## 5 Model Simulation and Discussion  
### Derivation of \(R_{a}\)-\(V\)B Relationship  
After the experimental verification of \(R_{a}\)-\(t\) model (9), Appling the same conditions of the experimental, numerical simulation discussions can be conducted based on both \(R_{a}\)-\(t\) model (9) and the \(VB\)-\(t\) model (11) verified in previous studies. The detailed simulation input parameters are shown in Table 4. The tool and workpiece material characteristic parameters \(T\&WP\), comprehensive wear coefficients \(C\!WC\), and the wear model error coefficients \(W\!EC\) applied in \(VB\)-\(t\) model (11) all come from wear study [11]. For the simulation discussion based on both \(VB\)-\(t\) model and Ra-t model, the tool geometric parameters \(T\!GP\) used in this experiment and simulation are also consistent with the wear study. The roughness error coefficients \(REC\) are corrected by the experiment in Section 4 of this study. Specifically, since the finishing temperature of the titanium alloy in this experiment is lower than 600 \({}^{\circ}\)C, the \(n_{0}\) in (12) should be set as zero [34; 35], which makes \(T\&WP\)\(c\) and \(d\) in \(V_{1}\) of (12) are meaningless in this model, so \(b\) and \(c\) are not selected by reference.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(N\) & **2** \\ \hline [PERSON]’s W identification coefficient & 0.897 \\ Asymptotic significance & 0.011 \textless{} 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Verification statistics of model.  
In addition, the simulation adopts the same three machining conditions as the orthogonal experiment. The above contents are used as the input parameters and condition employed of model (9), (11) and (12), so that the numerical simulation can be realized based on MATLAB software. As shown in Figure 14, the \(VB\)-\(R_{a}\)-\(t\) three-dimensional curve can be generated by combining \(R_{a}\)-\(t\) model and \(VB\)-\(t\) model. By projecting the three-dimensional curve on the \(t\)-O-\(R_{a}\) surface, the relationship of \(R_{a}\)-\(t\) can be obtained, which is consistent with the Figures 2 and 13, and by projecting the three-dimensional curve on \(t\)-\(O\)-\(VB\) surface, the relationship of \(VB\)-\(t\) can be obtained. Furthermore, when model \(VB\)-\(t\) and model \(R_{a}\)-\(t\) are merged without \(t\) variable and the three-dimensional curve is projected on the \(VB\)-\(O\)-\(R_{a}\) surface, the relationship curve of \(R_{a}\)-\(VB\) can be generated, as shown in Figure 14, which is consistent with the Figure 1.  
As shown in Figure 14, the distribution characteristics of \(R_{a}\) and \(VB\) are very similar but different. The main difference is that the starting points of \(R_{a}\) and \(VB\) are different, due to the geometric envelope relationship between the ball end milling tool and the curved surface, the machining surface created by the ball end tool without any wear still remains a residual height of material, which leads to \(R_{a}\) a non-zero initial value. In addition, the relationship of \(R_{a}\) and \(VB\) are not strictly linear, which is the key reason makes the relatively stable regions of \(VB\) and \(R_{a}\) do not coincide as described in Figure 1.