What happens in a mesh

Two involute gears touch along one line and nowhere else.

That line is the line of action: the common tangent to both base circles, passing through the pitch point. Every contact between every pair of teeth happens somewhere on it, and the contact point travels along it steadily as the gears turn.

Only part of the line is used #

Contact begins where the driven gear’s tip circle crosses the line of action and ends where the driver’s does. Outside that segment the teeth are not touching.

The length of the segment is the path of contact. Divide it by the base pitch — the distance between corresponding points on adjacent teeth, measured along the line of action — and the result is the contact ratio.

For the pair above it is 1.612. That number is the average count of tooth pairs in contact, and the interpretation is direct: for 61% of the cycle two pairs share the load, and for the remaining 39% one pair carries it alone.

Why it must exceed one #

If the contact ratio were below 1, there would be moments with no teeth touching at all. The driven gear would coast, then be struck by the next tooth arriving. That is not a subtle degradation — it is a hammer blow once per tooth.

The design minimum is usually quoted as 1.2, which leaves margin for manufacturing error and for the tooth tips being relieved slightly to avoid interference. Standard 20° proportions on ordinary tooth counts give 1.5 to 1.8, so the constraint is not usually binding — but it becomes binding for small tooth counts and for high pressure angles, both of which shorten the path of contact.

That is one of the trades hidden in the pressure angle. Twenty-five degrees gives a stronger tooth and a shorter path of contact; 14½° gives a longer path and a weaker tooth.

What contact ratio does not measure #

It counts pairs in contact. It says nothing about how the load divides between them, which depends on the stiffness of the teeth and on manufacturing accuracy — quantities outside this site’s model entirely.

In practice the sharing is not equal, and gear designers deliberately modify tooth tips so that load transfers smoothly rather than abruptly as pairs engage and disengage. None of that is geometry, and none of it appears here.

Backlash, which is also absent #

Two gears at their nominal centre distance with nominal tooth thickness would mesh with no clearance at all, and would jam as soon as anything expanded or any dirt got in.

Real gears are cut slightly thin, or run slightly further apart, leaving backlash — a small gap between the non-driving flanks. It is essential, and it means the driven gear can move through a small angle without the driver moving at all.

That matters for anything positioning rather than merely transmitting power, and it is the reason a gear train reversing direction has a dead zone. Nothing on this site models it: every mesh here is ideal, with teeth touching on both flanks at once, which is a gear that could not be built.

Interference #

If a tooth tip reaches beyond the point where the line of action is tangent to the other gear’s base circle, it is trying to touch a region where no involute exists. That is interference, and the tip gouges the other tooth’s root.

The cure is either fewer teeth on the small gear, a larger pressure angle, shortened addenda, or profile shift. It is the same geometric fact that causes undercutting, seen from the meshing side rather than the cutting side.

What the mesh figure asserts #

Every gear figure here generates its flanks from the involute and then checks them. The velocity ratio is required to equal the tooth-count ratio exactly; the contact ratio is required to fall in the usual range; no flank point may lie inside its base circle; and the normal to the drawn flank is required to be tangent to the base circle to within a tolerance bracketed by the sampling noise below and an arc-approximated flank above.

That last one is the check that a picture of a gear would otherwise pass by looking right, and it needed correcting twice before it measured anything.

Where the contact point goes, and why it matters #

The contact point does not sit still on either tooth. It sweeps along both flanks as the pair rolls through engagement, and the two flanks sweep at different rates — which means the surfaces slide against each other everywhere except at one instant.

That instant is at the pitch point. There the two pitch circles are rolling without slipping, so the surface velocities match and the sliding is zero. Everywhere else on the path of contact there is relative sliding, and the amount grows with distance from the pitch point.

Three consequences follow, and they are most of what distinguishes a real gear from the geometric one this site draws.

Wear is not uniform. The tooth wears at the tip and root, where sliding is greatest, and hardly at all near the pitch line. A worn gear has a characteristic profile with a ridge at the pitch circle, and it is diagnostic.

Efficiency is not perfect. Sliding against a normal force is friction, so a gear mesh loses power — typically one or two percent per mesh, which is small and compounds through a train.

Scuffing is a tip-and-root failure. The combination of high sliding velocity and high contact stress is worst at the extremes of the path of contact, which is where lubrication films break down.

None of that is computed here. This site draws the geometry of the contact and not the tribology of it, and the sliding velocity would be a natural thing for a later phase to measure since it follows directly from the same solved positions.

What the gear figures check #

Every gear figure here generates its flanks from the involute and then checks them, which is the site’s habit applied to a curve rather than to a linkage.

The base-radius ratio must equal the tooth-count ratio exactly. The contact ratio must fall in the usual range. No flank point may lie inside its base circle. And the normal to the drawn flank must be tangent to the base circle, to a tolerance bracketed below by the sampling noise of a correct involute and above by the error an arc-approximated flank produces — 2.0 × 10⁻⁵ against 7.6 × 10⁻², with the tolerance at 2 × 10⁻⁴ between them.

That last check is the one a picture of a gear would otherwise pass by looking right, and it needed correcting twice before it measured anything: once because it was circular, and once because the distance from a centre to a normal line is a cross product rather than a dot product. Both corrections are recorded where the law of gearing is derived.

Helical teeth, and buying contact ratio a second way #

Everything above concerns a spur gear, whose teeth are parallel to the axis and whose contact ratio comes entirely from the path of contact in the transverse plane. There is another source of overlap available, and it changes the mesh completely.

Cut the teeth at an angle to the axis and each tooth engages progressively along its width instead of all at once. A point at one end of the face starts contact before the point at the other end has arrived, so the engagement of a single tooth is spread over a range of rotation that has nothing to do with the transverse geometry.

That contribution is the overlap ratio, and it adds to the transverse contact ratio to give the total. Its size is set by the face width and the helix angle, both of which are free design variables — unlike the transverse contact ratio, which is fixed by the tooth counts, the addendum and the pressure angle, and is hard to push much above 1.8.

The result is that a helical pair can have a total contact ratio of three or four without any strain, which means three or four tooth pairs sharing the load at every instant and a load handover that is gradual instead of abrupt. That is why helical gearing is quieter, and the difference is large: a change of ten decibels or more for the same tooth counts and the same loads.

The cost is a thrust load. The angled teeth push each other along the axis as well as around it, and the bearings must take it. Double-helical or herringbone teeth cancel the thrust by having both hands on one gear, at a substantial manufacturing cost — cutting a herringbone requires either a groove between the two hands or a shaping process that can run out.

This site draws spur gears throughout, because the transverse geometry is the part where the law of gearing lives and a helical pair’s transverse section is a spur pair. What helical adds is a second, independent way of raising the overlap, and it is worth knowing that the quiet gearbox in most machinery is quiet because of the second one rather than the first.

Why contact ratio has a lower bound and no upper one #

The requirement that the contact ratio exceed one is absolute: below it, there are moments when no tooth pair is in contact, and the driven gear is free.

What happens then is not a smooth handover but a collision. The driven gear coasts, the next tooth arrives, and the two meet at a relative velocity. In an unloaded train that is a rattle; in a loaded one it is an impact whose force does not depend on the transmitted torque at all, and which will pit the flanks in service.

The convention is a minimum of 1.2 rather than 1.0, and the margin is for the same reasons the transmission-angle margin exists: manufacturing tolerance, tooth deflection under load, wear, and centre-distance variation, each of which erodes the true contact ratio below the nominal one. Tip relief erodes it further, deliberately.

There is no corresponding upper bound. More contact ratio is monotonically better for smoothness and load sharing, and the reason gears do not have contact ratios of five is that the geometry does not offer them cheaply: raising the transverse contact ratio means taller teeth, which means thinner tips and more sliding at the extremes of engagement, which brings back the wear and scuffing problems that tall teeth already have.

So the design lands where it does — around 1.4 to 1.7 for spur gears, more with helicals — because one end of the range is a hard constraint and the other is a soft cost.

What a gear pair is, restated #

Stripping the essay back to its claims: a gear pair transmits rotation at a constant velocity ratio because the common normal at the contact passes through a fixed point, the involute is a curve whose common normal does that, and the ratio is exactly the ratio of the tooth counts because those are proportional to the base radii.

Everything else in this essay is a qualification on that.

The contact ratio says how much of the path of contact is actually used and therefore how many tooth pairs share the load, and it must exceed one or the transmission is not continuous at all.

Backlash says the ratio holds in one direction of drive and there is a gap on reversal, which no amount of geometric correctness removes and which a preloaded or split gear is built to eliminate at a cost.

Interference and undercutting say the geometry is only available above a certain tooth count, or below it with corrections that change the tooth into something the simple theory does not describe. That is the seventeen-tooth rule and everything around it.

Sliding says the flanks rub everywhere except at the pitch point, so the pair loses power and wears unevenly, in a distribution the geometry predicts precisely.

The pattern is that the headline claim — a constant ratio, exactly — is true, and every practical property of the mesh lives in the qualifications rather than in the claim. That is nearly the opposite of the situation with a linkage, where the ratio is not constant and the variation is the first thing to understand rather than the last. Gears are the family where the simple statement is exact, which is precisely why they get used wherever an exact number is what the machine needs.

The fractional part is the whole trouble #

A contact ratio of 1.612 is described as more than one and less than two, and the useful reading is the fractional part, because that is what the mesh actually does over a cycle.

At 1.612 the pair spends 61.2 per cent of each mesh cycle with two tooth pairs in contact and 38.8 per cent with one. The load carried by an individual tooth therefore halves and doubles twice in every tooth pitch, and the transitions are abrupt — a tooth arrives, the load redistributes, another leaves. That alternation is the excitation, and its frequency is the mesh frequency, which is the tooth count times the shaft rate.

So the design tension is not keep the ratio above one. It is that any ratio which is not an integer produces alternation, and the amplitude of the alternation is set by how much of the cycle sits on each side. A ratio of 1.5 alternates as hard as it can — half the cycle with one pair and half with two. A ratio of 1.9 spends nine tenths of the cycle at two pairs and the alternation is a brief dip.

Which makes the ideal an integer, and an integer is not reachable. At exactly 2.0 there are always two pairs in contact, the load never redistributes, and the excitation from this source vanishes entirely. It is also a knife edge: a hair below and the drive drops to one pair for a moment of every cycle, so a design aimed at 2.0 is a design aimed at a boundary with two qualitatively different sides — the same shape of specification a removal cone that is a single ray has, and unwise for the same reason.

So real designs aim above the integer rather than at it, and that is the second argument for helical teeth alongside the one about smoothness. A total contact ratio of 2.4 sits comfortably clear of two with margin for manufacturing error, always has at least two pairs engaged, and never returns to one — which is a stronger property than high contact ratio and is the one worth specifying.

It also explains why the spur gear’s usual range of 1.4 to 1.7 is where it is. It cannot reach two without an impractical tooth count or pressure angle, so it sits in the middle of the alternating band, and every spur drive accepts the load alternation as a cost. Helical teeth are the arrangement that buys the way out of that band, and the overlap ratio is the currency.

Modules, diametral pitch, and the number that makes gears interchangeable #

One practical thread runs under everything above and rarely gets stated: two gears mesh only if they share a tooth size, and the way that size is specified is worth knowing because it is the whole basis of gear interchangeability.

The module is the pitch diameter divided by the tooth count — millimetres of diameter per tooth. Two gears of the same module and pressure angle mesh, whatever their tooth counts, and their centre distance is half the module times the sum of the counts. Anglophone practice uses diametral pitch, teeth per inch of diameter, which is the reciprocal idea in the opposite units.

The consequence is that a gear is specified by three numbers — module, pressure angle, tooth count — plus a face width and a quality grade, and any two gears agreeing on the first two will run together. That is why gears can be bought from a catalogue and why one hob cuts a whole family, which is the manufacturing property that decided the involute over the cycloid.

It also fixes what can be changed to solve a design problem. Centre distance is determined by module and tooth counts, so a required ratio and a required centre distance over-determine the gearset unless one of them is adjustable — which is where profile shift enters, since shifting both gears changes the operating centre distance without changing the tooth counts or the tool.

That is the same correction the seventeen-tooth rule uses for a different purpose, applied to a different constraint, with the same tool setting. Gear design has a small number of levers and most of them do more than one thing, which is why the standard tables exist and why the underlying geometry is worth having exact before the tables are consulted.