## 3 Stability Analysis and Experimental Validation
### Experimental Validation of Stability Lobes  
Experimental milling cutting tests were performed in a Makino F3 machining center. A monolithic cantilever beam of \(116\times 177\times 12.7\) mm was used to mimic a 1 dof thin-floor behavior in the \(z\)-direction defined as shown in Figure 5. A 7075T6 aluminum workpiece was positioned on the cantilevered plate using two screws to maintain its dynamic characteristics during several passes (total mass does not change substantially). In the thin-floor milling process, the cutting tool is much stiffer in comparison with the workpiece properties. For that reason, the dynamic response of a thin floor is the most important factor during machining. The workpiece FRF was measured with a compact laser vibrometer, model CLV-2534 (Polytec GmbH, Waldbronn, Germany) and a impulse force hammer, Model 9722A (Kistler Group, Winterthur, Switzerland). The fitted modal parameters listed in Table 3 were performed using CutPro Simulation Software (Manufacturing Automation Laboratories Inc., Vancouver, BC, Canada). In order to validate the dynamics of the milling process together with the proposed cutting force model, stability lobes for 2.5 mm radial immersion were computed and shown in Figure 6 by using the EMHPM of Equation (11). As described in Section 2.2, the cutting forces model was considered for a range from 0 to 3 mm in the \(z\)-direction. The stability lobes verification was studied through the measurement of the vibrations of the workpiece mounted on the cantilevered plate. An accelerometer type 8778A500M14 (Kistler Group, Winterthur, Switzerland). weighing 0.4 g was used to acquire the vibrations, and a tachometer model FS-V31 (Keyence Corporation, Osaka, Japan) was used to detect every single revolution of the spindle as shown in Figure 5.  
The signal processing consisted of two steps. First, raw signals were filtered with a low-pass filter with a cut-off frequency of 5 kHz. Additionally, the tachometer signal was manipulated to include a virtual pulse in the middle of two consecutive detected pulses that allows us to sample the accelerometer signal according to the tool passing period (two flutes impact the workpiece during each revolution). Secondly, the analysis of the accelerometer signal was performed via wavelet transform (CWT), power spectral density (PSD), and Poincare diagrams. By comparing experimental and theoretical chatter frequencies [37], all this information was used to identify and categorize every single cutting test as having stable or unstable cutting parameters. After completing the signal process and categorization, experimental data were superposed in Figure 6.  
The stable cuts in Figure 6 are denoted by circles, the unstable quasi-periodic cuts are denoted by squares, and the unstable double periodic cuts are denoted by triangle symbols. Experimental results denoted by diamonds represent experimental conditions without dominant chatter frequencies, which makes the classification as either unstable or stable challenging. From Figure 6, three cutting tests that describe typical stable/unstable behavior have been selected (points A, B, and C). Point A describes unstable period-doubling chatter, Point B describes stable cutting conditions, and Point C describes unstable double periodic. Since the CWT analysis allows us to observe how frequencies evolve in the time domain, then this analysis was applied to accelerometer data as shown in Figure 7a-c. To relate the detected frequency in relation to the excitation, the following normalized frequency is introduced:  
\[f_{n}=f_{w}/f_{TPE}. \tag{19}\]  
In Equation (19), \(f_{w}\) represents the wavelet transform frequency and \(f_{TPE}\) represents the tooth passing excitation frequency. The overlapped circles and cross white marks in Figure 7a-c identify normalized frequencies of \(f_{n}=1\) and \(f_{n}=0.5\), respectively. In Figure 7d,e, the acceleration