## 2 Description of the Proposed Method
### Simulation Model
#### 2.2.1 Cutting Process Dynamics
\(l\)-1,  
\(f_{2}\)--feed per tooth; \(f_{2}=\
u_{f}/(nz)\),  
\(z\)--number of milling cutter teeth. This model also takes into account the effects of the internal and external modulation of the layer thickness and handles the situation when the cutting edge loses contact with the workpiece material. It is worth noting here that the interaction of the tool and the workpiece may also result in a ploughing force. Unlike cutting force, it does not remove a layer of material but causes local plasticizing. It results from the nose radius \(r_{t}\) of assuming a perfectly sharp cutting edge, and in a simplified mathematical description, its value is proportional to the width of the cut layer. However, in the considerations presented in the article, the following arguments led to the of the dynamic model of the cutting process. 1. The influence of the ploughing force is significant when \(h_{DI}<r_{t}\)[32]. In the case of inserts of unblunt milling tools, the \(r_{t}\) value does not exceed 10 \(\upmu\)m, which in the case of the experiment in question (\(f_{2}=0.115\) mm) causes \(h_{DI}=f_{2}\cos(\varphi_{l})>r_{t}\) in the range of (\(-85^{\circ}\); \(+85^{\circ}\)), so in almost the entire range of possible immersion angles \(\langle-90^{\circ};+90^{\circ}\rangle\). 2. The ploughing force is important, especially in light of possible non-linearities, at low cutting speeds [33]. In the present experiment, at \(n=1300\) rpm and \(D=44\) mm, \(v_{c}\) values are estimated to be close to 200 m/min, which fully justifies the omission of the ploughing force. 3. The mentioned plasticizing effect makes the proportionality coefficient necessary to determine the ploughing force problematic and even difficult to estimate, especially in real production conditions. Due to the above, the authors decided on a more practical approach based on a proportional mechanistic model (in Equations (1)-(3), there is a minimum number of only 3 parameters: \(k_{d}\), \(\mu_{l2}\), and \(\mu_{l3}\)) and its validation with the actual machining process by comparing RMS values observed in the time domain during the real and simulated process. Although these three parameters do not have a direct physical interpretation, their accurately adjusted values during the simulation confirm the compliance of the adopted model of the face milling process. The description of the cutting forces for CE no. \(l\) in a 6-DOF space takes the following form [30]:  
\[\mathbf{F}_{l}(t)=\mathbf{F}_{l}^{0}(t)-\mathbf{D}_{Pl}(t)\Delta\mathbf{w}_{l} (t)+\mathbf{D}_{Ol}(t)\Delta\mathbf{w}_{l}(t-\tau_{l}), \tag{6}\]  
where  
\[\mathbf{F}_{l}(t)=col\left(F_{yl1}(t),F_{yl2}(t),F_{yl3}(t),0,\;0,\;0\right), \tag{7}\]  
\[\mathbf{F}_{l}^{0}(t)=col\left(k_{dl}a_{p}h_{Dl}(t),\mu_{l2}k_{dl}a_{p}h_{Dl}( t),\mu_{l3}k_{dl}a_{p}h_{Dl}(t),\;0,\;0,\;0\right), \tag{8}\]  
\[\mathbf{D}_{Pl}(t)= \begin{bmatrix}0&k_{dl}\left(a_{p}-\Delta a_{pl}(t)\right) &k_{dl}h_{Dl}(t)\\ 0&\mu_{l2}k_{dl}\left(a_{p}-\Delta a_{pl}(t)\right)&\mu_{l2}k_{dl}h_{Dl}(t)& \mathbf{0}_{3\times 3}\\ 0&\mu_{l3}k_{dl}\left(a_{p}-\Delta a_{pl}(t)\right)&\mu_{l3}k_{dl}h_{Dl}(t)& \mathbf{0}_{3\times 3}\\ \mathbf{0}_{3\times 3}&&\mathbf{0}_{3\times 3}\end{bmatrix} , \tag{9}\]  
\[\mathbf{D}_{Ol}(t)= \begin{bmatrix}0&k_{dl}\left(a_{p}-\Delta a_{pl}(t)\right) &0\\ 0&\mu_{l2}k_{dl}\left(a_{p}-\Delta a_{pl}(t)\right)&0&\mathbf{0}_{3\times 3}\\ 0&\mu_{l3}k_{dl}\left(a_{p}-\Delta a_{pl}(t)\right)&0&\\ \mathbf{0}_{3\times 3}&&\mathbf{0}_{3\times 3}\end{bmatrix} , \tag{10}\]  
\[\Delta\mathbf{w}_{l}(t)=col\left(q_{zl}(t),\Delta h_{l}(t),\Delta a_{pl}(t),0, \;0,\;0\right), \tag{11}\]  
\[\Delta\mathbf{w}_{l}(t-\tau_{l})=col\left(q_{zl}(t-\tau_{l}),\Delta h_{l}(t- \tau_{l}),\Delta a_{pl}(t-\tau_{l}),0,\;0,\;0\right), \tag{12}\]  
and \(q_{zl}(t)\)--relative displacement of edge tip and workpiece along direction \(y_{l1}\) at instant of time \(t\), and \(q_{zl}(t-\tau_{l})\)--relative displacement of edge tip and workpiece along direction \(y_{l1}\) at instant of time \(t-\tau_{l}\).