Contact locus equation of meshing point of face gear

In the meshing transmission, the rotation angles of the cylindrical gear and the face gear are  1 and  2, respectively. The tooth surface equation and the tooth surface normal equation of the cylindrical gear in its fixed coordinate system S10 are as follows:

Where: ψ 1 (28) ψ 1 ± ( 10 (10)  1); [M01] is the coordinate conversion matrix from the dynamic coordinate system S1 of the cylindrical gear to its fixed coordinate system S10; [l 01] is the matrix that removes the fourth row and the fourth column from [M01].

By substituting U1 (28) R1 Q12, the equation R1 ( 1) of meshing locus on cylindrical tooth surface is obtained as follows:

According to the relation of coordinate transformation, the meshing trace equation of face gear on its coordinate system S2 is as follows: