The tooth surface of long epicycloid bevel gear can be obtained by conjugate surface theory. Insert the tool into the cutting edge line Γ The potential vector of B is represented by Rb (U), as shown in the figure. In continuous indexing motion, Γ B the locus in the coordinate system scr0 constitutes the generating gear tooth surface. From the coordinate transformation matrix from SC to SCR, the tooth surface equation of the forming wheel can be obtained:
Similarly, from the coordinate transformation, the motion track of the forming wheel in the coordinate system S1 can be obtained:
According to the formula, the relative motion speed of the shaping wheel and the processed long epicycloid bevel gear is:
For the generating gear tooth surface, its normal vector can be expressed as follows under S1:
Thus, the meshing equation between the shaping wheel and the machined long epicycloid bevel gear can be obtained from the formula:
Namely:
According to the conjugate surface theory, the tooth surface equation of the long epicycloid bevel gear processed by the shaping wheel is composed of the formula, that is, the tooth surface of the processed long epicycloid bevel gear is described by the following equations:
