# A Study on the Effect of Gear Hobbing Process Parameters on the Residual Stress of the Tooth Root
## 4 Results and Discussion  
### Orthogonal Test Analysis  
Through the analyses of the above methods, the residual stress measurement results of the tooth root in the orthogonal test are shown in Table 7.  
Before the scientific tests, four levels of hob speeds, axial feed speeds, and radial cutting depths were selected in advance, as shown in Table 8. The significant degree of influence of each gear hobbing parameter on the residual stress at the root of the tooth was compared by range R. The analysis results are shown in Table 9, and the main effect trend is shown in Figure 6. The greater the range R is, the more significant the influence of the process parameters on the residual stress of the tooth root is. It can be concluded that the importance of the process parameters on the residual stress of the tooth root is, successively, hob speed, axial feed speed, and radial cutting depth. In each process parameter range, the best process parameter combination is low hob speed, low axial feed speed, and small path cutting depth. Just as the results of the analysis of factors affecting the quality of gear hobbing in reference [33], it is believed that the process parameters having a greater impact on the quality of gear hobbing are hob speed, axial feed speed, and radial cutting depth. In order to further quantify the importance of various process parameters on the residual root stress, the Least Absolute Shrinkage and Selection Operator (LASSO) is used to evaluate the process parameters. LASSO [34] is a statistical method for feature selection, which has the advantages of more efficient feature selection, better robustness, and stronger interpretation in data importance analysis. It is widely used and has significant effects in process parameter feature importance analysis, which could also provide the feature importance value. Therefore, LASSO is selected to analyze the importance of the process parameters of hobbing to the residual stress of the tooth root. In the LASSO [35] algorithm, for the sample data set \(D=\{x_{i},y_{i}\},i=1,2,3,\ldots,n\), the linear regression model is shown in Formula (6):  
\[y_{i}=\sum_{j=1}^{p}\beta_{j}x_{ij}+\varepsilon_{i} \tag{6}\]  
\(\beta=\big{(}\beta_{1},\beta_{2},\ldots,\beta_{p}\big{)}^{T}\) define the LASSO evaluation as Formula (7):  
\[\widehat{\beta}=\text{argmin}\left\{\sum_{i=1}^{n}\left(y_{i}-\sum_{j=1}^{p} \beta_{j}x_{ij}\right)^{2}\right\},s.t.\sum_{j=1}^{p}\big{|}\beta_{j}\big{|}\leq t \tag{7}\]  
\(t\geq 0\) is the adjustment parameter, and \(\widehat{\beta}\) is the least squares method, which through the control of the value achieves the objective function's least squares results. Restrict the condition function \(s.t.\sum\limits_{j=1}^{p}\big{|}\beta_{j}\big{|}\leq t\) as a penalty function, and add the regular term to the objective function, as shown in Formula (8). \[\widehat{\beta}_{N}(\lambda_{n})=\text{argmin}\left\{\sum_{i=1}^{n}\left(y_{i }-\sum_{j=1}^{p}\beta_{j}x_{ij}\right)^{2}+\lambda_{n}\sum_{j=1}^{p}\beta_{j}\right\} \tag{8}\]  
\(\lambda_{n}\geq 0\) is the penalty coefficient (regularization coefficient); therefore, the weight meeting the objective function can reflect the importance of the feature:  
\[LFI(i)=\beta_{j},j=1,2,3,\ldots,p \tag{9}\]  
As shown in Figure 7, the importance values of hob speed, axial feed speed, and radial cutting depth to the residual stress at the root of the tooth are, respectively, 0.460, 0.278, and 0.262, and the analysis results are consistent with the range analysis results. \begin{table}
\begin{tabular}{c c c c} \hline  & **Hob Speed/ (r-min\({}^{-1}\))** & **Axial Feed Velocity/ (mm/min)** & **Radial Cutting Depth/ (mm)** \\ \hline k1 & 106.43 & 138.23 & 114.96 \\ k2 & 201.72 & 153.54 & 163.60 \\ k3 & 194.03 & 150.83 & 191.08 \\ k4 & 154.78 & 214.36 & 187.33 \\ R & 95.28 & 76.13 & 76.12 \\ \hline \end{tabular}
\end{table} 
Table 9: Range analysis of residual stress of tooth root during hobbing.