## 5 Conclusions  
Chatter vibration during the milling process of thin-floor components is a key factor that limits productivity. Placing an MR fluid in contact with the thin floor modifies the damping ratio without changing the natural frequency and modal stiffness. Experimental measurements indicate that a good degree of accuracy is attained from the proposed cutting force model, which is a key factor to predict stability lobes during the machining of thin floors using a bull-nose end mill.  
* The proposed force model in which the tool edge is discretized in disks along the \(z\)-direction predicts the experimental cutting forces with high accuracy.
* The predicted EMHPM stability lobes of the cantilever beam closely follow experimental data.
* The use of an MR damper device located under the cantilever beam modifies modal damping depending on the magnitude of the magnetic flux density. For the higher magnetic flux density values, the damping ratio increases about 4 times, while the modal stiffness slightly varies.
* Under the effect of the MR damper device, the stability boundaries shift toward higher critical axial depth of cut values, which substantially enhances stable cutting conditions. In other words, experimental measurements indicate that when using the MR damper device, it is possible to increase the critical depth of cut from 0.5 to 1.5 mm in the range of spindle speed from 7000 to 10,000 rev/min, with an increase in the material removal rate and productivity by a factor of at least 3.  
Figure 11: Stability lobes diagrams for three scenarios of magnetic flux density applied to the MR damper device together with experimental data in a down-milling operation. (\(\Circle\) stable cut, \(\square\) quasi-periodic chatter, \(\Delta\) double periodic chatter, and \(\phi\) border cutting parameters).  
Figure 12: Analysis of the collected accelerometer signals (D, E and F) for different MR damper magnetic flux density values. Continuous wavelet transforms (**a**,**b**,**c**). Sampled acceleration signals (**d**,**e**,**f**).  
* The use of a semi-active MR damper device represents an alternative way to increase productivity while machining thin-floor components because of its robustness and adaptability to complex geometries. Since MR fluids can modify their yield and shear stresses as a function of the applied magnetic flux density, the damping ratio can be adjusted as needed in order to reach stable cutting conditions. Additionally, the semi-MR damper device is simple to operate versus its counterpart, the active MR damper device, which needs sensors and real-time data processing.  
Conceptualization, S.D.P.-A. and D.O.-T.; Methodology, S.D.P.-A.; Resources, O.M.-R., A.E.-Z. and L.N.L.d.L.; Supervision, O.M.-R., A.E.-Z. and L.N.L.d.L.; Validation, S.D.P.-A., D.O.-T. and G.U.; Writing--Original draft, S.D.P.-A. and D.O.-T.; Writing--Review and editing, O.M.-R., G.U. and A.E.-Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Tecnologico de Monterrey through the Research Group of Nanotechnology for Devices Design, and by the Consejo Nacional de Ciencia y Tecnologia de Mexico (Conacyt), Project Numbers 242269, 255837, 296176, and National Lab in Additive Manufacturing, 3D Digitizing and Computed Tomography (MADiT) LN299129.  
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.