# Separation Method for Installation Eccentricity Error of Workpiece
## 5 Experimental Study
### Experimental Equipment  
The experimental machine model is a commercial CNC6130 lathe (Yingtai Machine Tool, Wenling, China), equipped with four eddy current sensors (ML33-01-00-03, MIRAN, Shenzhen, China) for detecting displacement signals on the workpiece, as shown in Figure 6. Sensors S1 and S2 detect section a of the workpiece, while S3 and S4 detect section b. S1 and S3 detect the displacement signal in the \(y\) direction, while S2 and S4 detect the displacement signal in the \(x\) direction. The distance between the chuck end face and sensor S1 is 80 mm, while the distance between sensor S1 and S3 is 50 mm. After the spindle is started, the data acquisition card (NI-DAQ) inputs the sensor signal into the computer with a sampling frequency of 25.6 kHz.  
Figure 5: Comparison of theory and separation results of ellipse 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Error**} & \multicolumn{2}{c}{**First Harmonic Component**} & \multicolumn{2}{c}{**Installation Eccentricity**} \\ \cline{2-5}  & \(\mathbf{x}\) & \(\mathbf{y}\) & \(\mathbf{x}\) & \(\mathbf{y}\) \\ \hline SSE & 0 & 0.98 & 0.98 & 0.158 \\ RMSE & 0 & 0.002 & 0.044 & 0.008 \\ MAE & 0 & 0.040 & 0.039 & 0.008 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Calculation error of simulation for the spindle radial runout error detection.  
Eddy current displacement sensors save more than 80% in hardware costs compared to laser interferometers and coordinate measuring instruments while ensuring detection accuracy. Furthermore, the proposed algorithm does not require a dedicated computing platform, a factor that significantly reduces software costs. At the same time, detection can be carried out during processing, offering fast detection speed and reducing downtime cost in industrial production. Thus, this method balances detection accuracy and cost efficiency, making it suitable for cost-sensitive manufacturing scenarios.  
The specifications of workpieces 1 to 3 are shown in Figure 7. Workpieces 1 and 2, as shown in Figure 7a,b, are cylindrical parts with different radius. Workpiece 3, as seen in Figure 7c, has a blind hole machined into it. All workpieces are made of 45# steel with a bright machining surface on the outer surface. The mass of workpieces 1 to 3 are 2.56 kg, 3.22 kg, and 1.49 kg, respectively. The depth \(l_{\text{h}}\) of the blind hole of the workpiece 3 is 120 mm. The wall thickness \(h\) of workpiece 3 is 5 mm. The front face of the workpiece is in contact with the chuck, and the clamping mechanism of the chuck fixes the machining surface of the workpiece to ensure that each jaw has the same contact area with the workpiece.  
In subsequent experiments, displacement signals were measured under various operating conditions. Subsequently, the Donaldson reversal method was used to extract radial runout errors from the displacement signal [15]. This method obtains spindle motion error by adding or subtracting two sets of signals before and after reversing 180 degrees. Afterward, the separation method proposed in this study was applied to calculate the amplitude and phase of both workpiece installation eccentricity and the spindle's first harmonic component. Finally, the separation results under different operating conditions are discussed next.  
Figure 6: Experimental testing system.  
Figure 7: Workpiece.