## 2 Modeling of Dynamic Rotational Error for Spindle-Rolling Bearing System
### Nonlinear Bearing Model
#### 2.1.1 Modeling of Dynamic Supporting Stiffness  
The geometry of an angular contact ball bearing and the coordinate systems is shown in Figure 1. The global coordinate system (O-XYZ) is fixed at the rotating axis center of the outer ring. The inner ring rotates at a constant speed \(\omega_{0}\) with Z axis while the outer ring is fixed. \(\alpha_{0}\) is the initial contact angle of the bearing. The outer ring is fixed and assume that \(\delta_{x}\), \(\delta_{y}\), \(\delta_{z}\), \(\theta_{x}\) and \(\theta_{y}\) are the three translational and two angular displacements of the inner ring. The displacement vector of the inner ring is expressed as  
\[\delta_{i}=\left[\delta_{x},\delta_{y},\delta_{z},\theta_{x},\theta_{y}\right] ^{T}, \tag{1}\]  
There are \(\mathrm{N_{b}}\) rolling elements in the bearing and the position angle of the kth rolling element, \(\phi_{bk}\), could be obtained from the following formula:  
\[\mathrm{f_{bk}}=\frac{2\pi(k-1)}{\mathrm{N_{b}}}, \tag{2}\]  
Figure 1: The structure of the angular contact ball bearing. The local coordinate system (o-xyz), whose center is at the mass center of the rolling element, rotates at an angular speed \(\omega_{\rm bk}\) with the \(Z\) axis. \(\omega_{\rm bk}\) is the orbital angular speed of the rolling element and can be expressed as  
\[\omega_{\rm bk}=\frac{\omega_{0}}{2}\bigg{(}1-\frac{\rm D}{\rm d_{m}}\text{cos} \alpha^{\rm o}\bigg{)}, \tag{3}\]  
where \(\rm d_{m}\) is the pitch diameter of the bearing, \(\rm D\) is the diameter of the rolling element. The rolling elements also rotate at an angular speed \(\omega_{\rm sk}\) around the axis of itself, which is called the spinning axis. The self-rotation angular speed \(\omega_{\rm sk}\) can be easily obtained by [31]. \[\omega_{\rm sk}=\frac{\rm d_{m}\omega_{0}}{2\rm D}\bigg{(}\bigg{(}\frac{\rm D }{\rm d_{m}}\text{cos}\alpha^{\rm o}\bigg{)}^{2}-1\bigg{)}, \tag{4}\]  
The angle between the spinning axis of the rolling elements and \(Z\) axis is \(\beta_{\rm k}\), which is called as rolling element attitude angle. \(\beta_{\rm k}\) can be expressed as [31]. \[\beta_{\rm k}=\tan^{-1}\bigg{(}\frac{\text{sin}\alpha_{\rm ok}}{\text{cos} \alpha_{\rm ok}+\frac{\rm D}{\rm d_{m}}}\bigg{)}, \tag{5}\]  
When the rotational speed \(\omega_{0}\) and the axial preload are zero, both the inner contact angle and the outer contact angle are the same and equal to the initial contact angle \(\alpha^{\rm o}\). The inner raceway groove curvature center, outer raceway groove curvature center and the rolling element center are collinear (red line), as shown in Figure 2. The distance between the inner raceway groove curvature center and the rolling element center, as well as the distance between the outer raceway groove curvature center and the rolling element center before deformation are  
\[ \begin{split}&\rm L_{\rm ik}=r_{i}-0.5\rm D=(f_{i}-0.5)\rm D\\ &\rm L_{\rm ok}=r_{o}-0.5\rm D=(f_{o}-0.5)\rm D\end{split} , \tag{6}\]  
where \(\rm r_{i}\) and \(\rm r_{o}\) are the inner and outer raceway groove curvature radius, respectively, \(\rm f_{i}\) and \(\rm f_{o}\) are the inner and outer raceway groove curvature radius coefficient, respectively. With the rotational speed \(\omega_{0}\) and axial preload increase, the inner raceway groove curvature center, outer raceway groove curvature center and the rolling element center are not collinear (blue line). The inner contact angle and the outer contact angle are not equal to each other. The inner contact angle increase while the outer contact angle decreases due to the centrifugal force. Besides, the distance between the inner/outer raceway groove curvature center and rolling element center changes in the  
Figure 2: The relative positions of the kth rolling element center with the inner and outer race groove curvature centers. Y-Z plane. The distance between the inner raceway groove curvature center and the rolling element center, as well as the distance between the outer raceway groove curvature center and the rolling element center, after deformation are  
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