# Effect of the Lubrication Parameters on the Ceramic Ball Bearing Vibration in Starved Conditions
## Dynamic Model
### Model of the Ball at Starved Lubricated Conditions
\(K_{\text{rij}}\) refers to the radial component of the comprehensive stiffness between the ball and the inner ring, and \(K_{\text{aij}}\) refers to the axial component of the comprehensive stiffness between the ball and the inner ring. \[K_{\text{rij}}=\frac{\left(K_{oil}\right)_{ij}\times\left(K_{\text{c}}\right) _{ij}}{\left(K_{oil}\right)_{ij}+\left(K_{\text{c}}\right)_{ij}}\times\cos^{ 2}\alpha_{\text{ij}} \tag{39}\]  
Figure 4:0  
Figure 4:1  
Figure 4:2  
where \((K_{oil})_{ij}\) is the oil film stiffness of the \(j\)th ball and the inner ring and \((K_{\text{c}})_{ij}\) is the contact stiffness between the \(j\)th ball and the inner ring. Bearing axial stiffness \(K_{\text{a}}\) and radial stiffness \(K_{\text{r}}\) can be expressed as:  
Figure 4:3  
Figure 4:4  
The natural frequency of the bearing vibration can be expressed as:  
Figure 4:5  
Figure 4:6  
where \(M\) is the mass of the bearing system. When the thickness of the oil film was small, for example, \(L=1.1\), some balls could nor contact the raceway, and the contact stiffness \(K_{\text{c}}\) was 0. According to Equations (39)-(42), this shows that \(K_{\text{rij}}\), \(K_{\text{aij}}\), \(K_{\text{rej}}\), and \(K_{\text{aej}}\) were also 0. Therefore, by calculating Equations (43) and (44), we can get the total stiffness. When the thickness of the oil film was large, for example, \(L=1.9\), all the balls could contact the raceway. By calculating (41) and (42), this shows that the stiffness at this time was greater than the stiffness at \(L=1.1\). According to Equations (45) and (46) that the greater the stiffness, the greater the natural frequency of the bearing. This phenomenon can be seen in Figures 6-10. It can also be seen from Figures 6-10 that the bearing vibration natural frequency increased with radial load. As a result, the increase in load increased the contact force between the ball and the raceway. As a result, the minimum oil film thickness was reduced, and the oil film stiffness increased. Assuming the rotation speed of the inner ring was 15,000 r/min, the axial force (\(F_{x}\)) was 10 N, 20 N, and 30 N, respectively, and the radial force (\(F_{z}\)) was 0 N. Figures 11-15 show the relationship between the axial load of the bearing and the vibration of the bearing when the dimensionless inlet distance \(L\) was different. Figure 4:7 The bearing vibration under different axial loads when \(L=1.1\). Figure 4:8 The bearing vibration under different axial loads when \(L=1.3\). Figure 4:9 The bearing vibration under different axial loads when \(L=1.5\). It can be observed from Figures 11-15 that as the amount of oil increased, the amplitude of bearing vibration significantly reduced. The dominant frequency of the bearing increased as the oil quantity increased. These characteristics were the same as the changes in Figures 6-10. As shown, during the increase, the bearing vibration decreased with a certain amount of force. According to Equation (1), the force generated in the bearing could be balanced when by applying a certain axial load, and the vibration of the bearing reduced. When the load continued to increase, it increased the bearing vibration. Therefore, for bearings with a given structure, there was a reasonable range of axial loads that made the bearings vibrate less during operation. Figures 16 and 17 show the relationship between the inlet distance and the peak vibration frequency \(f_{p}\) of the bearing under different working conditions. Figure 5:0 Peak vibration frequency with different radial loads. Figure 5:1 The bearing vibration under different axial loads when \(L\) = 1.7. Figure 5:2 The bearing vibration under different axial loads when \(L\) = 1.9. As the oil increased, the contact stiffness of the ball with the inner ring increased, resulting in an increase in the natural frequency of the bearing. When the amount of oil increased and \(L>L^{*}\), the oil film thickness did not change substantially, and the influence of the increase of the oil quantity on the rigidity became small.