Higher order contact of spiral bevel gear tooth surface can expand the local characteristics of tooth surface, provide a powerful tool for global control of tooth surface, and effectively improve the running stability of spiral bevel gear pair. However, the current tooth surface design methods are mainly based on the second order contact. For example, Gleason Company, based on the “local conjugate method”, selects the reference point on the tooth surface of the big gear and uses the tooth surface conjugate principle to calculate the first and second order contact parameters of the small gear tooth surface at the reference point; Litvin proposed the “local synthesis method” to design the tooth surface by presetting the second order contact parameters, which avoided the solution of overdetermined equations through assumptions; Yuan et al. compensated the error between the CNC machine tool and the cutter to achieve the accurate design of the tooth surface through mathematical models such as gear cutting engagement and CNC machining; Shih defined ease-off as the normal relative modification of tooth surface, proposed the design method of ease-off tooth surface active modification of spiral bevel gears and hypoid gears based on face hobbing, and established the general mathematical model of face hobbing machining; Fan analyzed the easy off generation theory and applied it to the tooth contact analysis (TCA) of spiral bevel gears and hypoid gears; Wei Bingyang put forward the easy off surface synthesis method, constructed the easy off difference surface of the conjugate tooth surface of the large and small gears, simulated the meshing with the easy off close surface, and obtained the size and shape of the tooth surface contact area, the direction of the contact trace, and the parabola mismatch; Tang Jinyuan’s team put forward the design of tooth surface error compensation and modification aiming at ease-off correction and geometric error; Fang Zongde et al. proposed a tooth surface modification method combining ease-off and tooth surface high-order transmission error, and established a design method of tooth surface high-order transmission error through tooth surface curvature correction to achieve accurate design of tooth surface; Fong established a mathematical model of hyperboloid machine tool with high-order tooth surface modification movement; Stadtfeld proposed a simple method for high order tooth surface correction; Fan proposed a tooth surface error correction for high order spiral bevel gears to reduce noise. Design method of above tooth surface, even basis
Second order contact theory. In order to solve the high-order contact problem, Wang Xiaochun et al. theoretically analyzed the third-order contact state of the tooth surface through the meshing principle method, but did not consider the high-order contact conditions, nor formed a tooth surface design method considering the third-order contact parameters, and the calculation was difficult to achieve; Liu Huran et al. obtained the high-order contact conditions of the tooth surface based on the high-order Taylor expansion of the surface, but did not consider the related parameters of high-order contact; Ding et al. obtained high-order contact conditions along the characteristic curve direction by analyzing the relationship between the tool and the designed surface; Wu Baolin, Zheng Gang and others analyzed the relationship between higher-order contact parameters and applied the higher-order contact theory of surfaces in different directions.
With the more and more extensive application of spiral bevel gears, the requirements for the transmission performance of the tooth surface are getting higher and higher. The tooth surface design method based on the second-order contact analysis cannot guarantee the excellent global performance, so the accurate analysis of the high-order contact state of the tooth surface is particularly important. The higher order contact characteristics of tooth surface, such as the higher order transmission ratio derivative and the curvature of contact trace, have great influence on the stability of spiral bevel gear pairs. Therefore, in order to fill the gap of tooth surface high-order design theory, it is imperative to study tooth surface high-order contact.
To solve the above problems, a high-order contact analysis method of discrete spiral bevel gear tooth surface based on differential geometry and ease-off principle is proposed. The discrete tooth surface of spiral bevel gear is established, and the high-order contact parameter analysis and calculation method of discrete tooth surface based on ease-off is given. The calculation difficulty is reduced by solving and using the finite difference method. Finally, the proposed analysis method is analyzed by calculating the initial tooth surface high-order contact parameters to verify the feasibility of the method.