Determining the forging deformation force is a very important part of the metal plastic forming theory. The research on the precision forging forming force of spiral bevel gear is of great significance to select the forging equipment with appropriate tonnage, design the die to ensure the forging forming and sufficient strength, and formulate reasonable process regulations. The research methods of precision forging forming force mainly include mathematical analysis method, principal stress method, slip line field method, upper bound method, deformation work method, intuitive plastic method and plastic finite element method. The following is a brief introduction to each analysis method:

**Mathematical analysis**

Mathematical analysis method is the most accurate method for mechanical analysis of metal plastic forming, but 16 basic equations representing the stress-strain state during plastic deformation, such as differential equilibrium equation, constitutive relationship, volume invariant condition, yield criterion and boundary condition, need to be solved simultaneously. The solving process is very complex and difficult to realize. Therefore, this method can only deal with some relatively simple plane problems and is less used in practical engineering problems.

**Principal stress method**

The principal stress method is also called the cutting block method. Its analysis process is based on simplifying the stress state of the deformed body. The equilibrium differential equation and yield equation (plastic condition) are simplified, and then solved simultaneously to obtain the normal stress distribution and average deformation resistance on the contact surface. Because the solution process of this method is relatively simple and the solution result is relatively accurate, it is widely used in the analysis of plastic forming deformation force. The basic points of the solution of the principal stress method include:

1) The plastic deformation problem to be analyzed is simplified into plane problem or axisymmetric problem.

2) According to the flow trend of metal at a certain time and the coordinate system selected in the solution, the primitive body is cut along the whole section of the variable body. It is considered that the normal stress on the section is the main stress (ignoring the shear stress) and uniformly distributed (independent of a certain coordinate). Therefore, the equilibrium differential equation of the primitive body is simplified into an ordinary differential equilibrium equation.

3) In order to facilitate the simultaneous solution of plastic condition equation and equilibrium differential equation, it is considered that there is only normal stress in each coordinate plane, so the yield equation is simplified into linear equation, which makes the solution process easy to realize.

**Slip line field method**

Slip line field method is a relatively perfect forming force analysis method of plastic deformation. It was first proposed by M. levy in the 1870s and used by Prandtl to solve the compression problem of plane strain in 1923. The slip line field method holds that the plastic deformation of the material conforms to the Mises yield criterion. When the material is plastic deformed, there are two maximum shear stresses with mutually orthogonal directions at any particle inside. When the position of the particle changes along these two directions respectively, its trajectory is two groups of mutually orthogonal curve families, and the metal flows plastically along these two groups of curves. Slip line field theory includes velocity field theory and stress field theory. The slip line field method is used to solve the plane strain problem of ideal rigid plastic body with high accuracy, but there are still some deficiencies if it is used to solve the axisymmetric problem.

**Ceiling method**

For complex plastic forming problems, it is very difficult and unnecessary to obtain strict mathematical solutions. As an analytical method for approximate solutions, the upper bound method has been well applied in practical forming problems. Its theoretical basis is the variational extreme value principle and virtual work principle. The deformation power is obtained by solving the velocity field and velocity discontinuity, and then the deformation force is obtained. When the upper bound method is used to calculate the ultimate load, several velocity fields with movement permission should be set up in the plastic deformation area respectively. These velocity fields are assumed to meet the following conditions: ① meet the boundary conditions of velocity (or displacement); ② Maintain continuity in the deformation zone without overlapping and cracking; ③ Keep the volume constant. According to the principle of minimum power consumption, the velocity field satisfying the velocity boundary condition is considered to be the real velocity field. The formula is:

Using the upper bound method to solve the plastic forming problem can be carried out according to the following ideas: ① according to the law of metal flow and the needs of the problem, the whole deformation area is divided into several parts, and the dynamically admissible velocity field is established in each part; ② The strain velocity field and velocity discontinuity in each zone are obtained by using the volume invariant condition and the velocity boundary condition of each velocity field; ③ Calculate each upper limit power; ④ The optimal upper bound solution is obtained according to the minimum energy principle.

The forming load obtained by the upper bound method is equal to or greater than the real load, which leaves a margin for the selection of forging equipment and the design of die, which has a good reference value. In addition, the calculation amount of the upper bound method is small and simple, so it is widely used in plastic forming analysis.

**Deformation work method**

Deformation work method is a simple method to calculate plastic forming force directly by using the principle of work balance. This method can be used to calculate the deformation force of various forming processes such as upsetting, punching and extrusion, as well as the forging deformation force of rotary die forgings such as spiral bevel gear precision forgings.

**Visual shaping method**

Visual shaping method is to carve coordinate grids on the surface of the deformed body or the section to be analyzed by photolithography or other methods, measure and analyze the changes of these grids after plastic deformation, and then obtain the strain of each particle. The stress can be obtained through the stress-strain relationship, but the premise of using this method is to do a lot of experiments first, The process is cumbersome.

**Plastic finite element method**

Plastic finite element method is an analytical method with high accuracy developed with the development of computer in recent years. According to the different relationship between material strain and displacement and between strain and stress, it can be divided into elastic-plastic finite element method, rigid plastic finite element method and viscoplastic finite element method. Through numerical simulation technology, the plastic finite element method can intuitively reflect the metal flow and the distribution of stress field, strain field and temperature field in the whole plastic deformation process on the computer. It provides a powerful tool for studying the metal plastic forming process, optimizing die design and predicting product defects.

The combined method of principal stress and upper bound and deformation work method are mainly used to solve and analyze the forging deformation force required by open precision forging and non flash precision forging of spiral bevel gear. Firstly, the following assumptions are made for metal materials:

1) The internal structure of the material is uniform and continuous without cracking and folding.

2) The volume of the material remains unchanged during plastic deformation.

3) The material satisfies Mises condition in the process of plastic deformation.

4) The influence of elastic deformation and work hardening in the process of plastic deformation is ignored.