# Common methods to solve gear contact problems

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

Since the 1970s, people have used the classical elastic mechanics method to solve the problem of gear contact. Gear contact problem belongs to the boundary value problem in mathematics. Singular integral equation is the dominant equation in contact mechanics. The scope of this method to solve the problem is very limited, because it can only get analytical solutions for some regular shapes, but it can grasp the physical essence and mechanical characteristics of the object, so this field is still in fruitful development. With the rapid development of computer technology, various numerical calculation methods have been widely used and 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 bearing contact of gears.

Because the combination of geometric analysis and mechanical analysis has not been well solved, some studies mainly focus on geometric analysis, the mechanical model is too simplified, some studies are insufficient in geometric analysis, resulting in mechanical analysis distortion, and some studies are too cumbersome to be applied in engineering.

Now the numerical methods to solve the gear contact problems are: finite element method, boundary element method, mathematical programming method. Mathematical programming method includes: 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. The boundary element method (BEM) is based on the linear equations of unknown displacement and surface force components on all boundaries, which is stored in an asymmetric full rank. For many and multiple contact problems, the calculation takes a lot of time, and the scale of solution is limited. 