# Prediction of Machining Deformation for Split Equal-Base Circle Bevel Gear
## 3 Calculation of Bending Deformation  
To determine the precise numerical value of the bending deformation for the wheel blank with only one tooth groove and no helix angle, as depicted in Figure 2, an analysis of the additional torque is conducted. This analysis, combined with the machining characteristics of a single pitch, allows for the initial determination of the internal stress in each layer. Subsequently, the size parameters of the gear blank are substituted into Equation (1) to calculate the additional torque experienced by the gear blank during processing,  
\[M=\int_{0}^{z_{1}}\sigma_{il}(z)(h-z_{c}-z)*[x_{3}-(x_{3}-x_{2})(z_{1}/2.25m)  
For each tooth space in Figure 3, from the calculation method of equivalent stiffness derived, the stiffness of the gear blank when machining different tooth slots can be obtained, and it is set to \(K_{11}\), \(K_{12}\), \(K_{13}\), \(K_{14}\), and \(K_{15}\) in sequence according to the machining order. Based on the split structure of equal-base circle level gear and the machining process, the equivalent stiffness of the cutting gear blank is  
\[K_{i}=\int_{0}^{L}El_{k}(y_{1}(x)^{\prime\prime})^{2}dx \tag{4}\]  
where \(y_{1}(x)\) is the deflection curve equation of the equivalent equal section beam of the split gear blank. The stiffness is substituted into Equation (3) and the integration interval is adjusted; the deformation \(S_{i}\) in \(Z_{i}\) direction can be obtained via machining single tooth space at different positions. The deformation is located in the plane \(X_{i}O_{i}Z_{i}\), and the angle between the plane \(X_{i}O_{i}Z_{i}\) and the XOZ plane of the datum coordinate system is \(\theta_{i}\). Figure 4 shows the deformation projection diagram of the wheel blank. To solve the total deformation, the independent deformation of each tooth space is projected to the XOZ plane of the datum coordinate system, and then the deformation superposition calculation is performed. The principle of deformation projection is shown in Figure 5. After projection, the bending deformation \(S\) corresponding to \(S_{i}\) is obtained. As shown in Figure 5, from bending deformation \(S_{i}\) to \(S\) the amount of bending deformation remains unchanged, but the length of the analysis unit changes. When it is projected to the plane OXZ of the datum coordinate system, there is an additional projection relationship of \(cos(\theta_{i})\). \[L_{i}=L*\cos(\theta_{i}) \tag{5}\]  
According to the former finite element analysis of wheel blank deformation, the deformation has nothing to do with the cutting sequence, and the bending deformation follows the law of deformation superposition. Without losing generality, the first machined tooth space starts from the third tooth space as shown in Figure 6.  
The coordinate system \(S_{3}(o_{3}-x_{3},z_{3})\), as shown in the Figure 6, is established at the tooth space of the segmented gear blank, and \(z_{3}\) is the axis of symmetry in the middle of the tooth slot. When machining the first tooth space, on the XOZ plane of the coordinate system OXYZ, \(\theta_{11}\) and \(\theta_{12}\) are the angles between the bottom surface of the left and right sides without the processing areas and the line which is horizontal, respectively. Figure 4: Projection diagram of gear blank deformation. Because of the invariance of the bending deformation projection, \(S_{1}\) can be obtained directly according to Equation (6). \[S_{1}=\int_{0}^{L_{1}}x*M/(K_{1}L_{1})dx \tag{6}\]  
\[\theta_{11}\approx arcsin(S_{1}/l_{2}) \tag{7}\]  
\[\theta_{12}\approx arcsin(S_{1}/l_{1}) \tag{8}\]  
According to the above formulas and parameters, the additional torque of cutting one tooth space is calculated, which is substituted into the deflection calculation formula, and the calculation program is written in MATLAB (MATLAB R2014b (8.4.0.150421)) to obtain the gear blank deformation \(S_{1}\), \(\theta_{11}\), and \(\theta_{12}\).