*2.1. Model-Based Data Augmentation*

To analyze the structural vibration characteristics of the rolling element bearing, the contact between the outer race and other components can be considered as a spring-mass system, in which the outer race is fixed on a pedestal, and the inner race is fixed with the shaft. The sensor is placed on the pedestal with an outer race to detect high-frequency natural vibrations of the bearing. Thus, to provide the vibration response signals of the rolling bearings containing different working states, a vibration model with four degrees of freedom (DOFs) was constructed by considering the movements in the horizontal and vertical directions of the inner race and outer race, as shown in Figure 2.

**Figure 2.** Vibration model of rolling element bearing.

Considering the vibration of the outer race and inner race in a vertical direction, the dynamic equation of the bearing system can be described as:

$$\begin{array}{l}m\_{s}\ddot{\mathbf{x}}\_{s} + c\_{s}\dot{\mathbf{x}}\_{s} + k\_{s}\mathbf{x}\_{s} + F\_{\mathbf{x}} = \mathbf{0} \\ m\_{s}\ddot{\mathbf{y}}\_{s} + c\_{s}\dot{\mathbf{y}}\_{s} + k\_{s}\mathbf{y}\_{s} + F\_{\mathbf{y}} = F\_{r} \\ m\_{p}\ddot{\mathbf{x}}\_{p} + c\_{p}\dot{\mathbf{x}}\_{p} + k\_{p}\mathbf{x}\_{p} = F\_{\mathbf{x}} \\ m\_{p}\ddot{\mathbf{y}}\_{p} + c\_{p}\dot{\mathbf{y}}\_{p} + k\_{p}\mathbf{y}\_{p} = F\_{\mathbf{y}} \end{array} \tag{1}$$

where *xs* and *xp* denote the displacement of the inner race and outer race in the *x* direction, *ys* and *yp* represent the displacement of two raceways in the *y* direction, accordingly. *Fx* and *Fy* are the elastic contact force between the raceways and the rolling elements in the *x*

and *y* direction, and *Fr* is the radial load generally produced by the weight of the shaft and the rotor. Other parameters in Equation (1) can refer to the given nomenclature table.

According to the Hertz contact theory, the contact force between the raceways and the rolling elements can be given as

$$f = k\_b \delta^n \tag{2}$$

where *kb* represents the load-deflection factor which depends on the contact geometry and the elastic contacts of the material. *δ* is the overall contact deformation of the rolling elements, which is composed of the contact deformation of each rolling element. The exponent *n* = 1.5 for ball bearings and *n* = 1.1 for roller bearings.

When the bearings operate, part of the raceway will be in the load zone, and the other part of the raceway will be in the non-load zone, which is shown in Figure 3. The contact deformation of each rolling element is determined by the angular position of the rolling element, the relative displacement between the inner and outer races, and the bearing clearance. The calculation of the contact deformation of the *j*th rolling element can be given as

$$\delta\_{\dot{j}} = (\mathbf{x}\_s - \mathbf{x}\_p)\cos\phi\_{\dot{j}} + (y\_s - y\_p)\sin\phi\_{\dot{j}} - \mathbf{c}\_\prime \dot{j} = 1,\ 2,\ \dots,\ n\_b\tag{3}$$

where *nb* denotes the number of the rolling elements. According to the elasto-hydrodynamic lubrication (EHL) theory, the clearance value *c* is set as negative, owing to the effect of oil EHL film [28]. The angular positions of the *j*th rolling element *φj* can be described as

$$\begin{array}{l} \phi\_{\text{j}} = \frac{2\pi(j-1)}{n\_{\text{b}}} + \omega\_{\text{c}}dt + \phi\_{\text{0}}\\ \omega\_{\text{c}} = (1 - \frac{D\_{\text{b}}}{D\_{\text{P}}})\frac{\omega\_{\text{c}}}{2} \end{array} \tag{4}$$

where *ωc* is the angular velocity of the bearing cage, *φ*0 denotes the initial angular position of the bearing cage, *Db* is ball diameter and *Dp* is the pitch circle diameter of the bearing, *ωs* is the angular velocity of the shaft.

**Figure 3.** Load distribution of roller bearing.

According to Equations (2) and (3), summing up the contact forces of the *nb* rolling elements, the overall nonlinear elastic contact forces of the bearing in the *x* and *y* directions can be calculated as

$$\begin{aligned} F\_{\mathbf{x}} &= k\_b \sum\_{j=1}^{n\_b} \gamma\_j \left( (\mathbf{x}\_s - \mathbf{x}\_p) \cos \phi\_j + (y\_s - y\_p) \sin \phi\_j - c \right)^{1.5} \cos \phi\_j \\\ F\_{\mathbf{y}} &= k\_b \sum\_{j=1}^{n\_b} \gamma\_j \left( (\mathbf{x}\_s - \mathbf{x}\_p) \cos \phi\_j + (y\_s - y\_p) \sin \phi\_j - c \right)^{1.5} \sin \phi\_j \end{aligned} \tag{5}$$

where *γj* is a switch function which depends on the positive and negative values of the contact deformation *δj*, described as

$$\gamma\_j = \begin{cases} \begin{array}{c} 1 \ if \,\delta\_j > 0 \\ 0 \,\text{otherwise} \end{array} \tag{6} \\ \tag{7}$$

It is noticed that the vibration model of the bearings presented above does not take different fault types into consideration. To simulate the vibration of the localized faults on the different components of bearings, the effects of the localized faults will be considered in the vibration model. The dynamic equation of the bearing system with faults can still be given by Equation (1). The main difference lies in the expression of the contact deformation of the rolling elements.

If a bearing operates in the health state at a steady speed, all forces in the bearing are in quasi-equilibrium. Once a localized fault occurs in the inner and outer races or the rolling elements, a certain deformation will be suddenly released when the fault contacts other components. As a result, a rapid change will take place in the elastic deformation of the components, and the force equilibrium state will be disturbed. Considering the new variations in the model with localized faults, the contact deformation of the *j*th rolling element is rewritten as

$$\delta\_{\dot{j}} = (\mathbf{x}\_s - \mathbf{x}\_p)\cos\phi\_{\dot{j}} + (y\_s - y\_p)\sin\phi\_{\dot{j}} - c - \underbrace{\stackrel{\text{fault part}}{\beta\_{\dot{j}}c\_d}}\_{\text{(7)}} \tag{7}$$

where *cd* denotes the fault depth. *βj* is a switch function to describe whether there is a contact loss due to the fault depth, which is closely related to the angular position of the faults. In addition, different fault types bring about different expressions of switch functions *βj*. In what follows, the expressions of the switch functions for the different localized faults will be discussed in terms of the outer race fault, inner race fault, and the fault in the rolling elements.
