# Separation Method for Installation Eccentricity Error of Workpiece
## 4 Simulation Verification
The data of ellipse 2 are also similar, with a small error between the theoretical and calculated values, further verifying the reliability of the calculation method. The data of ellipse 3 exhibit a high degree of consistency, with minor errors arising from truncation errors in numerical calculations or limitations in calculation accuracy. These data indicate that the calculation error between the calculated and theoretical values is very small, further confirming the \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Ellipse** & \(\mathbf{R_{1}}\) (\(\mathbf{\mu}\)) & \(\mathbf{a}\) (\(\mathbf{\mu}\)) & \(\mathbf{b}\) (\(\mathbf{\mu}\)) & \(\mathbf{\lambda}\) (rad) & \(\mathbf{\phi_{1}}\) (rad) & \(\mathbf{\phi_{2}}\) (rad) \\ \hline
1 & 20 & 7 & 3 & 0 & 0.4\(\pi\) & 0.63\(\pi\) \\
2 & 20 & 8 & 4 & 0.1\(\pi\) & 0.45\(\pi\) & 0.76\(\pi\) \\
3 & 25 & 8 & 4 & 0.3\(\pi\) & 0.50\(\pi\) & 0.90\(\pi\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of first harmonic and workpiece installation eccentricity error. high accuracy and promising application prospects of the proposed separation method for workpiece installation eccentric error in this investigation. As shown in Figure 5, based on the parameters of ellipse 2 from Tables 1 and 2, the theoretical and calculated trajectories of the spindle's first harmonic component and the workpiece installation eccentricity error were obtained. Furthermore, the sum of squared errors (SSE), root mean square error (RMSE), and mean absolute error (MAE) were used as evaluation index to quantitatively evaluate the computational accuracy of the proposed error separation method. Table 3 shows the calculation errors between the calculated values and theoretical values of both the spindle's first harmonic component and the workpiece installation eccentricity error along different directions under different error indicators, where \(x\) and \(y\) represent directions. The slight values of each error indicate the high accuracy and reliability of the separation method proposed in this investigation for the first harmonic component of the spindle and the installation eccentricity error of the workpiece. In summary, the proposed separation method for workpiece installation eccentric error exhibits high computational accuracy and stability when dealing with the nonlinear vibration form of the first harmonic component of the spindle radial runout error. In addition, when the first harmonic component of the spindle radial runout error is in the form of linear vibration, that is, when the amplitude of the first harmonic component of  
Figure 3: Simulated motion. the spindle radial runout error in the \(x\) or \(y\) direction is zero, the calculation results of the separation method also maintain high calculation accuracy and stability. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ellipse** & **Parameter** & **Theoretical Value** & **Calculated Value** & **Error** \\ \hline \multirow{6}{*}{\(1\)} & \(R_{1}\) (\(\upmu\)m) & 20.00 & 19.99 & 0\% \\  & \(a\) (\(\upmu\)m) & 7.00 & 7.00 & 0\% \\  & \(b\) (\(\upmu\)m) & 3.00 & 2.99 & 0.07\% \\  & \(\lambda\) (rad) & 0.00 & 0.00 & 0\% \\  & \(\phi_{1}\) (rad) & 0.40 & 0.40 & 0\% \\  & \(\phi_{2}\) (rad) & 0.63 & 0.629 & 0.2\% \\ \hline \multirow{6}{*}{\(2\)} & \(R_{1}\) (\(\upmu\)m) & 20.00 & 19.99 & 0.05\% \\  & \(a\) (\(\upmu\)m) & 8.00 & 8.00 & 0 \\  & \(b\) (\(\upmu\)m) & 4.00 & 3.99 & 0.2\% \\  & \(\lambda\) (rad) & 0.100 & 0.101 & 0.1\% \\  & \(\phi_{1}\) (rad) & 0.450 & 0.450 & 0\% \\  & \(\phi_{2}\) (rad) & 0.760 & 0.759 & 0.1\% \\ \hline \multirow{6}{*}{\(3\)} & \(R_{1}\) (\(\upmu\)m) & 25.00 & 25.00 & 0\% \\  & \(a\) (\(\upmu\)m) & 8.00 & 8.00 & 0\% \\ \cline{1-1}  & \(b\) (\(\upmu\)m) & 4.00 & 4.00 & 0\% \\ \cline{1-1}  & \(\lambda\) (rad) & 0.30 & 0.30 & 0\% \\ \cline{1-1}  & \(\phi_{1}\) (rad) & 0.50 & 0.50 & 0\% \\ \cline{1-1}  & \(\phi_{2}\) (rad) & 0.90 & 0.90 & 0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation results of simulation parameters. Figure 4: The trajectory of simulated elliptical motion.