# Separation Method for Installation Eccentricity Error of Workpiece
## 5 Experimental Study
### Experimental Results and Discussions
#### 5.2.2 The Influence of Different Axial Distances  
In this section, the influence of different axial distances on the separation of eccentricity errors in a workpiece installation is studied. This experiment collected displacement data from four displacement sensors when the axial distance between sensor S1 and the chuck end face was 80 mm, 90 mm, and 100 mm, respectively, at a spindle speed of 300 r/min.  
The radial runout error of the spindle before separation at different axial distances is shown in Figure 13. Each figure consists of two parts: experimental measurement values and their first harmonic components. The separation results at an axial distance of 80 mm were provided earlier. Here are the radial runout errors and their first harmonic components measured at two other positions. The curve in the figure shows that the radial runout error and its first harmonic component at different axial positions are highly similar.  
\begin{table}
\begin{tabular}{c c c} \hline
**Speed** & **Along the \(x\) Direction** & **Along the \(y\) Direction** \\ \hline
300 r/min & 0.9816 & 0.9714 \\
600 r/min & 0.9973 & 0.9852 \\
900 r/min & 0.9982 & 0.9946 \\ \hline \end{tabular}
\end{table}
Table 7: \(R^{2}\) between measured values and simulated signals at different speeds.  
Figure 12: Measured values and simulated signals at different speeds.  
Next, the first harmonic component of the radial runout error of the spindle at different axial distances was substituted into the proposed method for separating eccentricity errors in workpiece installation, and the separation results of eccentricity errors in the workpiece installation at different axial distances were obtained. Table 8 shows results of both amplitude \(R_{1}\) 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 radial runout error. As the axial distance increases, the variation in the installation eccentricity error parameter becomes less than 0.5%, indicating that the increase in tilt angle swing error will not affect the calculation and separation of workpiece installation eccentricity error.  
Subsequently, the parameters in the table above were used to construct simulation signals and compared with experimental measurements to verify the accuracy of the separation results. Figure 14 compares experimental measurements and simulated signals of the first harmonic component of the spindle motion error measured at different axial distances. The curve in the figure clearly shows that the simulated values exhibit a high degree of coincidence with the measured values.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **80 mm** & **90 mm** & **100 mm** \\ \hline \(R_{1}\) (\(\upmu\)m) & 14.75 & 14.68 & 14.82 \\ \(a\) (\(\upmu\)m) & 23.17 & 23.13 & 23.18 \\ \(b\) (\(\upmu\)m) & 30.90 & 31.02 & 31.33 \\ \(\lambda\) (rad) & 1.43 & 1.42 & 1.40 \\ \(\phi_{1}\) (rad) & 0.83 & 0.81 & 0.83 \\ \(\phi_{2}\) (rad) & 0.31 & 0.31 & 0.31 \\ \hline \end{tabular}
\end{table}
Table 8: Calculation results of parameters at different axial distances.  
Figure 13: The radial runout error of the spindle at different axial distances.  
Table 9 shows the \(R^{2}\) between measured values and simulated signals at different axial distances, which are close to 1, indicating that the separation method proposed in this work demonstrates high accuracy and reliability.