## 2 Description of the Proposed Method
### Simulation Model
#### 2.2.2 Hybrid Model of the Milling Process  
The dynamics of the milling process is modeled as a hybrid system (Figure 2) which consists of the modal and structural subsystems connected by coupling elements. The modal subsystem represents a flexible workpiece moving with the feed speed \(y_{l}\). It is described by the vector of its modal coordinates \(\mathbf{a}\). Parameters of the modal model are identified during experimental modal tests. For a finite, selected number of normal modes \(mod\), the dynamic properties are defined by the following [30]:  
\(\mathbf{\Omega}\) = \(diag(\omega_{0i})\)--matrix of angular natural frequencies of the modal subsystem; \(i\) = 1,..., \(mod\). Its square is also called the _stiffness modal matrix_;  
\(\mathbf{\Psi}= \begin{bmatrix}\mathbf{\Psi}_{1}&\ldots&\mathbf{\Psi}_{mod} \end{bmatrix} \)--matrix of the considered mass-normalized normal modes of the modal subsystem; \(i\) = 1,..., \(mod\);  
\(\mathbf{Z}=diag(\zeta_{i})\)--matrix of dimensionless damping coefficients (also called _modal damping_) of the modal subsystem; \(i\) = 1,..., \(mod\). The structural subsystem represents a rotating milling tool. Its behavior is described by the vector of generalized displacements \(\mathbf{q}_{t}\) in a local Cartesian coordinate system, \(x_{r1}\), \(x_{r2}\), \(x_{r3}\) of the RFE, and subsequently, these displacements are transformed to the immovable coordinate system \(x_{1}\), \(x_{2}\), \(x_{3}\). The movable coordinate system rotates together with the milling tool. Thus dynamic properties of the structural subsystem are defined by inertia \(\mathbf{M}\), damping \(\mathbf{L}\), stiffness \(\mathbf{K}\) matrices [30], and additionally, matrices dependent on the effect of the tool rotation, i.e., \(\hat{\mathbf{L}}_{r}\) and \(\hat{\mathbf{K}}_{r}\)[31] The above results in the fact that the first double natural frequency \(f_{0}\) of decoupled transverse vibrations of the non-rotating system will be replaced by two natural frequencies \(f_{01,\ 02}=f_{0}\pm\frac{\mu}{60}\), corresponding to the coupled modes of vibrations. The way to define matrices \(\hat{\mathbf{L}}_{r}\) and \(\hat{\mathbf{K}}_{r}\), in the case of RFE no. \(r\), is described in Appendix A. The matrix equation of the dynamics of the non-stationary hybrid model of the milling process in hybrid coordinates is described as follows:  
\[ \begin{bmatrix}\mathbf{M}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}\end{bmatrix} \tilde{\xi}+ \begin{bmatrix}\mathbf{L}+\hat{ \mathbf{L}}_{r}-\hat{\mathbf{L}}_{r}^{T}&\mathbf{0}\\ \mathbf{0}&\mathbf{2Z}\Omega\end{bmatrix} \hat{\xi}+ \begin{bmatrix}\mathbf{K}- \hat{\mathbf{K}}_{r}+\sum\limits_{l=1}^{\hat{l}_{l}}\mathbf{T}_{l}^{T} \mathbf{D}_{Pl}(t)\mathbf{T}_{l}&-\sum\limits_{l=1}^{\hat{l}_{l}}\mathbf{T}_{ l}^{T}\mathbf{D}_{Pl}(t)\mathbf{W}_{l}(t)\\ -\sum\limits_{l=1}^{\hat{l}_{l}}\mathbf{W}_{l}^{T}(t)\mathbf{D}_{Pl}(t) \mathbf{T}_{l}&\mathbf{\Omega}^{2}+\sum\limits_{l=1}^{\hat{l}_{l}}\mathbf{W}_ {l}^{T}(t)\mathbf{D}_{Pl}(t)\mathbf{W}_{l}(t)\end{bmatrix}  \tag{13}\]  
\[ \begin{split}\mathbf{\xi}=\begin{bmatrix}\sum\limits_{l=1}^{\hat{ l}_{l}}\mathbf{T}_{l}^{T}\mathbf{F}_{l}^{0}(t)+\mathbf{T}_{l}^{T}\mathbf{D}_{Ol}(t) \Delta\mathbf{w}(t-\tau_{l})\\ -\sum\limits_{l=1}^{\hat{l}_{l}}\mathbf{W}_{l}^{T}(t)\mathbf{F}_{l}^{0}(t)- \mathbf{W}_{l}^{T}(t)\mathbf{D}_{Ol}(t)\Delta\mathbf{w}(t-\tau_{l})\end{bmatrix},\end{split} \]  
where  
\[ \begin{split}\mathbf{\xi}=\begin{cases}\mathbf{q}\\ \mathbf{a}\end{cases}\text{--vector of hybrid coordinates of the hybrid system},\end{split} \]  
\(\mathbf{T}_{l}\)--transformation matrix of displacement vector \(\mathbf{q}\) from the \(x_{r1}\), \(x_{r2}\), \(x_{r3}\) coordinates of the RFE, to the coordinate system \(y_{l1}\), \(y_{l2}\), \(y_{l3}\) of CE no. \(l\)[30],  
\(\mathbf{W}_{l}(t)\)--transformation matrix between the displacement vector in modal coordinates \(\mathbf{a}\) and displacements in the coordinate system \(y_{l1}\), \(y_{l2}\), \(y_{l3}\) of CE no. \(l\)[30],  
\(i_{l}\)--number of \"active\" CEs.