Machine Kinematics
solved, then drawn
Teeth

What happens in a mesh

Contact between two gear teeth happens only along one straight line, and only over part of it. How much of that line lies between the two tip circles, divided by the base pitch, is the contact ratio — and if it drops below one the drive periodically stops being driven.
16 min read 6 figures The constant that is not constantSolved, not drawn

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.

20 teeth driving 32Both flanks generated from the involute, not approximated. The orange line is the line of action — tangent to both base circles, and the only place contact happens. Its length between the two tip circles divided by the base pitch is the contact ratio, 1.612 here, which means that for 61% of the cycle two tooth pairs are carrying the load and for the rest just one. The velocity ratio is 0.6250, and it is constant because the common normal never moves.pitch pointline of actionmodule 1, 20° pressure angle, centre distance 26contact ratio 1.612
Fig. 1 Twenty teeth driving thirty-two. The orange line is the line of action; the contact point runs along it. Drag to turn the gears — the line does not move, which is the whole reason the ratio is constant.

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.

Ratios, and the gear that does not change oneThe ratio of a train is the product of its stages, so the middle gear of a simple three-gear train cancels: 30/20 × 40/30 = 40/20, exactly as if it were not there. What it does change is the direction, since every external mesh reverses — which is the entire reason idlers exist. A compound train, where two gears share a shaft, does not cancel, and that is how large reductions are built without absurdly large wheels.reduction ratio20 → 402.00 : 1reversed20 → 30 → 40 (idler)2.00 : 1same direction20 → 40, 15 → 45 compound6.00 : 1same direction20 → 60, 20 → 60, 20 → 6027.00 : 1reversedproduct of the stages · sign flips at every external meshthe idler cancels exactly
Fig. 2 Ratios in trains, where the geometry above is used but the mesh detail is not. The middle gear of a three-gear train cancels exactly — 30/20 × 40/30 = 40/20 — which is why idlers exist to change direction rather than ratio.

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.

Undercutting, either side of 17.10 teethFive gears, drawn whole and scaled to a common pitch circle so the tooth counts can be compared by counting them. The dedendum sits a fixed 1.25 modules below the pitch circle and the base circle sits at r·cos α, so as the tooth count falls the base circle rises relative to the root. Below N = 2/sin²α = 17.097 it rises above it, and the part of the flank between them lies where no involute exists — the cutter removes it. The familiar rule says seventeen; the exact figure is 17.10, so seventeen undercuts slightly and eighteen is the smallest count that does not. The teeth are not to a common scale, because at a common module the small gears would be unreadable.10 teethundercut14 teethundercut17 teethundercut18 teethclean24 teethcleanred: the root circle has risen above the base circlethreshold N = 2/sin²α = 17.097
Fig. 3 The cutting side of the same fact: below a threshold tooth count the root rises above the base circle and part of the flank is removed by the cutter.

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.

The involute, unwoundHold a string taut against a circle and unwind it: the end traces this curve. Two properties follow immediately and between them they are the whole of gear geometry. The string is always tangent to the base circle, and it is always perpendicular to the curve. So the normal to an involute at any point is a tangent to its base circle — measured on the generated points here, to 2.3e-5 — and when two involutes touch, the common normal is a line tangent to both base circles, which does not move as the gears turn.the traced pointtangencybase circle r = 9.4020 teeth, module 1, 20° pressure anglenormal meets the base circle to 2.3e-5
Fig. 4 The property being checked: the string is tangent to the base circle at every position, and the string is the normal to the curve. Everything in a mesh follows from that.
Sun 24, ring 72, planet 24An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. With the ring held, a sun input turns the carrier at 1 + 72/24 = 4.000 to one. Willis's equation and the tabular superposition method are independent derivations and both are computed here; the site requires them to agree, because planetary ratios are the most commonly mis-stated numbers in mechanism work and the sign errors are notorious.ringsunhold the ringsun in, carrier out4.000 : 1same directionhold the carriersun in, ring out−3.000 : 1output reverseshold the sunring in, carrier out1.333 : 1same directionWillis: (ω_s − ω_c)/(ω_r − ω_c) = −72/24both derivations agree, and the build requires it
Fig. 5 Meshes with a moving centre. Every engagement in an epicyclic is an ordinary involute mesh; what makes the train interesting is that one shaft centre orbits.
A ratio that is not a numberThe output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.-0.25000.2500.5000100200300crank angle (degrees)output ÷ input angular velocitya 20:40 gear pair, 0.500mean 0.000four-bar 4/1/3.5/3-0.40 to 0.33 through one turn
Fig. 6 The property being relied on. A gear pair holds its ratio at every instant; the linkage plotted here does not, and the difference is the whole reason for the involute.

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.

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.

Where this goes next

the traced pointtangencybase circle r = 9.4020 teeth, module 1, 20° pressure anglenormal meets the base circle to 2.3e-5 Why a tooth is an involute A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined. 10 teethundercut14 teethundercut17 teethundercut18 teethclean24 teethcleanred: the root circle has risen above the base circlethreshold N = 2/sin²α = 17.097 Undercutting, and the seventeen-tooth rule Below a certain tooth count a standard cutter eats into the flank it is supposed to be forming. The count is quoted as seventeen. The formula gives 17.097, which means seventeen undercuts slightly and eighteen is the smallest that does not — the rounding went the convenient way. ringsunhold the ringsun in, carrier out4.000 : 1same directionhold the carriersun in, ring out−3.000 : 1output reverseshold the sunring in, carrier out1.333 : 1same directionWillis: (ω_s − ω_c)/(ω_r − ω_c) = −72/24both derivations agree, and the build requires it Epicyclic ratios, two ways An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so this site computes every ratio by Willis's equation and by the tabular method and requires them to agree.