage followed by a four-to-one reduction contributes a quarter of its own angular backlash; a stage followed by two such reductions contributes a sixteenth. So the contributions form a geometric series running backwards from the output, with the last stage undivided.

Put three stages of four-to-one, each contributing $b$ of angular backlash at its own output, and the total at the final shaft is $b + b/4 + b/16 = 1.3125\,b$. The last stage is 76 per cent of it, the middle stage 19 per cent, and the first under five. Improving the input stage by half changes the total by two per cent.

The bound generalises immediately. For equal per-stage reductions $r$ the series sums to $b\,r/(r-1)$, so a train’s output backlash never exceeds $r/(r-1)$ times its final stage’s however many stages there are — 33 per cent more at four to one, 11 per cent at ten to one, and less the deeper the reduction. Adding stages at the input end adds nothing measurable.

That is the design rule stated exactly rather than as a preference. Spend the accuracy on the last mesh, use whatever the workshop finds convenient upstream, and stop worrying about the input stage entirely — its contribution is bounded by the same series and it is small. A gearbox whose first stage is cut to the same specification as its last has paid twice for a quarter of a per cent.

It also explains a fact about real gearboxes that looks like inconsistency. A precision reduction unit frequently has a coarse, cheap first stage and an expensive final one — different tooth qualities on one shaft train, sometimes different processes — and the reason is this series rather than any argument about speed or load. The input stage runs fast and carries little torque; the output stage runs slow and carries all of it; and on backlash, which is neither of those, the same asymmetry happens to hold for a third reason.

The general shape #

Backlash belongs in this field rather than in the gear field because it is the same object under a different name.

A pin in a hole is a short link of fixed length and free direction. A tooth in a space is the same thing: a fixed amount of freedom, whose direction reverses when the load does. Both are designed in, both are specified on a drawing, both produce a lost motion that is the freedom divided by a velocity ratio, and both are removed — never reduced, removed — by taking the direction away rather than the magnitude.

What distinguishes them is only that one subject named the quantity and gave it a table of values, and the other left it to be discovered on the finished machine.