# Objective tooth surface conjugate tooth surface of hypoid gear pinion

In order to obtain the ideal conjugate surface of the small wheel, according to the theory of the big wheel, the small wheel surface matched with it is developed. At this time, all the instantaneous transmission errors are zero. According to the generating principle, the conjugate surface of the pinion must satisfy the gear transmission law and meshing equation, as shown in the formula

Where, φ 1 and φ 2 represent the blade angle of the small wheel and the angle of the big wheel, where the rotation angle satisfies 1, φ 1 = Z1 / Z2 * φ 2, and Z1 and Z2 denote the number of pinion and gear teeth. VG is the relative velocity of the paired teeth at the instantaneous contact point in the conjugate coordinate system GS. In this case, S1 coincides with the conjugate coordinate system SG, and there is φ g = φ 1, then the position vector GR and normal vector ng of the conjugate tooth surface of the pinion are:

Where, M12 is the coordinate transformation matrix used for the transformation from the large wheel coordinate system to the conjugate small wheel coordinate system:

Among them:

V is the offset distance, H1 and H2 are the axial installation distance difference between the small wheel and the large wheel, and ∑ is the axle intersection angle. 