Japanese scholar Takao Sakai applied mathematical tools such as binary vector and tensor in the study of spiral bevel gear theory, which made the demonstration of relevant theories concise and rigorous. He also introduced the concept of medium gear, deduced the calculation formula of slip line curvature, and discussed the meshing boundary of two kinds of gears. Professor F.L. Litvin of Illinois State University has been committed to the research of gear meshing theory. Considering the complex influence of normal curvature correction on the contact characteristics of tooth surface, its corresponding relationship can be obtained only through contact analysis or actual trial cutting and rolling inspection. F. In the 1960s, L. Litvin proposed a local contact comprehensive calculation method based on Gleason’s local conjugate principle, which can accurately control the contact state of the calculation point. The basic idea of this method is to directly solve the curvature parameters of the contact reference point of the small gear tooth surface according to the first-order and second-order contact characteristic parameters of the tooth surface contact reference point after obtaining the curvature parameters of the calculation reference point of the large gear tooth surface. Then the cutting adjustment parameters and tool parameters of the small wheel are solved according to the diameter vector, normal vector and curvature parameters of the contact reference point of the small wheel tooth surface. Then the small gear tooth surface determined by this set of parameters can mesh with the large gear tooth surface, and obtain the required contact characteristics at the contact reference point. Local contact synthesis method combined with tooth surface contact analysis can effectively control the contact characteristics of gears.
The theoretical research on spiral bevel gear in China started relatively late. In the early 1970s, in order to break the foreign technological monopoly, the state organized relevant domestic universities, research institutes and factories to systematically study the end face milling theory and implementation technology of Gleason spiral bevel gear, and listed it as a key research topic. During this period, professors Wu Daren, Yan Zhida and Luo Jiashun of the Department of mathematics of Nankai University carried out research on gear meshing theory. They applied differential geometry to gear meshing theory, clarified many important concepts in gear meshing theory, and systematically expounded the geometric theory of conjugate tooth surface, such as the relative derivative method, the expression of two boundary functions and their relationship, and the general formula of induced curvature. It has established a unique theoretical system, which has become the mathematical basis of gear meshing theory, provided a powerful tool for the research of gear meshing theory in China, and promoted the scientific research of bevel gear and so on. Wu Daren and Luo Jiashun also co authored the theory of gear meshing (Science Press, 1985). Yan Zhida also made mathematical treatment on the secondary contact phenomenon and the secondary envelope theory based on the work of Sakai Gaonan and muchong, and applied this theory to the direct and indirect expansion method. Chen Zhixin of Shanghai University of technology has systematically and deeply studied the gear meshing theory and deduced the calculation formula of induced normal curvature of conjugate surface, which lays a theoretical foundation for breaking through Gleason technology [37,38]. The gear research group of the mechanism teaching and Research Department of Xi’an Jiaotong University has developed and improved the calculation method of spiral bevel gear face milling and cutting in the former Soviet Union, and used this method to analyze the calculation principle of the early cutting adjustment calculation card of Gleason company. Inspired by the curvature calculation theory of Baxter and Yan Zhida’s induction method, Zheng Changqi of Chongqing University and Zeng Tao of Central South University comprehensively studied the machine tool adjustment calculation and tooth surface contact analysis principle of Gleason’s various cutting methods, decoded the compilation principle of its cutting adjustment calculation card, and deduced the relevant formulas. After the “Seventh Five Year Plan” and “Eighth Five Year Plan”
According to the two five-year plans, China has basically mastered the theoretical basis and processing principle of Gleason spiral bevel gear face milling, designed and manufactured domestic mechanical spiral bevel gear processing machine tools, and formed China’s independent spiral bevel gear industry and technology system.
Relevant scholars have also developed the above theory. Dong Xuezhu proposed a new method for simple tooth cutting adjustment calculation, such as half generation and full generation of spiral bevel gear cutter inclination, with the center of tooth surface contact area as the reference point. Mao Shimin, Wu Xutang, etc. improved Gleason’s adjustment calculation method of cutter inclination and half generation cutting, which can arbitrarily specify the position of contact reference point. They also derived the calculation formula of the sensitivity of the contact point position of the point meshing tooth surface to the installation error, and analyzed the influence of the second-order parameters of the tooth surface on the error sensitivity. Guo Xiaodong and Zheng Changqi studied a new calculation method of spiral bevel gear cutting adjustment on the basis of local contact synthesis theory. This method can simultaneously control the position of contact reference point, the shape of contact area (length of contact area, diagonal direction of contact area) and the size of transmission error. In this method, the cutting tool can also be given according to the actual situation for calculation. The bevel gear application technology software system gshgears developed based on this method has been widely used in more than 100 processing enterprises at home and abroad, and has become a standard calculation tool for spiral bevel gears. Based on the local contact synthesis method, Wu Xutang and Wang Xiaochun used the optimization method to control the contact characteristic parameters of spiral bevel gears. They also established a set of third-order contact analysis system by using mathematical tools such as curvature tensor and movable frame method. Using this method, the instantaneous transmission ratio, acceleration, direction of contact trace and the length of instantaneous contact ellipse of contact reference point can be calculated. The higher-order acceleration, geodesic curvature of contact trace and the change of instantaneous contact ellipse can also be calculated. Therefore, the meshing characteristics of the tooth surface far from the tooth surface contact reference point can be controlled, and the shape of the tooth surface contact area can be avoided to a great extent. Yi Pei Shih studied the method of modifying conjugate difference surface to control transmission error and contact path in the tooth cutting adjustment calculation of spiral bevel gear. Cao Xuemei, Fang Zongde control the contact path and transmission error by controlling the contact trace to pass through three predetermined points on the tooth surface. F. L. Litvin, Alfonso Fuentes and others obtain spiral bevel gears with low noise level by designing the transmission error curve into parabola and limiting the maximum value of transmission error, adjusting the meshing trace and chamfering the top of matched gears; Emmanuel Mermoz, Julien astoul control the noise and vibration of spiral bevel gear transmission by optimizing the tooth surface contact pressure and transmission error; Su Jinzhan, Fang Zongde optimized the transmission error curve of spiral bevel gear to reduce its vibration and noise during operation and improve its adaptability to load. Deng Xiaozhong, Fang Zongde and others studied the method of reducing meshing noise by increasing the inclination of contact path and improving the coincidence degree of spiral bevel gear. Vilmos Simon studied the influence of tooth profile error and misalignment of installation on the contact state of spiral bevel gear.