The technical theory of spiral bevel gear and hypoid gear was first put forward by wildhaber, Baxter and others of Gleason company in the United States. Later, Oerlikon company in Switzerland (acquired by Klingeinberg company in 1991 and formed Klingeinberg Oerlikon group) and Klingeinberg company in Germany also owned spiral bevel gear technology, And formulated their own standards. The circular arc tooth, cycloid tooth bevel gear and hypoid gear are processed by end face milling and end face hobbing respectively. The relative position of the cutter head and the generating wheel during the processing is shown in the figure.
The early Gleason technology is based on the local conjugate principle. Its basic idea is: first cut the gear tooth surface, select a calculation reference point, and use the continuous tangent contact conditions to calculate the first-order and second-order contact parameters such as the position, normal vector and normal curvature of the small gear tooth surface in contact with the gear tooth surface at the reference point, Then the normal curvature of the pinion tooth surface at the reference point is corrected according to the requirements, so as to determine the pinion gear cutting adjustment parameters. From the 1940s to the 1960s, Gleason company developed a series of new technologies for machining spiral bevel gears based on the theory of local common vehicle, including tooth contact analysis (TCA), which realized the quantitative analysis of mismatched tooth contact and motion transmission. However, this early design method has obvious shortcomings. Because its tooth cutting calculation theory does not pre control the second-order contact parameters of the local common gear tooth surface, it is difficult to select the tooth surface curvature correction, and it needs to be cut for many times to obtain the ideal whistle quality, which largely depends on the experience of the operator, It is difficult to guarantee the machining accuracy.
In 1968, Professor FL Litvin proposed the local synthesis method, that is, specify three second-order contact parameters at the reference point: the tangent direction of the contact trace on the gear tooth surface, the first derivative of the transmission ratio function, and the length of the long half shaft of the instantaneous contact ellipse, and calculate the principal curvature and principal direction at the reference point of the small gear tooth surface by using the differential geometry theory, Thus, the adjustment parameters of the machine tool for machining small wheels are obtained. Later, Professor FL Litvin proposed to control the transmission error by presetting the parabolic transmission error function, so as to achieve the purpose of vibration and noise reduction, and further improve the local synthesis method.
In recent years, Gleason has developed high-order motion error design theory, namely universal motion concept (UMC) and universal motion graph (UMG), which provide a theoretical basis for high-precision grinding. UMC refers to the four movements of vertical, radial, spiral tooth and roll ratio correction realized on Gleason machine tool, and changes the variables from zero order to fourth order through NC six axis control. UMG shows UMC in grinding state through gear grinding. Generally, the first and second-order errors in transmission error measurement can be eliminated by gear grinding, but the transmission errors above the fourth and fifth order must be eliminated by gear grinding. After the technology is adopted, gear grinding can be avoided.
The above research theories are mainly aimed at circular arc bevel gears and hypoid gears. The processing method of Cycloid Bevel Gear and hypoid gear was first proposed by American and Italian scientists, and then developed by Oerlikon company in Switzerland. Gleason company of the United States also published a series of technical documents on the design and processing of Cycloid Bevel Gears and hypoid gears. Relevant scholars and technicians have made relevant research on the geometric forming and meshing characteristics of Cycloid Bevel Gears and hypoid gears. Among them, Litvin described the process of gear hobbing and cutting on the end face of Cycloid Bevel gear, and proposed a method to directly measure the relationship between pitch cone angle and spiral angle of cycloid hypoid gear; K. Kawasaki et al. Studied the restrained spiral bevel gear, gave its geometric description, detection method and tooth surface contact analysis, and analyzed the influence of installation error on the tooth surface contact trace and transmission error of restrained spiral bevel gear under the condition of small spiral angle; S. Kato and a.kubo determined the calculation formula of tooth surface error and transmission error under different cutter head diameters, and quantitatively analyzed the influence of cutter size on gear performance; Q. Fan introduced the machining principle of Cycloid Bevel Gear and hypoid gear end face hobbing, l.m.sung established the mathematical model and carried out the coupling simulation; The mathematical model of Cycloid Bevel gear is established, and the calculation method of gear pair coincidence degree is proposed.
With the continuous development of CNC machine tools, the theoretical research and design technology of Cycloid Bevel Gears and hypoid gears are also constantly developing and updating. Based on the multi axis linkage of CNC machine tools, k.kawasaki studied the machining method of large restrained spiral bevel gears; Based on the machining process of end face hobbing and end face gear combination of freeform NC machine tool, Qi fan established the general mathematical model of spiral bevel gear and hypoid gear, and gave the optimization algorithm of gear pair contact simulation, and then established the general mathematical model of machine tool adjustment during cycloid hypoid gear machining, The influence law of general motion factors on tooth surface error is analyzed, and a tooth surface modification method based on general motion model is proposed: Yi Pei Shih, Zhang Hua Fong, etc. establish a mathematical model based on general end face hobbing NC machine tool, and on this basis, propose a tooth surface modification method to meet the tooth surface performance, This mathematical model can be used to machine spiral bevel gears and hypoid gears with restrained, Austrian and lattice face hobbing.
In order to ensure the best transmission performance of spiral bevel gear in work engineering, the performance parameters such as tooth surface contact area and motion error must be calculated and analyzed. At present, the analysis methods widely used in the research of spiral bevel gears and hypoid gears include: tooth contact analysis (TCA) without considering load and loaded deformation, load tooth contact analysis (LTCA) considering load action Error Tooth Contact Analysis (etca) finite element analysis considering errors, etc. Based on the theoretical analysis of computer, the machining parameters are adjusted and the theoretical tooth surface is optimized according to the actual requirements. Martino Vimercati et al. Presented the calculation method of tooth surface geometry, contact analysis and stress analysis of cycloid hypoid gear; 5. Simon established the geometric forming and tooth surface contact analysis models of Cycloid Bevel Gears and hypoid gears, and on this basis, reduced the contact stress and transmission error by optimizing the tooth surface.