Investigation on the skidding dynamic response of rolling bearing with local defect under elastohydrodynamic lubrication

. The rolling element skidding may lead to the failure of the rolling bearing. The skidding characteristics can be effectively analyzed by using dynamic response of the rolling bearing. A dynamic model is established to investigate the vibration response of the rolling bearing with local defect on inner/outer race in this paper. In the proposed model, the rolling element skidding, contact stiffness and displacement, the interaction force between ball and race, the interaction force between cage and race, elastohydrodynamic lubrication are taken into consideration. The dynamic responses of the rolling bearing with the rolling element skidding are solved by the proposed model in the time and frequency domains. The effects of defect size, rotational speed, external load, and compound factors on skidding characteristics are investigated. The proposed model is veri ﬁ ed by the experiments. The results show that the rolling element skidding leads to the signi ﬁ cant difference of dynamic characteristics in the time and frequency domains, which aggravates the failure of the rolling bearings.


Introduction
The rolling bearing is one of the most important parts of the rotating machinery and widely used in automotive industry, aviation, spaceflight, navigation and other rotational machinery. It is a complex component of a machine and makes up of outer race, inner race, rolling element, cage, housing, oil film. The local defects (the crack, pitting and peeling, etc.) may generate when the rolling bearing operates for a long-time. When the local defects occur, the bearing will rotate unsteadily. In order to diagnose the bearing operation condition immediately, the condition monitoring and fault diagnosis are adopted in the field of bearing. The dynamics analysis method is an effective method for analyzing the bearing operation condition. The dynamics model can reflect the actual dynamic response characteristics with the time-varying. It is necessary to further research of the rolling bearing dynamics, which is important to explore the vibration mechanism of the rolling bearings.
The vibration response has been widely used to diagnose the rolling bearing defects. When the rolling bearing has local defects on the rolling element and the inner/outer race, the vibration response will change significantly. Jone [1] firstly put forward the quasi-static model of the rolling bearing. The slipping friction and motion were investigated based on the race control hypothesis. Gupta [2] established a three-dimensional dynamic model of the rolling bearings. In Gupta's model, the centrifugal force, gyroscopic, lubrication and the cage effects are all take into consideration. However, the localized defects are not considered in the Gupta's model. Due to the important role of dynamic response in the fault diagnosis of the rolling bearings, most scholars have established dynamic models to investigate the response characteristics of the rolling bearings. Garad et al. [3] predicted the presence of the defects on the outer and inner races based on 2-DOF (degree of freedom) model. The effect of defects on vibration behaviors of the rolling bearing was investigated by the 2-DOF model. Niu et al. [4] proposed a dynamic model to investigate vibration responses of the angular contact ball bearings with ball defects, the three dimensional motion of ball was considered based on Gupta's model. The result shows that the geometrical characteristics and operation conditions affect the frequency property of envelope spectrum, it also increases the difficulty of diagnosing the ball defects. Wang et al. [5] presented a method to analyze the friction of the deep-groove ball bearings with varying circumference radial preload, the experiment was conducted to verify the validity of the proposed method under different operation conditions. Based on the 4-DOF mathematical model, Yu et al. [6] analyzed the vibration characteristics of the deep groove ball bearing by the Runge-Kutta method. Moshrefzadeh et al. [7] investigated the planetary gears vibration properties under the healthy and faulty conditions. A lumped parameter model (LPM) of planetary gear trains was integrated with a comprehensive bearing model. By the proposed model, the dynamic responses of the system were identified and analyzed. The defect frequencies of inner/outer race, ball/roller and their side bands were discussed thoroughly. Jamadar et al. [8] presented a new fault diagnosis approach of the deep groove ball bearing with localized surface defects on the races. The experimental results show that the proposed approach is effective for diagnosing the surface defects on the race of the deep groove ball bearing. The above scholars established the dynamic model to analyze the characteristics of the vibration response. However, most scholars did not consider the effect of elastohydrodynamic lubrication (EHL) between the bearing race and rolling element. Dewangan et al. [9] proposed a resistance method to measure the lubricant film thickness based on Hertz contact theory. The resistive lubricant film thickness was calculated based on elliptical and circular contact area of the two lubricants. The result showed that the resistance method can predict the failure of the rolling bearing, and it also can be used for online condition monitoring of the rolling element bearings. Bizarre et al. [10] established a 5-DOF nonlinear model by considering the effect of the EHL, the equivalent stiffness and damping parameters were evaluated under different loading conditions. The result showed that the model is feasible to analyze the complex rotating systems supported by oil lubricant. Yang et al. [11] presented a semi-analytical method to investigate the defect characteristic frequency of the ball bearing system with/without defect on the outer race. The nonlinear characteristics of a rotor-ball bearing system were analyzed using their presented method. Wang et al. [12] presented an improved nonlinear dynamic model with consideration of the preload condition, surface waviness, Hertz contact and EHL. Servais et al. [13] proposed a new efficient method to compute the transverse loads without the race control hypothesis, and obtained a relationship between the ball bearing kinematics and transverse loads. The skidding of the rolling element is neglected in the above works.
However, the skidding characteristics affect the dynamic responses of the rolling bearings. Xu et al. [14] proposed an analytical method to calculate the optimum preload at different speeds. The results showed that the method can effectively calculate the optimum preload under the condition of the skidding and rolling of the rolling bearing. Randall et al. [15] presented a combined dynamic model of the gears-bearings system, in which an extended fault in the inner/outer race of the rolling element bearings can be studied in the presence of gear interaction. Wang et al. [16] developed a dynamic model for angular contact ball bearing to investigate the contact characteristics and motion of ball-race conformity. The results revealed that the contact pressure and skidding between the ball and race have some influence on the ball-race conformity. Wang et al. [17] established the dynamic model to investigate the skidding of angular contact ball bearing with the interaction between race and ball. The skidding and EHL are considered in the proposed model, the results showed that the different conditions significantly affect the behavior of skidding. Han et al. [18] presented a three-dimensional nonlinear dynamic model to predict the skidding between the ball and race under different load conditions. Niu et al. [19] proposed a new method for accurately calculating the ball passing frequencies based on their dynamic model of the rolling bearing with localized defects. The relative skidding, cage effect, local defect and relative motion factors were introduced in their model. The proposed model was verified by the experiment. Liu et al. [20] established a new dynamic model to formulate the local surface defect of the cylindrical roller bearing. The relationship between the time-varying contact stiffness and the load-displacement of the rolling element and the race were investigated in the proposed model. The effects of the radial load, defect sizes, contact deformation and contact force between roller and race were investigated using the proposed model.
In this paper, a dynamics model is proposed to investigate the vibration response of the rolling bearing with the local defect. In order to simulate the operation condition of the rolling element, the rolling element skidding, interaction force between ball and race, cage/ race interactions, damping and EHL are taken into consideration. Based on the proposed model, the dynamic responses of the rolling bearing with rolling element skidding are solved. The dynamic characteristics are investigated under different operation conditions. The proposed model is applied to diagnose and monitor the local defect on inner/outer race of the rolling bearing, and it is verified by the experiment results. In order to investigate the dynamic responses of the rolling bearing, the dynamic model of the rolling bearing is established based on the Hertz contact theory. The contact behaviors between the rolling bearing and rolling element can be analyzed by using a equivalent spring-stiffnessdamping model, as shown in Figure 1. In Figure 1, K is the contact stiffness, C is the damping, I is the inner race, O is the outer race, O j1 is the shaft center of the rolling bearing.
According to the Hertz contact theory, the contact force is expressed as where K is the contact stiffness of the rolling bearing, d is the contact displacement between the rolling element and rolling bearing. For the rolling bearing, n is equal to 3/2. The functions of the contact stiffness are written as where E is the material elastic modulus of the rolling bearing, s is the Poisson's ratio, P r i is the sum of the curvature of the inner race, P r o is the sum of the curvature of the outer race, d i * is the dimensionless contact deformation of the inner race, d o * is the dimensionless contact deformation of the outer race.

Calculation of contact deformation
The center of the rolling bearing varies with the load, and consequently changes the contact geometric relationship between the rolling element and rolling bearing, as shown in Figure 2. In Figure 2, O j2 is the center of the race after deformation. The contact deformation is described as follows where x, y are the deflections of the inner race along Xand Y-axes, u i is the contact angle between the ith rolling element and rolling race, c 0 is the radial clearance of the rolling bearing.
The expression of u i is written as where z is the total number of the rolling element, i is the rolling element number, v s is the shaft velocity, u 0 is the initial contact angle. v c is the cage velocity, and it is defined as where d is the diameter of the rolling element, D is the pitch diameter.

Local defect model of inner/outer race
The local defect can be occurred on the inner/outer race of the rolling bearing after continuously working. When the rolling elements pass the race with local defect, the contact deformation changes significantly. In this paper, the crack defect is taken as an example to investigate the influence of the local defect on the dynamic response of the rolling bearing based on the built model. The profile of the local defect is simplified to the rectangle, as shown in Figure 3. In Figure 3, L is the length of the local defect. The contact process of the rolling element is shown in Figure 4. In Figure 4, B is the width of the local defect, Fq is the defect angle, H max is the maximum additional displacement.
When the rolling element passes through the local defect, the contact displacement between the rolling element and bearing race changes, and consequently leads to an additional displacement. The additional where F 0 is the initial position angle of the ith rolling element to local defect, H max is determined by the sizes of the rolling element and local defect, which is given by The expression of F q is given by where r is the radius of the inner/outer race. F di is the angular position of the ith rolling element with respect to X-axis, and it can be expressed as The final contact displacement can be expressed by following equation

Race/cage interactions
The interaction between the cage and guide race occurs with the action of the hydrodynamic pressure of the lubricant. Figure 5 shows the interaction force between cage and race (outer race). The interaction friction between the cage and guide race (outer race) can be expressed as where h 0 is the viscosity of oil film, u 1 is the entrainment velocity of oil film, B 1 is the width of the centering surface of the cage, e is the relative eccentricity of the cage center, c 1 is the size of the cage guide clearance. The expressions of u 1 and e are given as follows  where R 1 is the radius of the centering surface of the cage, e is the relative variation of the cage center.

The calculation of point contact EHL
The parameters of the oil film changes with the temperature and pressure. In this paper, in order to simplify the calculation, the temperature and pressure are assumed to be constant. The contact model of the EHL between the rolling element and the race is shown in Figure 6.

Calculation the thickness and stiffness of the oil film
Based on the theoretical and experimental investigation, the Hamrock and Dowson [21] proposed a formula to calculate the minimum oil film thickness of the elliptical point contact. The minimum oil film thickness is written as where U 0 is the dimensionless velocity parameter, G is the dimensionless material parameter, k is the ellipticity, Q 0 is the dimensionless load parameter, and it can be written as where E 0 is the equivalent elastic modulus of material, R 0 is the equivalent radius of the rolling bearing in the rotational direction, Q is the external load. The expressions of E 0 and R 0 are written as where '-' indicates the rolling element contact with the inner race, '+' indicates the rolling element contact with the outer race, the g is calculated by g = (d cos a)/d m .
The central oil film thickness of contact area of the ellipse point contact is given as The minimum oil film thickness of the elliptical point contact can be expressed as According to the definition of stiffness, the oil film stiffness in the Hertz contact area can be expressed as In the Hertz contact area, the damping of the oil film is little effect on the dynamic response. Thus, the oil film damping is neglected in the Hertz contact area in this part.

Calculation of the stiffness and damping of oil film in the inlet region
In the inlet region of the oil film, it is assumed that the sinusoidal vibration has happened near the contact point of the rolling element in the normal direction. The supporting load in the contact area can be expressed as [22] where u x is the entrainment velocity of the rolling element/ race, a is the long semi-axis length of the contact ellipse, v is the oscillating frequency.  The stiffness and damping of the oil film in the inlet region can be expressed as

Calculation of total contact stiffness and damping
The relationship among the rolling element, bearing race and oil film is shown in Figure 7. According to the seriesparallel model, the expressions functions of the stiffness and damping can be written as 4 Skidding of the rolling element The speed difference between the rolling element and race leads to the skidding of the rolling element due to the uneven load distribution. Figure 8 shows the interaction between the inner race and rolling element. As shown in Figure 8, the linear velocity of the contact point M can be expressed as where R i is the radius of the inner race.
The rolling element operates around the outer race center of the rolling bearing. It also operates around the center of itself. The linear velocity of the contact point N can be expressed as where v m is the angular velocity of the rolling element, v r is the rotation angular velocity of the rolling element. The relative skidding speed between the point M and point N on the inner race can be given by Considering the EHL, the relative skidding speed can be obtained by

Calculation of oil film resistance
The frictional force of the oil film affects the friction of the rolling element directly. Hence, it is necessary to calculate the friction of the oil film. In this study, a typical traction curve of the EHL proposed by Gupta's [2] is used to calculate the friction of the oil film. The relationship between the friction coefficient and relative velocity is shown in Figure 9. As shown in Figure 9, the friction coefficient can be given by Hence, the oil film resistance can be expressed as

Dynamic equation
According to the analysis of the local defect model under EHL, the the external force is balanced with the inertial  force, damping force, contact force caused by the elastic deformation and oil film force. The dynamic equations are written as where M is the mass of inner race and shaft, C is the total damping coefficient, K z is the total stiffness, b is the load parameter for the ith rolling element, and it can be expressed as 6 Results and discussions The SKF 6205 deep groove ball bearing with local defect is investigated, the outer race is fixed with the house, the inner race is fixed with the shaft. The bearing parameters are shown in Table 1. The influence of defect length and depth on dynamic response can be neglected in this paper because the crack defect size is merely little. The width and depth of the defect sizes are set as 0.1778 mm. In order to verify the proposed model, the test data of Case Western Reserve University is adopted for the model verification.
The test rig consists of a 2 HP motors. The motor shaft supported by the test bearings. The faults are introduced into the test bearings with different fault diameters (the 7 mils, 14 mils, 21 mils). Vibration signal is collected using accelerometers, which are placed at the 12 o'clock position at both the drive end and fan end of the motor housing. Digital data is collected at 12 000 samples per second, and data is also collected at 48 000 samples per second for drive end bearing faults. Outer raceway faults are stationary faults, therefore placement of the fault relative to the load zone of the bearing has a direct impact on the vibration response of the motor/bearing system [23]. A fourth order Runge-Kutta method is used to calculate the equations (36) and (37). The rotational speed of the shaft is 1750 rpm. The fault characteristic frequency is a important signal to diagnose the rolling bearing, it also can recognize the position (inner race or out race) of the defect. The theoretical calculations of the Ball Pass Frequency Inner race (BPFI) and Ball Pass Frequency Outer race (BPFO) can be conducted by the following equations where f s is the rotational frequency.
The theory values of the characteristic frequency for shaft rotation, inner and outer race at the speed of 1750 rpm are shown in Table 2.

Reasonability of established model
In order to obtain the dynamic response of the proposed model, the dynamic equations are solved firstly. The radial load is 450 N, the time step used in the numerical investigation is 10 À5 s, the initial displacements in X and Y direction are chosen to be 10 À6 m.  By solving the equations (36) and (37), the dynamic responses of the acceleration with time-varying are acquired. Figure 10 is the simulated signal of the healthy bearing, Figure 10a plots the acceleration signal in the time domain, and Figure 10b gives the frequency domain signal which is obtained by the fast Fourier transform from Figure 10a. In Figure 10a, it can be seen that the response curve shows a stable periodic and no impulse signal. This means that the bearing is healthy at running time. As shown in Figure 10b, the unstable impulse signal is observed at the initial operation stage. With the continuing operation of the bearing, the amplitude of frequency signal tends to be stable, and then the rotational frequency (28.23 Hz) and doubling frequency (57.22 Hz) are appeared. The unstable signal in the frequency domain is caused by the start-up characteristics. Figure 11 shows the test signals of the healthy bearing in the time and frequency domains. From Figure 11a, it can be shown that there are some slight impacts during the operation of the bearing. The impact signal component is caused by the misalignment of the bearing and the environmental noise. By reducing the noise and envelope analysis of the original signals, the FFT method is used to obtain the spectrum. It can be concluded from the Figure 11a that the time-varying signal is the stable periodic signal approximately. The frequency domain signal of the healthy bearing is shown in Figure 11b. The rotational frequency and doubling frequency of the signal are 31.38 Hz and 62.47 Hz respectively. The test and simulation frequency are in good agreement by comparison of Figures 10b and 11b. Comparing the simulation with the test frequency, the error of the frequencies are very small. Therefore, the model is verified by the comparison. Figure 12 shows the signal of inner race with local defect in time and frequency domains. As shown in Figure 12a, the periodic pulses signal of the time domain is irregular. The FFT is applied to obtain the Figure 12b. The fault characteristic frequency and side band can be observed in    Figure 12b. The fault characteristic frequency is produced by the defect of the inner race, and the side-bands is caused by the modulation of the rotation frequency. Figures 13a  and b are the test signals of the inner race with local defect in the time and frequency domains, the envelope analysis and noise reduction are used to process the test signal. The periodic impact signal is also observed in Figure 13a. Due to the external interference, the time domain signal in Figure 13a is less stable than Figure 12a. As shown in Figure 13b, the characteristic frequency of the test signal is 156.6 Hz. The BPFI of the simulation signal (158.1 Hz) and experiment (156.6 Hz) signal is close to the fault characteristic frequency. Therefore, the proposed model is validated by comparing the dynamic response signals with the experimental results. Figures 14 and 15 show the simulation and test signals of the outer race with local defect in time and frequency domains. By analyzing the results of the Figures 14a and 15a, the impulse signal and periodic features are also observed. The same method is also used to acquire the fault characteristic frequency of the outer race. As shown in Figures 14b and 15b, the numerical value of the BPFO are 104.4 Hz and 104.8 Hz respectively. The side band disappears because of the fixed outer race. The low numerical value error of the fault characteristic frequency proved that the model is verified.

Effect of defect size and external load on dynamic response
In this section, the effect of different operation conditions on the vibration response of the SKF 6205 deep groove ball bearing is investigated by the proposed model. The influence of local defect size on the dynamic response is also studied when the rotating speed is 1750 rpm and the external load is 450 N. Figures 16a and b are the spectrum of inner and outer race with local defect size of 0.3556 mm. As shown in Figures 16a and b, the characteristic frequencies and double frequency have no change compared with Figures 12b and 14b. Figures 17a and b are the   increases with the increase of the defect size. However, the amplitude of BPFI (Ball Pass Frequency Inner) decreases with the increase of the defect size. The change of the amplitude is relatively large for the outer ring. The sideband of the frequency increases significantly for the inner ring. Therefore, the defect size has greater effect on the response amplitude of the outer race than the inner race. The effects of external loads on dynamic response are investigated, as shown in Figures 18 and 19. In Figures 18  and 19, the rotational speed is 1750 rpm, and defect size is 0.1778 mm, and the external loads are 500 N and 800 N respectively. Comparing the Figure 12b, Figure 14b with Figure 18 and Figure 19, it can be observed that the frequency characteristics are invariable. A conclusion can be drawn that increasing the load is disadvantage to stabilize operation of the rolling bearing with the local defect. However, comparing the Figure 18a with Figure 19a, the amplitudes of the one-frequency decreases with the increase of external load, and the amplitude of double frequency is increases with the increase of the external load. It indicates that the occurrence of the skidding is low when the rolling element under the heavy   load. As for the outer race, the first order frequency increases, the other characteristic frequencies relatively decrease. In particular, the increasing external load is disadvantage to the stable running of the deep groove ball bearing with local defect.

Effect of speed and compound factors on dynamic response
The characteristic frequencies are mainly controlled by the rotational speed. So the operational speed is an important factor for investigating the dynamic response of the rolling bearing. Figures 20 and 21 are the spectra of the inner and outer race with local defect. In Figures 20 and  21, the external load is 450 N, the defect size is 0.1778 mm, and the shaft rotational speeds are 1772 rpm and 1797 rpm respectively. Comparing the characteristic frequencies of Figures 20 and 21 with theoretical value, the results show that the relative error is very small. As shown in Figures 20 and 21, by discussing the effect of the rotational speed on dynamic response, it can be concluded that the rotational speed has a great influence on the vibration response of the deep groove ball bearing with the local defect. In order to improve the bearing life, it's necessary to control the rotational speed when the local defect occurs.
The dynamic response of the deep groove ball bearing under the combined factors (load and speed) is also studied, as shown in Figure 22. In Figure 22, the defect size is 0.1778 mm, the external load is 800 N, and the shaft rotational speed is 1797 rpm. The results in    Figures 21 and 22 show that the amplitudes of the inner race and out race vary greatly. It indicates that the occurrence of the skidding is relatively lower at lower rotational speed under the same external load.

Conclusion and prospect
A dynamic model is proposed to investigate the vibration response of the rolling element with local defect on inner/ outer race in this paper. To achieve the velocity and acceleration, the dynamics equations are solved by the fourth order Runge-Kutta method. The effects of defect size, rotational speed, external load, and compound factors on skidding characteristics are investigated by the proposed model. The main conclusions are as follows: -The proposed model is validated by comparing the dynamic response signals with the experimental results. The good consistency of the simulation and test signals verify that the proposed model can be used to investigate the dynamic response of the rolling bearing with the local defect. -The increase of defect size and external load can lead to the change of the amplitude of spectrum, and aggravate the vibration of the rolling bearing and its operation stability. At the low-speed stage, the skidding with heavy load is smaller than light. -The vibration response of the rolling bearings with the local defects is greatly affected by the change of velocity. At the same rotational speed, the external load has a great influence on skidding, and the effect of high speed and heavy load on skidding is less than that of high speed and light load.
The proposed model can be used to diagnose the fault of the rolling bearing in this paper. The evolution of the local defect of the rolling bearing is not considered. In the next research, we will devote ourselves to construct the expression of the contact relationship between the rolling element and the local defect edge at different stages of evolution. Investigating the influence of the local defect evolution on the vibration response of the rolling bearing.