## 4 Stability Analysis of Multivariable Milling Tools
### Stability Analysis of 1 DOF Milling with a Multivariable Tool
When the axial depth of cut was increased to 1.7 mm, the stability diagram predicted unstable cutting conditions according to the stability lobes for the regular tool. This case is denoted with cutting conditions B and the corresponding scalogram (shown in Figure 7d) illustrated how the intensity of the dominant frequency increased with time even when the excitation frequency was the same as the case in A.  
Figure 7: Analysis of cutting conditions A, B and C. Continuous wavelet transform (CWT) scalograms: (**a**,**d**,**g**); power spectral density (PSD): (**b**,**e**,**h**) and Poincaré maps (PM): (**c**,**f**,**i**) corresponds to the cutting conditions A, B and C respectively. The PM diagram shown in Figure 7f exhibited a vibration far from zero. In fact, the PM diagram shows that the vibration amplitude grows exponentially because our equation of motion did not consider nonlinear effects such as those that appeared when the tool lost contact with the workpiece. Both cutting conditions A and B agreed with the stability boundaries in Figure 6. Now, the cutting conditions B were used but with a multivariable tool, which was referred to as cutting conditions C. The CWT plotted in Figure 7g described completely different results since there were no single dominant frequencies in comparison with cutting conditions A, but appeared several frequencies around \(f_{n}=3.2\) and close to \(f_{n}=1\) that reduced in intensity with time, suggesting a stable cutting. Figure 7i illustrates how the vibration amplitude approached to zero when using a multivariable tool in contrast to the PM obtained for the regular tool and exhibited in Figure 7f. This can be explained by observing that there were several excitation frequencies due to the irregular pitch and helix angles that break a single excitation frequency avoiding regenerative chatter phenomena. Figure 8 illustrates the CWT, PSD and PM for simulated vibrations using the multivariable tool with different axial depths denoted as cutting conditions D, E, F and G for the axial depths of cut of 2.3, 3.0, 8.55 and 18 mm respectively. Notice that a stable case C was already validated when the axial depth was 1.7 mm in Figure 7g-i that corresponded to cutting conditions under the stability boundaries shown in Figure 6. For case D, a transient cutting condition was chosen very close to the critical axial depth of the cut. It is interesting to point out that transition cutting conditions in the CWT scalogram shown in Figure 8a not only shows frequencies with higher intensity in comparison with the stable case B, but also presents shifted frequencies that varied in intensity every single revolution. This shifting suggests a marginally stable cutting condition that was confirmed by the PM illustrated in Figure 8c, where circular trajectories were described close to the center point. Figure 8: _Cont._  
Unstable vibrations that appeared for case E were because of the intensity of frequencies increased exponentially with time, see Figure 8d. Notice that other frequencies arose with time close to the values of \(f_{n}\) = 0.5 and \(f_{n}\) = 1.5. These frequencies also occurred for cutting conditions D, which is an indication of the appearance of chatter phenomena. In contrast to Figure 7i for a stable case, Figure 8f exhibited few trajectories because the vibration amplitude was out of the range selected (\(\pm\)1 mm). The qualitative and quantitative dynamic behaviour due to cutting conditions F, and illustrated in Figure 8g-i, were classified as transition cutting behaviour. Here, a more severe shifting in frequencies was observed in the scalogram (Figure 8g). From Figure 8g, it is seen that drastic shifting occurred in the time domain in the range of normalized frequencies from 3.5 to 6.