## 2 Materials and Method
Figure 2: Types of installation of face Up and Down milling (\(a_{i}\), uncut chip thickness; \(\psi_{1}=\psi_{in}\), inlet angle; \(\psi_{2}=\psi_{out}\), out angle; \(\psi\), the contact angle; \(\theta\), tooth spacing angle; \(\psi_{i}\), angular coordinate of the i-th tooth; \(B\), depth of milling; \(e\), positions symmetry of cutter and workpiece; \(d^{\prime}\), positions the end of the cutter diameter and the beginning of the workpiece; \(D_{r}\), direction of rotary driven; \(D_{s}\), feed movement; 1, 2, 3, 1\({}^{\prime}\), 2\({}^{\prime}\), 3\({}^{\prime}\), points determining the amount of chip scraps removed per one rotation of the cutter); (**a)** Asymmetric circuit (Up milling), (**b)** Asymmetric circuit (Down milling), (**c)** Symmetric circuit (Up milling), (**d)** Symmetric circuit (Down milling). We show the formulas for calculating the parameters given in the table. Inlet angle:  
\[\psi_{1}=\arccos\left(\frac{B+2\cdot e}{D}\right) \tag{1}\]  
Out angle:  
\[\psi_{2}=\arccos\left(\frac{2\cdot e-B}{D}\right) \tag{2}\]  
In Equations (1) and (2), \(B\) is the depth of milling; \(D\) is diameter of milling; \(e\) is relative positions symmetry of cutter and workpiece. Through the \(e=R-a^{\prime}-B/2\), correlations (see Figure 2) and Equations (1) and (2) are converted to the form shown in Equations (3) and (4), respectively. The contact angle of a face mill with the workpiece \(\psi\) generally is determined:  
\[\psi=\psi_{2}-\psi_{1} \tag{3}\]  
The angle between adjacent teeth would be multiple to the contact angle (see Figure 2). \[\theta=2\cdot\pi/z, \tag{4}\]  
where \(\pi\) = 3.14159,..., i.e., Pi number--a mathematical constant that expresses the ratio of circumference to its diameter length [38]. The number of active teeth involved in the work is determined by the formula:  
\[z_{c}=\psi/\theta, \tag{5}\]  
Table 5 shows the experimental values of the roughness of the surfaces \(Ra\) and \(Rz\) (for four runs and average values) for different values of the relative position of the face mill and workpiece \(a^{\prime}\). Table 6 shows the experimental components of the cutting forces \(F_{x}\), \(F_{y}\), \(F_{z}\) (minimum, maximum, average, and RMS (Root-mean-square)) and accelerations of vibrations \(a_{x}\) and \(a_{y}\) (RMS) for different values of relative position of face mill and workpiece, \(a^{\prime}\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Inputs Parameters**} & \multicolumn{6}{c}{**Outputs Parameters**} \\ \cline{2-13}
**Exp.** & **Relative Position** & \multicolumn{6}{c}{**Roughness, \(Ra\), \(\mu\)m**} & \multicolumn{6}{c}{**Roughness, \(Rz\), \(\mu\)m**} \\ \cline{3-13}
**No.** & **Face Mill and** & **1** & **2** & **3** & **4** & **1** & **2** & **3** & **4** & **Av** \\  & **Workpiece** & \(a^{\prime}\) & **meas** & **meas** & **meas** & **meas** & **Av** & **meas** & **meas** & **meas** & **Av** \\ \hline
1 & 0 & 2.19 & 1.75 & 3.24 & 2.61 & 2.45 & 12.6 & 10.56 & 19.35 & 15.35 & 14.47 \\
2 & 0.3 & 2.11 & 2.45 & 3.36 & 2.78 & 2.68 & 12.15 & 15.25 & 19.01 & 14.13 & 15.14 \\
3 & 0.8 & 2 & 2.45 & 3.54 & 2.94 & 2.73 & 12.4 & 14.24 & 19.95 & 14.94 & 15.38 \\
4 & 3.3 & 2.25 & 2.28 & 2.87 & 3.24 & 2.66 & 14.63 & 12.26 & 16.87 & 15.65 & 14.85 \\
5 & 6.3 & 2.42 & 1.85 & 3.37 & 3.25 & 2.72 & 13.98 & 10.91 & 20.07 & 17.53 & 15.62 \\
6 & 9.3 & 1.92 & 2.11 & 2.92 & 3.37 & 2.58 & 10.2 & 11.62 & 18.62 & 22.04 & 15.62 \\
7 & 11.8 & 2.45 & 2.95 & 2.81 & 2.51 & 2.68 & 14.19 & 16.47 & 14.88 & 12.73 & 14.57 \\
8 & 12.3 & 2.88 & 2.37 & 3.25 & 3.38 & 2.97 & 15.85 & 13.7 & 18.46 & 19.1 & 16.78 \\
9 & 12.6 & 3.05 & 3.32 & 3.24 & 3.28 & 3.22 & 19.05 & 16.74 & 15.2 & 19.64 & 17.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The experimental values of the roughness of surfaces \(Ra\) and \(Rz\).