# Improving Stability Prediction in Peripheral
## 4 Discussion  
Chatter is an old enemy to manufacturers but still strongly affects machining times and workpiece quality. Together with mathematical models, this work proposed a numerical method to reproduce many effects arising in practice, which need to be introduced for an accurate representation of the system's stability. These parameters are common ones, such as the geometry of the end-mill (tool diameter, flutes, helix angle, etc.), but also many others, such as (1) runout, (2) cutting speed, and (3) helix angle.  
Particularly in the machining of the aluminum alloy family, the authors found that it is mandatory to incorporate the effects of cutting speed in the calculation of the stability lobes, as these may differ greatly depending on the constant or variable cutting coefficients.  
The helix angle did not have an important influence in this chart as the helix pitch is 43.5 mm, higher than the 10 mm limit plotted. However, the runout effect is significant not only in the chatter frequencies but also over the modulus of the multipliers. Notice that these values have been increased towards the stability boundaries, and a new region, or flip lobe (when the multipliers leave the unit circle for the real negative axis) appears in the middle of the Hopf bifurcation at around 3250 rpm. As runouts always appear in the workshop, this proposed model can be easily incorporated into any cutting force model because it is based on an experimental measurement.  
The cutting speed has a more predictable influence, since the cutting coefficients decrease while the cutting speed increases, and stability boundaries grow as the spindle speed increases. For instance, at 7500 rpm, the axial depth of the cut limit was predicted to be 5.6 mm using a typical analysis, but increased to 7.8 mm for the improved analysis.  
The boundary limits are especially sensitive in the stable zones between consecutive lobes, which are often used to improve chip removal rates. For instance, when the spindle speed was 2500 rpm in Figure 5, the prediction here failed due to the severe transition between stable and unstable behavior (see slope of the diagram).  
If the prediction model is to be put to good use, a convenient characterization should be done. This non-trivial aspect was not studied by the scientific literature. Models usually assume cutting coefficients to be constant by characterizing and validating the tool-material pair at the same cutting speeds. However, this is not realistic because the most interesting zones between lobes will vary depending on the system's dynamics. The resulting change in the cutting coefficient due to cutting speed cannot be neglected when milling aluminum alloys.  
The EMHP method was compared with other similar and competitive methods in the literature, such as the Chebyshev collocation, semi-discretization, and full-discretization methods. A single degree of freedom model was developed and validated with the mentioned updates. When considering computation time and convergence rate, the EMHP and Chebyshev collocation methods proved to be  
Figure 7: Computation time for each combination of parameters (\(n\), \(a_{p}\)) of all studied methods.  
the most efficient when \(N<60\). In terms of overall accuracy, all four methods resulted in the same stability lobes when \(N>30\), but this may not necessarily be the case for other milling conditions.