Mathematical model and surface deviation of cylindrical gears with curvilinear shaped teeth cut by a hob cutter

Helical gears are widely used as parallel-axis power transmission devices, however they transmit power with axial thrust forces. The cylindrical gear with curvilinear shaped teeth transmits power without inducing axial thrust forces, therefore, this kind of gear can be considered to substitute the helical gear. Liu 1 proposed the manufacture of the cylindrical gear with curvilinear shaped teeth by a face mill-cutter with a special machine, and stated that the merits of curvilinear-tooth gears included higher bending strength, lower noise, better lubrication effect, and no axial thrust force. Dai et al. 2 proposed the manufacture of a cylindrical gear with curved teeth by a CNC hobbing machine with an attachment for the hob head, male and female flying cutters. Tseng and Tsay 3 considered an imaginary rack cutter with a curved-tooth to develop the mathematical model of cylindrical gears with curvilinear shaped teeth, and investigated the tooth undercutting of curvilinear-tooth gears. Andrei et al. 4 developed a special cutting tool to generate the curved face width gears for nonmetallic materials. Owing to easy tool settings, high efficiency and reliable quality, hob cutters have been widely used for manufacturing a variety of gears such as spur, helical, and worm gears. A hob cutting mechanism is a mechanism with multiple degrees of freedom in the process of gear generation. Chang et al. 5 proposed a general gear mathematical model simulating the generation process of a 6-axis CNC hobbing machine when the hob’s swivel axis is fixed. Litvin and Seol 6 investigated the necessary and sufficient conditions of the envelope for a two-parameter family of surfaces. It is not easy to manufacture curvilinear-tooth gears by using an ordinary hobbing machine. When the curvilinear-tooth gear is generated by a hob cutter, the hob rotates with an angular velocity about the hob’s swivel axis, and the revolution center of the hob’s swivel translates with a velocity v along the worktable axis. However, the linear velocity v correlates with angular velocity . In this study, the authors first set up the cutting mechanism of a CNC hobbing machine and develop the mathematical model of a hob cutter. According to the cutting mechanism, the kinematic relationship between the hob cutter and work piece can be obtained. The mathematical model of the curvilinear-tooth gear hobbing simulation for a 6-axis CNC hobbing machine can be developed based on the proposed cutting mechanism, generation concept with multiple degrees of freedom, and theory of mechanisms. Using computer graphics, a three-dimensional tooth surface of curvilinear-tooth gears can be plotted. In addition to developing a mathematical model for cylindrical gears with curvilinear shaped teeth cut by a hob cutter, this study investigates the relationship between tooth surface deviations and machinetool settings. The proposed mathematical model, which is capable of simulating the gear cutting process for the CNC hobbing with a hob cutter, can facilitate gear manufacturers to design and manufacture curvilinear-tooth gears.