# Three dimensional simulation of hypoid gear with small number of teeth and large reduction ratio

Hypoid gear has the advantages of strong bearing capacity, high transmission efficiency and large overlap coefficient. It is widely used in vehicles, engineering machinery, electric tools and other fields. In recent years, the demand for large reduction ratio transmission is increasing. Therefore, the research and development of hypoid gear transmission with small number of teeth and large reduction ratio has important theoretical significance and application value. This paper is based on the theory of forming method and the principle of envelope method. The geometric design and three-dimensional modeling method of hypoid gear with small number of teeth and large reduction ratio are studied. Because the number of small gear teeth is less than 5, the means of undercutting inspection, orthogonal trial calculation and three-dimensional simulation are comprehensively applied in the design, and a variety of design examples of hypoid gears with large reduction ratio are completed. Finally, the machining of hypoid gear pair with tooth ratio of 4:41 is completed. The main conclusions of this paper are as follows:

The geometric design principle of hypoid gear with small number of teeth less than 5 is studied, and the geometric constraints of hypoid gear design with small number of teeth and large reduction ratio are given. These conditions include undercutting, sharpening of tooth top, meshing interference of tooth surface, restriction of surface cone size, and analysis of gear stress. When the transmission ratio reaches more than 10, radial displacement and tangential displacement must be considered comprehensively. Using the orthogonal analysis method, the main geometric parameters affecting the gear pitch cone angle are analyzed. It is found that the selection of cutter head radius and helix angle has the greatest influence on the gear pitch cone.

According to the meshing relationship between large and small wheels, the machining model of large wheel forming method and the tooth surface model of large wheel into small wheel are established. Based on the principle of gear meshing and coordinate transformation, the tooth surface equations of large wheel and small wheel are deduced.

Using MATLAB programming software and the method of tooth surface rotation projection, the three-dimensional points of tooth surface are solved. According to the solved three-dimensional coordinate points, the tooth surface of large and small wheels is established by using UG three-dimensional software, and then the three-dimensional modeling of large and small wheels is completed. Through three-dimensional modeling, it is found that the tooth profile does not change when the number of small teeth is reduced to 4, 3 or even 2 teeth, which verifies the feasibility of the hypoid gear with small number of teeth and large speed ratio proposed in this paper.

According to the geometric design of hypoid gear with few teeth, the machining parameters of large and small wheels are calculated, and the tooth contact analysis is carried out. Under the condition of ensuring the credibility of machining parameters, the tooth cutting test of hypoid gear with 4:41 teeth is completed on gh-35 gear milling machine. From the test results, the actual tooth profile and three-dimensional modeling results are known, which verifies the correctness of the theoretical model of hypoid gear with small number of teeth and large speed ratio and the feasibility of large speed ratio transmission.

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