TCA technology uses computer to simulate the meshing process of gear pair under no-load condition, and can obtain each instantaneous contact spot and transmission error from entering meshing to exiting meshing, so as to obtain the contact trace, contact mark and transmission error curve of hypoid gear tooth surface in the whole meshing process. TCA technology is of great significance in improving gear manufacturing technology and meshing quality. It was first proposed by Gleason company of the United States and used to analyze the tooth contact characteristics of spiral bevel gears and hypoid gears. Later, it was extended to other forms of gear transmission and gradually developed and improved.
Although domestic research on TCA technology started late, it has developed rapidly. Fang Zongde deduced all the calculation processes of tooth surface contact and edge contact of modified helical gears, compiled a complete set of computer programs to simulate the contact process of helical gears with different errors and modifications, and discussed the effect of tooth surface modification on improving the transmission performance of gears and the corresponding modification design method by taking cutter tooth profile modification and CNC machine tool modification as examples; Fang Zongde also expressed the tangent vector of the edge curve by the cross product of the normal vector of the tooth surface and the normal vector of the top cone, successfully solved the problem of edge contact analysis of spiral bevel gears, and gave a geometric analysis method of edge contact problem. Wang Sanmin proposed a tooth contact analysis method of spiral bevel gears considering manufacturing errors and tooth deformation under load, and analyzed the influence of errors on gear meshing performance. Su Jinzhan, Cai Xiangwei, etc. studied the machining method, meshing characteristics and tooth surface modification method of arc gear, and established the TCA model considering the installation error. The simulation results show that the machining parameters such as tool inclination, the number of teeth of the profiled gear, the radius of the tool tip and the installation error vector have a great influence on the parameters such as the contact trace and the instantaneous contact ellipse. Tang Jinyuan proposed the contact analysis method of spiral bevel gear transmission with errors. Through the comparative analysis with the TCA results without errors, it was found that the machine tool motion error and installation error had a great impact on the tooth surface contact quality of spiral bevel gear, and the degree of impact was quantitatively analyzed. Wang Xing analyzed the tooth contact of Gleason HGT hypoid gear, and studied the meshing process of gear pair under different parameters and errors. Li Jianhua studied the tooth contact analysis of herringbone gear pair under the condition of error. In terms of Cycloid Teeth, Wang Feng studied the tooth contact analysis of hypoid gears with Cycloid Teeth in the presence of errors. Nie Shaowu analyzed the tooth contact of Cycloid Bevel gear at the predetermined position, established the mathematical model of gear surface rolling inspection with axial position variables, deduced the simplified algorithm of meshing contact analysis, and gave the solution method of contact ellipse. In addition, Yan Hongzhi, Feng Yihan, Xu Chaode, Li Wei and others simulated the tooth profile of “Ke” spiral bevel gear and analyzed its contact performance. Meng Fanjing studied the contact zone correction method of “gram” spiral bevel gears. Li Haitao established the tooth surface contact analysis model of extended epicycloid bevel gear and hypoid gear based on Oricon SKM2 machine tool, proposed a new method that can predict the tooth surface contact quality and reduce the design and processing cost, and carried out the computer-aided analysis of the tooth surface contact area of hypoid gear. Chen Liangyu studied the motion optimization and Simulation of “gram” cycloidal bevel gear with the goal of minimizing the gear motion error. Liu Zhifeng established the rolling model of the “gram” Cycloid Bevel Gear and deduced the prediction formula of the contact area. Through computer simulation, the contact mark and transmission error curve of the gear were obtained, which laid a foundation for the loading contact analysis and strength analysis. Yan Hongzhi and Liu Ming studied the dynamic and static meshing characteristics of hypoid gears with Cycloid Teeth in the main reducer of automobile drive axle.
The traditional solution method of TCA is based on nonlinear iteration, especially the selection of normal vector equation, which leads to low convergence accuracy and speed in the calculation process, and is sensitive to the initial value. It is difficult to give the initial value of TCA calculation when the calculation point is unknown. Professor Litvin reduces the deviation of tooth surface position vector and normal vector through iterative method, so as to obtain the initial value very close to the contact point, On this basis, it is proposed to pair the tooth surface parameters in blocks, and determine the parameter pair with the minimum normal deviation by selecting the appropriate coefficient, so as to determine the initial value. The reference point data calculated by the local synthesis method can also be substituted into the TCA calculation as the initial value. Li Jingcai established an algorithm to automatically solve the initial value of TCA by finding a pair of points with the smallest space distance in the assembly coordinate system. The parameter value corresponding to the pair of points can be substituted into TCA as the initial value.
In some cases, even if the TCA equations can be successfully solved, the results have no practical geometric significance, that is, they are geometrically inaccurate, because the third component of the normal vector is not considered. In order to avoid this problem, Litvin improved the basic equation of TCA, and eliminated the error caused by the normal vector equation by using the relationship between the tangent vector and normal vector of one tooth surface and the normal vector of the other tooth surface in the tooth surface of paired hypoid gear. Fan also proposed an improved TCA model, which displays the machining rotation angle of large and small wheels through the vector rotation formula and the condition of normal vector equality, so as to reduce the number of equations and improve the accuracy and stability of the solution.
For the calculation of the instantaneous contact ellipse, the previous method is based on the quadratic approximation of the tooth surface of two matched hypoid gears near the reference point. The principal curvature and relative curvature of the contact surface need to be calculated through complex derivation, and then the long axis of the instantaneous contact ellipse can be calculated. In particular, it is difficult to calculate the principal curvature in the case of quadratic approximation surface envelope. In addition, the instantaneous contact area is generally a slender ellipse, so the second-order approximation surface near the contact can not truly reflect the characteristics of the entire instantaneous contact area. To solve this problem, Gleason company proposed an improved method, which first “finds” the direction of the long axis of the ellipse through iteration, and then obtains the length of the long axis of the ellipse through another iteration in this direction. However, this method can only obtain the long axis of the instantaneous contact ellipse. In some cases, we expect to obtain a complete instantaneous contact ellipse, that is, the complete boundary of the instantaneous contact ellipse on the tooth surface of hypoid gear.