# Research status of bevel gear design theory

For bevel gears, the most developed and widely used tooth surface design method is conjugate surface meshing and local synthesis method. Foreign researchers have conducted more in-depth research on conjugate surface meshing theory, such as Willis, Buckingham, wildhaber and Dudley in the UK, ketov, Kolchin and gavrilenko in the former Soviet Union, wildhaber in the United States, Baxter and krenzer et al. In terms of gear geometry, according to different tooth lines, bevel gears can be divided into two categories, including straight bevel gears and spiral bevel gears. Spiral bevel gear is widely used because it is more stable in operation and can bear large torque. As far as spiral bevel gear is concerned, the main research objects at home and abroad at present are circular arc bevel gear (based on Gleason machining method), long epicycloid bevel gear (based on Klingelnberg machining method) and logarithmic spiral bevel gear. Among them, logarithmic spiral bevel gear is still mainly in theoretical research.

Scholars have done a lot of research on the basic tooth profiles of these three tooth systems. Fong et al. Described the mathematical model of circular arc cutting bevel gear in detail, and compared the bevel gear with spherical involute bevel gear in theory [31]. Starting from the machining motion, fan deduced the theoretical geometric model of long epicycloid bevel gear in detail, and analyzed the theoretical model by using tooth surface contact analysis. Alves et al. Discussed the geometric design of logarithmic spiral bevel gear and analyzed its meshing situation by using the finite element method. At the same time, a set of logarithmic spiral bevel gear was processed and tested with 5-axis linkage machining center. In addition to these three kinds of spiral bevel gears, in order to obtain bevel gears with better performance, scholars at home and abroad have also explored other bevel gear tooth shapes. For example, Tsai and Hsu proposed a bevel gear with circular arc tooth line and circular arc normal tooth profile, which can be processed only by 4-axis linkage gear milling machine. Yao et al. Proposed the spiral bevel gear with double segment arc normal tooth profile, deduced the theoretical model, and processed the prototype with the machining center.

According to the conjugate surface meshing theory, the ideal theoretical tooth profile curve of bevel gear should be spherical involute. However, as far as the existing processing equipment is concerned, the spherical involute tooth profile is difficult to process. Moreover, the line contact bevel gear is sensitive to error and prone to edge contact. Therefore, at present, the research on bevel gear is mainly transformed into the research on point contact bevel gear. For circular arc bevel gears, Gleason technology is mainly based on the so-called “local conjugate principle”. According to this theory, the required second-order geometric parameters (normal curvature and geodesic torsion, etc.) must be obtained at the reference point on the bevel gear tooth surface. First cut out the large gear tooth surface, and then select a calculation reference point, The first-order and second-order contact parameters such as the position, normal vector and normal curvature of the conjugate small gear tooth surface in line contact with the gear tooth surface at the reference point are obtained. Finally, the normal curvature of the small gear tooth surface at the reference point is corrected, and the small gear cutting adjustment parameters are determined on this basis.

In order to make the point contact bevel gear have better mechanical properties, on the basis of Gleason technology theory, Litvin et al. Put forward the comprehensive methods of local synthesis and tooth contact analysis. This theory is mainly developed from the perspective of machining, In order to control the contact area in the middle of the tooth surface as much as possible, the adjustment method of machine tool motion is proposed. With the help of this set of theories, Litvin et al. Developed the processing and adjustment methods of various types of gears such as circular arc gear, face gear and spiral bevel gear. Medvedev and Volkov proposed a local synthesis algorithm for spiral bevel gears with small shaft angle. In addition to the local synthesis method, Simon proposed a method to change the tool shape and optimize the bevel gear tooth profile. Zhang et al. Proposed a generation method of spiral bevel gear with spherical involute profile, which is generated by a given motion of a straight line. Wang and Zhang introduced the tensor method to analyze the curvature relationship of spiral bevel gear. With the help of tensor theory, the local synthesis method can get a clearer explanation. Because the tensor has nothing to do with the selection of coordinate system, the application of tensor method can make the selection of reference coordinate system more convenient and flexible.

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