§1Why the involute
Gears exist to transmit rotation at an exact, constant ratio. The tooth profile that guarantees this is the involute — the curve a taut string traces as it unwinds from a circle.
The requirement is the fundamental law of gearing: for the velocity ratio to stay constant, the common normal at the point of tooth contact must always pass through a fixed point on the line of centres — the pitch point. The involute satisfies this automatically, and it brings two gifts besides: the ratio is unaffected by small errors in centre distance (the gears simply mesh a little differently but still constant-ratio), and every tooth can be cut with a straight-sided rack cutter or hob, which is why involute gears are cheap to make to precision. Almost every power-transmission gear in service is an involute.
Contents§2Module and pitch
Tooth size is set by one number. In metric practice it is the module; in inch practice, the diametral pitch — reciprocals of each other.
Here d is the pitch diameter and z the number of teeth. Two gears only mesh if they share a module (and pressure angle) — the module is the compatibility key. Standard proportions follow from it: addendum (tooth above the pitch circle) a = m, dedendum b = 1.25 m, so the whole depth is 2.25 m. For the worked pair below (m = 4 mm), the circular pitch is π × 4 = 12.566 mm, the addendum 4 mm and the whole depth 9 mm.
Contents§3The circles of a gear
A gear is described by a family of concentric circles; the pitch circle is the one that matters most, being the imaginary rolling circle on which the ratio is defined.
| Circle | Definition | Pinion value |
|---|---|---|
| Pitch | d = m z | 80 mm |
| Base (involute origin) | d_b = d cos φ | 75.18 mm |
| Addendum (outside) | d_a = d + 2m | 88 mm |
| Dedendum (root) | d_f = d − 2.5m | 70 mm |
| The involute exists only outside the base circle — below it the flank is a non-working fillet. This is why the base circle, not the pitch circle, is the true parent of the tooth, and why the hero flank is drawn unwinding from it. | ||
§4Pressure angle and the line of action
Teeth do not push along the tangent; they push along the line of action, inclined to it by the pressure angle φ — the standard value being 20°.
The line of action is tangent to both base circles and passes through the pitch point; all contact travels along it, and the force between teeth acts along it too. That force therefore has a component driving the gear round (the useful tangential load) and a component pushing the shafts apart (the separating, or radial, load = tangential × tan φ). A larger pressure angle gives stronger, stubbier teeth but higher separating force and bearing load; 20° is the near-universal compromise, with 14.5° surviving in older work and 25° used where tooth strength must be maximised.
Contents§5Ratio and centre distance
Two numbers fall straight out of the tooth counts — the ratio the gears provide and the distance their shafts must sit apart.
Pinion z₁ = 20, gear z₂ = 60, module 4 mm. Pitch diameters 80 mm and 240 mm; ratio i = 60/20 = 3 : 1 (a 1500 rev/min input leaves at 500 rev/min); centre distance C = (80 + 240)/2 = 160 mm. Because C = m(z₁+z₂)/2, the only centre distances available for a given module are fixed by the tooth totals — a real constraint when a gearbox housing is already cast.
§6Contact ratio
Smooth running needs more than one tooth pair engaged on average, so that load is always being handed over, never dropped. That average is the contact ratio.
Feeding the worked pair’s radii into the formula (addendum radii 44 and 124 mm, base radii 37.59 and 112.76 mm, C = 160, φ = 20°) gives a contact ratio of 1.67. It means that on average between one and two tooth pairs carry the load — for two-thirds of the mesh cycle two pairs share it, for the remaining third a single pair carries alone. A contact ratio below about 1.4 runs rough and noisy; below 1.0 the drive would momentarily disengage. More teeth, finer module or a smaller pressure angle all raise it.
§7Undercut and minimum teeth
Cut too few teeth on a pinion and the generating tool gouges the flank below the base circle — undercut — weakening the root and removing working profile.
Below the minimum, the standard cures are a larger pressure angle (25° drops the minimum to about 12), or profile shift — feeding the cutter outward to move the tooth off the undercut zone, at the cost of a modified tooth thickness and a corrected centre distance. Profile shift is also the standard trick for making a chosen tooth count land on a required centre distance. For a first design, keeping every pinion at 18 teeth or more sidesteps the problem entirely.
Contents§8Quick reference
The working core of the page on one card rack.
Size
m = d/z · p = πm
P = 25.4/m
Circles
d_b = d cos φ
d_a = d + 2m
Ratio
i = z₂/z₁ · C = m(z₁+z₂)/2
Meshing
contact ratio > 1.4
force ⟂ = F_t tan φ
Undercut
z_min = 2/sin²φ
20° → 18 teeth