# Establishment of gear geometric analysis and mechanical analysis

In the establishment of mechanical model, the gear tooth is simplified as a cantilever beam, and the contact state under load is simplified as a contact line along the long axis of the instantaneous contact ellipse. In the 1980s, the calculation method of gear contact strength was obtained based on Hertz theory, but the above method has no effect on the elastic deformation of the whole surface of the load-bearing gear tooth The load distribution of actual meshing teeth and the load distribution in each contact area can not be accurately determined, so it is an approximate method. In fact, the tooth contact area along the tooth direction is very long, but the tooth height is very short, and the tooth rigidity is large. Therefore, the plane elastic problem as an infinite cylinder is not handled accurately. The classical elastic contact theory can not give a good analytical expression. These theories can only be applied to some simple semi infinite bodies or uncomplicated symmetrical bodies.

Since the 1970s, people have used the classical elastic method to solve the gear contact problem. The gear contact problem belongs to the mathematical boundary value problem. The singular integral equation is the dominant equation of contact mechanics. The scope of this method to solve the problem is very limited, because it can only obtain analytical solutions for some regular shapes, but it can grasp the physical essence and mechanical characteristics of the object, so this field is still developing effectively. With the rapid development of computer technology, various numerical calculation methods have been widely used and continuously improved. The numerical method developed in recent years discretizes the boundary nonlinear contact problem into a linear problem, which provides a new idea for us to solve the gear bearing contact.

Because the combination between geometric analysis and mechanical analysis has not been well solved, some studies mainly focus on geometric analysis, and the mechanical model is too simplified. Some studies lack geometric analysis, resulting in the distortion of mechanical analysis. Some studies have too cumbersome calculation models, which are difficult to be applied and popularized in engineering.

Now the numerical methods to solve the gear contact problem are: finite element method, boundary element method and mathematical programming method. Mathematical programming methods include variational method, quadratic programming method, etc. Many solutions of contact nonlinear problems by finite element method have been quite mature and are widely used at present. Because the boundary element method is solved by the linear equations of unknown displacement and surface force components on all boundaries and stored in asymmetric full rank, it takes a lot of time to calculate for many contact pairs and multi contact problems, and the solution scale is limited. At present, it is not mature. 