# A Theoretical Model with the Effect of Cracks in the Local Spalling of Full Ceramic Ball Bearings
## 2 Nonlinear Multi-Body Dynamic Modelling of a Defective Bearing
### Modified Time-Varying Stiffness of Local Spalling Faults  
Unlike the steel outer ring, the surface of the ceramic outer ring will not undergo obvious local deformation when full ceramic bearings produce local spalling failure. Therefore, it is not suitable for the traditional Hertz contact theory in the area where the rolling element contacts spalling, and the contact stiffness value in this area needs to be revised. As can be seen in Figure 2, when the rolling element operates within the spalling fault range (the rolling element enters the edge and does not come into contact with another spalling edge), the position of the rolling element is \(\beta_{\text{en}}<\beta<\beta_{\text{f}}\). The new  
Figure 1: Load distribution diagram of outer ring during bearing operation.  
contact stiffness \(K_{\rm o1}\) is obtained by modifying the contact stiffness \(K_{\rm o}\) between the outer rings, and the outer rings can be obtained by:  
\[K_{\rm o1}=K_{\rm o}\bigg{[}1-R\bigg{(}1+\bigg{(}\frac{\beta_{\rm en}-\beta}{ \beta_{\rm en}}\bigg{)}\bigg{)}^{\rm S}\bigg{]} \tag{6}\]  
where \(K_{\rm o}\) is the contact stiffness between rolling element and outer ring--its calculation formula can be found in reference [35]--\(\beta_{\rm f}\) is the angular position of spalling fault center, \(R\) is the reduction factor, the reduced ratio of load-deformation contacts in the range \(\beta_{\rm en}<\beta<\beta_{\rm f}\), of which the parameter is usually expressed as a percentage, and \(R>50\%\) is taken; \(S\) is the shape factor and can be any value.  
As the rolling element operates within the range of spalling faults, the corresponding \(K_{\rm o1}\) values differ at different locations. When the rolling element is in a spalling fault (contacting both edges of the spalling), the position of the rolling element is \(\beta=\beta_{\rm f}\), and the modified contact stiffness between the rolling element and the outer ring of the ceramic bearing can be expressed as:  
\[K_{\rm o1}=K_{\rm o}(1-R). \tag{7}\]  
As that rolling element leaves the spalling area, the rolling element is restored to contact only one side of the spalling fault position, where the position of the rolling element is \(\beta_{\rm f}<\beta<\beta_{\rm ex}\), and the modified contact stiffness can be expressed as:  
\[K_{\rm o1}=K_{\rm o}\bigg{[}1-R\bigg{(}1+\bigg{(}\frac{\beta-\beta_{ex}}{ \beta_{ex}}\bigg{)}\bigg{)}^{\rm S}\bigg{]}. \tag{8}\]  
To sum up, by modifying the value \(K_{\rm o1}\) in different states, the time-varying stiffness of the outer ring can be modified to make the load-deformation relationship more consistent with the actual situation of the contact between the rolling element and the ring, and also make the subsequent modeling more authentic.  
According to the literature, the total contact stiffness \(K\) of the bearings can be obtained, and the calculation formula is expressed as follows:  
\[K=\Bigg{[}\frac{1}{\left(1/K_{\rm i}\right)^{2/3}+\left(1/K_{\rm o1}\right)^{ 2/3}}\Bigg{]}^{3/2} \tag{9}\]  
Figure 2: Angular diagram of the position of the rolling elements passing through the bearing outer ring spalling.  
where \(K_{\rm i}\) is the contact stiffness between the rolling element and the inner ring; the calculation formula of \(K_{\rm i}\) can be referred to reference [35].  
According to the Hertz contact theory, the contact deformation between the rolling element and the outer rings can be obtained when the rolling element passes through the stripping fault region:  
\[\delta_{\rm os}=\left(\frac{Q_{\varphi}}{K_{0\rm i}}\right)^{-3/2}. \tag{10}\]