## 4 Experimental Results and Discussion
### The Evolution of Frequency for Resultant Vibration in Beating Phenomenon
It is obvious that many other additional vibrations of the lathe headstock (some of them with high frequency) disturb the accuracy of the zero-crossing detections, as a main reason for the noisy evolution from Figure 11.  
The best information available in Figure 11 is the average value of the combination frequency \(f_{c}\) (\(\overline{f}_{c}=17.3830\) Hz, very close to the rotational frequency of the main spindle, \(f_{1}=17.3856\) Hz). Based on the previous conclusions from Figures 9 and 10 it is evident that the amplitude \(A_{1}\) of unbalanced vibration generated by the main spindle (shaft 1) is higher than the amplitude \(A_{2}\) of shaft 2 (so \(A_{1}=85\) um and \(A_{2}=33\) um, an item analysed before). As previously mentioned, the rotational frequency of shaft 2 is \(f_{2}=f_{1}-f_{b}=17.3752\) Hz or \(f_{2}=f_{1}+f_{b}=17.3959\) Hz. Figure 12 presents a low-pass filtered evolution of the instantaneous combination frequency from Figure 11, (using a multiple moving average filter [43], as a well-known method to attenuate the signal noise). Figure 11: The evolution of the instantaneous combination frequency \(f_{c}\) during the vibration beating phenomenon (displacement of headstock) described in Figure 7.  
Figure 12: The evolution of the low pass filtered instantaneous combination frequency \(f_{c}\) during the vibration beating phenomenon described in Figure 7 (a low pass filtering of Figure 11). Despite a relatively strong irregular variation of the combination frequency \(f_{c}\) (due to the variation of experimental conditions: e.g., the small variation of rotational speeds of shafts 1 and 2, caused mainly by the variation of frequency of the supplying voltages applied to the asynchronous driving motor, around a theoretical value of 50 Hz, as Figure 13 clearly indicates), the previous simulations and conclusions are fully experimentally confirmed. Three supplementary identical experiments confirm the evolution presented in Figure 12.  
Firstly, in Figure 12 there are two negative peaks (for the two nodes in Figure 3 or Figure 7; each node produces a negative peak on \(f_{c}\) evolution, an item already discussed in the simulation from Figure 10) at a time interval very close to the beat period value \(T_{b}\), already defined in Figure 3 (96.73 s here, compared with 96.6 s in Figure 3). Secondly, as shown in Figure 14, a superposition of filtered frequency \(f_{c}\) evolution from Figure 12 (here in a conventional blue coloured description) over the experimental envelopes of vibration displacement in beating (the same as those depicted in Figure 7) indicates that the negative peaks of \(f_{c}\) are placed, as expected, in nodes. Figure 14: The position of negative peaks on the combination frequency (formally represented) relative to the position of nodes on the vibration beating phenomenon. Figure 10:0 Low-pass filtered rotational frequency evolution of the main spindle and supplying voltage frequency evolution of the driving motor. The small displacement to the right of the negative peaks of the filtered combination frequency evolution (as against the nodes on Figure 14) is not related to the numerical filtering. This is proved by the result of the simulation of Figure 14, as given in Figure 15 (with addition of pure harmonic signals \(y_{1s}\) and \(y_{2s}\) in vibration beating simulation). This periodic pattern of filtered combination frequency \(f_{c}\) evolution experimentally revealed in Figures 12 and 14 (according to the simulations from Figures 10 and 15) is strongly attenuated if the amplitude \(A_{2}\) becomes significantly lower than \(A_{1}\) (and vice versa). An important question is in order here due to a very small variation of filtered frequencies revealed before (less than 50 mHz full scale evolution in Figures 9, 10, 12, 13 and 15): how accurate is this frequency measurement method [(42)]?