## 2 Modeling of Dynamic Rotational Error for Spindle-Rolling Bearing System
### Dynamic Model of the SRB System  
Figure 5 shows the layout of the SRB system. The spindle is supported by two pairs of angular contact bearings. The front and rear bearings are arranged back to back. The types of the front and rear bearing are 7017 and 7016 C, respectively. The light preload is exerted on the SRB system in assembling process. The system coordinate \(O\)-\(U_{\text{s}}V_{\text{s}}W_{\text{s}}\) is fixed at the mass center of the spindle. Table 1 shows the parameters of the SRB system. In this section, the dynamic model of the SRB system considering the dynamic supporting stiffness, time-varying bearing force caused by the surface waviness, dynamic unbalance force and the disturbance force, will be derived, which lays a foundation for evaluating the dynamic rotational error. The simplified dynamic model of the SRB system is shown in Figure 6. The spindle can be regarded as a rigid body and has five degrees of freedom, i.e., three translational freedom along the U\({}_{\text{s}}\)-, V\({}_{\text{s}}\)- and Ws-axes, and two rotational freedom with respect to the U\({}_{\text{s}}\)- and V\({}_{\text{s}}\)-axes. The dynamic displacement vector of the rotor is  
\[\text{q}=\left[\text{u}_{\text{s}},\text{v}_{\text{s}},\text{w}_{\text{s}}, \text{\theta}_{\text{s}}^{\text{u}},\text{\theta}_{\text{s}}^{\text{v}}\right] ^{\text{T}} \tag{28}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Parameters** & **Items** & **Parameters** \\ \hline \(m_{\text{s}}\)/kg & 30.735 & \(l_{\text{3}}\)/m & 0.144 \\ \(l_{\text{s}}\)/kg m\({}^{2}\) & 0.963 & \(l_{\text{4}}\)/m & 0.188 \\ \(l_{\text{s}}\)/kg m\({}^{2}\) & 0.035 & \(l_{\text{5}}\)/m & 0.312 \\ \(l_{\text{1}}\)/m & 0.109 & \(l_{\text{6}}\)/m & 0.446 \\ \(l_{\text{2}}\)/m & 0.153 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation parameters of the SRB system. Figure 5: Layout of the spindle-rolling bearing (SRB) system. The dynamic model for SRB system can be obtained based on the Lagrange equation. The final equations of motion of the SRB system can be expressed as  
\[[\mathrm{M}]\big{[}\ddot{\mathrm{q}}\big{]}+([\mathrm{C}]+\omega_{0}[\mathrm{G}] )\big{\langle}\dot{\mathrm{q}}\big{]}+\big{[}\mathrm{K}^{\mathrm{b}}\big{]}[ \mathrm{q}\big{]}=\{\mathrm{F}_{\mathrm{e}}\}+\{\mathrm{F}_{\mathrm{dr}}\}+\{ \mathrm{F}_{\mathrm{b}}\} \tag{29}\]  
where \([\mathrm{M}]=[\mathrm{m}_{\mathrm{s}},\mathrm{m}_{\mathrm{s}},\mathrm{m}_{ \mathrm{s}},\mathrm{I}_{\mathrm{s}},\mathrm{I}_{\mathrm{s}}]^{\mathrm{T}}\), is the mass matrix of the system, \(m_{\mathrm{s}}\) is the mass of the spindle and \(\mathrm{I}_{\mathrm{s}}\) is the inertia moment of the spindle about the \(\mathrm{U}_{\mathrm{s}}\)-axis and \(\mathrm{V}_{\mathrm{s}}\)-axis. \(\mathrm{K}^{\mathrm{b}}\) is the stiffness matrix which can be calculated in Section 2. C is the structural damping matrix constructed from modal damping ratios identified experimentally. G is the gyroscopic matrix and can be expressed as  
\[[\mathrm{G}]=\left[ \begin{array}{cccc}0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&-\mathrm{J}_{\mathrm{s}}\\ 0&0&0&\mathrm{J}_{\mathrm{s}}&0\end{array} \right] \tag{30}\]  
\(\mathrm{J}_{\mathrm{s}}\) is the inertia moment of the rotor about the \(\mathrm{W}_{\mathrm{s}}\)-axis. \(\mathrm{F}_{\mathrm{e}}\)\(\mathrm{F}_{\mathrm{dr}}\) and \(\mathrm{F}_{\mathrm{b}}\) are the dynamic unbalance force caused by unbalanced mass, disturbance force and the time-varying bearing force caused by surface waviness, respectively. \(\mathrm{F}_{\mathrm{e}}\) can be states as  
\[\{\mathrm{F}_{\mathrm{e}}\}=\mathrm{m}_{\mathrm{u}}\mathrm{e}_{\mathrm{s}} \omega_{0}^{2}[\cos(\omega_{0}\mathrm{t}),\sin(\omega_{0}\mathrm{t}),0,\mathrm{ I}_{\mathrm{3}}\mathrm{cos}(\omega_{0}\mathrm{t}),\mathrm{  
The displacement data express sinusoidal trends, with a deviation from zero level. In accordance with the ISO or ANSI/ASME test standard, different kinds of error motion values can be evaluated by using the displacement signals of the artifact [6; 7].