## 4 Stability Analysis of Multivariable Milling Tools
### Stability Analysis of 1 DOF Milling with a Multivariable Tool  
The stability lobes computed for the multivariable tool using the third-order EMHPM with a mesh of \(400\times 200\) (\(n\times a_{p}\)) are shown in Figure 6 together with stability lobes for a regular tool (angles of \(90^{\circ}\) and helix angles of \(30^{\circ}\) for all flutes). An approximation of order \(m=7\) was used with \(N=241\) and \(a_{d}=1\) mm. Notice from Figure 6 that the stable zone obtained for the multivariable tool was significantly larger, meaning that the critical depth of cut was higher in most spindle speeds, which allowed having more global productivity. It is also observed in the range of spindle speed between 2000 and 3000 rpm, a stable peninsula formed with axial depth ranging from 11 to 20 mm or higher values of critical depth of cut. For instance, for the multivariable cutter at 2500 rpm, the critical depth of cut \(a_{p}\) was 2.17 mm, however it became stable again as shown in Figure 6 for the interval values between 11 and 20 mm. To validate this unexpected behavior, we performed several time-domain simulations using the third-order EMHPM solution described by Equation (37). Furthermore, the simulated vibrations for the chosen cutting conditions were analyzed using the continuous wavelet transform (CWT), the power spectral density (PSD) and Poincare maps (PM). The CWT is a time-frequency representation of a signal that offers the capability to observe how frequencies evolve in time. The scalograms display the absolute value of CWT of the simulated vibration and therefore, they were used to detect chatter phenomena that appeared when milling with a multivariable tool. The PSD is based on the Fourier transform that provides the transformation from the time-domain to the frequency-domain. Additionally, PSD is defined as the squared value of the signal and describes the power of a signal or time series distributed over different frequencies [46]. Moreover, a PM represents points in phase space, which are sampled every spindle rotation [47]. \begin{table}
\begin{tabular}{c c} \hline \hline Diameter & 12.7 mm \\ \hline Cutting length & 25 mm \\ \hline Coating type & Uncoated \\ \hline Number of teeth & 4 \\ \hline Helix angles & \(39^{\circ},37^{\circ},39^{\circ},41^{\circ}\) \\ \hline Pitch angles & \(80^{\circ},100^{\circ},70^{\circ},110^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main geometric parameters of multivariable tool. Figure 6: **(a)** Comparison of stability lobes for regular (black solid line) and multivariable (red solid line) cutters by using the third-order EMHPM and **(b)** zoom in on chosen cutting conditions for time-domain simulations. The selected points are marked as follows: unstable (cross mark), stable (circle mark) and transition (plus mark) cutting conditions. The frequencies \(f\) of the CWT and PM were normalized \(f_{n}=f/f_{h}\) according to the spindle frequency \(f_{h}\). When milling with a regular milling tool the excitation frequency \(f_{e}\) is equal to \(z_{n}\) times frequencies of the spindle speed \(f_{h}\) but in a multivariable tool, there are several excitation frequencies since the angular spacing between teeth change as a function of the axial depth of cut. Figure 7 illustrates the CWT, PSD and PM for simulated vibrations using the multivariable tool with different axial depths denoted as cutting conditions A, B and C for the axial depths of cut of 1.0, 1.7 and 1.7 mm respectively. Figure 7a-c refers to the vibrations of the cutting conditions A marked in Figure 6, using a regular tool. The scalogram in Figure 7a identifies point A as a stable cutting since normalized cutting frequencies present a dominant value of \(f_{n}=3.2\), which corresponds to the natural frequency \(f_{m}=132\) Hz. This is also confirmed by the PSD analysis shown in Figure 7b. The PM illustrated in Figure 7c shows a vibration that decreased with time and sampled data concentrated in the center confirmed a typical stable case.