# Stability Analysis of Milling Process with Multiple Delays
## 2 Milling Model with Multiple Delays  
Milling vibration considering the regeneration effect is usually modeled by DDEs which can be written as  
\[\mathbf{M}\ddot{\mathbf{y}}(t)\mathbf{+}\mathbf{Cy}(t)\mathbf{+}\mathbf{K} \mathbf{y}(t)=\mathbf{f}(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are the mass, damping and stiffness matrices respectively, \(\mathbf{y}(t)\) is the vibration displacements vector and \(\mathbf{f}(t)\) is the time-varying cutting force vector. When using the variable pitch cutter (as described in Figure 1), the dynamics model of the milling process becomes DDEs with multiple delays. The cutting force \(\mathbf{f}(t)\) in Equation (1) can therefore be expressed as  
\[\mathbf{f}(t)=\sum_{j=1}^{N}\mathbf{H}_{j}(t)\cdot\left[\mathbf{y}(t)-\mathbf{ y}\big{(}t-T_{j}\big{)}\right] \tag{2}\]  
where \(N\) is the number of teeth, \(T_{j}\) is the time delay of the \(j\)-th tooth, \(\mathbf{y}(t)\) and \(\mathbf{y}\big{(}t-T_{j}\big{)}\) are the vibration displacements at the current moment and the previous tooth-passing period of the \(j\)-th tooth separately. \(\mathbf{H}_{j}(t)\) is the time-varying cutting force coefficient matrix of the \(j\)-th tooth, which satisfies \(\mathbf{H}_{j}(t)=\mathbf{H}_{j}(t+T)\), where \(T\) is the spindle period. \(\mathbf{H}_{j}(t)\) can be written as  
\[\mathbf{H}_{j}(t)=\left[ \begin{array}{cc}h_{xx,j}(t)&h_{xy,j}(t)\\ h_{yx,j}(t)&h_{yy,j}(t)\end{array} \right] \tag{3}\]  
where  
\[\left\{ \begin{array}{l}h_{xx,j}(t)=\int\limits_{z_{l,j}(t)}^{z_{u,j}(t)} \left(-K_{t}\cos\varphi_{j}(t,z)\sin\varphi_{j}(t,z)-K_{n}\sin\varphi_{j}(t,z) \sin\varphi_{j}(t,z)\right)\cdot dz\\ h_{xy,j}(t)=\int\limits_{z_{l,j}(t)}^{z_{u,j}(t)}\left(-K_{t}\cos\varphi_{j}( t,z)\cos\varphi_{j}(t,z)-K_{n}\sin\varphi_{j}(t,z)\cos\varphi_{j}(t,z)\right) \cdot dz\\ h_{yx,j}(t)=\int\limits_{z_{l,j}(t)}^{z_{u,j}(t)}\left(+K_{t}\sin\varphi_{j}( t,z)\sin\varphi_{j}(t,z)-K_{n}\cos\varphi_{j}(t,z)\sin\varphi_{j}(t,z)\right) \cdot dz\\ h_{yy,j}(t)=\int\limits_{z_{l,j}(t)}^{z_{u,j}(t)}\left(+K_{t}\sin\varphi_{j}( t,z)\cos\varphi_{j}(t,z)-K_{n}\cos\varphi_{j}(t,z)\cos\varphi_{j}(t,z)\right) \cdot dz\end{array} \right. \tag{4}\]  
where \(K_{t}\) and \(K_{n}\) are the tangential and radial cutting force coefficients, respectively. \(z_{u,j}(t)\) and \(z_{l,j}(t)\) are the upper and lower bounds of the cutting edge participating in cutting on the \(j\)-th tooth as shown in Figure 1c, respectively. \(\varphi_{j}(t,z)\) is the angular position of the differential element of cutting edge with a height \(z\) on the \(j\)-th tooth, which can be expressed as  
\[\varphi_{j}(t,z)=\varphi_{j}(0,0)+\frac{2\pi\Omega}{60}\cdot t-\phi_{lag}(z) \tag{5}\]  
where \(\varphi_{j}(0,0)\) is the angular position of the lowest differential element on the \(j\)-th tooth at the initial time, \(\Omega\) is the spindle speed, \(\phi_{lag}(z)\) is the lag angle of the differential element at height \(z\) as shown in Figure 1d, and \(\phi_{lag}(z)=(2\tan\beta/D)\cdot z\), where \(D\) is the diameter of the cutter, \(\beta\) is the helix angle. Equation (1) can be converted to the state-space form as  
\[\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\sum_{j=1}^{N}\mathbf{B}_{j}(t) \left[\mathbf{x}(t)-\mathbf{x}\big{(}t-T_{j}\big{)}\right] \tag{6}\]  
where \(\mathbf{A}\) is a constant matrix, representing the time-invariant characteristics of the milling system, while \(\mathbf{B}_{j}\) is a periodic matrix satisfying \(\mathbf{B}_{j}(t)=\mathbf{B}_{j}(t+T)\), which describes the periodic characteristics of the system. \(\mathbf{x}\) is known as the state vector  
\[\mathbf{A}=\left[ \begin{array}{cc}0&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array} \right], \mathbf{B}_{j}(t)=\left[ \begin{array}{cc}0&0\\ \mathbf{M}^{-1}\cdot\mathbf{H}_{j}(t)&0\end{array} \right],\mathbf{x}(t)=\left[ \begin{array}{c}\mathbf{y}(t)\\ \dot{\mathbf{y}}(t)\end{array} \right]\]  
where \(\mathbf{I}\) is an identity matrix with the same size as \(\mathbf{M}\).