An Improved Analytical Model of a Flexible–Rigid Combined Rolling Bearing with Elastohydrodynamic Lubrication

Abstract

In industrial applications, the operating state of bearings can directly affect the performance of mechanical equipment. Local defects, flexible raceways and lubrication have a great impact on the vibration characteristics of rolling bearings. However, previous works only considered the bearing as a rigid whole or unlubricated state. In order to solve this problem, a flexible–rigid combined bearing dynamics (FRBD) model considering grease lubrication is proposed in this paper. Both the flexible inner race and flexible outer race of a lubricated rolling bearing and a localized defect are formulated in the presented FRBD model. The influence of flexible–rigid combined and grease lubrication on the time- and frequency-domain responses of a rolling bearing with a local fault is analyzed. Based on the constructed model, the root mean square (RMS) value and maximum (MAX) value of time-domain statistical indicators are used to discuss the characteristic changes in acceleration vibration signals under different fault widths, inner-race speeds, and radial forces. In order to verify the accuracy of the FRBD model, the simulation results are compared with the experimental results. These results have a certain guiding role in fault diagnosis for the predictive maintenance of a rolling bearing.

1. Introduction

Rolling bearings have a crucial impact on the lifespan and overall performance of complex mechanical devices. In a rolling bearing, the surface of the raceway and rolling element bear load and roll, relatively. As the alternating load continues to increase, the spalling pits on the surface of the inner and outer races produce local defects, which affect the dynamics of the bearing rotation system. In the present study [1], 90% of rolling bearing failures are caused by local defects in the inner and outer races. Relevant research indicates [2] that 80% of premature failures of bearings are attributed to lubrication issues, which include insufficient lubrication, inadequate selection of lubricants, the aging of the lubricate, etc. Therefore, a comprehensive study on the dynamic characteristics of a defective rolling bearing is helpful for developing bearing fault detection methods.

Recently, a large number of modeling and simulation studies have focused on the dynamic analysis of bearings. For instance, Mochimaru et al. [3] used the fluid dynamics software FLUENT 2010 to study the static characteristics of the bearing and revealed the complexity of a mixed oil flow in the bearing. Bailey et al. [4] improved the Reynolds equation and derived an incompressible air flow model for a fluid film bearing. Mathematical and numerical models are applied to the coupling process between fluid flowing through the bearing and the axial movement of the rotor and stator. On this basis, Gao et al. [5] considered elastohydrodynamic lubrication (EHL) theory, radial clearance, and other factors to establish a four-degree-of-freedom dynamic model with multiple defects in the inner and outer raceways. Dipen et al. [6] proposed a dynamic model that considers the mass, stiffness, and damping of different components in rolling bearings. Liu et al. [7] proposed a new dynamic model that considers the coupling of time-varying displacement excitation and contact stiffness excitation in lubrication traction. Hong et al. [8] established a quasi-static analysis model for bearings with local defects. On the basis of the quasi-static model of bearings, local contour functions, including local defect depth and circumferential variation degree, are introduced into the analysis model. Kiral and Karagulle [9] regarded the outer race of the bearing as a rigid body and used I-DEA to analyze the forced vibration of the ball bearing. Then, the vibration response of a rolling bearing with local defects was simulated. Kai et al. [10] proposed a non-contact measuring method for both the oil film capacitance and resistance. This method achieved the continuous online monitoring of the lubricating condition of the bearing oil film. Zhu et al. [11] proposed an algorithm combining random matrix theory and a principal component analysis algorithm from a data-driven perspective to construct a bearing performance degradation index. The above papers have developed and perfected the theory of fluid lubrication but regarded the bearing as a pure rigid body without considering the structural deformation of the inner and outer races of the bearing. This assumption is not suitable for actual industrial conditions.

In practice, the coupling effects between the flexible inner race and flexible outer race can greatly affect the dynamics of the transmission system. Therefore, some articles have discussed the flexible–rigid combined characteristics of rolling bearings. Zhao et al. [12] used ABAQUS 2015 to establish a multi-body contact model and a dynamic model. As the key component of a harmonic driver, the flexible bearing was studied, and its fatigue life was analyzed. The results provide theoretical reference and practical guidance for the design and lifecycle estimation of flexible thin-walled bearings. Considering the coupling of axial and bending elastic motions, MAl-Solihat et al. [13] studied the three-dimensional nonlinear dynamics and force transfer characteristics of a flexible disk rotor system. Then, the nonlinear frequency response and force transmissibility curves caused by non-equilibrium force were constructed. Михаил Леoнтьев et al. [14] proposed a model of a radial rolling bearing that explained the deformation of its race structure and used the three-dimensional finite element software to simulate and verify the proposed hypothesis. Zhang et al. [15] proposed a semi-simplified finite element method to study the sealing performance of flexible–rigid combined seal grooves in order to solve the problem of wheel hub bearing failures. Their model was able to effectively study the flexible–rigid combined seal groove. Zhang et al. [16] used Pro/E and Adams to establish a flexible–rigid multi-body dynamic model of a bearing with an outer-race failure by studying the stress, contact force, and node displacement when the roller enters the defect area. This provides a theoretical basis for exploring the mechanism of bearing failure. On the basis of structural mechanics and rolling bearing dynamics, Sun et al. [17] proposed a dynamic model for the elastic support of a flexible outer-race cylindrical bearing and analyzed the load distribution and cage slip ratio of a cylindrical bearing. In recent years, some scholars have used deep learning to diagnose faulty bearings, but to some extent, it lacks interpretability [18,19]. Some scholars have proposed methods based on data-driven approaches, including numerical methods used in EHL calculations of radial bearings and the statistical analysis of fault characteristics from the perspective of fault defect size [20,21,22]. Tsuha et al. [22] optimized a dynamic model of EHL bearings by equating damping and stiffness coefficients.

Li et al. [23] used the theory of EHL to establish the dynamic equations of rigid elastohydrodynamically coupled ball bearings under high-speed and high-axial-load conditions. But they did not take into account the dynamic changes of bearings under fault conditions. Su et al. [24] proposed a dynamic model of cylindrical roller bearings with surface texture and studied the contact characteristics between rollers and raceways.

The necessity of this thesis stems from the significant theoretical deficiencies in existing bearing dynamics analysis and the urgent demand in engineering practice. Currently, research on the vibration characteristics of rolling bearings in the academic community mainly presents a two-polarized trend: on the one hand, the elastohydrodynamic lubrication theory under the traditional rigid assumption can effectively describe the formation mechanism of a lubricating oil film, but its failure to consider the elastic deformation of the bearing rings and rolling elements directly affects the reliability of bearing fatigue lifecycle prediction; on the other hand, the flexible multi-body dynamics model that has emerged in recent years can accurately characterize the elastic deformation characteristics of the bearing components, but it generally neglects the coupling mechanism between the lubrication effect and structural vibration. This theoretical disconnection has seriously restricted the application of high-speed precision bearings in cutting-edge fields such as aerospace and precision machine tools. To overcome this problem, the multi-stage coupling problem of lubrication stirring in a rolling system is studied in this paper. An FRBD model considering lubrication is proposed. Based on the introduction of EHL theory and the consideration of grease lubrication, both the flexible inner race, flexible outer race, and a localized defect in the rolling bearing are formulated in the FRBD model by using dynamics software. The impact of the vibration of the rolling bearing caused by the fault is analyzed. This model is more rigorous in calculating the comprehensive equivalent stiffness, damping effect, and displacement excitation caused by the fault. The vibration results are more suitable for real-life scenarios, which can provide some guidance for the incipient fault detection and diagnosis of rolling bearings.

The specific structure of the article is as follows: Section 2 explains the influence of three factors on bearing rotation dynamics: an inner-race or outer-race defect, inner-race and outer-race flexible–rigid combined lubrication, and grease lubrication. The general dynamic modeling process of a deep groove ball bearing is discussed. Among them, EHL theory, the Bingham plastic model, and the Reynolds equation are used to establish the EHL contact model of a rolling bearing, and a commercial multi-body system software application is used to establish a flexible–rigid combined model. Section 3 uses the method proposed in this article for a RB SKF 6203-RS rolling bearing to consider the vibration characteristics of the bearing under four conditions: rigid with lubrication (RBD with lubrication), rigid without lubrication (RBD without lubrication), flexible–rigid combined with lubrication (FRBD with lubrication), and flexible–rigid combined without lubrication (FRBD without lubrication). The simulation results for rolling bearings with inner- and outer-race defects are compared with the experimental results to verify the stability and accuracy of the proposed model. The influence of different inner-race speeds, defect positions, radial forces and spalling pit sizes on the time- and frequency-domain responses of the FRBD with lubrication model is investigated. Finally, Section 4 summarizes the main findings and conclusions.

2. FRBD with Lubrication Model

2.1. Grease Lubrication Model

Rolling bearings play an important role in the dynamic characteristics of the internal system of a machine. The theory of EHL has developed from classical EHL to modern EHL. Compared with base oil-lubricated bearings, research on grease lubrication is relatively weak. Compared with other lubrication with a similar viscosity, grease has a strong service life, strong retention on the friction surface, and a better bearing capacity and damping capacity. Based on the advantages of grease, about 90% of rolling bearings currently use grease as a lubricant, so it is necessary to study the dynamic analysis of grease-lubricated bearings. In this paper, the influence of the oil film on the interaction between the rolling elements and the raceway in grease lubrication is considered. EHL analysis is used to calculate the approximate actual elastohydrodynamic model film thickness, the oil film stiffness, and the equivalent comprehensive stiffness. This aims to provide more practical parameters for dynamic simulation later on and to make the bearing dynamic model more accurate and reasonable.

In order to study the dynamic characteristics of rolling bearings, the dynamic model proposed in this paper makes the following assumptions and considerations:

• Rolling bearings operate under isothermal conditions to avoid temperature interference with the model.
• The balls of the ball bearing model are equidistant from the surface of the inner race and the surface of the outer race, and there is no interaction between them.
• The contact form between the inner/outer race and the rolling elements of the bearing is Hertz contact. The contact between the cage and the rolling elements does not generate additional force.
• The center of mass of the rolling element is always maintained at the geometric center.
• There is no relative sliding between the inner race of the rolling bearing and the shaft. The angular velocity of the inner race remains the same as the rotational speed of the shaft.
• When the rolling element comes into contact with the seat ring, elastic deformation and local contact deformation will occur, but this will not change the overall shape and size of the bearing.

We first calculate the contact stiffness between a rolling element and the raceway in rolling bearing operation $K^{*}$, the damping parameter of the rolling bearing $C^{*}$, and the penetration depth between the rolling element and the inner race/outer race/cage $d_{\max }$. The shaft–bearing–pedestal system under EHL conditions could be simplified into the dynamic model shown in Figure 1a.

EHL theory, the Bingham plastic rheological model, and the Reynolds equation are used to establish a contact model of EHL for rolling bearings. It is assumed that grease is present at the point of contact between the rolling body and the raceway in isothermal conditions without an oil shortage. Combined with Hertz contact theory to express the contact between races and roller, the contact stiffness, oil film stiffness, and damping are shown in Figure 1b.

For a lubricated rolling bearing, the equivalent comprehensive stiffness between the inner race and roller is defined by

$${\displaystyle \frac{1}{{K_i}^{*}}}={\displaystyle \frac{1}{K_{iy}}}+{\displaystyle \frac{1}{K_i}}$$

(1)

where $K_{iy}$ is the stiffness of lubricating grease in the contact area between the inner race and roller and $K_i$ is the contact deformation stiffness between the inner race and roller.

$${\displaystyle \frac{1}{{K_o}^{*}}}={\displaystyle \frac{1}{K_{oy}}}+{\displaystyle \frac{1}{K_o}}$$

(2)

where $K_{oy}$ is the stiffness of lubricating grease in the contact area between the outer race and roller and $K_o$ is the contact deformation stiffness between the outer race and roller.

The stiffness of the raceway $K_s$ or $K_p$ can be formulated as

$$\left\{\begin{array}{l}{K}_s={({({\displaystyle \frac{\mathsf{\partial }{W}_i}{\mathsf{\partial }{h}_{i\min }}})}^{-1}+{K_i}^{*-1})}^{-1}\\ K_p={({({\displaystyle \frac{\mathsf{\partial }{W}_o}{\mathsf{\partial }{h}_{o\min }}})}^{-1}+{K_o}^{*-1})}^{-1}\end{array}\right.$$

(3)

For a lubricated rolling bearing, the time-varying damping coefficient between the inner race and roller is written as

$${C_i}^{*}={\displaystyle \frac{6\pi {\mu }_0{R_{ix}}^{1.5}a{\delta }_i}{\sqrt{2}{h_{\mathrm{i}\min }}^{1.5}}}$$

(4)

The time-varying damping coefficient between the outer race and roller is written as

$${C_o}^{*}={\displaystyle \frac{6\pi {\mu }_0{R_{ox}}^{1.5}a{\delta }_o}{\sqrt{2}{h_{o\min }}^{1.5}}}$$

(5)

where $a$ is the long half axis of contact between the roller and race in the contact area; ${\delta }_o$ is the total displacement between the race and roller; and $h$ is the thickness of the center of the grease oil film. The details of the calculation methods are listed in [25,26,27].

The structural damping coefficient is given as [28]

$$\left\{\begin{array}{l}{C}_s={\displaystyle \frac{{\eta }_b\cdot K_s}{f_{ext}}}\\ C_p={\displaystyle \frac{{\eta }_b\cdot K_p}{f_{ext}}}\end{array}\right.$$

(6)

where ${\eta }_b$ is the loss coefficient and $f_{ext}$ is the excitation frequency.

From this, the contact deformation stiffness between the roller and races is given by [29]

$$\left\{\begin{array}{l}{K}_i={\displaystyle \frac{Q_i}{{{\mu }_i}^{1.5}}}\\ K_o={\displaystyle \frac{Q_o}{{{\mu }_o}^{1.5}}}\end{array}\right.$$

(7)

The rolling bearing IMPACT function expression is defined by [27]

$$\left\{\begin{array}{l}{Q}_r-k{(B{d}_b)}^n{\displaystyle \sum _{\theta =0}^{2\pi }{[{({x_{1\theta }}^2+{x_{2\theta }}^2)}^{0.5}-1]}^{1.5}\frac{x_{2\theta }}{{({x_{1\theta }}^2+{x_{2\theta }}^2)}^{0.5}}\cos \theta =0}\\ Q_a-k{(B{d}_b)}^n{\displaystyle \sum _{\theta =0}^{2\pi }{[{({x_{1\theta }}^2+{x_{2\theta }}^2)}^{0.5}-1]}^{1.5}\frac{x_{1\theta }}{{({x_{1\theta }}^2+{x_{2\theta }}^2)}^{0.5}}=0}\\ M-0.5d_{m}k{(B{d}_b)}^n{\displaystyle \sum _{\theta =0}^{2\pi }{[{({x_{1\theta }}^2+{x_{2\theta }}^2)}^{0.5}-1]}^{1.5}\frac{x_{1\theta }}{{({x_{1\theta }}^2+{x_{2\theta }}^2)}^{0.5}}\cos \theta =0}\end{array}\right.$$

(8)

$$\left\{\begin{array}{l}{x}_{1\theta }={\displaystyle \frac{{\delta }_a}{B{d}_b}}+{\displaystyle \frac{{\phi }_{\omega }{r}_i}{B{d}_b}}\cos \theta \\ x_{2\theta }=1+{\displaystyle \frac{{\delta }_r}{B{d}_b}}\cos \theta \end{array}\right.$$

(9)

$$B=f_i+f_o-1$$

(10)

where $B$ is the total curvature; $f_i$ and $f_o$ represent the radii of curvature of the inner race and outer race, respectively, which is formulated in [28], $M$ is the moment acting on the bearing; $Q_r$ and $Q_a$ describe the radial and axial forces acting on the bearing; $n$ is the load–displacement index for ball bearings, $n=1.5$, ${\phi }_{\omega }$ is the angular displacement of the bearing, $\theta$ is the position angle of the bearing; $r_i$ is the radius of the inner raceway of the bearing, and $k$ is the load deformation coefficient considering contact deformation. The contact deformation coefficient of the inner race is defined by $k_i$, and the contact deformation coefficient of the outer race is defined by $k_o$; refer to [29] for the calculation process. The axial displacement ${\delta }_a$ and radial displacement ${\delta }_r$ of the bearing are calculated by the Newton–Raphson equation, substituting Equation (9) to calculate the inner-race contact parameters, $x_{1\theta i}$ and $x_{2\theta i}$, and the outer-race contact parameters, $x_{1\theta o}$ and $x_{2\theta o}$.

The load between the roller and the inner race or outer race at the position angle $\theta$ is calculated by

$$\left\{\begin{array}{l}{Q}_i=k_i{(B{d}_b)}^n{[{({x_{1\theta i}}^2+{x_{2\theta i}}^2)}^{0.5}-1]}^{1.5}\\ Q_o=k_o{(B{d}_b)}^n{[{({x_{1\theta o}}^2+{x_{2\theta o}}^2)}^{0.5}-1]}^{1.5}\end{array}\right.$$

(11)

The contact deformation between the roller and the inner race or outer race at the position angle $\theta$ is given as

$$\left\{\begin{array}{l}{\mu }_i=B{d}_b{[{({x_{1\theta i}}^2+{x_{2\theta i}}^2)}^{0.5}-1]}^{1.5}\\ {\mu }_o=B{d}_b{[{({x_{1\theta o}}^2+{x_{2\theta o}}^2)}^{0.5}-1]}^{1.5}\end{array}\right.$$

(12)

The grease film stiffness between the roller and the inner race or outer race is defined by [23]

$$\left\{\begin{array}{l}{K}_{iy}=6.4066\times {10}^8\times {h_{i\min }}^{-14.6986}{U}^{9.361507}{G}^{6.7123}E{R_{ix}}^{15.6986}{(1-e^{-0.68m_i})}^{13.6986}\\ K_{oy}=6.4066\times {10}^8\times {h_{o\min }}^{-14.6986}{U}^{9.361507}{G}^{6.7123}E{R_{ox}}^{15.6986}{(1-e^{-0.68m_o})}^{13.6986}\end{array}\right.$$

(13)

where $h_{i\theta \min }$ and $h_{o\theta \min }$ are the minimum grease film thicknesses between the lubricated races and roller.

When the thickener in the grease enters the friction contact zone, the drop in film thickness does not only depend on the base oil. Therefore, under the conditions of full-immersion grease lubrication, the results of Yang and Qian [30] showed that the yield stress has no effect on the film thickness. Under these conditions, the effect of grease and its base oil on film thickness only depends on their viscosity difference, which can be expressed as

$$\left\{\begin{array}{l}{h}_{i\min }=({\displaystyle \frac{K^{\prime }}{{\mu }_{oil}}})\times h_{ioil\min }\\ h_{o\min }=({\displaystyle \frac{K^{\prime }}{{\mu }_{oil}}})\times h_{ooil\min }\end{array}\right.$$

(14)

where $K^{\prime }$ is the bulk plastic viscosity and ${\mu }_{oil}$ is the kinematic viscosity of the base oil. The minimum oil film thickness in the contact area between the inner race and roller is defined by $h_{ioil\theta \min }$ and that between the outer race and roller is defined by $h_{ooil\theta \min }$.

In the Bingham model, $K^{\prime }$ is calculated by [31]

$$K^{\prime }={\displaystyle \frac{\tau -{\tau }_y}{{\gamma }_o}}$$

(15)

where $\tau$ is the shear stress; ${\tau }_y$ is the stress parameter, whose value is $2\times {10}^{10}{Pen}^{-2.95}$; and ${\gamma }_o$ is the shear rate, whose value is $10{\mathrm{s}}^{-1}$.

The equation for the minimum oil film thickness of the base oil, $h_{oil\min }$, is given as

$$h_{oil\min }=3.63U^{0.68}{G}^{0.49}{W}^{-0.073}(1-e^{-0.68k})R_x$$

(16)

The other parameters are expressed as [27]

$$\left\{\begin{array}{l}{U}_{i/o}={\displaystyle \frac{V_s{\mu }_0}{2E_{i/o}{R}_{i/ox}}}\\ G_{i/o}={\alpha }_{\tau }{E}_{i/o}\\ W_{i/o}={\displaystyle \frac{Q_{i/o}}{E_{i/o}{R_{i/ox}}^2}}\\ k_{i/o}=1.03{({\displaystyle \frac{R_{i/oy}}{R_{i/ox}}})}^{0.63}\end{array}\right.$$

(17)

where $U$ is a dimensionless speed parameter; $G$ is a dimensionless material parameter; and $W_{i/o}$ is the load parameter. ${\mu }_0$ is the operating viscosity at the reference temperature. In low-load bearing applications, ${\alpha }_{\tau }=2.0\times {10}^{-8}\mathrm{Pa}\cdot \mathrm{s}$, ${\mu }_0=0.009\mathrm{Pa}\cdot \mathrm{s}$, $E_{i/o}$ is the equivalent elastic modulus of the inner or outer races.

The overall velocity of the rolling element and raceway, $V_s$, is calculated as

$$V_s={\displaystyle \frac{d_m}{2}}\left[\left(1-{\gamma }_b\right)\left({\omega }_i-{\omega }_m\right)+{\gamma }_b{\omega }_b\right],$$

(18)

where ${\omega }_i$, ${\omega }_m$, and ${\omega }_b$ are the angular velocities of the inner-race raceway, cage, and rolling elements, respectively.

The equivalent radius of the ball and the inner/outer race in the x direction, $R_{i/ox}$, and the equivalent radius of the ball and the inner/outer race in the y direction, $R_{i/oy}$, are given as

$$\left\{\begin{array}{l}{R}_{ix}={\displaystyle \frac{d_b}{2}}\left(1-{\gamma }_b\right)\\ R_{ox}={\displaystyle \frac{d_b}{2}}\left(1+{\gamma }_b\right)\end{array}\right.$$

(19)

$$\left\{\begin{array}{l}{R}_{iy}={\displaystyle \frac{f_i{d}_b}{2f_i-1}}\\ R_{oy}={\displaystyle \frac{f_o{d}_b}{2f_o-1}}\end{array}\right.$$

(20)

$${\gamma }_b={\displaystyle \frac{d_b\cos \alpha }{d_m}}$$

(21)

2.2. Flexible–Rigid Combined Fault Dynamic Model

In the past, the bearing has often been regarded as a rigid whole in dynamic analysis. In the industry, the sensor is often set up on a relatively stationary outer race. If the bearing is a rigid one, the effect of the force is not enough to produce deformation. When failure occurs, the vibration signal obtained by the simulation of the outer race will be very weak. To keep the dynamic model highly consistent with the experimental model, this paper uses a multi-body system software application to simulate the rolling bearing and builds a flexible–rigid combined model. The specific process is shown in Figure 2. Firstly, the 3D model is imported into the multi-body dynamics software ADAMS 2020. Secondly, the inner and outer races are meshed (the mesh size does not exceed 20% of the minimum size parameter of the bearing). Then, the stiffness and damping of grease lubrication are set after the flexible– rigid combined boundary is set. Finally, the axial force, radial force, and driving force are applied to the rolling bearing. The inner race and cage of the bearing are set to rotate around the origin. The function of the set driving speed makes the inner race reach a stable speed from a static state within. After the model is built in the View module, the data is displayed and analyzed in the Post-Processor. Figure 3a,b plot the presented FRBD model and RBD model for the rolling bearing and races, respectively. The flexible races of the FRBD model are developed according to Figure 2 in the software. The difference between the RBD model and the FRBD model is that the former’s inner and outer races are rigid bodies, as shown in Figure 3b.

During the operation of the rolling bearing, it may be damaged in various ways. Even when the installation, lubrication, and maintenance processes are normal, the bearing will still suffer fatigue spalling and wear after a period of time, which will affect the operation of the rotating machinery. Therefore, it is necessary to conduct research on defective bearings. Different from the idealized fault model, the fault is regarded as a rectangular pit. When the rolling element comes into contact with the fault of the inner/outer race, deformation occurs instantly, and the deformation is reversed the moment this contact is broken. This paper establishes a fault model based on the circular edge fault studied by Liu J et al. [32], a local fault profile, as shown in Figure 4a, which maximizes the actual situation of a bearing spalling fault.

Table 1. Parameters for the bearing operation.

Defect Cases		Depth H (mm)	Length L (mm)	Circle Radius r (mm)	Inner-Race Speed	Defect Cases	Depth H (mm)
Inner-race defect	1	0.3	0.4	0.15	1397	50	6
	2	0.3	0.4	0.15	1797	50	6
	3	0.3	0.4	0.15	2197	50	6
	4	0.3	0.4	0.15	2597	50	6
	5	0.3	0.4	0.15	1797	100	6
	6	0.3	0.4	0.15	1797	150	6
	7	0.3	0.4	0.15	1797	200	6
	8	0.4	0.6	0.2	1797	50	13
	9	0.5	0.8	0.25	1797	50	18
Outer-race defect	10	0.3	0.4	0.15	1397	50	6
	11	0.3	0.4	0.15	1797	50	6
	12	0.3	0.4	0.15	2197	50	6
	13	0.3	0.4	0.15	2597	50	6
	14	0.3	0.4	0.15	1797	100	6
	15	0.3	0.4	0.15	1797	150	6
	16	0.3	0.4	0.15	1797	200	6
	17	0.4	0.6	0.2	1797	50	13
	18	0.5	0.8	0.25	1797	50	18

plots the studied local fault conditions. Such faults could affect the inner race, the outer race, the cage, or the rolling elements. An improved local fault was used to simulate the defect (whether located on the inner race or on the outer race), as shown in Figure 4b.

3. Results

3.1. Rolling Bearing SKF 6203-RS Dynamic Model

Based on the method of establishing a dynamic model proposed in Section 2, a vibration analysis of the SKF 6203-RS rolling bearing was carried out.

Table 2. The main parameters of RB SKF 6203-RS.

Parameters	Value
Number of balls (Z)	8
Pitch diameter (dm)	28.500 mm
Ball diameter (db)	6.746 mm
Inner race diameter (di)	17 mm
Outer race diameter (do)	40 mm
Width (B)	12 mm

gives the geometric parameters of the bearing.

3.2. Analysis of the Influence of Flexible–Rigid Combined and Grease Lubrication on Rolling Bearing Vibration

The rolling bearing is simulated under four different conditions: with a grease film or without, and under flexible–rigid combined conditions or not. The vertical vibration acceleration signal of a bearing outer race is extracted and analyzed. Figure 5, by comparing the accelerations of the four models (the pure rigid bearing dynamic without lubrication (RBD without lubrication) model, the pure rigid bearing dynamic with lubrication (RBD with lubrication) model, the flexible–rigid combined bearing dynamic without lubrication (FRBD without lubrication) model, and the flexible–rigid combined bearing dynamic with lubrication (FRBD with lubrication) model, the influence of lubrication and flexible–rigid combined bearings on the accelerations of a defective rolling bearing is obtained. It is assumed that there is an outer-race fault and the fault spalling pit is an arc-shaped edge, with depth H = 0.3 mm, width L = 0.4 mm, edge radius r = 0.15 mm, and inner-race speed n = 1797 r/min. It can be seen that the amplitude of the bearing acceleration vibration signal is obviously affected by lubrication and flexible–rigid combined bearings.

By comparing Figure 5a,b with Figure 5c,d, the pulse amplitude caused by the defective bearing without lubrication is more severe than that caused by the defective bearing with lubrication. This is because grease facilitates shock absorption and lubrication. By comparing Figure 5a,c with Figure 5b,d, note that the frequency-domain responses from the flexible–rigid and rigid models are very different. The pure rigid model causes more severe pulse amplitude than the flexible–rigid combined model. Because the flexible body has a large damping effect, it can absorb part of the energy and make the vibration acceleration value smaller relative to the rigidity. The changes in the acceleration signal characteristics obtained in this simulation can provide useful information for the evaluation of bearing degradation.

Using the envelope spectrum to analyze the simulation results in Figure 5a–d, it can clearly be seen from Figure 6a–d that the characteristic frequency of the fault gradually attenuated from one to five times its frequency. According to a calculation method developed by Harris [29], the theoretical value is obtained, as shown in

Table 3. Characteristic frequencies of bearing (for rotor speed 1797 r/min).

$\mathbf{Shaft} \mathbf{Frequency} ({\mathit{f}}_{\mathbf{r}}$)	$\mathbf{Rolling} \mathbf{Element} \mathbf{Passing} \mathbf{the} \mathbf{Outer} \mathbf{Race} \mathbf{Frequency} ({\mathit{f}}_{\mathbf{o}}$)	$\mathbf{Cage} \mathbf{Frequency} ({\mathit{f}}_{\mathbf{c}}$)
29.950 Hz	91.445 Hz	11.430 Hz

. The rolling element passing the outer race frequency ($f_o$) is 91.445 Hz. From the envelope spectrum, it can be seen that the four cases of fault frequency are 94.68 Hz, 93.13 Hz, 92.13 Hz, and 91.58 Hz, respectively. At the same time, the fault frequency of the FRBD model with lubrication is 91.58 Hz, which is closest to the theoretical value of the outer-race fault frequency, which is 91.445 Hz, and the deviation is within 0.1%. The proposed FRBD model with grease lubrication takes into account the elastic lubrication (EHL) theory. It is no longer just studying the simple motion hypothesis of pure rolling. On this basis, the inner and outer races of the bearing are treated flexibly. Therefore, this situation would be more suitable for actual fault diagnosis and analysis. It has higher accuracy and stability.

3.3. FRBD with Lubrication Model Validation

The cage angular velocity is an important index for evaluating the smoothness of bearing operation. A comparison between the simulation value and the theoretical value of cage angular velocity is shown in Figure 7. The bearing inner-race speed was set to 1797 rpm/min, and the theoretical angular velocity of the cage calculated was 11.4 rad/s. Through the comparison between the theoretical angular velocity of the cage and the simulated angular velocity, it can be seen that the numerical value of the simulated cage’s angular velocity is 11.25 rad/s to 11.62 rad/s, the theoretical rotational angular velocity value fluctuates up and down, and the error does not exceed 1%. Considering the calculation error of the theoretical value, the model establishment error, and the simulation error, the above error is within a reasonable range. This verifies the smooth running of the rolling bearing built in this paper and verifies the rationality of the model in terms of kinematics.

In order to analyze and verify the comparison between the defective rolling bearing dynamics model built in this paper and the actual application, the frequency-domain responses from an experiment performed in [33] were used for comparison. The experimental platform mainly consists of a 2-horsepower motor, a sensor for detecting torque, a power meter, and electronic control equipment. During the experiment, acceleration sensors were used to collect vibration signals from the high-speed end and motor end of the bearing supporting the motor shaft. The bearing model is RB SKF 6203-RS. A qualitative comparison between the experimental signal under the same fault conditions and the envelope diagram obtained from the simulation signal of the flexible–rigid combined model considering the grease lubrication results obtained in Figure 6 is shown in Figure 8.

The shaft rotational frequency, the defect frequency, and the decayed frequency components of the frequency are clearly seen in Figure 8. The main peak frequency difference between the simulated results of the FRBD with lubrication oil model in this paper and the experimental results proposed is shown in

Table 4. Numerical comparison between simulation results and experimental results of rolling bearing.

Numerical Comparison Between Simulation Results and Experimental Results of Outer-Race Defect Bearing
Frequency	Simulation (HZ)	Experiment (HZ)	Error
$f_{\mathrm{r}}$	27.94	30	6.87%
$f_{\mathrm{o}}$	91.58	91.8	0.24%
$2f_{\mathrm{o}}$	181.6	183.4	0.99%
$3f_{\mathrm{o}}$	274.7	275.2	0.18%
$4f_{\mathrm{o}}$	364.8	366.8	0.32%
Numerical Comparison Between Simulation Results and Experimental Results of Inner-Race Defect Bearing
Frequency	Simulation (HZ)	Experiment (HZ)	Error
$f_{\mathrm{r}}$	28.99	30	3.48%
$f_{\mathrm{i}}$	152.6	147.6	3.28%
$2f_{\mathrm{i}}$	296	299.2	1.08%
$3f_{\mathrm{i}}$	447.1	448.8	0.38%

. The maximum error comes from the first-order frequency of the inner-ring fault. Compared with the inner ring, the first-order fault frequency error of the outer ring is relatively small, but the overall error is less than 7%. The difference between the simulation results and the experimental results may be caused by the damping and vibration of the test bench structure. Due to some errors in the structure and modeling parameters of rolling bearings, the deviation between the simulation results and the experimental results is within an acceptable range. The good matching of vibration amplitude at the defect frequency for the simulated and experimental results proves the effectiveness of this model. It also shows that the proposed flexible–rigid combined dynamic model considering grease lubrication can simulate the behavior of actual defective bearings under experimental conditions.

For the same dataset, a comparison was made between the deep learning methods in the relevant literature [34] and the method proposed in this paper to more fully discuss the effectiveness of the proposed diagnostic framework. Compared to neural networks, the algorithm in this article is more interpretable, especially for the case with lubrication. It can be seen from the comparison of accuracy that with lubrication, the fault recognition accuracy is no less than that of neural network methods. Therefore, it can be concluded that if the dynamic model and fault model considering lubrication for defining bearings can be accurately described, then this method has more advantages than neural network methods.

3.4. Effect of the Rotor Speed, Defect Position, Radial Load, and Defect Size on the Time- and Frequency-Domain Vertical Vibration Accelerations of FRBD with Lubrication Model

Shaft rotational speed is a crucial physical parameter in bearing operation. We aimed to explore the influence of the inner-race speed, defect position, radial force, and the size of the fault pit on the time–frequency characteristics of the FRBD model’s vibrational acceleration. Here, a defect size for the inner or outer race of 0.1778 mm and an operating speed of 1 hp/1397 rpm/min, 1797 rpm/min, 2197 rpm/min, or 2597 rpm/min are used as a reference. Figure 9 shows the simulated vibration signals within 0.4 s to 0.6 s and the frequency of the inner defect bearing at different shaft rotation speeds. The vibrations generated by a bearing with a local defect on its inner race are complicated due to the rotation of the defect at the shaft speed. It can be seen that the speed of the inner race has a significant effect on the acceleration of the rolling bearing. We analyze the signal domain frequency through envelope analysis. The rotational frequency ($f_{\mathrm{r}}$), the fault characteristic frequency for the inner raceway ($f_{\mathrm{i}}$), and the sideband frequencies ($f_{\mathrm{i}}\pm 2\times f_{\mathrm{r}}$) are also visible due to modulation. The sideband peak in the figure represents the defect position variation caused by its rotation at the shaft speed. Due to the consideration of the influence of lubricants, the friction torque is independent of the applied load, and therefore the amplitude of $f_{\mathrm{i}}$ gradually increases with increasing speed, as shown in Figure 9b,d,f,h. The experimental results show that the amplitude of the defect frequency matches the theoretical results, indicating that the model has high accuracy.

The vibration spectra are plotted for different rotational speeds in Figure 10a,c,e,g. For the bearing with a defect on the outer race, because the outer race is fixed on a pedestal, the signal shows the characteristics of a small impact interval and a gentle amplitude change. In order to study the low-frequency characteristics of the bearing signal, the vibration signal is enveloped. The rotational frequency $f_r$, the frequency at which the rolling element passes the outer race $f_{\mathrm{o}}$, and the harmonic frequencies ($2\times f_{\mathrm{o}}$, $3\times f_{\mathrm{o}}$) can be observed in the simulated vibration spectra shown in Figure 10b,d,g,h. As the speed increases, the pulse interval gradually becomes shorter, and the amplitude gradually becomes larger. In the envelope spectrum of the simulated signal, it can be seen that the amplitude of other shock components in the low-frequency band increases with the increase in rotational speed.

In the case of the rotor bearing system, the vibration transmission ratio for an outer-race defect is adversely affected by defect position. The vibration amplitude is also affected by defect position. If a radial load is applied to the bearing, the significant amplitude generated when the defect is exactly below the top center of the test bearing is within a specific range ($\pi <{\theta }_{LOD}<3\pi /2$). Within this angle range, the defect is closest to the load application area, resulting in the maximum amplitude of vibration acceleration generated. Due to the interaction between rolling elements and defects, a constant amplitude pulse train is generated in the characteristic signal. Under the same inner-race speed and radial force, the simulated and experimental signal vibration characteristics of the bearing with defect positions of 0, π, and 2π are given in Figure 11. Periodical impulses occur in the time domain due to the defect. It can be seen that five impulses occur in Figure 11 when the defect on the outer race comes into contact with the rolling elements. As the defect moves away from the load zone, the amplitude decreases. Due to the random impact generated by the simulation, there is a certain error between the simulated signal and the experimental signal, but it is within an acceptable range.

Figure 12 and Figure 13 use time-domain statistical indicators, namely the RMS value and MAX value, to study the influence of radial load and defect size on the vibration amplitude of the FRBD with lubrication model. The radial force is assumed to be 50 N, 100 N, 150 N, and 200 N; the different inner- and outer-race fault lengths are assumed to be 0.4 mm, 0.6 mm, and 0.8 mm. The fault length under normal conditions is expressed as being acceptable. Due to the variation in radial force, the vibration amplitude changes. Figure 12 shows that the RMS and MAX values of the bearing vibration acceleration increase with the increase in the radial force on the bearing. And this increase is linear, with outer- and inner-ring defects having different effects on the acceleration in the time domain. In most cases, at the same speed or radial force, the outer-ring faults are greater than the inner-ring faults. This is due to the larger vibration acceleration impact caused by an outer-ring fault. But at smaller radial forces, the RMS and MAX values of the vibration acceleration of the inner- and outer-ring bearings are almost the same. This confirms the accuracy of our theoretical calculations from another perspective. The RMS and MAX values are smallest in a normal state, which is consistent with reality.

In addition, Figure 13 depicts the influence of defect size on bearing acceleration. Under the same inner-race speed and radial force, the RMS value and MAX value continue to increase with the increase in the length of the fault. Usually, the outer-race fault value is slightly greater than the inner-race fault value, and the vibration level caused by the inner-race fault is smaller than that caused by the outer-race fault. To analyze time-domain statistical indicators of vibration acceleration for the same type of fault, Figure 13c,d show the RMS value and MAX value of the experimental acceleration data, respectively. Similarly to the increasing trend in Figure 13a,b obtained from the simulated experiment, both increase with the increase in the fault length, L, and the statistical value of the inner-race fault is slightly smaller than the outer-race fault’s statistical value. The simulation results and experimental results have similar trends. This result verifies the accuracy of the flexible–rigid combined dynamics (FRBD) with lubrication model proposed in this paper.

4. Conclusions

This article focuses on the issue with the existing dynamic model in that it does not fit the actual real-world situation. After the improvement of the pure rigid bearing, a more complete flexible–rigid combined bearing dynamic (FRBD) model considering grease lubrication is proposed, which is helpful for more accurately expressing the bearing stiffness and the vibration response of the bearing. Assuming that the fault spalling pit has an arc-shaped edge and considering the combined effect of Hertz contact theory stiffness and grease film stiffness, a multi-flexible flexible–rigid combined model of the bearing is created and the contact parameters of the bearing are calculated and analyzed. There are significant differences in the time domain and frequency domain of the RBD with lubrication model, RBD without lubrication model, FRBD with lubrication model, and FRBD without lubrication model.

The simulation results of the FRBD with lubrication model proposed in this paper are closest to the theoretical values. In general, the dynamic response of flexible–rigid combined lubrication bearings is periodic and stable and has good dynamic performance. The simulation results are compared with the envelope spectrum of the experimental results for the rolling bearing with an inner- or outer-race fault. The error of the fault frequency and its frequency multiplication components do not exceed 7%, which verifies the accuracy of the model.

In addition, on the basis of the FRBD with lubrication model, the influence of the inner-race speed, defect position, radial force, and defect size on the vibration response of the bearing is analyzed. As the inner-race speed increases, the pulse interval gradually becomes shorter and the amplitude gradually becomes larger. Moreover, the vibration peaks at the shaft rotational frequency, defect frequency, and their harmonics were visible for the defect bearing. Also, the difference in defect position has an impact on the shock of the vibration signal for the same defect size, radial load, and shaft speed. Then, the influence of the radial force and fault size on the bearing vibration was studied. Note that the statistics (RMS value and MAX value) increase with the increase in radial force and fault size. Compared with inner-race defects of the same size, the vibration level is higher under the condition of local defects on the outer race. The RMS value and MAX value simulations for different fault sizes are compared with the experimental results, and a good agreement is obtained. The dynamic model proposed in this paper can be used to study the vibration characteristics of bearings with more complex defect geometry. Therefore, the dynamic model proposed in this article can predict the defect frequency and vibration amplitude of the bearing system in the presence of lubricating oil. Further contributions could be made to bearing failure prediction and health management in the future.

This paper only considers the influence of load on the oil film of rolling bearings for the time being. However, the dynamic model considered in this paper represents an ideal situation. In reality, when the bearing is working, the cage and balls will not be in an ideal position, which will cause the lubrication model to change. This point will be fully considered in subsequent research.