## 2 3-DOF Chatter Stability Model
### Calculation of SLD with SDM  
Equation (1) can be represented by the matrix equation as follows:  
\[\mathbf{M\tilde{q}}(t)+\mathbf{C\dot{q}}(t)+\mathbf{Kq}(t)=\mathbf{H}(t)[ \mathbf{q}(t)-\mathbf{q}(t-T)] \tag{26}\]  
where  
\[\mathbf{M}= \begin{bmatrix}m_{x}&0&0\\ 0&m_{y}&0\\ 0&0&m_{z}\end{bmatrix}  \tag{27}\]  
\[\mathbf{C}= \begin{bmatrix}c_{x}&0&0\\ 0&c_{y}&0\\ 0&0&c_{z}\end{bmatrix}  \tag{28}\]  
\[\mathbf{K}= \begin{bmatrix}k_{x}&0&0\\ 0&k_{y}&0\\ 0&0&k_{z}\end{bmatrix}  \tag{29}\]  
\[\mathbf{q}(t)= \begin{bmatrix}x_{c}(t)\\ y_{c}(t)\\ z_{w}(t)\end{bmatrix}  \tag{30}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) denote the modal mass matrix, the modal damping matrix, and the modal stiffness matrix of the 3-DOF milling system, respectively. \(\mathbf{q}(t)\) denotes the displacement vector. At present, the SDM serves as an effective and widely applied method to calculate SLD. According to [PERSON] and [PERSON]'s theory [11], Equation (26) is rewritten in the following state-space form:  
\[\dot{\mathbf{Q}}(t)=\mathbf{R}(t)\mathbf{Q}(t)+\mathbf{S}(t)\mathbf{Q}(t-\tau) \tag{31}\]  
with  
\[\mathbf{R}(t)=\mathbf{R}(t+T) \tag{32}\]  
\[\mathbf{S}(t)=\mathbf{S}(t+T) \tag{33}\]  
where  
\[\mathbf{Q}(t)=\left[ \begin{array}{c}\mathbf{q}(t)\\ \dot{\mathbf{q}}(t)\end{array} \right] \tag{34}\]\[\mathbf{R}(t)= \begin{bmatrix}\mathbf{0}&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}-\mathbf{M}^{-1}\mathbf{H}(t)&-\mathbf{M}^{-1} \mathbf{C}\end{bmatrix}  \tag{35}\]  
\[\mathbf{S}(t)= \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{M}^{-1}\mathbf{H}(t)&\mathbf{0}\end{bmatrix}  \tag{36}\]  
where \(\mathbf{Q}(t)\) denotes the time-state vector. \(\mathbf{R}(t)\) and \(\mathbf{S}(t)\) denote the periodic coefficient matrixes describing the dynamics of the 3-DOF milling system. \(\tau\) is the time delay. The time delay \(\tau\) is discretized into \(m\) numbers of equal time intervals \(\Delta t\), namely, \(\tau=m\Delta t\), where \(m\) denotes an integer and is determined as follows:  
\[m=\text{int}\left(\frac{\tau+\Delta t/2}{\Delta t}\right) \tag{37}\]  
where int is the function for rounding a number down to the closest integer. When the initial condition satisfies \(\mathbf{Q}(t)=\mathbf{Q}_{i}\), the solution of Equation (31) is stated as follows:  
\[\mathbf{Q}(t)=e^{\mathbf{R}_{i}(t-t_{i})}\Big{[}\mathbf{Q}_{i}+\mathbf{R}_{i} ^{-1}\mathbf{S}_{i}\mathbf{Q}_{\tau,i}\Big{]}-\mathbf{R}_{i}^{-1}\mathbf{S}_ {i}\mathbf{Q}_{\tau,i} \tag{38}\]  
Substituting \(\mathbf{Q}_{i+1}=\mathbf{Q}(t_{i+1})\) and Equation (38) into Equation (31) yields as follows:  
\[\mathbf{Q}_{i+1}=\mathbf{U}_{i}\mathbf{Q}_{i}+\mathbf{\alpha}\mathbf{V}_{i} \mathbf{Q}_{i-m+1}+\beta\mathbf{V}_{i}\mathbf{Q}_{i-m} \tag{39}\]  
where  
\[\mathbf{U}_{i}=e^{\mathbf{R}_{i}\Delta t} \tag{40}\]  
\[\mathbf{V}_{i}=\left(e^{\mathbf{R}_{i}\Delta t}-\mathbf{I}\right)\mathbf{R}_ {i}^{-1}\mathbf{S}_{i} \tag{41}\]  
\[\mathbf{\alpha}=\frac{m\Delta t+\frac{\Delta t}{2}-\tau}{\Delta t} \tag{42}\]  
\[\beta=\frac{\tau+\frac{\Delta t}{2}-m\Delta t}{\Delta t} \tag{43}\]  
According to Equation (39), a discrete map can be defined as follows:  
\[\mathbf{w}_{i+1}=\mathbf{D}_{i}\mathbf{w}_{i} \tag{44}\]  
where  
\[\mathbf{w}_{i}= \begin{bmatrix}\mathbf{Q}_{i}\\ \mathbf{Q}_{i-1}\\ \vdots\\ \mathbf{Q}_{i-m}\end{bmatrix}  \tag{45}\]  
\[\mathbf{D}_{i}= \begin{bmatrix}\mathbf{U}_{i}&\mathbf{0}&\cdots&\mathbf{ \alpha}\mathbf{V}_{i}&\beta\mathbf{V}_{i}\\ \mathbf{I}&\mathbf{0}&\cdots&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{I}&\cdots&\mathbf{0}&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \mathbf{0}&\mathbf{0}&\cdots&\mathbf{I}&\mathbf{0}\end{bmatrix}  \tag{46}\]  
For evaluating the stability of the 3-DOF milling system, Equation (44) is written as follows:  
\[\mathbf{w}_{k}=\mathbf{\Phi}\mathbf{w}_{0}=\mathbf{D}_{k-1}\mathbf{D}_{k-2} \cdots\mathbf{D}_{1}\mathbf{D}_{0}\mathbf{w}_{0} \tag{47}\]  
According to [PERSON]'s theory, the eigenvalues of the matrix \(\mathbf{\Phi}\) are employed to evaluate the 3-DOF milling stability.