## 3 Stability Analysis and Discussion
### Single DOF Model of VHCVSS
#### 3.1.1 Modeling  
Compared with a VPC, the pitches of a VHC vary with the axial height \(z\). Considering regeneration effect, the related time delay will be a function of \(z\). When the effect of VSS is also taken into account, the multiple and time-varying delays \(\tau_{j}(t,\phi)\) associated with VPCVSS (for more details, see [49]) will be changed to the multiple and distributed ones \(\tau_{j}(t,z)\) for VHCVSS. Starting with this idea, considering the nonlinear force model in [7], the 1-DOF dynamic equation of a VHCVSS milling shown in Figure 2a,c, can be expressed as  
\[m_{x}\ddot{x}(t)+c_{x}\dot{x}(t)+k_{x}x(t)\ =\ -\sum_{j\ =\ 1}^{N}\int_{0}^{a_{p}}h_{ j}(z,t)(x(t)-x(t-\tau_{j}(t,z)))dz \tag{17}\]  
where \(m_{x},c_{x}\ =\ m_{x}\omega_{x}^{2}\) and \(k_{x}\ =\ 2m_{x}\zeta_{x}\omega_{x}\) respectively are the mode mass, the damping coefficient and the spring stiffness where \(\zeta_{x}\) is the damping ratios and \(\omega_{x}\) is the natural frequencies, \(N\) is the number of the cutter teeth, \(h_{j}(t,z)\) are the dynamic chip thickness for the \(j\)-th tooth defined as  
\[h_{j}(t,z)\ =\ qf_{z}^{d-1}g_{j}(t,z)\sin^{q}\phi_{j}(t,z)(K_{t}\cos\phi_{j}(t,z)+K _{r}\sin\phi_{j}(t,z)) \tag{18}\]  
where \(f_{z}\) is the feed per tooth, \(q\) is the nonlinear factor related to the milling force, \(K_{t}\) and \(K_{r}\) are cutting force coefficients in tangential and radial direction, the angular position for the \(j\) tooth is  
\[\phi_{j}(t,z)\ =\ \left\{ \begin{array}{c}\frac{2\pi}{60}\int_{0}^{t}\Omega( s)ds-z\rho,\ if\ j\ =\ 1\\ \frac{2\pi}{60}\int_{0}^{t}\Omega(s)ds\ +\sum\limits_{i\ =\ 2}^{j}\psi_{i}-z\rho,\ if\ 1<j\leq N \end{array} \right. \tag{19}\]  
where \(\Omega(s)\) is the variable spindle speed and \(\rho\) is equal to \(\tan\beta_{j}/R\). For a sinusoidal modulation [7] as shown in Figure 1b, the time period is \(T\ =\ 60/\Omega_{0}/\text{RVF}\), the amplitude is \(\Omega_{1}\ =\ \text{RVA}\times\Omega_{0}\) and its function can be expressed by  
\[\Omega(t)\ =\ \Omega_{0}+\Omega_{1}\sin(\frac{2\pi}{T}t)\ =\ \Omega_{0}[1+\text{RVA} \cdot\sin(\text{RVF}\cdot\frac{2\pi}{60}\Omega_{0}t)] \tag{20}\]  
where \(\text{RVA}\ =\ \Omega_{1}/\Omega_{0}\) and \(\text{RVF}\ =\ 60/(\Omega_{0}T)\) are respectively used to characterize the ratio of the speed variation amplitude to nominal spindle speed and the ratio of the speed variation frequency to the nominal spindle speed in a VSS milling. Considering Equations (2) and (3), and slicing the axial depth of cut into\(f\) disks axially (see Figure 3), then Equation (17) can be changed into  
\[m_{x}\ddot{x}(t)+c_{x}\dot{x}(t)+k_{x}x(t)\ =\ -\sum_{j\ =\ 1}^{N}\sum_{k\ =\ 1}^{f}\widetilde{C}_{j,k}(t)(x(t)-x(t-\widetilde{\tau}_{j,k}(t))) \tag{21}\]  
where  
\[\widetilde{C}_{j,k}(t)\ =\ \left\{ \begin{array}{c}\int_{(k-1)h}^{kh}h_{j}(z,t)dz \quad k\ =\ 1,2,\ldots,f-1\\ \int_{(f-1)h}^{a_{p}}h_{j}(z,t)d\phi\quad\quad\quad\quad\quad k\ =\ f\\ \widetilde{\tau}_{j,k}(t)\ =\ \tau_{j}(t,a_{k})\\ \end{array} \right. \tag{22}\]In order to select the appropriate value of \(f\) to reduce the calculation cost, the values of \(h\) in Equation (22) can be calculated by \(2\pi R/\tan\beta_{1}/l\). Under the circumstances of VHCVSS, the time delay \(\widetilde{\tau}_{j,k}(t)\) in Equation (20) can be written as  
\[\int_{t-\widetilde{\tau}_{j,k}(t)}^{t}\frac{\Omega(s)}{60}ds\ =\ \frac{ \widetilde{\psi}_{j,k}}{360} \tag{23}\]  
Clearly, one cannot obtain a closed form of \(\widetilde{\tau}_{j,k}(t)\) in Equation (23), thus, only numerical calculation can be used [7]. It should be noted that because compared to \(\Omega_{0}\), the values of \(\Omega_{1}\) in Equation (20) can be small enough in practice, one can expand Equation (23) in series, and obtain the approximation of \(\widetilde{\tau}_{j,k}(t)\) (for more details, see [7,49]) as  
\[\widetilde{\tau}_{j,k}(t)\approx\widetilde{\psi}_{j,k}/\theta/6 \tag{24}\]  
where \(\widetilde{\psi}_{j,k}\ =\ \ \psi_{j}-a_{k}\gamma_{j}/R\), \(\gamma_{j}\ =\ \