## 2 Description of Ball Bearing VC Vibration  
Considering the nonlinearities of VC parametric excitation, bearing clearance, and the Hertzian contact between the rolling elements and raceway, a classic two degrees-of-freedom ball bearing model is studied here. This model has been verified for qualitative and quantitative analysis of the response characteristics of radial VC motions in ball bearings [14; 15; 16; 24], and the JIS6306 specification shown in Table 1 is adopted in this article. As shown in Figure 1, the instant angular location of the \(i\)th ball is denoted as:  
\[\theta_{i}=2\pi(i-1)/N_{b}+\Omega t \tag{1}\]  
with ball passing frequency (i.e., the cage angular velocity) of:  
\[\Omega=\omega_{S}(1-D_{b}/D_{h})/2 \tag{2}\]  
Then, the contact deformation between the \(i\)th ball and bearing raceways is expressed as:  
\[\delta_{i}=x\cos\theta_{i}+y\sin\theta_{i}-\delta_{0} \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline
**Item** & **Value** \\ \hline Contact stiffness \(C_{b}\) (N/m3/2) & \(1.334\times 10^{10}\) \\ Ball diameter \(D_{b}\) (mm) & 11.9062 \\ Pitch diameter \(D_{h}\) (mm) & 52.0 \\ Number of balls \(N_{b}\) & 8 \\ Equivalent mass \(m\) (kg) & 20 \\ Damping factor \(c\) (Ns/m) & 200 \\ Radial load \(W\) (N) & 196 \\ \hline \end{tabular}
\end{table} 
Table 1: Specifications of JIS6306 ball bearing [22]. where \(\omega_{\rm s}\) and \(2\delta_{0}\) denote the rotor shaft speed and the radial internal bearing clearance, respectively. Assuming only Hertzian contact deformations are considered, the nonlinear restoring forces \(F_{x}\) and \(F_{y}\) are:  
\[\left[ \begin{array}{c}F_{x}\\ F_{y}\end{array} \right]=C_{\rm b}\sum_{i=1}^{N_{\rm b}}\left(\delta_{i}G[\delta _{i}]\right)^{1.5}\left[ \begin{array}{c}\cos\theta_{i}\\ \sin\theta_{i}\end{array} \right] \tag{4}\]  
where \(G[\cdot]\) is a Heaviside function, taking 1 if \(\delta_{i}\geq 0\) represents the condition of contact, or else taking 0, which denotes the contact loss between the rolling element and raceway. The VC motion of a rigid Jeffcott rotor-ball bearing system can be expressed as:  
\[m\left[ \begin{array}{c}\ddot{x}\\ \ddot{y}\end{array} \right]+c\left[ \begin{array}{c}\dot{x}\\ \dot{y}\end{array} \right]+\left[ \begin{array}{c}F_{x}\\ F_{y}\end{array} \right]=\left[ \begin{array}{c}W\\ 0\end{array} \right] \tag{5}\]  
Due to the Hertzian contact deformation relationship, the system has hard spring characteristics [44; 45; 46]. The approximate linearized static stiffness of System (5) in the direction of static load \(W_{s}\) can be expressed as [46]:  
\[k_{W{\rm s}}=\frac{dW_{s}}{d\delta_{W{\rm s}}}=\frac{1.5W_{s}}{\delta_{W{\rm s }}} \tag{6}\]  
where \(\delta_{W{\rm s}}\) denotes the total deformation along the direction of \(W_{s}\). Equation (6) is clearly not applicable for the estimation of the dynamic stiffness of a ball bearing system during operation. The dynamic stiffness characteristics of rolling bearings have a significant impact on the nonlinear response characteristics of their supporting systems [12; 22; 26]. According to the restoring forces of Equation (4), the radial linearized dynamic stiffness of the system can be approximated as [15]:  
\[\left[ \begin{array}{c}k_{xx}(t)\\ k_{yy}(t)\end{array} \right]=\left[ \begin{array}{c}\partial F_{x}(t)/ \partial x\\ \partial F_{y}(t)/\partial y\end{array} \right] \tag{7}\]  
Then, the dynamic resonant frequencies of the system can be expressed as:  
\[\left[ \begin{array}{c}\omega_{xx}(t)\\ \omega_{yy}(t)\end{array} \right]=\left[ \begin{array}{c}\sqrt{k_{xx}(t)/m}\\ \sqrt{k_{yy}(t)/m}\end{array} \right] \tag{8}\]  
and the equivalent resonant frequencies \(\omega_{x0}\), \(\omega_{y0}\) can be estimated from the arithmetic mean of Equation (8) [22].