# Separation Method for Installation Eccentricity Error of Workpiece
## 4 Simulation Verification  
In this section, the accuracy and reliability of the proposed separation method in this investigation are verified through numerical simulation. Numerical simulation verification is divided into three steps:  
Step 1: Determine the parameters for simulating synthetic motion. Synthetic motion is generally elliptical motion and consists of two parts, one of which is the elliptical motion of the first harmonic component in the nonlinear vibration of the main axis, and the other is the circular motion caused by installation eccentricity error. The simulation parameters of  
Figure 2: Algorithm flowchart. the first harmonic of the radial runout error of the spindle and the eccentricity error of the workpiece installation are shown in Table 1.  
Step 2: As shown in Figure 3, simulation trajectories are constructed based on the parameters of ellipse 1 to ellipse 3 in Table 1 for the primary harmonic component of the spindle and the eccentricity error of the workpiece installation, as well as their composite motion. The sampling frequency is 1000 Hz, and the rotational speed is 20\(\pi\) rad/s. The simulation motion of the first harmonic of the spindle radial runout error, the workpiece installation eccentricity simulation motion, and their combined motion are shown in Figure 3a-f, respectively. Figure 3a,b show the simulated motion trajectory of the nonlinear vibration of the first harmonic component of the spindle along the \(x\) and \(y\) directions, where the amplitudes are \(a\), \(b\) and phases are \(\phi_{1}\), \(\phi_{2}\), directly affecting the shape and direction of the ellipse. Figure 3c,d depict the circular motion with installation eccentricity error, where the eccentricity \(R_{1}\) determines the radius of the circular motion, and the phase \(\lambda\) controls the direction of the center's offset. Finally, Figure 3e,f show the superposition results of the synthesized motion along the \(x\) and \(y\) directions, which are determined by the combination of nonlinear parameters. The elliptical trajectory of composite motion in space is shown in Figure 4. In the figure, _Oxy_ is a fixed coordinate system, and \(Ox^{\prime}y^{\prime}\), \(Ox^{\prime}y^{\prime\prime}\), and \(Ox^{\prime\prime\prime}y^{\prime\prime\prime}\) are new coordinate systems established along the three major and minor axes of the ellipse. Comparing the size parameters of three ellipses, as the amplitude of both the spindle's first harmonic component and the workpiece installation eccentricity error increase, the long and short axes of the synthesized ellipse also increase. Meanwhile, due to the increase in the phase of both the spindle's first harmonic component and the workpiece installation eccentricity error, the angle between the long half-axis and the \(x\)-axis of the synthesized elliptical trajectory is also different. In summary, composite motion is determined by the amplitude and phase of the spindle's first harmonic component and the workpiece installation's eccentricity error. Step 3: The simulated synthesized motion is substituted into the vector \(\mathbf{h}\) in Equation (14). Next, the PSO search method proposed in the previous section is applied to find the phase angles that minimize the residual \(\mathbf{v}\). After substituting the phase angles into matrix \(\mathbf{C}\), Equation (16) is used to calculate the amplitude of the first harmonic component \(a\) and \(b\) of the spindle and the amplitude \(R_{1}\) of the workpiece installation eccentricity error. Afterward, the computational accuracy of the proposed separation method is evaluated. The calculation results of different ellipse parameters are shown in Table 2, including the calculation results and errors of amplitudes \(R_{1}\), \(a\), and \(b\), and phases \(\lambda\), \(\phi_{1}\), and \(\phi_{2}\). For ellipse 1, the theoretical and calculated values are very close, with errors within 0.07%, indicating that the calculation results of ellipse 1 parameters have high accuracy.