## 2 Numerical Modelling  
### Bearing Vibration Model Considering Clearance  
To study the dynamic behaviors of bearings, a large number of works have focused on bearing models for bearing vibration characteristics or diagnostic signatures. [PERSON] [27] first proposed a two-DOF mathematical bearing vibrations model based on Hertz contact theory. [PERSON] et al. [28] considered the shaft-bearing assembly as a mass-spring system to investigate the nonlinear characteristic of rolling element vibrations. [PERSON] and [PERSON] [29] studied the influence of pedestal and proposed a five-DOF bearing-pedestal model to simulate localized bearing faults. [PERSON] et al. [30] investigated the vibration transmission characteristics through a dynamic model of the gear-shaft-bearing-housing system. [PERSON] and [PERSON] [31] took housing and shaft into consideration and proposed a lumped spring mass model for a ball bearing system to study the localized defect sizes on the vibration characteristics. Through these literatures, the shaft-bearing-housing model has better performance when describing bearing vibrations, and it is widely used in bearing studies. Thus, in this study, a lumped shaft-housing vibration model is developed with six DOF, which includes two sensors, as shown in Figure 1. In this shaft-housing system, \(M_{s}\) includes the mass of inner race and shaft and \(M_{h}\) includes the mass of outer race and house, which is a common configuration of bearing operations. Moreover, it allows the numerical simulations to be evaluated by corresponding experiments. In this system, \(X_{s}\), \(Y_{s}\), \(X_{h}\), \(Y_{h}\), \(X_{r}\) and \(Y_{r}\) represent the displacement of the shaft (s), housing (h) and sensor (r) in the \(X\) and \(Y\) directions, respectively. \(\dot{X}\) and \(\ddot{X}\) denote velocity and acceleration, respectively. The bearing vibration model is presented in Equations (1)-(6). \[M_{s}\ddot{X}_{s}+\sum\
olimits_{i=1}^{N_{b}}K[\delta_{i}]^{3/2}cos\phi_{i}+ \sum\
olimits_{i=1}^{N_{b}}C[v_{i}]cos\phi_{i}=F, \tag{1}\]  
\[M_{s}\ddot{Y}_{s}+\sum\
olimits_{i=1}^{N_{b}}K[\delta_{i}]^{3/2}sin\phi_{i}+ \sum\
olimits_{i=1}^{N_{b}}C[v_{i}]sin\phi_{i}=0, \tag{2}\]  
\[M_{h}\ddot{X}_{h}+C_{h}\dot{X}_{h}+K_{h}X_{h}-\sum\
olimits_{i=1}^{N_{b}}K[ \delta_{i}]^{3/2}cos\phi_{i}-\sum\
olimits_{i=1}^{N_{b}}C[v_{i}]cos\phi_{i}=0, \tag{3}\]  
\[M_{h}\ddot{Y}_{h}+C_{h}\dot{Y}_{h}+K_{h}Y_{h}-\sum\
olimits_{i=1}^{N_{b}}K[ \delta_{i}]^{3/2}sin\phi_{i}-\sum\
olimits_{i=1}^{N_{b}}C[v_{i}]sin\phi_{i}=0, \tag{4}\]  
\[M_{r}\ddot{X}_{r}+C_{r}\left(\dot{X}_{r}-\dot{X}_{h}\right)+K_{r}(X_{r}-X_{h}) =0, \tag{5}\]  
\[M_{r}\ddot{Y}_{r}+C_{r}\left(\dot{Y}_{r}-\dot{Y}_{h}\right)+K_{r}(Y_{r}-Y_{h}) =0, \tag{6}\]  
where \(M_{s}\), \(M_{h}\) and \(M_{r}\) denote the mass of shaft, housing and sensor, respectively. In addition, \(K_{h}\), and \(K_{r}\) represent the stiffness of housing and sensor; \(C_{h}\) and \(C_{r}\) are the damping of housing and sensor; \(K\) and \(C\), respectively, represent the stiffness and damping between raceways and balls; \(\delta_{i}\) and \(v_{i}\) represent the nonlinear deformation and nonlinear velocity of the \(i\)th ball, respectively; \(\phi_{i}\) denotes the angle position of the \(i\)th ball; and \(N_{b}\) denotes the number of rolling elements. The deformation \(\delta_{i}\) and velocity \(v_{i}\) for each element could be calculated by Equations (7) and (8)  
\[\delta_{i}=\left\{ \begin{array}{c}(X_{s}-X_{h})cos\phi_{i}+(Y_{s}-Y_{h})sin \phi_{i}-c/2\times(1-cos\phi_{i})\ \delta_{i}>0,\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \delta_{i}\leq 0,\end{array} \right. \tag{7}\]  
\[v_{i}=\left\{ \begin{array}{c}(\dot{X}_{s}-\dot{X}_{h})cos\phi_{i}+\left( \dot{X}_{s}-\dot{X}_{h}\right)sin\phi_{i}\ \delta_{i}>0,\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \delta_{i}\leq 0.\end{array} \right. \tag{8}\]  
where \(c\) denotes radial clearance.