# Improving Stability Prediction in Peripheral
## 2 Materials and Methods
### Effect of Runout and Cutting Speed  
In this section, the effects of runout and cutting speed in the peripheral milling of Al7075T6 alloy at high cutting speeds are investigated--or more specifically, their influence in the prediction and accuracy of stability lobes is analyzed. Because perfect cutting is impossible, runout appears naturally in every milling operation and unfortunately produces undesirable effects, such as uneven material removal, wear, and poor surface quality. It is defined as an offset of the tool's geometrical axis with respect to the spindle rotation axis, as shown in Figure 1. This is impossible to avoid because the total runout is a sum of the misalignments of the spindle, tool-holder, and milling tool itself. This makes difficult to predict the current flutes' trajectories, as the real cheap thickness is defined by the trajectory at every point in each flute, which changes not only in the \(x\)-\(y\) plane but also in space. When considering this effect during milling operations, the cutting force can be expressed as:  
\[F_{y}(t)=\sum\limits_{j=1}^{z_{n}}\rho_{j}\frac{1}{4k_{\beta}}\big{[}K_{tc} \cos 2\phi_{j}(z)+K_{rc}\big{(}\sin 2\phi_{j}(z)+2\phi_{j}(z)\big{)}\big{]}_{z_{j,1}(\phi_{j})}^{z_{j,2}(\phi_{j})} \tag{16}\]where \(\rho_{j}\) represents a weighted factor that is related to the cutting force magnitudes for each flute, \(j\). Here, we propose a different alternative to consider the runout effects, by assuming that relating \(\rho_{j}\) is given as:  
\[\rho_{j}=\big{[}1+\delta\big{(}\cos\phi_{p}(j-1)-\cos\phi_{p}(j-2)\big{)}\big{]},\ \delta=\frac{\widetilde{r}}{f_{z}} \tag{17}\]  
where \(\widetilde{r}\) represents the tool's apparent runout radius in the feed direction, \(\alpha\) is the runout angle, and \(f_{z}\) is the feed rate. Notice that \(\delta\) represents the chip variation in the feed direction (\(x\)-axis). To determine the value of \(\widetilde{r}\) experimentally, once the milling tool is attached to in the machine center, a deflection gauge fixed to the table was set to zero once the first flute pushed the gauge. Then, the apparent radius \(\widetilde{r}\) was estimated by the average deflection between \(180^{\circ}\)-spaced flutes. The proportional factor \(\rho_{j}\) for the chip load in the actual \(j\) teeth was then calculated as the nominal feed rate plus the ratio \(\delta=\widetilde{r}/f_{z}\), representing the percentage of the nominal feed rate. Note that the factor \(\delta\) is related to the amount of cutting force magnitude of each tooth that depends on the apparent runout radius. In the ideal case of no runout, i.e., \(\delta=0\) and \(\rho_{j}=1\), Equation (15) becomes periodic in \(\tau\). On the other hand, when the runout influences the machining processes, the periodic behavior arises, related to the spindle period \(\tau_{T}\). When the runout is zero, the dynamical system exhibits only quasi-periodic (Hopf bifurcation) and flip-bifurcation behaviors of period two, while chatter in period one never arises. In multi-flute machining processes, the runout practically always arises since the tool is not perfectly symmetric. In this case, the principal period is equal to the spindle rotation period \(\tau_{T}=60/n\). Thus, the determination of the stability lobes by the EMHP method can be done by computing the transition matrix \(\boldsymbol{\Phi}\) over the spindle rotation period, i.e., by considering that Equation (15) can be written as:  
\[\boldsymbol{\Phi}=\mathbf{D}_{z_{n}(N-1)}\mathbf{D}_{n_{z}(N-1)-1}\cdots \mathbf{D}_{2}\mathbf{D}_{1} \tag{18}\]  
Within this study, a 2-fluted, 16 mm-diameter, carbide end-mill (\(30^{\circ}\) helix) was used for the experiments. The apparent runout was measured by means of a digital gauge, and the apparent radius was estimated to be at 17 \(\upmu\)m. In order to validate the proposed runout model, this was compared with the chip thickness in the time domain by a time domain simulation similar to that developed in [19].