## 2 Description of the Proposed Method
### Simulation Model  
#### 2.2.1 Cutting Process Dynamics  
In order to select the best clamping conditions, simulations of the face milling process are performed. In the considered case, the workpiece is not perfectly rigid, and the tool is a multi-edge milling cutter. The dynamics of the milling process is analyzed using the model (Figure 2) and assumptions described in [30]. The most important of them are as follows:  
Figure 1: A scheme of the EAOVP with linear interpolation of the vibration shapes for selecting the best clamping conditions. The tool rotates at spindle speed \(n\) and the workpiece moves at feed speed \(v_{f}\). Only the tool (modeled as the Rigid Finite Element (RFE) and tool holder (connected with the tool by Spring Damping Element (SDE)) and the workpiece are modeled. Other elements of the milling machine and process dynamics are neglected [1; 30]. The dynamic properties of the workpiece are taken into account. The proportional and delayed feedback interactions caused by current and previous passes of the cutting edge along the cut layer are included so the effects of internal and external modulation of the layer thickness are accommodated [31]. In the conventional contact points of the tool edges with the workpiece, coupling elements (CEs) are located that model the dynamics of the cutting process [31]. The positions of CEs relative to the workpiece change over time as the tool rotates at speed \(n\). The actual position of the cutter edge no. \(l\) is described by the immersion angle \(\varphi_{l}=\varphi_{l}(t)\). It corresponds to the temporary position of CE no. \(l\). Edges having, at a given time, contact with materials that are called \"active\". Other symbols included in Figure 2 are as follows:  
\(\gamma_{0}\)--rake angle and \(\alpha_{0}\)--clearance angle as elements of the cutting edge geometry,  
\(\kappa_{r}\)--cutting edge angle, and in the presented case, \(\kappa_{r}=90^{\circ}\),  
\(a_{p}\)--depth of cutting,  
\(F_{y1,}F_{y2,}F_{y3}\)--forces acting in the direction of, respectively, the nominal cutting speed \(v_{c}\), the cutting layer thickness \(h_{l}\), and current depth of cutting \(a_{pl}\),  
Figure 2: A scheme of a face milling of a flexible workpiece. * \(D\)--tool diameter and milling widths \(B_{1}\) and \(B_{2}\),
* local rotating coordinate system \(x_{r1}\), \(x_{r2}\), \(x_{r3}\) of the RFE,
* conventional point S of the tool--workpiece contact [28; 30; 31] and the non-rotating coordinate system \(x_{1}\), \(x_{2}\), \(x_{3}\) for this point, which moves linearly with respect to the workpiece. For the temporary point of tool edge--workpiece contact--modeled as CE no. \(l\), a proportional model of the dynamics of the cutting process was adopted [30; 31]:  
\[F_{yl1}(t)=\left\{ \begin{matrix}k_{d}a_{pl}(t)h_{l}(t),&h_{l}(t)>0\wedge a_{pl} (t)>0,\\ 0,&h_{l}(t)\leq 0\lor a_{pl}(t)\leq 0,\end{matrix} \right. \tag{1}\]  
\[F_{yl2}(t)=\left\{ \begin{matrix}\mu_{l2}k_{d}a_{pl}(t)h_{l}(t),&h_{l}(t)>0\wedge a _{pl}(t)>0,\\ 0,&h_{l}(t)\leq 0\lor a_{pl}(t)\leq 0,\end{matrix} \right. \tag{2}\]  
\[F_{yl3}(t)=\left\{ \begin{matrix}\mu_{l3}k_{d}a_{pl}(t)h_{l}(t),&h_{l}(t)>0\wedge a _{pl}(t)>0,\\ 0,&h_{l}(t)\leq 0\lor a_{pl}(t)\leq 0,\end{matrix} \right. \tag{3}\]  
where  
\[a_{pl}(t)=a_{p}-\Delta a_{pl}(t), \tag{4}\]  
\[h_{l}(t)=h_{DI}(t)-\Delta h_{l}(t)+\Delta h_{l}(t-\tau_{l}), \tag{5}\]  
\(a_{p}\)--desired cutting depth,  
\(\Delta a_{pl}(t)\)--dynamic change in cutting depth for CE no. \(l\),  
\(h_{DI}(t)\)--desired cutting layer thickness for CE no. \(l\); \(h_{DI}(t)\cong f_{2}\sin\kappa_{r}\cos\varphi_{l}(t)\),  
\(\Delta h_{l}(t)\)--dynamic change in cutting layer thickness for CE no. \(l\),  
\(k_{d}\)--average dynamic specific cutting pressure for CE no. \(l\),  
\(\mu_{l2}\), \(h_{l3}\)--cutting force ratios for CE no. \(l\), as quotients of forces \(F_{yl2}\) and \(F_{yl1}\), and forces \(F_{yl3}\) and \(F_{yl1}\),  
\(\tau_{l}\)--time delay between the same position of CE no. \(l\) and of CE no.