*3.2. Vibration FFT Analysis*

The Fast Fourier Transform (FFT) of the vibration signal is a useful and representative indicator of the dynamic behavior of the system, which can be employed for determining the appearance of chatter vibration. In the current study, an FFT of the vibration of the samples, measured by the accelerometer placed below them during machining, has been obtained per cutting pass, namely, 33 FFT per sample. They are shown in three-dimensional plots in Figure 7. In these plots, the cutting passes where chatter takes place can be observed, as well as the frequency at which chatter occurs. Also, the vibration content of low frequencies is remarkable.

Three main conclusions can be obtained from these plots. Firstly, it is noticeable that there are two zones where chatter can be initialized. The first zone is around cutting passes 3–5, and the second zone is around cutting pass 21. For TP10, chatter only appears in the first zone. For TP08 and TP04, chatter appears in both zones, but disappears at the middle. For TP02, chatter appears almost continuously between these two zones.

The existence of these two zones is due to the complex interrelation of different facts. On the one hand, at the beginning of the machining in the center of the sample, despite starting far from fixtures, the rigidity of the sample is still high, and chatter does not occur. On the other hand, the presence of modal nodes, where rigidity is higher, and antinodes, where it is lower, in addition to the progressive variation of modal parameters, leads to the alternative appearance and disappearance of chatter along the tool path [31].

Secondly, it is noticeable that chatter appearance is higher for lower material removals; namely, it affects more cutting zones. As Campa et al. [32] have stated, this phenomenon happens in the milling of thin floors with bull-nose end mills, and it is related to the fact that low depths of cut also involve low lead edge angles, that ease chatter appearance.

**Figure 7.** FFT of vibration signal of samples during machining. (**a**) TP10. (**b**) TP08. (**c**) TP04. (**d**) TP02.

And thirdly, it has been confirmed that chatter appears for all cases, as stated in the SLD analysis, but it disappears as long as the tool is reaching the fixtures. From cutting pass 25 the milling is stable for any depth of cut. That is to say that milling should be avoided in the central area between fixtures, but that milling is feasible and stable outside this area. In this case, the central area is approximately a central square of 30 × 30 mm2.

The accurate chatter frequency can be more clearly seen in Figure 8, where only the highest vibration frequencies are shown. This chatter frequency is close to the first natural frequency of the samples, that according to the conducted FRF tests it is around 1730 Hz before machining. A second chatter frequency, close to the second natural frequency of 4000 Hz, is also excited. Both chatter frequencies are excited simultaneously, which discards the possibility of each zone being created by the vibration of different modes, as happened in thin wall studies [25,33].

Besides, in Figure 8 the harmonics of chatter frequency can be seen. These harmonics can be expressed by the following binomial:

$$f\_{chatter} + n \cdot f\_{tooth} \ (n \in \mathcal{N}) \tag{4}$$

It is remarkable that the first and second harmonics are higher than the chatter frequency. There are also low frequencies with high amplitude. These are the tooth passing frequency (133.33 Hz) and its harmonics, or they are related to the shape of the signal. In any case, they are higher for passes with chatter appearance, which indicates that chatter also affects the shape of the vibration signal.

Chatter frequencies decrease due to material removal, as natural frequency does, although this decrease is very slight and it is only appreciable for the highest material removals. The detailed image in Figure 8 illustrates this effect, as it shows the variation of the chatter frequency of TP02 during machining.

**Figure 8.** Frequencies with maximum vibration content per cutting pass, and TP02 detailed image.

#### *3.3. Roughness Analysis*

The final thin floors after milling are shown in Figure 9. In the four cases the first cutting pass does not exhibit chatter, but in the figure, due to the radial immersion of the successive passes, cutting pass 1 cannot be seen. In addition to chatter, there are ploughing marks at some changes of cutting direction.

**Figure 9.** Samples after milling. Chatter marks in red, ploughing marks in blue square. (**a**) TP10. (**b**) TP08. (**c**) TP04. (**d**) TP02.

The average roughness (*Ra*) analysis (Figure 10) shows two types of passes clearly delimited. In the passes where chatter happens, the average roughness is higher than 0.9 μm, whereas in passes without chatter this value is lower than 0.6 μm. It must be noted that these values correspond to the middle of each cutting pass, where roughness has been measured.

**Figure 10.** Average roughness per cutting pass.

Additionally, the roughness does not perceptibly change during machining. At most, it decreases very softly. This fact shows that, in absence of chatter, the roughness is affected by the *ap* employed, but it is not affected by the change of modal parameters, and it is only very slightly affected by the tool position and the proximity to fixtures. So, it can be concluded that the change of modal parameters affects chatter appearance, as stated in the previous SLD analysis, but once stability is reached surface roughness is constant and depends primarily on the machining parameters. Del Sol et al. [13] consider the idea that this happens due to the influence that machining parameters have on the cutting forces. In any case, given the current experimental setup, this achieved final surface quality is under 0.6 μm *Ra* and thus it meets the industrial tolerances that are typically imposed on thin floors [13,24,34].

These results have also been compared to the available bibliography. Del Sol et al. [16] analyzed very similar kind of samples milled with the same mill, a feed rate of 0.08 mm/tooth and an axial depth of cut of 0.4 mm, although directly threaded to a plating sheet. Consequently, chatter does not appear in them. They exhibit an average roughness value of 0.25 μm measured in the direction of the milling. This value is lower than the average roughness value measured in the present case TP04 with a feed rate of 0.1 mm/tooth, which is around 0.4 μm. In a following experiment of Del Sol et al. [13], also with the same samples directly threaded to a plating sheet, average roughness varies between 0.2 μm and 0.4 μm. However, in this case roughness was not measured only in the direction of the milling, but also in the normal direction.

Rubio-Mateos et al. [14] studied the same type of samples milled with the same mill, clamped to a rubber-based vacuum fixture. Due to this fact, they do not exhibit chatter, and their average roughness value is between 0.4 and 0.6 μm for the same cutting conditions.

In another case, Campa et al. [32] milled various blocks of aluminum without back support from 30 mm thickness to 1 mm, employing a bull-nose end mill of 16 mm diameter, a feed of 0.05 mm/tooth, and depths of cut higher than 5 mm, as well as a parallel strategy (instead of helicoidal). The final average roughness was between 0.3 mm and 1.4 mm in absence of chatter. These results are consistent with the ones obtained in the present study, although the milling conditions were significantly different. Campa et al. [11] also conducted another similar experiment, where chatter marks were evident in some sections of the final part, as well as ploughing marks, a phenomenon in which the tool engages and penetrates on the part and that is related to the lack of stiffness. Arnaud et al. [25] consider ploughing as a type of process damping, in which the clearance face of the mill contacts the sample and thus leads to a more stable machining, a fact that can be confirmed in the present cases TP08 and TP04, where ploughing happens just before chatter disappearance.

In the present study, in addition to the lack of stiffness and to damping, it can be noticed in Figure 9 that ploughing is also related to the change of cutting direction. Milling strategies without sudden changes in the cutting direction, as circular ones, should be considered as they may avoid ploughing.

The FFT analysis of the roughness profile of several cutting passes (Figures 11 and 12) leads to two conclusions. On the one hand, the roughness caused by passes without chatter is mainly dominated by the tooth passing frequency (133.33 Hz, namely, 10 impacts/mm), the tool spin frequency (66.67 Hz, namely, 5 impacts/mm) and their harmonics. This phenomenon indicates that some tool runout is present during milling. These results also show that in the absence of chatter, surface roughness depends on the machining conditions. On the other hand, the roughness caused by passes with chatter appearance is higher and more chaotic. It does not appear at chatter frequencies (128 impacts/mm), but at low frequencies, even lower than the tooth passing frequency. It may be related to the previously described influence on the shape of the vibration signal caused by chatter.

**Figure 11.** Roughness profile.

**Figure 12.** Roughness profile FFT.

Surface roughness is the result of the combination of various factors acting simultaneously, as chatter and deflection, being difficult to discern between them, as López de Lacalle et al. [35] have pointed out. With the purpose of determining the origins of surface roughness, the roughness model proposed has been applied and its results are shown in

Figure 13. In the case of chatter appearance, it causes up to 70% of roughness. In absence of chatter, the floor theoretical component of roughness (*Rh*) is the smallest one and it varies from 36% in the lowest roughness case (TP08) to 25% in the highest one (TP10). In these cases, the displacement of fixtures and relative movement of thin plates (*Rf*) cause at least 65% of *Ra*.

Figure 14 shows the mean and standard deviation values of *Ra* for each thin plate, as well as their components. Low axial depths of cut lead to a high and variable roughness, whereas higher ones achieve smaller *Ra* values. As previously stated while analyzing SLDs, this phenomenon happens because high axial depths of cut also entail high lead edge angles in bull-nose end mills, which make chatter appearance more difficult. Regarding *Rh*, it is inherent to the employed mill and machining conditions, so it is constant in all thin plates, because the applied feed per tooth and mill are the same. The remaining roughness is mainly caused by forced vibrations and the axial displacement of the tool (*Rf*); namely, it is attributable to the setup employed and the milled samples.

**Figure 13.** Average roughness components per cutting pass. (**a**) TP10. (**b**) TP08. (**c**) TP04. (**d**) TP02.

**Figure 14.** Components of the mean *Ra* for each thin plate.
