## 4 Semi-Active Magnetorheological Damper Device for Chatter Mitigation
### Experimental Determination of Stability Lobes under the Action of the MR Damper Device  
Notice when the MR damper device was placed on the cantilever beam, the damping ratio of the beam change substantially, as listed in Table 4. Once the modal parameters were computed, the stability lobes were plotted in Figure 11 following the same methodology described in Section 3.1. Since the MR damper device increases the damping ratio in the system, the stability boundaries not only shift toward higher axial depth but also shrink, which substantially allows more productive cutting combinations. For instance, when the MR damper device is turned off and cuts are performed at 7000 rev/min, the critical axial depth of cut is close to 0.25 mm, but this value increases to 0.75 and to 1.4 mm when the current in the MR damper is adjusted at 1.7 and to 3.4 A, respectively. In order to validate the effect of the MR damper over the stability behavior, three cutting tests were selected, and these were named points D, E, F as illustrated in Figure 11. All of them correspond to spindle speeds of 7000 rpm but with different axial depth-of-cut values and subjected to specific scenarios. The accelerometer signal of point D was acquired when the MR damper was turned off, while the acceleration for points E and F were collected when the MR damper was turned on with 1.7 and 3.4 A, respectively.  
Figure 12a shows the CWT obtained from the acceleration signal collected for point D (\(a_{p}\) = 0.5 mm). Notice that this CWT plot illustrates dominant frequencies (red colors) lower than the normalized \(f_{n}\) = 1, which indicates that cutting is an unstable quasi-periodic chatter test, as predicted by the stability lobes in Figure 11. This is also confirmed with the sampled acceleration signal plotted in Figure 12d, since the amplitudes of the samples are not similar. The same combination of cutting parameters (\(a_{p}\) = 0.5 mm and \(n\) = 7000 rev/min) are now analyzed when the MR damper device is turned on with 1.7 A. Notice from Figure 12b that for this cutting test (point E), the CWT describes dominant frequencies that are very close to normalized frequency \(f_{n}\) = 1, and thus, this is an indication of stable cutting conditions. This result agrees with the stability lobes plotted in Figure 11, since the location of point E indicates stable cutting conditions because it is below the critical axial depth of cut value of 0.75 mm. Similarly, point F (\(a_{p}\) = 1.25 mm) exhibits stable cutting conditions, since the critical depth of cut has now the value of 1.5 mm. It is important to observe that the acceleration amplitude for experimental test F (Figure 12f) is even lower than the value recorded for point E (Figure 12e), confirming that the MR damper device offers higher damping and is an effective technique to avoid unstable vibrations.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Electrical** & **Average Magnetic** & **Yield Stress** & **Natural** & **Stiffness \(k\)** & **Damping \(\xi\)** \\
**Current [A]** & **Flux Density [G]** & **[KPa]** & **Frequency \(f\) [Hz]** & **[N/m]** & **Damping \(\xi\)** \\ \hline No Damper & - & - & 190 & \(2.40\times 10^{6}\) & 0.007 \\ Off–0 & - & 0 & 180 & \(2.18\times 10^{6}\) & 0.011 \\ On–1.7 & 372 & 6 & 178 & \(2.04\times 10^{6}\) & 0.023 \\ On–3.4 & 695 & 12 & 175 & \(2.07\times 10^{6}\) & 0.036 \\ \hline \end{tabular}
\end{table}
Table 4: Modal parameters of the cantilever beam under the action of a semi-active MR damper device.  
Figure 10: Measured Frequency Response Function (FRF) of the thin floor with the MR damper.