## 2 Shaft-Bearing System Model  
### Quasi-Static Model of Angular Contact Ball Bearing  
The scheme diagram of a shaft supported by multiple ACBBs is shown in Figure 1. \(O\)-\(xyz\) denotes the inertial coordinate system where the \(z\)-axis is coincident with the shaft axis, and \(O^{k}\)-\(x^{k}y^{k}z^{k}\) denotes the local coordinate system for the \(k\)th bearing in which the forward direction of the \(z^{k}\)-axis is defined as from the small side of the bearing to its big side. Here, the quasi-static bearing model is presented in the bearing local coordinate system. A ball bearing is taken as an example to analyze the interactions between ball elements and raceways. The kinematics of the ball element is shown in Figure 2. Here, the inner ring is assembled with a shaft that rotates at angular speed \(\omega_{\rm i}\) about the bearing axis while the outer ring is fixed. For a bearing with pitch diameter \(d_{\rm m}\) and ball diameter \(D\), one can get the angular speed \(\omega_{\rm c}\) at which the ball element orbits around the bearing axis and the spinning speed \(\omega_{\rm b}\) at which the ball element rotates around its own axis as given in Equations (1) and (2), respectively. \[\omega_{\rm c}=\omega_{\rm i}\frac{(1-\gamma_{\rm i})\cos(\alpha_{\rm e}-\beta )}{(1+\gamma_{\rm e})\cos(\alpha_{\rm i}-\beta)+(1-\gamma_{\rm i})\cos(\alpha_ {\rm e}-\beta)} \tag{1}\]  
\[\omega_{\rm b}=\omega_{\rm i}\frac{d_{\rm m}}{D}\frac{(1-\gamma_{\rm i})(1+ \gamma_{\rm e})}{(1+\gamma_{\rm e})\cos(\alpha_{\rm i}-\beta)+(1-\gamma_{\rm i })\cos(\alpha_{\rm e}-\beta)} \tag{2}\]  
where \(\alpha_{\rm i}\) and \(\alpha_{\rm e}\) are the contact angles between the ball and inner/outer raceway. \(\gamma_{\rm i}\) and \(\gamma_{\rm e}\) are equal to \(D\cos\alpha_{\rm i}/d_{\rm m}\) and \(D\cos\alpha_{\rm e}/d_{\rm m}\), respectively.\(\beta\) is the ball pitch angle. Based on d'Alembert's inertia force principle, [PERSON] [16] derived the following pitch angle formula as shown in Equation (3). \[\tan\beta=\frac{\frac{M_{\rm bi}^{\rm S}}{M_{\rm bi}^{\rm S}}\Big{(}\frac{1+ \gamma_{\rm e}}{1-\gamma_{\rm i}}+1\Big{)}\sin\alpha_{\rm i}+2\sin\alpha_{\rm e}} {\frac{M_{\rm bi}^{\rm S}}{M_{\rm bi}^{\rm S}}\Big{(}\frac{1+\gamma_{\rm e}}{ 1-\gamma_{\rm i}}+1\Big{)}\cos\alpha_{\rm i}+2(\cos\alpha_{\rm e}+\gamma^{ \prime})+\gamma^{\prime}\cdot\frac{M_{\rm bi}^{\rm S}}{M_{\rm bi}^{\rm S}} \Big{(}\cos(\alpha_{\rm i}-\alpha_{\rm e})-\frac{1+\gamma_{\rm e}}{1-\gamma_ {\rm i}}\Big{)}} \tag{3}\]  
in which \(\gamma^{\prime}\) is equal to \(D/d_{\rm m}\), \(M_{\rm bi}^{\rm S}\) and \(M_{\rm be}^{\rm S}\) are the friction moments for the ball and inner/outer raceways, respectively, and:  
\[\frac{M_{\rm bi}^{\rm S}}{M_{\rm be}^{\rm S}}=\frac{Q_{\rm i}a_{\rm i}L_{2}( \kappa_{\rm i})}{Q_{\rm e}a_{\rm e}L_{2}(\kappa_{\rm e})} \tag{4}\]  
where \(Q_{\rm i}\) and \(Q_{\rm e}\) are the contact forces between the ball and inner/outer raceway. \(a_{\rm i}\) and \(a_{\rm e}\) are the semi-major axes of the contact ellipses for the ball and inner/outer raceways, respectively. \(L_{2}(\cdot)\) is the second kind of elliptic integral function. \(\kappa_{\rm i}\) and \(\kappa_{\rm e}\) are the ratios of the semi-major axis to the semi-minor axis of the contact ellipses at the ball inner and ball outer raceway contacts, respectively. The detailed derivation of the pitch angle is omitted here and can be found in [PERSON]'s work [16]. The contact angle and contact elastic deformation between the ball element and raceways are shown in Figure 3. When the external loads are applied on the static bearing, the ball center O and raceway groove curvature center O\({}_{\text{e}}\), O\({}_{\text{i}}\) are collinear. These three points are no longer collinear due to the centrifugal force of the ball element when the rotating speed is imposed.