# Separation Method for Installation Eccentricity Error of Workpiece
## 3 Separation Method of Workpiece Installation Eccentricity Error  
In this section, a separation method was proposed to solve the problem of inseparability between the first harmonic caused by the nonlinear vibration of the spindle and the installation eccentricity error in Equations (5)-(7). The combined motion of the first harmonic of the radial runout error of the spindle and the installation eccentricity error of the workpiece is generally elliptical motion, as shown in Equation (5). The correspondence between the radial runout error separated from experimental data using the reversal method and the variables in Equation (5) is as follows:  
\[x=\overline{\delta}_{x}(\varphi_{i}),y=\overline{\delta}_{y}(\varphi_{i}) \tag{8}\]  
Assuming the spindle rotates once and uniformly samples \(N\) points, when the spindle angular velocity is \(\Omega^{l}\), the discrete Equation (5) at position \(\varphi_{i}\) and time \(t_{i}\) can be expressed as  
\[\left\{ \begin{array}{l}x_{f}^{l}(\varphi_{i})=R_{1}\cos\Bigl{(}\Omega^{l}t_ {i}+\lambda\Bigr{)}+a\cos\Bigl{(}\Omega^{l}t_{i}+\phi_{1}\Bigr{)}\\ y_{f}^{l}(\varphi_{i})=R_{1}\sin\Bigl{(}\Omega^{l}t_{i}+\lambda\Bigr{)}+b\sin \Bigl{(}\Omega^{l}t_{i}+\phi_{2}\Bigr{)}\end{array} \right.\quad i=1,2,\cdots,N. \tag{9}\]  
Therefore, there are three coefficients and three initial phases to be determined. If Equation (9) is rearranged, it can be expressed as  
\[ \begin{cases}R_{1}c_{1}^{li}+ac_{2}^{li}=b_{1}^{li}\\ R_{1}c_{3}^{li}+bc_{4}^{li}=b_{2}^{li}\end{cases} \quad i=1,2,\cdots,N;l=1,2,\cdots,M. \tag{10}\]  
with  
\[ \begin{split} c_{1}^{li}&=\cos\left(\Omega^{l}t_{i}+\lambda \right),c_{2}^{li}=\cos\left(\Omega^{l}t_{i}+\phi_{1}\right),c_{3}^{li}=\sin \left(\Omega^{l}t_{i}+\lambda\right),c_{4}^{li}=\sin\left(\Omega^{l}t_{i}+ \phi_{2}\right)\\ b_{1}^{li}&=x(\varphi_{i}),b_{2}^{li}=y(\varphi_{i})\end{split}  \tag{11}\]  
Equation (10) can be expressed in matrix form as follows:  
\[\mathbf{C}^{li}\Big{\{}R_{1}\quad a\quad b\Big{\}}^{\mathrm{T}}=\mathbf{b}^{li}\quad i =1,2,\ldots,N. \tag{12}\]  
with  
\[\mathbf{C}^{li}= \begin{bmatrix}c_{1}^{li}&c_{2}^{li}&0\\ c_{3}^{li}&0&c_{4}^{li}\end{bmatrix} ,\mathbf{b}^{li}= \begin{cases}b_{1}^{li}\\ b_{2}^{li}\end{cases}  \tag{13}\]  
As shown in Equation (12), 3\(N\) equations can be obtained. In addition, the phase \(\lambda\) of the radial runout error of the spindle and the initial phases \(\phi_{1}\) and \(\phi_{2}\) of the workpiece installation eccentricity error are also unknown. Currently, the number of equations should be greater than the number of unknowns. In Equations (10)-(13), index \(l\) denotes the experiment results at rotational speed \(\Omega^{l}(l=1,2,\cdots,J_{l})\). It assumes that  
\[\mathbf{\chi}=\Big{\{}R_{1}\quad a\quad b\Big{\}}^{\mathrm{T}}\quad\mathbf{h}=\Big{\{} b_{1}^{1i}\quad b_{2}^{1i}\quad b_{1}^{2i}\quad b_{2}^{2i}\quad\cdots\quad b_{1}^{ Mi}\quad b_{2}^{li}\Big{\}}^{\mathrm{T}}\]  
All the experiment results can be expressed as  
\[\mathbf{C}\mathbf{\chi}=\mathbf{h} \tag{14}\]  
with  
\[\mathbf{C}= \begin{bmatrix}c_{1}^{11}&c_{1}^{12}&\cdots&c_{1}^{1N}&c_{3}^{11}&c_{ 3}^{12}&\cdots&c_{3}^{1N}&\cdots&c_{1}^{l1}&c_{1}^{l2}&\cdots&c_{3}^{lN}\\ c_{2}^{11}&c_{2}^{12}&\cdots&c_{2}^{1N}&0&0&\cdots&0&\cdots&c_{2}^{l,1}&c_{2}^ {l,2}&\cdots&0\\ 0&0&\cdots&0&c_{4}^{11}&c_{4}^{12}&\cdots&c_{4}^{1N}&\cdots&0&0&\cdots&c_{4}^{ lN}\\ \end{bmatrix}  \tag{15}\]  
where matrix \(\mathbf{C}\) and vector \(\mathbf{h}\) are constructed by vertically stacking matrices and vectors obtained from experiments under different operating conditions (such as multiple spindle speeds). If the least squares method is applied to determined vector \(\mathbf{\chi}\), one has  
\[\mathbf{\chi}=\Big{(}\mathbf{C}^{\mathrm{T}}\mathbf{C}\Big{)}^{-1}\mathbf{C}\mathbf{h} \tag{16}\]  
Due to the nonlinear parameters of the phase \(\lambda\) of the radial runout error of the spindle and the initial phases \(\phi_{1}\) and \(\phi_{2}\) of the workpiece installation eccentricity error, they cannot be directly solved.