# Stability Analysis of Milling Process with Multiple Delays
## 3 AVSNIM Considering the Helix Angle
### Algorithm Description
Next, the discrete number \(m_{i}(w)\) and angle step \(\Delta\phi_{p,i}(w)\) of each forced vibration angle interval \(\phi_{fo,i}(w)\) can be obtained by the following formula  
\[\left\{ \begin{array}{l}m_{i}(w)=round\Big{(}\frac{\phi_{fo,i}(w)}{\Delta \phi_{p}(w)}\Big{)}\\ \Delta\phi_{p,i}(w)=\frac{\phi_{fo,i}(w)}{m_{i}(w)}\end{array} \right. \tag{12}\]  
where \"round (*)\" means rounds \"*\" towards the nearest integer, \(i=1,2,\cdots,N_{d}\). The discrete time step can be easily converted from the discrete angle step through the following formula  
\[\left\{ \begin{array}{l}\Delta t_{fr,i}(w)=\frac{60\cdot\Delta\phi_{fr,i}(w )}{2\pi\Omega}\\ \Delta t_{p,i}(w)=\frac{60\cdot\Delta\phi_{p,i}(w)}{2\pi\Omega}\end{array} \right. \tag{13}\]  
To date, we have got all the discretization information in one spindle period, including the number of discrete intervals and the length of each interval. The number of discrete intervals is \(m(w)=\sum\limits_{i=1}^{N_{d}}[1+m_{i}(w)]\) and therefore the index of the discrete time nodes in one spindle period are \(\{1,2,\cdots,m(w),m(w)+1\}\). Let \(n(i,k)\) be the index of the \(k\)-th node of the \(i\)-th segment in the discrete time nodes of one spindle period, where \(k=1,2,\cdots,1+m_{i}(w)+1\), \(n(i,k)\in\{1,2,\cdots,m(w),m(w)+1\}\) and can be expressed as  
\[n(i,k)=\left\{ \begin{array}{l}k(i=1)\\ \sum\limits_{r=1}^{i-1}[1+m_{r}(w)]+k(1<i\leq N_{d})\end{array} \right. \tag{14}\]  
According to the theory of differential equation, the state vector at the \(k\)-th time node of the \(i\)-th segment \(\textbf{x}\Big{(}t_{n(i,k)}\Big{)}\) can be deduced as  
\[\textbf{x}\Big{(}t_{n(i,k)}\Big{)}=\left\{\begin{array}{l}\textbf{x}_{1}(i =1,k=1)\text{ Initial Value }\\ e^{\textbf  
Let \(n\big{(}i,k,T_{j}\big{)}\) be the index of the time node on two adjacent spindle periods and satisfy the following relationship  
\[\left\{\begin{array}{l}t_{n(i,k-1,T_{j})}\leq t_{n(i,k-1)}-T_{j}<t_{n(i(k-1,T_{ j}))+1}\\ t_{n(i,k,T_{j})}\leq t_{n(i,k)}-T_{j}<t_{n(i  
Then, the Floquet transition matrix \(\mathbf{\Phi}\) can be expressed as  
\[\mathbf{\Phi}=\left(\mathbf{P}-\sum_{j=1}^{N}\left(\mathbf{R}_{1j}+\mathbf{R}_{2j }\right)\right)^{-1}\cdot\sum_{j=1}^{N}\left(\mathbf{Q}_{1j}+\mathbf{Q}_{2j}\right) \tag{23}\]  
where the superscript '\(-1\)' indicate the operation of matrix inversion. For details of the coefficients of the state vectors in Equation (21) and matrices \(\mathbf{P}\), \(\mathbf{R}_{1j}\), \(\mathbf{R}_{2j}\), \(\mathbf{Q}_{1j}\) and \(\mathbf{Q}_{2j}\) in Equations (22) and (23), please refer to Appendix A. The stability of milling process can be determined by the spectral radius of the Floquet transition matrix \(\mathbf{\Phi}\) according to the Floquet theory. It can be observed from the derivation process that the proposed method can adaptively complete the discrete of the spindle period and get the time step of each discrete interval only by simply setting a discretization parameter in advance. It is worth noting that under some machining parameters, the cutter will not experience free vibration, and then the variable-step method degenerates to the equal-step algorithm. Meanwhile, although the trapezoid rule is used to approximate the integral term in the algorithm derivation, other numerical methods, such as the SBM [27], the SSM [28], etc. can also be adopted to deal with Equation (15).