Mechanism and Method for Outer Raceway Defect Localization of Ball Bearings

The localization of outer raceway defect plays a significant role in malfunction elimination, failure cause analysis as well as the residual life prediction of ball bearings. Based on the nonlinear dynamic model for a ball bearing and the outer raceway defect model considering the ball finite size, this article employs the detailed mathematical derivation and theoretical analysis of the load distribution for the bearing system with an outer raceway defect located at the different angular positions. Therefore, the essential mechanism of the approximate linear relationship between the proposed indices, namely horizontal–vertical synchronized Peak (HVSPeak) and horizontal–vertical synchronized RMS (HVSRMS), and the defect angular position is explained. More importantly, it is theoretically demonstrated that HVSRMS is approximately a cotangent function with the defect angular position as the only variable, which indicates that the index has excellent anti-interference and practicability. In addition, the superiority and necessity of the HVSRMS index can be seen when compared to HVSPeak, RMS, SampEn and Lemple-Ziv. It is validated through simulation and experiment results that HVSRMS index can efficiently diagnose the angular position of outer raceway defect. Finally, the signal denoising methods for HVSRMS are compared and studied, which indicates the direction for subsequent research.


I. INTRODUCTION
The mechanism and method of fault diagnosis for ball bearings is a continuous research focus. With the development of theory and technology, the research hotspot from qualitative diagnosis to quantitative diagnosis is gradually becoming a trend. The quantitative diagnosis for an outer raceway defect mainly includes two aspects: defect size estimation and defect angular position estimation. The angular position estimation of the outer raceway defect, namely the localization diagnosis of outer raceway defect mainly has the following important significance or application value [1]: 1) The defect angular position is one of the main factors affecting the residual life prediction of the bearing; for example, in the case of all other factors held constant, the defect closer to the load center may have a faster expansion speed making the residual life of the bearing also shorter. 2) Different angular positions of outer The associate editor coordinating the review of this manuscript and approving it for publication was Dazhong Ma . raceway defects may correspond to different failure causes; for example, if a defect happens close to the load center, it mainly caused by fatigue damage; but if the defect occurs far away from the load center, the cause is more likely to be the machining defects, maintenance defect, or others.
In recent years, scholars have carried out a large number of systematic studies on the quantitative diagnosis mechanism and method of the defect size of rolling bearing and have obtained some remarkable achievements that enhanced the scholarship. If a rolling element bearing has a local fault, a series of impulses with certain laws will be generated in its time-domain waveform [2], [3], so vibration signals are used widely for the fault diagnosis of bearing [4]- [9]. However, bearing fault signatures are usually contaminated or even overwhelmed by interfering noise [10]. In such circumstances, a method for noise reduction and feature extraction is needed, such as wavelet transform, sparse representation and empirical mode decomposition [11]- [14]. A technique based on decomposition Symlet wavelet was employed for VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ estimating the inner raceway defect width [15]. To separate the entry-exit events, and to calculate the size of the fault, the approximate entropy method and empirical mode decomposition were applied by Ref. [16]. Zhao et al. [17] presented a methodology for the detection and recovery of fault impulses, with which the double impact phenomenon caused by a distributed defect was extracted successfully, so the defect size of a bearing can be estimated from its vibration signal without dismantling the component. An averaged dual-impulse interval determining method was used to evaluate the spall size [18]. For the fault diagnosis, the mechanism of fault is indispensable, which gives reasons and characteristics of fault changes. Many models of ball bearing have been established by scholars to research and reveal the defect generation mechanism and characteristic. A dynamic model with six DOFs was developed to investigate vibrations of high-speed rolling ball bearings with localized surface defects on raceways [19]. Khanam et al. [20] proposed an analytical force modeling approach based on the principles of engineering mechanics to explain the mechanism of excitation generation due to the impact of ball mass against the defect edge. Cui et al. [21] established a nonlinear dynamic model of rolling element bearings for assessment of the severity of an outer race fault, and analyzed quantitatively the correlation between vibration responses and fault sizes based on the model. Liu et al. [22] proposed a local fault model which includes the time-varying displacement impulse and contact stiffness, and the relationship between the contact stiffness and fault sizes was obtained through this method. Petersen et al. [23] pointed out that faults which differ in size by natural multiples of the rolling element angular spacing have the same vibration time interval based on the established dynamic model, and then proposed a valid method to distinguish those faults. The above research are carried out around the mechanism and method of outer raceway defect size estimation. However, the research aiming at the localization diagnosis of outer raceway defect is almost unreported. The main reasons linked to two factors, the first is sparse breakthroughs in the mechanism research. The second is the lack of the single factor characteristics of the localization diagnosis, which is not affected by the noise and the inherent characteristics of the system such as the defect characteristic frequency in the qualitative diagnosis and the time interval between double impact points in quantitative diagnosis. Cui et al. [1], [24] preliminarily explored the localization diagnosis of the outer raceway defect based on the proposed horizontal-vertical synchronized RMS (HVSRMS) index. However, the inherent mechanism of the index has not been excavated. Therefore, based on the nonlinear dynamic model for a ball bearing and the outer raceway defect model considering ball finite size, the detailed mathematical derivation and theoretical analysis of dynamic behavior for ball bearings with an outer raceway defect located at different angular position are researched in this paper. Hence, the essential mechanism of the approximate linear relationship between HVSRMS and the defect angular position is explained. Furthermore, the superiority and necessity of HVSRMS index is validated through the comparison to other indices, including horizontal-vertical synchronized Peak (HVSPeak), RMS, SampEn and Lemple-Ziv.
This paper is organized as follows. Section II establishes the static model of a ball bearing with an outer raceway defect and introduces the theory of horizontal-vertical synchronization eigenvalue. In Section III, the dynamic model of a ball bearing with an outer raceway defect is established and the horizontal -vertical synchronization signal analysis is implemented, where the mechanism of the approximate linear relationship between HVSRMS and the defect angular position is explained. In Section IV, the experiment is carried out to evaluate the performance of the proposed method. A comparison between the proposed index and other indices, such as RMS, SampEn and Lemple-Ziv is also presented. Concluding remarks are presented in Section V.   where N b is the number of balls,Q j,x andQ j,y are the static contact forces between ball j and raceway in the x and y directions, respectively. ( ) + represents negative setting zero. The angular position φ j of the ball j can be given by:

II. STATIC MODEL AND THE THEORY OF HORIZONTAL -VERTICAL SYNCHRONIZATION INDEX
δ j is the contact deformation of the ball j. It can be defined as: where δ x = x o -x i and δ y = y o -y i are the relative displacements between the inner and outer race in the x and y directions, r l is the radial clearance, d(φ j ) is the effective defect depth. For the defect model shown in Fig. 2, d(φ j ) is defined as: Q j,x (φ j ) andQ j,y (φ j ) loaded on the ball j in random angular position φ j can be calculated by Eqs. (1)-(4). Taking NSK6308 bearing as an example, set W x = 0, W y = −100, The other parameter values of the bearing system can be found in [1]. Fig. 3 shows the variation curve ofQ j,x (φ j ) and Q j,y (φ j ) in a complete rotation period in the failure-free case which illustrates that the load interval is [206.3 • , 333.8 • ] in this case. Furthermore, the curves ofQ j,x andQ j,y varying with φ j are approximate the sine and versine curves centered at 270 • , respectively, and both have the same frequency f q and phase ϕ q . The sine and versine function curves obtained by the data fitting using the lsqcurvefit function of MATLAB are shown as a red line in Fig. 3, and its expressions can be described as follows where A x , A y , B y , f q , ϕ q are the coefficients of the fitting function. It is revealed by combining with Fig. 3 and Eq. (5) that in the load interval and regarding φ j = 270 • as a starting point, there is no monotonic mapping relationship betweenQ j,x and φ j , and the inflection points are 239.4 • and 300.7 • , while there is a monotonic mapping relationship betweenQ j,y and φ j , but the curve around 270 • changes slightly. To excavate the characteristic indices with a simple and effective mapping relationship with φ j , horizontal-vertical synchronization contact force HVSQ j is proposed as follows Fig. 4 presents the results of HVSQ j in the load interval. The comparison between Fig. 3 and Fig. 4 shows that the relationship between HVSQ j and φ j is the simplest, especially in the range of φ j = 240 • ∼ 300 • , HVSQ j values are approximately two straight lines with symmetrical about φ j = 270 • . Since the contact force between the ball and raceway is the main excitation source of faulty bearing, it can be expected that this feature will provide a very effective support for excavating localization diagnosis characteristics for outer raceway defect. VOLUME 8, 2020 To verify the applicability scope of this characteristic, the mapping relationship between HVSQ j and φ j is simulated and analyzed in the case of W y = 30 ∼ 3000N, as shown in Fig. 5. It can be seen from Fig. 5 (a) and (b) thatQ j,x (φ j ) and Q j,y (φ j ) increase with W y varying from 30N to 3000N, which matches with Eq. (1). In addition, the increase of the load also leads to the increase of the load interval. However, it can be seen from Fig. 5 (c) that the mapping relationship between HVSQ j and φ j is completely unaffected by the change of load. Actually, it can be known from Eq. (6) that HVSQ j is only affected by φ j , and it is most remarkable that the relationship between HVSQ j and φ j is almost linear in the main load interval (240 • ∼300 • ), as shown in Fig. 5 (c).

B. THE MAPPING RELATIONSHIP BETWEEN HVS Q f AND THE OUTER RACEWAY DEFECT ANGULAR POSITION
This section investigates the influence of the outer raceway defect angular position φ f on the contact forces. The static contact forces between the ball j and raceways in the x and y directions for a defective bearing with an outer raceway defect are named asQ d j,x andQ d j,y in order to distinguish them from the forces for failure-free bearings. The circumferential extent and depth were set to φ f = 1 • and h = 0.1mm, respectively. The load is still set as W x = 0 and W y = −100. Fig. 6 presents the variation curve ofQ d j,x andQ d j,y in the case of φ f = 250 • . As observed in Fig. 6, irrespective of the x or y direction, once the ball j passes through the defect zone, the contact force of the ball j will appear mutation.
x andQ d f ,y when the ball j is arriving at the defect center, where δ f is the contact deformation of the ball j located at the defect center, namely, . That is to say, the ball located at φ f is unloaded. It is caused by two factors: 1) the two intervals are close to the unloaded interval. Combined with Fig. 4, it can be concluded that even for a failure-free bearing system, the ball contact force is very small in the two intervals, that is, the contact deformation δ f is very small. 2) when the defect is located in the two intervals, it can be obtained from Eq. (3) that δ f becomes to be nonpositive influenced by the effective defect depth d,

FIGURE 6.
Static contact forces of the ball j for a defective bearing with an outer raceway defect: Most noteworthy, it can be known from Eq. (7) thatQ d f ,x andQ d f ,y are affected by many factors, such as the load, the defect angular position and the ball number. However, HVSQ f is a univariate function with the defect angular position φ f as the variable. In addition, HVSQ f distributes symmetrically by the angle position 270 • for the center, and has the relation of approximate linear change with defect angle position φ f in the main load interval (240 • ∼300 • ), as shown in Fig. 7(c). Q f ,x and Q f ,y represent the value of the difference between the contact forces of ball j for a failurefree bearing and a defective bearing of ball angular position φ j = φ f . Combined with Eq. (6) and (7), HVS Q f can be expressed as follows From Eq. (8), it can be noted that HVS Q f is a cotangent function with φ f as the only variable. The value of the difference for the contact force caused by a ball passing the defect zone is the main excitation source of the vibration acceleration response of the bearing system, which reveals the advantage and the necessity of the horizontal-vertical synchronization signal analysis for defect localization of the outer raceway defect in essence.

III. DYNAMIC MODEL AND THE HORIZONTAL-VERTICAL SYNCHRONIZATION SIGNAL ANALYSIS
In Section II, the relationships between the contact force and the defect angular position of the static model in the cases of failure-free and defective are studied. To reveal the mechanism of the corresponding relationships for the actual system more realistically, the horizontal-vertical synchronization signal analysis based on the dynamic model of a bearing system is carried out in this section. The dynamic equations of the ball bearing shown in Fig. 1 can be expressed as where Q x and Q y are the dynamic contact forces in the x and y directions, Q dx and Q dy are the dynamic contact damping forces in the x and y directions. Eq. (9) shows a typical multi-freedom nonlinear self-excited vibration system. It is difficult to obtain its analytical solution directly. This paper will analyze it from three aspects: high-precision numerical solution, simplified analysis solution and linearization analysis. Firstly, Eq. (9) was solved numerically by the ODE 45 solver in MATLAB. The main parameters of the bearing system can be found in Ref. [23]. Hence, the dynamic contact force and vibration acceleration of the outer raceway were obtained. The dynamic contact force and vibration acceleration response of a ball bearing system with an outer raceway defect of the angular position φ f = 270 • , circumferential extent φ f = 1 • and depth h = 0.3mm are numerically solved with the ODE function in Matlab by substituting Eqs. (1) -(4) into Eq. (9), as shown in Fig. 8. According to Fig. 8 (a) and (b), it can be seen that the peak of the resultant contact force increases significantly when the ball passes the outer raceway defect. This results in a significant increase of the peak of the vibration acceleration. Therefore, this indicates that the peak of vibration acceleration of the outer raceway defect bearing depends on the difference value of the contact force caused by a ball passing the defect. To obtain the approximate analytical relationship between the defect angular position and the vibration acceleration, the multi-degree freedom dynamic model of the bearing system represented in Fig. 1 and Eq. (9) is simplified as a single-degree-freedom system. Taking the vibration in the x direction of the outer ring as an example, the contact damping term is ignored, and for the failure-free system, the dynamic contact force can be approximately replaced by the static contact force. Therefore, the dynamic equation is defined as According to Section I, it can be seen that the external excitation forceQ x of the single-degree-freedom system is the linear superposition of the contact forcesQ j,x . It can be VOLUME 8, 2020 seen from Eq. (5) that for the failure-free case,Q j,x is approximately a sine function. Therefore, the vibration response of the single-degree-freedom system under the excitationQ j,x can be expressed as Similarly, for the single-degree-freedom system of the outer race in the y direction, the vibration response under the excitation ofQ j,y can be calculated here as follows Eq. (11) and Eq. (13) represent the vibration responses of a failure-free system. When a defect of φ f = 1 • and h = 0.3mm occurs in the outer raceway, the contact force of the ball j passing through the defect is defined by where Q j,x and Q j,y represent the value of the difference between the contact forces of the ball j for a failure-free bearing and a defective bearing when the ball j locates at the defect zone. The effects of Q j,x and Q j,y on the single-degreefreedom systems can be expressed as impulse excitations, that is where t is the duration of the ball passing through the defect zone. The response of the single-degree-freedom system of the outer race in the x and y directions under the excitation of Q f ,x and Q f ,y can be expressed as (16) where The total responses of the outer race with a defect in the x and y directions are defined as Substituting Eqs. (11)-(17) into Eq. (18) and gaining the quadric derivative, the vibration acceleration responses can be expressed as where (20) According to ISO standard, this paper makes a general comparison and analysis of the peak value ofā j,x (t) and a ∼ x,o (t) in the rated load range of the selected NSK6308 ball bearing, that is Eq. (21) illustrates that the peak ofā j,x (t) is far less than the peak of a According to the above analysis, the horizontal-vertical synchronization peak (HVSPeak) proposed in this paper is defined as Then, combining Eqs. (8), (15), (22) and (23), the theoretical relationship between HVSPeak and φ f can be obtained as: Eq. (24) shows that HVSPeak is approximately a cotangent function that takes φ f as the only variable and factor. In section II, it has been proved based on Eq. (7) and Fig. 6(c) that for the static system, when the ball passes through the defect zone, although the value ofQ d j,x andQ d j,y are affected by multiple factors, HVSQ f and HVS Q f are both the cotangent functions with φ f as the only variable. To verify the applicability of this property in the nonlinear dynamic system, the relationship between HVSQ f of the dynamic system and φ f is simulated by the numerical solution of Eq. (9), as the red line shown in Fig. 9. It can be seen that the relationship of HVSQ f and φ f is also approximately a cotangent function. With Eq. (23) and the numerical solution of Eq. (9), the variation curve of HVSPeak with the change of the defect angular position φ f was simulated, as the black line shown in Fig. 9. It can be seen that the relationship between HVS-Peak and φ f is approximately a cotangent function, which verifies the accuracy of Eq. (24).
However, the stability of the peak is highly susceptible to the noise and other factors in actual signals. To overcome this problem, and on account of the linear relationship between the peak and RMS for the exponential attenuation function, the horizontal-vertical synchronization RMS (HVSRMS) is proposed as HVSRMS = RMS ax RMS ay (25) and the theoretical relationship between HVSRMS and φ f is shown as: where RMS ax and RMS ay are the root-mean-square values of the horizontal and vertical vibration acceleration signal for the bearing system, respectively. With Eq. (25) and the numerical solution of Eq. (9), the curve of HVSRMS changing with φ f is calculated as the blue line shown in Fig. 9, which indicates that HVSRMS is approximately a cotangent function that takes φ f as the variable, which verifies the accuracy of Eq. (26). By now, the mathematical and physical nature of the approximate linear relationship between HVSRMS and φ f on both sides of 270 • , proposed by reference [1] through qualitative analysis and numerical simulation, is revealed, and a more accurate functional relationship between them is obtained. Predictably, HVSRMS is a practical diagnostic feature for the angular position estimation of the outer raceway defect.

IV. EXPERIMENTAL VERIFICATION AND METHOD COMPARISON
The experimental apparatus, applied to acquire actual signals of a ball bearing with an outer raceway defect and verify the effectiveness of the method proposed in this paper, is shown in Fig. 10. The bearing is NSK6308, and has N b = 8 balls. The shaft frequency and sample frequency were set to f s = 7Hz and F s = 65536, respectively. The sampling points is N = 131073. The wire cutting method is used to process an outer raceway defect of circumferential extent φ f = 1 • , depth h = 0.3mm. The vertical (y) and horizontal (x) direction vibration acceleration signal for the defect angular position φ f = 240 • ∼ 300 • are measured at intervals of 10 • . Fig. 11 shows the time-domain waveform of the measured vibration acceleration signal for φ f = 240 • . It can be seen that due to the existence of the outer raceway defect, the acceleration signals in the x and y directions both vibrate at the same period. Moreover, the peak of the vibration in the x direction is significantly lower than that of the y direction. These characteristics are consistent with the simulation results shown in Fig. 8. To verify the applicability of HVSPeak and HVSRMS in actual signals, the results of the measured signals were solved.
With the definitions in section III, HVSRMS and HVSPeak at different defect angular positions are solved and normalized for observation, as shown in Fig. 12. It can be seen from Fig. 12 that the curve of HVSPeak for the measured vibration signal varying with φ f is not only non-linear, but also non-monotonic. It confirms the prediction in section III: although the relationship between HVSPeak and φ f is an approximate linear relationship in theoretically, actual signals VOLUME 8, 2020  are susceptible to noise and other factors, which affects the applicability and accuracy of HVSPeak.
Comparing with Fig. 9 and Fig. 12, it can be seen that the HVSRMS of the measured signal gradually exhibits the symmetrical distribution centered on φ f = 270 • , and has the relation of approximate linear change with φ f , which verifies the simulation results and Eq. (26) in section III.
To further verify the necessity and superiority of HVS-RMS, Fig. 13 shows curves of some common indices varying with φ f = 240 • ∼300 • for the measured horizontal and vertical acceleration signals, including RMS, Kurtosis, CF, Sr, Sα [1], SampEn [25] and LempelZiv [26]. Since the magnitudes of the different indicators are different, all indicators are normalized in Figure 13 for ease of observation. It can be seen from Fig. 13 (a) and (b) that all of these indices do not have any monotonicity variation relationship with φ f , so they have no ability to locate the angular position of the outer raceway defect. This adequately approves the effectiveness and superiority of the HVSRMS method for localization diagnosis of outer raceway defect in ball bearings.
It should be noticed that the noise in the measured signals may interfere the calculation results of HVSRMS, thus affecting the diagnostic accuracy. For example, there is an obvious deviation of the HVSRMS curve on the left side of 270 • in Fig. 12, especially at φ f = 250 • . To this end, three methods of lifting wavelet, morphological filtering and matching pursuit are applied to the measured signal preprocess, where the signal processing method suitable for HVSRMS is highlighted to improve the localization accuracy. Due to the space limit, only the original signal and the processed signals by the three methods in the case of φ f = 260 • are given here,  as shown in Fig. 14. It can be seen from Fig. 14 that the signalto-noise ratio of the signals processed by the three methods is significantly improved, and the mature spectrum analysis methods such as Hilbert envelope spectrum is sufficient to qualitatively diagnose the bearing outer ring fault.
However, it does not mean that all three methods can effectively improve the accuracy of the HVSRMS index of the measured signal. Fig. 15 shows the curves of HVSRMS varying with φ f of the measured signals processed by the three methods. It can be found that the morphological filtering method can significantly improve the accuracy of HVSRMS, followed by the lifting wavelet method, while the result of the matching pursuit method is totally ineffective. It can be explained combined with the definition of HVSRMS and its mathematical nature in the previous section that although the matching pursuit method can greatly improve the signal-to-noise ratio, it cannot accurately maintain the amplitude information of the fault impulse waveform, as shown in Fig. 14(d). This is a common problem in existing sparse decomposition methods. It can be seen from Fig. 14(b) that the lifting wavelet method maintains the amplitude of the fault impact waveform while reducing noise, thus improving the accuracy of HVSRMS. As can be seen from Fig. 14(c), the morphological filtering method significantly increases the amplitude of the fault impulse waveform while suppressing noise. Besides, the most critically, the fault impulse waveform in the x-and y-direction is scaled up meanwhile, which makes the morphological filtering method achieve the best results. This discovery can be used to indicate the direction of the subsequent signal processing methods for HVS-RMS: 1. Research a noise reduction method that does not cause amplitude distortion of the fault impulse waveform. 2. Develop a method that can proportionally enhance the fault impulse waveform of the x-direction and y-direction signals. 3. Improve the existing sparse decomposition and other methods with the goal that reserve the amplitude information of the fault impulse waveform accurately.

V. CONCLUSION
In-depth discussion and demonstration about the localization of the outer raceway defect on the ball bearing were carried out in this paper. Firstly, based on the mathematical derivation and solution of the static model, it shows that the extremums of the contact component forcesQ d f ,x andQ d f ,y of the ball passing through the defect zone are respectively the sine and versine functions that take the defect angular position φ f , the external load and bearing parameters as variables. However, HVSQ f and HVS Q f are both cotangent functions that take φ f as the only variable. These findings reveal the mechanism and basis of the HVS-index method to realize the localization diagnosis of outer raceway defect in essence.
On that basis, the numerical simulation and approximate analytical analysis of the nonlinear dynamic model of the bearing system are carried out. It is confirmed that both HVSQ f and HVS Q f in the dynamic system are approximately cotangent functions that take φ f as the only variable.
Then two localization diagnosis indices: horizontal-vertical synchronized Peak (HVSPeak) and horizontal-vertical synchronized RMS (HVSRMS) were proposed. Theoretical analysis and simulation results show that HVSPeak and HVSRMS are also approximately cotangent functions that take φ f as the only variable, that is to say, both curves of HVSPeak and HVSRMS varying with φ f are approximately two fixed gradient lines with symmetrical about φ f = 270 • in the interval of [240 • , 300 • ]. In addition, the function is not affected by any other factors except φ f in theory. This feature is similar to the defect characteristic frequency that the qualitative diagnosis relies on, which provides support for the localization of outer raceway defect. Then, based on the analysis of the measured signal acquired from the experimental apparatus, the applicability and accuracy of HVSRMS in localization of outer raceway defect are verified. More importantly, the necessity and superiority of HVSRMS is highlighted by comparing with HVSPeak, RMS, Kurtosis, CF, S r , S α , SampEn and LempelZiv.
Finally, the effects of three signal processing methods, lifting wavelet, morphological filtering and matching pursuit, are compared to solve the problem that noises in actual signals interfere the accuracy of localization diagnosis based on HVSRMS. The results show that the morphological filtering method has the best extraction effect, and the matching pursuit method has the worst effect, even counterproductive. Combining with the mathematical mechanism of HVSRMS and the characteristics of the three methods, the principle is explained, and the following directions are pointed out for the research of signal processing methods on HVSRMS: 1. Research a noise reduction method that does not cause amplitude distortion of the impact waveform. 2. Develop a method that can proportionally enhance the fault impulse waveform of the x-direction and y-direction signals. 3. Improve the existing sparse decomposition and other methods with the goal that reserve the amplitude information of the fault impulse waveform accurately.