## 2 Shaft-Bearing System Model
### Shaft Model  
In order to consider shaft flexibility, the finite element method is adopted here, and the shaft is discretized into \(N\) segments containing \(N+1\) nodes using [PERSON]'s beam element, as shown in Figure 5. _O-xyz_ is the inertial coordinate system of the whole shaft-bearing system (mentioned in Section 2.1). The torsional deformation of the shaft is ignored; only the bending and axial deformations are considered. The \(i\)th node on beam element has three translational DOFs \(\delta_{\mathrm{x}}^{i}\), \(\delta_{\mathrm{y}}^{i}\), and \(\delta_{\mathrm{z}}^{i}\) along the  
Figure 4: Forces on the ball element.  
\(x\)-, \(y\)-, and \(z\)-axis, respectively, and two rotational DOFs \(\theta_{x}^{i}\), \(\theta_{y}^{i}\) around the \(x\)- and \(y\)-axis, respectively. In the present shaft-bearing system, the bearing inner ring is close fit with the shaft; therefore, the inner ring of the bearing has the same displacements as the shaft node at the bearing mounting position. The displacement vector at the \(i\)th node location is:  
\[\{\delta_{i}\}=\left[\delta_{x}^{i},\theta_{y}^{i},\delta_{x}^{i},\delta_{y}^ {i},\theta_{x}^{i}\right]^{\mathrm{T}}\!\!i=1,2,\cdots,N+1 \tag{22}\]  
The corresponding load vector is:  
\[\{F_{i}\}=\left\{F_{x}^{i},M_{y}^{i},F_{x}^{i},F_{y}^{i},M_{x}^{i}\right\}^{ \mathrm{T}}\!\!i=1,2,\cdots,N+1 \tag{23}\]  
The stiffness matrix for each beam element can be obtained as follows:  
\[\left[K^{\mathrm{e}}\right]=\frac{E\!\!\!I}{\left(1\!\!\left(1\!\!\left(1\! \!\left(1\!\!\left(1\!\!\left(1\!\!\left(1\!\!\left(1\!\!\left(1\!\!\left(1\!\! \right)\!\!\right)\!\!\right)\!\right)\!\right)\!\right)\!\right)}\right. \tag{24}\]  
where \(g\) is a parameter including the transverse shear effect of beam, and its detailed expression can be found in [18].  
The finite element equilibrium equation of the shaft can be written as:  
\[K\delta=F \tag{25}\]  
where \(K\) is the global stiffness matrix obtained by assembling all the beam element stiffness matrix \(K^{\mathrm{e}}\), \(\delta\) is the node displacement vector of the shaft and can be expressed as \(\{\delta\}=\left\{\delta_{1}^{\mathrm{T}},\delta_{2}^{\mathrm{T}},\cdots, \delta_{i}^{\mathrm{T}},\cdots\delta_{\mathrm{N+1}}^{\mathrm{T}}\right\}^{ \mathrm{T}}\), and \(F\) is the load vector applied on the shaft nodes given as \(\{F\}=\left\{F_{1}^{\mathrm{T}},F_{2}^{\mathrm{T}},\cdots F_{i}^{\mathrm{T}}, \cdots F_{\mathrm{N+1}}^{\mathrm{T}}\right\}^{\mathrm{T}}\). The load vector \(F\) consists of two parts: one part is attributed to the external loads, and the other part is the supporting loads provided by the ball bearings and also the counter force of the bearing loads calculated in Equation (21). Based on the transformation relation between the bearing local coordinate system and the global inertial coordinate system, the displacements of the bearing inner ring in the bearing local coordinate system can be easily represented by the node displacements of the shaft in the inertial coordinate system at the location where the shaft is supported by the ball bearing. Therefore, the supporting loads applied on the shaft nodes provided by the ball bearings can be expressed as a nonlinear function of the node displacements of the shaft.  
Figure 5: Finite element model for the shaft.