Integrated Design Symmetry Method for Point Meshing Tooth Surfaces Based on Surface Envelope Approximation Theory

Based on the idea of a surface moving frame in differential geometry, a surface envelopment approximation method is proposed for the integrated design of point-contact tooth surfaces. This method utilizes the envelopment characteristic curve of the first tooth surface as the spline curve and adopts the local structure of the second tooth surface along a predesigned contact path as the surface interpolation condition. Through motion transformation described by the motion invariants of the first tooth surface, a conjugate motion space for the second tooth surface is fully defined by the motion invariants of the first tooth surface. This constitutes the basis of the integrated optimization design space and ensures the global optimization and machinability of the tooth surface design method. Using the experimental data of the point meshing tooth surface loading contact, the gap between the two tooth surfaces during no-load meshing is used as the design target parameter to predict and control the shape and size of the contact area under heavy load and further the symmetry requirements of the tooth surface design. Consequently, a variational inequality model for the global optimal design of the point meshing tooth surface is established.