Designing Gear Cutters

Making gears requires making a cutter with the appropriate profile. The amateur machinist usually does this using the button method developed by Ivan Law and further improved by John Stevenson. They have both given tables for the button diameter, button spacing and infeed required to produce the cutter. This method relies on the fact that the involute curve can be closely approximated to a circle over the limited distance of a gear tooth.

Relatively little information is available on the methods used to create the tables of button diameter, button spacing and infeed for the button method. On this page Mike (see credits below) has derived generalised equations for these parameters based on the geometry. With these equations it is straightforward to design cutters for any gear given the number of teeth, the module and the pressure angle.

Some Definitions

Number of teeth (N) This is simply the number of teeth on the gear.

Pitch circle diameter (pcd) This is the diameter of an imaginary circle running through the gear teeth that is used to calculate the tooth profile.

Module (M) This is the pitch circle diameter divided by the number of teeth. Thus pcd = NM.

Crest circle diameter (ccd) This is the circle that touches the extremities of all the teeth. This is the outside diameter of the disc from which the gear is made. The ccd = (N+2)M

Root circle diameter (rcd) This is the circle that touches the base of all the teeth. The rcd is equal to the ccd minus twice the full depth of the teeth(D) + clearance (f). The full depth of the tooth + clearance is usually expressed as D+f and this is the depth the gear cutter must be fed into the gear blank when cutting the teeth. By convention D+f = 0.6866 x circular pitch = 0.6866 (PI) NM/N =2.157M. Thus the root circle diameter = (N+2)M - 4.314M = (N-2.314)M

Base circle diameter (bcd) This is the circle that is used to generate the involute profile of the teeth.

Pressure angle (p) This is the angle AOC in the diagram above. A line drawn from the intercept of the gear tooth with the pitch circle forms a tangent to the base circle at the pressure angle.

Angular pitch (ap) This is the angle between any point on one tooth to a similar point on the next tooth.It is given by 360/N in degrees or 2(PI)/N in radians where PI = 3.142. The angle BOA in the diagram above is equal to 1/2 the circular pitch.

Button diameter (bd) This is the diameter of the button used to form the cutter. The button radius AC extends from the pitch point A on the tooth to a tangent C on the base circle.

Geometry

In the diagram the line AC is a tangent to the base circle and it is thus at right angles to the radius OC.

Calculation of the base circle diameter

$\mathrm{OC}=\mathrm{AO}\cos <!--FUNCTION\; APPLICATION-->\left(p\right)\text{but}\mathrm{OC}=\frac{\mathrm{bcd}}{2}\text{and}\mathrm{AO}=\frac{\mathrm{pcd}}{2}$

Thus

$\mathrm{bcd}=\mathrm{pcd}\cos <!--FUNCTION\; APPLICATION-->\left(p\right)\text{and}\mathrm{bcd}=\mathrm{NM}\cos <!--FUNCTION\; APPLICATION-->\left(p\right)$

Calculation of the button diameter

$\mathrm{AC}=\mathrm{AO}\sin <!--FUNCTION\; APPLICATION-->\left(p\right)\text{but}\mathrm{AC}=\frac{\mathrm{bd}}{2}\text{and}\mathrm{AO}=\frac{\mathrm{pcd}}{2}$

Thus

$\mathrm{bd}=\mathrm{pcd}\sin <!--FUNCTION\; APPLICATION-->\left(p\right)\text{and}\mathrm{bd}=\mathrm{NM}\sin <!--FUNCTION\; APPLICATION-->\left(p\right)$

Calculation of the button separation

The triangle CEO is a right angled triangle. Hence:

$\mathrm{CE}=\mathrm{CO}\sin <!--FUNCTION\; APPLICATION-->\left(\text{angle}\mathrm{COE}\right)$

$\mathrm{CE}=\mathrm{CO}\sin <!--FUNCTION\; APPLICATION-->(\text{angle}\mathrm{AOC}+\text{angle}\mathrm{AOG})$

$\mathrm{CE}=\mathrm{CO}\sin <!--FUNCTION\; APPLICATION-->\left(p+\frac{\mathrm{ap}}{4}\right)$

$\mathrm{CE}=\mathrm{CO}\sin <!--FUNCTION\; APPLICATION-->\left(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\right)$

But $\mathrm{CE}=\frac{1}{2}$ the button separation and $\mathrm{CO}=\frac{\mathrm{bcd}}{2}$. Hence:

$\mathrm{bs}=\mathrm{bcd}\sin <!--FUNCTION\; APPLICATION-->\left(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\right)$

but $\mathrm{bcd}=\mathrm{NM}\cos <!--FUNCTION\; APPLICATION-->p$, thus

$\mathrm{bs}=\mathrm{NM}(\cos <!--FUNCTION\; APPLICATION-->p)\{\sin <!--FUNCTION\; APPLICATION-->(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\left)\right\}$

Calculation of the infeed

Mike measures infeed from the outer edge of the buttons ie. point G. The infeed (inf) is the distace GF ie from point G to the root circle.

$\mathrm{GF}=\mathrm{CH}-\mathrm{EF}=\mathrm{CH}-(\mathrm{FO}-\mathrm{EO})$

$\mathrm{CH}=\frac{\text{button diameter}}{2}$

$\mathrm{FO}=\frac{\text{root circle diameter}}{2}$

$\mathrm{EO}=\frac{\mathrm{bcd}}{2}\cos <!--FUNCTION\; APPLICATION-->\left(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\right)$

but $\mathrm{bcd}=\mathrm{pcd}\cos <!--FUNCTION\; APPLICATION-->p$. Hence

$\mathrm{EO}=\frac{\mathrm{NM}}{2}(\cos <!--FUNCTION\; APPLICATION-->p)\{\cos <!--FUNCTION\; APPLICATION-->(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\left)\right\}$ Hence:

$\mathrm{CE}=\frac{\text{button separation}}{2}$ and $\mathrm{CO}=\frac{\mathrm{bcd}}{2}$. Hence:

$\inf =\frac{\text{button diameter}}{2}-\frac{\text{root circle diameter}}{2}+\frac{\mathrm{NM}}{2}(\cos <!--FUNCTION\; APPLICATION-->p)\{\cos <!--FUNCTION\; APPLICATION-->(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\left)\right\}$

$\inf =\frac{\mathrm{NM}\sin <!--FUNCTION\; APPLICATION-->p-(N-2.314)M+\mathrm{NM}(\cos <!--FUNCTION\; APPLICATION-->p\{\cos <!--FUNCTION\; APPLICATION-->\left(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\right)\}}{2}$

$\inf =\frac{M[N\sin <!--FUNCTION\; APPLICATION-->p-(N-2.314)+N(\cos <!--FUNCTION\; APPLICATION-->p\{\cos <!--FUNCTION\; APPLICATION-->(p+\frac{\mathrm{\pi <!--GREEK\; SMALL\; LETTER\; PI-->}}{2N}\left)\right\}]}{2}$