Because the “cantilever beam” model is quite different from the real spur gear, the material mechanics method is difficult to truly reflect the influence of complex tooth profile and transmission system on the strength and deformation of spur gear. According to the plane elasticity theory, many scholars apply appropriate mapping function to study the stress distribution in spur gears, and draw many valuable conclusions. For the conformal mapping method, the tooth shape boundary is mapped into a straight line, and the displacement field of the half plane is calculated according to the complex variable function of the concentrated force acting on the half plane, so as to obtain the stress and deformation of the loaded spur gear. The key to the accuracy of conformal mapping method lies in the active selection of mapping function and scientific selection of various coefficients. In 1982, cardou and TORION began to use the “direct search method” computer program to optimize the coefficients of the mapping function and improve the solution accuracy. Xu Lizhong obtained the analytical solution of tooth burning degree of spur cylindrical gear by conformal mapping method. Conformal mapping method has made some achievements in the study of strength and deformation of spur gears, but it also has certain circumferential limitations:
① The accuracy of the mapping function is high and the calculation formula is complex, so it is difficult to generalize to the problem of spatial elasticity;
② The load moves from the action point on the tooth gallery to the symmetry line of spur gear. In the process of the action point getting closer to the tooth root, the influence of the movement on the local maximum stress of the tooth root will increase;
③ It is difficult to consider the complex tooth profile, boundary conditions, support conditions and the modeling and calculation of the whole system;
④ It is inconvenient to embed factors such as machining error, assembly deviation and wear of spur gear.