## 1 Introduction  
As a critical component in rotating machinery, rolling bearings directly influence the operational safety and reliability of the entire mechanical system. However, rolling bearings are prone to various faults, e.g., pitting, wear, cracks, etc., due to their prolonged operation in complex conditions such as high speeds and heavy loads. These abnormalities will lead to equipment downtime and even cause safety incidents if not detected and handled in time [1, 2]. Furthermore, rolling bearings' dynamic response characteristics are critical to the operational reliability of the machinery on which they are installed. Especially when a bearing continues to operate after a minor fault occurs, the ongoing expansion of the fault size poses a significant threat to the stable operation of the equipment. In previous studies, scholars established dynamic models to analyze the fault characteristics of rolling bearings. For instance, [PERSON] et al. [3] developed dynamic models to analyze the vibration characteristics of spindle-bearing systems considering pedestal tilt due to assembly errors. [PERSON] et al. [4] developed a four-degrees-of-freedom bearing dynamic model based on time-varying displacement to investigate the dynamic characteristics of bearings with localized faults under different conditions. [PERSON] et al. [5] studied the sliding mechanism of angular contact ball bearings from the perspective of rotor characteristics, discovering resonance sliding phenomena and examining the effect of foreign objects on cage slip rates. [PERSON] et al. [6] used the GStiff algorithm to solve the nonlinear dynamic differential equations of cylindrical roller bearings under dynamic imbalance and provided an effective evaluation of the nonlinear dynamic response and stability of the cage. [PERSON] et al. [7] established a dynamic model of bearings to study the effect of raceway defects on the nonlinear dynamics of bearings. [PERSON] et al. [8] analyzed the vibration response of rolling bearings with localized defects subjected to external dynamic loads. [PERSON] [PERSON] et al. [9] simulated the vibration response signals of the bearing dynamics model under different failure degrees by changing the defect parameters of the outer ring of the bearing, analyzed the relationship between the fault size and the vibration response, and revealed the vibration response mechanism of the bearing under different failure degrees. However, the above studies primarily use dynamic methods to analyze the dynamic characteristics of faulty bearings under various conditions, with limited focus on the relationship between these characteristics and crack propagation of the bearing. For the research of bearing crack propagation, [PERSON] et al. [10] established a dynamic model of an axlebox bearing cage. On this basis, the dynamic characteristics of the cage with cracks and the variation patterns of contact force are investigated. [PERSON] et al. [11] developed a finite element model to investigate the impact of crack depth and inclination angle on bearing contact deformation, contact width, and stiffness. Also, they determined the relationship between contact characteristics and crack size. After that, an improved sub-model was introduced by [PERSON] et al. [12] to propose a crack-initiated contact finite element model. Meanwhile, FRANC3D is employed to embed the crack for analyzing the distribution and extension characteristics of the stress intensity factor at the crack front. [PERSON] et al. [13] developed a finite element model to analyze the effect of horizontal and inclined subsurface cracks in roller bearings on contact characteristics, and they explored the relationship between contact deformation, crack size, and contact area width. Furthermore, a two-degrees-of-freedom nonlinear dynamics model was used as a basis for evaluating the effect of cracks on bearings through the variation in strain energy in the literature [14].