## 2 Modeling of Dynamic Rotational Error for Spindle-Rolling Bearing System
### Nonlinear Bearing Model
#### 2.1.2 Modeling of Time-Varying Bearing Force  
The machining errors of the bearing parts, which has great important effects on the bearing force and dynamic rotational error, are mainly include shape errors, surface waviness and roughness. The surface roughness is not considered in modeling the bearing force for the wavelength of the surface roughness is smaller than the width of the Hertz contact [31]. The outer ring is fixed and the displacements for the inner raceway groove curvature center corresponding to the kth rolling element, including two translations along the \(y\)-, \(z\)-axes, and one rotation with respect to the \(x\)-axis, are expressed as  
\[\delta_{\rm ik}=\left[\rm u_{yk},\rm u_{zk},\rm\theta_{\rm kk}\right]^{T} \tag{18}\]  
Under the assumption of small deflection, \(\delta_{\rm ik}\) could be obtained through coordinate transformations  
\[\delta_{\rm ik}=\rm T_{\rm k}\delta_{i}+\Delta w_{\rm k}+\Delta D_{\rm k} \tag{19}\]  
where \(\rm T_{\rm k}\) is the transformation matrix, and can be expressed as  
\[\rm T_{\rm k}=\left[ \begin{array}{cccc}\rm sinf_{\rm bk}&\rm cosf_{\rm bk}&0& 0&0\\ 0&0&1&\rm\eta_{i}cosf_{\rm bk}&-\rm\eta_{i}sinf_{\rm bk}\\ 0&0&0&\rm cosf_{\rm bk}&-sinf_{\rm bk}\end{array} \right] \tag{20}\]  
\(\Delta w_{\rm k}\) denotes the additional displacements caused by the waviness of the bearing rings and rolling elements. Similarly, \(\Delta D_{\rm k}\) represents the additional displacements due to roundness error of the bearing rings and rolling elements. As illustrated in Figure 4, the waviness exists on each bearing components. \(\rm p_{ik}\), \(\rm P_{\rm ok}\) and \(\rm q_{ik}\), \(\rm q_{\rm ok}\) represent radial and axial waviness of inner and outer raceways, \(\rm w_{ik}\) and \(\rm w_{ok}\) denote the waviness of rolling elements. The harmonic functions are utilized to express the surface waviness due to the waviness vary with time periodically. \[\rm p_{ik}=\sum_{n_{in}=1}^{N}A_{in}cos\left(n_{in}\left((\omega_{0}-\omega_{ \rm bk})t-\frac{2\pi(k-1)}{N_{b}}\right)+f_{in}\right) \tag{21}\] \[\rm q_{ik}=\sum_{n_{in}=1}^{N}B_{in}cos\left(n_{in}\left((\omega_ {0}-\omega_{\rm bk})t-\frac{2\pi(k-1)}{N_{b}}\right)+\psi_{in}\right)\] \[\rm P_{ok}=\sum_{n_{out}=1}^{N}A_{out}cos\left(n_{out}\left( \omega_{\rm bk}t+\frac{2\pi(k-1)}{N_{b}}\right)+f_{out}\right)\] \[\rm q_{ok}=\sum_{n_{out}=1}^{N}B_{out}cos\left(n_{out}\left( \omega_{\rm bk}t+\frac{2\pi(k-1)}{N_{b}}\right)+\psi_{out}\right)\] \[\rm w_{ik}=\sum_{n_{out}=1}^{N}C_{ba}cos\left(n_{ba}\omega_{\rm sk }t+\varphi_{ba}\right)\] \[\rm w_{ok}=\sum_{n_{out}=1}^{N}C_{ba}cos\left(n_{ba}\omega_{\rm sk }t+\pi+\varphi_{ba}\right)\]  
where \(\rm A_{in}\), \(\rm A_{out}\), \(\rm B_{in}\), \(\rm B_{in}\), \(\rm C_{ba}\), \(\rm C_{ba}\) are the amplitudes of the surface waviness; \(\rm n_{in}\), \(\rm n_{out}\), \(\rm n_{ba}\), are the harmonic orders of the surface waviness; \(\rm\phi_{in}\), \(\rm\psi_{in}\), \(\rm\phi_{out}\), \(\rm\psi_{out}\), \(\rm\varphi_{ba}\) are the initial phases angle of the surface waviness. In accordance with the geometric relations, one can obtain  
\[\Delta w_{k}=\left[ \begin{array}{c}p_{ok}-p_{ik}\\ q_{ok}-q_{ik}\\ 0\end{array} \right] \tag{22}\]  
The effects of the surface waviness on the dynamic supporting stiffness are very small; however, the effects of the surface waviness on the dynamic rotational error can be reflected in the form of time-varying bearing force. The distances between the inner/outer raceway groove curvature center and the rolling element center, considering the effect of surface waviness, are  
\[ \begin{array}{c}L_{ik}=(f_{i}-0.5)D+p_{ik}-w_{ik}\\ L_{ok}=(f_{0}-0.5)D+p_{ok}-w_{ok}\end{array}  \tag{23}\]  
The geometric relationships can be written as follows:  
\[ \begin{array}{c}\sin\alpha_{ik}=\frac{L_{ik}\sin\alpha_{0}+v_{xk}-u_{ik}}{l_ {ik}}\\ \cos\alpha_{ik}=\frac{L_{ik}\cos\alpha_{0}-v_{yk}+u_{yk}}{l_{ik}}\\ \sin\alpha_{ok}=\frac{L_{ik}\sin\alpha_{0}-v_{yk}}{l_{ik}}\\ \cos\alpha_{ok}=\frac{L_{ik}\cos\alpha_{0}+v_{yk}}{l_{ok}}\end{array}