When deriving the helical gear equation, the main parameters of the tool need to be determined are: normal surface modulus: MNS, number of teeth: NS, normal surface pressure angle: α NS, pitch circle helix angle: β， Tooth top height: ha, tooth root height: HF, indexing circle radius: RBS.
The tool coordinate system is shown in Fig. 1. Corresponding to the coordinate system in Fig. 2, the involute tooth surface vector RS equation of the tool can be obtained:
Where RBS is the radius of the base circle, θ S is the angle parameter of a point on the tool involute, θ So the angle parameter from the symmetry line of the tool slot to the starting point of the involute. The “±” symbol in the formula corresponds to the tool tooth surface I and tooth surface II respectively. ZS is the helix angle of helical motion in the coordinate system ZS, and PS is the helix parameter. among θ So is determined by the following formula:
Where ns is the number of teeth of the tool; α NS is the normal pressure angle of the tool; inv α NS is the involute function of pressure angle, and its value is:
Set in the tool coordinate system SS, the tool tooth surface equation is expressed as:
The normal vector ns of the tool tooth surface is:
Where NXS, NYS and NZS are the three coordinate components of the normal vector NX respectively.
The unit normal vector of the tool tooth surface is:
β B is the helix angle of the tool base circle.