# Separation Method for Installation Eccentricity Error of Workpiece
## 1 Introduction
To prevent displacement during workpiece processing, the clamping torque between the chuck and the workpiece is relatively high, so the distance between the workpiece center and the spindle center is kept constant, that is, the installation eccentricity and phase are both fixed values. The eccentric motion of the workpiece is a circle in the Cartesian coordinate system, and its trajectory is projectedon the \(x\)- and \(y\)-axes as the first harmonic, which is equal in amplitude and has a phase difference of \(\pi/2\), that is:  
\[\left\{ \begin{array}{ll}x_{w}(\theta)=R_{1}\cos(\theta+\lambda)\\ y_{w}(\theta)=R_{1}\cos(\theta+\lambda-\frac{\pi}{2})=R_{1}\sin(\theta+\lambda) \end{array} \right. \tag{1}\]  
where \(x_{w}(\theta)\) and \(y_{w}(\theta)\) are components of eccentric motion at the \(x\)- and \(y\)-axis, respectively. \(R_{1}\) is the eccentricity. \(\lambda\) is the initial phase of eccentric motion. The first harmonic motion of the spindle can be expressed by the parametric equation  
\[\left\{ \begin{array}{ll}x_{s}(\theta)=a\cos(\theta+\phi_{1})\\ y_{s}(\theta)=b\sin(\theta+\phi_{2})\end{array} \right. \tag{2}\]  
where \(x_{s}(\theta)\) and \(y_{s}(\theta)\) are first harmonic components of spindle motion error at the \(x\)- and \(y\)-axis, respectively. \(a\) and \(b\) are the amplitude of the first harmonic at the \(x\)-and \(y\)-axes, respectively. \(\phi_{1}\) and \(\phi_{2}\) are the initial phases of the first harmonic. The motion form of the first harmonic of the radial runout error of the spindle exists in various forms. If \(\phi_{1}=\phi_{2}\), the initial phase is consistent, it is a linear vibration, and its expression is  
\[y(\theta)=kx(\theta);\;k=b/a,a\
eq 0. \tag{3}\]  
If the difference between \(\phi_{1}\) and \(\phi_{2}\) is \(\pi/2\) and \(a=b\), then the first harmonic component is circular motion, that is:  
\[y^{2}(\theta)+x^{2}(\theta)=a^{2}\quad\text{or}\quad y^{2}(\theta)+x^{2}( \theta)=b^{2} \tag{4}\]  
If none of the above conditions are met, the first harmonic component is a general elliptical motion centered around the coordinate origin. Therefore, the trajectory of the synthesized motion of the first harmonic of the spindle and the eccentricity of the workpiece are generally elliptical and can be expressed as  
\[\left\{ \begin{array}{ll}x_{f}(\theta)=R_{1}\cos(\theta+\lambda)+a\cos( \theta+\phi_{1})\\ y_{f}(\theta)=R_{1}\sin(\theta+\lambda)+b\sin(\theta+\phi_{2})\end{array} \right. \tag{5}\]  
Equation (5) is a system of nonlinear equations. In this case, the trajectory image of the synthesized motion is shown in Figure 1, where \(O\) is the center of the spindle, \(R_{1}\) is the eccentricity error of the workpiece installation, and \(\tau\) is the angle between the long half-axis and the \(x\)-axis. Figure 1: Synthesized motion. Assuming that the motion form of the first harmonic component of the radial runout error of the spindle is circular motion, and the combined motion with the installation eccentricity error of the workpiece is still circular motion, that is, the initial phase of the eccentricity of the spindle and workpiece installation is the same, Equation (5) can be written as follows:  
\[\left\{ \begin{array}{l}x(\theta)=R_{1}\cos(\theta+t_{1})+a\cos(\theta+t_{1}) \\ y(\theta)=R_{1}\sin(\theta+t_{1})+a\sin(\theta+t_{1})\end{array} \right. \tag{6}\]  
In Equation (6), there are three unknown variables, a, \(R_{1}\), and \(t_{1}\), and only two equations. Therefore, the amplitude \(R_{1}\) of the workpiece installation eccentricity error cannot be separated from the amplitude \(a\) of the first harmonic component of the spindle vibration. In addition, when the first harmonic component of the radial runout error of the spindle is still circular motion, but is different from the initial phase of the installation eccentricity error of the workpiece, Equation (6) can be rewritten as Equation (7), that is:  
\[\left\{ \begin{array}{l}x(\theta)=R_{1}\cos(\theta+\lambda)+a\cos(\theta+t_{1 })\\ y(\theta)=R_{1}\sin(\theta+\lambda)+a\sin(\theta+t_{1})\end{array} \right.