## 4 Analysis of Results
### Optimization Model Approach  
The machining process optimization model proposed to minimize machining time takes the form proposed by Equation (45):  
\[Min\Bigg{\{}t=\Bigg{(}\frac{3l_{mcc}}{2F_{v}}\Bigg{)}\Bigg{[}R_{sup}\Bigg{(} \frac{D_{c}-d_{m}}{2a_{c}}\Bigg{)}\Bigg{]}\Bigg{\}} \tag{45}\]  
The restrictions requested for the calculation of this process are of the structural, energy, and surface finish type, defined by the relationships given in Equation (46):\[K_{1}a_{c}\leq P_{\varepsilon} \tag{46}\] \[K_{\sigma}\left[\frac{D_{c}a_{c}k_{c}F_{\mu}}{V_{c}(d_{m}+2a_{c})^{2 }}\right]\left[4K_{F}+\frac{32K_{M}}{d_{m}+2a_{c}}\right]\leq S_{y}\] \[R_{c}=max\Big{\{}0.05d_{m}\ \frac{K_{2}}{V_{c}^{2}}\Big{\}}\]  
For the coefficients defined from Equation (47):  
\[\begin{array}{l}K_{1}=\frac{\pi D_{c}F_{\mu}k_{c}}{60.000P_{T}}\\ K_{2}=\frac{0.008}{K_{a}}\left(\frac{\pi D_{c}F_{\mu}}{1000}\right)^{2}\\ K_{F}\approx 0.001073\\ K_{M}=\sqrt{\left(\frac{d_{m}}{1000}\right)^{2}+\left(\frac{l_{T}}{3000} \right)^{2}+\left(\frac{d_{m}}{5000}-\frac{l_{T}}{3000}\right)^{2}}\\ K_{\sigma}=K_{\sigma}(R_{c},D_{c},d_{m})\in Max\Big{\{}0.05d_{m},\ \frac{K_{2}}{V_{c}^{2}}\Big{\}}\cap\ \frac{D_{c}}{d_{m}}\end{array} \tag{47}\]  
In this proposed model, the cutting depth \(a_{c}\) will depend on the process parameters: material strength, the machining process to be performed, and the power available to carry out the process, delivered by the machine tool motor.  
Table 2 shows the dynamic parameters used to define the process provided by cutting tool manufacturers for work on conventional machine tools, in this case, a parallel lathe. All parameters are constant and fixed except for the cutting speed, which takes values defined by intervals according to the roughing or finishing operation.  
Table 3 shows the typical surface roughness values, requested during the chip removal machining process depending on the application of the component to be machined.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{2}{c}{**Cutting Speed**} & \multirow{2}{*}{**Feed Speed**} & \multirow{2}{*}{**Specific Cutting Pressure**} \\  & \(V_{c}\) [m/min] & & & \(F_{\mathrm{p}}\) [mm/min] & \\  & **Roughing** & **Finish** & & \(k_{\mathrm{s}}\) [MPa] \\ \hline Copper-aluminum & 60–80 & 80–100 & 120 & 500 \\ Aluminum & 60–80 & 80–100 & 80 & 450 \\ Common bronze & 25–28 & 35–40 & 80 & 800 \\ Mild steel & 16–20 & 25–30 & 60 & 1700 \\ semi-hard steel & 12–16 & 18–22 & 30 & 2100 \\ hard steel & 10–15 & 16–18 & 25 & 3000 \\ very hard steel & 8–12 & 13–15 & 20 & 3500 \\ gray cast iron & 12–15 & 18–20 & 45 & 1250 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cutting speeds, feeds, and specific cutting pressure for high-speed steel tools [14; 15].  
\begin{table}
\begin{tabular}{c c} \hline \hline \(R_{a}\) [mm] & **Typical Surface Roughness Applications** \\ \hline
0.0003 & Machined surfaces for guides such as piston heads. \\
0.0004 & Machined surfaces for seating tracks. \\
0.0015 & Machined surfaces for precision shaft-hole fits. \\
0.0020 & Machined surfaces in general: shafts, keys, housings, etc. \\
0.0030 & Machined reference surfaces: support surfaces. \\
0.0040 & Rough machined surfaces \\ \hline \hline \end{tabular}
\end{table}
Table 3: Common roughness values for machining parts [14; 15].  
Based on the available technical parameters and the requested geometric data, it is possible to begin the process of selecting the optimal speed, which will be linked to the cutting depth, and will comply with the proposed machining time indicator, subject to the imposed restrictions.