# Separation Method for Installation Eccentricity Error of Workpiece
## 5 Experimental Study
### Experimental Results and Discussions
#### 5.2.1 The Influence of Different Spindle Speeds  
In this section, the influence of different spindle speeds on the separation results of workpiece installation eccentricity errors is investigated. This experiment collects displacement data of workpiece 1 measured by four displacement sensors at the same position at spindle speeds of 300 r/min, 600 r/min, and 900 r/min.  
The radial runout error of the spindle before separation at different speeds is shown in Figure 11. Each figure consists of two parts: experimental measurement values and their first harmonic components. The separation results at a speed of 300 r/min were provided earlier. It is obvious that although there are differences in the local characteristics of radial runout errors, their first harmonic components demonstrate a high degree of similarity by comparing the first harmonic components of radial runout errors in the same direction in Figure 11.  
Figure 10: Measured values and simulated signals of different sensors.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Error Index** & **Along the \(x\) Direction** & **Along the \(y\) Direction** \\ \hline \(R^{2}\) & 0.9816 & 0.9714 \\ \hline \hline \end{tabular}
\end{table}
Table 5: \(R^{2}\) between experimental measurement values and simulated signals.  
The first harmonic component of the spindle radial runout error at different speeds was substituted into the proposed separation method, and the separation results at three different speeds were obtained. Table 6 shows the results of both amplitude \(R_{1}\) and phase \(\lambda\) of the installation eccentricity error, and amplitudes \(a\), \(b\), and phases \(\phi_{1}\), \(\phi_{2}\) of the spindle radial runout error at different speeds. As the rotational speed increases, the variation in the installation eccentricity error parameters becomes less than 0.3%, indicating that increased spindle speed minimally impacts the amplitude and phase of workpiece installation eccentricity error.  
The parameters in Table 6 exhibit that the workpiece installation eccentricity calculated at different speeds is consistent, indicating that the workpiece installation eccentricity always exists and does not vary with speed. Subsequently, it is necessary to use the parameters in the table to construct simulation signals and compare them with experimental measurements to verify the accuracy of the separation results. Figure 12 compares the experimentally measured value of spindle motion error measured by different sensors and the simulation signal.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **300 r/min** & **600 r/min** & **900 r/min** \\ \hline \(R_{1}\) (\(\mu\)m) & 14.75 & 14.79 & 14.81 \\ \(a\) (\(\mu\)m) & 23.17 & 25.37 & 28.47 \\ \(b\) (\(\mu\)m) & 30.90 & 33.90 & 37.15 \\ \(\lambda\) (rad) & 1.43 & 1.42 & 1.41 \\ \(\phi_{1}\) (rad) & 0.83 & 0.80 & 0.81 \\ \(\phi_{2}\) (rad) & 0.31 & 0.30 & 0.30 \\ \hline \end{tabular}
\end{table}
Table 6: Calculation results of parameters at different speeds.  
Figure 11: The radial runout error of the spindle at different speeds.  
Figure 12 can be clearly observed that the simulated values show a high degree of coincidence with the measured values. Table 7 shows the \(R^{2}\) between experimental measurement values and simulation signals at different speeds. The results show that the \(R^{2}\) is close to 1, indicating that curve similarity exceeds 97%, further verifying the high accuracy and reliability of the separation method proposed in this work.