## 2 Materials and Methods
### Measurement Data Processing
The second group, tagged with blue circles, includes the results for which the frequency exceeded the limit value. The results are presented in Figures 12 and 13. Figure 11: Box-and-whisker plot for particular samples (arranged according to increasing frequency). Box size indicates 25 th and 75 th percentiles. Figure 12: Line profile shape offset in the domain of the drive unit’s operating frequency: (**a**) NCDT; (**b**) coordinate measuring machine CMM. The results demonstrate a strong correlation between the frequency and the profile deviations for trials conducted at operating frequency \(f\) higher than the limit frequency \(f_{S}\). Depending on the measurement method, the determination coefficient \(R^{2}\) for the line profile offset is higher than 0.8991 and its minimum value of standard deviation amounts to 0.9191. The linear growth in the error is potentially caused by system inertia which is critical for higher frequency and the limitation of the piezoelectric stack. Due to the use of a conventional turning lathe (with a limited number of spindle speed settings), for frequencies higher than the limit, the obtained FTS operating frequencies are grouped around three values: 8.2, 11.5, and 16.5 Hz. Changes in the drive amplitude of motion \(\Delta x\) were registered for these frequencies and compared with the profile deviations \(\overline{e}\) averaged for the specified frequency ranges. The results are presented in Table 4.  
For the trials conducted at the frequency below the limit value, both the profile shape deviation and the standard deviation were approximately constant and no correlations between them and the frequency were observed. The mean values \(\overline{e}\), maximum values \(\epsilon_{\text{max}}\), and their standard deviations \(\mathbf{s_{e}}\) for both the profile deviation and its SD are presented in Table 5. \begin{table}
\begin{tabular}{c c c} \hline \(f\) [Hz] & \(\overline{e}\) [\(\mathbf{\mu}\)] & \(\Delta x\) [\(\mathbf{\mu}\)] \\ \hline
8.2 & 55 & \(-10\) \\
11.5 & 274 & \(-140\) \\
16.5 & 413 & \(-420\) \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of the averaged line profile shape offsets with amplitude values from the frequency specification. Figure 13: Outline shape offset standard deviation in the domain of the drive unit’s operating frequency: (**a**) NCDT; (**b**) coordinate measuring machine CMM. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & & **NCDT [\(\mathbf{\mu}\)m]** & **CMM [\(\mathbf{\mu}\)m]** \\ \hline \multirow{2}{*}{Line profile shape deviation} & \(\overline{e}\) & 46.7 & 39.8 \\  & \(\epsilon_{\text{max}}\) & 56.4 & 57.6 \\  & \(\mathbf{s_{e}}\) & 10.3 & 9.9 \\ \hline \multirow{3}{*}{Standard deviation} & \(\overline{e}\) & 12.2 & 10.1 \\  & \(\epsilon_{\text{max}}\) & 15.5 & 11.7 \\ \cline{1-1}  & \(\mathbf{s_{e}}\) & 3.1 & 1.4 \\ \hline \end{tabular}
\end{table} 
Table 5: Line profile offset standard deviations and the line profile offset standard deviation for attempts with the frequency lower than the limit value. For the FTS drive operating frequency--lower than the limit frequency--the mean values, maximum values, and standard deviation for both the profile deviation and its standard deviation are similar in numbers. Therefore, the determination coefficient \(R^{2}\) was designated for the results obtained using the optical transducer testing the tool position during processing and compared with the results obtained during the coordinate measuring machine test. This test was aimed at checking whether the online tool position measurement during processing can be the basis for a preliminary determination of the obtained profile deviation. The results in the form of a plot are presented in Figure 14.  
Despite the fact that the spindle position measurement during the processing is influenced by errors deriving from the turning process (vibration, build-up edge, etc. ), the measurement allows for the satisfactory determination of as-machined contour geometry without additional off-line measurements (e.g., by CMM).