# Prediction of Machining Deformation for Split Equal-Base Circle Bevel Gear
## 2 Analysis of Gear Blank Deformation and Additional Torque  
The profile of the tooth blank of a large equal-base bevel gear closely resembles a rectangle. With a substantial root angle and a large outer circle diameter, the tooth blank, post-splitting, can be effectively regarded as a cuboid. This study defines the length direction of the semi-finished product as the X direction (rolling direction), the lateral direction as the Y direction, and the thickness direction as the Z-axis, oriented vertically upward. For this processing and measurement test, 7075-7451 aluminum alloy is employed. To ensure the finite element model accurately represents the residual stress distribution of the wheel blank, an appropriate function is designed to express the relationship between the thickness of the wheel blank and the calculated initial residual stress distribution characteristics [17]. Subsequently, the subroutine is designed, and the corresponding subroutine is invoked by ABAQUS software, establishing the finite element analysis model for the automatic continuous loading of residual stress.  
The internal stress within the corresponding part dissipates when the material in the tooth space is removed, representing the fundamental aspect of the processing deformation of the split gear blank. Applying additional stress to the left and right sides of the tooth slot allows for an equivalent simulation of the influence of cutting on deformation. This approach facilitates the determination of torque and wheel blank deformation resulting from material removal. Figure 1 illustrates a cloud diagram depicting deformation during simulated processing. It can be observed from the cloud diagram that the processing area is small, and the deformation is the largest in other places. Furthermore, as one moves farther away from the cutting position, the deformation diminishes. Upon completion of the machining process, the gear blank adopts an overall curved state, with deformation symmetrically distributed in the X direction of the wheel blank.  
If the curvature of the tooth line is not considered, the shape of the tooth blank after machining a tooth pitch is almost U-shaped in the X direction. The research shows that the maximum change in internal stress occurs in the length direction of the processed gear blank. At this time, the specific distribution of additional stress and the established coordinate system are shown in Figure 2. After removing the material, the initial support stress \(\sigma\) on the left and right sides of the tooth slot also disappears. In this case, applying stress \(-\sigma\) to the sidewall of the tooth slot achieves the force-equivalent effect.  
In Figure 2, \(z_{c}\) is the position of the neutral axis, on which the centroid of the section is located. During the gear cutting process, the position of the centroid of the section changes continuously. Different additional stresses are created when the material is removed layer by layer from the tooth slot. The accumulation of additive stress and the initial residual stress of the part lead to the occurrence of the gear blank bending deformation. With the help of the relevant theory of material mechanics, the curvature can be calculated and combined with the additive stress distribution of bending in the process of gear processing.  
Figure 1: Deformation nephogram of simulated machining. Notes: (**a**) deformation nephogram of machining middle cogging; (**b**) deformation nephogram of finished machining.  
Figure 2: Diagram of additional bending stress.  
The curvature change law, additional stress calculation formula, and residual stress balance expression of the gear before and after processing are solved in sequence. Finally, the formula for the additive torque due to the redistributed internal stress \(\sigma_{i1}\) located on the instantaneous neutral axis is calculated.  
\[M=\int_{0}^{z_{1}}\sigma_{i1}(z)(z-z_{c})\chi_{2}dz \tag{1}\]