Meshing equation and tooth profile equation of noncircular gear

In order to get the meshing equation and the tooth profile equation of non-circular gear, it is necessary to establish the tooth surface equation of gear shaper cutter. Take the OS XSYS plane as the projection plane, as shown in the figure, project the end face of the gear shaper cutter’s involute profile onto it.

In the figure, θ OS is the parameter used to determine the slot width of the gear shaper cutter on the base circle. The solution formula of θ OS in the figure is shown in the formula


ζ s — number of teeth of cutter;

α – pressure angle of tool.

When solving the tooth profile equation of the nth tooth, the angle variable θ KS of any point on the tooth profile involute is:

It can be seen from the figure that in the moving coordinate system SS, the cutter tooth surface vector RRRS can be expressed as a formula. The tooth profile equations of the left tooth profile and the right tooth profile of the tooth are not exactly the same, and there is a difference of “±”. The involute tooth profile of any tooth is composed of the left tooth profile β and the right tooth profile γ.


θ s — angle variable (RAD) of a point on the involute of cutter tooth profile;

Us is the projection of a point on the involute tooth profile of the tool in the z-axis direction of the coordinate system SS (mm);

Rb is the base circle radius of the tool (mm).

The unit normal vector ns of gear shaper cutter tooth surface is as follows: