Gear module ($m=d/N$)

The gear module specifies the size of metric gear teeth through the gear reference circle. For a spur gear with reference diameter $d$ and $N$ teeth,The tooth count is dimensionless, so gear module has units of length, conventionally millimetres. A larger gear module means a larger reference diameter at fixed tooth count. It does not mean a larger gear ratio, which depends on the tooth counts of the meshing pair.^[ref]

The gear circular pitch $p$ is the arc length between equivalent points on adjacent teeth, measured along the gear reference circle. Dividing the circumference by the tooth count givesFor example, a spur gear with $m=1.5\mathrm{mm}$ and $N=24$ has reference diameter $36\mathrm{mm}$ and gear circular pitch $1.5\pi \mathrm{mm}\approx 4.712\mathrm{mm}$. A 48-tooth gear of the same gear module has diameter $72\mathrm{mm}$ but the same tooth spacing. Their standard axis separation is $(36+72)/2=54\mathrm{mm}$, and their reduction gear ratio is $48/24=2$.

Conversely, two gears can have the same diameter without compatible tooth spacing: $d=36\mathrm{mm}$ can describe 24 teeth of gear module $1.5\mathrm{mm}$ or 18 teeth of gear module $2\mathrm{mm}$. Such teeth do not form the specified standard pair. For ordinary compatible spur gears, matching gear module is necessary but does not by itself specify the complete tooth geometry: the gear pressure angle and other tooth details must also agree. The definition uses reference diameter, not the diameter through the tooth tips or the operating pitch circle of a particular mounting.

Gear circular pitch is the arc distance between corresponding points on successive gear teeth, measured on the gear reference circle. If that circle has diameter $d$ and tooth count $N$, its circumference contains $N$ repetitions, givingwhere $m$ is gear module. The length is measured along the reference arc, rather than as a straight chord between tooth positions.^[ref]

For a spur gear with $m=2\mathrm{mm}$ and $N=30$, reference diameter is $60\mathrm{mm}$, gear circular pitch is $2\pi \mathrm{mm}\approx 6.283\mathrm{mm}$, and angular tooth spacing is $2\pi /30$. The corresponding chord on the reference circle isThe small numerical difference matters when a dimensional inspection is intended to measure reference arc spacing. Increasing the tooth count while holding gear module fixed enlarges the gear reference circle but preserves gear circular pitch.

For an involute tooth profile, the gear base circle provides another spacing, the gear base pitch. Since $d_b=d\cos \alpha$, with reference gear pressure angle $\alpha$,At $\alpha =20{}^{\circ }$, the preceding example gives $p_b\approx 5.904\mathrm{mm}$. Compatible contact must preserve the spacing of successive tooth flanks along their gear line of action; this explains why equal gear circular pitch with incompatible gear pressure angles is not enough to specify a mating pair. Tooth thickness and backlash are additional quantities: gear circular pitch includes an entire tooth-and-space repetition.

Gear base pitch is the spacing of successive involute tooth profiles referred to the gear base circle. For base diameter $d_b$, tooth count $N$, gear module $m$ and reference gear pressure angle $\alpha$,It also gives the spacing between successive corresponding tooth contacts measured along the gear line of action. The gear circular pitch uses the reference circle instead, giving $p=\pi m$ and $p_b=p\cos \alpha$.^[ref]

For $m=2\mathrm{mm}$ and $\alpha =20{}^{\circ }$, $p=6.2832\mathrm{mm}$ but $p_b\approx 5.9043\mathrm{mm}$. A second gear with the same module but $25{}^{\circ }$ reference angle has $p_b\approx 5.6945\mathrm{mm}$. Matching reference tooth spacing alone therefore does not make these the same specified mating geometry.

Changing the mounting separation of a compatible involute pair does not change its cut profiles, base circles or gear base pitch. In operating-circle notation, $p_w=2\pi r_w/N$ changes with mounted radius, whilestays fixed. This links the mounted operating pressure angle to an invariant tooth-profile spacing.

For a tooth rotating at angular speed $\omega$, base-circle tangential speed is $r_b|\omega |$. Dividing by gear base pitch gives the gear mesh frequency,A 30-tooth gear at $600$ revolutions per minute gives $f_m=300\mathrm{Hz}$. Finally, the gear contact ratio divides usable gear path of contact length by this base pitch, so profile spacing and available contact length jointly determine overlap between successive tooth engagements.