Mathematical model of S-shaped gear surface

In this study, an imaginary S-shaped surface rack cutter was used to create a gear pair with S-shaped surface. First, a mathematical model of the imaginary S-shaped surface rack cutter was constructed by using the geometry. Then, a family of the imaginary rack cutter surfaces was obtained through the homogeneous coordinate transformation matrix. The equation of meshing was calculated by using differential geometry. A mathematical model for the S-shaped surface gear pair was determined by substituting the equation of meshing into the family of imaginary rack-cutter surfaces. The kinematic errors of the gear pair were calculated by setting the assembly error and using tooth contact analysis. Contact between the gear and the pinion was simulated by using the interference function of a computer-aided design software package. To investigate the real contact between the pinion and the gear, a rapid prototype machine was used to manufacture a real gear pair. Three of the teeth on the real gear were coated with dyes. After running the gear for long time, the dyes on the teeth were scraped and rubbed, and they were evenly rolled over the teeth.