Using a planar rack cutter with variable modulus to generate a spherical gear pair

In this study, an imaginary planar rack cutter with variable modulus and discrete conical teeth was used to generate a spherical gear pair with double degree of freedom. First, a geometric method was used to design the mathematical model of the imaginary planar rack cutter with variable modulus and discrete conical teeth. Next, the relationship of coordinate systems between the generating and generated surfaces was established. Then, the family of the imaginary planar rack cutter surfaces was obtained by homogeneous coordinate transformation matrix. Further, two equations of meshing between the generating and generated surfaces were determined by the two-parameter envelope theory. The mathematical model of spherical gear pair with variable modulus and discrete ring-involute teeth can be created by using the two equations of meshing and the family of the imaginary planar rack cutter surfaces. The mathematical models of the spherical gear pair with double degree of freedom and tooth contact analysis method were used to investigate the assembly errors of the gear pair in four cases.

Meshing principle analysis of 3-DOF involute spherical gear

In this paper, a mathematical model of meshing motion of three degree of freedom involute spherical gear pair is constructed. The mathematical model can realize continuous meshing transmission between gear pairs without transmission principle error. Based on the meshing principle and motion analysis of the gear, the tooth profile of the spherical gear is designed by combining the two tooth surfaces of the involute ring gear and the hemispherical bevel gear. According to the conjugate motion relationship of spherical gear pair, a mathematical model of arc tooth surface of hemispherical bevel gear is established, and the mathematical description of the tooth profile of spherical gear is completed by combining the equation of ring tooth surface. It provides the basis and Reference for the meshing design of ball gear.