# Separation Method for Installation Eccentricity Error of Workpiece
## 5 Experimental Study
### Experimental Results and Discussions  
The results for when the spindle speed is set to 300 r/min and the displacement signal is obtained by detecting workpiece 1 through four sensors (S1, S2, S3, and S4) are shown in Figure 8. The calculation of radial runout error requires measuring the displacement signals of the workpiece before and after reversal. The displacement measurement signal of the initial position of the workpiece is shown in Figure 8a. The displacement measurement signal of the workpiece after being loosened, reversed 180 degrees, and re-clamped is shown in Figure 8b. During the experiment, a working speed with fewer interfering frequency components other than the fundamental frequency was deliberately selected to reduce the interference of other frequency components on the separation results, thereby improving the accuracy and reliability of subsequent calculations. The radial runout error obtained from the original data of the displacement sensor is shown in Figure 9. Among them, the radial runout error along the \(x\) direction and its first harmonic component are shown in Figure 9a, and their frequency spectrum is shown in Figure 9c. The first harmonic component includes the first harmonic component of spindle vibration and the installation eccentricity error of the workpiece. The radial runout error along the \(y\) direction and its first harmonic component are shown in Figure 9b, and their frequency spectrum is shown in Figure 9d. Based on the spectral analysis results of Figure 9c,d, it can be observed that the radial runout errors along the \(x\) and \(y\) directions contain rich frequency components, including integer multiples of the operating frequency and other harmonic frequencies. Figure 8: Displacement signals of four sensors. Next, the first harmonic component of the radial runout error of the spindle and the spindle speed were substituted into the proposed separation method. The parameters of the PSO algorithm are set as follows: population size \(M\) is 500; the max iterations \(I_{max}\) is 200. Table 4 shows the calculated results of amplitudes \(R_{1}\) and phase \(\lambda\) of the installation eccentricity error separated from the first harmonic component of the radial runout error, as well as the amplitudes \(a\), \(b\), and phases \(\phi_{1}\), \(\phi_{2}\) of the spindle's radial runout error. Based on the above results, it is necessary to use the separated parameters to construct simulation signals and compare them with experimental measurements to verify the accuracy of the separation results. Figure 10 compares the experimentally measured value of the radial runout error of the spindle measured by different sensors and the simulation signal, where the simulation signal is constructed according to the parameters in Table 4. It can be clearly observed that the simulated values have a high degree of coincidence with the measured values. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Calculated Value** & **Parameter** & **Calculated Value** \\ \hline \(R_{1}\) (\(\mu\)m) & 14.75 & \(\lambda\) (rad) & 1.43 \\ \(a\) (\(\mu\)m) & 23.17 & \(\phi_{1}\) (rad) & 0.83 \\ \(b\) (\(\mu\)m) & 30.90 & \(\phi_{2}\) (rad) & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation results of experimental parameters. Figure 9: The radial runout error of spindle. Due to the presence of low amplitude interference harmonics in the radial runout error of the spindle, error indicators such as SSE, RMSE, and MAE are inevitably affected by them, which in turn affects the quantitative evaluation of the calculation results. In this work, the coefficient of determination \(R^{2}\) is applied as the quantitative evaluation index to evaluate the separation results [33; 34]. In regression analysis, \(R^{2}\) is commonly used to measure the fit of a model. The model fits the data better as the indicator value approaches 1, and conversely, the model fits the data the worse as the value nears 0.