## 4 Effect of Fault Size Extension on Bearing Dynamic Characteristics
### Variation in Rolling Body Rotational Angular Velocity with Different Failure Dimensions  
During the operation of rolling bearings, the contact between the cage and the rolling elements is discontinuous. When the rolling elements rotate around the bearing's centroid in the load zone and non-load zone, their revolution angular velocities differ. To investigate the impact of different fault sizes on the change in the revolution angular velocity of the rolling elements as they pass through the fault position, this study divides the rotational process of the rolling elements in the load zone into three stages, Stage I to II, Stage II, and Stage II to III. The angular range for these three stages is \(-\pi/6\) to \(\pi/6\), as detailed in Figure 5.  
Selecting one rolling element, the angular velocity during its transition from position I through position II to position III was measured. Figure 6 illustrates the variation in angular velocity for bearings made of GCr15 and AISI304 materials, comparing fault-free  
Figure 5: Schematic of rotation angle range. conditions and five different degrees of faults, clearly showing the portions where the rolling element speed fluctuations exceed 5% under different fault severities. From Figure 6, it can be observed that the angular velocity of fault-free bearings made from both materials fluctuates around the theoretical values as the rolling element moves from position I through position II to position III, with the maximum fluctuation amplitude remaining within 3%. The impacts on the rolling element's angular velocity are attributed to intermittent collisions between the rolling elements and the cage. For the GCr15 material, fault types 1-3 show that during the transition from positions I to II and II to III, the rotational speeds near positions I and III remain within a normal fluctuation range of less than 5% of the theoretical values. However, the speeds near position II exhibit significant impacts, with maximum values exceeding 5% of the theoretical values. In contrast, for the AISI304 material, the rotational speeds of fault types 1-3 remain  
Figure 6: The speed changes when the rolling element passes the fault point. (**a**) Material GCr15; (**b**) material AISI304. within a normal fluctuation range, with maximum impact values not exceeding 5% of the theoretical values. Additionally, both materials show an increasing trend in rotational speed fluctuation as the fault size increases. For fault types 4-5, the rotational speeds of the rolling elements for both GCr15 and AISI304 materials exhibit severe fluctuations during the transitions from positions I to II and II to III, with speeds consistently exceeding 5% of the theoretical values. Notably, the speed fluctuations for GCr15 are greater during the transition from position II to III compared to that from position I to II, while for AISI304, the fluctuations during position II to III are smaller than those from position I to II. In both materials, significant impacts on speed occur at position II, far exceeding the theoretical values by more than 5%. This phenomenon is attributed to macro-slipping of the rolling elements at position II for fault types 4 and 5. During the operation of the fault-free bearing and bearings with varying degrees of outer ring faults, the trend of the sliding rate of the rolling elements in the load zone is shown in Figure 7. The formula for calculating the sliding rate is as follows:  
\[\mathrm{S}=\frac{w_{o}-w}{w_{o}} \tag{15}\]  
where \(w\) is the simulated speed and \(w_{o}\) is the theoretical speed. Figure 7 clearly shows that as the fault size increases, the slip rate of bearings made from both materials exhibits a gradual upward trend. In the case of fault-free bearings and outer ring fault types 1-3, the slip rates are relatively low, remaining below 2%.