## 5 Experimental Verification
### Analysis of the Transverse Vibration Signal of the Shaft  
A CZ6132A lathe was used to process the flexible shafts. The relevant shaft parameters are listed in Table 1. Cutting depth is an important parameter affecting the turning process [26]. Three different cutting depths were used in this experiment: 0.25, 0.5, and 0.75 mm. The transverse vibration signals of the shaft were collected during turning. As turning progressed, the diameter of the shaft decreased continuously according to the cutting depth. A fast Fourier transform was used to convert the time-domain signal into a frequency-domain signal, which was then smoothed in the frequency domain. The processed signals are shown in Figures 13 and 14, and the red arrow indicated the change in diameter after the workpiece material was turned. It can be seen that for both clamping modes, there were peaks in the four frequency bands 420-455 Hz, 970-1000 Hz, 1100-1200 Hz, and 1360-1380 Hz, regardless of diameter. It can be inferred that these four frequencies are the natural frequencies of the lathe system (or multiples of them). In particular, the peak value in the 1100-1200 Hz frequency band increased with the cutting depth. It's well known that cutting depth is related to cutting force [35]: greater cutting depth implies greater cutting force and, thus, a larger vibration peak. Therefore, it can be inferred that the amplitude of the peak value of vibration in the 1100-1200 Hz frequency band is caused by the cutting force. Additionally, this frequency band can be estimated to be twice the frequency of the slender shaft with the effect of the support stiffness from the tool holder, shown in Figure 12b. Figure 12: Natural frequency of the lathe system by hammer test: frequency response functions for (**a**) chuck; (**b**) tool holder; (**c**) tailstock. Figure 14: Fixed–free on-rotor-sensing vibration signal at depth of cutting (DOC) (**a**) 0.25 mm; (**b**) 0.5 mm; (**c**) 0.75 mm. Figure 13: Fixed–pinned on-rotor-sensing vibration signal at depth of cutting (DOC) (**a**) 0.25 mm; (**b**) 0.5 mm; (**c**) 0.75 mm. The peak value in the 970-1000 Hz frequency band was very stable for both clamping methods; therefore, it is judged to be twice the natural frequency of the three-jaw chuck, which was obtained by the hammer test. The peak value in the 420-455 Hz frequency band was very stable in the fixed-free clamping mode, but in the fixed-pinned clamping mode it changed significantly as the turning process continued; therefore, it is considered to be the first-order natural frequency of the flexible workpiece with the effect of tailstock supported stiffness. The location of the peak in the 500-900 Hz frequency band in the fixed-free model was stable (Figure 14) but fluctuated significantly in the fixed-pinned model (Figure 13). The diameter of the shaft was initially 22 mm and decreased continuously during the turning process. In the fixed-pinned model, the frequency corresponding to the peak value of 900 Hz decreased continuously; however, when the diameter was less than 14 mm, the corresponding frequency increased rapidly, and the rate of the increase was much faster than that of the decrease. According to the numerical results in Figure 9, when \(Kt=6\times 10^{5}\) N/m, the natural frequency first decreased and then increased as the diameter decreased. When the diameter was greater than 15 mm, the natural frequency decreased very slowly, whereas when the diameter was less than 12 mm, the natural frequency increased very quickly as the diameter decreased. The dynamic model of the fixed-pinned system established in this study was thus verified to be correct and consistent with the experimental results. Figure 15 demonstrated the surface quality of the workpiece with different diameters and clamping modes. As can be seen from the figure, when the diameter of the workpiece at the start of cutting is relatively large, the cutting process was stable and the shaft surface was of high quality, as shown in Figure 15a,b.