# Separation Method for Installation Eccentricity Error of Workpiece
## 6 Conclusions  
In this work, a mathematical model of spindle first order harmonics and workpiece installation eccentricity error motion was established. A separation method combining PSO and the least squares method was proposed. It can separate the first order harmonic of spindle radial runout error from workpiece installation eccentricity error, which is crucial for improving machine tool accuracy. Then, numerical simulations validated the method's accuracy and effectiveness. The effects of different speeds, axial distances, and workpieces were analyzed and discussed in detection experiments. The similarity between  
Figure 16: Measured values and simulated signals of different workpieces. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Workpiece** & **Along the \(x\) Direction** & **Along the \(y\) Direction** \\ \hline
1 & 0.9816 & 0.9714 \\
2 & 0.9959 & 0.9843 \\
3 & 0.9867 & 0.9848 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The \(R^{2}\) between measured values and simulated signals of different workpieces. the simulated signal generated using identification parameters and the first harmonic of the measured signal exceeded 97%. The conclusions are as follows:  
1. Workpiece installation eccentricity error remains consistent at different spindle speeds. As the speed increases, the spindle radial motion error amplitude rises, but the variation of amplitude and phase of the separated workpiece installation eccentricity error varies less than 0.3%. This indicates that the error does not vary with spindle speed, and increased spindle vibration response does not affect its calculation and separation. 2. The workpiece installation eccentricity error separated at different axial distances is completely the same. When the detection section of the workpiece gradually moves away from the chuck end face, the variation in the separation result of the workpiece installation eccentricity error is less than 0.5%. This indicates that the increase in tilt swing error will not affect the calculation and separation of workpiece installation eccentricity error, further verifying the effectiveness of the proposed installation eccentricity error separation method. 3. The variation in workpiece installation eccentricity error calculated when installing different workpieces is within 0.6%. This indicates that the installation eccentricity error introduced when clamping different workpieces is very small. The surface morphology of different workpieces will not affect the calculation and separation of installation eccentricity errors. The installation eccentricity error stems from the misalignment between the chuck rotation center and the spindle geometric center. 4. The proposed method demonstrates robust generalizability across diverse machining conditions, with calculation errors of installation eccentricity error remaining below 0.6% under varying spindle speeds, axial distances, and workpiece geometries. This enables its application in diverse CNC machining scenarios and eccentricity detection of rotating machinery and provides a reliable and easily adoptable solution for precision manufacturing applications. In summary, this investigation proposes a separation method for workpiece installation eccentricity error that combines PSO algorithm and least squares method. The advantages of this method are that it only requires displacement sensors, resulting in low cost, and that it eliminates the need for complex spectral processing, thereby ensuring high computational efficiency. At varying speeds, the fluctuation in error calculations of this method remains below 0.3%, significantly enhancing its engineering applicability. This study is significant for improving product quality, reducing waste, and enhancing productivity in the production environment. Conceptualization, G.Q. and H.M.; Methodology, G.Q. ; Software, Y.C. ; Validation, G.Q. ; Formal analysis, Y.C. ; Investigation, H.M.; Resources, C.Z. ; Data curation, G.Q. ; Writing--original draft, G.Q. ; Writing--review & editing, C.Z. ; Supervision, C.Z.