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A hobbing machine's cutting mechanism is a mechanism with multi-degree of freedom during the cutting process. In this paper, we propose a general gear mathematical model simulating the generation process of a 6-axis CMC hobbing machine based on the cutting mechanism of CNC hobbing machine and worm-type hob cutter. The proposed gear mathematical model can be applied to simulate different types of gear cutting. Some examples are included to verify the mathematical model. Also, a novel type of gear named "Helipoid" which can be used in crossed axes transmission is proposed. The proposed general gear mathematical model can facilitate a more thorough understanding of generation processes and toward the development of novel types of gears.7848
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Introduction
Hobbing, shaping, and other special purpose machines are widely used in industry to manufacture different types of gears.Owing to easy tool setting, high efficiency and reliable quality,the hobbing machine is conventionally used in manufacturing spur, helical, and worm gears. The development of CNC hobbing machines has recendy made it possible for cutting gears with high-productivity and high-accuracy. The loading and unloading time of a gear blank are also significantly reduced. By using the CNC hobbing machine with different manufacturing processes, gears with novel shapes can be manufactured for power transmissions in parallel, intersected and crossed axes.However, the CNC hobbing process is complicated owing to its complex tool geometry, tool setting, and the cutting motion of multi-degree of freedom. Until now, this topic has received only limited attention. Most investigations involving tooth geometry have based on the rack cutter generation with one degree of freedom (Litvin and Tsay, 1985; Litvin, 1989). The generation with multi-degree of freedom has been seldom studied.Litvin et al. (1975, 1994) proposed the concept of multi-degree of freedom to apply to the theory of gearing. Chakraborty and Dhande (1977) investigated the geometry of spatial cams with two degrees of freedom, i.e., Camoid and Conoid. Also, Tsay and Hwang (1994) applied the envelope theory to study the geometry of Camoid. Moreover, Mitome (1981) used the envelope theory to study the hobbing of a conical gear. Wu (1982) investigated the hobbing process of a hobbing machine with multi-degree of freedom. However, the above models can not be applied to a 6-axis CNC hobbing machine and therefore, can not adequately simulate and develop new types of gears.
In this paper, we first set up the cutting mechanism of a CNC hobbing machine and mathematical model of a hob cutter. The kinematic relationship can be obtained on the basis of the cutting mechanism and transformation matrices. Based on the cutting mechanism, generation concept with multi-degree of freedom,and theory of mechanisms, a general mathematical model for the gear hobbing simulation on a 6-axis CNC hobbing machine can be developed. By properly selecting the parameters of the developed general gear mathematical model, the equations of the tooth surfaces for different types of gears can be obtained and the respective gear tooth surfaces can also be cut by using the CNC hobbing machine.
The crossed helical-gear is used in crossed axes power transmission.However, the load capacity of the crossed helical-gear is comparatively lower than the hypoid gear due to its point contact and low contact ratio. In this paper, we propose a general gear mathematical model to derive a novel type of gear named "Helipoid" (as invented by the third author Professor Nagata).An illustrative example demonstrates the effectiveness of the proposed mathematical model and a gear cut by a CNC hobbing machine with multi-degree of freedom.
The general gear mathematical model, capable of simulating the gear cutting process of CNC hobbing, can facilitate the manufacturer in gear design and manufacturing. The general gear mathematical model can also be applied to the design and manufacturing of spur, helical, worm gears and noncircular gears. Results shown in this study also provide the industry an important software for design, analysis, and manufacturing of various types of gears.