According to the meshing theory and installation relationship of hyperboloid gear, the gear tooth surface is regarded as an imaginary hyperboloid gear. The gear cutter and the small wheel workpiece are rolled according to the nominal transmission ratio, and the small wheel tooth surface which is completely conjugate with the big wheel of hyperboloid gear is obtained. The ease off modification tooth surface changes the inter tooth clearance and the normal clearance of the conjugate tooth surface. The inter tooth clearance is shown in figure (a); The normal clearance of the tooth surface is shown in figure (b), which can be mapped by the tooth profile modification curve in figure (c) through rotation transformation. In the figure ε 0 ～ ε 4 and λ 1 ～ λ 2 is the contact clearance parameter between teeth, D1, D2, Q1, Q2 and θ A is the normal contact clearance parameter of tooth surface.
The normal clearance of the tooth surface is superimposed on the tooth surface of the hypoid gear pinion which only contains the transmission error δ 1. The analytical tooth surface of the modified hypoid gear pinion is obtained, and its ease off surface expression is:
Where: R10 and N10 are the position vector and normal vector of the tooth surface of the small gear completely conjugate with the large gear of the hyperboloid gear; R1 and N1 are the position vector and normal vector of the tooth surface of the small gear of the hyperboloid gear containing only the transmission error; Ｒ1 γ、 N1 γ It is the position vector and normal vector of the tooth surface for ease off modification of the hypoid gear pinion; δ Is ease off modification amount; u、 β Is the theoretical tooth surface parameter of hypoid gear pinion.