Gear reference circle ($d=mN$)

A gear reference circle is the imaginary circle used to specify an individual gear's tooth geometry. For a spur gear, its diameter $d$ is defined by gear module $m$ and tooth count $N$ through $d=mN$. It supplies a common datum for tooth spacing and other dimensions, whether or not that gear is currently meshing with another one. The operating pitch circle, in contrast, describes the mounted pair's motion.^[ref]

For a 30-tooth spur gear with gear module $2\mathrm{mm}$, the reference diameter is $60\mathrm{mm}$. Thus successive teeth repeat at an angular interval of $2\pi /30$ and a gear circular pitch of $2\pi \mathrm{mm}$ on that circle. In the standard full-depth, unshifted geometry specified by KHK, the tooth-tip diameter is $d_a=d+2m$, giving $64\mathrm{mm}$ in this example. Measuring the outside diameter and calling it the reference diameter would therefore produce the wrong gear module.^[ref]

The gear base circle is another distinct geometric datum. For a tooth profile constructed as the involute of a circle, the gear base circle generates the curved tooth flank, whereas the gear reference circle fixes its nominal sizing. Their diameters obey $d_b=d\cos \alpha$, with reference gear pressure angle $\alpha$. With $d=60\mathrm{mm}$ and $\alpha =20{}^{\circ }$, the base diameter is approximately $56.382\mathrm{mm}$. These three diameters—base, reference and tip—answer different questions about the same physical gear.^[ref]

A gear base circle is the fixed circle from which an involute tooth flank is generated. Imagining a taut string unwinding from that circle, the free end traces the involute. It is a geometric construction for the tooth shape, distinct from the gear reference circle used for sizing and the operating pitch circle describing a particular meshing pair.^[ref]

For base radius $r_b$ and unwinding parameter $u\ge 0$, one orientation of the involute isSquaring and adding gives $r=\sqrt{x^2+y^2}=r_b\sqrt{1+u^2}$. The local pressure-angle relation $\tan \alpha =u$ therefore gives $r_b=r\cos \alpha$. At the gear reference circle, this becomesFor reference diameter $d=60\mathrm{mm}$ and reference gear pressure angle $20{}^{\circ }$, the base diameter is $56.382\mathrm{mm}$, approximately. A change in mounting separation does not change this base diameter or the cut tooth profile; it changes which operating geometry describes the mesh.

In a compatible external pair, a common tangent to the two gear base circles is the gear line of action. Their fixed geometry gives $r_{b1}+r_{b2}=a\cos {\alpha }_w$, explaining why the operating pressure angle changes when $a$ changes. Repeating a tooth profile every $2\pi /N$ also gives gear base pitch $p_b=\pi d_b/N$ along the base circumference. Finally, an ideal involute generated this way lies on or outside its gear base circle; a real tooth's lower connecting portion can have a different shape. The gear base circle is not a physical boundary of the gear material.