Rolling bearings are precision components widely used in rotating machinery. They convert sliding friction between rotating and stationary components into rolling friction and serve to support loads and guide motion [3] . The basic structure consists of four components: inner ring, outer ring, rolling elements, and cage, as shown in Figure 1 . Here, r₁ and r₂ represent the inner and outer ring raceway radii, respectively; d is the diameter of the rolling elements; D is the bearing pitch diameter; and $\alpha$ is the contact angle.

Under normal circumstances, the outer ring of a rolling bearing is fixed to the bearing seat hole, while the inner ring is fixed to the shaft and rotates with it. The rolling elements roll within the cavity formed by the inner and outer rings to maintain relative motion between them. The cage serves to separate the rolling elements, reduce friction and collision, and distribute them uniformly along the circumference, enabling the bearing to bear loads.

This model takes the rolling bearing as the research object. In order to better extract the internal displacement fault signals, the bearing is simplified as a two-degree-of-freedom lumped mass-spring system, as shown in Figure 2 . The rotor rotates at a constant speed along the axis, and the rolling bearing is installed on the bearing housing to support the rotation of the rotor [4] [5] . In Figure 2 , $K_s$, $K_p$, $C_s$ and $C_p$ represent the combined stiffness and damping of the inner and outer rings of the rolling bearing, respectively. The horizontal and vertical vibration displacements of the inner and outer rings are selected as the generalized coordinates of the system.

The following assumptions are made, taking into account the main contact relationships within the system:

1) The inner ring rotates with the shaft, while the outer ring is fixed to the bearing housing.

2) The inertial forces and gyroscopic torques of the rolling elements are neglected.

3) The bearing is well lubricated, and relative sliding occurs only within the load-bearing zone.

4) The thermal effects caused by friction are neglected.

5) Contact friction is neglected.

6) Only isothermal elastic-fluid lubrication is considered.

In the two-degree-of-freedom dynamic model of rolling bearings, the bearing is simplified as a spring-damping system, and the rolling elements are simplified as nonlinear elastic bodies [6] . Based on Newton’s second law, the two-degree-of-freedom nonlinear dynamic equations for angular contact ball bearings can be established as:

$\{\begin{array}{l}m{\ddot{\delta }}_x+C_b{\stackrel{\dot{}}{\delta }}_x+K_b{\delta }_x+F_x=W_x\\ m{\ddot{\delta }}_y+C_b{\stackrel{\dot{}}{\delta }}_y+K_b{\delta }_y+F_y=W_y\end{array}$

In the equation, m is the equivalent mass of the rotating component; $C_b$ is the equivalent damping system of the bearing; $K_b$ is the equivalent stiffness of the bearing; ${\delta }_x$ and ${\delta }_y$ are the composite displacements of the bearing in the x-direction and y-direction, respectively; $F_x$ and $F_y$ are the interaction forces in the x-direction and y-direction within the bearing, respectively; $W_x$ and $W_y$ are the external loads applied to the bearing in the x-direction and y-direction, respectively.

As shown in Figure 3 , the stiffness of the inner and outer rings of a well-lubricated rolling bearing can be considered to consist of two parts: Hertz contact stiffness and oil film stiffness. At the same time, when lubrication is good, a lubricating oil film will form between the rolling elements and the raceway, so the damping part will also consist of two parts: internal damping and oil film damping.

Under load, the normal approach between the two raceways separated by the rolling elements and the lubricating oil film is equal to the sum of the approaches between the rolling elements and each raceway. Therefore, the comprehensive stiffness expression obtained by connecting the inner ring stiffness and the outer ring stiffness in series is:

$K_b={\left[\frac{1}{{\left(1/K_s\right)}^{1/n}+{\left(1/K_p\right)}^{1/n}}\right]}^n$

In the formula, for ball bearings, n = 3/2.

Damping is an important parameter in mechanical systems, mainly serving to reduce vibration and amplitude. According to the definition of damping, the damping of the inner and outer rings is expressed by the following formula:

$\{\begin{array}{l}{C}_s={\left(C_i^{-1}+C_{fi}^{-1}\right)}^{-1}\\ C_p={\left(C_o^{-1}+C_{fo}^{-1}\right)}^{-1}\end{array}$

Further, the comprehensive damping is derived as:

$C_b=\frac{1}{C_s^{-1}+C_p^{-1}}$

Important parameters of rolling bearings include the number of balls $N_b$, ball diameter d, bearing diameter D, and contact angle $\alpha$. Typically, the outer ring of a rolling bearing is fixed to the bearing housing, while the inner ring is fixed to the shaft and rotates with the shaft. The balls roll purely in the raceway. The total contact deformation of the i-th ball is a function of the relative displacement between the inner and outer rings and the angular position ${\phi }_i$ of the i-th ball, specifically expressed as:

${\delta }_i=\left(x_n-x_w\right)\cos {\phi }_i+\left(y_n-y_w\right)\sin {\phi }_i$

In the formula, $i=1,2,\cdots ,N_b$； $x_n$、 $x_w$ are the horizontal displacements of the inner ring and outer ring of the bearing, respectively, ${\delta }_x=x_n-x_w$; $y_n$, $y_w$ are the vertical displacements of the inner ring and outer ring of the bearing, respectively, ${\delta }_y=y_n-y_w$.

Since only the balls in the bearing load zone will experience contact deformation, the following switch function is defined:

${\gamma }_i=\{\begin{array}{l}1{\delta }_i>0\\ 0\text{otherwise}\end{array}$

The angular position of the i-th ball is a function of the time interval $dt$, the initial angular position ${\phi }_0$ of the cage, and the angular velocity ${\omega }_c$ of the cage, expressed as:

${\phi }_i=\frac{2\pi \left(i-1\right)}{N_b}+{\omega }_{c}dt+{\phi }_0$

${\omega }_c=\left(1-\frac{d}{D}\right)\frac{\omega }{2}$

In the formula, $\omega$ is the rotational speed of the axis, $\omega =\frac{2\pi N}{60}$.

In summary, it can be concluded that the total nonlinear contact force of the bearing in the horizontal and vertical directions can be expressed as follows:

$F_x=k_b{\displaystyle \sum {\gamma }_i{\delta }_i^{1.5}\cos {\phi }_i}$

$F_y=k_b{\displaystyle \sum {\gamma }_i{\delta }_i^{1.5}\sin {\phi }_i}$

Bearings that rotate under heavy loads for extended periods of time experience periodic loads at fixed positions, leading to localised fatigue damage. Bearing damage often exhibits a certain width and depth, manifesting as multiple local defects on the raceway surface. The time-varying displacement of rolling bearings during operation may also be altered due to the influence of local defects, as shown in Figure 4 . When the defect is located on the inner raceway surface, the defect will sequentially enter and exit the load-bearing zone as the inner ring rotates; when the defect is located on the outer raceway, its position remains unchanged.

In Figure 4 , the fault width is set to L, the depth is h, the crossing angle is $\Delta {\phi }_d$, and the fault angle position is ${\phi }_d$.

When the ball rolls over a local fault, it releases a certain amount of deformation. Define the switch function ${\beta }_i$ for the occurrence of the fault. At this point, the deformation of the i-th rolling element entering the defect can be expressed as:

${\delta }_i=\left(x_n-x_w\right)\cos {\phi }_i+\left(y_n-y_w\right)\sin {\phi }_i-c-{\beta }_i{c}_d$

${\beta }_i=\{\begin{array}{l}1{\phi }_d<{\phi }_i<{\phi }_d+\Delta {\phi }_d\\ 0\text{otherwise}\end{array}$

Using the above equation, calculate the nonlinear contact force of the bearing. Substitute the nonlinear contact force into the given dynamic differential equation of the bearing to obtain the vibration response of the rolling bearing.

To investigate the fault response, local defect characteristics were added to the bearing dynamics model. The simulation under a 500 N radial load produced a vibration response, which was analyzed in the frequency domain using MATLAB’s ODE solver and the Fourier method [7] .

The resulting signal for a defect size of 0.5 mm wide by 0.25 mm deep is presented in Figure 5 .

As shown in Figure 5(a) , the time-domain plots of horizontal and vertical accelerations within the 0 - 2 s range clearly exhibit oscillatory curves.

Figure 5(b) shows two acceleration pulse signals generated when the rolling body passes over defects 1 and 2 within the 0.2 - 0.25 s time interval. Based on the rotational direction of the rolling body and the time interval between the acceleration signals, it can be determined that the acceleration pulse signals generated when the rolling body passes over defect 1 are A1 and A2, and those generated when passing over defect 2 are B1 and B2. Additionally, the time intervals between pulses A1 and A2, and B1 and B2 are approximately 16.89 ms, while the time interval between B1 and A2 is approximately 8.43 ms.

As shown in Figure 5(c) , by performing envelope analysis on the simulated acceleration signals, the vibration signals in the frequency domain are obtained. Under the initially set fault magnitude, the bearing exhibits a frequency of 89.4727 Hz, which exactly corresponds to the characteristic frequency of an outer ring fault in the bearing. Therefore, based on the demodulated spectrum, it can be determined that the bearing fault is an outer ring fault.

As shown in Figure 5 , the amplitude of the acceleration pulse signals generated when the rolling element rolls over the same defect is almost the same. Additionally, the amplitude of the acceleration pulse signal generated when rolling over defect 2 is greater than that generated when rolling over defect 1. This is because when the rolling element rolls over the two defects sequentially, the acceleration pulse signals A1 and A2 generated when rolling over defect 2 have not yet fully decayed, while the acceleration pulse signals B1 and B2 appear when rolling over defect 1. The undecayed A1 and A2 are superimposed on B1 and B2, respectively, resulting in the amplitudes of pulses B1 and B2 being greater than those of A1 and A2, respectively.

Four different sizes of failures were set using the failure switch function, with outer ring failure sizes of 0.5, 2.0, 3.5, and 5.0 mm, respectively.

From Figure 5 to Figure 8 , it can be seen that as the defect becomes larger, the impact oscillation becomes more and more obvious, and the fault becomes more and more serious, requiring manual replacement to avoid affecting normal actual use.