## 2 Shaft-Bearing System Model
### Quasi-Static Model of Angular Contact Ball Bearing
The ball center will be shifted from O to O\({}^{\prime}\), and the inner raceway groove curvature center moves from O\({}_{\text{i}}\) to O\({}^{\prime}\)\({}_{\text{i}}\), while the groove curvature center of the outer raceway remains unchanged since the outer ring is fixed. Figure 1: Geometrical configuration of a shaft-bearing system. Figure 3: Positions of ball centers and raceway groove curvature centers. Figure 2: Kinematic relation of a ball element. In Figure 3, \(D_{1j}\), \(D_{2j}\) denote the axial and radial distance between the inner and outer raceway groove curvature centers at the \(j\)th ball element position and can be written as:  
Figure 2.0  
Figure 2.1  
where \(\delta_{\rm x}\), \(\delta_{\rm y}\), and \(\delta_{\rm z}\) are the translational displacements of the inner ring along the \(x\)-, \(y\)-, and \(z\)-axis, respectively, \(\theta_{\rm x}\) and \(\theta_{\rm y}\) are the angular displacements around the \(x\)- and \(y\)-axis, respectively. \(\alpha_{0}\) is the bearing initial contact angle. \(f_{\rm i}\) and \(f_{\rm e}\) are the inner and outer raceway groove curvature coefficients, respectively. The radius of the locus of the inner raceway groove curvature centers \(R_{\rm i}\) and the \(j\)th ball element azimuth angle \(\varphi_{j}\) are determined by:  
Figure 2.2  
Figure 2.3  
where \(Z\) is the ball element number in the bearing. The loaded contact angles \(\alpha_{\rm i}\) and \(\alpha_{\rm e}\) at ball-raceway contacts can be obtained:  
Figure 2.4  
Figure 2.5  
The contact elastic deformations \(\delta_{ij}\) and \(\delta_{\rm ej}\) can be given as:  
Figure 2.6  
Figure 2.7  
where \(X_{1j}\) and \(X_{2j}\) are the axial and radial distances between ball center \(\rm O^{\prime}\) and outer raceway groove curvature center \(\rm O_{e}\), respectively. As shown in Figure 4, considering the centrifugal force \(F_{cj}\), gyroscopic moment \(M_{\rm gj}\), contact forces \(Q_{\rm ij}\) and \(Q_{\rm ej}\), and friction forces \(F_{\rm ij}\) and \(F_{\rm ej}\), the force and moment equilibrium equations for the \(j\)th ball element can be presented as:  
\[\left\{ \begin{array}{l}Q_{\rm ij}\sin\alpha_{\rm ij}-Q_{\rm ej}\sin\alpha_ {\rm ej}+F_{\rm ij}\cos\alpha_{\rm ij}-F_{\rm ej}\cos\alpha_{\rm ej}=0\\ Q_{\rm ij}\cos\alpha_{\rm ij}-Q_{\rm ej}\cos\alpha_{\rm ej}-F_{\rm ij}\sin \alpha_{\rm ij}+F_{\rm ej}\sin\alpha_{\rm ej}+F_{cj}=0\\ F_{\rm ej}+F_{\rm ij}-2M_{\rm gj}/D=0\end{array} \right. \tag{13}\]  
Here, an approximate relation between friction forces \(F_{ij}\), \(F_{\rm ej}\) and contact forces \(Q_{\rm ij}\), \(Q_{\rm ej}\), at ball-raceway contacts is applied, as follows:  
\[F_{ij}=\mu_{j}Q_{ij} \tag{14}\]  
\[F_{\rm ej}=\mu_{j}Q_{\rm ej} \tag{15}\]  
where \(\mu_{j}\) is the friction coefficients at ball-raceway contacts. The centrifugal force and gyroscopic moment for the \(j\)th ball element can be written, respectively, as follows:  
\[F_{cj}=0.5d_{\rm m}m\omega_{cj}^{2} \tag{16}\]  
\[M_{\rm gj}=J\omega_{\rm bj}\omega_{\rm cj}\sin\beta_{j} \tag{17}\]where \(m\) is the mass of the ball element and \(J\) is the mass moment of inertia:  
\[m=\rho\pi D^{3}/6 \tag{18}\]  
\[J=\rho\pi D^{5}/60 \tag{19}\]  
Based on [PERSON]'s work, the ball-raceway contact forces can be related to the elastic contact deformation presented in Equations (11) and (12). \[Q_{ij/ej}=K_{i/e}\varrho_{ij/ej}^{1.5} \tag{20}\]  
in which \(K_{i}\), \(K_{e}\) are the contact stiffness coefficients and can be calculated based on [PERSON] and [PERSON] work [17]. The bearing loads applied on the inner ring from the shaft include the translational forces \(F_{\mathrm{x}}\), \(F_{\mathrm{y}}\), and \(F_{\mathrm{z}}\) and moments \(M_{\mathrm{x}}\), \(M_{\mathrm{y}}\). The total bearing loads should be balanced by contact forces and friction forces between ball elements and the inner raceway, and the equilibrium relations of the inner ring can be presented as:  
\[\left\{