A method (envelope-approximation theory) based on the surface moving frame was presented to design the global properties of point-contact tooth surface. The presented method proposed a universal approach to reconstruct the second tooth surface with motion invariants of the first tooth surface. The proposal reconstruction approach was formulated for approximating the conjugate surface of the first tooth surface to the first tooth surface under the movement transformation described with motion invariants of the first tooth surface. The basis function of the previously mentioned approximation was taken as the characteristic curve on the first tooth surface

the interpolation condition was designated as the local structure of the second tooth surface along the contact curve. Based on the experiment data of tooth contact

a variational inequality model for the global optimal design of tooth surfaces was established to determine the loaded meshing characteristics of the second tooth surface according to the position

orientation and dimension of the contact area on the first tooth surface under load-deformation condition. Different from the previous studies about tooth contact analysis

the presented method

freeing the tooth surface optimization design from unnecessary restrictions of the specific machine tool processing parameters

can satisfy the predesigned loaded meshing characteristics and ensure that the tooth surface can be generated by multi-axis flank milling with a conical cutter. A numerical example was given to validate that above-mentioned design method can construct unbroken spiral bevel gear tooth surface with excellent loaded meshing characteristics

for exanple

the loaded contact ratio was increased to 2.122 and the loaded transmission errors amplitude was lowered to 0.063°.