# Research on Dynamic Modeling and Vibration Characterization of Integrated Bearings
## 1 Dynamic Modeling of Integrated Bearing Systems  
### Specialized Structure of Integrated Bearings  
In radar motors, rolling bearings are not only required to have a long service life but also to meet stringent vibration and noise criteria. The noise generated during bearing operation is a result of vibrations propagating as sound waves through the surrounding medium, thereby establishing vibration as the primary source of noise. The factors influencing bearing vibration are multifaceted, encompassing structural parameters, operating conditions, and lubrication, making the underlying mechanisms highly complex. Research on this specialized structure within the bearing field is limited. Therefore, it is crucial to undertake research on integrated bearings. Integrated bearings, which feature a unique design devoid of an inner ring, incorporate the inner race into the shaft itself. This configuration enables them to exhibit superior vibration and noise characteristics compared to conventional bearings. The structural and operating parameters of bearings are critical factors influencing their vibration characteristics. Researchers have also conducted studieson dynamic models of rolling bearings [1; 2; 3; 4] and the influence of structural parameters on vibration [5; 6]. About how bearing structural parameters affect bearing vibration, [PERSON] et al. [7] simulated local defects in the inner and outer raceways using rectangular, triangular, and half-sine functions with specified widths. They investigated the vibration responses of bearings under pure axial and pure radial load conditions. [PERSON] et al. [8] examined how the radial clearance in rolling bearings affects the vibration of the aero-engine integrated system. Finite element methods were utilized to model both the bearing-rotor and casing systems. The study examined how radial clearance in bearings affects contact forces and overall vibrations under typical support conditions. [PERSON] et al. [9] created a dynamic model for a rotor system with deep-groove ball bearings, examining how bearing clearance and inner race acceleration influence the rotor systems' dynamics. [PERSON] et al. [10] examined how radial and axial waviness in raceways, ball waviness, and uneven cage pocket hole distribution affect bearing-rotor system vibrations. [PERSON] et al. [11] studied the effects of the number of balls and axial preload on bearing vibrations based on raceway control theory. The research indicates that appropriately applying axial preload and selecting the appropriate number of balls can effectively reduce bearing vibrations. [PERSON] et al. [12] proposed a dynamic model for deep-groove ball bearings that includes the surface waviness of bearing components. They analyzed the effects of various structural parameters, operating conditions, and harmonic parameters on bearing vibrations. [PERSON] et al. [5] developed a nonlinear vibration model for bearings with six degrees of freedom, examining how clearance and load influence their vibration characteristics. [PERSON] et al. [13] performed a theoretical and experimental analysis on how radial load affects the vibration amplitude of deep-groove ball bearings under oil lubrication. [PERSON] et al. [14] developed a dynamic model for a bearing-rotor system with inner race misalignment, examining how misalignment angle, radial clearance, and rotational speed influence system dynamics. [PERSON] et al. [15] developed nonlinear dynamic equations for a deep groove ball bearing featuring a crowned cage. A variable step size predictor-corrector method was used to solve the equations, and the impact of rotational speed on bearing vibrations was analyzed. [PERSON] et al. [16] established a dynamic model of the deep-groove ball bearing-rotor system to investigate the effects of radial load and bearing clearance on the vibration characteristics of the system, considering misalignment in the bearings. About the establishment of bearing dynamics modeling, [PERSON] et al.