## 2 Numerical Modelling
### Numerical Simulation Implementation and Analysis  
To concentrate the study on the influences of bearing clearance on vibration responses, the model was based on the following assumptions and considerations. 1. The vibration model is developed for an ideal bearing under perfect operating conditions, i.e., geometric errors and assembly errors are ignored;
2. The bearing operates under isothermal conditions, i.e., the influence of temperature is not considered;
3. The lubrication is sufficient and appropriate during the operation at a constant speed, i.e., abnormal lubrication and raceway roughness and waviness are not taken into account;
4. The motion between raceways and balls is regarded as pure rolling without any sliding and skidding between bearing components;
5. The sensor is considered as a mass-damping system. Taking 6206 deep groove ball bearing as an example, the main geometry parameters of the adopted 6206 deep groove ball bearing are listed in Table 1. The model parameters for simulation are listed in Table 2. It was noted that the increase in clearances was realized through the variation of raceway diameter. In addition, the rotational speed of the shaft was 1500 rpm and the vibration model was iteratively solved in MATLAB 2018a with the sub-function of ode15 s.  
Taking the load of 1600 N and the clearance of 10 um as an example, the vibration acceleration, spectrum and envelope spectrum of the sensor in load and non-load direction are shown in Figure 3. As can be seen, the waveform displayed periodic impulses and two peaked regions could be clearly distinguished from the spectrum in both directions. Through envelope analysis, it was found that the ball pass frequency of outer race (BPFO) and its harmonics dominated the entire spectrum. Coincidentally, the amplitude of the non-load direction was higher than the load direction. The higher vibration level in the non-load direction could be attributed to the more complicated motion of the bearing system and the influence of dynamic force. \begin{table}
\begin{tabular}{c c c} \hline
**Notation** & **Description** & **Value** \\ \hline \(d\) & Nominal bore diameter (mm) & 30 \\ \(D\) & Nominal outside diameter (mm) & 62 \\ \(d_{i}\) & Inner race diameter (mm) & 37.48 \\ \(d_{o}\) & Outer race diameter (mm) & 56.45 \\ \(d_{m}\) & Pitch diameter (mm) & 46.96 \\ \(d_{b}\) & Ball diameter (mm) & 9.485 \\ \(d_{b}\) & Original radial clearance (μm) & 0 \\ \(N_{b}\) & Number of rollers & 9 \\ \(\alpha\) & Contact angle (\({}^{\circ}\)) & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Main geometry parameters of deep groove ball bearing 6206. \begin{table}
\begin{tabular}{c c c} \hline
**Notation** & **Description** & **Value** \\ \hline \(m_{s}\) & Mass of shaft (kg) & 1.32 \\ \(m_{h}\) & Mass of house (kg) & 0.46 \\ \(m_{r}\) & Mass of sensor (kg) & 0.02 \\ \(m_{b}\) & Mass of each ball (kg) & \(2.95\times 10^{-3}\) \\ \(K_{l}\) & Stiffness of house (N/m) & \(1.2\times 10^{9}\) \\ \(K_{r}\) & Stiffness of sensor (N/m) & \(4\times 10^{7}\) \\ \(C_{h}\) & Damping of house & 939.62 \\ \(C_{r}\) & Damping of sensor & 17.89 \\ \hline \end{tabular}
\end{table} 
Table 2: Physical properties for simulation study. Without loss of generality, dynamic force of the first ball was calculated according to Equation (15). Figure 4a,b depict the spring and damping force, respectively. In Figure 4, \(F_{51x}\) and \(F_{51y}\) represent the spring force in the load direction and non-load direction of the first ball, respectively; \(F_{d1x}\) and \(F_{d1y}\) denote the damping force in the load direction and non-load direction of the first ball, respectively; \(F_{s1}\), \(F_{d1}\) and \(F_{r1}\) denote the damping force, damping force and resultant force of the first ball, respectively; \(F_{r}\) represents the resultant force of all balls. As can be seen, although there was no external load in the non-load direction, due to the difference in the contact angle of each element, the force in the non-load direction fluctuated as well.