# Improving Stability Prediction in Peripheral
## 2 Materials and Methods
### Effect of Runout and Cutting Speed
The results of this are shown in Figure 2. Note that the model matches the maximum thickness in every single flute, but it does not capture the discontinuity when one of the flutes is not cutting. In spite of this, the analytical model is a good approximation for the stability analysis. When considering runout effects, new chatter frequencies appeared due to the one-period bifurcation, which was many times the spindle rotation frequency of a two-flute end mill (see Figure 3). The plotted chatter frequencies were also used when comparing with experimental chatter frequencies for validation later on. Figure 1: (**Left**) geometrical runout; (**Right**) approximated runout. With regard to cutting speed, it is commonly assumed in the literature that the stability analysis and mechanistic force models are independent of this parameter [20,21]. However, during the machining of aluminum workpieces, cutting speed has a strong influence on the magnitude of the cutting forces and on the stabilities' boundaries. According to [PERSON] [22], cutting speed affects the chip formation process in two major ways. Firstly, it changes the strain rate in the deformation zone, which influences the resistance of the workpiece material and the thermal energy generated during its deformation processes. Secondly, the cutting speed influences the tool-chip's relative speed and the natural length of the tool-chip interface. Therefore, a proper characterization should consider variable cutting coefficients. In this work, cutting coefficients in the tangential and normal directions were obtained as functions of the cutting speed. Up to 120 full-immersion cutting tests were done by using tools with 1 mm-4 mm depth of cut, 5 feed rates (0.05-0.25 mm), and 6 cutting velocities (2000-10,000 rpm), as shown in Figure 4. These tests were performed on a [PERSON] dynamometer (9255 B). The solid end-mill was a 30\({}^{\circ}\) helix, which was 16 mm in diameter with two flutes, and the material workpiece selected was aluminum 7075-T6. It is important to mention that the associated spindle speed was not greater than 1/5 of the natural frequency of the chain table dynamometer workpiece (\(\approx\)1.6 KHz) in order to obtain reliable measurements [23]. During the interpretation of the experimental data, shear cutting coefficients showed significant variations due to the cutting speed in the radial and tangential directions. However, the fact that friction effects contributed to the total magnitude of the cutting forces due to cutting speed is negligible. We used a simple exponential model to fit the experimental data:  
\[\mathbf{K}_{qc}(\mathbf{v_{c}})=4(A_{q}e^{-\mathbf{a_{q}v_{c}}}+B_{q}) \tag{19}\]  
Figure 3: Chatter frequencies, (**a**) no runout, and (**b**) runout. Figure 2: Proposed runout model versus a time domain simulation runout. where the parameters for the fitting are \(A_{q}\), \(B_{q}\), and \(\alpha_{q}\) are obtained for the tangential and normal directions, as: \(A_{t}\) = 53.3, \(B_{t}\) = 201, \(\alpha_{t}\) = 0.0067, \(A_{r}\) = 84.4, \(B_{t}\) = 20, and \(\alpha_{t}\) = 0.0045.